WO2007051078A2 - System and method of computing and rendering the nature of polyatomic molecules and polyatomic molecular ions - Google Patents

System and method of computing and rendering the nature of polyatomic molecules and polyatomic molecular ions Download PDF

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WO2007051078A2
WO2007051078A2 PCT/US2006/042692 US2006042692W WO2007051078A2 WO 2007051078 A2 WO2007051078 A2 WO 2007051078A2 US 2006042692 W US2006042692 W US 2006042692W WO 2007051078 A2 WO2007051078 A2 WO 2007051078A2
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energy
given
atom
bond
electrons
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PCT/US2006/042692
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WO2007051078A3 (en
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Randell L. Mills
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Blacklight Power, Inc.
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Publication of WO2007051078A3 publication Critical patent/WO2007051078A3/en

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    • G—PHYSICS
    • G16—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR SPECIFIC APPLICATION FIELDS
    • G16C—COMPUTATIONAL CHEMISTRY; CHEMOINFORMATICS; COMPUTATIONAL MATERIALS SCIENCE
    • G16C20/00—Chemoinformatics, i.e. ICT specially adapted for the handling of physicochemical or structural data of chemical particles, elements, compounds or mixtures
    • G16C20/80—Data visualisation
    • G—PHYSICS
    • G16—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR SPECIFIC APPLICATION FIELDS
    • G16C—COMPUTATIONAL CHEMISTRY; CHEMOINFORMATICS; COMPUTATIONAL MATERIALS SCIENCE
    • G16C10/00—Computational theoretical chemistry, i.e. ICT specially adapted for theoretical aspects of quantum chemistry, molecular mechanics, molecular dynamics or the like

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  • This invention relates to a system and method of physically solving the charge, mass, and current density functions of polyatomic molecules, polyatomic molecular ions, diatomic molecules, molecular radicals, molecular ions, or any portion of these species, and computing and rendering the nature of these species using the solutions.
  • the results can be displayed on visual or graphical media.
  • the displayed information provides insight into the nature of these species and is useful to anticipate their reactivity, physical properties, and spectral absorption and emission, and permits the solution and display of other species.
  • CQM classical quantum mechanics
  • Applicant's previously filed WO2005/067678 discloses a method and system of physically solving the charge, mass, and current density functions of atoms and atomic ions and computing and rendering the nature of these species using the solutions.
  • the complete disclosure of this published PCT application is incorporated herein by reference.
  • Applicant's previously filed WO2005/116630 discloses a method and system of physically solving the charge, mass, and current density functions of excited states of atoms and atomic ions and computing and rendering the nature of these species using the solutions.
  • the complete disclosure of this published PCT application is incorporated herein by reference.
  • Applicant's previously filed U.S. Published Patent Application No. 20050209788A1 relates to a method and system of physically solving the charge, mass, and current density functions of hydrogen-type molecules and molecular ions and computing and rendering the nature of the chemical bond using the solutions.
  • the complete disclosure of this published application is incorporated herein by reference.
  • derivations consider the electrodynamic effects of moving charges as well as the Coulomb potential, and the search is for a solution representative of the electron wherein there is acceleration of charge motion without radiation.
  • the mathematical formulation for zero radiation based on Maxwell's equations follows from a derivation by Haus [18].
  • the function that describes the motion of the electron must not possess spacetime Fourier components that are synchronous with waves traveling at the speed of light.
  • nonradiation is demonstrated based on the electron's electromagnetic fields and the Poynting power vector.
  • the current and charge density functions of the electron may be directly physically interpreted.
  • spin angular momentum results from the motion of negatively charged mass moving systematically, and the equation for angular momentum, r x p , can be applied directly to the wave function (a current density function) that describes the electron.
  • the Schr ⁇ dinger equation gives a vague and fluid model of the electron.
  • Schr ⁇ dinger interpreted e ⁇ * (x) ⁇ (x) as the charge-density or the amount of charge between x and x + dx ( ⁇ * is the complex conjugate of ⁇ ). Presumably, then, he pictured the electron to be spread over large regions of space.
  • Max Born who was working with scattering theory, found that this interpretation led to inconsistencies, and he replaced the Schr ⁇ dinger interpretation with the probability of finding the electron between x and x + dx as
  • the radii are determined from the force balance of the electric, magnetic, and centrifugal forces that corresponds to the minimum of energy of the system.
  • the ionization energies are then given by the electric and magnetic energies at these radii.
  • the spreadsheets to calculate the energies from exact solutions of one through twenty- electron atoms are given in '06 Mills GUT [1] and are available from the internet [19]. For 400 atoms and ions, as well as hundreds of molecules, the agreement between the predicted and experimental results is remarkable.
  • CQM classical quantum mechanics
  • the present invention stems from a new fundamental insight into the nature of the atom.
  • Applicant's new theory of Classical Quantum Mechanics reveals the nature of atoms and molecules using classical physical laws for the first time.
  • traditional quantum mechanics can solve neither multi-electron atoms nor molecules exactly.
  • CQM produces exact, closed-form solutions containing physical constants only for even the most complex atoms and molecules.
  • the present invention is the first and only molecular modeling program ever built on the CQM framework. All the major functional groups that make up most organic molecules have been solved exactly in closed-form solutions with CQM. By using these functional groups as building blocks, or independent units, a potentially infinite number of organic molecules can be solved. As a result, the present invention can be used to visualize the exact 3D structure and calculate the heat of formation of almost any organic molecule.
  • the significant building-block molecules of chemistry have been successfully solved using classical physical laws in exact closed-form equations having fundamental constants only.
  • the major functional groups have been solved from which molecules of infinite length can be solved almost instantly with a computer program. The predictions are accurate within experimental error for over 375 exemplary molecules. Applicant's CQM is the theory that physical laws (Maxwell's Equations, Newton's
  • the present invention's advantages over other models includes: Rendering true molecular structures; Providing precisely all characteristics, spatial and temporal charge distributions and energies of every electron in every bond, and of every bonding atom; Facilitating the identification of biologically active sites in drugs; and Facilitating drug design.
  • An objective of the present invention is to solve the charge (mass) and current-density functions of polyatomic molecules, polyatomic molecular ions, diatomic molecules, molecular radicals, molecular ions, or any portion of these species from first principles.
  • the solution for the polyatomic molecules, polyatomic molecular ions, diatomic molecules, molecular radicals, molecular ions, or any portion of these species is derived from Maxwell's equations invoking the constraint that the bound electron before excitation does not radiate even though it undergoes acceleration.
  • Another objective of the present invention is to generate a readout, display, or image of the solutions so that the nature of polyatomic molecules, polyatomic molecular ions, diatomic molecules, molecular radicals, molecular ions, or any portion of these species be better understood and potentially applied to predict reactivity and physical and optical properties.
  • Another objective of the present invention is to apply the methods and systems of solving the nature of polyatomic molecules, polyatomic molecular ions, diatomic molecules, molecular radicals, molecular ions, or any portion of these species and their rendering to numerical or graphical form to all atoms and atomic ions.
  • composition of matter comprising a plurality of atoms
  • the improvement comprising a novel property or use discovered by calculation of at least one of a bond distance between two of the atoms, a bond angle between three of the atoms, and a bond energy between two of the atoms, orbital intercept distances and angles,charge-density functions of atomic, hybridized, and molecular orbitals, the bond distance, bond angle, and bond energy being calculated from, physical solutions of the charge, mass, and current density functions of atoms and atomic ions, which solutions are derived from Maxwell's equations using a constraint that a bound electron(s) does not radiate under acceleration.
  • the presented exact physical solutions for known species of the group of polyatomic molecules, polyatomic molecular ions, diatomic molecules, molecular radicals, molecular ions, or any functional group therein, can be applied to other species. These solutions can be used to predict the properties of other species and engineer compositions of matter in a manner which is not possible using past quantum mechanical techniques.
  • the molecular solutions can be used to design synthetic pathways and predict product yields based on equilibrium constants calculated from the heats of formation. Not only can new stable compositions of matter be predicted, but now the structures of combinatorial chemistry reactions can be predicted.
  • Pharmaceutical applications include the ability to graphically or computationally render the structures of drugs that permit the identification of the biologically active parts of the specie to be identified from the common spatial charge-density functions of a series of active species. Novel drugs can now be designed according to geometrical parameters and bonding interactions with the data of the structure of the active site of the drug.
  • the system can be used to calculate conformations, folding, and physical properties, and the exact solutions of the charge distributions in any given specie are used to calculate the fields. From the fields, the interactions between groups of the same specie or between groups on different species are calculated wherein the interactions are distance and relative orientation dependent. The fields and interactions can be determined using a finite-element- analysis approach of Maxwell's equations.
  • Embodiments of the system for performing computing and rendering of the nature of the polyatomic molecules, polyatomic molecular ions, diatomic molecules, molecular radicals, molecular ions, or any portion of these species using the physical solutions may comprise a general purpose computer. Such a general purpose computer may have any number of basic configurations.
  • such a general purpose computer may comprise a central processing unit (CPU), one or more specialized processors, system memory, a mass storage device such as a magnetic disk, an optical disk, or other storage device, an input means, such as a keyboard or mouse, a display device, and a printer or other output device.
  • CPU central processing unit
  • specialized processors such as a central processing unit (CPU)
  • system memory such as a magnetic disk, an optical disk, or other storage device
  • an input means such as a keyboard or mouse
  • a display device such as a printer or other output device.
  • a system implementing the present invention can also comprise a special purpose computer or other hardware system and all should be included within its scope.
  • any of the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
  • Fig. 1 illustrates an elliptical current element of the prolate spheroidal MO
  • Fig. 2 illustrates the ellipsoidal current-density surface obtained by stretching Y° ( ⁇ , ⁇ ) along the semimajor axis;
  • Fig. 3 illustrates the angular momentum components of the MO and S ;
  • Fig. 4 illustrates cross section of an atomic orbital;
  • Fig. 5 illustrates A. Prolate spheroid MO;
  • Fig. 6 illustrates the equilateral triangular H 3 + (1//»);
  • Fig. 7 illustrates the cross section of the OH MO
  • Fig. 8 illustrates OH MO comprising the superposition of the H 2 -type ellipsoidal MO and the O2p y AO with a relative charge-density of 0.75 to 1.25;
  • Fig. 9 illustrates Tf 2 O MO comprising the linear combination of two O- H -bond MOs
  • Fig. 10 illustrates the cross section of the NH MO showing the axes, angles, and point of intersection of the H 2 -type ellipsoidal MO with the NIp x AO;
  • Fig. 11 illustrates NH MO comprising the superposition of the Tf 2 -type ellipsoidal MO and the NIp x AO with a relative charge-density of 0.75 to 1.25;
  • Fig. 12 illustrates JVH 2 MO comprising the linear combination of two N-H -bond MOs
  • Fig. 13 illustrates NH 3 MO comprising the linear combination of three N- H -bonds
  • Fig. 14 illustrates the cross section of the CH MO showing the axes, angles, and point of intersection of the H 2 -type ellipsoidal MO with the dsp* HO;
  • Fig. 15 illustrates CH MO comprising the superposition of the H 2 -type ellipsoidal MO and the C2sp 3 HO with a relative charge-density of 0.75 to 1.25;
  • Fig. 16 illustrates CH 2 MO comprising the linear combination of two C- H -bond MOs
  • Fig. 17 illustrates CH 3 MO comprising the linear combination of three C-H-bond MOs
  • Fig. 18 illustrates CH 4 MO comprising the linear combination of four C-H-bond MOs formed by the superposition of a H 2 -type ellipsoidal MO and a C2sp l HO;
  • Fig. 19 illustrates the cross section of the N 2 MO;
  • Fig. 20 illustrates N 2 MO comprising the ⁇ MO ( H 2 -type MO) with N atoms at the foci;
  • Fig. 21 illustrates the cross section of the O 2 MO
  • Fig. 22 illustrates O 2 MO comprising the ⁇ MO (H 2 -type MO);
  • Fig. 23 illustrates the cross section of the F 2 MO
  • Fig. 24 illustrates F 2 MO comprising the ⁇ MO (H 2 -type MO) with F atoms at the foci;
  • Fig. 25 illustrates the cross section of the CZ 2 MO
  • Fig. 26 illustrates Cl 2 MO comprising the superposition of the H 2 -type ellipsoidal MO and the two Cl3sp 3 ⁇ Os;
  • Fig. 27 illustrates the cross section of the CN MO
  • Fig. 28 illustrates CN MO
  • Fig. 29 illustrates the cross section of the CO MO
  • Fig. 30 illustrates CO MO
  • Fig. 31 illustrates the cross section of the NO MO
  • Fig. 32 illustrates NO MO
  • Fig. 33 illustrates the cross section of the CO 2 MO
  • Fig. 34 illustrates CO 2 MO
  • Fig. 35 illustrates the cross section of the NO 2 MO
  • Fig. 36 illustrates NO 2 MO
  • Fig. 37 illustrates the cross section of the C-C -bond MO ( ⁇ MO) and one C-H -bond MO of ethane
  • Fig. 38 illustrates the cross section of one C -H -bond MO of ethane showing the axes, angles, and point of intersection of the H 2 -type ellipsoidal MO with the C elhane 2sp 3
  • Fig. 39 illustrates CH 3 CH 3 MO comprising the linear combination of two sets of three
  • Fig. 41 illustrates the cross section of one C- H -bond MO of ethylene showing the axes, angles, and point of intersection of the H 2 -type ellipsoidal MO with the C ejhylene 2sp 3
  • Fig. 42 illustrates CH 2 CH 2 MO comprising the linear combination of two sets of two
  • FIG. 43 illustrates the cross section of the C ⁇ C -bond MO ( ⁇ MO) and one C-H -bond MO of acetylene showing the axes, angles, and point of intersection of each H 2 -type ellipsoidal MO with the corresponding C acelylem 2sp 3 HO;
  • Fig. 44 illustrates CHCH MO comprising the linear combination of two C-H-bond MOs and a C s C -bond MO
  • Fig. 46 illustrates the cross section of one C-H -bond MO of benzene showing the axes, angles, and point of intersection of the H 2 -type ellipsoidal MO with the C benzene 2sp 3 HO;
  • Fig. 47 illustrates C 6 H 6 MO comprising the linear combination of six sets of C-H -bond
  • Fig. 48 illustrates the cross section of one C-C -bond MO ( ⁇ MO) and one C -H -bond MO of C n H 2n+2 showing the axes, angles, and point of intersection of each H 2 -type ellipsoidal MO with the corresponding C alkam 2sp 3 HO;
  • Fig. 49 illustrates the cross section of one C- H -bond MO of C n H 2n+2 showing the axes, angles, and point of intersection of the H 1 -type ellipsoidal MO with the C alkam 2sp 3
  • Fig. 50 illustrates C 3 H 8 MO comprising a linear combination of C-H -bond MOs and C-C -bond MOs of the two methyl groups and one methylene group;
  • Fig. 51 illustrates C 4 H 10 MO comprising a linear combination of C-H-bond MOs
  • Fig. 52 illustrates C 5 H 12 MO comprising a linear combination of C-H -bond MOs
  • Fig. 53 illustrates C 6 H 14 MO comprising a linear combination of C-H -bond MOs
  • Fig. 54 illustrates C 7 H 16 MO comprising a linear combination of C-H-bond MOs
  • Fig. 55 illustrates C 8 H 18 MO comprising a linear combination of C- H-bond MOs and C-C -bond MOs of the two methyl and six methylene groups;
  • Fig. 56 illustrates C 9 H 20 MO comprising a linear combination of C-H -bond MOs
  • Fig. 58 illustrates C 11 H 24 MO comprising a linear combination of C-H-bond MOs
  • Fig. 59 illustrates C 12 H 26 MO comprising a linear combination of C - H -bond MOs
  • Fig. 60 illustrates C 18 H 38 MO comprising a linear combination of C -H-bond MOs
  • Fig. 6 LA illustrates 1,3 Butadiene
  • Fig. 61. B illustrates 1,3 Pentadiene
  • Fig. 61. C illustrates 1,4 Pentadiene
  • Fig. 61. D illustrates 1,3 Cyclopentadiene
  • Fig. 6 IE illustrates Cyclopentene
  • Fig. 62 illustrates Naphthalene
  • Fig. 63 illustrates Toluene
  • Fig. 64 illustrates Benzoic acid
  • Fig. 65 illustrates Pyrrole
  • Fig. 66 illustrates Furan
  • Fig. 67 illustrates Thiophene
  • Fig. 68 illustrates Imidazole
  • Fig. 69 illustrates Pyridine
  • Fig. 70 illustrates Pyrimidine
  • Fig. 71 illustrates Pyrazine
  • Fig. 72 illustrates Quinoline
  • Fig. 73 illustrates Isoquinoline
  • Fig. 74 illustrates Indole
  • Fig. 75 illustrates Adenine
  • Fig. 76 illustrates a block diagram of an exemplary software program
  • Figs. 77 and 78 illustrate pictures of an exemplary software program.
  • Hydrogen molecules form hydrogen molecular ions when they are singly ionized.
  • dihydrino molecules form dihydrino molecular ions when they are singly ionized.
  • Each hydrogen-type molecular ion comprises two protons and an electron where the equation of motion of the electron is determined by the central field which is p times that of a proton at each focus (p is one for the hydrogen molecular ion, and p is an integer greater than one for each H ⁇ (l/ p), called dihydrino molecular ion).
  • Eq. (11.19) permits the classification of the orbits according to the total energy, E, as follows:
  • the central force equation, Eq. (11.14) has orbital solutions, which are circular, elliptical, parabolic, or hyperbolic.
  • the former two types of solutions are associated with atomic and molecular orbitals. These solutions are nonradiative.
  • A ⁇ ab (11.25) where b and 2 ⁇ are the lengths of the semiminor and minor axes, respectively, and a aa ⁇ 2a are the lengths of the semimajor and major axes, respectively.
  • the geometry of molecular hydrogen is ellipsoidal with the internuclear axis as the principal axis; thus, the electron orbital is a two-dimensional ellipsoidal-time harmonic function.
  • the mass follows an elliptical path, time harmonically as determined by the central field of the protons at the foci.
  • Rotational symmetry about the internuclear axis further determines that the orbital is a prolate spheroid.
  • ellipsoidal orbits of molecular bonding hereafter referred to as ellipsoidal molecular orbitals (MOs)
  • MOs ellipsoidal molecular orbitals
  • the semiprincipal axes of the ellipsoid are a, b, c .
  • the Laplacian is (11.27)
  • Excited states of orbitspheres are discussed in the Excited States of the One-Electron Atom (Quantization) section.
  • excited electronic states are created when photons of discrete frequencies are trapped in the ellipsoidal resonator cavity of the MO.
  • the photon changes the effective charge at the MO surface where the central field is ellipsoidal and arises from the protons and the effective charge of the "trapped photon" at the foci of the MO.
  • Force balance is achieved at a series of ellipsoidal equipotential two- dimensional surfaces confocal with the ground state ellipsoid.
  • the "trapped photons" are solutions of the Laplacian in ellipsoidal coordinates, Eq. (11.27). As is the case with the orbitsphere, higher and lower energy states are equally valid.
  • the photon standing wave in both cases is a solution of the Laplacian in ellipsoidal coordinates.
  • AaE allowed circumference
  • ⁇ photon standing wavelength
  • the potential, ⁇ , and distribution of charge, ⁇ , over the conducting surface of an ellipsoidal MO are sought given the conditions: 1.) the potential is equivalent to that of a charged ellipsoidal conductor whose surface is given by Eq. (11.26), 2.) it carries a total charge g ⁇ -e , and 3.) initially there is no external applied field. To solve this problem, a potential function must be found which satisfies Eq. (11.27), which is regular at infinity, and which is constant over the given ellipsoid. The solution is well known and is given after Stratton [2].
  • the equipotential surfaces are the ellipsoids ⁇ - constant .
  • Eq. (11.37) is an elliptic integral and its values have been tabulated [3]. Since the distance along a curvilinear coordinate u 1 is measured not by du x but by
  • the surface density at any point on a charged ellipsoidal conductor is proportional to the perpendicular distance from the center of the ellipsoid to the plane tangent to the ellipsoid at the point.
  • the charge is thus greater on the more sharply rounded ends farther away from the origin.
  • Eq. (11.46) for an isolated electron MO the electric field inside is zero as given by Gauss' Law
  • E 2 is the electric field inside which is zero.
  • the force balance equation between the protons and the electron MO is solved to give the position of the foci, then the total energy is determined including the repulsive energy between the two protons at the foci to determine whether the original assumption of an elliptic orbit was valid. If the condition that E ⁇ 0 is met, then the problem of the stable elliptic orbit is solved. In any case that this condition is not found to be met, then a stable orbit can not be formed.
  • T 1 and r 2 are the radial vectors of the central forces from the corresponding focus to the point ⁇ x,y,z) on the ellipsoidal MO.
  • the polar-coordinate elliptical orbit of a point charge due to its motion in a central inverse-squared-radius field is given by Eqs. (11.10-11.12) as the solution of the polar- coordinate-force equations, Eqs. (11.5-11.19) and (11.68-11.70).
  • the orbit is also completely specified in Cartesian coordinates by the solution of Eqs. (11.5-11.19) and (11.68-11.70) for the semimajor and semiminor axes.
  • the corresponding polar-coordinate elliptical orbit is given as a plane cross section through the foci of the Cartesian-coordinate-system ellipsoid having the same axes given by Eq.
  • Eq. (11.56) is based on a single point charge e .
  • the ⁇ -dependence must vanish.
  • the polar-coordinate elliptical orbit is also completely specified by the total constant total energy E and the angular momentum which for the electron is the constant h .
  • Eq. (11.56) the corresponding total energy of the electron is conserved and is determined by the integration over the MO to give the average:
  • Eq. (11.57) is transformed from a two-centered-central force to a one-centered-central force to match the form of the potential of the ellipsoidal MO.
  • n ⁇ r 2 a (11.59) then, r (11.60) and the one-centered-central force is in the L -direction.
  • Eq. (11.57) transforms as
  • Eq. (11.61) has the same form as that of the electric field of the ellipsoidal MO given by Eq. (11.49), except for the scaling factor of two-centered coordinates A 2n , :
  • the charge-density distribution corrects the angular variation in central force over the surface such that a solution of the central force equation of motion and the Laplacian MO are solved simultaneously. It can also be considered as a multipole normalization factor such those of the spherical harmonics and the spherical geometric factor of atomic electrons that gives the central force as a function of ⁇ only.
  • ground state hydrogen-type molecular ion is an integer p .
  • the integer is one in the case of the hydrogen molecular ion and an integer greater than one in the case of each dihydrino molecular ion.
  • the central-electric-force constant, k from the two protons that includes the central-field contribution due photons of lower-energy states is
  • the mass and charge density along the ellipse is such that the magnitudes of the radial and transverse forces components at point (0,b) are equivalent.
  • the central force of each proton at a focus is separable and symmetrical to that at the other focus.
  • the transverse forces of the two protons are in opposite directions and the radial components are in the same direction. But, the relationship between the magnitudes must still hold wherein at point (0,b) the transverse force is equivalent to that due to the sum of the charges at one focus.
  • the sum of the magnitudes of the transverse forces which is equivalent to a force of 2e at each focus is
  • the centrifugal force along the radial vector from each proton at each focus of the ellipsoid is given by the mr ⁇ 2 term of Eq. (11.5).
  • the tangent plane at any point on the ellipsoid makes equal angles with the foci radii at that point and the sum of the distance to the foci is a constant, 2a .
  • the normal is the bisector of the angle between the foci radii at that point as shown in Figure 1.
  • the transverse component of the central force of one foci at any point on the elliptic orbit due to the central force of the other (Eq. (11.5)) must cancel on average and vice versa.
  • the centrifugal force due to the superposition of the central forces in the direction of each foci must be normal to an ellipsoidal surface in the direction perpendicular to the direction of motion. Thus, it is in the ⁇ -direction. This can be only be achieved by a time rate of change of the momentum density that compensates for the variation of the distances from each focus to each point on an elliptical cross section. Since the angular momentum must be conserved, there can be no net force in the direction transverse to the elliptical path over each orbital path. The total energy must also be conserved; thus, as shown infra, the distribution of the mass must also be a solution of Laplace's equation in the parameter ⁇ only.
  • the mass-density constraint is the same as the charge-density constraint.
  • the distribution and concomitantly the centrifugal force is a function of D , the time-dependent distance from the center of the ellipsoid to a tangent plane given by Eq. (11.44) where D and the Cartesian coordinates are the time-dependent parameters.
  • Each point or coordinate position on the continuous two-dimensional electron MO defines an infinitesimal mass-density element which moves along an orbit comprising an elliptical plane cross section of the spheroidal MO through the foci.
  • the kinetic energy of the electron is conserved.
  • the current density J is given by the product of the constant frequency (Eq. (11.24)) and the charge density (Eq. (11.40)):
  • F c has an equivalent dependence on D as the electric force based on the charge distribution
  • V i (t) m l r(t)xv(t)
  • Eqs. (11.63-11.65) From Eqs. (11.63-11.65), the result of Eq. (11.113) can be used to the obtain the electric force F ele between the protons and the ellipsoidal MO as
  • the potential energy is doubled due to the transverse electric force.
  • the force normal to the MO is given by the dot product of the sum of the force vectors from each focus with d
  • V 2 ⁇ 2pe D ab V ⁇
  • T is given by the corresponding integral of the centrifugal force (LHS of Eq. (11.115)) with the constraint that the current motion allows the equipotential and equal energy condition with a central field due to the protons; thus, it is corrected by the scale factor Ti 100 given by Eq. (11.62).
  • the Jt 200 correction can be considered the scaling factor of the moment of inertial such that the kinetic energy is equivalent to the rotational energy for constant angular frequency ⁇ .
  • the kinetic energy, T of the electron MO is given by
  • V p ⁇ (11.123)
  • the total energy, which includes the proton-proton-repulsion term is negative which justifies the original treatment of the force balance using the analytical-mechanics equations of an ellipse that considered only the binding force between the protons and the electron and the electron centrifugal force.
  • T is one-half the magnitude of V e as required for an inverse- squared force [1] wherein V e is the source of T .
  • J(k, ⁇ ) ⁇ J m (kcos ⁇ d) ⁇ [ ⁇ -(m + l) ⁇ 0 ] + ⁇ [ ⁇ -(m- ⁇ ) ⁇ 0 ] ⁇ (11.126)
  • J n 's are Bessel functions of order m . These Fourier components can, and do, acquire phase velocities that are equal to the velocity of light [10].
  • the protons of hydrogen-type molecular ions and molecules oscillate as simple harmonic oscillators; thus, vibrating protons will radiate.
  • non-oscillating protons may be excited by one or more photons that are resonant with the oscillatory resonance frequency of the molecule or molecular ion, and oscillating protons may be further excited to higher energy vibrational states by resonant photons.
  • the energy of a photon is quantized according to Planck's equation
  • the energy of a vibrational transition corresponds to the energy difference between the initial and final vibrational states.
  • Each state has an electromechanical resonance frequency, and the emitted or absorbed photon is resonant with the difference in frequencies.
  • quantization of the vibrational spectrum is due to the quantized energies of photons and the electromechanical resonance of the vibrationally excited ion or molecule. It is shown by Fowles [11] that a perturbation of the orbit determined by an inverse- squared force results in simple harmonic oscillatory motion of the orbit. In a circular orbit in spherical coordinates, the transverse equation of motion gives
  • An apsis is a point in an orbit at which the radius vector assumes an extreme value
  • the apsidal angle in this case is just the amount by which the polar angle ⁇ increases during the time that r oscillates from a minimum value to the succeeding maximum value which is
  • ⁇ ⁇ (3 + n) ⁇ m (11.140)
  • the apsidal angle is independent of the size of the orbit in this case.
  • a prolate spheroid MO and the definition of axes are shown in Figures 5A and 5B, respectively.
  • the two nuclei A and B each at focus of the prolate spheroid MO. From Eqs. (11.115), (11.117), and (11.119), the attractive force between the electron and each nucleus at a focus is
  • the distance from the position of the electron MO at the semimajor axis to the opposite nuclear repelling center at the opposite focus is given by the sum of the semimajor axis, a , and c ⁇ 111 the internuclear distance.
  • the contribution from the repulsive force between the two protons is
  • E TotaMb The total energy of the oscillating molecular ion, E TotaMb , is given as the sum of the kinetic and potential energies
  • the resonance condition between these frequencies is to be satisfied in order to have a net change of the energy field [13].
  • the bound electrons are excited with the oscillating protons.
  • the mechanical resonance frequency, ⁇ 0 is only one-half that of the electromechanical frequency which is equal to the frequency of the free space photon, ⁇ , which excites the vibrational mode of the hydrogen molecule or hydrogen molecular ion.
  • the vibrational energy, E v ⁇ corresponding to the photon is given by
  • Ar(O) ZlOS-SI iVw- 1 (11.162)
  • the spring constant and vibrational frequency for the formed molecular ion are then obtained from Eqs. (11.136) and (11.141-11.145) using the increases in the semimajor axis and internuclear distances due to vibration in the transition state.
  • E vib (I) P 2 0.27O eV (11.166) where ⁇ is the vibrational quantum number.
  • a harmonic oscillator is a linear system as given by Eq. (11.146).
  • the predicted resonant vibrational frequencies and energies, spring constants, and amplitudes for Hj (l/p) for vibrational transitions to higher energy U 1 ⁇ ⁇ f are given by ( ⁇ f -o ⁇ times the corresponding parameters given by Eq. (11.160) and Eqs. (11.162-11.164).
  • excitation of vibration of the molecular ion by external radiation causes the semimajor axis and, consequently, the internuclear distance to increase as a function of the vibrational quantum number ⁇ .
  • the vibrational energies of hydrogen-type molecular ions are nonlinear as a function of the vibrational quantum number ⁇ .
  • the lines become more closely spaced and the change in amplitude, ⁇ A reduced , between successive states becomes larger as higher states are excited due to the distortion of the molecular ion in these states.
  • the energy difference of each successive transition of the vibrational spectrum can be obtained by considering nonlinear terms corresponding to anharmonicity.
  • the harmonic oscillator potential energy function can be expanded about the internuclear distance and expressed as a Maclaurin series corresponding to a Morse potential after Karplus and Porter (K&P) [14] and after Eq. (11.134). Treating the Maclaurin series terms as anharmonic perturbation terms of the harmonic states, the energy corrections can be found by perturbation methods.
  • the electron orbiting the nuclei at the foci of an ellipse may be perturbed such that a stable reentrant orbit is established that gives rise to a vibrational state corresponding to time harmonic oscillation of the nuclei and electron.
  • the perturbation is caused by a photon that is resonant with the frequency of oscillation of the nuclei wherein the radiation is electric dipole with the corresponding selection rules.
  • Oscillation may also occur in the transition state.
  • the perturbation arises from the decrease in internuclear distance as the molecular bond forms.
  • the reentrant orbit may give rise to a decrease in the total energy while providing a transient kinetic energy to the vibrating nuclei.
  • radiation must be considered.
  • the nuclei may be considered point charges. A point charge undergoing periodic motion accelerates and as a consequence radiates according to the Larmor formula (cgs units) [15]:
  • the radiation has a corresponding force that can be determined based on conservation of energy with radiation.
  • the radiation reaction force, F rad given by Jackson [16] is
  • the spectroscopic linewidth arises from the classical rise-time band- width relationship, and the Lamb Shift is due to conservation of energy and linear momentum and arises from the radiation reaction force between the electron and the photon.
  • the radiation reaction force in the case of the vibration of the molecular ion in the transition state corresponds to a Doppler energy, E D , that is dependent on the motion of the electron and the nuclei.
  • E D Doppler energy
  • E R is the recoil energy which arises from the photon's linear momentum given by Eq. (2.141)
  • E x is the vibrational kinetic energy of the reentrant orbit in the transition state
  • M is the mass of the electron m e .
  • the coefficient of x in Eq. (11.135) is positive, and the equation is the same as that of the simple harmonic oscillator. Since the electron of the hydrogen molecular ion is perturbed as the internuclear separation decreases with bond formation, it oscillates harmonically about the semimajor axis given by Eq. (11.116), and an approximation of the angular frequency of this oscillation is
  • the total energy of the molecular ion is decreased by E D .
  • the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency given in the Vibration of Hydrogen-Type Molecular Ions section.
  • the total energy of vibration is equally distributed between kinetic energy and potential energy [17].
  • the average kinetic energy of vibration corresponding to the Doppler energy of the electrons, E Kv ⁇ b is 1/2 of the vibrational energy of the molecular ion given by Eq. (11.166).
  • the total energy of the hydrogen molecular ion which is equivalent to the negative of the ionization energy is given by the sum of E ⁇ (Eqs. (11.121) and (11.125)) and E osc given by Eqs. (11.185-11.188).
  • E ⁇ Eqs. (11.121) and (11.125)
  • E osc Eqs. (11.185-11.188)
  • the bond dissociation energy, E D is the difference between the total energy of the corresponding hydrogen atom or H(I/ p) atom [18-19], called hydrino atom having a principal quantum number ⁇ l p where p is an integer, and E 7 , .
  • E D deuterium molecular ion bond energy
  • Hydrogen-type molecules comprise two indistinguishable electrons bound by an elliptic field. Each electron experiences a centrifugal force, and the balancing centripetal force (on each electron) is produced by the electric force between the electron and the elliptic electric field and the magnetic force between the two electrons causing the electrons to pair.
  • the angular frequency given by Eq. (11.24) corresponds to a Lorentzian invariant magnetic moment of a Bohr magneton, ⁇ B , as given in the Magnetic Moment of an
  • the internal field is uniform along the major axis, and the far field is that of a dipole as shown in the Magnetic Field of an Ellipsoidal MO section.
  • the magnetic force is derived by first determining the interaction of the two electrons due to the field of the outer electron 2 acting on the magnetic moments of electron 1 and vice versa. Insight to the behavior is given by considering the physics of a single bound electron in an externally applied uniform magnetic field as discussed in the Two-Electron Atoms section.
  • the electron spin angular momentum gives rise to a trapped photon with % of angular momentum along an S -axis.
  • the dipole spins about the S -axis at the angular velocity given by Eq.(1.55) with h of angular momentum.
  • S rotates about the z-axis at the Larmor frequency
  • the orbitsphere can serve as a basis element to form a molecular orbital (MO).
  • MO molecular orbital
  • the total magnitude of the angular momentum of h is conserved for each member of the linear combinations of F 0 0 ( ⁇ , ⁇ ) 's in the transition from the 7 0 ° ( ⁇ , ⁇ ) 's to the MO. Since the charge and current densities are equivalent by the ratio of the frequency, the solution of Laplace's equation for the charge density that is an equipotential energy surface also determines the current density.
  • the frequency and the velocity are given by Newton's laws.
  • the further constraint from Newton's laws that the orbital surface is a constant total energy surface and the condition of nonradiation provide that the angular velocity of each point on the surface is constant, the current is continuous and constant, and determines the corresponding velocity function.
  • the nonuniform charge distribution given by Laplace's equation is compensated by a nonuniform velocity distribution such that the constant current condition is met.
  • the conservation of the angular momentum is provided by symmetrically stretching the current density along an axis perpendicular to the plane defined by the orthogonal components of angular momentum.
  • the angular momentum projection may be determined by first considering the case of the hydrogen molecular ion.
  • the angular momentum must give the results of the Stern-Gerlach experiment as shown for atomic electrons and free electrons in the Resonant Precession of the Spin- 1/2- Current-Density Function Gives Rise to the Bohr Magneton section and Stern-Gerlach Experiment section, respectively.
  • the hydrogen-molecular-ion MO, and all MOs in general, have cylindrical symmetry along the bond axis.
  • the two orthogonal semiminor axes are equivalent and interchangeable.
  • Y Q ( ⁇ , ⁇ ) can serve as a basis element for an MO having equal angular momentum projections along each of the semiminor axes. This defines the plane and the orthogonal axis for stretching the Y Q ( ⁇ , ⁇ ) basis element to form the MO.
  • ⁇ J, 0 ⁇ , ⁇ ) is stretched along the semimajor axis as shown in Figure 2.
  • the Larmor-excitation photon carries Ti of angular momentum that gives rise to a prolate spheroidal dipole current about an S -axis in the same manner as in the case of the spherical dipole of the Larmor excited orbitsphere shown in Figures 1.15 and 1.16 in Chapter 1.
  • the S -axis is the direction of the magnetic moment of each unpaired electron of a molecule or molecular ion.
  • the magnetic moment of S of ⁇ B corresponding to its h of angular momentum is consistent with the Stern-Gerlach experiment wherein the Larmor excitation can only be parallel or antiparallel to the magnetic field in order to conserve the angular momentum of the electron, the photon corresponding to the Larmor excitation, and the h of angular momentum of the photon that causes a 180° flip of the direction of S .
  • the hydrogen-type molecule is formed by the binding of an electron 2 to the hydrogen-type molecular ion comprising two protons at the foci of the prolate spheroidal MO of electron 1.
  • the ellipsoids of electron 1 and electron 2 are confocal; thus, the electric fields and the corresponding forces are normal to the each MO of electron 1 and electron 2.
  • the two electrons are bound by the central field of the two protons as in the case of the molecular ion. Since the field of the protons is only ellipsoidal on average, the field of the hydrogen- type molecular ion is not equivalent to an ellipsoid of charge +1 outside of the electron MO. In addition there is a spin pairing force between the two electrons.
  • molecule is not predicted to be infrared active. However, it is predicted to be Raman active due to the quadrupole moment.
  • the liquefaction temperature of H 2 is also predicted to be significantly higher than isoelectronic helium.
  • each energy component is the total for the two equivalent electrons with the central-force action at the position of the electron MO where the parameters a and b are given by Eqs. (11.202) and (11.205), respectively.
  • V e potential energy of the two-electron MO comprising equivalent electrons in the field of magnitude p times that of the two protons at the foci is
  • T is one-half the magnitude of V e as required for an inverse-squared force [1] wherein V e is the source of T .
  • the vibrational energy levels of hydrogen-type molecules may be solved in the same manner as hydrogen-type molecular ions given in the Vibration of Hydrogen-type Molecular Ions section.
  • the corresponding central force terms of Eq. (11.136) are and
  • the distance for the reactive nuclear-repulsive terms is given by the sum of the semimajor axis, a , and c ⁇ 1/2 the internuclear distance.
  • the contribution from the repulsive force between the two protons is
  • the spring constant and vibrational frequency for the formed molecule are then obtained from Eqs. (11.136) and (11.213-11.222) using the increases in the semimajor axis and internuclear distances due to vibration in the transition state.
  • the vibrational energies of successive states are given by Eqs. (11.167) and (11.223-11.224).
  • the radiation reaction force in the case of the vibration of the molecule in the transition state also corresponds to the Doppler energy, E D , given by Eq. (11.181) that is dependent on the motion of the electrons and the nuclei.
  • E D Doppler energy
  • a nonradiative state must also be achieved after the emission due to transient vibration wherein the nonradiative condition given by Eq. (11.24) must be satisfied.
  • a third body is required to form hydrogen-type molecules. For example, the exothermic chemical reaction of H + H to form H 2 does not occur with the emission of a photon.
  • the reaction requires a collision with a third body, M , to remove the bond energy — H + H +M ⁇ H 2 +M* [21].
  • the third body distributes the energy from the exothermic reaction, and the end result is the H 2 molecule and an increase in the temperature of the system.
  • a third body removes the energy corresponding to the additional force term given by Eq. (11.180). From Eqs. (11.200), (11.207) and (11.209), the central force terms between the electron MO and the two protons are
  • the total energy of the molecule is decreased by E D .
  • the nuclei In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency given in the Vibration of Hydrogen-Type
  • K sc -/0.326469 eV+-p 2 (0.401380 eV) (11.238) TOTAL, IONIZATION, AND BOND ENERGIES OF HYDROGEN AND DEUTERIUM MOLECULES
  • the total energy of the hydrogen molecule is given by the sum of E 7 . (Eqs. (11.211-11.212)) and E osc given Eqs. (11.233-11.236).
  • E 7 . Eqs. (11.211-11.212)
  • E osc Eqs. (11.233-11.236)
  • the total energy, which includes the proton-proton-repulsion term is negative which justifies the original treatment of the force balance using the analytical mechanics equation of an ellipse that considered only the binding force between the protons and the electrons, the spin- pairing force, and the electron centrifugal force.
  • IP 2 /16.180 eV + p 3 0.imileV (11.248)
  • the bond dissociation energy, E 0 is the difference between the total energy of the corresponding hydrogen atoms and E 1 ,
  • E D /4.151 eF + /0.326469 eV
  • the experimental internuclear distance is 2a 0 .
  • Eqs. (11.262-11.267) the radius of the hydrogen atom a H (Eq. (1.287)) was used in place of ⁇ 0 to account for the corresponding electrodynamic force between the electron and the nuclei as given in the case of the hydrogen atom by Eq. (1.231).
  • the negative of Eq. (11.267) is the ionization energy of H 2 + and the second ionization energy, IP 2 , of H 2 .
  • the total energy, E 7 . for the deuterium molecular ion (the ionization energy of D 2 and the second ionization energy, IP 2 , of D 2 ) is
  • the bond dissociation energy, E D is the difference between the total energy of the corresponding hydrogen atom and E 1 , .
  • the experimental internuclear distance is -J2a o .
  • the bond dissociation energy, E D is the difference between the total energy of two of the corresponding hydrogen atoms and E ⁇ .
  • the results of the determination of the bond, vibrational, total, and ionization energies, and internuclear distances for hydrogen and deuterium molecules and molecular ions are given in Table 11.1.
  • the calculated results are based on first principles and given in closed form equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
  • the experimental total energy of the hydrogen molecule is given by adding the first (15.42593 eV) [28] and second (16.2494 eV) ionization energies where the second ionization energy is given by the addition of the ionization energy of the hydrogen atom (12.59844 eV) [18] and the bond energy of H 2 (2.651 eV) [22].
  • the experimental total energy of the deuterium molecule is given by adding the first (15.466 eV) [23] and second (16.294 eV) ionization energies where the second ionization energy is given by the addition of the ionization energy of the deuterium atom (12.603 eV) [19] and the bond energy of D 2 (2.692 eV) [23].
  • IP 2 The experimental second ionization energy of the hydrogen molecule, IP 2 , is given by the sum of the ionization energy of the hydrogen atom (12.59844 eV) [18] and the bond energy of H 2 (2.651 eV) [22].
  • the experimental second ionization energy of the deuterium molecule, IP 2 is given by the sum of the ionization energy of the deuterium atom (12.603 eV) [19] and the bond energy of D 2 (2.692 eV) [23].
  • the internuclear distances are not corrected for the reduction due to E osc .
  • the internuclear distances are not corrected for the increase due to E n ⁇ .
  • the energy, V , of the magnetic force is
  • E D E T (2H(I Ip)) -E x (H 2 (Vp))
  • the internuclear distance can also be determined geometrically.
  • the spheroidal MO of the hydrogen molecule is an equipotential energy surface, which is an energy minimum surface.
  • the electric field is zero for ⁇ > 0.
  • two hydrogen atoms A and B approaching each other Consider that the two electrons form a spheroidal MO as the two atoms overlap, and the charge is distributed such that an equipotential two- dimensional surface is formed.
  • the electric fields of atoms A and B add vectorially as the atoms overlap.
  • the energy at the point of intersection of the overlapping orbitspheres decreases to a minimum as they superimpose and then rises with further overlap. When this energy is a minimum the internuclear distance is determined. It can be demonstrated [33] that when two hydrogen orbitspheres superimpose such that the radial electric field vector from nucleus A and B makes a 45° angle with the point of intersection of the two original orbitspheres, the electric energy of interaction between orbitspheres given by
  • f act i on ' (11.350) is a minimum ( Figure 7.1 of [33]).
  • the MO is a minimum potential energy surface; therefore, a minimum of energy of one point on the surface is a minimum for the entire surface of the MO.
  • the experimental internuclear bond distance is 0.746 A .
  • the first ionization energy, IP 1 of the dihydrino molecule
  • IP 1 E 7 [H ⁇ (IZp))- E 1 . (H 2 (IZp)) (11.353)
  • a hydrino atom can react with a hydrogen, deuterium, or tritium nucleus to form a dihydrino molecular ion that further reacts with an electron to form a dihydrino molecule.
  • the energy released is the
  • E E(H(V p))-E T (11.357) where E 7 . is given by Eq. (11.241).
  • a hydrino atom can react with a hydrogen, deuterium, or tritium atom to form a dihydrino molecule.
  • the energy released is the
  • E E(H (Hp)) + E(H)- E T (11.359) where E 1 . is given by Eq. (11.241).
  • He Helium Atom
  • Each proton of hydrogen-type molecules possesses a magnetic moment, which is derived in the Proton and Neutron section and is given by
  • the frequency, / can be determined from the energy using the Planck relationship, Eq. (2.18).
  • the NMR frequency, / is the product of the proton gyromagnetic ratio given by Eq. (11.366) and the magnetic flux, B .
  • a typical flux for a superconducting NMR magnet is 1.5 T . According to Eq. (11.367) this corresponds to a radio frequency (RF) of 63.86403 MHz .
  • RF radio frequency
  • the frequency is scanned to yield the spectrum where the frequency scan is typically achieved using a Fourier transform on the free induction decay signal following a radio frequency pulse.
  • the radiofrequency is held constant
  • the frequency of energy absorption is recorded at the various values for H 0 .
  • the spectrum is typically scanned and displayed as a function of increasing H 0 .
  • the protons that absorb energy at a lower H 0 give rise to a downfield absorption peak; whereas, the protons that absorb energy at a higher H 0 give rise to an upfield absorption peak.
  • the electrons of the compound of a sample influence the field at the nucleus such that it deviates slightly from the applied value.
  • the value of H 0 at resonance with the radiofrequency held constant at 60 MHz is ⁇ L. (2»)(60 iffl . ) 3 ⁇ o ⁇ P // 0 42.57 '602 MHz T '1 °
  • the current of hydrogen-type molecules is along elliptical orbits parallel to the semimajor axis.
  • the electronic interaction with the nuclei requires that each nuclear magnetic moment is in the direction of the semiminor axis.
  • the electric field, E along a perpendicular elliptic path of the dihydrino MO at the plane z - 0 is given by
  • a is the semimajor axis given by Eq. (11.202)
  • b is the semiminor axis given by Eq. (11.205)
  • e is the eccentricity given by Eq. (11.206).
  • the acceleration along the path, dv/dt , during the application of the flux is determined by the electric force on the charge density of the electrons: dv pursue e ⁇ ab dB ,. . occidental__.
  • the average current, I of a charge moving time harmonically along an ellipse is
  • Eq. (11.383) is an ellipse with semimajor axis, a ' , and semiminor axis, b ' , given by
  • the eccentricity, e ' is given by a where e is given by Eq. (11.372).
  • the area, A ' is given by
  • X ⁇ -+Y ⁇ -+Z- ⁇ - 1 (11.393) a 2 b e
  • X, Y, Z are running coordinates in the plane.
  • the surface density at any point on the ellipsoidal MO is proportional to the perpendicular distance from the center of the ellipsoid to the plane tangent to the ellipsoid at the point.
  • the charge is thus greater on the more sharply rounded ends farther away from the origin.
  • the MO is an equipotential surface, and the current must be continuous over the two- dimensional surface.
  • the charge density is spheroidally symmetrical about the semimajor axis.
  • X the charge density per unit length along each elliptical path cross section of Eq. (11.383) is given by distributing the surface charge density of Eq.
  • the two electrons are spin-paired and the velocities are mirror opposites.
  • the change in velocity of each electron treated individually (Eq. (10.3)) due to the applied field would be equal and opposite.
  • the two paired electrons may be treated as one with twice the mass where m e is replaced by 2m e in Eq. (11.399).
  • the paired electrons spin together about the applied field axis, the z-axis, to cause a reduction in the applied field according to Lenz's law.
  • the change in magnetic moment is given by
  • the opposing diamagnetic flux is uniform, parallel, and opposite the applied field as given by Stratton [36]. Specifically, the change in magnetic flux, ⁇ B , at the nucleus due to the change in magnetic moment, ⁇ m , is
  • the ratio of the radius of the hydrino hydride ion H ' (l/ p) to that of the hydride ion H ⁇ (l/l) is the reciprocal of an integer p . It follows from Eqs. (7.90-7.96) that compared to a proton with no chemical shift, the ratio of AH 0 for resonance of the proton of the hydrino hydride ion H ⁇ (l / p) to that of the hydride ion H ⁇ (l/l) is a positive integer.
  • the absorption peak of the hydrino hydride ion occurs at a value of AH 0 that is a multiple of p times the value that is resonant for the hydride ion compared to that of a proton with no shift.
  • a hydrino hydride ion is equivalent to the ordinary hydride ion except that it is in a lower energy state. The source current of the state must be considered in addition to the reduced radius.
  • the ratio of the total charge distributed over the surface at the radius of the hydrino hydride ion H ⁇ (l/ p) to that of the hydride ion H ⁇ (l/l) is an integer p
  • the corresponding total source current of the hydrino hydride ion is equivalent to an integer p times that of an electron.
  • the "trapped photon" obeys the phase-matching condition given in Excited States of the One-Electron Atom (Quantization) section, but does not interact with the applied flux directly. Only each electron does; thus, ⁇ v of Eq.
  • Eqs. (2.159-2.160) Eq. (2.166) may be written as , ⁇ ⁇ ⁇ ⁇ _ N (11.415)
  • the relativistic stored magnetic energy contributes a factor of a2 ⁇
  • the stored magnetic energy term of the electron g factor of each electron of a ⁇ dihydrino molecule is the same as that of a hydrogen atom since — is invariant and the m e invariant angular momentum and magnetic moment of the former are also fi and ⁇ B , respectively, as given in the Magnetic Moment of an Ellipsoidal MO and Magnetic Field of an Ellipsoidal MO sections.
  • the corresponding correction in ellipsoidal coordinates follows from Eq. (2.166) wherein the result of the length contraction for the circular path in spherical coordinates is replaced by that of the elliptical path.
  • the parametric radius, r(t) is a minimum at the position of the semiminor axis of length b , and the motion is transverse to the radial vector. Since the angular momentum of h is constant, the electron wavelength without relativistic correction is given by
  • H 2 has been characterized by gas phase 1 H NMR.
  • the experimental absolute resonance shift of gas-phase TMS relative to the proton's gyromagnetic frequency is -28.5 ppm [30].
  • H 2 was observed at 0.48 ppm compared to gas phase TMS set at 0.00 ppm [31].
  • Non-hydrogen diatomic and polyatomic molecular ions and molecules can be solved using the same principles as those used to solve hydrogen molecular ions and molecules wherein the hydrogen molecular orbitals (MOs) and hydrogen atomic orbitals serve as basis functions for the MOs of the general diatomic and polyatomic molecular ions or molecules.
  • the MO must (1) be a solution of Laplace's equation to give a equipotential energy surface, (2) correspond to an orbital solution of the Newtonian equation of motion in an inverse-radius- squared central field having a constant total energy, (3) be stable to radiation, and (4) conserve the electron angular momentum of h .
  • Energy of the MO must be matched to that of the outermost atomic orbital of a bonding heteroatom in the case where a minimum energy is achieved with a direct bond to the atomic orbital (AO).
  • the AO force balance causes the remaining electrons to be at lower energy and a smaller radius.
  • the atomic orbital may hybridize in order to achieve a bond at an energy minimum. At least one molecule or molecular ion representative of each of these cases was solved.
  • the polyatomic molecular ion H 3 ( ⁇ /p) is formed by the reaction of a proton with a hydrogen-type molecule
  • the surface density at any point on a charged ellipsoidal conductor is proportional to the perpendicular distance from the center of the ellipsoid to the plane tangent to the ellipsoid at the point.
  • the charge is thus greater on the more sharply rounded ends farther away from the origin. This distribution places the charge closest to the protons to give a minimum energy.
  • the balanced forces also depend on D as shown in the Nature of the Chemical Bond of Hydrogen-Type Molecules section.
  • H 3 + (l/p) MO comprising the superposition of three H 2 (I/ ⁇ ) -type ellipsoidal MOs is shown in Figure 6.
  • the outer surface of the superposition comprises charge density of the MO.
  • the equilateral triangular structure was confirmed experimentally [I].
  • the H 3 + (l/ p) MO having no distinguishable electrons is consistent with the absence of strong excited stated observed for H 3 + [I]. It is also consistent with the absence of a permanent dipole moment [I].

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Abstract

A method and system of physically solving the charge, mass, and current density functions of polyatomic molecules, polyatomic molecular ions, diatomic molecules, molecular radicals, molecular ions, or any portion of these species using Maxwell's equations and computing and rendering the physical nature of the chemical bond using the solutions. The results can be displayed on visual or graphical media. The display can be static or dynamic such that electron motion and specie's vibrational, rotational, and translational motion can be displayed in an embodiment. The displayed information is useful to anticipate reactivity and physical properties. The insight into the nature of the chemical bond of at least one specie can permit the solution and display of those of other species to provide utility to anticipate their reactivity and physical properties.

Description

SYSTEM AND METHOD OF COMPUTING AND RENDERING THE NATURE OF POLYATOMIC MOLECULES AND POLYATOMIC MOLECULAR IONS
This application claims priority to U.S. Application Nos.: 60/730,882, filed October 28, 2005; 60/732,154, filed November 2, 2005; 60/737,744, filed November 18, 2005;
60/758,528, filed January 13, 2006; 60/780,518, filed March 9, 2006; 60/788,694, filed April 4, 2006; 60/812,590, filed June 12, 2006; and 60/815,253, June 21, 2006, the complete disclosures of which are incorporated herein by reference.
Field of the Invention:
This invention relates to a system and method of physically solving the charge, mass, and current density functions of polyatomic molecules, polyatomic molecular ions, diatomic molecules, molecular radicals, molecular ions, or any portion of these species, and computing and rendering the nature of these species using the solutions. The results can be displayed on visual or graphical media. The displayed information provides insight into the nature of these species and is useful to anticipate their reactivity, physical properties, and spectral absorption and emission, and permits the solution and display of other species.
Rather than using postulated unverifiable theories that treat atomic particles as if they were not real, physical laws are now applied to atoms and ions. In an attempt to provide some physical insight into atomic problems and starting with the same essential physics as Bohr of the e~ moving in the Coulombic field of the proton with a true wave equation, as opposed to the diffusion equation of Schrδdinger, a classical approach is explored which yields a model that is remarkably accurate and provides insight into physics on the atomic level. The proverbial view deeply seated in the wave-particle duality notion that there is no large-scale physical counterpart to the nature of the electron is shown not to be correct. Physical laws and intuition may be restored when dealing with the wave equation and quantum atomic problems.
Specifically, a theory of classical quantum mechanics (CQM) was derived from first principles as reported previously [reference Nos. 1-8] that successfully applies physical laws to the solution of atomic problems that has its basis in a breakthrough in the understanding of the stability of the bound electron to radiation. Rather than using the postulated Schrδdinger boundary condition: "Ψ — » 0 as r —* ∞ ", which leads to a purely mathematical model of the electron, the constraint is based on experimental observation. Using Maxwell's equations, the classical wave equation is solved with the constraint that the bound n = \ -state electron cannot radiate energy. Although it is well known that an accelerated point particle radiates, an extended distribution modeled as a superposition of accelerating charges does not have to radiate. A simple invariant physical model arises naturally wherein the predicted results are extremely straightforward and internally consistent requiring minimal math, as in the case of the most famous equations of Newton, Maxwell, Einstein, de Broglie, and Planck on which the model is based. No new physics is needed; only the known physical laws based on direct observation are used.
Applicant's previously filed WO2005/067678 discloses a method and system of physically solving the charge, mass, and current density functions of atoms and atomic ions and computing and rendering the nature of these species using the solutions. The complete disclosure of this published PCT application is incorporated herein by reference.
Applicant's previously filed WO2005/116630 discloses a method and system of physically solving the charge, mass, and current density functions of excited states of atoms and atomic ions and computing and rendering the nature of these species using the solutions. The complete disclosure of this published PCT application is incorporated herein by reference.
Applicant's previously filed U.S. Published Patent Application No. 20050209788A1, relates to a method and system of physically solving the charge, mass, and current density functions of hydrogen-type molecules and molecular ions and computing and rendering the nature of the chemical bond using the solutions. The complete disclosure of this published application is incorporated herein by reference.
Background of the Invention The old view that the electron is a zero or one-dimensional point in an all-space probability wave function Ψ(x) is not taken for granted. The theory of classical quantum mechanics (CQM), derived from first principles, must successfully and consistently apply physical laws on all scales [1-8]. Stability to radiation was ignored by all past atomic models. Historically, the point at which QM broke with classical laws can be traced to the issue of nonradiation of the one electron atom. Bohr just postulated orbits stable to radiation with the further postulate that the bound electron of the hydrogen atom does not obey Maxwell's equations — rather it obeys different physics [1-12]. Later physics was replaced by "pure mathematics" based on the notion of the inexplicable wave-particle duality nature of electrons which lead to the Schrδdinger equation wherein the consequences of radiation predicted by Maxwell's equations were ignored. Ironically, Bohr, Schrδdinger, and Dirac used the Coulomb potential, and Dirac used the vector potential of Maxwell's equations. But, all ignored electrodynamics and the corresponding radiative consequences. Dirac originally attempted to solve the bound electron physically with stability with respect to radiation according to Maxwell's equations with the further constraints that it was relativistically invariant and gave rise to electron spin [13]. He and many founders of QM such as Sommerfeld, Bohm, and Weinstein wrongly pursued a planetary model, were unsuccessful, and resorted to the current mathematical-probability-wave model that has many problems [9- 16]. Consequently, Feynman for example, attempted to use first principles including Maxwell's equations to discover new physics to replace quantum mechanics [17].
Physical laws may indeed be the root of the observations thought to be "purely quantum mechanical", and it was a mistake to make the assumption that Maxwell's electrodynamic equations must be rejected at the atomic level. Thus, in the present approach, the classical wave equation is solved with the constraint that a bound n = 1 -state electron cannot radiate energy.
Herein, derivations consider the electrodynamic effects of moving charges as well as the Coulomb potential, and the search is for a solution representative of the electron wherein there is acceleration of charge motion without radiation. The mathematical formulation for zero radiation based on Maxwell's equations follows from a derivation by Haus [18]. The function that describes the motion of the electron must not possess spacetime Fourier components that are synchronous with waves traveling at the speed of light. Similarly, nonradiation is demonstrated based on the electron's electromagnetic fields and the Poynting power vector. It was shown previously [1-8] that CQM gives closed form solutions for the atom including the stability of the n = 1 state and the instability of the excited states, the equation of the photon and electron in excited states, and the equation of the free electron and photon, which predict the wave particle duality behavior of particles and light. The current and charge density functions of the electron may be directly physically interpreted. For example, spin angular momentum results from the motion of negatively charged mass moving systematically, and the equation for angular momentum, r x p , can be applied directly to the wave function (a current density function) that describes the electron. The magnetic moment of a Bohr magneton, Stern Gerlach experiment, g factor, Lamb shift, resonant line width and shape, selection rules, correspondence principle, wave particle duality, excited states, reduced mass, rotational energies, and momenta, orbital and spin splitting, spin-orbital coupling, Knight shift, and spin-nuclear coupling, and elastic electron scattering from helium atoms, are derived in closed-form equations based on Maxwell's equations. The calculations agree with experimental observations.
The Schrδdinger equation gives a vague and fluid model of the electron. Schrδdinger interpreted eΨ * (x)Ψ(x) as the charge-density or the amount of charge between x and x + dx (Ψ * is the complex conjugate of Ψ ). Presumably, then, he pictured the electron to be spread over large regions of space. After Schrδdinger' s interpretation, Max Born, who was working with scattering theory, found that this interpretation led to inconsistencies, and he replaced the Schrδdinger interpretation with the probability of finding the electron between x and x + dx as
Figure imgf000006_0001
Born' s interpretation is generally accepted. Nonetheless, interpretation of the wave function is a never-ending source of confusion and conflict. Many scientists have solved this problem by conveniently adopting the Schrδdinger interpretation for some problems and the Born interpretation for others. This duality allows the electron to be everywhere at one time — yet have no volume. Alternatively, the electron can be viewed as a discrete particle that moves here and there (from r = 0 to r = ∞), and ΨΨ * gives the time average of this motion. In contrast to the failure of the Bohr theory and the nonphysical, adjustable-parameter approach of quantum mechanics, multielectron atoms [1, 4] and the nature of the chemical bond [1, 5] are given by exact closed-form solutions containing fundamental constants only. Using the nonradiative wave equation solutions that describe the bound electron having conserved momentum and energy, the radii are determined from the force balance of the electric, magnetic, and centrifugal forces that corresponds to the minimum of energy of the system. The ionization energies are then given by the electric and magnetic energies at these radii. The spreadsheets to calculate the energies from exact solutions of one through twenty- electron atoms are given in '06 Mills GUT [1] and are available from the internet [19]. For 400 atoms and ions, as well as hundreds of molecules, the agreement between the predicted and experimental results is remarkable.
The background theory of classical quantum mechanics (CQM) for the physical solutions of atoms and atomic ions is disclosed in R. Mills, The Grand Unified Theory of Classical Quantum Mechanics, January 2000 Edition, BlackLight Power, Inc., Cranbury, New Jersey, (" 1OO Mills GUT"), provided by BlackLight Power, Inc., 493 Old Trenton Road, Cranbury, NJ, 08512; R. Mills, The Grand Unified Theory of Classical Quantum Mechanics, September 2001 Edition, BlackLight Power, Inc., Cranbury, New Jersey, Distributed by Amazon.com (" 01 Mills GUT"), provided by BlackLight Power, Inc., 493 Old Trenton Road, Cranbury, NJ, 08512; R. Mills, The Grand Unified Theory of Classical Quantum Mechanics, July 2004 Edition, BlackLight Power, Inc., Cranbury, New Jersey, (" '04 Mills GUT"), provided by BlackLight Power, Inc., 493 Old Trenton Road, Cranbury, NJ, 08512; R. Mills, The Grand Unified Theory of Classical Quantum Mechanics, January 2005 Edition, BlackLight Power, Inc., Cranbury, New Jersey, (" '05 Mills GUT"), provided by BlackLight Power, Inc., 493 Old Trenton Road, Cranbury, NJ, 08512 (posted at www.blacklightpo wer. com) ; R. L. Mills, "The Grand Unified Theory of Classical Quantum Mechanics", June 2006 Edition, Cadmus Professional Communications-Science Press Division, Ephrata, PA, ISBN 0963517171, Library of Congress Control Number 2005936834, (" O6 Mills GUT"), provided by BlackLight Power, Inc., 493 Old Trenton Road, Cranbury, NJ, 08512 (posted at http://www.blacklightpo wer. com/bookdo wnload. shtmD ; in prior PCT applications PCT/US02/35872; PCT/US02/06945; PCT/US 02/06955; PCT/US01/09055; PCT/USOl/ 25954; PCT/USOO/20820; PCT/USOO/20819; PCT/USOO/09055; PCT/US99/17171; PCT/US99/17129; PCT/US 98/22822; PCT/US98/14029; PCT/US96/07949;
PCT/US94/02219; PCT/US91/08496; PCT/US90/01998; and PCT/US89/05037 and U.S. Patent No. 6,024,935; the entire disclosures of which are all incorporated herein by reference (hereinafter "Mills Prior Publications").
The following list of references, which are also incorporated herein by reference in their entirety, are referred to in the above sections using [brackets]:
1. R. L. Mills, "The Grand Unified Theory of Classical Quantum Mechanics", June 2006 Edition, Cadmus Professional Communications-Science Press Division, Ephrata, PA, ISBN 0963517171, Library of Congress Control Number 2005936834; posted at http://www.blacklightpower.com/bookdownload.shtml.
2. R. L. Mills, "Classical Quantum Mechanics", Physics Essays, Vol. 16, No. 4, December, (2003), pp. 433-498; posted with spreadsheets at www.blacWightpower.com/techpapers.shtrnl. 3. R. Mills, "Physical Solutions of the Nature of the Atom, Photon, and Their Interactions to Form Excited and Predicted Hydrino States", submitted.
4. R. L. Mills, "Exact Classical Quantum Mechanical Solutions for One- Through Twenty- Electron Atoms", in press, posted with spreadsheets at http://www.blacklightpower.com/techpapers.shtml.
5. R. L. Mills, "The Nature of the Chemical Bond Revisited and an Alternative Maxwellian Approach", Physics Essays, Vol. 17, (2004), pp. 342-389, posted with spreadsheets at http^/www.blacklightpower.com/techpapers.shtml.
6. R. L. Mills, "Maxwell's Equations and QED: Which is Fact and Which is Fiction", in press, posted with spreadsheets at http^/www.blacklightpower.com/techpapers.shtml.
7. R. L. Mills, "Exact Classical Quantum Mechanical Solution for Atomic Helium Which Predicts Conjugate Parameters from a Unique Solution for the First Time", submitted, posted with spreadsheets at http://www.blacklightpower.com/theory/theory.shtml..
8. R. Mills, "The Grand Unified Theory of Classical Quantum Mechanics", Int. J. Hydrogen Energy, Vol. 27, No. 5, (2002), pp. 565-590.
9. R. L. Mills, "The Fallacy of Feynman's Argument on the Stability of the Hydrogen Atom According to Quantum Mechanics", Annales de Ia Fondation Louis de Broglie, Vol. 30, No. 2, (2005), pp. 129-151, posted athttp^/www.blacklightpower.com/techpapers.shtml.
10. R. Mills, The Nature of Free Electrons in Superfluid Helium — a Test of Quantum Mechanics and a Basis to Review its Foundations and Make a Comparison to Classical
Theory, Int. J. Hydrogen Energy, Vol. 26, No. 10, (2001), pp. 1059-1096.
11. R. Mills, "The Hydrogen Atom Revisited", Int. J. of Hydrogen Energy, Vol. 25, Issue 12, December, (2000), pp. 1171-1183.
12. F. Laloe, Do we really understand quantum mechanics? Strange correlations, paradoxes, and theorems, Am. J. Phys. 69 (6), June 2001, 655-701.
13. P. Pearle, Foundations of Physics, "Absence of radiationless motions of relativistically rigid classical electron", Vol. 7, Nos. 11/12, (1977), pp. 931-945.
14. V. F. Weisskopf, Reviews of Modern Physics, Vol. 21, No. 2, (1949), pp. 305-315.
15. H. Wergeland, "The Klein Paradox Revisited", Old and New Questions in Physics, Cosmology, Philosophy, and Theoretical Biology, A. van der Merwe, Editor, Plenum
Press, New York, (1983), pp. 503-515.
16. A. Einstein, B. Podolsky, N. Rosen, Phys. Rev., Vol. 47, (1935), p. 777.
17. F. Dyson, "Feynman's proof of Maxwell equations", Am. J. Phys., Vol. 58, (1990), pp. 209-211.
18. Haus, H. A., "On the radiation from point charges", American Journal of Physics, 54, (1986), pp. 1126-1129.
19. http://www.blacklightpower.com/new.shtml.
SUMMARY OF THE INVENTION
The present invention, an exemplary embodiment of which is also referred to as Millsian software, stems from a new fundamental insight into the nature of the atom. Applicant's new theory of Classical Quantum Mechanics (CQM) reveals the nature of atoms and molecules using classical physical laws for the first time. As discussed above, traditional quantum mechanics can solve neither multi-electron atoms nor molecules exactly. By contrast, CQM produces exact, closed-form solutions containing physical constants only for even the most complex atoms and molecules.
The present invention is the first and only molecular modeling program ever built on the CQM framework. All the major functional groups that make up most organic molecules have been solved exactly in closed-form solutions with CQM. By using these functional groups as building blocks, or independent units, a potentially infinite number of organic molecules can be solved. As a result, the present invention can be used to visualize the exact 3D structure and calculate the heat of formation of almost any organic molecule. For the first time, the significant building-block molecules of chemistry have been successfully solved using classical physical laws in exact closed-form equations having fundamental constants only. The major functional groups have been solved from which molecules of infinite length can be solved almost instantly with a computer program. The predictions are accurate within experimental error for over 375 exemplary molecules. Applicant's CQM is the theory that physical laws (Maxwell's Equations, Newton's
Laws, Special and General Relativity) must hold on all scales. The theory is based on an often overlooked result of Maxwell's Equations, that an extended distribution of charge may, under certain conditions, accelerate without radiating. This "condition of no radiation" is invoked to solve the physical structure of subatomic particles, atoms, and molecules. In exact closed-form equations with physical constants only, solutions to thousands of known experimental values arise that were beyond the reach of previous outdated theories. These include the electron spin, g-factor, multi-electron atoms, excited states, polyatomic molecules, wave-particle duality and the nature of the photon, the masses and families of fundamental particles, and the relationships between fundamental laws of the universe that reveal why the universe is accelerating as it expands. CQM is successful to over 85 orders of magnitude, from the level of quarks to the cosmos. Applicant now has over 65 peer-reviewed journal articles and also books discussing the CQM and supporting experimental evidence. The molecular modeling market was estimated to be a two-billion-dollar per year industry in 2002, with hundreds of millions of government and industry dollars invested in computer algorithms and supercomputer centers. This makes it the largest effort of computational chemistry and physics.
The present invention's advantages over other models includes: Rendering true molecular structures; Providing precisely all characteristics, spatial and temporal charge distributions and energies of every electron in every bond, and of every bonding atom; Facilitating the identification of biologically active sites in drugs; and Facilitating drug design.
An objective of the present invention is to solve the charge (mass) and current-density functions of polyatomic molecules, polyatomic molecular ions, diatomic molecules, molecular radicals, molecular ions, or any portion of these species from first principles. In an embodiment, the solution for the polyatomic molecules, polyatomic molecular ions, diatomic molecules, molecular radicals, molecular ions, or any portion of these species is derived from Maxwell's equations invoking the constraint that the bound electron before excitation does not radiate even though it undergoes acceleration.
Another objective of the present invention is to generate a readout, display, or image of the solutions so that the nature of polyatomic molecules, polyatomic molecular ions, diatomic molecules, molecular radicals, molecular ions, or any portion of these species be better understood and potentially applied to predict reactivity and physical and optical properties.
Another objective of the present invention is to apply the methods and systems of solving the nature of polyatomic molecules, polyatomic molecular ions, diatomic molecules, molecular radicals, molecular ions, or any portion of these species and their rendering to numerical or graphical form to all atoms and atomic ions. These objectives and other objectives are obtained by a system of computing and rendering the nature of at least one specie selected from a group of diatomic molecules having at least one atom is other than hydrogen, polyatomic molecules, molecular ions, polyatomic molecular ions, or molecular radicals, or any functional group therein, comprising physical, Maxwellian solutions of charge, mass, and current density functions of said specie, said system comprising processing means for processing physical, Maxwellian equations representing charge, mass, and current density functions of said specie; and an output device in communication with the processing means for displaying said physical, Maxwellian solutions of charge, mass, and current density functions of said specie.
Also provided is a composition of matter comprising a plurality of atoms, the improvement comprising a novel property or use discovered by calculation of at least one of a bond distance between two of the atoms, a bond angle between three of the atoms, and a bond energy between two of the atoms, orbital intercept distances and angles,charge-density functions of atomic, hybridized, and molecular orbitals, the bond distance, bond angle, and bond energy being calculated from, physical solutions of the charge, mass, and current density functions of atoms and atomic ions, which solutions are derived from Maxwell's equations using a constraint that a bound electron(s) does not radiate under acceleration.
The presented exact physical solutions for known species of the group of polyatomic molecules, polyatomic molecular ions, diatomic molecules, molecular radicals, molecular ions, or any functional group therein, can be applied to other species. These solutions can be used to predict the properties of other species and engineer compositions of matter in a manner which is not possible using past quantum mechanical techniques. The molecular solutions can be used to design synthetic pathways and predict product yields based on equilibrium constants calculated from the heats of formation. Not only can new stable compositions of matter be predicted, but now the structures of combinatorial chemistry reactions can be predicted.
Pharmaceutical applications include the ability to graphically or computationally render the structures of drugs that permit the identification of the biologically active parts of the specie to be identified from the common spatial charge-density functions of a series of active species. Novel drugs can now be designed according to geometrical parameters and bonding interactions with the data of the structure of the active site of the drug.
The system can be used to calculate conformations, folding, and physical properties, and the exact solutions of the charge distributions in any given specie are used to calculate the fields. From the fields, the interactions between groups of the same specie or between groups on different species are calculated wherein the interactions are distance and relative orientation dependent. The fields and interactions can be determined using a finite-element- analysis approach of Maxwell's equations. Embodiments of the system for performing computing and rendering of the nature of the polyatomic molecules, polyatomic molecular ions, diatomic molecules, molecular radicals, molecular ions, or any portion of these species using the physical solutions may comprise a general purpose computer. Such a general purpose computer may have any number of basic configurations. For example, such a general purpose computer may comprise a central processing unit (CPU), one or more specialized processors, system memory, a mass storage device such as a magnetic disk, an optical disk, or other storage device, an input means, such as a keyboard or mouse, a display device, and a printer or other output device. A system implementing the present invention can also comprise a special purpose computer or other hardware system and all should be included within its scope. Although not preferred, any of the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
BRIEF DESCRIPTION OF THE DRAWINGS
Fig. 1 illustrates an elliptical current element of the prolate spheroidal MO;
Fig. 2 illustrates the ellipsoidal current-density surface obtained by stretching Y° (θ, φ) along the semimajor axis;
Fig. 3 illustrates the angular momentum components of the MO and S ; Fig. 4 illustrates cross section of an atomic orbital; Fig. 5 illustrates A. Prolate spheroid MO; Fig. 6 illustrates the equilateral triangular H3 + (1//»);
Fig. 7 illustrates the cross section of the OH MO;
Fig. 8 illustrates OH MO comprising the superposition of the H2 -type ellipsoidal MO and the O2py AO with a relative charge-density of 0.75 to 1.25;
Fig. 9 illustrates Tf2O MO comprising the linear combination of two O- H -bond MOs; Fig. 10 illustrates the cross section of the NH MO showing the axes, angles, and point of intersection of the H2 -type ellipsoidal MO with the NIpx AO;
Fig. 11 illustrates NH MO comprising the superposition of the Tf2 -type ellipsoidal MO and the NIpx AO with a relative charge-density of 0.75 to 1.25;
Fig. 12 illustrates JVH2 MO comprising the linear combination of two N-H -bond MOs; Fig. 13 illustrates NH3 MO comprising the linear combination of three N- H -bonds;
Fig. 14 illustrates the cross section of the CH MO showing the axes, angles, and point of intersection of the H2 -type ellipsoidal MO with the dsp* HO;
Fig. 15 illustrates CH MO comprising the superposition of the H2 -type ellipsoidal MO and the C2sp3 HO with a relative charge-density of 0.75 to 1.25;
Fig. 16 illustrates CH2 MO comprising the linear combination of two C- H -bond MOs;
Fig. 17 illustrates CH3 MO comprising the linear combination of three C-H-bond MOs;
Fig. 18 illustrates CH4 MO comprising the linear combination of four C-H-bond MOs formed by the superposition of a H2 -type ellipsoidal MO and a C2spl HO; Fig. 19 illustrates the cross section of the N2 MO;
Fig. 20 illustrates N2 MO comprising the σ MO ( H2 -type MO) with N atoms at the foci;
Fig. 21 illustrates the cross section of the O2 MO;
Fig. 22 illustrates O2 MO comprising the σ MO (H2 -type MO);
Fig. 23 illustrates the cross section of the F2 MO; Fig. 24 illustrates F2 MO comprising the σ MO (H2 -type MO) with F atoms at the foci;
Fig. 25 illustrates the cross section of the CZ2 MO;
Fig. 26 illustrates Cl2 MO comprising the superposition of the H2 -type ellipsoidal MO and the two Cl3sp3 ΗOs;
Fig. 27 illustrates the cross section of the CN MO; Fig. 28 illustrates CN MO;
Fig. 29 illustrates the cross section of the CO MO;
Fig. 30 illustrates CO MO;
Fig. 31 illustrates the cross section of the NO MO;
Fig. 32 illustrates NO MO; Fig. 33 illustrates the cross section of the CO2 MO;
Fig. 34 illustrates CO2 MO; Fig. 35 illustrates the cross section of the NO2 MO; Fig. 36 illustrates NO2 MO; Fig. 37 illustrates the cross section of the C-C -bond MO ( σ MO) and one C-H -bond MO of ethane; Fig. 38 illustrates the cross section of one C -H -bond MO of ethane showing the axes, angles, and point of intersection of the H2 -type ellipsoidal MO with the Celhane2sp3
HO; Fig. 39 illustrates CH3CH3 MO comprising the linear combination of two sets of three
C-H -bond MOs and a C - C -bond MO;
Fig. 40 illustrates the cross section of the C = C -bond MO (σ MO) and one C -H -bond MO of ethylene showing the axes, angles, and point of intersection of each H2 -type ellipsoidal MO with the corresponding Cethylene2sp3 HO; Fig. 41 illustrates the cross section of one C- H -bond MO of ethylene showing the axes, angles, and point of intersection of the H2 -type ellipsoidal MO with the Cejhylene2sp3
HO;
Fig. 42 illustrates CH2CH2 MO comprising the linear combination of two sets of two
C-H -bond MOs and a C = C -bond MO; Fig. 43 illustrates the cross section of the C ≡ C -bond MO ( σ MO) and one C-H -bond MO of acetylene showing the axes, angles, and point of intersection of each H2 -type ellipsoidal MO with the corresponding Cacelylem2sp3 HO;
Fig. 44 illustrates CHCH MO comprising the linear combination of two C-H-bond MOs and a C s C -bond MO; Fig. 45 illustrates the cross section of one C = C -bond MO (σ MO) and one C-H -bond MO of benzene showing the axes, angles, and point of intersection of each H2 -type ellipsoidal MO with the corresponding Cbemem2sp3 HO;
Fig. 46 illustrates the cross section of one C-H -bond MO of benzene showing the axes, angles, and point of intersection of the H2 -type ellipsoidal MO with the Cbenzene2sp3 HO;
Fig. 47 illustrates C6H6 MO comprising the linear combination of six sets of C-H -bond
MOs bridged by C = C -bond MOs;
Fig. 48 illustrates the cross section of one C-C -bond MO ( σ MO) and one C -H -bond MO of CnH2n+2 showing the axes, angles, and point of intersection of each H2 -type ellipsoidal MO with the corresponding Calkam2sp3 HO; Fig. 49 illustrates the cross section of one C- H -bond MO of CnH2n+2 showing the axes, angles, and point of intersection of the H1 -type ellipsoidal MO with the Calkam2sp3
HO;
Fig. 50 illustrates C3H8 MO comprising a linear combination of C-H -bond MOs and C-C -bond MOs of the two methyl groups and one methylene group;
Fig. 51 illustrates C4H10 MO comprising a linear combination of C-H-bond MOs and
C-C -bond MOs of the two methyl and two methylene groups; Fig. 52 illustrates C5H12 MO comprising a linear combination of C-H -bond MOs and
C -C -bond MOs of the two methyl and three methylene groups; Fig. 53 illustrates C6H14 MO comprising a linear combination of C-H -bond MOs and
C-C -bond MOs of the two methyl and four methylene groups; Fig. 54 illustrates C7H16 MO comprising a linear combination of C-H-bond MOs and
C-C -bond MOs of the two methyl and five methylene groups; Fig. 55 illustrates C8H18 MO comprising a linear combination of C- H-bond MOs and C-C -bond MOs of the two methyl and six methylene groups;
Fig. 56 illustrates C9H20 MO comprising a linear combination of C-H -bond MOs and
C-C -bond MOs of the two methyl and seven methylene groups. (A) Opaque view of the charge-density of the C-C -bond and C-H -bond MOs; Fig. 57 illustrates C10H22 MO comprising a linear combination of C -H -bond MOs and C-C -bond MOs of the two methyl and eight methylene groups;
Fig. 58 illustrates C11H24 MO comprising a linear combination of C-H-bond MOs and
C-C -bond MOs of the two methyl and nine methylene groups; Fig. 59 illustrates C12H26 MO comprising a linear combination of C - H -bond MOs and
C -C -bond MOs of the two methyl and ten methylene groups; Fig. 60 illustrates C18H38 MO comprising a linear combination of C -H-bond MOs and
C-C -bond MOs of the two methyl and sixteen methylene groups;
Fig. 6 LA illustrates 1,3 Butadiene;
Fig. 61. B illustrates 1,3 Pentadiene;
Fig. 61. C illustrates 1,4 Pentadiene; Fig. 61. D illustrates 1,3 Cyclopentadiene; Fig. 6 IE illustrates Cyclopentene;
Fig. 62 illustrates Naphthalene;
Fig. 63 illustrates Toluene;
Fig. 64 illustrates Benzoic acid; Fig. 65 illustrates Pyrrole;
Fig. 66 illustrates Furan;
Fig. 67 illustrates Thiophene;
Fig. 68 illustrates Imidazole;
Fig. 69 illustrates Pyridine; Fig. 70 illustrates Pyrimidine;
Fig. 71 illustrates Pyrazine;
Fig. 72 illustrates Quinoline;
Fig. 73 illustrates Isoquinoline;
Fig. 74 illustrates Indole; Fig. 75 illustrates Adenine;
Fig. 76 illustrates a block diagram of an exemplary software program; and
Figs. 77 and 78 illustrate pictures of an exemplary software program.
Section I
THE NATURE OF THE CHEMICAL BOND
OF HYDROGEN-TYPE MOLECULES
AND MOLECULAR IONS
With regard to the Hydrino Theory — BlackLight Process section, the possibility of states with n = Up is also predicted in the case of hydrogen molecular species wherein H{\l p) reacts a proton or two H(\l p) atoms react to form H^ (I/ p) and H2 [I/ p) , respectively. The natural molecular-hydrogen coordinate system based on symmetry is ellipsoidal coordinates. The magnitude of the central field in the derivations of molecular hydrogen species is taken as the general parameter p wherein p may be an integer which may be predictive of new possibilities. Thus, p replaces the effective nuclear charge of quantum mechanics and corresponds to the physical field of a resonant photon superimposed with the field of the proton. The case with p = 1 is evaluated and compared with the experimental results for hydrogen species in Table 11.1, and the consequences that p = integer are considered in the Nuclear Magnetic Resonance Shift section.
Two hydrogen atoms react to form a diatomic molecule, the hydrogen molecule.
2H[aH] →H2[2c' = J2ao] (11.1) where 2c' is the internuclear distance. Also, two hydrino atoms react to form a diatomic molecule, a dihydrino molecule.
Figure imgf000017_0001
where p is an integer.
Hydrogen molecules form hydrogen molecular ions when they are singly ionized.
Figure imgf000017_0002
Also, dihydrino molecules form dihydrino molecular ions when they are singly ionized.
#, 2c' = Sa → H, (11.4)
Figure imgf000017_0003
HYDROGEN-TYPE MOLECULAR IONS
Each hydrogen-type molecular ion comprises two protons and an electron where the equation of motion of the electron is determined by the central field which is p times that of a proton at each focus (p is one for the hydrogen molecular ion, and p is an integer greater than one for each H\ (l/ p), called dihydrino molecular ion). The differential equations of motion in the case of a central field are [1] m(r -rθ2) = f(r) (11.5) m(2rθ + rθ) = 0 (11.6)
The second or transverse equation, Eq. (11.6), gives the result that the angular momentum is constant. r2θ = constant - LIm (11.7) where L is the angular momentum ( % in the case of the electron). The central force
equations can be transformed into an orbital equation by the substitution, u = — . The r differential equation of the orbit of a particle moving under a central force is
(11.8)
Figure imgf000018_0001
Because the angular momentum is constant, motion in only one plane need be considered; thus, the orbital equation is given in polar coordinates. The solution of Eq. (11.8) for an inverse-squared force
f(r) = ~ (ii.9) is
r = r0 l + \ (11.10) l + ecosθ
L2 e = A~^- (11.11)
m- 2 ^o =— ~ (H.12) where e is the eccentricity of the ellipse and A is a constant. The equation of motion due to a central force can also be expressed in terms of the energies of the orbit. The square of the speed in polar coordinates is v 2 = (f2 +rΨ) (11.13) Since a central force is conservative, the total energy, E , is equal to the sum of the kinetic, T , and the potential, V , and is constant. The total energy is
~m(r2 + rΨ) + V(r) = E = constant (11.14)
Substitution of the variable u = — and Eq. (11.7) into Eq. (11.14) gives the orbital energy r equation.
Figure imgf000019_0001
Because the potential energy function V(r) for an inverse-squared force field is
V(r) = -- = -ku (11.16) r the energy equation of the orbit, Eq. (11.15),
which has the solution
Figure imgf000019_0002
where the eccentricity, e , is
Figure imgf000019_0003
Eq. (11.19) permits the classification of the orbits according to the total energy, E, as follows:
E < 0, e < \ closed orbits (ellipse or circle)
E = O, e = \ parabolic orbit
E > 0, e > 1 hyperbolic orbit Since E = T + V and is constant, the closed orbits are those for which T <| V \ , and the open orbits are those for which T >| V \ . It can be shown that the time average of the kinetic energy, < T > , for elliptical motion in an inverse-squared field is 1 / 2 that of the time average of the magnitude of the potential energy, < V > . <T >= \ll\< V > [I]. As demonstrated in the One-Electron Atom section, the electric inverse-squared force is conservative; thus, the angular momentum of the electron, h , and the energy of atomic orbitspheres are constant. In addition, the orbitspheres are nonradiative when the boundary condition is met.
The central force equation, Eq. (11.14), has orbital solutions, which are circular, elliptical, parabolic, or hyperbolic. The former two types of solutions are associated with atomic and molecular orbitals. These solutions are nonradiative. The boundary condition for nonradiation given in the One-Electron Atom section, is the absence of components of the spacetime Fourier transform of the current-density function synchronous with waves traveling at the speed of light. The boundary condition is met when the velocity for the charge density at every coordinate position on the orbitsphere is v = — (11.20)
WΛ
The allowed velocities and angular frequencies are related to rn by v» = W, (11.21) ωn = -\ (11.22) mern As demonstrated in the One-Electron Atom section and by Eq. (11.22), this condition is met for the product function of a radial Dirac delta function and a time harmonic function where the angular frequency, o? , is constant and given by Eq. (11.22).
(11.23)
Figure imgf000020_0001
where L is the angular momentum and A is the area of the closed orbit. Consider the solution of the central force equation comprising the product of a two-dimensional ellipsoid and a time harmonic function. The spatial part of the product function is the convolution of a radial Dirac delta function with the equation of an ellipsoid. The Fourier transform of the convolution of two functions is the product of the individual Fourier transforms of the functions; thus, the boundary condition is met for an ellipsoidal-time harmonic function when πh h . .. ωB = — τ = r (11-24)
where the area of an ellipse is
A = πab (11.25) where b and 2δ are the lengths of the semiminor and minor axes, respectively, and a aaά2a are the lengths of the semimajor and major axes, respectively. The geometry of molecular hydrogen is ellipsoidal with the internuclear axis as the principal axis; thus, the electron orbital is a two-dimensional ellipsoidal-time harmonic function. The mass follows an elliptical path, time harmonically as determined by the central field of the protons at the foci.
Rotational symmetry about the internuclear axis further determines that the orbital is a prolate spheroid. In general, ellipsoidal orbits of molecular bonding, hereafter referred to as ellipsoidal molecular orbitals (MOs), have the general equation
Figure imgf000021_0001
The semiprincipal axes of the ellipsoid are a, b, c . In ellipsoidal coordinates the Laplacian is (11.27)
Figure imgf000021_0002
An ellipsoidal MO is equivalent to a charged perfect conductor (i.e. no dissipation to current flow) whose surface is given by Eq. (11.26). It is a two-dimensional equipotential membrane where each MO is supported by the outward centrifugal force due to the corresponding angular velocity, which conserves its angular momentum of h . It satisfies the boundary conditions for a discontinuity of charge in Maxwell's equations, Eq. (11.48). It carries a total charge q = -e , and it's potential is a solution of the Laplacian in ellipsoidal coordinates, Eq.
(11.27).
Excited states of orbitspheres are discussed in the Excited States of the One-Electron Atom (Quantization) section. In the case of ellipsoidal MOs, excited electronic states are created when photons of discrete frequencies are trapped in the ellipsoidal resonator cavity of the MO. The photon changes the effective charge at the MO surface where the central field is ellipsoidal and arises from the protons and the effective charge of the "trapped photon" at the foci of the MO. Force balance is achieved at a series of ellipsoidal equipotential two- dimensional surfaces confocal with the ground state ellipsoid. The "trapped photons" are solutions of the Laplacian in ellipsoidal coordinates, Eq. (11.27). As is the case with the orbitsphere, higher and lower energy states are equally valid.
The photon standing wave in both cases is a solution of the Laplacian in ellipsoidal coordinates. For an ellipsoidal resonator cavity, the relationship between an allowed circumference, AaE , and the photon standing wavelength, λ , is 4aE = nλ (11.28) where n is an integer and where the elliptic integral E of Eq. (11.28) is given by
Figure imgf000022_0001
Applying Eqs. (11.28) and (11.29-11.30), the relationship between an allowed angular frequency given by Eq. (11.24) and the photon standing wave angular frequency, ω , is: πh % ft 1 /-1 1 i -ι \ meA Tn^a1TIb1 meanbn n where n = 1,2,3,4,...
1 1 1 n = — ,—,—,...
2 3 4
O1 is the allowed angular frequency for n = 1 ar and bx are the allowed semimajor and semiminor axes for n = 1
The potential, φ , and distribution of charge, σ, over the conducting surface of an ellipsoidal MO are sought given the conditions: 1.) the potential is equivalent to that of a charged ellipsoidal conductor whose surface is given by Eq. (11.26), 2.) it carries a total charge g ~ -e , and 3.) initially there is no external applied field. To solve this problem, a potential function must be found which satisfies Eq. (11.27), which is regular at infinity, and which is constant over the given ellipsoid. The solution is well known and is given after Stratton [2]. Consider that the Laplacian is solved in ellipsoidal coordinates wherein ξ is the parameter of a family of ellipsoids all confocal with the standard surface ξ = 0 whose axes have the specified values a, b, c . The variables ζ and η are the parameters of confocal hyperboloids and as such serve to measure position on any ellipsoid ξ = constant . On the surface ξ = 0 ; therefore, φ must be independent of ζ and η . Due to the uniqueness property of solutions of the Laplacian, a function which satisfies Eq. (11.27), behaves properly at infinity, and depends only on ξ , can be adjusted to represent the potential correctly at any point outside the ellipsoid ξ = 0.
Thus, it is assumed that φ = φ{ξ) . Then, the Laplacian reduces to
Figure imgf000023_0001
which on integration leads to (11.33)
Figure imgf000023_0003
where C1 is an arbitrary constant. The upper limit is selected to ensure the proper behavior at infinity. When ξ becomes very large, Rξ approaches ξyi and
(ξ → ∞) (11.34)
Figure imgf000023_0004
Furthermore, the equation of an ellipsoid can be written in the form
Figure imgf000023_0005
If r2 = x2 ÷y2 +z2 is the distance from the origin to any point on the ellipsoid ξ , it is apparent that as ξ becomes very large ξ -> r2. Thus, at great distances from the origin, the potential becomes that of a point charge at the origin: φ ~ ?2. (11.36) r
The solution Eq. (11.33) is, therefore, regular at infinity, and the constant C1 is then determined. It has been shown by Stratton [2] that whatever the distribution, the dominant term of the expansion at remote points is the potential of a point charge at the origin equal to the total charge of the distribution — in this case a . Hence C1 = -^- , and the potential at
Sπεn any point is
Figure imgf000023_0002
The equipotential surfaces are the ellipsoids ξ - constant . Eq. (11.37) is an elliptic integral and its values have been tabulated [3]. Since the distance along a curvilinear coordinate u1 is measured not by dux but by
\d^ , the normal derivative in ellipsoidal coordinates is given by
Figure imgf000024_0001
where
1 J ^(ξ-η){ξ-ζ) (11.39)
2 Rξ
The density of charge, σ , over the surface ξ = 0 is
Figure imgf000024_0002
Defining x, 3;, z in terms of |, 77, ζ we put ξ - 0 , ύ' may be easily verified that
z44=J? «=o) (ii-4i) Consequently, the charge density in rectangular coordinates is
Figure imgf000024_0003
(The mass-density function of an MO is equivalent to its charge-density function where m replaces q of Eq. (11.42)). The equation of the plane tangent to the ellipsoid at the point
Figure imgf000024_0004
where X, Y, Z are running coordinates in the plane. After dividing through by the square root of the sum of the squares of the coefficients of X, Y, and Z , the right member is the distance D from the origin to the tangent plane. That is,
Figure imgf000024_0005
so that for an electron MO
<*=- Afπaibrc D (1L45)
In other words, the surface density at any point on a charged ellipsoidal conductor is proportional to the perpendicular distance from the center of the ellipsoid to the plane tangent to the ellipsoid at the point. The charge is thus greater on the more sharply rounded ends farther away from the origin.
In the case of hydrogen-type molecules and molecular ions, rotational symmetry about the internuclear axis requires that two of the axes be equal. Thus, the MO is a spheroid, and Eq. (11.37) can be integrated in terms of elementary functions. If a > b = c , the spheroid is prolate, and the potential is given by
Figure imgf000025_0001
SPHEROIDAL FORCE EQUATIONS
Electric Force
The spheroidal MO is a two-dimensional surface of constant potential given by Eq. (11.46) for ξ = 0. For an isolated electron MO the electric field inside is zero as given by Gauss' Law
J EcM= J — dV (11.47)
S V Eo where the charge density, p , inside the MO is zero. Gauss' Law at a two-dimensional surface with continuity of the potential across the surface according to Faraday's law in the electrostatic limit [4-6] is
11.(E1 -E2) = - (11.48) εo
E2 is the electric field inside which is zero. The electric field of an ellipsoidal MO with semimajor and semiminor axes a and b = c , respectively, is given by substituting σ given by Eq. (11.38-11.42) into Eq. (11.48).
Figure imgf000025_0002
(11.49) wherein the ellipsoidal-coordinate parameter ξ = 0 at the surface of the MO and D is the distance from the origin to the tangent plane given by Eq. (11.44). The electric field and thus the force and potential energy between the protons and the electron MO can be solved based on three principles: (1) Maxwell's equations require that the electron MO is a equipotential energy surface that is a function of ξ alone; thus, it is a prolate spheroid, (2) stability to radiation, and conservation first principles require that the angular velocity is constant and given in polar coordinates with respect to the origin by Eq. (11.24), and (3) the equations of motion due to the central force of each proton (Eqs. (11.5-11.19) and Eqs. (11.68-11.70)) also determine that the current is ellipsoidal, and based on symmetry, the current is a prolate spheroid. Thus, based on Maxwell's equations, conservation principles, and Newton's Laws for the equations of motion, the electron MO constraints and the motion under the force of the protons both give rise to a prolate spheroid. Since the energy of motion is determined from the Coulombic central field (Eqs. (11.5-11.19), the protons give rise to a prolate spheroidal energy surface (a surface of constant energy) that is matched to the equipotential, prolate spheroidal electron MO.
The force balance equation between the protons and the electron MO is solved to give the position of the foci, then the total energy is determined including the repulsive energy between the two protons at the foci to determine whether the original assumption of an elliptic orbit was valid. If the condition that E < 0 is met, then the problem of the stable elliptic orbit is solved. In any case that this condition is not found to be met, then a stable orbit can not be formed.
The force and energy equations of a point charge(mass) (Eqs. (11.5-11.24)) are reformulated in term of densities for charge, current, mass, momentum, and potential, kinetic, and total energies. Consider an elliptical orbit shown in Figure 1 that applies to a point charge(mass) as well as a point on a continuous elliptical current loop that comprises a basis element of the continuous current density of the ellipsoidal MO. The tangent plane at any point on the ellipsoid makes equal angles with the foci radii at that point and the sum of the distance to the foci is a constant, 2a . Thus, the normal is the bisector of the angle between the foci radii at that point as shown in Figure 1. The unit vector normal to the ellipsoidal MO at a point (x, y, z) is
Figure imgf000026_0001
-^i (KO) an(^ -^ (KO) dXQ defined as the components of the central forces centered on F1 and F2. The components of the central forces that are normal to the ellipsoidal MO in the direction of d, the unit vector in the \ξ -direction are defined as F11 (r(t)) and F21 (r (Y)) . The normalized projections or projection factor of the sum of these central forces in the d- direction at the point (x, y, z) is
Figure imgf000027_0001
where T1 and r2 are the radial vectors of the central forces from the corresponding focus to the point {x,y,z) on the ellipsoidal MO.
The polar-coordinate elliptical orbit of a point charge due to its motion in a central inverse-squared-radius field is given by Eqs. (11.10-11.12) as the solution of the polar- coordinate-force equations, Eqs. (11.5-11.19) and (11.68-11.70). The orbit is also completely specified in Cartesian coordinates by the solution of Eqs. (11.5-11.19) and (11.68-11.70) for the semimajor and semiminor axes. Then, the corresponding polar-coordinate elliptical orbit is given as a plane cross section through the foci of the Cartesian-coordinate-system ellipsoid having the same axes given by Eq. (11.26) where c = b . Thus, the Columbic central force can be determined in terms of the general Cartesian coordinates from the polar-coordinate central force equations (Eqs. (11.5-11.19)). Consider separately the elliptical solution at each focus given in polar coordinates by Eq. ( 11.10) :
Figure imgf000027_0002
where
ro = a-c' = a(l-— j = α(l-e) (11.54) The magnitude of the sum of the central forces centered on Fx and F2 that are normal to the ellipsoidal MO are
(1 + e cos θf + (1 - e cos 6>)2
Figure imgf000028_0001
The vector central forces centered on F1 and F2 that are normal to the ellipsoidal MO are then given by the product of the corresponding magnitude and vector projection given by Eqs. (11.55) and (11.51), respectively:
_, / ,Λ Λ . ,.
Fi i ('1 (11.56)
Figure imgf000028_0002
Eq. (11.56) is based on a single point charge e . For a charge-density distribution that is given as an ellipsoidal equipotential, the θ -dependence must vanish. In addition to the elliptical orbit being completely specified in Cartesian coordinates by the solution of Eqs. (11.5-11.19) and Eqs. (11.68-11.70) for the semimajor and semiminor axes in Eq. (11.26), the polar-coordinate elliptical orbit is also completely specified by the total constant total energy E and the angular momentum which for the electron is the constant h . Considering Eq. (11.56), the corresponding total energy of the electron is conserved and is determined by the integration over the MO to give the average:
Figure imgf000028_0003
Eq. (11.57) is transformed from a two-centered-central force to a one-centered-central force to match the form of the potential of the ellipsoidal MO. In this case,
Figure imgf000028_0004
In the case that n ~ r2 = a (11.59) then, r (11.60) and the one-centered-central force is in the L -direction. Thus, Eq. (11.57) transforms as
Figure imgf000029_0001
Eq. (11.61) has the same form as that of the electric field of the ellipsoidal MO given by Eq. (11.49), except for the scaling factor of two-centered coordinates A2n, :
As shown in the case of the derivation of the Laplacian charge-density and electric field, if r2 = x2 +y2 + z2 is the distance from the origin to any point on the ellipsoid ξ s it is apparent that as ξ becomes very large ξ -» r2 . Thus, at great distances from the origin, the potential becomes that of a point charge at the origin as given by Eq. (11.36). The same boundary condition applies to the potential and field of the protons. The limiting case is also given as e — » 0 . Then, to transform the scale factor to that of one-centered coordinates for an ellipsoidal MO, the reciprocal of the scaling factor multiplies the Laplacian-MO-electric-fϊeld term. The reciprocal of Eq. (11.62) is
Figure imgf000029_0003
such that as e → 0 , fζ]c -> — . This transform scale factor corresponds to the interchange of
the points of highest and lowest velocity on the surface and the distribution of the charge- density in the opposite manner as shown infra. The charge-density distribution corrects the angular variation in central force over the surface such that a solution of the central force equation of motion and the Laplacian MO are solved simultaneously. It can also be considered as a multipole normalization factor such those of the spherical harmonics and the spherical geometric factor of atomic electrons that gives the central force as a function of ξ only.
The reciprocal of the Ji100 form-factor with the dependence of the charge density on the distance parameter r (7) gives
(11.64)
Figure imgf000030_0001
From Eq. (11.31), the magnitude of the ellipsoidal field corresponding to a below
"ground state" hydrogen-type molecular ion is an integer p . The integer is one in the case of the hydrogen molecular ion and an integer greater than one in the case of each dihydrino molecular ion. The central-electric-force constant, k , from the two protons that includes the central-field contribution due photons of lower-energy states is
ft, — Ze2 _ . —_ pie2 ' (11.65)
4πε0 4πs0
Substitution of Eq. (11.65) for k in Eq. (11.64) gives the one-center-coordinate electric force ¥ele between the protons and the ellipsoidal MO:
** -*u{'i'))+*» oi*)
Figure imgf000030_0002
where e is the charge and w th the distance from the origin to a nucleus at a focus defined as c ' , the eccentricity, e , is
e = - (11.67) a
From the orbital equations in polar coordinates, Eqs. (11.10-11.12), the following relationship can be derived [I]:
Figure imgf000030_0003
For any ellipse,
Figure imgf000030_0004
Thus, charge (mass) in polar coordinates) (11.70)
Figure imgf000031_0001
From, the equal energy condition, it can be shown that b for the motion of a point charge (mass) in polar coordinates due to a proton at one focus corresponds to c'^ ja2 -b2 (11.71) of the MO in ellipsoidal coordinates, and kx of one attracting focus is replaced by k = Ikx of ellipsoidal coordinates with two attracting foci. In ellipsoidal coordinates, k is given by Eq. (11.65) and L for the electron equals % .
Consider the force balance equation for the point on the ellipse at the intersection of the semiminor axis b with the ellipse. At this point called [Q, b) , the distances from each focus, rx and r2, to the ellipse are equal. The relationship for the sum of the distances from the foci to any point on the ellipse is η + r2 = 2α (11.72)
Thus, at point (θ,b) ,
Tx = V2 = a (11.73) Using Eq. (11.5), the magnitude of the force balance in the radial (r(r)) direction, from the origin, is given by
A2 2pe2 . 2pe2 b mrθ nι ~ = r --Ssiinn## == - r— (11.74)
4πε Wna A 4ππεεnaac ' a wherein the mr term is zero and θ is the angle from the focus to point (0,b) . Using Eqs. (11.24), (11.94), and (11.95), Eq. (11.74) becomes
Figure imgf000031_0002
In order for the prolate spheroidal MO to be an equipotential surface, the mass and charge density must be according to Eq. (11.45). In this case, the mass and charge density along the ellipse is such that the magnitudes of the radial and transverse forces components at point (0,b) are equivalent. Furthermore, according to Eq. (11.5), the central force of each proton at a focus is separable and symmetrical to that at the other focus. Based on symmetry, the transverse forces of the two protons are in opposite directions and the radial components are in the same direction. But, the relationship between the magnitudes must still hold wherein at point (0,b) the transverse force is equivalent to that due to the sum of the charges at one focus. The sum of the magnitudes of the transverse forces which is equivalent to a force of 2e at each focus in turn is
Figure imgf000032_0001
c Thus, using the mass and charge-density scaling factor, -f- = — , to match the equipotential b_ h a condition in Eq. (75) gives
Figure imgf000032_0002
Using Eq. (1.235)
Figure imgf000032_0003
Then, the length of the semiminor axis of the prolate spheroidal MO, δ = c , is
Figure imgf000032_0004
Correspondingly, c1 is given by Eq. (11.71).
Substitution of Eq. (11.79) into Eq. (11.66) gives the electric force:
Figure imgf000033_0001
Centrifugal Force The centrifugal force along the radial vector from each proton at each focus of the ellipsoid is given by the mrθ2 term of Eq. (11.5). The tangent plane at any point on the ellipsoid makes equal angles with the foci radii at that point and the sum of the distance to the foci is a constant, 2a . Thus, the normal is the bisector of the angle between the foci radii at that point as shown in Figure 1. In order to satisfy the equation of motion for an equal energy surface for both foci, the transverse component of the central force of one foci at any point on the elliptic orbit due to the central force of the other (Eq. (11.5)) must cancel on average and vice versa. Thus, the centrifugal force due to the superposition of the central forces in the direction of each foci must be normal to an ellipsoidal surface in the direction perpendicular to the direction of motion. Thus, it is in the ξ -direction. This can be only be achieved by a time rate of change of the momentum density that compensates for the variation of the distances from each focus to each point on an elliptical cross section. Since the angular momentum must be conserved, there can be no net force in the direction transverse to the elliptical path over each orbital path. The total energy must also be conserved; thus, as shown infra, the distribution of the mass must also be a solution of Laplace's equation in the parameter ξ only. Thus, the mass-density constraint is the same as the charge-density constraint. As further shown infra. , the distribution and concomitantly the centrifugal force is a function of D , the time-dependent distance from the center of the ellipsoid to a tangent plane given by Eq. (11.44) where D and the Cartesian coordinates are the time-dependent parameters.
Each point or coordinate position on the continuous two-dimensional electron MO defines an infinitesimal mass-density element which moves along an orbit comprising an elliptical plane cross section of the spheroidal MO through the foci. The kinetic energy of the electron is conserved. Then, the corresponding radial conservative force balance equation is m(r + C1O = O (11.82)
The motion is such that eccentric angle, θ , changes at a constant rate at each point. That is θ = ωt at time t where the angular velocity ω is a constant. The solution of the homogeneous equation with C1 = ω2 is
Figure imgf000034_0001
where a is the semimajor axis, b is semiminor axis, and the boundary conditions of r (t) = a
for (Dt = O and r{t) = b for ωt = — were applied. Eq. (11.83) is the parametric equation of
J^ the ellipse of the orbit. The velocity is given by the time derivative of the parametric position vector: v(t) = f(t) = -\aω sin ωt + \bω cos ωt (11.84)
The velocity is — out of phase with the charge density at r(t) = a (cot = 0 ) and r(t) = b
(ωt = —) such that the lowest charge density has the highest velocity and the highest charge
density has the lowest velocity. In this case, it can be shown that the current is constant along each elliptical path of the MO. Recall that nonradiation results when ω = constant given by Eq. (11.24) that corresponds to a constant current, which further maintains the current continuity condition.
Consider Eq. (11.32) for the prolate spheroidal MO. From this equation, the mass and current-densities, the angular momentum, and the potential and kinetic energies are a function of ξ alone, and any dependence on the orthogonal coordinate parameters averages to unity. From Eq. (11.32),
^JI = C1 (11.85)
Substitution of Eq. (11.40) into Eq. (11.85) gives
Figure imgf000035_0001
where C1 is from Eq. (11.36). Substitution of Eq. (11.39) into Eq. (11.86) gives
Figure imgf000035_0002
Comparison of Eq. (11.86) with Eq. (11.87) demonstrates that the
(11.88)
Figure imgf000035_0006
The current density J is given by the product of the constant frequency (Eq. (11.24)) and the charge density (Eq. (11.40)):
(11.89)
Figure imgf000035_0007
The total constant current is dependent on ξ alone according to Eq. (11.32). Then, applying the result of Eq. (11.88) to Eq. (11.89) gives
Figure imgf000035_0003
the constant current that is nonradiative.
If a(t) denotes the acceleration vector, then
Α(t) = -ω2r(t)ir . (11.91) In other words, the acceleration is centrifugal as in the case of circular motion with constant angular speed ω . The dot product of r \t) with d , the unit vector normal to the ellipsoidal MO at a point {x, y, z) given by Eq. (11.50), is
Figure imgf000035_0004
Using Eq. (11.26), the normal Component projection is
Figure imgf000035_0005
where D , the distance from the origin to the tangent plane, is given by Eq. (11.44). The centrifugal force, Fd , on mass element m, [7] given by the second term of Eq.
(11.82) is
Fd = m.a = ~m(ω2r(t) (11.94)
Substitution of the angular velocity given by Eq. (11.24) and me for m into Eq. (11.94) gives the centrifugal force Fc on the electron that is normal to the MO surface according to Eq. (11.93):
Figure imgf000036_0001
Fc has an equivalent dependence on D as the electric force based on the charge distribution
Q
(Eq. (11.45)). This is expected based on the invariance of — which results in the same me distribution of the mass and charge.
The equipotential charge-density distribution gives rise to the constant current condition. It also gives rise to a constant total kinetic energy condition wherein the angular velocity given by Eq. (11.24) is a constant. Recall from Eq. (11.32), that on the surface ξ = Q ; φ must be independent of ζ and η and depend only on ξ at any point outside the ellipsoid ξ = 0. Since the current and total kinetic energy are also constant on the surface ξ = 0 , the total kinetic energy depends only on ξ . Thus, the centrifugal force on the mass of the electron, me, must be in the same direction as the electric field corresponding to φ , normal to the electron surface wherein any tangential component in Eq. (11.94) averages to zero over the electron MO by the mass distribution given by Eqs. (11.40) and (11.45) with me replacing e .
The cancellation of tangential acceleration over each elliptical path maintains the charge density distribution given by Eq. (11.40) with constant current at each point on each elliptical path of the MO. Since the centrifugal force is given by Eq. (11.94), the multiplication of the mass density by the scaling factor Jtx and integration with respect to ξ gives a constant net centrifugal force. Thus, the result matches those of the determination of the constant current (Eq. (11.90)) and angular momentum shown infra. (Eq. (11.101)) wherein the charge and mass densities given in Eqs. (11.90-11.91) and (11.100), respectively, were integrated over.
Specifically, consider the normal-directed centrifugal force, Fc/ , on mass element m. : Fc, = -m,ω2Diξ (11.96)
The mass density is given by Eq. (11.40) with me replacing e . Then, the substitution of the mass density for m;. in Eq. (11.96) and using Eq. (11.24) for ω gives the centrifugal force density Fcfl :
Fcα = . mr- f 2, 2 Diξ (11.97)
4πylηζ m^a b
Eq. (11.32) determines that the centrifugal force is a function of ξ alone, and any dependence on the transverse coordinate parameters averages to zero. Using the result of Eq. (11.88) gives the net centrifugal force Fc :
(11.98)
Figure imgf000037_0001
In the limit as the ellipsoidal coordinates go over into spherical coordinates, Eq.
(11.95) reduces to the centrifugal force of the spherical orbitsphere given by Eq. (1.232) with Eq. (1.47). This condition must be and is met as a further boundary condition that parallels that of Eqs. (11.32-11.37). Using the same dependence of the total mass(charge) on the scale factor \ according to Eqs. (11.32-11.40), the further boundary conditions on the angular momentum and kinetic energy are met.
Specifically, the constant potential and current conditions and the use of Eq. (11.32) in the derivation of Eq. (11.95) also satisfy another condition, the conservation of fr of angular momentum of the electron. The angular momentum p( at each point / of mass mt is
Vi (t) = mlr(t)xv(t)
= mμhω (cos2 ωt + sin2 cat J i x j = mμbωk The mass density is given by Eq. (11.40) with me replacing e . Then, substitution of mi in Eq. (11.99) by the mass density and using Eq. (11.24) for ω gives the angular momentum density p (t) :
(11.100)
Figure imgf000037_0002
Using the result of Eq. (11.88) gives the total constant angular momentum L :
Figure imgf000038_0001
Eq. (11.101) demonstrates conservation of angular momentum that is a function of ξ alone that parallels the case of atomic electrons where L conservation is a function of the radius r alone as given by Eq. (1.57). Similarly, the kinetic energy T(t) at each point / of mass m, is
Figure imgf000038_0002
= — mt (-iαωsinωt + jόωcosωt) (11.102)
= —mp2 (a2 sin2 ωt + b2 cos2 ωi\
In Eqs. (11.96-11.98), m, was replaced by the mass density and the ξ integral was determined to give the centrifugal force in terms of the mass of the electron. The kinetic energy can also be determined from the ξ integral of the centrifugal force:
Figure imgf000038_0003
The result is given in Eq. (11.119). From Eq. (11.102), the kinetic energy is time (position) dependent, but the total kinetic energy corresponding to the centrifugal force given by Eq. (11.95) satisfies the condition that the time-averaged kinetic energy is 1/2 the time-averaged potential energy for elliptic motion in an inverse-squared central force [I]. (Here, the potential and total kinetic energies are constant and correspond to the time-averaged energies of the general case.) Thus, as shown by Eqs. (11.122) (11.124), (11.262), and (11.264) energy is conserved.
Force Balance of Hydrogen-type Molecular Ions Consider the case of spheroidal coordinates based on the rotational symmetry about the semimajor axis [2]. In the limit, as the focal distance 2c and the eccentricity of the series of confocal ellipses approaches zero, spheroidal coordinates go over into spherical coordinates with ξ -» r and η -> cos θ . The field of an equipotential two-dimensional charge surface of constant radius r = R is equivalent to that of a point charge of the total charge of the spherical shell at the origin. The force balance between the centrifugal force and the central Coulomb force for spherical symmetry is given by Eq. (1.232). Similarly, the centrifugal force is the direction of ξ and balances the central
Coulombic force between the protons at the foci and the electron MO. In the case of the prolate spheroidal MO, the inhomogeneous equation given by Eq. (11.5) must hold for each fixed position of r (t) since the MO is static in time due to the constant current condition. With r(t) fixed, the mr term of Eq. (11.5) is zero, and the force balanced equation is the balance between the centrifugal force and the Coulombic force which are both normal to the surface of the elliptic orbit: mrθ2 = f(r) (11.104)
Substitution of Eq. (11.81) and Eq. (11.95) into Eq. (11.104) gives the force balance between the centrifugal and electric central forces:
(11.105)
Figure imgf000039_0001
Figure imgf000039_0002
Figure imgf000039_0003
Figure imgf000039_0004
Figure imgf000039_0005
α = 2^ (11.109)
P
Substitution of a given by Eq. (11.109) into Eq. (11.79) gives
c' = 2± (11.110)
P
The internuclear distance from Eq. (11.110) is
2c' = ^ (11.111)
P Substitution of α = — - and c' = — into Eq. (11.80) gives the length of the semiminor axis
P P of the prolate spheroidal MO, b = c :
Figure imgf000040_0001
Substitution of a = — - and c ' = — into Eq. (11.67) gives the eccentricity, e : P P
e = \ (11.113)
From Eqs. (11.63-11.65), the result of Eq. (11.113) can be used to the obtain the electric force Fele between the protons and the ellipsoidal MO as
Fele = ZeMξ = %c -P¥-Diξ ={^Diξ (11.114)
where the electric field E of the MO is given by Eq. (11.49). Then, the force balance of the hydrogen-type molecular ion is given by
-^-TD ^ ^—D (11.115) mea b Sπε0 which has the parametric solution given by Eq. (11.83) when
a = ^ (11.116)
P
The solutions for the prolate spheroidal axes and eccentricity are given by Eqs. (11.109- 11.113).
ENERGIES OF HYDROGEN-TYPE MOLECULAR IONS
From Eq. (11.31), the magnitude of the ellipsoidal field corresponding to a below "ground state" hydrogen-type molecule is an integer, p . The force balance equation (Eq. (11.115)) applies for each point of the electron MO having non-constant charge (mass)-density and velocity over the equipotential and equal energy surface. The electron potential and kinetic energies are thus determined from an ellipsoidal integral.
The potential energy is doubled due to the transverse electric force. The force normal to the MO is given by the dot product of the sum of the force vectors from each focus with d
TV where the angle β is β = a, and the transverse forces are given by the cross product
with d . As shown in Figure 1, equivalently, the transverse projection is given with the angle a replacing β where the range of a is the same as β . The two contributions to the potential energy doubles it. The potential energy, Ve , of the electron MO in the field of magnitude p times that of the two protons at the foci is
„2 „1.2 ∞
V = 2~2pe Dab Vξ
4πεo ID I Rξ
Figure imgf000041_0001
where 4 a2 -b2 = c' (11.118)
2c ' is the distance between the foci which is the internuclear distance. The kinetic energy, T , of the electron MO follows from the same type of integral as Ve using Eqs. (7-14) of
Stratton [8], Eqs. (11.37-11.46), and integral #147 of Lide [9]. T is given by the corresponding integral of the centrifugal force (LHS of Eq. (11.115)) with the constraint that the current motion allows the equipotential and equal energy condition with a central field due to the protons; thus, it is corrected by the scale factor Ti100 given by Eq. (11.62). The Jt200 correction can be considered the scaling factor of the moment of inertial such that the kinetic energy is equivalent to the rotational energy for constant angular frequency ω . The kinetic energy, T , of the electron MO is given by
Figure imgf000041_0002
(11.119)
The potential energy, VP , due to proton-proton repulsion in the field of magnitude p times that of the protons at the foci (ξ = 0 ) is
Figure imgf000041_0003
The total energy, Eτ , is given by the sum of the energy terms
ET = Ve +Vp +T (11.121)
Substitution of a and b given by Eqs. (11.109) and (11.112), respectively, into Eqs. (11.117), (11.119), (11.120), and (11.121) gives -4»V Ve = P In 3 (11.122)
8πε0a0
Vp = ^~ (11.123)
Figure imgf000042_0001
ET = -13.6 eV(4p2 ln3-p2 -2p2 ln3) = -p21628 eV (11.125) The total energy, which includes the proton-proton-repulsion term is negative which justifies the original treatment of the force balance using the analytical-mechanics equations of an ellipse that considered only the binding force between the protons and the electron and the electron centrifugal force. T is one-half the magnitude of Ve as required for an inverse- squared force [1] wherein Ve is the source of T .
VIBRATION OF HYDROGEN-TYPE MOLECULAR IONS A charge, q , oscillating according to ro(t) = dsinø/ has a Fourier spectrum
J(k,ω) =^^Jm(kcosθd){δ[ω -(m + l)ω0] + δ[ω-(m-ϊ)ω0]} (11.126)
where Jn 's are Bessel functions of order m . These Fourier components can, and do, acquire phase velocities that are equal to the velocity of light [10]. The protons of hydrogen-type molecular ions and molecules oscillate as simple harmonic oscillators; thus, vibrating protons will radiate. Moreover, non-oscillating protons may be excited by one or more photons that are resonant with the oscillatory resonance frequency of the molecule or molecular ion, and oscillating protons may be further excited to higher energy vibrational states by resonant photons. The energy of a photon is quantized according to Planck's equation
E = hω (11.127)
The energy of a vibrational transition corresponds to the energy difference between the initial and final vibrational states. Each state has an electromechanical resonance frequency, and the emitted or absorbed photon is resonant with the difference in frequencies. Thus, as a general principle, quantization of the vibrational spectrum is due to the quantized energies of photons and the electromechanical resonance of the vibrationally excited ion or molecule. It is shown by Fowles [11] that a perturbation of the orbit determined by an inverse- squared force results in simple harmonic oscillatory motion of the orbit. In a circular orbit in spherical coordinates, the transverse equation of motion gives
θ = ^- (11.128) r where L is the angular momentum. The radial equation of motion is m[r -rθ2) = f (r) (11.129)
Substitution of Eq. (11.128) into Eq. (11.129) gives
Figure imgf000043_0001
For a circular orbit, r is a constant and r = 0. Thus, the radial equation of motion is given by
JϋHdpL = f{a) 01.131)
where a is the radius of the circular orbit for central force f(a) at r = a . A perturbation of the radial motion may be expressed in terms of a variable x defined by x = r -a (11.132)
The differential equation can then be written as mx-m(Llm)2 (x + aγ = f(x + a) (11.133)
Expanding the two terms involving x + a as a power series in x , gives
mx-m(L/m)2 a-3 (l-3- + Δ = f(a) + f '(a)x + ... (11.134)
Substitution of Eq. (11.131) into Eq. (11.134) and neglecting terms involving x2 and higher powers of x gives
mx + \ —f(a)-f '(a) x = 0 (11.135)
For an inverse-squared central field, the coefficient of x in Eq. (11.135) is positive, and the equation is the same as that of the simple harmonic oscillator, hi this case, the particle, if perturbed, oscillates harmonically about the circle r = a , and an approximation of the angular frequency of this oscillation is
Figure imgf000043_0002
An apsis is a point in an orbit at which the radius vector assumes an extreme value
(maximum or minimum). The angle swept out by the radius vector between two consecutive apsides is called the apsidal angle. Thus, the apsidal angle is π for elliptical orbits under the inverse-squared law of force. In the case of a nearly circular orbit, Eq. (11.135) shows that r oscillates about the circle r = a , and the period of oscillation is given by
Figure imgf000044_0001
The apsidal angle in this case is just the amount by which the polar angle θ increases during the time that r oscillates from a minimum value to the succeeding maximum value which is
τr . From Eq. (11.128), θ = — — ; therefore, θ remains constant, and Eq. (11.131) gives rr
Figure imgf000044_0002
Thus, the apsidal angle is given by
Figure imgf000044_0003
Thus, the power force of /(?") = —cr" gives
ψ = π(3 + n)~m (11.140) The apsidal angle is independent of the size of the orbit in this case. The orbit is re-entrant, or repetitive, in the case of the inverse-squared law (n = -2) for which ψ = π .
A prolate spheroid MO and the definition of axes are shown in Figures 5A and 5B, respectively. Consider the two nuclei A and B, each at focus of the prolate spheroid MO. From Eqs. (11.115), (11.117), and (11.119), the attractive force between the electron and each nucleus at a focus is
Figure imgf000044_0004
and
Figure imgf000044_0005
In addition to the attractive force between the electron and the nuclei, there is a repulsive force between the two nuclei that is the source of a corresponding reactive force on the reentrant electron orbit. Consider an elliptical orbital plane cross section of the MO in the xy-plane with a nucleus A at (-c1, 0) and a nucleus B at (c'5 0). For B acting as the attractive focus, the reactive repulsive force at the point (a, 0), the positive semimajor axis, depends on the distance from (a, 0) to nucleus A at (-c1, 0) (i.e. the distance from the position of the electron MO at the semimajor axis to the opposite nuclear repelling center at the opposite focus). The distance is given by the sum of the semimajor axis, a , and c\ 111 the internuclear distance. The contribution from the repulsive force between the two protons is
Figure imgf000045_0001
and
Figure imgf000045_0002
Thus, from Eqs. (11.136) and (11.141-11.144), the angular frequency of this oscillation is
Figure imgf000045_0003
= p24.44865 X 1014 rad I s
where the semimajor axis, a , is a = — — according to Eq. (11.116) and c' is c' = —
P P according to Eq. (11.110). In the case of a hydrogen molecule or molecular ion, the electrons which have a mass of 1/1836 that of the protons move essentially instantaneously, and the charge density is that of a continuous membrane. Thus, a stable electron orbit is maintained with oscillatory motion of the protons. Hydrogen molecules and molecular ions are symmetrical along the semimajor axis; thus, the oscillatory motion of protons is along this axis. Let x be the increase in the semimajor due to the reentrant orbit with a corresponding displacement of the protons along the semimajor axis from the position of the initial foci of the stationary state. The equation of proton motion due to the perturbation of an orbit having a central inverse- squared central force [1] and neglecting terms involving x2 and higher is given by μx + kκ = 0 (11.146) which has the solution in terms of the maximum amplitude of oscillation, A , the reduced nuclear mass, μ , the restoring constant or spring constant, k , the resonance angular frequency, ωQ , and the vibrational energy, Evlb , [12] Λcosαy (11.147) where
ωo = β- (11.148)
For a symmetrical displacement x , the potential energy corresponding to the oscillation EPvib is given by
Figure imgf000046_0001
The total energy of the oscillating molecular ion, ETotaMb , is given as the sum of the kinetic and potential energies
Eτolalvib = +kκ2 (11.150)
Figure imgf000046_0002
The velocity is zero when x is the maximum amplitude, A . The total energy of the oscillating molecular ion, ETotahib , is then given as the potential energy with x = A
ETolam = M2 • (11.151)
Thus,
Figure imgf000046_0003
It is shown in the Excited States of the One-Electron Atom (Quantization) section that the change in angular frequency of the electron orbitsphere (Eq. (2.21)) is identical to the angular frequency of the photon necessary for the excitation, (ophoton (Eq. (2.19)). The energy of the photon necessary to excite the equivalent transition in an electron orbitsphere is one- half of the excitation energy of the stationary cavity because the change in kinetic energy of the electron orbitsphere supplies one-half of the necessary energy. The change in the angular frequency of the orbitsphere during a transition and the angular frequency of the photon corresponding to the superposition of the free space photon and the photon corresponding to the kinetic energy change of the orbitsphere during a transition are equivalent. The correspondence principle holds. It can be demonstrated that the resonance condition between these frequencies is to be satisfied in order to have a net change of the energy field [13]. The bound electrons are excited with the oscillating protons. Thus, the mechanical resonance frequency, ω0 , is only one-half that of the electromechanical frequency which is equal to the frequency of the free space photon, ω , which excites the vibrational mode of the hydrogen molecule or hydrogen molecular ion. The vibrational energy, Evώ , corresponding to the photon is given by
Figure imgf000047_0001
where Planck's equation (Eq. (11.127)) was used. The reduced mass is given by m,m7
(11.154) W1 + m2 Thus,
A = , (11.155)
2k
Since the protons and electron are not fixed, but vibrate about the center of mass, the maximum amplitude is given by the reduced amplitude, Areduced , given by
Figure imgf000047_0002
where An is the amplitude n ifthe origin is fixed. Thus, Eq. (11.155) becomes
Aeduced m (.1 1.11537/NJ
Figure imgf000047_0003
and from Eq. (11.148), Areduced is
Figure imgf000047_0004
Then, from Eq. (11.67), Ac,, the displacement of c' is the eccentricity e given by Eq. (11.113) times Areduced (Eq. (11.158)):
/| 1 1 C(Vv U 1 - 1 -3^
Figure imgf000047_0005
Thus, during bond formation, the perturbation of the orbit determined by an inverse- squared force results in simple harmonic oscillatory motion of the orbit, and the corresponding frequency, fl>(θ) , for a hydrogen-type molecular ion H\ (l/ p) given by Eqs.
(11.136) and (11.145) is (1 U6O)
Figure imgf000048_0003
where the reduced nuclear mass of hydrogen given by Eq. (11.154) is μ = 0.5mp (11.161) and the spring constant, k(θ) , given by Eqs. (11.136) and (11.145) is
Ar(O) = ZlOS-SI iVw-1 (11.162)
The transition-state vibrational energy, Evjb (0) , is given by Planck's equation (Eq. (11.127)): Evib (O) = Hω = tip24.44865 X lO14 radls = p20.2928 eV (11.163) The amplitude of the oscillation, Areduced (O) , given by Eq. (11.158) and Eqs. (11.161-11.162) is
Figure imgf000048_0001
Then, from Eq. (11.67), Ac, (O) , the displacement of c' is the eccentricity e given by Eq. (11.113) times Areduced (θ) (Eq. (11.164)):
The spring constant and vibrational frequency for the formed molecular ion are then obtained from Eqs. (11.136) and (11.141-11.145) using the increases in the semimajor axis and internuclear distances due to vibration in the transition state. The vibrational energy, Evjb (l) , for the Hl (l/ p) ϋ = l → y = 0 transition given by adding 4, (0) (Eq. (11.159)) to the distances a and a + c' in Eqs. (11.145) and (11.163) is
Evib (I) = P20.27O eV (11.166) where υ is the vibrational quantum number.
A harmonic oscillator is a linear system as given by Eq. (11.146). In this case, the predicted resonant vibrational frequencies and energies, spring constants, and amplitudes for Hj (l/p) for vibrational transitions to higher energy U1 → υf are given by (υf -oλ times the corresponding parameters given by Eq. (11.160) and Eqs. (11.162-11.164). However, excitation of vibration of the molecular ion by external radiation causes the semimajor axis and, consequently, the internuclear distance to increase as a function of the vibrational quantum number υ . Consequently, the vibrational energies of hydrogen-type molecular ions are nonlinear as a function of the vibrational quantum number υ . The lines become more closely spaced and the change in amplitude, ΔAreduced , between successive states becomes larger as higher states are excited due to the distortion of the molecular ion in these states. The energy difference of each successive transition of the vibrational spectrum can be obtained by considering nonlinear terms corresponding to anharmonicity.
The harmonic oscillator potential energy function can be expanded about the internuclear distance and expressed as a Maclaurin series corresponding to a Morse potential after Karplus and Porter (K&P) [14] and after Eq. (11.134). Treating the Maclaurin series terms as anharmonic perturbation terms of the harmonic states, the energy corrections can be found by perturbation methods. The energy vυ of state υ is vυ = υω0 -υ(υ-\) ωoxQ , υ = 0,1,2,3... (11.167) where heal ,Λ Λ Λ ,o^ ωoxo = ^fT (11.168)
ω0 is the frequency of the υ = 1 -» υ = 0 transition corresponding to Eq. (11.166), and D0 is the bond dissociation energy given by Eq. (11.198). From Eqs. (11.166), (11.168), and (11.198),
Figure imgf000049_0001
The vibrational energies of successive states are given by Eqs. (11.166-11.167) and (11.169).
Using Eqs. (11.145), (11.158-11.160), (11.162-11.169), and (11.199) the corresponding parameters for deuterium-type molecular ions with μ = mp (11.170) are
(11.171)
Figure imgf000049_0002
k(0) = p"\65.65 Nm'1 (11.172)
Evib (0) = /0.20714 eV (11.173) OU74,
Figure imgf000050_0001
Eφ (l) = p20Λ93 eV (11.175)
Figure imgf000050_0002
The vibrational energies of successive states are given by Eqs. (11.167) and (11.175-11.176).
THE DOPPLER ENERGY TERM OF HYDROGEN-TYPE MOLECULAR
IONS
As shown in the Vibration of Hydrogen-type Molecular Ions section, the electron orbiting the nuclei at the foci of an ellipse may be perturbed such that a stable reentrant orbit is established that gives rise to a vibrational state corresponding to time harmonic oscillation of the nuclei and electron. The perturbation is caused by a photon that is resonant with the frequency of oscillation of the nuclei wherein the radiation is electric dipole with the corresponding selection rules.
Oscillation may also occur in the transition state. The perturbation arises from the decrease in internuclear distance as the molecular bond forms. Relative to the unperturbed case given in the Force Balance of Hydrogen-type Molecular Ions section, the reentrant orbit may give rise to a decrease in the total energy while providing a transient kinetic energy to the vibrating nuclei. However, as an additional condition for stability, radiation must be considered. Regarding the potential for radiation, the nuclei may be considered point charges. A point charge undergoing periodic motion accelerates and as a consequence radiates according to the Larmor formula (cgs units) [15]:
P ~ v|2 (11.177)
3c3 where e is the charge, v is its acceleration, and c is the speed of light. The radiation has a corresponding force that can be determined based on conservation of energy with radiation. The radiation reaction force, Frad , given by Jackson [16] is
Frad =|4* (H-178)
Then, the Abraham-Lorentz equation of motion is given by [16]
Figure imgf000051_0001
where Fcxt is the external force and m is the mass. The external force for the vibrating system is given by Eq. (11.146).
Fcxt = Ax (11.180) where x is the displacement of the protons along the semimajor axis from the position of the initial foci of the stationary state in the absence of vibration with a reentrant orbit of the electron. A nonradiative state must be achieved after the emission due to transient vibration wherein the nonradiative condition given by Eq. (11.24) must be satisfied.
As shown in the Resonant Line Shape and Lamb Shift section, the spectroscopic linewidth arises from the classical rise-time band- width relationship, and the Lamb Shift is due to conservation of energy and linear momentum and arises from the radiation reaction force between the electron and the photon. The radiation reaction force in the case of the vibration of the molecular ion in the transition state corresponds to a Doppler energy, ED , that is dependent on the motion of the electron and the nuclei. The Doppler energy of the electron is given by Eq. (2.146)
Figure imgf000051_0002
where ER is the recoil energy which arises from the photon's linear momentum given by Eq. (2.141), Ex is the vibrational kinetic energy of the reentrant orbit in the transition state, and M is the mass of the electron me . As given in the Vibration of Hydrogen-Type Molecular Ions section, for inverse- squared central field, the coefficient of x in Eq. (11.135) is positive, and the equation is the same as that of the simple harmonic oscillator. Since the electron of the hydrogen molecular ion is perturbed as the internuclear separation decreases with bond formation, it oscillates harmonically about the semimajor axis given by Eq. (11.116), and an approximation of the angular frequency of this oscillation is
Figure imgf000051_0003
From Eqs. (11.115), (11.117), and (11.119), the central force terms between the electron MO and the two protons are
Figure imgf000052_0001
and
/1W = ^F (H.184)
Thus, the angular frequency of this oscillation is
/2.06538 X 1016 radls (11.185)
Figure imgf000052_0002
where the semimajor axis, α, is a = — — according to Eq. (11.116) including the reduced
P electron mass. The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)):
Eκ = hω = hp22.06538 X 1016 radls = /13.594697 e V (11.186)
In Eq. (11.181), substitution of the total energy of the hydrogen molecular ion, Eτ , (Eq. (11.125)) for E1n, , the mass of the electron, me , for M , and the kinetic energy given by Eq. (11.186) for Eκ gives the Doppler energy of the electron for the reentrant orbit.
ED ■ -/ 0.118755 eV
Figure imgf000052_0003
(11.187) The total energy of the molecular ion is decreased by ED . In addition to the electron, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency given in the Vibration of Hydrogen-Type Molecular Ions section. On average, the total energy of vibration is equally distributed between kinetic energy and potential energy [17]. Thus, the average kinetic energy of vibration corresponding to the Doppler energy of the electrons, EKvιb , is 1/2 of the vibrational energy of the molecular ion given by Eq. (11.166). The decrease in the energy of the hydrogen molecular ion due to the reentrant orbit in the transition state corresponding to simple harmonic oscillation of the electron and nuclei, Eosc , is given by the sum of the corresponding energies, ED and EKvΛ . Using Eq. (11.187) and Evώ from Eq. (11.166) gives (11.188)
Figure imgf000053_0002
1
Eosc = VO- 118755 eV + -p2 (0.29282 eV) (11.189)
To the extent that the MO dimensions are the same, the electron reentrant orbital energies Eκ are the same independent of the isotope of hydrogen, but the vibrational energies are related by Eq. (11.148). Thus, the differences in bond energies are essentially given by 1/2 the differences in vibrational energies. Using Eq. (11.187) with the deuterium reduced electron mass for E1, and E0 , and Evώ for D^ (Hp) given by Eq. (11.173), that corresponds to the deuterium reduced nuclear mass (Eq. (11.170)), the corresponding Eosc is
E0n = -/0.118811 eV +-p2 (0.20714 eV) (11.190)
TOTAL, IONIZATION, AND BOND ENERGIES OF HYDROGEN AND DEUTERIUM MOLECULAR IONS
The total energy of the hydrogen molecular ion which is equivalent to the negative of the ionization energy is given by the sum of Eτ (Eqs. (11.121) and (11.125)) and Eosc given by Eqs. (11.185-11.188). Thus, the total energy of the hydrogen molecular ion having a central field of +pe at each focus of the prolate spheroid molecular orbital including the Doppler term is
ET = Ve + Vp +T + E0SC (11.191)
Figure imgf000053_0001
From Eqs. (11.189) and (11.191-11.192), the total energy for hydrogen-type molecular ions is ET = ~p2\6.28033 eV + Eosc
= -/16.28033 eV -p3O.mi55 eV + - p2 (0.29282 eV) (11.193)
= -/16.13392 eV - p30.118755 eV
The total energy of the deuterium molecular ion is given by the sum of Eτ (Eq. (11.125)) corrected for the reduced electron mass of D and Eosc given by Eq. (11.190): ET = -p216.284 eV + Eosc
= -/16.284 eF-/0.118811 <?F+-/ (0.20714 eF) (11.194)
= -/16.180 eF-/0.118811 eV The bond dissociation energy, ED , is the difference between the total energy of the corresponding hydrogen atom or H(I/ p) atom [18-19], called hydrino atom having a principal quantum number \l p where p is an integer, and E7, .
En = E(H(I/ p))-ET (11.195) where [18] E(H(IZp)) = -/13.59844 eV (11.196) and [19]
E(D(I/ p)) = -p2n.603 eV (11.197)
The hydrogen molecular ion bond energy, ED , is given by Eq. (11.193) with the reduced electron mass and Eqs. (11.195-11.196):
Figure imgf000054_0001
= -/l3.59844-(-/l6.13392 eF -/0.118755 eF) (11.198)
= /2.535 eF + /0.118755 eF
The deuterium molecular ion bond energy, ED , is given by Eq. (11.194) with the reduced electron mass of D and Eqs. (11.195) and (11.197):
Figure imgf000054_0002
= -/l3.603-(-/l6.180 eF-/0.118811 eV) (11.199)
= /2.5770 eF + /0.118811 eV
HYDROGEN-TYPE MOLECULES FORCE BALANCE OF HYDROGEN-TYPE MOLECULES
Hydrogen-type molecules comprise two indistinguishable electrons bound by an elliptic field. Each electron experiences a centrifugal force, and the balancing centripetal force (on each electron) is produced by the electric force between the electron and the elliptic electric field and the magnetic force between the two electrons causing the electrons to pair. In addition to nonradiation, the angular frequency given by Eq. (11.24) corresponds to a Lorentzian invariant magnetic moment of a Bohr magneton, μB , as given in the Magnetic Moment of an
Ellipsoidal MO section. The internal field is uniform along the major axis, and the far field is that of a dipole as shown in the Magnetic Field of an Ellipsoidal MO section. The magnetic force is derived by first determining the interaction of the two electrons due to the field of the outer electron 2 acting on the magnetic moments of electron 1 and vice versa. Insight to the behavior is given by considering the physics of a single bound electron in an externally applied uniform magnetic field as discussed in the Two-Electron Atoms section. The orbitsphere-cvf and the uniform current- (charge-) density function 70° (θ,φ) was given in the Orbitsphere Equation of Motion for £= 0 section and Appendix III. The resultant angular momentum projections of the spherically-symmetric orbitsphere current density, Y° (θ,φ) ,
are L^ = — and Lz = — . As shown in the Resonant Precession of the Spin-l/2-Current-
Density Function Gives Rise to the Bohr Magneton section, the electron spin angular momentum gives rise to a trapped photon with % of angular momentum along an S -axis. Then, the spin state of an orbitsphere comprises a photon standing wave that is phase- matched to a spherical harmonic source current, a spherical harmonic dipole Y"' (θ,φ) = sin θ with respect to the S -axis. The dipole spins about the S -axis at the angular velocity given by Eq.(1.55) with h of angular momentum. S rotates about the z-axis at the Larmor frequency
TV at θ = ~ such that it has a static projection of the angular momentum of
Tt Ti S1I = ±hcos— = ±— iz as given by Eq. (1.85), and from Eq. (1.84), the projection of S onto
K ϊϊ the transverse plane (xy-plane) is Sx = % sin— = ±J— h IYR . Then, the vector projection of the
radiation-reaction-type magnetic force of the Two Electron Atom section given by Eqs. (7.24) and (7.31) contain the factor J— % . This represents the maximum projection of the time-
dependent magnetic moment onto an axis of the spherical-central-force system.
The orbitsphere can serve as a basis element to form a molecular orbital (MO). The total magnitude of the angular momentum of h is conserved for each member of the linear combinations of F0 0 (θ, φ) 's in the transition from the 70° (θ,φ) 's to the MO. Since the charge and current densities are equivalent by the ratio of the frequency, the solution of Laplace's equation for the charge density that is an equipotential energy surface also determines the current density. The frequency and the velocity are given by Newton's laws. Specifically, the further constraint from Newton's laws that the orbital surface is a constant total energy surface and the condition of nonradiation provide that the angular velocity of each point on the surface is constant, the current is continuous and constant, and determines the corresponding velocity function. In non-spherical coordinates, the nonuniform charge distribution given by Laplace's equation is compensated by a nonuniform velocity distribution such that the constant current condition is met. Then, the conservation of the angular momentum is provided by symmetrically stretching the current density along an axis perpendicular to the plane defined by the orthogonal components of angular momentum. The angular momentum projection may be determined by first considering the case of the hydrogen molecular ion. Specifically, the angular momentum must give the results of the Stern-Gerlach experiment as shown for atomic electrons and free electrons in the Resonant Precession of the Spin- 1/2- Current-Density Function Gives Rise to the Bohr Magneton section and Stern-Gerlach Experiment section, respectively.
The hydrogen-molecular-ion MO, and all MOs in general, have cylindrical symmetry along the bond axis. Thus, for the hydrogen molecular ion, the two orthogonal semiminor axes are equivalent and interchangeable. Then, in general, YQ (θ, φ) can serve as a basis element for an MO having equal angular momentum projections along each of the semiminor axes. This defines the plane and the orthogonal axis for stretching the YQ (Θ,Φ) basis element to form the MO. Thus, to conserve angular momentum, ΪJ,0 {θ, φ) is stretched along the semimajor axis as shown in Figure 2. This gives rise to an ellipsoidal surface comprised of the equivalent of elliptical-orbit, plane cross sections in the direction parallel to the semimajor axis with equal angular momentum projections along the orthogonal semiminor axes when the basis element has equal orthogonal angular momentum components. As shown in the Exact Generation of JJ,0 (θ, φ) from the Orbitsphere-cvf section, the orbitsphere is comprised of the uniform function Y° (θ,φ) corresponding to STEP ONE h having the angular momentum components hxy = 0 and L. = — and the uniform function 4
YQ (θ,φ) corresponding to STEP TWO having the angular momentum components
Figure imgf000057_0001
and L2 = — . These components are separable. Then, the basis element YQ (θ,φ) for the
construction of an MO that conserves the total magnitude of the angular momentum of h (Eq. (1.57)) that matches the MO conditions of equal orthogonal components of angular momentum along each semiminor axis is a single YQ° {θ,φ) that is generated according to STEP TWO but with twice the angular momentum in each great-circle basis element to give
L T = — ^ andJ T L7 = — ^ .
2 1
Now consider the behavior of the hydrogen molecular ion in a magnetic field. As shown in the Resonant Precession of the Spin-l/2-Current-Density Function Gives Rise to the Bohr Magneton section, the photon angular momentum corresponding to the resonant excitation of the Larmor excited state is ft , and the angular momentum change corresponding to the spin-flip transition is also fι . Furthermore, torque balance for the orbitsphere was determined by considering the energy minimum due to the interaction of the magnetic moments corresponding to the components of angular momentum. In the case of the hydrogen molecular ion, the Larmor-excitation photon carries Ti of angular momentum that gives rise to a prolate spheroidal dipole current about an S -axis in the same manner as in the case of the spherical dipole of the Larmor excited orbitsphere shown in Figures 1.15 and 1.16 in Chapter 1. The former are given by the prolate angular function, which comprises an associated Legendre function P"' (η) [20], and the latter comprises the spherical harmonic dipole Ye m (θ,φ) = sin ^ . Both are with respect to the S -axis. For hydrogen molecular ion,
— of intrinsic spin is along each of the semiminor axes of the prolate spheroidal MO. Torque
Z* balance is achieved with S along the semimajor axis as shown in Figure 3. Thus, the Larmor excitation is along the semimajor axis. In general, all bonds are cylindrically symmetrical about the internuclear or semimajor axis; thus, the Larmor precession occurs about the bond axis of an MO wherein the intrinsic angular momentum components rotate about S at the
Larmor frequency.
In the coordinate system rotating at the Larmor frequency (denoted by the axes labeled XR , YR , and ZR in Figure 2), the angular momentum of S of magnitude % is stationary. The
YR -component of magnitude — and the ZR -component of magnitude — rotate about S at the
Larmor frequency. The rotation occurs due to a resonant excitation that results in a balance between the magnetic moment of S of μB corresponding to its angular momentum of ft (Eq.
(28) of Box 1.3 and Eq. (2.65)) and those of the orthogonal — angular momentum
components along ZR and YR of — .
Then, the S -axis is the direction of the magnetic moment of each unpaired electron of a molecule or molecular ion. The magnetic moment of S of μB corresponding to its h of angular momentum is consistent with the Stern-Gerlach experiment wherein the Larmor excitation can only be parallel or antiparallel to the magnetic field in order to conserve the angular momentum of the electron, the photon corresponding to the Larmor excitation, and the h of angular momentum of the photon that causes a 180° flip of the direction of S . The result is the same as that for the atomic electron and the free electron given in the Resonant Precession of the Spin-l/2-Current-Density Function Gives Rise to the Bohr Magneton section and Stern-Gerlach Experiment section, respectively. The magnetic field is given in the Magnetic Field of an Ellipsoidal Molecular Orbital section.
Next, consider the magnetic-pairing force of the hydrogen molecule due to the spin- angular-momentum components. The magnetic moments of electrons 1 and 2 of the hydrogen molecule cancel as they are spin paired to form an energy minimum at the radius (i.e. rλ = r2 ). The magnetic force follows the derivation for that between the electrons of two- electron atoms as given in the Two-Electron Atoms section. The latter force was derived by first determining the interaction of the two electrons due to the field of the outer electron 2 acting on the magnetic moments of electron 1 and vice versa. It was also given by the relationship between the angular momentum, energy, and frequency for the transition of electron 2 from the continuum to the ground state. The magnitude of the magnetic force given by Eqs. (7.24) and (7.31) is equivalent to that of the centrifugal force given by Eqs. (7.1-7.2) multiplied by — times the magnitude of the photon, angular momentum vector that
Zh precesses at the Lamior frequency given by Eq. (7.4). In the present case of hydrogen-type molecules, the radiation-reaction-type magnetic force arises between the electrons, each having the components shown in Figure 3. With the photon angular momentum projection of h and the total nuclear (non-photon-field) of 2, the magnitude of the magnetic force between the two electrons is 1/2 that of the centrifugal force given by Eq. (11.95).
The hydrogen-type molecule is formed by the binding of an electron 2 to the hydrogen-type molecular ion comprising two protons at the foci of the prolate spheroidal MO of electron 1. The ellipsoids of electron 1 and electron 2 are confocal; thus, the electric fields and the corresponding forces are normal to the each MO of electron 1 and electron 2. The two electrons are bound by the central field of the two protons as in the case of the molecular ion. Since the field of the protons is only ellipsoidal on average, the field of the hydrogen- type molecular ion is not equivalent to an ellipsoid of charge +1 outside of the electron MO. In addition there is a spin pairing force between the two electrons. Due to the force between electron 2 and electron 1 as well as the central force of the protons, the balance between the centrifugal force and the central field of electron 2 of the hydrogen-type molecule formed by electron 2 binding to a hydrogen-type molecular ion also given by Eq. (11.115). The force balance between the centrifugal force and the sum of the Coulombic and magnetic spin- pairing forces to solve for the semimajor axis is
Figure imgf000059_0001
UL — 1 (11.201) pa pa
a = <h (11.202) P
Substitution of Eq. (11.202) into Eq. (11 .79) is
1 cC'' == ~^2 a° (11-203)
The internuclear distance given by multiplying Eq. (11.203) by two is
2c' = M±. (11.204)
P
Substitution of Eqs. (11.202-11.203) into Eq. (11.80) is (1U05)
Figure imgf000060_0001
Figure imgf000060_0002
For hydrogen, r{t) = D for θ = n — , n = 0, 1, 2, 3, 4. Thus, there is no dipole moment and the
molecule is not predicted to be infrared active. However, it is predicted to be Raman active due to the quadrupole moment. The liquefaction temperature of H2 is also predicted to be significantly higher than isoelectronic helium.
ENERGIES OF HYDROGEN-TYPEMOLECULES The energy components defined previously for the molecular ion, Eqs. (11.117), (11.119), (11.120), and (11.121), apply in the case of the corresponding molecule except that all of the field lines of the protons must end on the MO comprising two-paired electrons. With spin pairing of the mirror-image-current electrons, the scaling factors due to the non-ellipsoidal variation of the electric field of the protons is unity as in the case of the sum of squares of spherical harmonics. Thus, the hydrogen-type molecular energies are given by the integral of the forces without correction. Then, each molecular-energy component is given by the integral of corresponding force in Eq. (11.200) where each energy component is the total for the two equivalent electrons with the central-force action at the position of the electron MO where the parameters a and b are given by Eqs. (11.202) and (11.205), respectively. The potential energy, Ve , of the two-electron MO comprising equivalent electrons in the field of magnitude p times that of the two protons at the foci is
Figure imgf000060_0003
which is equivalent to Ze = 2pe times the potential of the MO given by Eq. (11.46) after Eq. (11.114). The potential energy, VP , due to proton-proton repulsion in the field of magnitude p times that of the protons at the foci (ξ = 0 ) is
Figure imgf000061_0001
The kinetic energy, T , of the two-electron MO of total mass 2me is
Figure imgf000061_0002
The magnetic energy, V1n , of the two-electron MO of total mass 2me corresponding to the magnetic force of Eq. (11.200) is
Figure imgf000061_0003
The total energy, Eτ , is given by the sum of the energy terms (Eqs. (11.207-11.210)):
ET = Ve +T + Vm +Vp (11.211)
Figure imgf000061_0004
where a and b are given by Eqs. (11.202) and (11.205), respectively. The total energy, which includes the proton-proton-repulsion term is negative which justifies the original treatment of the force balance using the analytical mechanics equation of an ellipse that considered only the binding force between the protons and the electrons and the electron centrifugal force. As shown by Eqs. (11.290) and (11.292), T is one-half the magnitude of Ve as required for an inverse-squared force [1] wherein Ve is the source of T .
VIBRATION OF HYDROGEN-TYPE MOLECULES
The vibrational energy levels of hydrogen-type molecules may be solved in the same manner as hydrogen-type molecular ions given in the Vibration of Hydrogen-type Molecular Ions section. The corresponding central force terms of Eq. (11.136) are
Figure imgf000062_0001
and
Figure imgf000062_0002
The distance for the reactive nuclear-repulsive terms is given by the sum of the semimajor axis, a , and c\ 1/2 the internuclear distance. The contribution from the repulsive force between the two protons is
Figure imgf000062_0003
and
Figure imgf000062_0004
Thus, from Eqs. (11.136) and (11.213-11.216), the angular frequency of the oscillation is
Figure imgf000062_0005
(11.217)
where the semimajor axis, a, is α = — according to Eq. (11.202) and c1 is c' = — %=
P i?V2 according to Eq. (11.203). Thus, during bond formation, the perturbation of the orbit determined by an inverse-squared force results in simple harmonic oscillatory motion of the orbit, and the corresponding frequency, ω(θ) , for a hydrogen-type molecule H2 (l/j?) given by Eqs. (11.136) and (11.145) is
ω(0) W* radians! s (11.218)
Figure imgf000062_0006
where the reduced nuclear mass of hydrogen is given by Eq. (11.161) and the spring constant, k(0) , given by Eqs. (11.136) and (11.217) is k (Q) = p4621.98 NnC1 (11.219)
The transition-state vibrational energy, Evib (θ) , is given by Planck's equation (Eq. (11.127)): Evib (0) = hω = np28.62385 X 10u radls = /0.56764 eV (11.220)
The amplitude of oscillation, Areduced (θ) , given by Eqs. (11.158), (11.161), and (11.219) is
Figure imgf000063_0001
Then, from Eq. (11.67), _4C. (θ) , the displacement of c' is the eccentricity e given by Eq. (11.206) times Areduced (θ) (Eq. (11.221)):
(11,22)
Figure imgf000063_0004
The spring constant and vibrational frequency for the formed molecule are then obtained from Eqs. (11.136) and (11.213-11.222) using the increases in the semimajor axis and internuclear distances due to vibration in the transition state. The vibrational energy, Evjb (l) , for the H2 (IZp) υ = 1 -» υ = 0 transition given by adding Ac, (θ) (Eq. (11.222)) to the distances a and a + c' in Eqs. (11.213-11.220) is
Evib (l) = p20.517 eV (11.223) where v is the vibrational quantum number. Using Eq. (11.176) with Eqs. (11.223) and (11.252), the anharmonic perturbation term, ωoxo , of H2 (l / p) is
Figure imgf000063_0002
where ω0 is the frequency of the υ = 1 -» L> = 0 transition corresponding to Eq. (11.223) and D0 is the bond dissociation energy given by Eq. (11.252). The vibrational energies of successive states are given by Eqs. (11.167) and (11.223-11.224).
Using the reduced nuclear mass given by Eq. (11.170), the corresponding parameters for deuterium-type molecules D2 (l / p) (Eqs. (11.213-11.224) and (11.253)) are
(11.225)
Figure imgf000063_0003
/c(0) = /621.98 iVm-1 (11.226)
Evib (0) = /0.4014 eV (11.227) m 7>X\ (11.2IX)
Figure imgf000064_0004
£vjfc (l) = /0.371 eF (11.229)
Figure imgf000064_0001
The vibrational energies of successive states are given by Eqs. (11.167) and (11.229-11.230).
THE DOPPLER ENERGY TERM OF HYDROGEN-TYPE MOLECULES The radiation reaction force in the case of the vibration of the molecule in the transition state also corresponds to the Doppler energy, ED , given by Eq. (11.181) that is dependent on the motion of the electrons and the nuclei. Here, a nonradiative state must also be achieved after the emission due to transient vibration wherein the nonradiative condition given by Eq. (11.24) must be satisfied. Typically, a third body is required to form hydrogen-type molecules. For example, the exothermic chemical reaction of H + H to form H2 does not occur with the emission of a photon. Rather, the reaction requires a collision with a third body, M , to remove the bond energy — H + H +M → H2 +M* [21]. The third body distributes the energy from the exothermic reaction, and the end result is the H2 molecule and an increase in the temperature of the system. Thus, a third body removes the energy corresponding to the additional force term given by Eq. (11.180). From Eqs. (11.200), (11.207) and (11.209), the central force terms between the electron MO and the two protons are
Figure imgf000064_0002
and
2pe 2
(11.232)
J K } 44ππεεoιa"
Thus, the angular frequency of this oscillation is
Figure imgf000064_0003
where the semimajor axis, α , is α = — according to Eq. (11.202). The kinetic energy, E κ>
P is given by Planck's equation (Eq. (11.127)):
Eκ = hω = hp24Λ34U Z lO16 rad/s = /27.2116 eV (11.234)
In Eq. (11.181), substitution of the total energy of the hydrogen molecule, Eτ , (Eq. (11.212)) for Ehv , the mass of the electron, me , for M , and the kinetic energy given by Eq. (11.234) for Eκ gives the Doppler energy of the electrons for the reentrant orbit.
Figure imgf000065_0001
The total energy of the molecule is decreased by ED .
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency given in the Vibration of Hydrogen-Type
Molecules section. On average, the total energy of vibration is equally distributed between kinetic energy and potential energy [17]. Thus, the average kinetic energy of vibration corresponding to the Doppler energy of the electrons, EKvιb , is 1/2 of the vibrational energy of the molecule given by Eq. (11.148). The decrease in the energy of the hydrogen molecule due to the reentrant orbit in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, E030 , is given by the sum of the corresponding energies, ED and
EKvib . Using Eq. (11.235) and Evib from Eq. (11.220) gives
E0SC = ED +ΕKvib = ED +Up2β- (11.236)
E0n = -/0.326469 eV+-p2 (0.56764 eV) (11.237)
To the extent that the MO dimensions are the same, the electron reentrant orbital energies, Eκ , are the same independent of the isotope of hydrogen, but the vibrational energies are related by Eq. (11.148). Thus, the differences in bond energies are essentially given by 1/2 the differences in vibrational energies. Using Eq. (11.235) and Evlb for
D2 (l/ p) given by Eq. (11.227), that corresponds to the deuterium reduced nuclear mass (Eq. (11.170)), the corresponding Eosc is
Ksc = -/0.326469 eV+-p2 (0.401380 eV) (11.238) TOTAL, IONIZATION, AND BOND ENERGIES OF HYDROGEN AND DEUTERIUM MOLECULES
The total energy of the hydrogen molecule is given by the sum of E7. (Eqs. (11.211-11.212)) and Eosc given Eqs. (11.233-11.236). Thus, the total energy of the hydrogen molecule having a central field of +pe at each focus of the prolate spheroid molecular orbital including the Doppler term is
Eτ = ve +τ+vm +vp +εosc (11.239)
Figure imgf000066_0001
(11.240)
From Eqs. (11.237) and (11.239-11.240), the total energy for hydrogen-type molecules is
Figure imgf000066_0002
= -p231.635 eV - p'0.326469 eV + - p2 (0.56764 eV) (11.241)
= -/31.351 eV -/0.326469 eV
The total energy of the deuterium molecule is given by the sum of Eτ (Eq. (11.212)) and Eosc given by Eq. (11.238):
Figure imgf000066_0003
= -/31.6354 eF-/0.326469 eV + -p2 (0.401380 eV) (11.242) z
= -/31.4345 eF-/0.326469 eV
The total energy, which includes the proton-proton-repulsion term is negative which justifies the original treatment of the force balance using the analytical mechanics equation of an ellipse that considered only the binding force between the protons and the electrons, the spin- pairing force, and the electron centrifugal force.
The first ionization energy of the hydrogen molecule, IP1 , H2(\/p)→H;(Up) + e- (11.243) is given by the difference of Eqs. (11.193) and (11.241): /P1=E, (H2 + (l/p))-ET(H2{l/p))
= -/16.13392 eV- /θ.118755 eV-(-p23l.351 eV ~ /0.326469 eV) (11.244) = /15.2171 eV + p30.2077 UeV
The second ionization energy, IP2 , is given by the negative of Eq. (11.193). IP2 =/16.13392 eV + p3O.UB755 eV (11.245)
The first ionization energy of the deuterium molecule, IP1 ,
D2(l/p)→Z)J(l/^) + e" (11.246) is given by the difference of Eqs. (11.194) and (11.242): /P1=E3, (A+ (l/p))-ET(D2(l/p))
= -/16.180 eF-/0.118811 eV-{-p23ΪA345 eV-p30.326469 eV) (11.247) = /15.255 eF + /0.2077 eV The second ionization energy, IP2 , is given by the negative of Eq. (11.194).
IP2 =/16.180 eV + p30.imileV (11.248)
The bond dissociation energy, E0, is the difference between the total energy of the corresponding hydrogen atoms and E1,
ED = E(2H (l/p))-ET (11.249) where [18]
E(2H(l/p)) = -p227.20eV (11.250) and [19]
E(2D(l/p)) = -p227.206eV (11.251)
The hydrogen bond energy, E0 , is given by Εqs. (11.249-11.250) and (11.241):
ED=-p227.2OeV-Ex =-/27.20 eV-(-p23\.35leV -p30.326469 eV) (11.252)
= /4.151 eF + /0.326469 eV The deuterium bond energy, ED , is given by Εqs. (11.249), (11.251), and (11.242): ED = -p227.206 eV- ET
= -/27.206 eV-(-p23lΛ345 eV- p30326469 eV) (11.253)
= p24.229 eV + p30.326469 eV
THE HYDROGEN MOLECULAR ION H2 [2c = 2ao]+
FORCE BALANCE OF HYDROGEN MOLECULAR ION
Force balance between the electric and centrifugal forces is given by Eq. (11.115) where p = l
Figure imgf000068_0001
which has the parametric solution given by Eq. (11.83) when a = 2a0 (11.255)
The semimajor axis, a , is also given by Eq. (11.116) where p = l . The internuclear distance, 2c ' , which is the distance between the foci is given by Eq. (11.111) where p = 1.
2c' = 2α0 (11.256)
The experimental internuclear distance is 2a0 . The semiminor axis is given by Eq. (11.112) where p = \ .
Figure imgf000068_0002
The eccentricity, e , is given by Eq. (11.113).
e = - (11.258)
ENERGIES OF THE HYDROGEN MOLECULAR ION
The potential energy, Ve , of the electron MO in the field of the protons at the foci (ξ = 0) is given by Eq. (11.117) where p = l
Figure imgf000068_0003
The potential energy, V , due to proton-proton repulsion is given by Eq. (11.120) where p = l (11.260)
Figure imgf000069_0004
The kinetic energy, T , of the electron MO is given by Eq. (11.119) where p = \
Figure imgf000069_0001
Substitution of a and b given by Eqs. (11.255) and (11.257), respectively, into Eqs. (11.259- 11.261) is
-4e2
V = -hi3 = -59.7575 eV (11.262)
8πεoaH
V = ■ = 13.5984 eV (11.263)
Sπε a H
T = 29.8787 eV (11.264)
Figure imgf000069_0002
The Doppler term, Eosc , for hydrogen and deuterium are given by Eqs. (11.189) and (11.190), respectively, where p = 1
E0SC (HZ ) = ^D + &KVΛ = -0.118755 eV +-(0.29282 eV) = 0.027655 (11.265)
Eosc (p\) = -0.11881 l eF+-(0.20714 eF) = -0.01524 eV (11.266)
The total energy, E7, , for the hydrogen molecular ion given by Eqs. (11.191-11.193) is
Figure imgf000069_0003
1
= -16.2803 eF-0.118811 eF + -(0.29282 eF) (11.267) z
= -16.2527 eV where in Eqs. (11.262-11.267), the radius of the hydrogen atom aH (Eq. (1.287)) was used in place of α0 to account for the corresponding electrodynamic force between the electron and the nuclei as given in the case of the hydrogen atom by Eq. (1.231). The negative of Eq. (11.267) is the ionization energy of H2 + and the second ionization energy, IP2 , of H2. From Eqs. (11.191-11.192) and (11.194), the total energy, E7. , for the deuterium molecular ion (the ionization energy of D2 and the second ionization energy, IP2 , of D2) is
Eτ = -16.284 eV- O.nm i eV+-(0.207U eV) = -l6.299 eV (11.268)
The bond dissociation energy, ED , is the difference between the total energy of the corresponding hydrogen atom and E1, . The hydrogen molecular ion bond energy, ED , including the reduced electron mass given by Eq. (11.198) where p = 1 is
ED = 2.535 eF + 0.118755 eV = 2.654 eV (11.269)
The experimental bond energy of the hydrogen molecular ion [22] is
ED = 2£5\ eV (11.270) From Eq. (11.199) where p = 1, the deuterium molecular ion bond energy, ED , including the reduced electron mass of D is
ED = 2.5770 eV + 0.11881 I eF = 2.6958 eV (11.271)
The experimental bond energy of the deuterium molecular ion [23] is
ED = 2.691 eV (11.272)
VIBRATION OF THE HYDROGEN MOLECULAR ION
It can be shown that a perturbation of the orbit determined by an inverse-squared force results in simple harmonic oscillatory motion of the orbit [H]. The resonant vibrational frequency for H\ given by Eq. (11.160) is
4.449 X 1014 radiansls (11.273)
Figure imgf000070_0001
wherein p = 1. The spring constant, k(θ) , for H2 given by Eq. (11.162) is
£(0) = 165.51 Nm"1 (11.274)
The vibrational energy, Evιb (θ) , of H2 during bond formation given by Eq. (11.163) is
Evlb (0) = 0.29282 eV (11.275) The amplitude of oscillation given by Eq. (11.164) is
A(O) = — J7T- = 5.952 X 10-12 rø = 0.1125« (11.276)
V } 2m (165.51 NnT1 μf The vibrational energy for the H^ o = 1 -> υ = 0 transition given by Eq. (11.166) is
Evib (\) = 0.270 eV (11.277)
The experimental vibrational energy of H^ [14, 19] is
Evlb = 027l eV (11.278) The anharmonicity term of H^ given by Eq. (11.169) is
O)0X0 = 55.39 cm'1 (11.279)
The experimental anharmonicity term of H^ from NIST [19] is ωexe = 66.2 cm-1 (11.280)
The vibrational energy for the D^ υ = 1 -> υ = 0 transition given by Eq. (11.175) is Evib = 0Λ93 eV (11.281)
The vibrational energy of the D\ [19] based on calculations from experimental data is
Evώ = 0.196 eV (11.282)
The anharmonicity term of D\ given by Eq. (11.176) is ωojco = 27.86 cm'1 (11.283)
The experimental anharmonicity term of £>2 for the state X ]T +1sσ is not given, but the
term for state B 2^ +3dσ from NIST [19] is ωexe = 2.62 cm-χ (11.284)
THE HYDROGEN MOLECULE H2 [2c1 = V2a0]
FORCE BALANCE OF THE HYDROGEN MOLECULE
The force balance equation for the hydrogen molecule is given by Eq. (11.200) where p = \
Figure imgf000071_0001
which has the parametric solution given by Eq. (11.83) when a = a0 (11.286)
The semimajor axis, a , is also given by Eq. (11.202) where p = 1. The internuclear distance, 2c ' , which is the distance between the foci is given by Eq. (11.204) where p = 1.
Figure imgf000072_0001
The experimental internuclear distance is -J2ao . The seminiinor axis is given by Eq. (11.205) where p = \ .
(11.288) h 'Tτ a° The eccentricity, e , is given by Eq. (11.206). 1 e - (11.289)
^
The finite dimensions of the hydrogen molecule are evident in the plateau of the resistivity versus pressure curve of metallic hydrogen [24].
ENERGIES OF THE HYDROGEN MOLECULE
The energies of the hydrogen molecule are given by Eqs. (11.207-11.210) where p = 1
V (11.290)
Figure imgf000072_0002
Figure imgf000072_0003
The energy, V1n , of the magnetic force is
-h2
V = ■ -In- ■ = -16.9589 eV (11.293)
4meaΛia2 -b2 a-sja2 -b2 The Doppler terms, Eosc , for hydrogen and deuterium molecules are given by Eqs. (11.237) and (11.238), respectively, where p = 1
Ksc [Hi) = ED +EKvιb = -0.326469 eF+-(0.56764 eV) = -0.042649 eV (11.294)
E0111 (D2) = -0.326469 eV +-(0.401380 eV) = -0.125779 eV (11.295)
The total energy, E1, , for the hydrogen molecule given by Eqs. (11.239-11.241) is
Figure imgf000073_0001
= -31.635 eF-0.326469 eF+-(0.56764 eF) (11.29
= -31.6776 e V
6)
From Eqs. (11.239-11.240) and (11.242), the total energy, E7. , for the deuterium molecule is
E7 = -31.635 eF -0.326469 eF+-(θ.4O138O eF) = -31.7608 eF (11.297)
The first ionization energies of the hydrogen and deuterium molecules, IP1 , (Eqs. (11.243) and (11.246)) are given by the differences in the total energy of corresponding molecular ions and molecules which are given by Eqs. (11.244) and (11.247), respectively, where p = 1 :
7i> (/f2) = 15.2171 eF + 0.207714 eF = 15.4248 eF (11.298) n\ (D2) = 15255 eV + 0.2077 eV = l5M27 eV (11.299) The bond dissociation energy, ED , is the difference between the total energy of two of the corresponding hydrogen atoms and Eτ . The hydrogen molecular bond energy, ED , given by Eq. (11.252) where p = 1 is
ED = 4.151 eF + 0.326469 eF = 4.478 eV (11.300)
The experimental bond energy of the hydrogen molecule [22] is ED = 4.478 eV (11.301)
The deuterium molecular bond energy, ED , given by Eq. (11.253) where p = 1 is
ED = 4.229 eV + 0.326469 eV = 4.556 eV (11.302)
The experimental bond energy of the deuterium molecule [22] is
ED =4.556 eV (11.303)
VIBRATION OF THE HYDROGEN MOLECULE
It can be shown that a perturbation of the orbit determined by an inverse-squared force results in simple harmonic oscillatory motion of the orbit [H]. The resonant vibrational frequency for H2 given by Eq. (11.218) is 8 62385 Z lQ14 radians Is (11.304)
Figure imgf000074_0001
The spring constant, k(θ) , for H2 given by Eq. (11.219) is k(0) = 621.98 Nm-1 (11.305) wherein /? = 1. The vibrational energy, Evib (θ) , of H2 during bond formation given by Eq. (11.220) is
Evib (0) = 0.56764 e V (11.306)
The amplitude of oscillation given by Eq. (11.221) is
10"12 rø = 0.08079αo (11.307)
Figure imgf000074_0002
The vibrational energy for the H2 υ = l -^- υ = 0 transition given by Eq. (11.223) is Evib (\) = 0.517 eV (11.308)
The experimental vibrational energy of H2 [25-26] is
Evib (\) = 0.5159 eV (11.309)
The anharmonicity term of H2 given by Eq. (11.224) is
O)0X0 = 120.4 cm'1 (11.310) The experimental anharmonicity term of H2 from Huber and Herzberg [23] is ωβxe = 121.33 cm-1 (11.311)
The vibrational energy for the D2 u = l -> ι; = 0 transition given by Eq. (11.229) is
Evib = 0.371 eV (11.312)
The experimental vibrational energy of D2 [14, 19] is Ev/A = 0.371 eF (11.313)
The anharmonicity term of D2 given by Eq. (11.230) is
(D0X0 = 60.93 cm'1 (11.314)
The experimental anharmonicity term of D2 from NIST [19] is ωexe = 61.82 cm~x (11.315) The results of the determination of the bond, vibrational, total, and ionization energies, and internuclear distances for hydrogen and deuterium molecules and molecular ions are given in Table 11.1. The calculated results are based on first principles and given in closed form equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
Despite the predictions of standard quantum mechanics that preclude the imaging of a molecule orbital, the full three-dimensional structure of the outer molecular orbital of N2 has been recently tomographically reconstructed [27]. The charge-density surface observed is similar to that shown in Figure 5 for H2 which is direct evidence that electrons are not point- particle probability waves that have no form until they are "collapsed to a point" by measurement. Rather they are physical, two-dimensional equipotential charge density surfaces.
Figure imgf000076_0001
Figure imgf000077_0001
The experimental total energy of the hydrogen molecule is given by adding the first (15.42593 eV) [28] and second (16.2494 eV) ionization energies where the second ionization energy is given by the addition of the ionization energy of the hydrogen atom (12.59844 eV) [18] and the bond energy of H2 (2.651 eV) [22].
The experimental total energy of the deuterium molecule is given by adding the first (15.466 eV) [23] and second (16.294 eV) ionization energies where the second ionization energy is given by the addition of the ionization energy of the deuterium atom (12.603 eV) [19] and the bond energy of D2 (2.692 eV) [23].
The experimental second ionization energy of the hydrogen molecule, IP2 , is given by the sum of the ionization energy of the hydrogen atom (12.59844 eV) [18] and the bond energy of H2 (2.651 eV) [22].
The experimental second ionization energy of the deuterium molecule, IP2 , is given by the sum of the ionization energy of the deuterium atom (12.603 eV) [19] and the bond energy of D2 (2.692 eV) [23]. The internuclear distances are not corrected for the reduction due to Eosc . The internuclear distances are not corrected for the increase due to En^ .
THE DIHYDRINO MOLECULAR ION H2 [2c = ao]+
FORCE BALANCE OF THE DIHYDRINO MOLECULAR ION
Force balance between the electric and centrifugal forces of H2 (1/2) is given by Eq. (11.115) where p = 2
* -D = D (11.316) mea2h2 8πεoab2 which has the parametric solution given by Eq. (11.83) when a = a (11.317)
The semimajor axis, a , is also given by Eq. (11.116) where p = 2 . The internuclear distance, 2c ' , which is the distance between the foci is given by Eq. (11.111) where p = 2 .
2c' = a0 (11.318) The semiminor axis is given by Eq. (11.112) where p = 2.
Figure imgf000078_0001
The eccentricity, e , is given by Eq. (11.113).
1 e = — (11.320) 2
ENERGIES OF THE DIHYDRINO MOLECULAR ION
The potential energy, Ve , of the electron MO in the field of magnitude twice that of the protons at the foci ( ξ = 0 ) is given by Eq. ( 11.117) where p = 2
Figure imgf000078_0002
The potential energy, Vp , due to proton-proton repulsion in the field of magnitude twice that ofthe protons at the foci (£ = 0) is given by Eq. (11.120) where p = 2 2e2
K = SπεΛa 2 -b 7 2- (11.322)
The kinetic energy, T , ofthe electron MO is given by Eq. (11.119) where p = 2
Figure imgf000078_0003
Substitution of a and b given by Eqs. (11.317) and (11.319), respectively, into Eqs. (11.321- 11.323) and using Eqs. (11.191-11.193) with p = 2 gives
-I6e2
K - ■In3 = -239.16 eV (11.324)
%πεoao
Figure imgf000078_0004
8e2
T = In3 = 119.58 eV (11.326)
%πεoao ET = Ve + Vp +T + E0SC (11.327)
Figure imgf000079_0001
= -22 (l6.13392 eF)-23 (0.118755 eF) (11.328)
= -65.49 eV where Eqs. (11.324-11.326) are equivalent to Eqs. (11.122-11.124) with p = 2 . The bond dissociation energy, ED , given by Eq. (11.198) with p = 2 is the difference between the total energy of the corresponding hydrino atom and E1 given by Eq. (11.328):
ED = ET(H(l/p))-ET(H2 + (Vp))
= 22 (2.535 eV) + 23 (0.118755 eV) (11.329)
= 11.09 eV
VIBRATION OF THE DIHYDRINO MOLECULAR ION
It can be shown that a perturbation of the orbit determined by an inverse-squared force results in simple harmonic oscillatory motion of the orbit [H]. The resonant vibrational frequency for H\ (1/2) from Eq. (11.160) is
ω(0) = 22 /165.51 Nm'1 = L?g χ χQl5 mdiam js (11.330)
wherein p = 2 . The spring constant, k(θ) , for H2 + (1/2) from Eq. (11.162) is
£(0) = 24165.51 JViW-1 = 2648 NnC1 (11.331) The amplitude of oscillation from Eq. (11.164) is
(11.332)
Figure imgf000079_0002
The vibrational energy, Evιb (l) , for the υ = 1 -» υ = 0 transition given by Eq. (11.166) is
Evώ (1) = 22 (0.270 eV) = 1.08 eV (11.333) THE DIHYDRINO MOLECULE H2 2c'
Figure imgf000080_0001
FORCE BALANCE OF THE DIHYDRINO MOLECULE
The force balance equation for the dihydrino molecule H2 (1/2) is given by Eq. (11.200) where p = 2
Figure imgf000080_0002
which has the parametric solution given by Eq. (11.83) when
«„
(11.335)
The semimajor axis, a , is also given by Eq. (11.202) where p = 2 . The internuclear distance, 2c ' , which is the distance between the foci is given by Eq. (11.204) where p = 2.
2c' = ——a (11.336)
S °
The semiminor axis is given by Eq. (11.205) where p = 2 .
Figure imgf000080_0003
The eccentricity, e , is given by Eq. (11.206).
Figure imgf000080_0004
ENERGIES OF THE DIHYDRINO MOLECULE
The energies of the dihydrino molecule H2 (l/2) are given by Eqs. (11.207-11.210) and Eqs. (11.239-11.241) with p = 2
Figure imgf000080_0005
The energy, V , of the magnetic force is
Figure imgf000081_0001
ET = Ve +T + Vm +Vp +E0SC (11.343)
Figure imgf000081_0002
= -22 (31.35 X eV)-I1 (0.326469 eV) (11.344) = -128.02 eV where Eqs. (11.339-11.342) are equivalent to Eqs. (11.207-11.210) with p = 2 . The bond dissociation energy, ED , given by Eq. (11.252) with p = 2 is the difference between the total energy of the corresponding hydrino atoms and E1. given by Eq. (11.344).
ED = ET (2H(I Ip)) -Ex(H2 (Vp))
= 22 (4.151 eV) + 23 (0.326469 eF) (11.345)
= 19.22 eV
VIBRATION OF THE DIHYDRINO MOLECULE It can be shown that a perturbation of the orbit determined by an inverse-squared force results in simple harmonic oscillatory motion of the orbit [H]. The resonant vibrational frequency for the
(11.346)
Figure imgf000081_0003
wherein /? = 2 . The spring constant, Jt(O) , for H2 (1/2) from Eq. (11.219) is ^(0) = 24621.98 Nm-1 = 9952 iVm-1 (11.347)
The amplitude of oscillation from Eq. (11.221) is
(11.348)
Figure imgf000081_0004
The vibrational energy, EvΛ (l) , of H2 (l / 2) from Eq. (11.223) is Kb (0 = 22 (0.517) eV = 2.07 eV (11.349) GEOMETRY
The internuclear distance can also be determined geometrically. The spheroidal MO of the hydrogen molecule is an equipotential energy surface, which is an energy minimum surface. For the hydrogen molecule, the electric field is zero for ξ > 0. Consider two hydrogen atoms A and B approaching each other. Consider that the two electrons form a spheroidal MO as the two atoms overlap, and the charge is distributed such that an equipotential two- dimensional surface is formed. The electric fields of atoms A and B add vectorially as the atoms overlap. The energy at the point of intersection of the overlapping orbitspheres decreases to a minimum as they superimpose and then rises with further overlap. When this energy is a minimum the internuclear distance is determined. It can be demonstrated [33] that when two hydrogen orbitspheres superimpose such that the radial electric field vector from nucleus A and B makes a 45° angle with the point of intersection of the two original orbitspheres, the electric energy of interaction between orbitspheres given by
faction ' (11.350)
Figure imgf000082_0001
is a minimum (Figure 7.1 of [33]). The MO is a minimum potential energy surface; therefore, a minimum of energy of one point on the surface is a minimum for the entire surface of the MO. Thus,
Figure imgf000082_0002
The experimental internuclear bond distance is 0.746 A .
DIHYDRINO IONIZATION ENERGIES
The first ionization energy, IP1 , of the dihydrino molecule
H2 (I /p) → H2 + (I /p) + e~ (11.352) is given by Eq. (11.244) with p = 2. IP1 = E7 [H^ (IZp))- E1. (H2 (IZp)) (11.353)
/P1 = 22 (15.2171 eV) + 23 (0.2077 eV) = 62.53 eV (11.354) The second ionization energy, IP2 , is given by Eq. (11.245) with p = 2.
IP2 = 22 (16.13392 eV) + 23 (0.118755 eV) = 65.49 eV (11.355)
A hydrino atom can react with a hydrogen, deuterium, or tritium nucleus to form a dihydrino molecular ion that further reacts with an electron to form a dihydrino molecule. H(llp) + H+ +e- → H2 (llp) (11.356)
The energy released is
E = E(H(V p))-ET (11.357) where E7. is given by Eq. (11.241). A hydrino atom can react with a hydrogen, deuterium, or tritium atom to form a dihydrino molecule.
H(H p) + H → H2 (Hp) (11.358)
The energy released is
E = E(H (Hp)) + E(H)- ET (11.359) where E1. is given by Eq. (11.241).
SIZES OF REPRESENTATIVE ATOMS AND MOLECULES
ATOMS
Helium Atom (He) .
Helium comprises the nucleus at the origin and two electrons as a spherical shell at r = 0.56Ia0.
Hydrogen Atom (H [aH])
Hydrogen comprises the nucleus at the origin and the electron as a spherical shell at r = aH .
Hydrino Atom (H
2
Hydrino atom (1/2) comprises the nucleus at the origin and the electron as a spherical shell at α. r = =*- .
MOLECULES
Hydrogen Molecular Ion (H2 [2c1 = 2ao]+ ) a = 2aQ
Figure imgf000084_0005
c =α0 2c' = 2α0
Figure imgf000084_0006
a = a0
Figure imgf000084_0001
Figure imgf000084_0007
a = a
Figure imgf000084_0002
2c' = αn
Figure imgf000084_0003
a = —an
Figure imgf000084_0004
Figure imgf000084_0008
ORTHO-PARA TRANSITION OF HYDROGEN-TYPE MOLECULES
Each proton of hydrogen-type molecules possesses a magnetic moment, which is derived in the Proton and Neutron section and is given by
Figure imgf000085_0001
The magnetic moment, m , of the proton is given by Eq. (11.360), and the magnetic field of the proton follows from the relationship between the magnetic dipole field and the magnetic moment, m , as given by Jackson [34] where m = μPiz .
H = -&-(ir2cos0 -iβ sin0) (11.361)
Multiplication of Eq. (11.361) by the permeability of free space, μQ , gives the magnetic flux, B , due to proton one at proton two.
B =^L(ir2cos0 -i, sin0) (11.362)
Δis°^o/para , the energy to flip the orientation of proton two's magnetic moments, μp , from ortho (parallel magnetic moments) to para (antiparallel magnetic moments) with respect to the direction of the magnetic moment of proton one with corresponding magnetic flux B is
AEm° 20/para = -2μpB = ~~2μfp (11.363)
where r is the internuclear distance 2c' where c' is given by Eq. (11.204). Substitution of the internuclear distance into Eq. (11.363) for r gives
The frequency, / , can be determined from the energy using the Planck relationship, Eq. (2.18).
Figure imgf000085_0003
From Eq. (11.365) with p = 2 , the ortho-para transition energy of the dihydrino molecule is 14.4 MHz . NUCLEAR MAGNETIC RESONANCE SHIFT
The proton gyromagnetic ratio, γp 12π , is γp /2π = 42.57602 MHz T~λ (11.366)
The NMR frequency, / , is the product of the proton gyromagnetic ratio given by Eq. (11.366) and the magnetic flux, B .
/ = γp 12τrB = 42.57602 MHz T 1B (11.367)
A typical flux for a superconducting NMR magnet is 1.5 T . According to Eq. (11.367) this corresponds to a radio frequency (RF) of 63.86403 MHz . With a constant magnetic field, the frequency is scanned to yield the spectrum where the frequency scan is typically achieved using a Fourier transform on the free induction decay signal following a radio frequency pulse. Or, in a less common type of NMR spectrometer, the radiofrequency is held constant
(e.g. 60 MHz ), the applied magnetic field, H0 (H0 = — ), is varied over a small range, and
Mo the frequency of energy absorption is recorded at the various values for H0. The spectrum is typically scanned and displayed as a function of increasing H0. The protons that absorb energy at a lower H0 give rise to a downfield absorption peak; whereas, the protons that absorb energy at a higher H0 give rise to an upfield absorption peak. The electrons of the compound of a sample influence the field at the nucleus such that it deviates slightly from the applied value. For the case that the chemical environment has no NMR effect, the value of H0 at resonance with the radiofrequency held constant at 60 MHz is ≥L. (2»)(60 iffl.) 3 μoγP //042.57 '602 MHz T'1 °
In the case that the chemical environment has a NMR effect, a different value of H0 is required for resonance. This chemical shift is proportional to the electronic magnetic flux charge at the nucleus due to the applied field, which in the case of each dihydrino molecule is a function of its semimajor and semiminor axes as shown infra. Consider the application of a z-axis-directed uniform external magnetic flux, B2 , to a dihydrino molecule comprising prolate spheroidal electron MOs with two spin-paired electrons. The dianiagnetic reaction current increases or decreases the MO current to counteract any applied flux according to Lenz's law as shown in the Ηydrino Hydride Ion Nuclear Magnetic Resonance Shift section. The current of hydrogen-type molecules is along elliptical orbits parallel to the semimajor axis. Thus, the electronic interaction with the nuclei requires that each nuclear magnetic moment is in the direction of the semiminor axis. Thus, the nuclei are NMR active towards B2 when the orientation of the semimajor axis, a, is along the x-axis, and the semiminor axes, b = c , are along the y-axis and z-axis, respectively. The flux is applied over the time interval At = t, -tf such that the field increases at a rate dB/dt . The electric field, E , along a perpendicular elliptic path of the dihydrino MO at the plane z - 0 is given by
Figure imgf000087_0001
The induced electric field must be constant along the path; otherwise, compensating currents would flow until the electric field is constant. Thus, Eq. (11.369) becomes
Figure imgf000087_0002
where E{k) is the elliptic integral given by
E (k) = jyjl - k sin2 φdφ = 1.2375 (11.371)
0
Figure imgf000087_0003
the area of an ellipse, A , is
A = πab (11.373) the perimeter of an ellipse, s , is s = 4aE(k) (11.374) a is the semimajor axis given by Eq. (11.202), b is the semiminor axis given by Eq. (11.205), and e is the eccentricity given by Eq. (11.206). The acceleration along the path, dv/dt , during the application of the flux is determined by the electric force on the charge density of the electrons: dv „ eπab dB ,. . „__. m = eE = -———— (11.375) dt 4aE(k) dt
Thus, the relationship between the change in velocity, v , and the change in B is dv = eπf dB (11.376)
AaE{k)me Let Δv represent the net change in v over the time interval At = tt -tf of the application of the flux. Then, (11.377)
Figure imgf000088_0001
The average current, I , of a charge moving time harmonically along an ellipse is
/ = e/ = eV, , (11.378)
J AaE (Jc) K J where / is the frequency. The corresponding magnetic moment is given by
m = AI = πabI = -≡*f- (11.379)
Thus, from Eqs. (11.377) and (11.379), the change in the magnetic moment, Δm , due to an applied magnetic flux, B , is [35]
Am (11.380)
Figure imgf000088_0002
Next, the contribution from all plane cross sections of the prolate spheroid MO must be integrated along the z-axis. The spheroidal surface is given by x2 v2 z2 s +¥+¥=l (1L381)
The intersection of the plane z = z} (-b ≤ z' ≤ b) with the spheroid determines the curve
Figure imgf000088_0003
or
Figure imgf000088_0004
Eq. (11.383) is an ellipse with semimajor axis, a ' , and semiminor axis, b ' , given by
Figure imgf000088_0005
The eccentricity, e ' , is given by a
Figure imgf000089_0001
where e is given by Eq. (11.372). The area, A ' , is given by
A' = πa'b ' (11.387) and the perimeter, s ' , is given by
(11.388)
Figure imgf000089_0002
where s is given by Eq. (11.374). The differential magnetic moment change along the z-axis is
Figure imgf000089_0003
Using Eq. (11.385) for the parameter b' , the change in magnetic moment for the dihydrino molecule is given by the integral over -b ≤ b' ≤ b :
Figure imgf000089_0004
Then, integral to correct for the z-dependence of b ' is
C1 = _ -b = ±b2 = ^ (11.391)
2b 3p
Ctn where the semiminor axis, b = — %= , given by Eq. (11.205) was used.
W2 The change in magnetic moment would be given by the substitution of Eq. (11.391) into Eq. (11.390), if the change density were constant along the path of Eqs. (11.370) and (11.378), but it is not. The charge density of the MO in rectangular coordinates (Eq. (11.42)) is
Figure imgf000089_0005
(The mass-density function of an MO is equivalent to its charge-density function where m replaces q of Eq. (11.42)). The equation of the plane tangent to the ellipsoid at the point
X^-+Y^-+Z-^- = 1 (11.393) a2 b e where X, Y, Z are running coordinates in the plane. After dividing through by the square root of the sum of the squares of the coefficients of X, Y, and Z , the right member is the distance D from the origin to the tangent plane. That is,
Figure imgf000090_0001
so that σ = _.L_£> (11.395)
Aπabc
In other words, the surface density at any point on the ellipsoidal MO is proportional to the perpendicular distance from the center of the ellipsoid to the plane tangent to the ellipsoid at the point. The charge is thus greater on the more sharply rounded ends farther away from the origin. In order to maintain current continuity, the diamagnetic velocity of Eq. (11.377) must be a constant along any given path integral corresponding to a constant electric field. Consequently, the charge density must be the minimum value of that given by Eq. (11.392). The minimum corresponds to y = b and x = z = 0 such that the charge density is
Figure imgf000090_0002
The MO is an equipotential surface, and the current must be continuous over the two- dimensional surface. Continuity of the surface current density, K , due to the diamagnetic effect of the applied magnetic field on the MO and the equipotential boundary condition require that the current of each elliptical curve determined by the intersection of the plane z = z' (-b ≤ z' ≤ b ) with the spheroid be the same. The charge density is spheroidally symmetrical about the semimajor axis. Thus, X , the charge density per unit length along each elliptical path cross section of Eq. (11.383) is given by distributing the surface charge density of Eq. (11.396) uniformly along the z-axis for -b ≤ z' ≤ b . So, λ {z ' = θ) , the linear charge density λ in the plane z' = 0 , is (11.397)
Figure imgf000091_0005
And, the linear charge density must be equally distributed over each elliptical path cross section corresponding to each plane z = z\ The current is independent of z ' when the linear charge density, λ(z ') , is normalized for the path length:
* (11.398)
Figure imgf000091_0001
where the equality of the eccentricities of each elliptical plane cross section given by Eq. (11.386) was used. Substitution of Eq. (11.388) for the corresponding charge density,
, of Eq. (11.390) and using Eq. (11.391) gives
AaxE(k)
Figure imgf000091_0004
The two electrons are spin-paired and the velocities are mirror opposites. Thus, the change in velocity of each electron treated individually (Eq. (10.3)) due to the applied field would be equal and opposite. However, as shown in the Three Electron Atom section, the two paired electrons may be treated as one with twice the mass where me is replaced by 2me in Eq. (11.399). In this case, the paired electrons spin together about the applied field axis, the z-axis, to cause a reduction in the applied field according to Lenz's law. Thus, from Eq. (11.399), the change in magnetic moment is given by
Figure imgf000091_0002
The opposing diamagnetic flux is uniform, parallel, and opposite the applied field as given by Stratton [36]. Specifically, the change in magnetic flux, ΔB , at the nucleus due to the change in magnetic moment, Δm , is
AB = μ0A2Am (11.401) where μ0 is the permeability of vacuum,
Figure imgf000091_0003
is an elliptic integral of the second kind given by Whittaker and Watson [37], and Rs = (s + b2),j<is + a2) (11.403) Substitution of Eq. (11.403) into Eq. (11.402) gives
Figure imgf000092_0001
From integral 154 of Lide [38]:
Figure imgf000092_0002
The evaluation at the limits of the first integral is
Figure imgf000092_0003
From integral #147 of Lide [9], the second integral is:
Figure imgf000092_0004
Evaluation at the limits of the second integral gives
Figure imgf000092_0005
Combining Eq. (11.406) and Eq. (11.408) gives
A (11.409)
Figure imgf000092_0006
where the semimajor axis, a =
Figure imgf000092_0007
given by Eq. (11.205) were used. Substitution of Eq. (11.400) and Eq. (11.409) into Eq. (11.401 ) gives
Figure imgf000092_0008
Additionally, it is found both theoretically and experimentally that the dimensions, r2 , of the molecule corresponding to the area in Eqs. (11.369) and (11.379) used to derived Eq.
(11.410) must be replaced by an average, (r2) , that takes into account averaging over the orbits isotropically oriented. The correction of 2/3 is given by Purcell [35]. In the case of hydrogen-type molecules, the electronic interaction with the nuclei require that each nuclear magnetic moment is in the direction of the semiminor axis. But free rotation about each of three axes results in an isotropic averaging of 2/3 where the rotational frequencies of hydrogen-type molecules are much greater than the corresponding NMR frequency (e.g.
1012 Hz versus 108 Hz). Thus, Eq. (11.410) gives the absolute upfield chemical shift, — ,
B of H2 relative to a bare proton:
Figure imgf000093_0001
= -p2%ti\ppm where p - 1 for H2.
It follows from Eqs. (11.202) and (11.411) that the diamagnetic flux (flux opposite to
the applied field) at each nucleus is inversely proportional to the semimajor radius, a = — - .
P
For resonance to occur, AH0 , the change in applied field from that given by Eq. (11.368), must compensate by an equal and opposite amount as the field due to the electrons of the dihydrino molecule. According to Eq. (11.202), the ratio of the semimajor axis of the dihydrino molecule H2 (I/ p) to that of the hydrogen molecule H2 is the reciprocal of an integer p . Similarly it is shown in the Hydrino Hydride Ion Nuclear Magnetic Resonance Shift section and previously [39], that according to Eq. (7.87) the ratio of the radius of the hydrino hydride ion H' (l/ p) to that of the hydride ion H~ (l/l) is the reciprocal of an integer p . It follows from Eqs. (7.90-7.96) that compared to a proton with no chemical shift, the ratio of AH0 for resonance of the proton of the hydrino hydride ion H~ (l / p) to that of the hydride ion H~ (l/l) is a positive integer. That is, if only the radius is considered, the absorption peak of the hydrino hydride ion occurs at a value of AH0 that is a multiple of p times the value that is resonant for the hydride ion compared to that of a proton with no shift. However, a hydrino hydride ion is equivalent to the ordinary hydride ion except that it is in a lower energy state. The source current of the state must be considered in addition to the reduced radius.
As shown in the Stability of "Ground" and Hydrino States section, for the below "ground" (fractional quantum number) energy states of the hydrogen atom, σpholon , the two- dimensional surface charge due to the "trapped photon" at the electron orbitsphere and phase- locked with the electron orbitsphere current, is given by Eqs. (5.08) and (2.11).
Figure imgf000094_0001
--U i.i l,.., , (11.4I2) And, σeleclr0)l , the two-dimensional surface charge of the electron orbitsphere is
σ electron (11.413)
Figure imgf000094_0002
The superposition of σphoton (Eq. (11.412)) and σelectron , (Eq. (11.413)) where the spherical harmonic functions satisfy the conditions given in the Angular Function section is
Figure imgf000094_0003
n = - = !,-,-,-,..., (11.414) p 2 3 4 J
The ratio of the total charge distributed over the surface at the radius of the hydrino hydride ion H~ (l/ p) to that of the hydride ion H~ (l/l) is an integer p , and the corresponding total source current of the hydrino hydride ion is equivalent to an integer p times that of an electron. The "trapped photon" obeys the phase-matching condition given in Excited States of the One-Electron Atom (Quantization) section, but does not interact with the applied flux directly. Only each electron does; thus, Δv of Eq. (11.377) must be corrected by a factor of 1/ p corresponding to the normalization of the electron source current according to the invariance of charge under Gauss' Integral Law. As also shown by Eqs. (7.17-7.23) and (7.87), the "trapped photon" gives rise to a correction to the change in magnetic moment due to the interaction of each electron with the applied flux. The correction factor of \l p consequently cancels the NMR effect of the reduced radius which is consistent with general observations on diamagnetism [40]. It follows that the same result applies in the case of Eq. (11.411) for H2 (l / p) wherein the coordinates are ellipsoidal rather than spherical.
The cancellation of the chemical shift due to the reduced radius or the reduced semiminor and semimajor axes in the case of H~ (l/ /?) and H2 (1/ p) , respectively, by the corresponding source current is exact except for an additional relativistic effect. The relativistic effect for H~ (l/ p) arises due to the interaction of the currents corresponding to the angular momenta of the "trapped photon" and the electrons and is analogous to that of the fine structure of the hydrogen atom involving the 2P3/2 — 2P112 transition. The derivation follows that of the fine structure given in the Spin-Orbital Coupling section.
of the electron, the electron angular momentum of % , and the electron magnetic mΛ momentum of μB are invariant for any electronic state. The same applies for the paired electrons of hydrino hydride ions. The condition that flux must be linked by the electron in units of the magnetic flux quantum in order to conserve the invariant electron angular momentum of % gives the additional chemical shift due to relativistic effects. Using Eqs. (2.159-2.160), Eq. (2.166) may be written as ,Λ Λ Λ Λ _N (11.415)
Figure imgf000095_0001
From Eq. (11.415) and Eq. (1.205), the relativistic stored magnetic energy contributes a factor of a2π In spherical coordinates, the relativistic change in flux ΔB5Λ may be calculated using Eq. (7.95) and the relativistic factor of γSR = 2πa which is the same as that given by Eq. (1.229):
AB57, = ~TSRJU O ~~ T (K cosθ-iθ sinθ) = -2πaμo —^-(i cos#- L sin#) (11.416) r n r n for r < rn.
The stored magnetic energy term of the electron g factor of each electron of a θ dihydrino molecule is the same as that of a hydrogen atom since — is invariant and the me invariant angular momentum and magnetic moment of the former are also fi and μB, respectively, as given in the Magnetic Moment of an Ellipsoidal MO and Magnetic Field of an Ellipsoidal MO sections. Thus, the corresponding correction in ellipsoidal coordinates follows from Eq. (2.166) wherein the result of the length contraction for the circular path in spherical coordinates is replaced by that of the elliptical path.
The only position on the elliptical path at which the current is perpendicular to the radial vector defined by the central force of the protons is at the semimajor axis. It was shown in the Special Relativistic Correction to the Ionization Energies section that when the condition that the electron's motion is tangential to the radius is met, the radius is Lorentzian invariant. That is, for the case that k is the lightlike k0 , with k = ωn /c, a is invariant. In the case of a spherically symmetrical MO such as the case of the hydrogen atom, it was also
shown that this condition determines that the electron's angular momentum of % , — of Eq. m»
(1.110), and the electron's magnetic moment of a Bohr magneton, μB , are invariant. The effect of the relativistic length contraction and time dilation for constant spherical motion is a change in the angle of motion with a corresponding decrease in the electron wavelength. The angular motion becomes projected onto the radial axis which contracts, and the extent of the decrease in the electron wavelength and radius due to the electron motion in the laboratory inertial frame are given by
Figure imgf000096_0001
and
Figure imgf000096_0002
respectively. Then, the relativistic factor γ* is
Figure imgf000096_0003
where the velocity is given by Eq. (1.56) with the radius given by Eq. (1.233). Each point or coordinate position on the continuous two-dimensional electron MO of the dihydrino molecule defines an infinitesimal mass-density element which moves along an elliptical orbit of a spheroidal MO in such a way that its eccentric angle, θ , changes at a constant rate. That is θ = ωt at time t where ω is a constant, and r(t) = ia cos ωt + jb sin cot (11.420) is the parametric equation of the ellipse. Next, special relativistic effects on distance and time are considered. The parametric radius, r(t) , is a minimum at the position of the semiminor axis of length b , and the motion is transverse to the radial vector. Since the angular momentum of h is constant, the electron wavelength without relativistic correction is given by
2πb = λ = — (11.421) mv such that the angular momentum, L , is given by L = rxmv = bmv = h (11.422)
The nonradiation and the h , — , and μB invariance conditions require that the angular m frequencies, ωs and ωe , for spherical and ellipsoidal motion, respectively, are πL h ma ω. =- (11.423) mer and πh n ωe = (11.424) meA meab where A is the area of the closed orbit, the area of an ellipse given by Eq. (11.373). Since the angular frequency ωe has the form as ωs , the time dilation corrections are equivalent, where the correction for ωs is given in the Special Relativistic Correction to the Ionization Energies section. Since the semimajor axis, a , is invariant, but b undergoes length contraction, the relationship between the velocity and the electron wavelength at the semiminor axis from Eq. (11.417) and Eq. (11.421) is
Figure imgf000097_0001
where λ — > a as v -> c replaces the spherical coordinate result of λ -> r ' as v — > c . Thus, in the electron frame at rest v = 0 , and, Eq. (11.425) becomes λ' = 2πb (11.426)
In the laboratory inertial frame for the case that v = c in Eq. (11.425), λ is λ = a (11.427)
Thus, using Eqs. (11.426) and (11.427), the relativistic factor, γ* , is
* — i = a
(11.428) λ' 2πb
From Eqs. (11.417-11.419) and Eq. (11.428), the relativistic diamagnetic effect of the inverse integer radius of H2 (l/ p) compared to H2, each with ellipsoidal MOs, is equivalent to the ratio of the semiminor and semimajor axes times the correction for the spherical orbital case given in Eq. (11.416). From the mass (Eq. (2.165)) and radius corrections (Eq. (2.163)) in Eq. (2.166), the relativistic stored magnetic energy contributes a factor γSR of γSR = 2πa(-) = πa (11.429)
Thus, from Eqs. (11.401), (11.416), and (11.429), the relativistic change in flux, ΔBSβ , for the dihydrino molecule H2 (1 / p) is
ΔB OT = -rSΛ/"oΛΔm = -πaμ^Am (11.430)
Thus, using Eq. (11.411) and Eq. (11.430), the upfield chemical shift, — &-, due to the
B relativistic effect of the molecule H2 (IZp^ corresponding to the lower-energy state with principal quantum energy state p is given by
Figure imgf000098_0001
A n
The total shift, — τ- , for H2 (l/ p) is given by the sum of that of H2 given by Eq. (11.411) B with p = 1 plus that given by Eq. (11.431):
Figure imgf000098_0002
(2S.01 + 0.64p)ppm (11.433 )
Figure imgf000098_0003
where p = integer > 1 .
H2 has been characterized by gas phase 1H NMR. The experimental absolute resonance shift of gas-phase TMS relative to the proton's gyromagnetic frequency is -28.5 ppm [30]. H2 was observed at 0.48 ppm compared to gas phase TMS set at 0.00 ppm [31].
Thus, the corresponding absolute H2 gas-phase resonance shift of -28.0 ppm (-28.5 + 0.48) ppm was in excellent agreement with the predicted absolute gas-phase shift of -28.01 ppm given by Eq. (11.411).
References for Section I
1. G. R. Fowles, Analytical Mechanics, Third Edition, Holt, Rinehart, and Winston, New York, (1977), pp. 145-158. 2. J. A. Stratton, Electromagnetic Theory, McGraw-Hill Book Company, (1941), pp. 38-59;
195-267.
3. Jahnke-Emde, Tables of Functions, 2nd ed., Teubner, (1933).
4. J. D. Jackson, Classical Electrodynamics, Second Edition, John Wiley & Sons, New York, (1975), pp. 17-22.
5. H. A. Haus, J. R. Melcher, "Electromagnetic Fields and Energy", Department of Electrical engineering and Computer Science, Massachusetts Institute of Technology, (1985), Sec. 5.3.
6. J. A. Stratton, Electromagnetic Theory, McGraw-Hill Book Company, (1941), p. 195. 7. G. R. Fowles, Analytical Mechanics, Third Edition, Holt, Rinehart, and Winston, New York, (1977), pp. 119-124.
8. J. A. Stratton, Electromagnetic Theory^ McGraw-Hill Book Company, (1941), pp. 38-54; 207-209.
9. D. R. Lide, CRC Handbook of Chemistry and Physics, 79th Edition, CRC Press, Boca Raton, Florida, (1998-9), p. A-29.
10. H. A. Haus, "On the radiation from point charges", American Journal of Physics, 54, (1986), pp. 1126-1129.
11. G. R. Fowles, Analytical Mechanics, Third Edition, Holt, Rinehart, and Winston, New York, (1977), pp. 161-164. 12. G. R. Fowles, Analytical Mechanics, Third Edition, Holt, Rinehart, and Winston, New York, (1977), pp. 57-66.
13. M. Mizushima, Quantum Mechanics of Atomic Spectra and Atomic Structure, W. A. Benjamin, Inc., New York, (1970), p.17.
14. M. Karplus, R. N. Porter, Atoms and Molecules an Introduction for Students of Physical Chemistry, The Benjamin/Cummings Publishing Company, Menlo Park, California,
(1970), pp. 447-484.
15. J. D. Jackson, Classical Electrodynamics, Second Edition, John Wiley & Sons, New York, (1975), p. 659.
16. J. D. Jackson, Classical Electrodynamics, Second Edition, John Wiley & Sons, New York, (1975), pp. 780-786.
17. D. A. McQuarrie, Quantum Chemistry, University Science Books, Mill Valley, CA, (1983), p. 172.
18. D. R. Lide, CRC Handbook of Chemistry and Physics, 79th Edition, CRC Press, Boca Raton, Florida, (1998-9), p. 10-175. 19. NIST Atomic Spectra Database, www.physics.nist.gov/cgi-bin/AtData/display.ksh.
20. M. Abramowitz, I. A. Stegun (Editors), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc, New York, (1970), pp. 753-759 21. N. V. Sidgwick, The Chemical Elements and Their Compounds, Volume I, Oxford, Clarendon Press, (1950), p.17.
22. P. W. Atkins, Physical Chemistry^ Second Edition, W. H. Freeman, San Francisco, (1982), p. 589.
23. K. P. Huber, G, Herzberg, Molecular Spectra and Molecular Structure, IV. Constants of Diatomic Molecules, Van Nostrand Reinhold Company, New York, (1979).
24. W. J. Nellis, "Making Metallic Hydrogen", Scientific American, May, (2000), pp. 84-90.
25. H. Beutler, Z. Physical Chem., "Die dissoziationswarme des wasserstoffmolekuls H2, aus einem neuen ultravioletten resonanzbandenzug bestimmt", Vol. 27B, (1934), pp. 287-302.
26. G. Herzberg, L. L. Howe, "The Lyman bands of molecular hydrogen", Can. J. Phys., Vol. 37, (1959), pp. 636-659.
27. J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin, J. C. Kieffer, P. B. Corkum, D. M. Villeneuve, "Tomographic imaging of molecular orbitals", Nature, Vol. 432, (2004), pp. 867-871.
28. D. R. Lide, CRC Handbook of Chemistry and Physics, 79th Edition, CRC Press, Boca Raton, Florida, (1998-9), p. 10-181.
29. R. Loch, R. Stengler, G. Werth, "Measurement of the electronic g factor of H^ ", Phys.
Rev. A, Vol. 38, No. 11, (1988), pp. 5484-5488.
30. C. Suarez, E. J. Nicholas, M. R. Bowman, "Gas-phase dynamic NMR study of the internal rotation in N-trifluoroacetlypyrrolidine", J. Phys. Chem. A, Vol. 107, (2003), pp. 3024- 3029.
31. C. Suarez, "Gas-phase NMR spectroscopy", The Chemical Educator, Vol. 3, No. 2, (1998).
32. D. R. Lide, CRC Handbook of Chemistry and Physics, 79th Edition, CRC Press, Boca Raton, Florida, (1998-9), p. 9-82. 33. R. L. Mills, J. J. Farrell, The Grand Unified Theory, Science Press, (1989), pp. 46-47;
117-119.
34. J. D. Jackson, Classical Electrodynamics, Second Edition, John Wiley & Sons, New York, (1975), p. 178. 35. E. Purcell, Electricity and Magnetism, McGraw-Hill, New York, (1965), pp. 370-389.
36. J. A. Stratton, Electromagnetic Theory, McGraw-Hill Book Company, (1941), pp. 211- 215, 257-258.
37. Whittaker and Watson, Modern Analysis, 4th Edition, Cambridge University Press, (1927), pp. 512ff.
38. D. R. Lide, CRC Handbook of Chemistry and Physics, 79th Edition, CRC Press, Boca Raton, Florida, (1998-9), p. A-30.
39. R. Mills, P. Ray, B. Dhandapani, W. Good, P. Jansson, M. Nansteel, J. He, A. Voigt, "Spectroscopic and NMR Identification of Novel Hydride Ions in Fractional Quantum Energy States Formed by an Exothermic Reaction of Atomic Hydrogen with Certain Catalysts", European Physical Journal- Applied Physics, Vol. 28, (2004), pp. 83-104.
40. E. Purcell, Electricity and Magnetism, McGraw-Hill, New York, (1985), pp. 417-418.
Section II
GENERAL DIATOMIC AND POLYATOMIC MOLECULAR IONS
AND MOLECULES
Non-hydrogen diatomic and polyatomic molecular ions and molecules can be solved using the same principles as those used to solve hydrogen molecular ions and molecules wherein the hydrogen molecular orbitals (MOs) and hydrogen atomic orbitals serve as basis functions for the MOs of the general diatomic and polyatomic molecular ions or molecules. The MO must (1) be a solution of Laplace's equation to give a equipotential energy surface, (2) correspond to an orbital solution of the Newtonian equation of motion in an inverse-radius- squared central field having a constant total energy, (3) be stable to radiation, and (4) conserve the electron angular momentum of h . Energy of the MO must be matched to that of the outermost atomic orbital of a bonding heteroatom in the case where a minimum energy is achieved with a direct bond to the atomic orbital (AO). In the case that an independent MO is formed, the AO force balance causes the remaining electrons to be at lower energy and a smaller radius. The atomic orbital may hybridize in order to achieve a bond at an energy minimum. At least one molecule or molecular ion representative of each of these cases was solved. Specifically, the results of the determination of bond parameters of H3 + , D3 + , OH , OD , H2O, D2O , NH , ND , NH2 , ND2 , NH3 , ND3 , CH , CD, CH2 , CH3 , CH4 , N2 , O2, F2, Cl2, CN , CO, and NO are given in Table 13.1. The calculated results for homo- and hetero-diatomic radicals and molecules, and polyatomic molecular ions and molecules are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
TRIATOMIC MOLECULARHYDROGEN-TYPE ION (H3 +)
The polyatomic molecular ion H3 (\/p) is formed by the reaction of a proton with a hydrogen-type molecule
H2 [Vp) + H+ → Ht {\lp) (13.1) and by the exothermic reaction H2 + (V p) + H2 (V p) → H^ (V p)+H(V p) (13.2)
FORCE BALANCE OF H3 + -TYPE MOLECULAR IONS
#3+ (l//0-tyPe molecular ions comprise two indistinguishable spin-paired electrons bound by three protons. The ellipsoidal molecular orbital (MO) satisfies the boundary constraints as shown in the Nature of the Chemical Bond of Hydrogen-Type Molecules section. Since the protons are indistinguishable, ellipsoidal MOs about each pair of protons taken one at a time are indistinguishable. Ht, (l//>) is then given by a superposition or linear combinations of three equivalent ellipsoidal MOs that form a equilateral triangle where the points of contact between the prolate spheroids are equivalent in energy and charge density. The outer perimeter of the superposition of three prolate spheroids is the H3 + (I//?) MO with the protons at the foci that bind and maintain the electron MO.
As in the case for Hl (II p) and' H2 (IZp) shown in the Nature of the Chemical
Bond of Hydrogen-Type Molecules section, the stability of H^ (l/p) is due to the dependence of the charge density of the distance D from the origin to the tangent plane. That is,
Figure imgf000103_0001
so that σ = D (13.4)
Aπab In other words, the surface density at any point on a charged ellipsoidal conductor is proportional to the perpendicular distance from the center of the ellipsoid to the plane tangent to the ellipsoid at the point. The charge is thus greater on the more sharply rounded ends farther away from the origin. This distribution places the charge closest to the protons to give a minimum energy. The balanced forces also depend on D as shown in the Nature of the Chemical Bond of Hydrogen-Type Molecules section. The D -dependence of the charge density as well as the centrifugal and Coulombic central field of two nuclei at the foci of the ellipsoid applies to each ellipsoid which is given from any other by a rotation of [^ | = — about an axis at a focus that is perpendicular to the plane of the equilateral triangle defined by the three foci. Since the centrifugal, Coulombic, and magnetic forces relate mass and charge densities which are interchangeable by the ratio e/me , the conditions at any point on any given ellipsoid is applicable to any other point on the ellipsoid. Furthermore, this condition can be generalized to any point of the other members of the set of three ellipsoids due to equivalence. As a further constraint to maintain the force balance between the three protons and the H3 + (l//?)
MO comprising the superposition of the three H2 (l/j?) -type ellipsoidal MOs, the total charge of the two electrons must be normalized over the three basis set H2 (I/ p) -type ellipsoidal MOs. In this case, the parameters of each basis element H2 (l / p) -type ellipsoidal MO is solved, and the energies are given by the electron charge where it appears multiplied by a factor of 3 / 2 (three MOs normalized by the total charge of two electrons).
Consider each H2 (I/ j>) -type ellipsoidal MO. At each point on the H3 + (I//?) MO, the electron experiences a centrifugal force, and the balancing centripetal force (on each electron) is produced by the electric force between the electron and the ellipsoidal electric field and the radiation-reaction-type magnetic force between the two electrons causing the electrons to pair. The force balance equation derived in Force Balance of Ηydrogen-Type Molecules section is given by Eq. (11.200):
Figure imgf000104_0001
^s.- 2L = 1 (13.6) pa pa
a = %- (13.7)
P
Substitution of Eq. (13.7) into Eq. (11.79) is
The inte cr'n~ucklear dai'stance given by multiplying Eq. (13.8) by two is (13-8)
Figure imgf000104_0002
Substitution of Eqs. (13.7-13.8) into Eq. (11.80) is h = c = -^=a0 (13.10) pV2 Substitution of Eqs. (13.7-13.8) into Eq. (11.67) is
Using the parameters given by Eqs. (13.7-13.11), the resulting H3 + (l/p) MO comprising the superposition of three H2 (I/^) -type ellipsoidal MOs is shown in Figure 6. The outer surface of the superposition comprises charge density of the MO. The equilateral triangular structure was confirmed experimentally [I]. The H3 + (l/ p) MO having no distinguishable electrons is consistent with the absence of strong excited stated observed for H3 + [I]. It is also consistent with the absence of a permanent dipole moment [I].
ENERGIES OF H3 + -TYPE MOLECULAR IONS
The due to the equivalence of the H2 (l / p) -type ellipsoidal MOs and the linear superposition of their energies, the energy components defined previously for the molecule, Eqs. (11.207-11.212) apply in the case of the corresponding H3 + (l/ j?) molecular ion. And, each molecular energy component is given by the integral of corresponding force in Eq. (13.5). Each energy component is the total for the two equivalent electrons with the exception that the total charge of the two electrons is normalized over the three basis set
H2 (l / p) -type ellipsoidal MOs. Thus, the energies are those given for H2 (IZp) in the
Energies of Ηydrogen-Type Molecules section with the electron charge, where it appears, multiplied by a factor of 3/2. In addition, the three sets of equivalent proton-proton pairs give rise to a factor of three times the proton-proton repulsion energy given by Eq. (11.208). The parameters a and b are given by Eqs. (13.7) and (13.10), respectively.
Figure imgf000105_0001
The energy, V1n , corresponding to the magnetic force of Eq. (13.5) is
Figure imgf000106_0001
= V + T + Vm (13.16)
Figure imgf000106_0002
(13.17) where the charge e appears in the magnetic energy Vm according to Eqs. (7.14-7.24) as discussed in the Force Balance of Hydrogen-Type Molecules section.
VIBRATION OF Tf3 + -TYPE MOLECULAR IONS
The vibrational energy levels of H3 + -type molecular ions may be solved as three equivalent coupled harmonic oscillators by developing the Lagrangian, the differential equation of motion, and the eigenvalue solutions [2] wherein the spring constants are derived from the central forces as given in the Vibration of Hydrogen-Type Molecular Ions section and the Vibration of Hydrogen-Type Molecules section.
THE DOPPLERENERGY TERM OF H3 +-TYPE MOLECULAR IONS
As shown in the Vibration of Ηydrogen-type Molecular Ions section, the electron orbiting the nuclei at the foci of an ellipse may be perturbed such that a stable reentrant orbit is established that gives rise to a vibrational state corresponding to time harmonic oscillation of the nuclei and electron. The perturbation is caused by a photon that is resonant with the frequency of oscillation of the nuclei wherein the radiation is electric dipole with the corresponding selection rules.
Oscillation may also occur in the transition state. The perturbation arises from the decrease in internuclear distance as the molecular bond forms. Relative to the unperturbed case given in the Force Balance of Ηydrogen-type Molecular Ions section, the reentrant orbit may give rise to a decrease in the total energy while providing a transient kinetic energy to the vibrating nuclei. However, as an additional condition for stability, radiation must be considered. A nonradiative state must be achieved after the emission due to transient vibration wherein the nonradiative condition given by Eq. (11.24) must be satisfied. The radiation reaction force due to the vibration of H2 (l/ p) and H2 (l/p) in the transition state was derived in the Doppler Energy Term of Hydrogen-type Molecular Ions section and the
Doppler Energy Term of Hydrogen-type Molecules section, respectively, and corresponds to a Doppler energy, ED , that is dependent on the motion of the electron and the nuclei. The radiation reaction force in the case of the vibration of H^ (l/p) in the transition state also corresponds to the Doppler energy, E0 , given by Eq. (11.181) that is dependent on the motion of the electrons and the nuclei. Here, a nonradiative state must also be achieved after the emission due to transient vibration wherein the nonradiative condition given by Eq. (11.24) must be satisfied. Typically, a third body is required to form H3 + -type molecular ions. For example, the exothermic chemical reaction of H + H to form H2 does not occur with the emission of a photon. Rather, the reaction requires a collision with a third body, M , to remove the bond energy- H + H + M → H2 +M * [3]. The third body distributes the energy from the exothermic reaction, and the end result is the H2 molecule and an increase in the temperature of the system. Thus, a third body removes the energy corresponding to the additional force term given by Eq. (11.180). The kinetic energy of the transient vibration is derived from the corresponding central forces. From Eqs. (13.5) and (13.12), the central force terms between the electron MO and the protons of each of the three H2 (l/ p) -type ellipsoidal MOs are
Figure imgf000107_0001
and
/-(O) = ^-EfL. (13.19)
2 4πεoa
Thus, using Eqs. (11.136) and (13.18-13.19), the angular frequency of this oscillation is
Figure imgf000107_0002
where the semimajor axis, a, is a = — according to Eq. (13.7). The kinetic energy, Eκ , is
P given by Planck's equation (Eq. (11.127)): Eκ = hω = hp25.06326 X IO16 rad/ s = p233.3273 eV (13.21) In Eq. (11.181), substitution of the total energy of the .H3 + -type molecular ion, Eτ % (Eq.
(13.17)) for E1n, , the mass of the electron, me , for M , and the kinetic energy given by Eq. (13.21) for Eκ gives the Doppler energy of the electrons for the reentrant orbit.
E0 s -/0.406013 eV (13.22)
Figure imgf000108_0001
The total energy of the H3 + -type molecular ion is decreased by ED .
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. On average, the total energy of vibration is equally distributed between kinetic energy and potential energy [4]. Thus, the average kinetic energy of vibration corresponding to the Doppler energy of the electrons, EKvlb , is 1/2 of the vibrational energy of the H3 + -type molecular ion given by Eq. (11.148). The decrease in the energy of the molecular ion due to the reentrant orbit in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED and EKvjb . Using Eq. (13.22) and the experimental vibrational energy H3 + of Evib = 2521.31 crn x = 0.312605 eV [1] gives
EOSe (13.23)
Figure imgf000108_0002
Eosc = -/ 0.406013 eV + -p2 (0.312605 eV) (13.24)
The reentrant orbit for the binding of a proton to H1 (1/ p) causes two bonds to oscillate by increasing and decreasing in length along opposite sides of the equilateral triangle at a relative phase angle of 180° . Since the vibration and reentrant oscillation is along two lengths of the equilateral triangular MO with E symmetry, Eosc for H3 + (I/^),
Eosc (m (l/p)), is:
Figure imgf000108_0003
To the extent that the MO dimensions are the same, the electron reentrant orbital energies, Eκ , are the same independent of the isotope of hydrogen, but the vibrational energies are related by Eq. (11.148). Thus, the differences in bond energies are essentially given by 1/2 the differences in vibrational energies per bond. Using Eq. (13.22), Eq. (13.25), and the experimental vibrational energy D3 + of Evlb = 1834.67 cm"1 = 0.227472 eV [1], the corresponding £osc (D3 + (I/ /?)) is
KSc (Ds (l /P)) = 2[ -V0-406013 eV+-p2 (0.227472 eV) (13.26)
TOTAL AND BOND ENERGIES OF H3 +(I//?)- AND Dt(llp)-ΥY?Ε
MOLECULARIONS
The total energy of the H3 + (l/p)-type molecular ion is given by the sum of Eτ (Eqs. (13.16- 13.17)) and EOSC (H+ (1 / pj) given Eqs. (13.20-13.25). Thus, the total energy of H3 + (Up) having a central field of +ρe at each focus of the prolate spheroid molecular orbital including the Doppler term is
ET = Ve +T + Vm +Vp +E0SC (H; (\/p)) (13.27)
Figure imgf000109_0001
(13.28) From Eqs. (13.24-13.25) and (13.27-13.28), the total energy of the H3 + -type molecular ion is
Figure imgf000109_0002
( \ _ . Λ
= -/35.54975 - 2/0.406013 eV + l( -/ (0.312605 eV) (13.29)
\ Δ J
= -/35.23714 eF -/0.812025 eV The total energy of the Z)3 + -type molecular ion is given by the sum of Eτ (Eq. (13.17)) and Eosc (A+ (Vp)) given by Eq. (13.26): Eτ = -/35.54975 eV + E^ (D* (Up))
= -/35.54975-2/0.406013 eV + lf-p2 (0.227472 eV) \ (13.30)
= -/35.32227 eV - /0.812025 eF
The bond dissociation energy, ED , is the difference between the total energy of the corresponding hydrogen molecule and Eτ
ED = E{H2 (\lp)) -ET (13.31) where E(H2 (Hp)) is given by Eq. (11.241):
E(H2 (l/p)) = -/31.351 eF-/0.326469 eV (13.32) and E(D2 (IIp)) is given by Eq. (11.242):
E(D2 (l/p)) = -p23lA345 eV-p30.326469 eV (13.33)
The H3 + bond dissociation energy, ED , is given by Eqs. (13.31-13.32) and (13.29):
ED = -/31.351 eF-/0.326469 eV -ET = -/31.351 eF-/0.326469 eF-(-/35.23714 eF -/0.812025 eV) (13.34)
= /3.88614 eF + /0.485556 eV The D3 + bond dissociation energy, ED , is given by Eqs. (13.31), (13.33), and (13.30):
ED = -/31.4345 eV-p30.326469 eV-ET
= -/31.4345 eF-/0.326469 eV- (-/35.32227 eV-p30.8\2025 eV) = /3.88777 eF + /0.485556 eF i (13.35)
THE H3 + MOLECULAR ION
FORCE BALANCE OF THE H3 + MOLECULAR ION
The force balance equation for H3 + is given by Eq. (13.5) where p = 1
(13.36)
Figure imgf000110_0001
which has the parametric solution given by Eq. (11.83) when a = a0 (13.37) The semimajor axis, a , is also given by Eq. (13.7) where p = 1. The internuclear distance,
2c\ which is the distance between the foci is given by Eq. (13.9) where p = 1.
2c' = V2α0 (13.38)
The semiminor axis is given by Eq. (13.10) where p = 1.
(13.39)
Figure imgf000111_0005
The eccentricity, e , is given by Eq. (13.11).
Figure imgf000111_0001
ENERGIES OF THE H3 + MOLECULAR ION The energies of H3 + are given by Eqs. (13.12-13.15) where p = 1
(13.41)
Figure imgf000111_0002
Figure imgf000111_0003
The energy, Vm , of the magnetic force is
Figure imgf000111_0004
The Doppler terms, Eosc (H3 + (I/^)) and Iosc (D3 + (I/;?)) are given by Eqs. (13.25) and (13.26), respectively, where p = \ E0SC (H;) = 2(ΕD +EKvώ)
1 λ
= 2[ -0.406013 eF+-(0.312605 eV) (13.45) = -0.499420 eF
Eosc (D2) = 2[ -0.406013 eF+-(0.227472 eV) j
(13.46)
= -0.584553 eF The total energy, Eτ , ϊox H3 + given by Eqs. (13.27-13.29) is
Figure imgf000112_0001
= -35.54975 -2(0.406013 eF) + 2[ -(0.31260516 eVU
= -36.049167
(13.47) From Eqs. (13.27-13.28) and (13.30), the total energy, Eτ , for D* is
Eτ = -35.54975 -2(0.406013 eV) + 2 [-(0.227472 eVU
= -36.134300 eV The bond dissociation energy, ED , is the difference between the total energy of H2 or D2 and E7. . The H\ molecular bond dissociation energy, ED , given by the difference between the experimental total energy of H2 [5-7] l and the total energy of H3 + (Eqs. (13.29) where p = l and (13.47)) is
ED = -31.675 eV -(-36.049167 eV)
(13.49)
= 4314X61 eV The Jf3 + bond dissociation energy, ED , given by Eq. (13.34) where p = 1 is
£D = 3.88614 eF + 0.485556 eV
(13.50)
= 4.37170 eF The experimental bond dissociation energy of H3 + [8] is
ED = 4.373 eV (13.51)
The difference between the results of Eqs. (13.49) and (13.50) is within the experimental and propagated errors in the different calculations. The calculated results are based on first principles and given in closed-form equations containing fundamental constants only. The agreement between the experimental and calculated results for the H3 + bond dissociation energy is excellent.
1 The experimental total energy of the hydrogen molecule is given by adding the first (15.42593 eV) [5] and second (16.2494 eV) ionization energies where the second ionization energy is given by the addition of the ionization energy of the hydrogen atom (13.59844 eV) [6] and the bond energy of H2 * (2.651 eV) [7]. The predicted D3 + molecular bond dissociation energy, ED , given by the difference between the total energy of .D3 + (Eqs. (13.30) where p = 1 and (13.48)) and the experimental total energy of D2 [9-10]2 is
ED = -31.76 βK-(-36.134300 eV) = 4.374300 eF The /J) 3 + bond dissociation energy, ED , given by Eq. (13.35) where p = 1 is
ED = 3.88777 eV + 0.485556 eV = 4.373331 eV
The results of the determination of bond parameters of H3 + are given in Table 13.1.
The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
HYDROXYL RADICAL (OH)
The water molecule can be solved by first considering the solution of the hydroxyl radical which is formed by the reaction of a hydrogen atom and an oxygen atom: H + O → OH (13.54)
The hydroxyl radical OH can be solved using the same principles as those used to solve the hydrogen molecule wherein the diatomic molecular orbital (MO) developed in the Nature of the Chemical Bond of Hydrogen-Type Molecules and Molecular Ions section serves as basis function in linear combination with an oxygen atomic orbital (AO) to form the MO of OH . The MO must (1) be a solution of Laplace's equation to give a equipotential energy surface, (2) correspond to an orbital solution of the Newtonian equation of motion in an inverse- radius-squared central field having a constant total energy, (3) be stable to radiation, and (4) conserve the electron angular momentum of Ti . A further constraint with the substitution of a heteroatom (O) for one of the hydrogen atoms is that the constant energy of the MO must match the energy of the heteroatom.
2 The experimental total energy of the deuterium molecule is given by adding the first (15.466 eV) [9] and second (16.294 eV) ionization energies where the second ionization energy is given by the addition of the ionization energy of the deuterium atom (13.603 eV) [10] and the bond energy of D* (2.692 eV) [9]. FORCE BALANCE OF OH
OH comprises two spin-paired electrons in a chemical bond between the oxygen atom and the hydrogen atom such that one electron on O remains unpaired. The OH radical MO is determined by considering properties of the binding atoms and the boundary constraints. The prolate spheroidal H2 MO developed in the Nature of the Chemical Bond of Hydrogen-Type
Molecules section satisfies the boundary constraints; thus, the H -atom electron forms a H2 - type ellipsoidal MO with one of the O -atom electrons. The O electron configuration given in the Eight-Electron Atoms section is \s12s22pd> , and the orbital arrangement is
(13.55)
Figure imgf000114_0001
corresponding to the ground state 3P2 .
In determining the central forces for O in the Radius and Ionization Energy of the Outer Electron of the Oxygen Atom section, it was shown that the energy is minimized with conservation of angular momentum by the cancellation of the orbital angular momentum of a Px electron by that of the py electron with the pairing of electron eight to fill the px orbital. Then, the diamagnetic force is given by Eq. (10.156) is that of atomic nitrogen (Eq. (10.136) corresponding to the pz -orbital electron (Eq. (10.82) with m = 0) as the source of diamagnetism with an additional contribution from the uncanceled px electron (Eq. (10.82) with m = 1). From Eqs. (10.83) and (10.89), the paramagnetic force, Fmαg 2, is given by Eq.
(10.157) corresponding to the spin-angular-momentum contribution alone of the px electron and the orbital angular momentum of the pz electron, respectively. The diamagnetic and paramagnetic forces cancel such that the central force is purely the Coulombic force. This central force is maintained with bond formation such that the energy of the O2p shell is unchanged. Thus, the angular momentum of each electron of the O2ρ shell is conserved with bond formation. The central paramagnetic force due to spin is provided by the spin- pairing force of the OH MO that has the symmetry of an s orbital that superimposes with the 2/7 orbitals such that the corresponding angular momenta of the O2p orbitals are unchanged.
The O2py electron combines with the His electron to form a molecular orbital. The proton of the H atom is along the internuclear axis. Due to symmetry, the other O electrons are equivalent to point charges at the origin. (See Eqs. (19-38) of Appendix IV.) Thus, the energies in the OH MO involve only the O2py and His electrons and the change in the magnetic energy of the O2py electron with the other O electrons (Eq. (13.152)) with the formation of the OH MO. The forces are determined by these energies. As in the case of H2, the MO is a prolate spheroid with the exception that the ellipsoidal MO surface cannot extend into O atom for distances shorter than the radius of the 2p shell. Otherwise, the electric field of the other O2p electrons would be perturbed, and the 2 p shell would not be stable. The corresponding increase in energy of O would not be offset by any energy decrease in the OH MO based on the distance from the O nucleus to the Hl* electron compared to those of the O2p electrons. Thus, the MO surface comprises a prolate spheroid at the H proton that is continuous with the 2p shell at the O atom. The energy of the prolate spheroid is matched to that of the 02 p shell.
The orbital energy E for each elliptical cross section of the prolate spheroidal MO is given by the sum of the kinetic T and potential V energies. E = T + V is constant, and the closed orbits are those for which T <\ V \ , and the open orbits are those for which T >| V | . It can be shown that the time average of the kinetic energy, < T > , for elliptic motion in an inverse-squared field is 1/2 that of the time average of the magnitude of the potential energy,
< V >\ . < T >= 1 / 21< V >| [11]. In the case of an atomic orbital (AO), E = T + V , and for all points on the AO, E =T =
Figure imgf000115_0001
. As shown in the Ηydrogen-type Molecular Ions section, each point or coordinate position on the continuous two-dimensional electron MO defines an infinitesimal mass-density element which moves along an orbit comprising an elliptic plane cross section of the spheroidal MO through the foci. The motion is such that eccentric angle, θ , changes at a constant rate at each point. That is θ = ωt at time t where ω is a constant, and r (t) = \a cos cot + ]b$\nωt (13.56)
Consider the boundary condition that the MO of OH comprises a linear combination of an oxygen AO and a H2 -type ellipsoidal MO. The charge density of H2 -type ellipsoidal MO given by Eq. (13.4) maintains that the surface is an equipotential; however, the potential and kinetic energy of a point on the surface changes as it orbits the central field. The potential energy is a maximum and the kinetic energy is a minimum at the semimajor axis, and the reverse occurs at the semiminor axis. Since the time average of the kinetic energy, < T > , for elliptic motion in an inverse-squared field is 1/2 that of the time average of the magnitude of the potential energy, by symmetry, the < T >= l/2 <\v\ > condition holds for 1/2 of the H2- type ellipsoidal MO having the H focus and ending at the plane defined by the semiminor axes. The O nucleus comprises the other focus of the OH MO. The 02p AO obeys the energy relationship for all points. Thus, the linear combination of the H2 -type ellipsoidal MO with the 02p AO must involve a 25% contribution from the H2 -type ellipsoidal MO to the 02p AO in order to match the energy relationships. Thus, the OH MO must comprise 75% of a H2 -type ellipsoidal MO (1/2 +25%) and an oxygen AO:
1 02py AO + 0.75 H2 MO → OH MO (13.57)
The force balance of the OH MO is determined by the boundary conditions that arise from the linear combination of orbitals according to Eq. (13.57). The force constant k of a H2 -type ellipsoidal MO due to the equivalent of two point charges of at the foci is given by Eq. (11.65):
Jt = - (13.58)
4πε0
Since the H2 -type ellipsoidal MO comprises 75% of the OH MO, the electron charge density in Eq. (13.58) is given by -0.75e . Thus, k' of the H2 -type-ellipsoidal-MO component of the OH MO is k.= <21f}» O3.59)
4πε0
L for the electron equals h ; thus, the distance from the origin of the OH MO to each focus c1 is given by Eqs. (11.79) and (13.59):
Figure imgf000116_0001
The internuclear distance from Eq. (13.60) is
2c' a2. K (13.61)
The length of the semiminor axis of the prolate spheroidal OH MO b - c given by Eq. (11.80) is
Figure imgf000117_0001
The eccentricity, e , is
e = - (13.63) a
Then, the solution of the semimajor axis a allows for the solution of the other axes of the prolate spheroidal and eccentricity of the OH MO.
The general equation of the ellipsoidal MO having semiprincipal axes a, b, c given by
Figure imgf000117_0002
is also completely determined by the total energy E given by Eq. (11.18):
Figure imgf000117_0003
The energy of the oxygen 2p shell is the negative of the ionization energy of the oxygen atom given by Eq. (10.163). Experimentally, the energy is [12]
£(2;? shell) - -E(ionization; O) = -13.6181 eV (13.66)
Since the prolate spheroidal MO transitions to the O AO, the energy E in Eq. (13.66) adds to that of the H2 -type ellipsoidal MO to give the total energy of the OH MO. From the energy equation and the relationship between the axes given by Eqs. (13.60-13.63), the dimensions of the OH MO are solved.
The energy components derived previously for the hydrogen molecule, Eqs. (11.207- 11.212), apply in the case of the H2 -type ellipsoidal MO. As in the case of the energies of Hs (IZp) given by Eqs. (13.12-13.16), each energy component of the H2 -type ellipsoidal
MO is the total for the two equivalent electrons with the exception that the total charge and energies of the two electrons is normalized by the percentage composition given by Eq. (13.57):
Figure imgf000117_0004
VP = — %πεQsj τan -bτ <13-68>
Figure imgf000118_0001
E = V +T + V +V (13.71)
Figure imgf000118_0002
Since the prolate spheroidal MO transitions to the O AO and the energy of the O2p shell must remain constant and equal to the negative of the ionization energy given by Eq. (13.66), the total energy E7. (OH) of the OH MO is given by the sum of the energies of the orbitals corresponding to the composition of the linear combination of the O AO and the H2 -type ellipsoidal MO that forms the OH MO as given by Eq. (13.57):
ET {0H) = E1. +E{2p shell)
= E7, - EQonization; O) (13.74)
Figure imgf000118_0003
To match the boundary condition that the total energy of the entire the H2 -type ellipsoidal MO is given by Eq. (11.212):
E7 (H2) eV (13.75)
Figure imgf000118_0004
E7. (OH) given by Eq. (13.74) is set equal to Eq. (13.75):
E7- (OH) = -
Figure imgf000118_0005
(13.76)
From the energy relationship given by Eq. (13.76) and the relationship between the axes given by Eqs. (13.60-13.63), the dimensions of the OH MO can be solved. Substitution of Eq. (13.60) into Eq. (13.76) gives
Figure imgf000119_0001
The most convenient way to solve Eq. (13.77) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is α = 1.2643Oan = 6.69039 X 10"" m (13.78) Substitution of Eq. (13.78) into Eq. (13.60) gives c' = 0.91808a0 = 4.85826 X 10"u m (13.79) The internuclear distance given by multiplying Eq. (13.79) by two is
2c' = 1.83616a0 = 9.71651 X 10~u m (13.80) The experimental bond distance is [13] 2c' = 9.71 X lO"11 m (13.81)
Substitution of Eqs. (13.78-13.79) into Eq. (13.62) gives b = c = 0.86925α0 = 4.59985 X 10'11 m (13.82)
Substitution of Eqs. (13.78-13.79) into Eq. (13.63) gives e = 0.72615 (13.83) The nucleus of the H atom and the nucleus of the O atom comprise the foci of the
H2 -type ellipsoidal MO. The parameters of the point of intersection of the H2 -type ellipsoidal MO and the O2py AO can be determined from the polar equation of the ellipse
(Eq. (11.10)):
1 + e r = rn (13.84)
1 + ecosø1 The radius of the O2py AO given by Eq. (10.162) is rs = α0, and the polar radial coordinate of the ellipse and the radius of the 02 py AO are equal at the point of intersection. Thus, Eq. (13.84) becomes
(13.85)
Figure imgf000119_0002
such that the polar angle θ ' is given by
Figure imgf000120_0001
Substitution of Eqs. (13.78-13.79) into Eq. (13.86) gives
#' = 123.65° (13.87)
Then, the angle θ02p Ao the radial vector of the O2py AO makes with the internuclear axis is 0O2Vo = 180o-123.65° = 56.35o (13.88) as shown in Figure 7.
The distance from the point of intersection of the orbitals to the internuclear axis must be the same for both component orbitals. Thus, the angle ωt = θH^MQ between the internuclear axis and the point of intersection of the H2 -type ellipsoidal MO with the O radial vector obeys the following relationship: a0 Sin θO2PyAO = b Sin 6H1MO 0- 3 -89) such that
Figure imgf000120_0002
with the use of Eq. (13.88). Substitution of Eq. (13.82) into Eq. (13.90) gives
(9^ = 73.27° (13.91)
Then, the distance dHiM0 along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by
JH1MO = a cos θH2uo (13-92) Substitution of Eqs. (13.78) and (13.91) into Eq. (13.92) gives d H,Mo = 0.36397α0 =1.92606 X 10~n m (13.93)
The distance dO2pAO along the internuclear axis from the origin of the O atom to the point of intersection of the orbitals is given by do2PAo = c'-dHlMO (13-94) Substitution of Eqs. (13.79) and (13.93) into Eq. (13.94) gives doipAo = 0.55411 a0 = 2.93220 X 10"11 m (13.95) As shown in Eq. (13.57), in addition to the p -orbital charge-density modulation, the uniform charge-density in the py orbital is increased by a factor of 0.25 and the H -atom density is decreased by a factor of 0.25. The internuclear axis of the O- H bond is perpendicular to the bonding py orbital. Using the orbital composition of OH (Eq. (13.57)), the radii of Ols = 0.12739α0 (Eq. (10.51)), 02s = 0.5902Oa0 (Eq. (10.62)), and O2p = a0
(Eq. (10.162)) shells, and the parameters of the OH MO given by Eqs. (13.3-13.4), (13.78- 13.80), (13.82-13.83), and (13.87-13.95), the dimensional diagram and charge-density of the OH MO comprising the linear combination of the H2 -type ellipsoidal MO and the O AO according to Eq. (13.57) are shown in Figures 7 and 8, respectively.
ENERGIES OF OH
The energies of OH given by the substitution of the semiprincipal axes (Eqs. (13.78-13.80) and (13.82)) into the energy equations (Eqs. (13.67-13.73)) are
Ve = -40.92709 eV (13.96)
Figure imgf000121_0001
Figure imgf000121_0002
h2
-- -3 τhχ a^a2 -b2 = ι6Λ8567 eV (13.98)
A) 2mea4a2 -b2 a-yja2 -b2
Figure imgf000121_0003
where E1, (OH) is given by Eq. (13.74) which is reiteratively matched to Eq. (13.75) within five-significant-figure round-off error.
VIBRATION AND ROTATION OF OH
The vibrational energy of OH may be solved in the same manner as that of hydrogen-type molecular ions and hydrogen molecules given in the Vibration of Hydrogen-type Molecular Ions section, and the Vibration of Hydrogen-type Molecules section, respectively, except that the orbital composition and the requirement that the O2p shell remain at the same energy and radius in the OH MO as it is in the O atom must be considered. Eachp-orbital comprises the sum of a constant function and a spherical harmonic function as given by Eq. (1.65). In addition to the j?-orbital charge-density modulation, the uniform charge-density in py orbital is increased by a factor of 0.25, and the H-atom electron density is decreased by a factor of 0.25. The force between the electron density of the H2 -type ellipsoidal MO and the nuclei determines the vibrational energy. With the radius of the orbit at the oxygen atom fixed at r& = a0 (13.101) according to Eq. (10.162), the central-force terms for the reentrant orbit between the electron density and the nuclei of the H2 -type ellipsoidal MO are given by Eqs. (11.213-11.214), except that the corresponding charge of -0.75e replaces the charge of -e of Eqs. (11.213- 11.214). Furthermore, due to condition that the O2p shell remain at the same energy and radius in the OH MO as it is in the O atom, the oscillation of H2 -type ellipsoidal is along the semiminor axis with the apsidal angle of Eq. (11.140) given by ψ = π . Thus, the semimajor axis a of Eqs. (11.213-11.214) is replaced by the semiminor axis b :
f{p) = -ψ^ (13.102)
and
Figure imgf000122_0001
Here, the force factor of 0.75 is equal to the equivalent term of Eq. (13.59). As the H2 -type ellipsoidal oscillates along b , the intemuclear distance changes 180° out of phase. Thus, the distance for the reactive nuclear-repulsive terms is given by intemuclear distance 2c ' (Eq. (13.80)). Similar to that of Eqs. (11.215-11.216), the contribution from the repulsive force between the two nuclei is
8πεo (2c') and
Figure imgf000122_0002
Thus, from Eqs. (11.136), (11.213-11.217), and (13.102-13.105), the angular frequency of the oscillation is
Figure imgf000123_0001
(13.106)
Figure imgf000123_0004
= 6.96269 X \0w rad/ s where δ is given by Eq. (13.82), 2c1 is given by Eq. (13.80), and the reduced mass of 16OH is given by:
Figure imgf000123_0002
where mp is the proton mass. Thus, during bond formation, the perturbation of the orbit determined by an inverse-squared force results in simple harmonic oscillatory motion of the orbit, and the corresponding frequency, ω(θ) , for 16OH given by Eqs. (11.136), (11.148), and (13.106) is radians / s (13.108)
Figure imgf000123_0003
where the reduced nuclear mass of 16OH is given by Eq.(13.107) and the spring constant, k(0) , given by Eqs. (11.136) and (13.106) is
Jc(O) = 763.18 Nm-1 (13.109)
The 16OH transition-state vibrational energy, Ev,6 (θ) , given by Planck's equation (Eq.
(11.127)) is: Evώ (0) = %ω = ^6.96269 X \0u rad/ s = 0.4583 eV = 3696.38 ctn 1 (13.110)
Zero-order or zero-point vibration is not physical and is not observed experimentally as discussed in the Diatomic Molecular Vibration section; yet, there is a term ωe of the old point-particle-probability-wave-mechanics that can be compared to Evώ (θ) . From Herzberg [14], ωe , from the experimental curve fit of the vibrational energies of 16OH" is ωe = 3735.21 COT"1 (13.111)
As shown in the Vibration of Hydrogen-type Molecular Ions section, the harmonic oscillator potential energy function can be expanded about the internuclear distance and expressed as a Maclaurin series corresponding to a Morse potential after Karplus and Porter
(K&P) [15] and after Eq. (11.134). Treating the Maclaurin series terms as anharmonic perturbation terms of the harmonic states, the energy corrections can be found by perturbation methods. The energy vυ of state υ is vυ = υω0 -υ(υ -Y)O)0X0 , υ = 0,1,2,3... (13.112) where
Figure imgf000124_0001
ω0 is the frequency of the ϋ = l -> u = 0 transition, and D0 is the bond dissociation energy given by Eq. (13.162). From Eq. (13.112), ω0 is given by ωo = Evib (θ)-2ωoxo (13.114)
Substitution of Eq. (13.113) into Eq. (13.114) gives
« (0) - _.*3L ■ (13.115)
Eq. (13.115) can be expressed as
Figure imgf000124_0002
which can be solved by the quadratic formula:
Figure imgf000124_0003
Only the positive root is real, physical; thus,
Figure imgf000124_0004
= 3522.02 cm -1
(13.118) where Evjb (θ) is given by Eq. (13.110) and D0 is given by Eq. (13.156). The corresponding
16 O/ H ϋ = l -> ϋ = 0 vibrational energy, Evtb (l) , in electron volts is:
Ev,, (l) = 0.43666 eV (13.119)
The experimental vibrational energy of 16OH is [16-17] Evib (\) = 0A424 eV (3568 cm-1) (13.120)
Using Εqs. (13.118-13.119) with Eq. (13.113), the anharmonic perturbation term,
Figure imgf000125_0001
The experimental anharmonic perturbation term, ωoxo , of 16OH [14] is 6Vt0 = 82.81 COT"1 (13.122)
The vibrational energies of successive states are given by Eqs. (13.110), (13.112), and (13.121).
Using the reduced nuclear mass of 16OD given by
(13.123)
Figure imgf000125_0003
where m is the proton mass, the corresponding parameters for deuterated hydroxyl radical 16OD (Eqs. (13.102-13.121) and (13.162)) are
l oi4 radiam ls (13- 124)
Figure imgf000125_0002
k{0) = 162>.\% Nm-1 (13.125)
Ev,i (0) = ^ = /i5.06610X 1014 radls = 0.33346 eV = 2689.51 COT"1 (13.126)
Figure imgf000126_0001
= 2596.02 cm"1
(13.127) EvΛ (l) = 0.3219 eF (13.128)
Figure imgf000126_0002
From Herzberg [14], ωe , from the experimental curve fit of the vibrational energies of 16OD is ωe = 2720.9 cm'1 (13.130)
The experimental vibrational energy of 16OD is [16-17] Evώ (ϊ) = 03263 eV (2632.1 cm-1) (13.131) and the experimental anharmonic perturbation term, ωQxQ , of 16OD [14] is
O)0X0 = 44.2 cm'1 (13.132) which match the predictions given by Eqs. (13.126), (13.127-13.128), and (13.129), respectively. The Be rotational parameters for 16OH and 16OD are given by Eq. (12.65):
Figure imgf000126_0003
where
7 = μr1 (13.134)
Using the internuclear distance, r = 2c\ and reduced mass of 16OH given by Eqs. (13.80) and (13.107), respectively, the corresponding Be is
Figure imgf000126_0004
The experimental Be rotational parameter of 16OH is [14] 5^ 18.871 CJW-1 (13.136)
Using the internuclear distance, r = 2c\ and reduced mass of 16OD given by Eqs. (13.80) and (13.123), respectively, the corresponding Be is
^ =9.971 ^' (13.137) The experimental Be rotational parameter of 16OD is [14]
5, S lOOl CIfI"1 (13.138)
THE DOPPLERENERGY TERMS OF 16OH AND 16OD
The radiation reaction force in the case of the vibration of 16OH in the transition state corresponds to the Doppler energy, E0 , given by Eq. (11.181) and Eq. (13.22) that is dependent on the motion of the electrons and the nuclei. The kinetic energy of the transient vibration is derived from the corresponding central forces. Following the same consideration as those used to derive Eqs. (13.102-13.103) and Eqs. (11.231-11.232), the central force terms between the electron density and the nuclei of 16OH MO with the radius of the orbit at the oxygen atom fixed at rs = a0 (13.139) according to Eq. (10.162) are
/(») = -££ (".HO and
Figure imgf000127_0001
wherein the oscillation of H2 -type ellipsoidal MO is along the semiminor axis b with the apsidal angle of Eq. (11.140) given by ψ = π due to condition that the GIp shell remain at the same energy and radius in the OH MO as it is in the O atom. Thus, using Eqs. (11.136) and (13.140-13.141), the angular frequency of this oscillation is
4.41776 X lθι6 rad/s (13.142)
Figure imgf000127_0002
The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)): Eκ = /ky = #4.41776 X 1016 rod Is = 29.07844 eV (13.143)
In Eq. (11.181), substitution of the total energy of OH, E7 (OH), (Eq. (13.76)) for Elw , the mass of the electron, me, for M , and the kinetic energy given by Eq. (13.143) for Eκ gives the Doppler energy of the electrons for the reentrant orbit.
E. s El (B.144)
Figure imgf000128_0001
The total energy of OH is decreased by ED .
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. On average, the total energy of vibration is equally distributed between kinetic energy and potential energy [4]. Thus, the average kinetic energy of vibration corresponding to the Doppler energy of the electrons, EKvΛ , is 1/2 of the vibrational energy of OH given by Eq. (13.120). The decrease in the energy of the OH due to the reentrant orbit in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED and EKvib . Using Eq. (13.144) and the experimental 16OH ωe of 3735.21 cm"1 (0.46311 I eF) [16-17] gives
(13.145)
Figure imgf000128_0002
EOM(16OH) = -0.33749 eF+-(0.463111 eV) = -0.10594 eV (13.146)
To the extent that the MO dimensions are the same, the electron reentrant orbital energies, Eκ , are the same independent of the isotope of hydrogen, but the vibrational energies are related by Eq. (11.148). Thus, the differences in bond energies are essentially given by 1/2 the differences in vibrational energies per bond. Using Eq. (13.144), Eqs. (13.145-13.146), and the experimental 16OD ωe of 2720.9 cm'1 (0.33735 eV) [16-17], the corresponding E011A 16ODj is
E0n (16OD) = -0.33749 eF+-(0.33735 eF) = -0.16881 eV (13.147) TOTAL AND BOND ENERGIES OF 16OH AND 16OD RADICALS
ET±O∞ (16OH) , the total energy of the 16OH radical including the Doppler term, is given by
the sum of E7. (OH) (Eq. (13.76)) and EOSC ( 16OH) given by Eqs. (13.142-13.146):
ET+osc (16OH) = Ve +T + Vm +Vp + E(2p Shell) + Eosc(l6OH)
(13.148) = Ex (OH) + I050 (16OH)
Figure imgf000129_0001
(13.149)
From Eqs. (13.145-13.146) and (13.148-13.149), the total energy of 16OH is ET+0SC ( 16OH) = -31.63537 eV + Eosc ( 16OH)
= -31.63537 eF-0.33749 eF + -(0.463111 eF) (13.150)
= -31.74130 eV
where the experimental ωe was used for the term. Eτ+osc (16OD) , the total energy of
Figure imgf000129_0003
10 16O£> including the Doppler term, is given by the sum of E7. (OD) = E7. (OH) (Eq. (13.76)) and Eosc (16OD) given by Eq. (13.147):
Eτ÷osc (16OD) = -3l.63537 eV+ Eosc (16OD)
= -31.63537 eF-0.33749 eF+-(0.33735 eF) (13.151) = -31.80418 eV
where the experimental ω was used for the term. The dissociation of the bond of the
Figure imgf000129_0002
hydroxyl radical forms a free hydrogen atom with one unpaired electron and an oxygen atom 15 with two unpaired electrons as shown in Eq. (13.55) which interact to stabilize the atom as shown by Eq. (10.161-10.162). The lowering of the energy of the reactants decreases the bond energy. Thus, the total energy of oxygen is reduced by the energy in the field of the two magnetic dipoles given by Eq. (7.46) and Eq. (13.101): E(magnetic) = 2π^ff = ^ΞhA = 0.114411 eV (13.152) m;a0 an
The corresponding bond dissociation energy, ED , is given by the sum of the total energies of the oxygen atom and the corresponding hydrogen atom minus the sum of Eτ+osc (16OH) and E(magnetic) : ED = E(16O) + E(H)- ET+OSC (16OH)-E(magnetic) (13.153)
E(16O) is given by Eq. (13.66), ED (H) [18] is
E(H) = -13.59844 eV (13.154) and ED (£») [19] is
E(D) = -13.603 eV (13.155) The 16OH bond dissociation energy, E0 (16OH) , is given by Εqs. (13.150) and (13.152- 13.155):
ED (16OH) = -(13.6181 eV + 13.59844 eV) - (E (magnetic) + ET+0SC (16OH))
= -27.21654 eF-(0.114411 eF-31.74130 eF) (13.156)
= 4.4104 eV
The experimental 16OH bond dissociation energy is [20]
E0 (16OH) = 4.41174 eF (13.157)
The 16OD bond dissociation energy, ED (16OTJ)) , is given by Εqs. (13.151-13.153):
E0 (16OD) = -(13.6181 eV + 13.603 e V) -(E (magnetic) + ET+OSC (16OD))
= -27.2211 eF-(0.114411 eF-31.804183 eV) (13.158)
= 4.4687 e V
The experimental 16OD bond dissociation energy is [21-22]
ED (16OD) = 4.454 e V (13.159)
The results of the determination of bond parameters of OH and OD are given in Table 13.1. The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent. WATER MOLECULE (H2O)
The water molecule H2O is formed by the reaction of a hydrogen atom with a hydroxyl radical:
OH+H → H2O (13.16Ο) The water molecule can be solved using the same principles as those used to solve the hydrogen molecule, H3 + and OH wherein the diatomic molecular orbital (MO) developed in the Nature of the Chemical Bond of Ηydrogen-Type Molecules and Molecular Ions section serves as basis function in a linear combination with an oxygen atomic orbital (AO) to form the MO of H2O . The solution is very similar to that of OH except that there are two OH bonds in water.
FORCE BALANCE OF H2O
H2O comprises two chemical bonds between oxygen and hydrogen. Each O-H bond comprises two spin-paired electrons with one from an initially unpaired electron of the oxygen atom and the other from the hydrogen atom. The H2O MO is determined by considering properties of the binding atoms and the boundary constraints. The H2 prolate spheroidal MO satisfies the boundary constraints as shown in the Nature of the Chemical Bond of Ηydrogen-Type Molecules section; thus, each H -atom electron forms a H2 -type ellipsoidal MO with one of the initially unpaired O -atom electrons. The initial O electron configuration given in the Eight-Electron Atoms section is Is22s22p4 , and the orbital arrangement is given by Eqs. (10.154) and Eq. (13.55).
As shown in the case of OH in the Force Balance of OH section, the forces that determine the radius and the energy of the O2p shell are unchanged with bond formation. Thus, the angular momentum of each electron of the 02 p is conserved with bond formation. The central paramagnetic force due to spin of each O-H bond is provided by the spin- paring force of the H2O MO that has the symmetry of an s orbital that superimposes with the O2p orbitals such that the corresponding angular momenta are unchanged.
Each of the O2pz and O2px electron combines with a His electron to form a molecular orbital. The proton of the H atom is along the internuclear axis. Due to symmetry, the other O electrons are equivalent to point charges at the origin. (See Eqs. (19- 38) of Appendix IV.) Thus, the energies in the H2O MO involve only each 02 p and each
His electron with the formation of each O-H bond. The forces are determined by these energies.
As in the case of H2, each of two O-H -bond MOs is a prolate spheroid with the exception that the ellipsoidal MO surface cannot extend into the O atom for distances shorter than the radius of the 2p shell. Otherwise, the electric field of the other O2p electrons would be perturbed, and the 2p shell would not be stable. The corresponding increase in energy of O would not be offset by any energy decrease in the O-H -bond MO based on the distance from the O nucleus to the His electron compared to those of the O2p electrons. Thus, the MO surface comprises a prolate spheroid at each H proton that is continuous with the 2p shell at the O atom. The sum of the energies of the prolate spheroids is matched to that of the 2p shell.
The orbital energy E for each elliptical cross section of the prolate spheroidal MO is given by the sum of the kinetic T and potential V energies. E = T + V is constant, and the closed orbits are those for which T <| V \ , and the open orbits are those for which T >| V \ . It can be shown that the time average of the kinetic energy, < T > , for elliptic motion in an inverse-squared field is 1/2 that of the time average of the magnitude of the potential energy,
< V >\ . <T >=II2\< V >\ [H]. In the case of an atomic orbital (AO), E = T + V, and for all points on the AO, E| = T = 1/2|F . As shown in the Ηydrogen-type Molecular Ions section, each point or coordinate position on the continuous two-dimensional electron MO defines an infinitesimal mass-density element which moves along an orbit comprising an elliptic plane cross section of the spheroidal MO through the foci. The motion is such that eccentric angle, θ , changes at a constant rate at each point. That is θ = ωt at time t where ω is a constant, and r(t) = iacosωt + }bύnωt (13.161)
Consider the boundary condition that the MO of H2O comprises a linear combination of an oxygen AO and two H2 -type ellipsoidal MOs, one for each O-H -bond. The charge density of each H2 -type ellipsoidal MO given by Εqs. (11.44-11.45) and (13.3-13.4) maintains that the surface is an equipotential; however, the potential and kinetic energy of a point on the surface changes as it orbits the central field. The potential energy is a maximum and the kinetic energy is a minimum at the semimajor axis, and the reverse occurs at the semiminor axis. Since the time average of the kinetic energy, < T > , for elliptic motion in an inverse- squared field is 1/2 that of the time average of the magnitude of the potential energy, by symmetry, the < 7" >= 1/2 < |F| > condition holds for 1/2 of each H2 -type ellipsoidal MO having the H focus and ending at the plane defined by the semiminor axes. The O nucleus comprises the other focus of each OH-MO component of the H2O MO. The 02p AO obeys the energy relationship for all points. Thus, the linear combination of the H2 -type ellipsoidal MO with the 02p AO must involve a 25% contribution from the H2 -type ellipsoidal MO to the O2p AO in order to match the energy relationships. Thus, the H2O MO must comprise two O-H -bonds with each comprising 75% of a H2 -type ellipsoidal MO (1/2 +25%) and an oxygen AO:
[1 02pz AO + 0.75 H2 M9]+[l 02py AO + 0.75 H2 MO] → H2O MO (13.162)
The force balance of the H2O MO is determined by the boundary conditions that arise from the linear combination of orbitals according to Eq. (13.162). The force constant k of a H2 -type ellipsoidal MO due to the equivalent of two point charges of at the foci is given by Eq. (11.65):
2e2 k = ^— (13.163)
4πεo
Since the each H2 -type ellipsoidal MO comprises 75% of the O-H -bond MO, the electron charge density in Eq. (13.163) is given by -0.75e . Thus, Jc' of the each H2 -type-ellipsoidal- MO component of the H2O MO is
*■ - 03.164,
4πε0
L for the electron equals h ; thus, the distance from the origin of each O-H -bond MO to each focus c' is given by Eqs. (11.79) and (13.164):
Figure imgf000133_0001
The internuclear distance from Eq. (13.165) is
2c1 = 2.pϊ (13.166) The length of the semiminor axis of the prolate spheroidal O- H -bond MO b = c given by
Eq. (11.80) is b = ja2 -c'2 (13.167)
The eccentricity, e , is
e = — (13.168) a
The solution of the semimajor axis a then allows for the solution of the other axes of the prolate spheroid and eccentricity of the O- H -bond MO.
The general equation of the ellipsoidal MO having semiprincipal axes a, b, c given by
£+£+£-1 03.169) is also completely determined by the total energy E given by Eq. (11.18):
Figure imgf000134_0001
The energy of the oxygen 2p shell is the negative of the ionization energy of the oxygen atom given by Eqs. (10.163) and (13.66). Experimentally, the energy is [12] E(Ip shell) = -E(ionization; O) = -13.6181 eV (13.171)
Since each of the two prolate spheroidal O- H -bond MOs comprises a Tf2 -type-ellipsoidal MO that transitions to the O AO, the energy E in Eq. (13.171) adds to that of the two corresponding H2 -type ellipsoidal MOs to give the total energy of the H2O MO. From the energy equation and the relationship between the axes given by Eqs. (13.165-13.168), the dimensions of the H2O MO are solved.
The energy components defined previously for the molecule, Eqs. (11.207-11.212), apply in the case of H2O. Since the H2O MO comprises two equivalent O-H -bond MOs, each a linear combination of a H2 -type-ellipsoidal MO and an O2p AO, the corresponding energy component of the H2O MO is given by the linear superposition of the component energies. Thus, the energy scale factor is given as two times the force factor, the term in parentheses in Eq. (13.164). In addition to the equivalence and linearity principles, this factor also arises from the consideration of the nature of each bond and the linear combination that forms the H2O MO. Each O - H -bond-energy component is the total for the two equivalent electrons with the exception that the total charge of the two electrons is normalized over the three basis set functions, two O-H -bond MOs (OH -type ellipsoidal MOs given in the Energies of OH section) and one O2p AO. Thus, the contribution of the O-H -bond MOs to the H2O MO energies are those given for H2 (I//?) in the Energies of Ηydrogen-Type Molecules multiplied by a factor of 3/2 as in the case with H3 + (Eqs. (13.12), (13.15),
13.18-13.20)). In addition, the two sets of equivalent nuclear-point-charge pairs give rise to a factor of two times the proton-proton repulsion energy given by Eq. (11.208). Thus, the component energies of the H2O MO are twice the corresponding energies of the OH MO given by Eqs. (13.67-13.73). The parameters a, b , andc' are given by Eqs. (13.165-13.167), respectively.
Figure imgf000135_0001
■ - 2 l (13.174)
*<)
Figure imgf000135_0002
Figure imgf000135_0003
ET = Ve +T + Vm +Vp (13.176)
Figure imgf000135_0004
Since the each prolate spheroidal H2 -type MO transitions to the O AO and the energy of the O2p shell must remain constant and equal to the negative of the ionization energy given by
Eq. (13.171), the total energy Eτ (H2O) of the H2O MO is given by the sum of the energies of the orbitals corresponding to the composition of the linear combination of the O AO and the two H2-type ellipsoidal MOs that forms the H2O MO as given by Eq. (13.162): E7 (H2O) = ET +E( 2p shell)
= E1. - EQonization; O) (13.179)
Figure imgf000136_0001
The two hydrogen atoms and the oxygen atom can achieve an energy minimum as a linear combination of two H2 -type ellipsoidal MOs each having the proton and the oxygen nucleus as the foci. Each O-H -bond MO comprises the same O2p shell of constant energy given by Eq. (13.171). Thus, the energy of the H2O MO is also given by the sum of that of the two H2 -type ellipsoidal MOs given by Eq. (11.212) minus the energy of the redundant oxygen atom of the linear combination given by Eq. (13.171):
Figure imgf000136_0002
= 2 (-31.63536831 eF)-(-13.6181 eF) (13.180)
= -49.652637 eV
E7 [H2Oi) given by Eq. (13.179) is set equal to two times the energy of the H2 -type ellipsoidal MO minus the energy of the O2p shell given by Eq. (13.180):
E7 (H2O) = - eV = -49.652637 eV
Figure imgf000136_0003
(13.181)
From the energy relationship given by Eq. (13.181) and the relationship between the axes given by Eqs. (13.165-13.167), the dimensions of the H2O MO can be solved. Substitution of Eq. (13.165) into Eq. (13.181) gives
1 = e36.034537 (13.182)
Figure imgf000136_0004
The most convenient way to solve Eq. (13.182) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is a = 1.264Ia0 = 6.68933 X 10"11 m (13.183) Substitution of Eq. (13.183) into Eq. (13.165) gives c' = 0.918005α0 = 4.85787 X 10"11 m (13.184)
The internuclear distance given by multiplying Eq. (13.184) by two is 2c' = 1.8360Ia0 = 9.71574 X 10~n m (13.185)
The experimental bond distance is [23]
2c' = 9.70 ±.005 X 10~n m (13.186)
Substitution of Eqs. (13.177-13.176) into Eq. (13.167) gives ό = c = 0.86903 Ia0 = 4.59871 X \QTn m (13.187)
Substitution of Eqs. (13.177-13.176) into Eq. (13.168) gives e = 0.726212 (13.188)
The nucleus of the H atom and the nucleus of the O atom comprise the foci of each
H2 -type ellipsoidal MO. The parameters of the point of intersection of each H2 -type ellipsoidal MO and the 02 py AO or 02 pz AO can be determined from the polar equation of the ellipse (Eq. (11.10)):
1 +p r = r0 - (13.189)
1 + ecosø '
The radius of the 02 p shell given by Eq. (10.162) is rs = α0 , and the polar radial coordinate of the ellipse and the radius of the O2p shell are equal at the point of intersection. Thus, Eq. (13.189) becomes
I + ^ ao =(a-c') ^- (13.190)
1 + -COS01 a such that the polar angle Θ ' is given by
Figure imgf000137_0001
Substitution of Eqs. (13.177-13.176) into Eq. (13.191) gives #' = 123.66° (13.192)
Then, the angle θ02pA0 the radial vector of the O2p AO makes with the internuclear axis is
Θ O2PAO = 180°-123.66° = 56.33° (13.193) as shown in Figure 7. The distance from the point of intersection of the orbitals to the internuclear axis must be the same for both component orbitals. Thus, the angle ωt = ΘH MO between the internuclear axis and the point of intersection of each H2 -type ellipsoidal MO with the O radial vector obeys the following relationship: ao sinθ02pΛ0 = bsmθHiM0 (13.194) such that
_ s. -i α° sin ΘO2PΛO _ . , Ct0 sin 56.33°
with the use of Eq. (13.193). Substitution of Eq. (13.188) into Eq. (13.195) gives θHiM0 = 73.28° (13.196)
Then, the distance dH M0 along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by dH2M0 = a C0S 0H2MO (13.197)
Substitution of Eqs. (13.183) and (13.196) into Eq. (13.197) gives d HlMo = O.3637αo = 1.9244 X 1(T11 m (13.198)
The distance d02pA0 along the internuclear axis from the origin of the O atom to the point of intersection of the orbitals is given by do2PAo = c'-dHιMO (13.199)
Substitution of Eqs. (13.184) and (13.198) into Eq. (13.199) gives doiPAo = 0.5543α0 = 2.93343 X 1(T11 m (13.200)
In addition to the intersection of the H2 -type MO with the 02 p shell, two adjoining ellipsoidal H2 -type MOs intersect at points of equipotential. The angle and distance parameters are given by Eqs. (13.595-13.600) for the limiting methane case wherein four adjoining intersecting H2 -type MOs have the possibility of forming a self-contained two- dimension equipotential surface of charge and current. Charge continuity can be obeyed for the H2O MO if the current is continuous between the adjoining H2 -type MOs. However, in the limiting case of methane, the existence of a separate linear combination of the H2 -type
MOs comprising four-spin paired electrons, not connected to the bonding carbon heteroatom requires that the electron be divisible. It is possible for an electron to form time-dependent singular points or nodes having no charge as shown by Eqs. (1.65a- 1.65b), and two- dimensional charge distributions having Laplacian potentials and one-dimensional regions of zero charge are possible for macroscopic charge densities and currents as given in Ηaus and Melcher [24]. However, it is not possible for single electrons to have two dimensional discontinuities in charge based on internal forces and first principles discussed in Appendix IV. Thus, at the points of intersection of the H2 -type MOs of methane, symmetry, electron indivisibility, current continuity, and conservation of energy and angular momentum require that the current between the points of mutual contact and the carbon atom be projected onto and flow along the radial vector to the surface of the C2sp3 shell. This current designed the bisector current (BC) meets the C2sp3 surface and does not travel to distances shorter than its radius. The methane result must also apply in the case of other bonds including that of the water molecule. Here, the H2 -type MOs intersect and the ellipsoidal current is projected onto the radial vector to the O2p shell and does not travel to distances shorter than its radius as in the case of a single O- H bond. As shown in Eq. (13.162), in addition to the p -orbital charge-density modulation, the uniform charge-density in the pz and py orbitals is increased by a factor of 0.25 and the H atoms are each decreased by a factor of 0.25. Using the orbital composition of H2O (Eq. (13.162)), the radii of OLs = 0.12739α0 (Eq. (10.51)), 02s = 0.5902Oa0 (Eq. (10.62)), and O2p = aQ (Eq. (10.162)) shells, and the parameters of the H2O MO given by Eqs. (13.3- 13.4), (13.183-13.185), (13.187-13.188), and (13.192-13.200), the charge-density of the H2O
MO comprising the linear combination of two O-H -bond MOs ( OH -type ellipsoidal MOs given in the Energies of OH section) according to Eq. (13.162) is shown in Figure 9. Each O-H -bond MO comprises a H2 -type ellipsoidal MO and an O2p AO having the dimensional diagram shown in Figure 8.
ENERGIES OF H2O
The energies of H2O given by the substitution of the semiprincipal axes (Eqs. (13.183- 13.185) and (13.187)) into the energy equations (Eqs. (13.172-13.180)) are
(13.201)
(13.202)
(13.203)
Figure imgf000139_0001
Figure imgf000140_0001
, (H2O) = - -49.6558 eV (13.205)
Figure imgf000140_0002
where Eτ (H2O) is given by Eq. (13.179) which is reiteratively matched to Eq. (13.180) within five-significant-figure round-off error.
VIBRATION OF H2O
The vibrational energy levels of H2O may be solved as two equivalent coupled harmonic oscillators by developing the Lagrangian, the differential equation of motion, and the eigenvalue solutions [2] wherein the spring constants are derived from the central forces as given in the Vibration of Ηydrogen-Type Molecular Ions section and the Vibration of Ηydrogen-Type Molecules section.
THE DOPPLER ENERGY TERM OF H2O
The radiation reaction force in the case of the vibration of H2O in the transition state corresponds to the Doppler energy, ED , given by Eq. (11.181) and Eqs. (13.22) and (13.144) that is dependent on the motion of the electrons and the nuclei. The kinetic energy of the transient vibration is derived from the corresponding central forces. As in the case of H3 + , the water molecule is a linear combination of three orbitals. The water MO comprises two H2 -type ellipsoidal MOs and the O AO. Thus, the force factor of water in the determination of the Doppler frequency is equivalent to that of the H3 + ion given in Eqs. (13.18-13.20) and given by Eq. (13.164). From Eqs. (11.231-11.232) and (13.18-13.20), the central force terms between the electron density and the nuclei of each O-H -bond MO with the radius of the orbit at the oxygen atom fixed at rs = a0 (13.206) according to Eq. (10.162) with the oscillation along the semiminor axis are
Figure imgf000140_0003
and
Figure imgf000141_0001
Thus, using Eqs. (11.136) and (13.207-13.208), the angular frequency of this oscillation is
Figure imgf000141_0002
The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)): Eκ = hω = h6.24996 X lO16 rod I s = 41.138334 eV (13.210)
The three basis elements of water, H , H , and O , all have the same Coulombic energy as given by Eqs. (1.243) and (10.163), respectively, such that the Doppler energy involves the total energy of the H2O MO. Thus, in Eq. (11.181), substitution of the total energy of H2O5
E7. (H2O), (Eqs. (13.179-13.180) and Eq. (13.181)) for E1n, , the mass of the electron, me, for M , and the kinetic energy given by Eq. (13.210) for Eκ gives the Doppler energy of the electrons for the reentrant orbit:
E° ϊL "l∞ϋffi . _0.630041 eV (13.211)
Figure imgf000141_0003
The total energy of H2O is decreased by ED ■
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. On average, the total energy of vibration is equally distributed between kinetic energy and potential energy [4]. Thus, the average kinetic energy of vibration corresponding to the Doppler energy of the electrons, EKvtb , is 1/2 of the vibrational energy of H2O . The decrease in the energy of H2O due to the reentrant orbit in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED and EKvib . Using Eq.
(13.211) and the experimental H16OH vibrational energy of Evib = 3755.93 cm'1 = 0.465680 eV [25] gives
E'osc = ED +EKvib = ED +U fc (13.212)
E\sc = -0.630041 eV + -(0.4656SO eV) = -0.397201 eV (13.213) per bond. As in the case for H3 + (I/;?) shown in the Doppler Energy Term of H3 + -type
Molecular Ions section, the reentrant orbit for the binding of a hydrogen atom to a hydroxyl radical causes the bonds to oscillate by increasing and decreasing in length along the two O- H bonds at a relative phase angle of 180°. Since the vibration and reentrant oscillation is along two bonds for the asymmetrical stretch ( V3 ), Eosc for H16OH , Eosc (H16OH") , is:
Figure imgf000142_0001
= 2[ -0.630041 eV + -(0Λ65680 eV)) (13.214)
= -0.794402 e V
To the extent that the MO dimensions are the same, the electron reentrant orbital energies, Eκ , are the same independent of the isotope of hydrogen, but the vibrational energies are related by Eq. (11.148). Thus, the differences in bond energies are essentially given by 1/2 the differences in vibrational energies per bond. Using Eq. (13.211), Eqs. (13.212-13.214), and the experimental D16OD vibrational energy of Evιb = 2787.92 cm"1 = 0.345661 eV [25], the corresponding Eosc [D16OD) is
Eosc (D16OD) = 2^-0.630041 eV+±(0345661 eV)j (13 215)
= -0.914421 eV
TOTAL AND BOND ENERGIES OF H16OH AND D16OD
Eτ+osc f H2 16Oj , the total energy of the H 16OH including the Doppler term, is given by the
sum of Eτ (H2O) (Eq. (13.181)) and Eosc (H16OH) given Eqs. (13.207-13.214):
Eτ+osc (H2 16θ) = Ve +T + Vm +Vp +E(θ2p) + Eosc (H16OH) = ET {H2O) + EOSC (H16OH)
Figure imgf000143_0001
(13.217)
From Eqs. (13.214) and (13.216-13.217), the total energy of H16OH is Eτ+osc [H2 16O) = -49.652637 eV + Eosc (H16OH)
eV — (0.46568O eF) (13.218)
Figure imgf000143_0002
= -50.447039 eV k where the experimental vibrational energy was used for the h — term. E7. [D2 16O ] , the
total energy of D16OD including the Doppler term is given by the sum of Eτ (D2O) = Eτ ( H2O) (Eq. (13.181)) and Eosc (DUOD) given by Eq. (13.215):
Er.0Sc [D2 16O) = -49.652637 eV + Eosc (D16OD)
= -49.652637 eF-2[ 0.630041 eF--(0.345661 eVU (13.219)
= -50.567058 eV k where the experimental vibrational energy was used for the Ti \ — term. As in the case of the
hydroxyl radical, the dissociation of the bond of the water molecule forms a free hydrogen atom and a hydroxyl radical, with one unpaired electron each. The lowering of the energy of the reactants due to the magnetic dipoles decreases the bond energy. Thus, the total energy of oxygen is reduced by the energy in the field of the two magnetic dipoles given by Eq. (13.152). The corresponding bond dissociation energy, ED , is given by the sum of the total energies of the corresponding hydroxyl radical and hydrogen atom minus the total energy of water, Eτ+osc (H16OH) , and E(magnetic) . Thus, ED of H16OH is given by:
E0 [H16OH) = E(H) + EC6OH) - Eτ÷osc (H16OH) - E(magnetic) (13.220)
where E7(16OH) is given by the of the sum of the experimental energies of 16O (Eq.
(13.171)), H (Eq. (13.154)), and the negative of the bond energy of 16OH (Eq. (13.157)):
E(16OH) = -13.59844 eF-13.6181 eF-4.41174 eF = -31.62828 eF (13.221)
From Εqs. (13.154), (13.218), and (13.220-13.221), Eβ (H16OH) is
E0(H16OH) = E(H) + E(16OH) -^(magnetic) + ET+OSC (H16OH))
= -13.59844 eF-31.62828 eF-(0.114411 eF-50.447039 eF) = 5.1059 eV (13.222)
The experimental H16OH bond dissociation energy is [26]
E0(H16OH) = 5.0991 eV (13.223)
Similarly, E D of D16OD is given by:
ED [D16OH] = E(D) + E(16OD) -[^(magnetic) + Eτ+osc [D16OD)) (13.224)
where E7(16OD) is given by the of the sum of the experimental energies of 16O (Eq.
(13.171)), D (Eq. (13.155)), and the negative of the bond energy of 16OD (Eq. (13.159)):
E(16OD) = -13.603 eF-13.6181 eF -4.454 eF = -31.6721 eV (13.225)
From Εqs. (13.155), (13.220), and (13.224-13.225), E0 [D16OD) is
Eo(D16OD) = -13.603 eF-31.6721 eF -(0.114411 eF -50.567058 eF) (l3 226)
= 5.178 eV The experimental D16OD bond dissociation energy is [27]
E0(D16OD) = 5.191 eV (13.227)
BOND ANGLE OF H2O
The H2O MO comprises a linear combination of two O-H -bond MOs. Each O-H -bond MO comprises the superposition of a H2 -type ellipsoidal MO and the O2pz AO or the O2py AO with a relative charge-density of 0.75 to 1.25; otherwise, the O2p orbitals are the same as those of the oxygen atom. A bond is also possible between the two H atoms of the O- H bonds. Such H- H bonding would decrease the O- H -bond strength since electron density would be shifted from the O- H bonds to the H-H bond. Thus, the bond angle between the two O- H bonds is determined by the condition that the total energy of the H2 - type ellipsoidal MO between the terminal H atoms of the O- H bonds is zero. Since the two H2 -type ellipsoidal MOs comprise 75% of the H electron density of H2 ; the energies and the total energy E7 of the H-H bond is given by Eqs. (13.67-13.73). From Eq. (11.79), the distance from the origin to each focus of the H-H ellipsoidal MO is
Figure imgf000145_0001
The internuclear distance from Eq. (13.228) is
Figure imgf000145_0002
The length of the semiminor axis of the prolate spheroidal H-H MO b = c is given by Eq. (13.167). Substitution of Eq. (13.228) into Eq. (13.73) gives
Figure imgf000145_0003
The radiation reaction force in the case of the vibration of H-H in the transition state corresponds to the Doppler energy, ED , given by Eq. (11.181) that is dependent on the motion of the electrons and the nuclei. The total energy E1. that includes the radiation reaction of the H-H MO is given by the sum of E1. (Eq. (13.73)) and Eosc (H2) given Eqs. (11.213-11.220), (11.231-11.236), and (11.239-11.240). Thus, the total energy E7 (H-H) of the H-H MO including the Doppler term is
ET = Ve +T + Vm + Vp +E0SC (H-H) (13.231)
Figure imgf000146_0001
To match the boundary condition that the total energy of the H -H ellipsoidal MO is zero, Eτ (H - H) given by Eq. (13.232) is set equal to zero:
Figure imgf000146_0002
From the energy relationship given by Eq. (13.233) and the relationship between the axes given by Eqs. (13.165-13.167), the dimensions of the H-H MO can be solved.
The most convenient way to solve Eq. (13.233) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is α = 4.300αn = 2.275 X 10 ,-10 m (13.234) Substitution of Eq. (13.234) into Eq. (13.228) gives c' = 1.466α0 = 7.759 X 10"u m (13.235) The internuclear distance given by multiplying Eq. (13.235) by two is
2c' = 2.933αn = 1.552 JST lO" -10 m (13.236)
Substitution of Eqs. (13.234-13.235) into Eq. (13.167) gives δ = c = 4.042α0 = 2.139X 10-10 m (13.237)
Substitution of Eqs. (13.234-13.235) into Eq. (13.168) gives e - 0.341 (13.238)
Using, distance between the two H atoms when the total energy of the corresponding MO is zero, the corresponding bond angle can be determined from the law of cosines:
A2 + B2 - 2ABcosinQθ = C2 (13.239) With A = B = 2c0' _H , the internuclear distance of each O- H bond given by Eq. (13.185), and C = 2c'H_H , the internuclear distance of the two H atoms, the bond angle between the O- H bonds is given by
(2c '0_H )2 +(2c 'O_H )2 - 2 (2c > 0_H )2 cosine θ = (2c \_H f (13.240)
Substitution of Eqs. (13.185) and (13.236) into Eq. (13.241) gives
, f 2(l.836)2 -(2.933)2 Nl
^ = COS-1 -i f—\2- L
{ 2(1.836)2 J
= cos"1 (-0.2756) (13.242)
= 105.998° The experimental internuclear distance of the two H atoms, 2c'H_H , is [23]
2c' = 1.55 ±0.01 X IQ-10 m (13.243) which matches Eq. (13.236) very well. The experimental angle between the O- H bonds is [23]
0 = 106° (13.244) which matches the predicted angle given by Eq. (13.242).
The results of the determination of bond parameters of H2O and D2O are given in Table 13.1. The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
HYDROGEN NITRIDE (NH )
The ammonia molecule can be solved by first considering the solution of the hydrogen and dihydrogen nitride radicals. The former is formed by the reaction of a hydrogen atom and a nitrogen atom:
H + N → NH (13.245) The hydrogen nitride radicals, NH and NH2 , and ammonia, NH3 , can be solved using the same principles as those used to solve OH and H2O .
FORCE BALANCE OF NH NH comprises two spin-paired electrons in a chemical bond between the nitrogen atom and the hydrogen atom such that two electrons on N remain unpaired. The NH radical molecular orbital (MO) is determined by considering properties of the binding atoms and the boundary constraints. The prolate spheroidal H2 MO developed in the Nature of the Chemical Bond of Ηydrogen-Type Molecules section satisfies the boundary constraints; thus, the H -atom electron forms a H2 -type ellipsoidal MO with one of the N -atom electrons.
The N electron configuration given in the Seven-Electron Atoms section is \s22s22pi , and the orbital arrangement is
(13.246)
Figure imgf000148_0001
corresponding to the ground state 4^2 . The N2px electron combines with the His electron to form a molecular orbital. The proton of the H atom is along the internuclear axis. Due to symmetry, the other N electrons are equivalent to point charges at the origin. (See Eqs. (19-38) of Appendix IV.) Thus, the energies in the NH MO involve only the N2 px and HIs electrons and the change in the magnetic energy of the N2px electron with the other N electrons (Eq. (13.305)) with the formation of the NH MO. The forces are determined by these energies.
As in the case of H2, the MO is a prolate spheroid with the exception that the ellipsoidal MO surface cannot extend into N atom for distances shorter than the radius of the 2 p shell. Thus, the MO surface comprises a prolate spheroid at the H proton that is continuous with the 2p shell at the N atom' whose nucleus serves as the other focus. The energy of the prolate spheroid is matched to that of the N2ρ shell. As in the case with OH, the linear combination of the H2 -type ellipsoidal MO with the N2p AO must involve a 25% contribution from the H2 -type ellipsoidal MO to the N2ρ atomic orbital (AO) in order to match potential, kinetic, and orbital energy relationships. Thus, the NH MO must comprise 75% of a H2 -type ellipsoidal MO and a nitrogen AO: 1 NIpx AO + 0.75 H2 MO → NH MO (13.247)
The force balance of the NH MO is determined by the boundary conditions that arise from the linear combination of orbitals according to Eq. (13.247) and the energy matching condition between the hydrogen and nitrogen components of the MO.
Similar to the OH case given by Eq. (13.59), the H2 -type ellipsoidal MO comprises
75% of the NH MO; so, the electron charge density in Eq. (11.65) is given by -0.75e . Based on the condition that the electron MO is an equipotential energy surface, Eq. (11.79) gives the ellipsoidal parameter c' in terms of the central force of the foci, the electron angular momentum, and the ellipsoidal parameter a . To meet the equipotential condition of the union of the H2 -type-ellipsoidal-MO and the N AO, the force constant used to determine the ellipsoidal parameter c' is normalized by the ratio of the ionization energy of N 14.53414 eV [6] and 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243). This normalizes the force to match that of the Coulombic force alone to met the force matching condition of the TVH MO under the influence of the proton and the N nucleus. Thus, k1 of Eq. (11.79) to determine c' is
Figure imgf000149_0001
L for the electron equals h ; thus, the distance from the origin of the iVH MO to each focus c1 is given by Eqs. (11.79) and (13.248):
(13.249)
Figure imgf000149_0002
The internuclear distance from Eq. (13.249) is
2c' = 2Λ/0.712154«α0 (13.250)
The length of the semiminor axis of the prolate spheroidal NH MO b = c is given by Eqs. (11.80) and (13.62). The eccentricity, e , is given by Eq. (13.63). Then, the solution of the semimajor axis a allows for the solution of the other axes of the prolate spheroidal and eccentricity of the NH MO.
The energy of the nitrogen 2p shell is the negative of the ionization energy of the nitrogen atom given by Eq. (10.143). Experimentally, the energy is [6] E(2p shell) = -E (ionization; N) = -14.53414 eV (13.251)
Since the prolate spheroidal MO transitions to the JV AO, the energy E in Eq. (13.251) adds to that of the H2 -type ellipsoidal MO to give the total energy of the JVH MO. From the energy equation and the relationship between the axes given by Eqs. (13.249-13.250) and (13.62-13.63), the dimensions of the JVH MO are solved.
The energy components of Ve , Vp, T , Vn, , and Eτ are the same as those of OH given by Eqs. (13.67-13.73). Similarly to OH , the total energy E7 (NH) of the JVH MO is given by the sum of the energies of the orbitals corresponding to the composition of the linear combination of the JV AO and the H2 -type ellipsoidal MO that forms the JVH MO as given by Eq. (13.247):
E7 (NH) = Eτ + E (2 p shell)
= E7 - EQonization; N) (13.252)
Figure imgf000150_0001
To match the boundary condition that the total energy of the entire the H2 -type ellipsoidal MO is given by Eqs. (11.212) and (13.75), E7 (JVH) given by Eq. (13.252) is set equal to Eq. (13.75):
Figure imgf000150_0002
(13.253)
From the energy relationship given by Eq. (13.252) and the relationship between the axes given by Eqs. (13.249-13.250) and (13.62-13.63), the dimensions of the JVH MO can be solved. Substitution of Eq. (13.249) into Eq. (13.253) gives
Figure imgf000150_0003
The most convenient way to solve Eq. (13.254) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is α = 1.36275αo = 7.21136X 10'11 m (13.255) Substitution of Eq. (13.255) into Eq. (13.249) gives d = 0.98513α0 = 5.21310Z 10~u m (13.256) The internuclear distance given by multiplying Eq. (13.256) by two is
Id = 1.97027Λ0 = 1.04262 X 10"10 m (13.257)
The experimental bond distance is [28]
2c' = 1.0362 X lO"10 m (13.258) Substitution of Eqs. (13.255-13.256) into Eq. (13.62) gives b = c = 0.94159α0 = 4.98270 X lO"11 m (13.259)
Substitution of Eqs. (13.255-13.256) into Eq. (13.63) gives e = 0.72290 (13.260)
The nucleus of the H atom and the nucleus of the N atom comprise the foci of the H2 -type ellipsoidal MO. The parameters of the point of intersection of the H2 -type ellipsoidal MO and the N2px AO are given by Eqs. (13.84-13.95). The polar intersection angle θ' is given by
Figure imgf000151_0001
where rn = rη = 0.93084α0 is the radius of the JV atom. Substitution of Eqs. (13.255-13.256) into Eq. (13.86) gives
0' = 114.61° (13.262)
Then, the angle θN2pχAO the radial vector of the NIpx AO makes with the internuclear axis is
9 NIpxAo =180°-l 14.61° = 65.39° (13.263) as shown in Figure 10. The distance from the point of intersection of the orbitals to the internuclear axis must be the same for both component orbitals. Thus, the angle ®t = θH M0 between the internuclear axis and the point of intersection of the H2 -type ellipsoidal MO with the N radial vector obeys the following relationship: rη ήκθN2pA0 = 0.93084α0 sin0wo = ^sin^0 (13.264) such that
Figure imgf000151_0002
with the use of Eq. (13.263). Substitution of Eq. (13.259) into Eq. (13.265) gives 0^ = 64.00° (13.266)
Then, the distance dH^M0 along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by dH2M0 = a∞S θH2MO (13.267) Substitution of Eqs. (13.255) and (13.266) into Eq. (13.267) gives d H2uo = 0.59747α0 = 3.16166X 1(T11 m (13.268)
The distance dN2pA0 along the internuclear axis from the origin of the N atom to the point of intersection of the orbitals is given by
JN2PAO ^ C - JH1MO (13-269) Substitution of Eqs. (13.79) and (13.93) into Eq. (13.94) gives d N2PΛo = 0.38767α0 = 2.05144 X lO"11 m (13.270)
As shown in Eq. (13.247), in addition to the p -orbital charge-density modulation, the uniform charge-density in the px orbital is increased by a factor of 0.25 and the H -atom density is decreased by a factor of 0.25. The internuclear axis of the N-H bond is perpendicular to the bonding px orbital. Using the orbital composition of TVTf (Eq. (13.27)), the radii of Ms = 0.14605α0 (Eq. (10.51)), NIs = 0.69385α0 (Eq. (10.62)), and N2p = 0.930S4a0 (Eq. (10.142)) shells, and the parameters of the NH MO given by Eqs. (13.3-13.4) and (13.255-13.270), the dimensional diagram and charge-density of the NH MO comprising the linear combination of the H2 -type ellipsoidal MO and the N AO according to Eq. (13.247) are shown in Figures 10 and 11, respectively.
ENERGIES OF NH
The energies of NH given by the substitution of the semiprincipal axes (Eqs. (13.255- 13.256) and (13.259)) into the energy equations (Eqs. (13.67-13.73)) are
eV (13.271)
Figure imgf000152_0001
Figure imgf000152_0002
_31.63544 eK
Figure imgf000153_0001
(13.275) where E7, (NH) is given by Eq. (13.253) which is reiteratively matched to Eq. (13.75) within five-significant-figure round-off error.
VIBRATION AND ROTATION OF NH
The vibrational energy of NH may be solved in the same manner as that of OH . From Eqs. (13.102-13.106) with the substitution of the NH parameters, the angular frequency of the oscillation is
Figure imgf000153_0002
(13.276)
Figure imgf000153_0004
= 6.18700X 1014 radls where b is given by Eq. (13.259), 2c' is given by Eq. (13.257), and the reduced mass of 14NH is given by:
In1Jn2 (1)(14) ,n .™
^NH = = 1 \ Λ mp (13.277)
NH Ynx I- M1 1 + 14 p where mp is the proton mass. Thus, during bond formation, the perturbation of the orbit determined by an inverse-squared force results in simple harmonic oscillatory motion of the orbit, and the corresponding frequency, ω(θ) , for 14NH given by Eqs. (11.136), (11.148), and (13.276) is
ω (o) X 1014 radians I s (13.278)
Figure imgf000153_0003
where the reduced nuclear mass of 14NH is given by Eq. (13.277) and the spring constant, Jfc(0) , given by Eqs. (11.136) and (13.276) is k (0) = 597.59 Nm~l ' (13.279)
The 14NH transition-state vibrational energy, Evjb (θ) , given by Planck's equation (Eq. (11.127)) is:
Evib (0) = hω = H6.18700 X 1014 radls = 0.407239 eV = 3284.58 cm~l (13.280) ωe , from the experimental curve fit of the vibrational energies of 14TVH is [28] ωe = 3282.3 cm'1 (13.281)
Using Eqs. (13.112-13.118) with Evib (θ) given by Eq. (13.280) and D0 given by Eq.
(13.311), the 14NH u = l → υ = 0 vibrational energy, Evib (\) is
£v/A (l) = 0.38581 eF (3111.84 CTw"1) (13.282) The experimental vibrational energy of 14NH using ωe and ωexe [28] according to K&P [15] is
Evib (1) = 0.38752 e V (3125.5 Cm"1) (13.283)
Using Eq. (13.113) with Evib (\) given by Eq. (13.282) and D0 given by Eq. (13.311), the anharmonic perturbation term, ωoxo , of 14NH is CO0X0 = 86.37 cm'1 (13.284)
The experimental anharmonic perturbation term, (Q0X0 , of 14NH [28] is fi)oxo = 78.4 cm'1 (13.285)
The vibrational energies of successive states are given by Eqs. (13.280), (13.112), and (13.284). Using b given by Eq. (13.259), 2c' given by Eq. (13.257), D0 given by Eq. (13.314), and the reduced nuclear mass of UND given by
Figure imgf000154_0001
where mp is the proton mass, the corresponding parameters for deuterium nitride 14ND (Eqs. (13.102-13.121)) are
ω(0) Z lO14 radians Is (13.287)
Figure imgf000154_0002
£(0) = 579.59 JVm"1 (13.288) Evib (0) = hω = h4.51S35X 1014 radls = 0.29741 eV = 22>9%.12 crrf1 (13.289)
£v/A (l) = 0.28710 eF (2305.35 cm"1) (13.290)
O0X0 = 47.40 cnC1 (13.291) ωe , from the experimental curve fit of the vibrational energies of 14M) is [28] 6), = 2398 cm"1 (13.292)
The experimental vibrational energy of 14M) using ωe and ωexe [28] according to K&P [15] is
Evib (1) = 0.2869 e V (2314 Cw"1) (13.293) and the experimental anharmonic perturbation term, ωoxo , of 14JVD [28] is O0X0 = 42 cm~l (13.294) which match the predictions given by Eqs. (13.289), (13.290) and (13.291), respectively.
Using Eqs. (13.133-13.134) and the internuclear distance, r = 2c' , and reduced mass of 14NH given by Eqs. (13.257) and (13.277), respectively, the corresponding Be is
5^ 16.495 CTT1 (13.295) The experimental Be rotational parameter of 14NH is [28]
5^ 16.6993 CTw"1 (13.296)
Using the internuclear distance, r = 2c\ and reduced mass of 14ZVD given by Eqs. (13.257) arid (13.286), respectively, the corresponding Be is
Be = %.191 cnfl (13.297) The experimental Be rotational parameter of 14ND is [28]
£e = 8.7913 cwT1 (13.298)
THE DOPPLER ENERGY TERMS OF 14NH AND 14JVO
The equations of the radiation reaction force of hydrogen and deuterium nitride are the same as those of the corresponding hydroxyl radicals with the substitution of the hydrogen and deuterium nitride parameters. Using Eqs. (11.136) and (13.140- 13.141), the angular frequency of the reentrant oscillation in the transition state is
Figure imgf000156_0001
where b is given by Eq. (13.259). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)):
Z^ = /ky = /a91850X 1016 rod Is = 25.79224 e V (13.300) In Eq. (11.181), substitution of the total energy of NH , E7 (NH) , (Eq. (13.253)) for E1n, , the mass of the electron, me , for M , and the kinetic energy given by Eq. (13.300) for Eκ gives the Doppler energy of the electrons for the reentrant orbit:
e
Figure imgf000156_0002
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of TVH due to the reentrant orbit in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by
Eq. (13.301) and EKvιb , the average kinetic energy of vibration which is 1/2 of the vibrational energy of NH . Using the experimental 14NH ωe of 3282.3 cm~l (0.40696 eV) [28] Eosc (HNH) is
E0Sc [14NH) = ED +EKvώ = ED +U - (13.302)
I815 (14M) = -0.31785 eV + -(0.40696 eV) = -0.11437 eV (13.303)
Using Eqs. (13.301) and the experimental 14ND ωe of 2398 cm"1 (0.29732 eV) [28] Eosc (14ND) is
Eosc ( 14ND) = -0.31785 eV+ -(0.29732 eV) = -Q.16919 eV (13.304)
TOTAL AND BOND ENERGIES OF 14NH AND 14ND
ET+OSC (NH) , the total energy of the 14NH radical including the Doppler term, is given by the sum of E7, (NH) (Eq. (13.253)) and E0SC ( UNH) given by Eq. (13.303): Eτ+osc (NH) = Ve +T + Vm +Vp +E(2p shell) + E0SC (UNH)
(13.305)
= E7. (NH) + Eo,c (14NH)
Figure imgf000157_0001
= -31.63537 er-0.31785
Figure imgf000157_0002
(13.306)
From Eqs. (13.302-13.303) and (13.305-13.306), the total energy of 14NH is ET*- (NH) = -31.63537 eV + Eosc (14NH)
= -31.63537 eF-0.31785 eF+-(0.40696 eF) (13.307) = -31.74974 e F
where the experimental ωe was used for the term. ET+OSC (ND), the total energy of
Figure imgf000157_0005
14ND including the Doppler term, is given by the sum of Eτ (ND) = E7. (NH) (Eq. (13.253)) and EOSC (UND) given by Eq. (13.304):
ET+CC (ND) = -31.63537 eV + E0SC (14M))
= -31.63537 eF-0.31785 eF + -(0.29732 eF) (13.308)
= -31.80456 eV
where the experimental ω was used for the term. The dissociation of the bond of the
Figure imgf000157_0003
hydrogen nitride forms a free hydrogen atom with one unpaired electron and a nitrogen atom with three unpaired electrons as shown in Eq. (13.246). The px and py fields cancel and the magnetic energy (Eq. (7.46) with r7 = 0.93084α0 is subtracted due to the one component of Emag given by Eq. (10.137): τr eV (13.309)
Figure imgf000157_0004
The corresponding bond dissociation energy, ED , is given by the sum of the total energies of the nitrogen atom and the corresponding hydrogen atom minus the sum of Eτ+osc (NH) and E(magnetic) :
ED = E(u N) + E(H) ~ET+0SC (NH) -E (magnetic) (13.310) E(14N) is given by Eq. (13.251), ED (H) is given by Eq. (13.154), and ED (D) is given by Eq. (13.155). The 14TVH bond dissociation energy, ED (14NH), is given by Eqs. (13.154), (13.251), (13.307), and (13.309-13.310):
ED (14NH) = -(14.53414 eF + 13.59844 eV)-(E (magnetic) + Eτ+osc (NH))
= -28.13258 βF-(0.14185 -31.74974 e7) (13.311)
= 3.47530 eF
The experimental 14NH bond dissociation energy from Ref. [29] and Ref. [30] is ED (14JVH) = 3.42 eV (13.312)
ED (14NH) < 3.47 e V (13.313)
The 14ND bond dissociation energy, E0 (14ND), is given by Eqs. (13.155), (13.251), (13.308), and (13.309-13.310):
E0 (14ND) = -(14.53414 eV + 13.603 eV) -(E (magnetic) + ET+OSC (ND))
= -28.13714 eF-(0.14185-31.80456 eF) (13.314)
= 3.5256 eV The experimental 14ND bond dissociation energy from Ref. [31] and Ref. [30] is
EDm (14M)) < 339 kJlmol = 3.513 eV (13.315)
ED (14ND) ≤ 3.54 eV (13.316)
The results of the determination of bond parameters of NH and ND are given in Table 13.1. The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
DIΗYDROGEΝ NITRIDE (NH2)
The dihydrogen nitride radical NH2 is formed by the reaction of a hydrogen atom with a hydrogen nitride radical: NH + H → NH2 (13.317)
NH2 can be solved using the same principles as those used to solve H2O . Two diatomic molecular orbitals (MOs) developed in the Nature of the Chemical Bond of Ηydrogen-Type Molecules and Molecular Ions section serve as basis functions in a linear combination with two nitrogen atomic orbitals (AOs) to form the MO of NH2 . The solution is very similar to that of NH except that there are two NH bonds in NH2 .
FORCE BALANCE OF NH2
NH2 comprises two chemical bonds between nitrogen and hydrogen. Each N-H bond comprises two spin-paired electrons with one from an initially unpaired electron of the nitrogen atom and the other from the hydrogen atom. Each H -atom electron forms a H2 - type ellipsoidal MO with one of the initially unpaired N -atom electrons, Ipx or 2py , such that the proton and the N nucleus serve as the foci. The initial N electron configuration given in the Seven-Electron Atoms section is Xs1Is2Ip3 , and the orbital arrangement is given by Eqs. (10.134) and (13.246). The radius and the energy of the N2p shell are unchanged with bond formation. The central paramagnetic force due to spin of each N-H bond is provided by the spin-pairing force of the NH2 MO that has the symmetry of an s orbital that superimposes with the N2p orbitals such that the corresponding angular momenta are unchanged. As in the case of H2 , each of two N-H -bond MOs is a prolate spheroid with the exception that the ellipsoidal MO surface cannot extend into N atom for distances shorter than the radius of the 2p shell since it is energetically unfavorable. Thus, the MO surface comprises a prolate spheroid at each H proton that is continuous with the 2p shell at the N atom. The energies in the NH2 MO involve only each N2p and each HXs electron with the formation of each N-H bond. The sum of the energies of the prolate spheroids is matched to that of the 2p shell. The forces are determined by these energies. As in the case of NH , the linear combination of each H2 -type ellipsoidal MO with each N2p AO must involve a 25% contribution from the H2 -type ellipsoidal MO to the N2p AO in order to match potential, kinetic, and orbital energy relationships. Thus, the NH2 MO must comprise two N-H bonds with each comprising 75% of a H2 -type ellipsoidal MO (1/2 +25%) and a nitrogen AO:
[1 NIpx AO+0J5 H2 MO] + [l N2py AO + 0.75 H2 Mθ] → NH2 MO (13.318)
The force constant k' of the each H2 -type-ellipsoidal-MO component of the NH2
MO is given by Eq. (13.248). The distance from the origin of each JV-H -bond MO to each focus c' is given by Eq. (13.249). The internuclear distance is given by Eq. (13.250). The length of the semiminor axis of the prolate spheroidal N-H -bond MO b = c is given by Eq. (13.62). The eccentricity, e , is given by Eq. (13.63). The solution of the semimajor axis a then allows for the solution of the other axes of each prolate spheroid and eccentricity of each iV-H -bond MO. Since each of the two prolate spheroidal N-H -bond MOs comprises a H2 -type-ellipsoidal MO that transitions to the N AO, the energy E in Eq. (13.251) adds to that of the two corresponding H2 -type ellipsoidal MOs to give the total energy of the NH2 MO. From the energy equation and the relationship between the axes, the dimensions of the NH2 MO are solved.
The energy components of Ve , Vp , T , V1n , and E7, are twice those of OH and NH given by Eqs. (13.67-13.73) and equal to those of H2O given by Eqs. (13.172-13.178).
Similarly to H2O , since the each prolate spheroidal H2 -type MO transitions to the N AO and the energy of the N2p shell must remain constant and equal to the negative of the ionization energy given by Eq. (13.251), the total energy E7, (NH2) of the NH2 MO is given by the sum of the energies of the orbitals corresponding to the composition of the linear combination of the N AO and the two H2 -type ellipsoidal MOs that forms the NH2 MO as given by Eq. (13.318):
E7, (NH2 ) = ET +E( 2p shell) = E7, - Eiioni∑ation; N) (13.319)
Figure imgf000160_0001
The two hydrogen atoms and the nitrogen atom can achieve an energy minimum as a linear combination of two H2 -type ellipsoidal MOs each having the proton and the nitrogen nucleus as the foci. Each N-H -bond MO comprises the same N2p shell of constant energy given by Eq. (13.251). Thus, the energy of the NH2 MO is also given by the sum of that of the two H2 -type ellipsoidal MQs given by Eq. (11.212) minus the energy of the redundant nitrogen atom of the linear combination given by Eq. (13.251):
Figure imgf000161_0001
= 2 (-31.63536831 eF)-(-14.53414 eV) (13.320)
= -48.73660 eV E7- (NH2) given by Eq. (13.319) is set equal to two times the energy of the H2 -type ellipsoidal MO minus the energy of the N2p shell given by Eq. (13.320):
eV
Figure imgf000161_0002
(13.321)
From the energy relationship given by Eq. (13.321) and the relationship between the axes given by Eqs. (13.248-13.250) and (13.62-13.63), the dimensions of the NH2 MO can be solved.
Substitution of Eq. (13.249) into Eq. (13.321) gives
Figure imgf000161_0003
The most convenient way to solve Eq. (13.322) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is α = 1.36276α0 = 7.21141 X 10"11 m (13.323)
Substitution of Eq. (13.323) into Eq. (13.249) gives c' = 0.98514α0 = 5.21312X 10"" m (13.324)
The internuclear distance given by multiplying Eq. (13.324) by two is 2c' = 1.97027øo = 1.04262 X 10~w m (13.325)
The experimental bond distance is [32]
2c' = 1.024X 10-10 m (13.326)
Substitution of Eqs. (13.323-13.324) into Eq. (13.62) gives b = c = 0.94l60o0 = 4.98276 X 10"11 m (13.327) Substitution of Eqs. (13.323-13.324) into Eq. (13.63) gives e = 0.72290 (13.328) The nucleus of the H atom and the nucleus of the N atom comprise the foci of the H2 -type ellipsoidal MO. The parameters of the point of intersection of each H2 -type ellipsoidal MO and the N2px AO or N2py AO are given by Eqs. (13.84-13.95) and (13.261-13.270).
Using Eqs. (13.323-13.325) and (13.327-13.328), the polar intersection angle θ' given by Eq. (13.261) with Yn = Y1 = 0.93084α0 is
0' = 114.61° (13.329)
Then, the angle θN2pAO the radial vector of the NIpx AO or NIp y AO makes with the internuclear axis is
^o = 180o-114.61o = 65.39° (13.330) as shown in Figure 10. The angle θHiU0 between the internuclear axis and the point of intersection of each H2 -type ellipsoidal MO with the N radial vector given by Eqs. (13.264- 13.265), (13.327), and (13.330) is θHiMO = 64.00° (13.331)
Then, the distance dH MO along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals given by Eqs. (13.267), (13.323), and (13.331) is
JH7UO = 0.59748α0 = 3.16175 X 10"π m (13.332)
The distance dN2pΛ0 along the internuclear axis from the origin of the N atom to the point of intersection of the orbitals given by Eqs. (13.269), (13.324), and (13.332) is dN2pAO = 0.m65a0 = 2.05UlX \0-n m (13.333)
As shown in Eq. (13.318), in addition to the p -orbital charge-density modulation, the uniform charge-density in the px and py orbitals is increased by a factor of 0.25 and the H
, atoms are each decreased by a factor of 0.25. Using the orbital composition of NH2 (Eq.
(13.318)), the radii of MJ = 0.14605O0 (Eq. (10.51)), N2s = 0.69385α0 (Eq. (10.62)), and ^2^ = 0.93084«,, (Eq. (10.142)) shells, and the parameters of the NH2 MO given by Eqs.
(13.3-13.4) and (13.323-13.333), the charge-density of the NH2 MO comprising the linear combination of two N- H -bond MOs (JVH -type ellipsoidal MOs given in the Energies of iVH section) according to Eq. (13.318) is shown in Figure 12. Each N-H-bond MO comprises a H2 -type ellipsoidal MO and an N2p AO having the dimensional diagram shown in Figure 10.
ENERGIES OF NH2 The energies of NH2 given by the substitution of the semiprincipal axes ((Eqs. (13.323- 13.325) and (13.327)) into the energy equations (Eqs. (13.172-13.176)) are
Figure imgf000163_0001
V = 2 = 27.62216 eV (13.335)
8πεja2 -b2
T 7.77974 eV (13.336)
V1 -13.88987 gF (13.337)
Figure imgf000163_0002
Figure imgf000163_0003
(13.338) where E7 [NH7) is given by Eq. (13.319) which is reiteratively matched to Eq. (13.320) within five-significant-figure round-off error.
VIBRATION OF NH2
The vibrational energy levels of TVH2 may be solved as two equivalent coupled harmonic oscillators by developing the Lagrangian, the differential equation of motion, and the eigenvalue solutions [2] wherein the spring constants are derived from the central forces as given in the Vibration of Hydrogen-Type Molecular Ions section and the Vibration of Hydrogen-Type Molecules section.
THE DOPPLER ENERGY TERM OF NH2
The radiation reaction force in the case of the vibration of NH2 in the transition state corresponds to the Doppler energy, ED , given by Eq. (11.181) and Eqs. (13.22) and (13.144) that is dependent on the motion of the electrons and the nuclei. The kinetic energy of the transient vibration is derived from the corresponding central forces. The equations of the radiation reaction force of dihydrogen and dideuterium nitride are the same as those of the corresponding water molecules with the substitution of the dihydrogen and dideuterium nitride parameters. Using Eqs. (11.136) and (13.207-13.209), the angular frequency of the reentrant oscillation in the transition state is
Figure imgf000164_0001
where b is given by Eq. (13.327). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)): Eκ = hω = /J5.54150X IO16 rad/s = 36.47512 eV (13.340)
In Eq. (11.181), substitution of Ex (H2) (Eqs. (11.212) and (13.75)), the maximum total energy of each Tf2 -type MO, for Ehv , the mass of the electron, me , for M , and the kinetic energy given by Eq. (13.340) for Eκ gives the Doppler energy of the electrons for the reentrant orbit:
ED S eV (13.341)
Figure imgf000164_0002
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of NH2 due to the reentrant orbit in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by Eq. (13.341) and EKvtb , the average kinetic energy of vibration which is 1/2 of the vibrational energy of NH2 . Using the experimental 14NH2 vibrational energy of Evib = 3301.110 cm"1 = 0.40929 eV [33] gives
E\sc = ED +EKvlb = ED +-U £ (13.342) z \j μ
E'osc = -0.37798 eV +-(0.40929 eV) = -0.17334 eV (13.343)
per bond. As in the case for H2O , the reentrant orbit for the binding of a hydrogen atom to a NH radical causes the bonds to oscillate by increasing and decreasing in length along the two N-H bonds at a relative phase angle of 180°. Since the vibration and reentrant oscillation is along two bonds for the asymmetrical stretch (v3), E0n for 14NH2 ,
EOsc (U NH2 ) , is:
Figure imgf000165_0001
= 2 -0.37798 eV + -(0.40929 eV) (13.344) = -0.34668 eV
Using Eq. (13.341), Eqs. (13.342-13.344), and the 14JVD2 vibrational energy of Evώ = 2410.79 cm~l = 0.2989O eF , calculated from the experimental 14NH2 vibrational energy using Eq. (11.148), the corresponding Eosc ( 14ND2 ) is
Eosc (14ND2) = 2| -0.37798 e F +-(0.29890 eVU
(13.345)
= -0.45707 e V
TOTAL AND BOND ENERGIES OF 14NH2 AND 14ND2
Eτ+osc T14NH2I , the total energy of the 14NH2 including the Doppler term, is given by the sum
of E7 (NH2) (Eq. (13.321)) and Eosc (14NH2 ) given Eqs. (13.339-13.344):
ET+osc {i4NH2) = Ve +T + Vm +Vp + E(N2p) + Eosc ("NH2)
(13.346) = E7 [NH2) + EOSC (UNH2)
Figure imgf000165_0002
From Eqs. (13.344) and (13.346-13.347), the total energy of 14NH2 is ET+osc {uNH2) = -4H.73660 eV + Eosc (uNH2)
= -48.73660 eF-2[ 0.37798 eV --(0.40929 eF) J (13.348)
= -49.08328 eV
where the experimental NH2 vibrational energy was used for the h i— term.
ET4.OSC (14N-D2) , the total energy of 14ND2 including the Doppler term is given by the sum of
E7 (ND2) = E7 ( NH2) (Eq. (13.321)) and EOSC ( UND2) given by Eq. (13.345):
Eτ+osc [14ND2) = -48.73660 eV + Eosc ("ND2)
= -48.73660 eV -li 0.37798 eV --(0.2989O eF) ] (13.349)
= -49.19366 e V where the experimental 14NH2 vibrational energy corrected for the reduced mass difference
of hydrogen and deuterium was used for the h i — term. The corresponding bond
dissociation energy, ED , is given by the sum of the total energies of the corresponding hydrogen nitride radical and hydrogen atom minus the total energy of dihydrogen nitride, ET+osc (uNH2 ) .
Thus, ED of 14NH2 is given by:
ED [14NH2) = E(H)+ E(14NH)-E^ (14NH2) (13.350) where E7-(14NH) is given by the of the sum of the experimental energies of 14N (Eq.
(13.251)), H (Eq. (13.154)), and the negative of the bond energy of 14NH (Eq. (13.312)): E(14NH) = -13.59844 eF -14.53414 eF-3.42 eF = -31.55258 eF (13.351)
From Εqs. (13.154), (13.348), and (13.350-13.35I)5E0 (14NH2) is
ED (14NH2) = E(H)+ E(14NH) -ετ+osc (14NH2)
= -13.59844 eF-31.55258 eF-(-49.08328 eF) (13.352)
= 3.9323 eV
The experimental 14NH2 bond dissociation energy from Ref. [34] and Ref. [35] is
E0(14NH2) = 88 + 4 fcα//τwo/e = 3.816O eF (13.353) E0(14NH2) = 91.0± 0.5 kcallmole = 3.9461 eV (13.354)
Similarly, E0 of 14ND2 is given by:
ED (14ND2) = E(D) + EC4ND) - (ET+0SC { UND2)) (13.355)
where ET(UND) is given by the of the sum of the experimental energies of 14N (Eq. (13.251)), D (Eq. (13.155)), and the negative of the bond energy of 14ND (Eq. (13.315)):
E(14JVD) = -13.603 eK-14.53414 eF-3.513 eV = -31.6506 eV (13.356)
From Eqs. (13.155), (13.349), and (13.355-13.356), E23 (14ND2) is
E0C4ND2) = -13.603 eV -31.6506 eV -(-49.19366 eV) = 3.9401 eV
The 14ND2 bond dissociation energy calculated from the average of the experimental bond energies [34-35] and vibrational energy of 14NH2 [33] is
E0(14ND2) = E0(14NH2) +i(Ev/, o^2)-Ew, (^2))
=—(3.8160 eF + 3.9461 eF)+-(θ.4O929 eF-0.29890 eF) (13.358) = 3.9362 eF
BOND ANGLE OF NH2
The NH2 MO comprises a linear combination of two N-H -bond MOs. Each N-H -bond MO comprises the superposition of a H2 -type ellipsoidal MO and the NIpx AO or the
N2py AO with a relative charge density of 0.75 to 1.25; otherwise, the N2p AOs are the same as those of the nitrogen atom. A bond is also possible between the two Η atoms of the N-H bonds. Such H-H bonding would decrease the N-H bond strength since electron density would be shifted from the N-H bonds to the H-H bond. Thus, the bond angle between the two N-H bonds is determined by the condition that the total energy of the H2 -type ellipsoidal MO between the terminal H atoms of the N-H bonds is zero.
From Eqs. (11.79) and (13.228), the distance from the origin to each focus of the H-H ellipsoidal MO is
Figure imgf000167_0001
The internuclear distance from Eq. (13.229) is
Figure imgf000168_0001
The length of the semiminor axis of the prolate spheroidal H -H MO b ~ c is given by Eq. (13.167). Since the two H2 -type ellipsoidal MOs comprise 75% of the H electron density of
H2 and the energy of each H2 -type ellipsoidal MO is matched to that of the N2p AO; the component energies and the total energy E7, of the H- H bond are given by Eqs. (13.67- 13.73) except that Ve , T, and V1n are corrected for the energy matching factor of 0.93613 given in Eq. (13.248). Substitution of Eq. (13.359) into Eq. (13.233) with the energy- matching factor gives
Figure imgf000168_0002
(13.361)
From the energy relationship given by Eq. (13.361) and the relationship between the axes given by Eqs. (13.359-13.360) and (13.167-13.168), the dimensions of the H -H MO can be solved.
The most convenient way to solve Eq. (13.361) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is α = 4.9500αn = 2.6194X 10 -10 m (13.362)
Substitution of Eq. (13.362) into Eq. (13.359) gives c' = 1.5732ar0 = 8.3251 X 10~" m (13.363)
The internuclear distance given by multiplying Eq. (13.363) by two is
2c' = 3.1464tf0 = 1.6650 JST 10~10 m (13.364) Substitution of Eqs. (13.362-13.363) into Eq. (13.167) gives b = c = 4.6933αn = 2.4836 X 10 ,-10 m (13.365) Substitution of Eqs. (13.362-13.363) into Eq. (13.168) gives e = 0.3178 (13.366)
Using, 2c'H_H (Eq. (13.364)), the distance between the two H atoms when the total energy of the corresponding MO is zero (Eq. (13.361)), and 2c\_H (Eq. (13.325)), the internuclear distance of each N-H bond, the corresponding bond angle can be determined from the law of cosines. Using, Eq. (13.242), the bond angle θ between the N-H bonds is
Figure imgf000169_0001
= cos~1 (-0.2751) = 105.969°
The experimental angle between the N-H bonds is [32]
0 = 103.3° (13.368)
The results of the determination of bond parameters of NH2 and ND2 are given in Table 13.1. The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
AMMONIA (NH3) Ammonia (NH3) is formed by the reaction of a hydrogen atom with a dihydrogen nitride radical:
NH2 + H -> NH3 (13.369)
NH3 can be solved using the same principles as those used to solve NH2 except that three rather than two H2 -type prolate spheroidal molecular orbitals (MOs) serve as basis functions in a linear combination with nitrogen atomic orbitals (AOs) to form the MO of NH3.
FORCE BALANCE OF NH3
NH3 comprises three chemical bonds between nitrogen and hydrogen. Each N-H bond comprises two spin-paired electrons with one from an initially unpaired electron of the nitrogen atom and the other from the hydrogen atom. Each H -atom electron forms a H2- type ellipsoidal MO with one of the initially unpaired N -atom electrons, 2px , 2py , or 2 pz , such that the proton and the N nucleus serve as the foci. The initial N electron configuration given in the Seven-Electron Atoms section is Is22s22p3 , and the orbital arrangement is given by Eqs. (10.134) and (13.246). The radius and the energy of the N2p shell are unchanged with bond formation. The central paramagnetic force due to spin of each N -H bond is provided by the spin-paring force of the NH3 MO that has the symmetry of an 5 orbital that superimposes with the N2p orbitals such that the corresponding angular momenta are unchanged.
As in the case of H2 , each of three N- H -bond MOs is a prolate spheroid with the exception that the ellipsoidal MO surface cannot extend into the TV atom for distances shorter than the radius of the 2p shell since it is energetically unfavorable. Thus, the MO surface comprises a prolate spheroid at each H proton that is continuous with the 2p shell at the N atom. The energies in the NH3 MO involve only each N2p and each His electron with the formation of each N-H bond. The sum of the energies of the prolate spheroids is matched to that of the 2p shell. The forces are determined by these energies. As in the cases of NH and NH2 , the linear combination of each H2 -type ellipsoidal MO with each N2p AO must involve a 25% contribution from the H2 -type ellipsoidal MO to the N2p AO in order to match potential, kinetic, and orbital energy relationships. Thus, the NH3 MO must comprise three N-H bonds with each comprising 75% of a H2 -type ellipsoidal MO (1/2 +25%) and a nitrogen AO:
[1 NIpx AO + 0.15 H2 Mθ] + [l N2py AO + 0.75 H2 Mθ] + [l N2p2 AO + 0.75 H2 MO]
→ NH3 MO
(13.370) The force constant k ' of the each H2 -type-ellipsoidal-MO component of the NH3
MO is given by Eq. (13.248). The distance from the origin of each N-H -bond MO to each focus c' is given by Eq. (13.249). The internuclear distance is given by Eq. (13.250). The length of the semiminor axis of the prolate spheroidal N-H -bond MO b = c is given by Eq. (13.62). The eccentricity, e , is given by Eq. (13.63). The solution of the semimajor axis a then allows for the solution of the other axes of each prolate spheroid and eccentricity of each N-H -bond MO. Since each of the three prolate spheroidal N-H -bond MOs comprises a H2 -type-ellipsoidal MO that transitions to the N AO, the energy E in Eq. (13.251) adds to that of the three corresponding H2 -type ellipsoidal MOs to give the total energy of the NH3 MO. From the energy equation and the relationship between the axes, the dimensions of the
NH3 MO are solved.
The energy components of Ve , Vp > T , V1n , and E7, are three times those of OH and
NH given by Eqs. (13.67-13.73) and 1.5 times those of H2O given by Eqs. (13.172-13.178). Similarly to H2O , since the each prolate spheroidal H2 -type MO transitions to the N AO and the energy of the N2p shell must remain constant and equal to the negative of the ionization energy given by Eq. (13.251), the total energy E7, (NH3) of the NH3 MO is given by the sum of the energies of the orbitals corresponding to the composition of the linear combination of the N AO and the three H2 -type ellipsoidal MOs that forms the NH3 MO as given by Eq. (13.370):
E7, (NH2) = Eτ + E( 2p shell)
= E1, - E(ionization; N) (13.371)
Figure imgf000171_0001
The three hydrogen atoms and the nitrogen atom can achieve an energy minimum as a linear combination of three H2 -type ellipsoidal MOs each having the proton and the nitrogen nucleus as the foci. Each N-H -bond MO comprises the same N2p shell of constant energy given by Eq. (13.251). Thus, an energy term of the NH3 MO is given by the sum of the three H2 -type ellipsoidal MOs given by Eq. (11.212) minus the energy of the redundant nitrogen atom of the linear combination given by Eq. (13.251). The total sum is determined by the energy matching condition of the binding atoms.
In Eq. (13.248), the equipotential condition of the union of each H2 -type-ellipsoidal- MO and the N AO was met when the force constant used to determine the ellipsoidal parameter c1 was normalized by the ratio of the ionization energy of N 14.53414 eF [6] and 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243). This normalized the force to match that of the Coulombic force alone to meet the force matching condition of the NH MO under the influence of the proton and the N nucleus. The minimum total energy of the NH3 MO from the sum of energies of a linear combination from four atoms is determined using the energy matching condition of Eq. (13.248). Since each of the three prolate spheroidal N-H -bond MOs of NH3 comprises a H2 -type-ellipsoidal MO that transitions to the N AO and the energy matching condition is met, the nitrogen energy E (Eq. (13.251)) and the energy (Eq. (1.243)) of a hydrogen atomic orbital (H AO), ECoulomb (H)5 corresponding to the Coulombic force of +e from the nitrogen nucleus is subtracted from the sum of the energies of the three corresponding H2 -type ellipsoidal MOs to given an energy minimum. From another perspective, the electron configuration of JVH2 is equivalent to that of OH and is given by Eq. (10.174). JVH2 serves as a one-electron atom that is energy matched by the H AO as a basis element to minimize the energy of JVH3 in the formation of the third N-H -bond.
Figure imgf000172_0001
= 3(-31.63536831 eF)-(-14.53414 eF-13.605804eF) (13.372) = -66.76616 e V
Ex (NH3) given by Eq. (13.371) is set equal to Eq. (13.372), three times the energy of the H2 -type ellipsoidal MO minus the energy of the N2p shell and the H AO:
Eτ (JVH3) = -3 eV
Figure imgf000172_0002
(13.373)
From the energy relationship given by Eq. (13.373) and the relationship between the axes given by Eqs. (13.248-13.250) and (13.62-13.63), the dimensions of the JVH3 MO can be solved.
Substitution of Eq. (13.249) into Eq. (13.373) gives
(13.374)
Figure imgf000172_0003
The most convenient way to solve Eq. (13.374) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is α = 1.3475Oa0 = 7.13066 X lO"11 m (13.375)
Substitution of Eq. (13.375) into Eq. (13.249) gives c' = 0.9796Ia0 = 5.18385 X 10"11 m (13.376)
The internuclear distance given by multiplying Eq. (13.376) by two is 2c' =1.9592Ian = 1.03677 X 10"10 m (13.377) The experimental bond distance is [32]
2C' = 1.012 X 1(T10 OT (13.378)
Substitution of Eqs. (13.375-13.376) into Eq. (13.62) gives b = c = 0.92527α0 = 4.89633 X 10"11 m (13.379) Substitution of Eqs. (13.375-13.376) into Eq. (13.63) gives e = 0.72698 (13.380)
The nucleus of the H atom and the nucleus of the JV atom comprise the foci of the H2 -type ellipsoidal MO. The parameters of the point of intersection of each H2 -type ellipsoidal MO and the NIpx , N2py , or N2pz AO are given by Eqs. (13.84-13.95), (13.261-13.270), and (13.261-13.270). Using Eqs. (13.375-13.377) and (13.379-13.380), the polar intersection angle θ' given by Eq. (13.261) with rn = rη = 0.93084«0 is
6>' = 115.89° (13.381)
Then, the angle θN2pA0 the radial vector of the NIpx , N2py , or N2pz AO makes with the internuclear axis is Θ N2PAO = 180° -115.89° = 64.11° (13.382) as shown in Figure 10. The angle ΘH MO between the internuclear axis and the point of intersection of each H2-type ellipsoidal MO with the N radial vector given by Eqs. (13.264-
13.265), (13.379), and (13.382) is έ^o = 64.83° (13.383) Then, the distance dHιMO along the internuclear axis from the origin of H2 -type ellipsoidal
MO to the point of intersection of the orbitals given by Eqs. (13.267), (13.375), and (13.383) is dHiM0 = 0.57314α0 = 3.03292 X 10"11 m (13.384)
The distance dN2pA0 along the internuclear axis from the origin of the JV atom to the point of intersection of the orbitals given by Eqs. (13.269), (13.376), and (13.384) is dN2pA0 = 0.40647α0 = 2.15093 X 10~u m (13.385)
As shown in Eq. (13.370), in addition to the p -orbital charge-density modulation, the uniform charge-density in the px , py, and p2 orbitals is increased by a factor of 0.25 and the
H atoms are each decreased by a factor of 0.25. Using the orbital composition of JVH3 (Eq. (13.370)), the radii of ΛOs = 0.14605α0 (Eq. (10.51)), N2s = 0.69385α0 (Eq. (10.62)), and
N2p = 0.93084α0 (Eq. (10.142)) shells, and the parameters of the NH3 MO given by Eqs. (13.3-13.4) and (13.375-13.385), the charge-density of the NH3 MO comprising the linear combination of three N -H -bond MOs (NH -type ellipsoidal MOs given in the Energies of NH section) according to Eq. (13.370) is shown in Figure 13. Each N-H -bond MO comprises a H2 -type ellipsoidal MO and an N2p AO having the dimensional diagram shown in Figure 10.
ENERGIES OF NH3 The energies of NH3 given by the substitution of the semiprincipal axes ((Eqs. (13.375- 13.377) and (13.379)) into the energy equations (Eqs. (13.67-13.73)) multiplied by three are
Ve -115.28799 eV (13.386)
Figure imgf000174_0001
Figure imgf000174_0002
Eτ [NH3) eF
Figure imgf000174_0003
(13.390) where E7 [NH3) is given by Eq. (13.371) which is reiteratively matched to Eq. (13.372) within five-significant-figure round-off error. '
VIBRATION OF NH3
The vibrational energy levels of NH3 may be solved as three equivalent coupled harmonic oscillators by developing the Lagrangian, the differential equation of motion, and the eigenvalue solutions [2] wherein the spring constants are derived from the central forces as given in the Vibration of Hydrogen-Type Molecular Ions section and the Vibration of
Hydrogen-Type Molecules section.
THE DOPPLER ENERGY TERM OF NH3 The radiation reaction force in the case of the vibration of NH3 in the transition state corresponds to the Doppler energy, ED , given by Eq. (11.181) and Eqs. (13.22) and (13.144) that is dependent on the motion of the electrons and the nuclei. The kinetic energy of the transient vibration is derived from the corresponding central forces. The equations of the radiation reaction force of ammonia are the same as those of the corresponding water and dihydrogen and dideuterium nitride radicals with the substitution of the ammonia parameters. Using Eqs. (11.136) and (13.207-13.209), the angular frequency of the reentrant oscillation in the transition state is
Figure imgf000175_0001
where b is given by Eq. (13.379). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)):
Eκ = hω = h5.6SmX 1016 radls = 37.44514 eV (13.392)
In Eq. (11.181), substitution of ET (H2) (Eqs. (11.212) and (13.75)), the maximum total energy of each H2 -type MO acting independently due to the D3h symmetry point group, for Ehv , the mass of the electron, me, for M , and the kinetic energy given by Eq. (13.392) for Eκ gives the Doppler energy of the electrons of each of the three bonds for the reentrant orbit:
(13.393)
Figure imgf000175_0002
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of NH3 due to the reentrant orbit in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by
Eq. (13.393) and EKvlb , the average kinetic energy of vibration which is 1/2 of the vibrational energy of NH3. Using the experimental 14NH3 vibrational energy of
Evib = 3443.59 cm"1 = 0.426954 eV [36] gives
(13.394)
Figure imgf000176_0003
E \sc = -0.38298 eV +-(0.426954 eV) = -0.16950 eV (13.395)
per bond. The reentrant orbit for the binding of a hydrogen atom to a NH2 radical involves three N-H bonds. Since the vibration and reentrant oscillation is along three bonds, Eosc for 14NH3 , EMC (14NH3) , is:
Figure imgf000176_0001
= 3[ -0.38298 eF+-(0.426954 eF) | (13.396)
= -0.50850 eF
Using Eq. (13.393), Eqs. (13.394-13.396), and the 1144N7 D3 experimental vibrational energy of Evlb = 2563.96 cm'1 = 0.317893 eV [36], the corresponding Eosc (14ND2) is
Figure imgf000176_0002
= -0.67209 e V
TOTAL AND BOND ENERGIES OF 14NH3 AND 14NA
Eτ+osc f 14NH3I, the total energy of the 14NH3 including the Doppler term, is given by the sum
of E7 (NH3) (Eq. (13.373)) and E0x (14NH3) given Eqs. (13.391-13.396):
ET+0SC {UNH3] = Ve +T + Vm +Vp + E{N2p) + Eosc (uNH3)
— , \ (13.398)
= Er (NH3)+ Eosc (14NH3)
Figure imgf000177_0001
From Eqs. (13.396) and (13.398-13.399), the total energy of 14NH2 is ET+0SC [uNH3) = -66.766l6 eV + E0SC (14NH3)
= -66.76616 eF - -(0.426954 eF) (13.400)
Figure imgf000177_0002
= -67.27466 e F
where the experimental 14NH3 vibrational energy was used for the term.
Figure imgf000177_0003
Eτ+os (14ND3 ) , the total energy of 14ND3 including the Doppler term is given by the sum of
Ex (ND3) = E7 ( NH3) (Eq. (13.373)) and EOSC (UND3) given by Eq. (13.397):
^+ωe (14^A) = -66.766i6 eF +E(KC (14NA)
= -66.76616 eVs( 0.38298 eF--(0.317893 eF) ] (13.401)
= -67.43780 eF
where the experimental 14ND3 vibrational energy was used for the term. The
Figure imgf000177_0004
corresponding bond dissociation energy, ED , is given by the sum of the total energies of the corresponding dihydrogen nitride radical and hydrogen atom minus the total energy of ammonia, Eτ+osc ( 14NH3 ) .
Thus, ED of 14NH3 is given by:
ED (14NH3) = E(H)+E(14NH2)-Er+wc (14NH3) (13.402) where E7(14NH2) is given by the of the sum of the experimental energies of 14N (Eq. (13.251)), two H (Eq. (13.154)), and the negative of the bond energies of 14NH (Eq. (13.312)) and 14NH2 (Eq. (13.354)): EC4NH2) = 2(-13.59844 eF)-14.53414 eK-3.42 eV -3.946 eV = -49.09709 eF
(13.403) From Eqs. (13.154), (13.400), and (13.402-13.403), E0 (14NH2) is
E0 (14NH3) = E(H)+ E(UNH2) - ET+OSC (14NH3)
= -13.59844 eF-49.09709 eF-(-67.27466 eF) (13.404) ■
= 4.57913 eF The experimental 14NH3 bond dissociation energy [37] is
E0(14NH3) = 4.60155 eV (13.405)
Similarly, ED of 14ND3 is given by:
ED {UND3) = E(D) + EC4ND2) - (ET+0SC { UND3)) (13.406)
where E7(14ND2) is given by the of the sum of the experimental energies of 14N (Eq. (13.251)), two times the energy of D (Eq. (13.155)), and the negative of the bond energies of 14ND (Eq. (13.315)) and 14ND2 (Eq. (13.358)):
E(14ND2) = 2(-13.603 eK)-14.53414 eF-3.5134 eF-3.9362 eV = -49.18981 eV
(13.407) From Εqs. (13.155), (13.401), and (13.406-13.407), ED (14ND3) is E0(14ND3) = -13.603 eF-49.18981 eV - (-67.43780 eV) = 4.64499 eV (13.408)
The experimental 14ND3 bond dissociation energy [37] is
E0(14ND3) = 4.71252 eV (13.409)
BOND ANGLE OF NH3 Using, 1cy H_H (Eq. (13.364)), the distance between the two H atoms when the total energy of the corresponding MO is zero (Eq. (13.361)), and 2c\_H , the internuclear distance of each N- H bond (Eq. (13.377)), the corresponding bond angle can be determined from the law of cosines. Using Eq. (13.367), the bond angle θ between the N-H bonds is
(13.410)
Figure imgf000178_0001
The experimental angle between the N-H bonds is [36] 0 = 106.67° (13.411)
The NH3 molecule has a pyramidal structure with the nitrogen atom along the z-axis at the apex and the hydrogen atoms at the base in the xy-plane. Since any two N-H bonds form an isosceles triangle, the distance dorigln_H from the origin to the nucleus of a hydrogen atom is given by
Substitu dti—on of E -q. (13.364) into Eq. (13.412) gives (13'412) dorigil,_H = LS1659a0 (13.413)
The height along the z-axis of the pyramid from the origin to N nucleus dhelght is given by
(13.414)
Figure imgf000179_0002
Substitution of Eqs. (13.377) and (13.413) into Eq. (13.414) gives
4^ = 0.73383^ (13.415)
The angle θv of each N-H bond from the z-axis is given by
Θv (13.416)
Figure imgf000179_0001
Substitution of Eqs. (13.413) and (13.415) into Eq. (13.417) gives θv = 68.00° (13.417)
The NH3 MO shown in Figure 13 was rendered using these parameters.
The results of the determination of bond parameters of NH3 and ND3 are given in
Table 13.1. The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
HYDROGEN CARBIDE (CH)
The methane molecule can be solved by first considering the solution of the hydrogen carbide, dihydrogen carbide, and methyl radicals. The former is formed by the reaction of a hydrogen atom and a carbon atom:
H + C → CH (13.418) The hydrogen carbide radicals, CH and CH2 , methyl radical, CH3, and methane, CH4 , can be solved using the same principles as those used to solve OH, H2O5 NH , NH2 , and NH3 with the exception that the carbon 2s and 2p shells hybridize to form a single 2sp3 shell as an energy minimum.
FORCE BALANCE OF CH
CH comprises two spin-paired electrons in a chemical bond between the carbon atom and the hydrogen atom. The CH radical molecular orbital (MO) is determined by considering properties of the binding atoms and the boundary constraints. The prolate spheroidal H2 MO developed in the Nature of the Chemical Bond of Ηydrogen-Type Molecules section satisfies the boundary constraints; thus, the H -atom electron forms a H2 -type ellipsoidal MO with one of the C -atom electrons. However, such a bond is not possible with the outer C electrons in their ground state since the resulting H2 -type ellipsoidal MO would have a shorter internuclear distance than the radius of the carbon 2p shell, which is not energetically stable. Thus, when bonding the carbon 2s and 2p shells hybridize to form a single 2spi shell as an energy minimum.
The C electron configuration given in the Six-Electron Atoms section is \s22s22p2 , and the orbital arrangement is
(13.419)
Figure imgf000180_0001
corresponding to the ground state 3P0 . The radius r6 of the 2p shell given by Eq. (10.122) is r6 = 1.20654α0 (13.420)
The energy of the carbon 2p shell is the negative of the ionization energy of the carbon atom given by Eq. (10.123). Experimentally, the energy is [12]
E(C,2p shell) = -E (ionization; C) = -11.2603 eV (13.421) The C2s atomic orbital (AO) combines with the C2p AOs to form a single 2sp3 hybridized orbital (HO) with the orbital arrangement 2sp3 state (13.422)
Figure imgf000181_0003
where the quantum numbers (£, me) are below each electron. The total energy of the state is given by the sum over the four electrons. The sum E7, (C, 2sp3 ) of calculated energies of C , C+ , C2+, and C3+ from Εqs. (10.123), (10.113-10.114), (10.68), and (10.48), respectively, is ET (c,2sp3 ) = 643921 eV + 4S.3125 eV + 2A.2762 eV + U276n eV = US.25751 eV
(13.423) which agrees well with the sum of 148.02532 eV from the experimental [6] values. The orbital-angular-momentum interactions cancel such that the energy of the Eτ (c,2sp3) is purely Coulombic. By considering that the central field decreases by an integer for each successive electron of the shell, the radius r2s 3 of the C2sp3 shell may be calculated from the Coulombic energy using Eq. (10.102):
Figure imgf000181_0001
Using Eqs. (10.102) and (13.424), the Coulombic energy ECoulomb (C,2sp3) of the outer electron of the C2,yp3 shell is
(13.425) v ;
Figure imgf000181_0004
During hybridization, one of the spin-paired 2s electrons is promoted to C2sp3 shell as an unpaired electron. The energy for the promotion is the magnetic energy given by Eq. (13.152) at the initial radius of the 2s electrons. From Eq. (10.62) with Z = 6 , the radius r3 of C2s shell is r3 = 0.84317α0 (13.426)
Using Eqs. (13.152) and (13.426), the unpairing energy is
Eimagnetic) eV (13.427)
Figure imgf000181_0002
Using Eqs. (13.425) and (13.427), the energy E[C,2sp3) of the outer electron of the C2sp3 shell is
Figure imgf000182_0001
= -14.82575 eK + 0.19086 eV (13.428)
= -14.63489 eV
The nitrogen atom's 2j? -shell electron configuration given by Eq. (10.134) is the same as that of the C2sp3 shell, and nitrogen's calculated energy of 14.61664 eV given by Eq. (10.143) is a close match with E(c,2.y/?3) . Thus, the binding should be very similar except that four bonds to hydrogen can occur with carbon.
The carbon C2sp3 electron combines with the His electron to form a molecular orbital. The proton of the H atom and the nucleus of the C atom are along the internuclear axis and serve as the foci. Due to symmetry, the other C electrons are equivalent to point charges at the origin. (See Eqs. (19-38) of Appendix IV.) Thus, the energies in the CH MO involve only the C2sp3 and His electrons. The forces are determined by these energies.
As in the case of H2 , the MO is a prolate spheroid with the exception that the ellipsoidal MO surface cannot extend into the C2sp3 HO for distances shorter than the radius of the C2sp3 shell. Thus, the MO surface comprises a prolate spheroid at the H proton that is continuous with the C2sp3 shell at the C atom whose nucleus serves as the other focus. The energy of the H2 -type ellipsoidal MO is matched to that of the C2sp3 shell. As in the case with OH and NH , the linear combination of the H2 -type ellipsoidal MO with the C2sp3 HO must involve a 25% contribution from the H2 -type ellipsoidal MO to the C2sp3 HO in order to match potential, kinetic, and orbital energy relationships. Thus, the CH MO must comprise 75% of a H2 -type ellipsoidal MO and a C2sp3 HO:
1 C2sp3 + 0.75 H2 MO → CH MO (13.429)
The force balance of the CH MO is determined by the boundary conditions that arise from the linear combination of orbitals according to Eq. (13.429) and the energy matching condition between the hydrogen and C2sp3 HO components of the MO. As in the case with OH (Eq. (13.57)), the H2 -type ellipsoidal MO comprises 75% of the CH MO; so, the electron charge density in Eq. (11.65) is given by -0.75e . The force constant k' to determine the ellipsoidal parameter c' in terms of the central force of the foci is given by Eq. (13.59). The distance from the origin to each focus c' is given by Eq. (13.60). The internuclear distance is given by Eq. (13.61). The length of the semiminor axis of the prolate spheroidal C -H -bond MO b = c is given by Eq. (13.62). The eccentricity, e , is given by Eq. (13.63). The solution of the semimajor axis a then allows for the solution of the other axes of each prolate spheroid and eccentricity of the CH MO. Since the CH MO comprises a Tf2 -type-ellipsoidal MO that transitions to the C2sp3 HO, the energy E(c,2sp3) in Eq. (13.428) adds to that of the H2 -type ellipsoidal MO to give the total energy of the CH MO. From the energy equation and the relationship between the axes, the dimensions of the CH MO are solved.
The energy components of Ve , Vp, T , and V1n are those of H2 (Eqs. (11.207-11.212)) except that they are corrected for electron hybridization. Hybridization gives rise to the C2sp3 HO-shell Coulombic energy ECmlomb (C,2sp3 ) given by Eq. (13.425). To meet the equipotential condition of the union of the H2-type-ellipsoidal-MO and the C2sp3 HO, the electron energies are normalized by the ratio of 14.8257S eF , the magnitude of Ecouhmb (C,2sp3 ) given by Eq. (13.425), and 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243). This normalizes the energies to match that of the Coulombic energy alone to meet the energy matching condition of the CH MO under the influence of the proton and the C nucleus. The hybridization energy factor CC2sp3rø is
(13.430)
Figure imgf000183_0001
The total energy E7, (CH) of the CH MO is given by the sum of the energies of the orbitals, the H2 -type ellipsoidal MO and the C2sp3 HO, that form the hybridized CH MO.
Eτ (CH) follows from by Eq. (13.74) for OH , but the energy of the C2sp3 HO given by Eq.
(13.428) is substituted for the energy of O and the H2 -type-ellipsoidal-MO energies are those of H2 (Eqs. (11.207-11.212)) multiplied by the electron hybridization factor rather than by the factor of 0.75 :
Figure imgf000184_0001
(13.431)
To match the boundary condition that the total energy of the entire the H2 -type ellipsoidal
MO is given by Eqs. (11.212) and (13.75), E7 (CH) given by Eq. (13.431) is set equal to Eq. (13.75):
-31.6353683I eF
Figure imgf000184_0002
(13.432)
From the energy relationship given by Eq. (13.432) and the relationship between the axes given by Eqs. (13.60-13.63), the dimensions of the CH MO can be solved. Substitution of Eq. (13.60) into Eq. (13.432) gives
Figure imgf000184_0003
The most convenient way to solve Eq. (13.433) is by the reiterative technique using a computer. The result to, within the round-off error with five-significant figures is α = 1.67465αn = 8.86186X 10 ,-π m (13.434)
Substitution of Eq. (13.434) into Eq. (13.60) gives c' = l.O5661αo = 5.59136X 10~π m (13.435) The internuclear distance given by multiplying Eq. (13.435) by two is
2c' = 2.11323α0 = 1.11827X 10~10 m (13.436) The experimental bond distance is [14]
2c' = 1.1198 X 10-10 m (13.437) Substitution of Eqs. (13.434-13.435) into Eq. (13.62) gives b = c = l.2992Aa0 = 6.87527 X 10"11 m (13.438)
Substitution of Eqs. (13.434-13.435) into Eq. (13.63) gives e = 0.63095 (13.439) The nucleus of the H atom and the nucleus of the C atom comprise the foci of the H2 -type ellipsoidal MO. The parameters of the point of intersection of the H2 -type ellipsoidal MO and the C2sp3 HO are given by Eqs. (13.84-13.95) and (13.261-13.270). The polar intersection angle θ' is given by Eq. (13.261) where rn = rls 3 = 0.9177Ia0 is the radius of the
C2sp3 shell. Substitution of Eqs. (13.434-13.435) into Eq. (13.261) gives
#' = 81.03° (13.440)
Then, the angle θC2s 3flO the radial vector of the C2sp3 HO makes with the internuclear axis is θc2ψ>m = 180o-81.03o = 98.97° (13.441) as shown in Figure 14.
The distance from the point of intersection of the orbitals to the internuclear axis must be the same for both component orbitals. Thus, the angle G>t = θH M0 between the internuclear axis and the point of intersection of the H2 -type ellipsoidal MO with the C2sp3 radial vector obeys the following relationship:
V sin*cV*0 = 0-9l7na° sin θc2sP^o = b s™θH2M0 (13-442) such that (13 443)
Figure imgf000185_0001
with the use of Eq. (13.441). Substitution of Eq. (13.438) into Eq. (13.443) gives 0^ = 44.24° (13.444)
Then, the distance dH M0 along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by d H2M0 = a C0S 6H2MO (13.445)
Substitution of Eqs. (13.434) and (13.444) into Eq. (13.445) gives d H2uo = l-19968α0 = 6.34845 X 10~u m (13.446)
The distance dC2s ,HQ along the internuclear axis from the origin of the C atom to the point of intersection of the orbitals is given by dc2s^o = d H2Mo - C (13.447)
Substitution of Eqs. (13.435) and (13.446) into Eq. (13.447) gives drn 3, = 0.14307^0 = 7.57090 X lO"12 m (13.448)
As shown in Eq. (13.429), the uniform charge-density in the C2sp3 HO is increased by a factor of 0.25 and the H -atom density is decreased by a factor of 0.25. Using the orbital composition of CH (Eq. (13.429)), the radii of CIs = 0.17113α0 (Eq. (10.51)) and C2sp3 = 0.9177Ia0 (Eq. (10.424)) shells, and the parameters of the CH MO given by Eqs. (13.3-13.4), (13.434-13.436), and (13.438-13.448), the dimensional diagram and charge- density of the CH MO comprising the linear combination of the H2 -type ellipsoidal MO and the C2sp3 HO according to Eq. (13.429) are shown in Figures 14 and 15, respectively.
ENERGIES OF CH
The energies of CH are given by the substitution of the semiprincipal axes (Eqs. (13.434- 13.435) and (13.438)) into the energy equations, (Eq. (13.431) and Eqs. (11.207-11.211) that are corrected for electron hybridization using Eq. (13.430):
V eV (13.449)
Figure imgf000186_0001
Figure imgf000186_0002
(13.451)
Figure imgf000186_0003
Vn, eV (13.452)
Figure imgf000186_0004
eV
Figure imgf000186_0005
(13.453) where E1. [CH) is given by Eq. (13.431) which is reiteratively matched to Eq. (13.75) within five-significant-figure round-off error. VIBRATION AND ROTATION OF CH
The vibrational energy of CH may be solved in the same manner as that of OH and NH except that the force between the electrons and the foci given by Eq. (13.102) is doubled due to electron hybridization of the two shells of carbon after Eq. (11.141). From Eqs. (13.102- 13.106) with the substitution of the CH parameters, the angular frequency of the oscillation is
(13.454)
Figure imgf000187_0001
= 5.39828 X 1014 rod I s where b is given by Eq. (13.438), 2c1 is given by Eq. (13.436), and the reduced mass of 12CH is given by:
Figure imgf000187_0002
where mp is the proton mass. Thus, during bond formation, the perturbation of the orbit determined by, an inverse-squared force results in simple harmonic oscillatory motion of the orbit, and the corresponding frequency, θ)(θ) , for 12CH given by Eqs. (11.136), (11.148), and (13.454) is
10M radians/ s (13.456)
Figure imgf000187_0003
where the reduced nuclear mass of 12CH is given by Eq.(13.455) and the spring constant, Jfc(0) , given by Eqs. (11.136) and (13.454) is jfc(0) = 449.94 iVm-1 (13.457)
The 12CH transition-state vibrational energy, Evjb (Oi) , given by Planck's equation (Eq. (11.127)) is:
Evlb (Q) = Λω = ft5.39828 X 10w rad/s = 0.35532 eV = 2865.86 cm'1 (13.458)
G) , from the experimental curve fit of the vibrational energies of 12CH is [14] ωe = 2861.6 cm'1 (13.459)
Using Eqs. (13.112-13.118) with Evιb (θ) given by Eq. (13.458) and D0 given by Eq. (13.488), the 12CH u = l → ϋ = 0 vibrational energy, Evib (l) is
Evib (l) = 0.33879 eV (2732.61 cm"1) ' (13.460) The experimental vibrational energy of 12CH using ωe and ωexe [14] according to K&P [15] is
Evib (1) = 0.33885 e V (2733 cm'1) (13.461)
Using Eq. (13.113) with Evib (l) given by Eq. (13.460) and D0 given by Eq. (13.488), the anharmonic perturbation term, ωoxo , of 12CH is G)0X0 = 66.624 cw"1 (13.462)
The experimental anharmonic perturbation term, ωoxo , of 12CH" [14] is
CO0X0 = 64.3 cm-1 (13.463)
The vibrational energies of successive states are given by Eqs. (13.458), (13.112), and (13.462). Using h given by Eq. (13.438), 2c1 given by Eq. (13.436), D0 given by Eq. (13.490), and the reduced nuclear mass of 12CD given by
Figure imgf000188_0002
(13-464) where mp is the proton mass, the corresponding parameters for deuterium carbide 12CD (Eqs. (13.102-13.121)) are
ω(0) 3.96126X 10" radians/ s (13.465)
Figure imgf000188_0001
£(0) = 449.94 iV«r1 (13.466)
Evlb (0) = hω = £3.96126 X 1014 rad/s = 0.26074 eV = 2102.97 cm'1 (13.467)
£V,A (1) = 0.25173 eV (2030.30 cmT1) (13.468) ωoxo = 36335 cm-1 (13.469) ωe , from the experimental curve fit of the vibrational energies of 12CD is [14] ωe = 2101.0 cw"1 (13.470) The experimental vibrational energy of 12CD using ωe and ωexe [14] according to K&P [15] is
Evib (l) = 0.25189 eV (2031.6 C^"1) (13.471) and the experimental anharmonic perturbation term, ωoxo , of 12CD is [14] O)0X0 = 34.7 cm~l (13.472) which match the predictions given by Eqs. (13.467), (13.468) and (13.469), respectively.
Using Eqs. (13.133-13.134) and the internuclear distance, r = 2c\ and reduced mass of 12CH given by Eqs. (13.436) and (13.455), respectively, the corresponding B0 is
Be = 14.498 cm'1 (13.473) The experimental Be rotational parameter of 12CH is [14]
£e = 14.457 cm'1 (13.474)
Using the internuclear distance, r = 2c' , and reduced mass of 12CD given by Eqs. (13.436) and (13.464), respectively, the corresponding Be is
B6 = IMl Cm-1 (13.475) The experimental Be rotational parameter of 12CD is [14]
5^ 7.808 Cm"1 (13.476)
THE DOPPLER ENERGY TERMS OF 12CH AND 12CD
The equations of the radiation reaction force of hydrogen and deuterium carbide are the same as those of the corresponding hydroxyl and hydrogen nitride radicals with the substitution of the hydrogen and deuterium carbide parameters. Using Eqs. (11.136) and (13.140-13.142), the angular frequency of the reentrant oscillation in the transition state is
(13.477)
Figure imgf000189_0001
where b is given by Eq. (13.438). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)):
Eκ = hω = £2.41759 X 1016 rod Is = 15.91299 e V (13.478) In Eq. (11.181), substitution of the total energy of CH , Eτ (CH) , (Eq. (13.432)) for Ehv , the mass of the electron, me, for M , and the kinetic energy given by Eq. (13.478) for Eκ gives the Doppler energy of the electrons for the reentrant orbit:
Figure imgf000190_0001
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of CH due to the reentrant orbit in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by
Eq. (13.479) and EKvlb , the average kinetic energy of vibration which is 1/2 of the vibrational energy of CH . The experimental 12CH ωe is 2861.6 cm"1 (0.35480 eV) [14] which matches the predicted ωe of 2865.86 cm'1 (0.35532 eV) given by Eq. (13.458). Using the predicted ωe for EKvib of the transition state, Emc (11CHj is
(13.480)
Figure imgf000190_0002
E0n (12CH) = -0.24966 βF+-(θ.35532 eV) = -0.07200 eV (13.481)
The experimental 12CD ωe is 2101.0 cm"1 (0.26049 eV) [14] which matches the predicted ωe of 2102.97 cm"1 (0.26074 eV) given by Eq. (13.467). Using Eq. (13.479) and the predicted <ue for ϋ^ of the transition state, Eos (12CD) is
EOSC (UCD) = -0.24966 eF+-(0.26074 eV) = -0.11929 eV (13.482)
TOTAL AND BOND ENERGIES OF 12CH AND 12CD
ET+OSC ( 12CH) , the total energy of the 12CH radical including the Doppler term, is given by
the sum of Eτ (CH) (Eq. (13.432)) and EBSC (12CH) given by Eq. (13.481):
Er^ {nCH) = Ve +T^Vm + Vp ^ E(c,2sp^ Eosc (-CH) = Ex (CH) + Eosc (12CH)
Figure imgf000191_0001
(13.484)
From Eqs. (13.480-13.481) and (13.483-13.484), the total energy of 12CH" is Eτ+osc ( 12CH) = -31.63537 eV + Eosc ( 12CH)
= -31.63537 eF-0.24966 eF+-(0.35532 eV) (13.485) = -31.70737 eV
where the predicted ω (Eq. (13.458)) was used for the h \ — term. E7. (12CD], the total
energy of 12CD including the Doppler term, is given by the sum of E1. (CD) = E1. (CH) (Eq. (13.432)) and E0SC ( UCD) given by Eq. (13.482):
ET+0SC ( 12CD) = -31.63537 eV + Eosc ( 12CD)
= -31.63537 eV -0.24966 eV + -(0.26074 eV) (13.486)
= -31.75462 e V
where the predicted ω (Eq. (13.467)) was used for the h I — term.
The CH bond dissociation energy, E0 (12CH) , is given by the sum of the total
energies of the C2sp3 HO and the hydrogen atom minus Eτ+osc ( 12CH)3:
ED ( 12CH) = E (C52sp3 ) + E(H) - ET+0SC ( 12CH) (13.487)
3 The hybridization energy is the difference between and E[C,2sp3 ) given by Eq. (13.428) and E[C,2p shell ) given by Eq. (13.421). Since this term adds to E[C,2p shell) to give the total energy from which £rw C2CHj is subtracted to give E0 O2CHj, it is more convenient to simply use E[C,2sp3 ^directly in Eq. (13.487). E(c,2sp3) is given by Eq. (13.428), and ED (H) is given by Eq. (13.154). Thus, the 12CH
bond dissociation energy, E^ (12CH) , given by Eqs. (13.154), (13.428), (13.485), and (13.487) is
E0 (12CH) = -(14.63489 eV + 13.59844 eV)-ET+osc (CH)
= -28.23333 eF-(-31.70737 eV) (13.488)
= 3.47404 e V The experimental 12CH" bond dissociation energy is [14]
ED (12CH) = 3.47 e V (13.489) which is a close match to that of NH as predicted based on the match between the N and C2sp3 HO energies and electron configurations.
The nCD bond dissociation energy, ED ( 12CD) , is given by the sum of the total energies of the C2sp3 HO and the deuterium atom minus Eτ+osc (CD) :
ED (nCD) = E(C,2sp") + E(D)-ET+0SC (12CD) (13.490)
E(C3ISp3) is given by Eq. (13.428), and ED (D) is given by Eq. (13.155). Thus, the 12CD
bond dissociation energy, E0 (12CD), given by Eqs. (13.155), (13.428), (13.486), and (13.490) is
ED (12CD) = -(14.63489 eV +13.603 eV) - Eτ+osc ( 12CD) = -28.23789 eV -(-31.75462 eV) (13.491)
= 3.51673 eV
The experimental 12CD bond dissociation energy is [14]
ED (12CD) = 3.52 eV (13.492)
The results of the determination of bond parameters of CH and CD are given in Table 13.1. The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
DiHYDROGEN CARBIDE (CH2)
The dihydrogen carbide radical CH2 is formed by the reaction of a hydrogen atom with a hydrogen carbide radical: CH + H → CH2 (13.493)
CH2 can be solved using the same principles as those used to solve H2O and NH2 with the exception that the carbon 2s and 2p shells hybridize to form a single 2sp3 shell as an energy minimum. Two diatomic molecular orbitals (MOs) developed in the Nature of the Chemical Bond of Ηydrogen-Type Molecules and Molecular Ions section serve as basis functions in a linear combination with two carbon 2sp3 hybridized orbitals (ΗOs) to form the MO of CH2 . The solution is very similar to that of CH except that there are two CH bonds in CH2 . >
FORCE BALANCE OF CH2
CH2 comprises two chemical bonds between carbon and hydrogen atoms. Each
C-H bond comprises two spin-paired electrons with one from an initially unpaired electron of the carbon atom and the other from the hydrogen atom. Each H -atom electron forms a H2 -type ellipsoidal MO with an unpaired C -atom electrons. However, such a bond is not possible with the outer two C electrons in their ground state since the resulting H2 -type ellipsoidal MO would have a shorter internuclear distance than the radius of the carbon 2p shell, which is not energetically stable. Thus, when bonding the carbon 2s and 2p shells hybridize to form a single 2sp3 shell as an energy minimum. The electron configuration and the energy, E(c,2sp3) , of the C2sp3 shell is given by Eqs. (13.422), and (13.428), respectively.
For each C-H bond, a C2sp3 electron combines with the Hb electron to form a molecular orbital. The proton of the H atom and the nucleus of the C atom are along each internuclear axis and serve as the foci. As in the case of H2 , each of the two C -H -bond MOs is a prolate spheroid with the exception that the ellipsoidal MO surface cannot extend into the C2sp3 HO for distances shorter than the radius of the C2sp3 shell since it is energetically unfavorable. Thus, each MO surface comprises a prolate spheroid at the H proton that is continuous with the C2sp3 shell at the C atom whose nucleus serves as the other focus. The radius and the energy of the C2sp3 shell are unchanged with bond formation. The central paramagnetic force due to spin of each C-H bond is provided by the spin-pairing force of the CH2 MO that has the symmetry of an s orbital that superimposes with the C2sp3 orbitals such that the corresponding angular momenta are unchanged.
The energies in the CH2 MO involve only each C2sp3 and each His electron with the formation of each C-H bond. The sum of the energies of the H2 -type ellipsoidal MOs is matched to that of the C2sp3 shell. As in the cases with of OH , H2O, NH , NH2 , NH3 , and CH the linear combination of each H2 -type ellipsoidal MO with the C2sp3 HO must involve a 25% contribution from the H2 -type ellipsoidal MO to the C2sp* HO in order to match potential, kinetic, and orbital energy relationships. Thus, the CH2 MO must comprise two C-H bonds with each comprising 75% of a H2 -type ellipsoidal MO and a C2sp3 HO:
[l C2sp3 + 0.75 H2 MO] + [l C2sp3 + 0.75 H2 MO] → CH2 MO (13.494)
The force balance of the CH2 MO is determined by the boundary conditions that arise from the linear combination of orbitals according to Eq. (13.494) and the energy matching condition between the hydrogen and C2^p3 HO components of the MO.
The force constant k ' to determine the ellipsoidal parameter c ' of the each H2 -type- ellipsoidal-MO component of the CH2 MO in terms of the central force of the foci is given by Eq. (13.59). The distance from the origin of each C-H-bond MO to each focus c' is given by Eq. (13.60). The internuclear distance is given by Eq. (13.61). The length of the semiminor axis of the prolate spheroidal C-H -bond MO b = c is given by Eq. (13.62). The eccentricity, e , is given by Eq. (13.63). The solution of the semimajor axis a then allows for the solution of the other axes of each prolate spheroid and eccentricity of each C-H -bond MO. Since each of the two prolate spheroidal C -H-bond MOs comprises a H2-type- ellipsoidal MO that transitions to the C2sp3 HO, the energy E(C,2sp3) in Eq. (13.428) adds to that of the two corresponding H2 -type ellipsoidal MOs to give the total energy of the CH2 MO. From the energy equation and the relationship between the axes, the dimensions of the CH2 MO are solved. The energy components of Ve , Vp, T , and V1n are twice those of CH corresponding to the two C-H bonds. Since the each prolate spheroidal H2 -type MO transitions to the C2sp3 HO and the energy of the C2sp3 shell must remain constant and equal to the E(C,2sp3) given by Eq. (13.428), the total energy Eτ (CH2) of the CH2 MO is given by the sum of the energies of the orbitals corresponding to the composition of the linear combination of the C2sp3 HO and the two H2 -type ellipsoidal MOs that forms the CH2 MO as given by Eq. (13.494). Using Eq. (13.431), E7, (CH2) is given by
ET (CH2) = ET + E(c,2sp3)
Figure imgf000195_0001
The two hydrogen atoms and the hybridized carbon atom can achieve an energy minimum as a linear combination of two H2 -type ellipsoidal MOs each having the proton and the carbon nucleus as the foci. Hybridization gives rise to the C2sp3 HO-shell Coulombic energy ^coulomb (C,2sp3) given by Eq. (13.425). To meet the equipotential condition of the union of the H2 -type-ellipsoidal-MO and the C2.sp3 HO, the electron energies in Eq. (13.495) were normalized by the ratio of 14.82575 eV , the magnitude of ECoulomb (c,2sp3 ) given by Eq. (13.425), and 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243). The factor given by Eq. (13.430) normalized the energies to match that of the Coulombic energy alone to meet the energy matching condition of each C-H -bond MO under the influence of the proton and the C nucleus. Each C-H- bond MO comprises the same C2sp3 shell having its energy normalized to that of the Coulombic energy between the electron and a charge of +e at the carbon focus of the CH2 MO. Thus, the energy of the CH2 MO is also given by the sum of that of the two H2 -type ellipsoidal MOs given by Eq. (11.212) minus the Coulombic energy, Ecouhmb (H) = -13.605804 eV , of the redundant +e of the linear combination:
Figure imgf000195_0002
= 2(-31.63536831 eF)-(-13.605804 eF) (13.496)
= -49.66493 e V E7 [CH2) given by Eq. (13.495) is set equal to two times the energy of the H2 -type ellipsoidal MO minus the Coulombic energy of H given by Eq. (13.496):
Er (CH2) = - eV = -49.66493 eV
4πε0c'
Figure imgf000196_0001
(13.497) From the energy relationship given by Eq. (13.497) and the relationship between the axes given by Eqs. (13.60-13.63), the dimensions of the CH2 MO can be solved. Substitution of Eq. (13.60) into Eq. (13.497) gives
Figure imgf000196_0002
The most convenient way to solve Eq. (13.498) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is β = 1.6401 Oa0 = 8.67903 AT 10"11 m (13.499) Substitution of Eq. (13.499) into Eq. (13.60) gives c' = l.O4566βo = 5.53338 X 10"11 m (13.500) The internuclear distance given by multiplying Eq. (13.500) by two is
2c' = 2.09132α0 = 1.10668 X 10"10 m (13.501) The experimental bond distance is [38]
2c' = l.l l l X 10-10 m (13.502) Substitution of Eqs. (13.499-13.500) into Eq. (13.62) gives b = c = l .26354αn = 6.68635 X 10~n m (13.503) Substitution of Eqs. (13.499-13.500) into Eq. (13.63) gives e = 0.63756 (13.504) The nucleus of the H atom and the nucleus of the C atom comprise the foci of each H2 - type ellipsoidal MO. The parameters of the point of intersection of each H2 -type ellipsoidal
MO and the C2sp3 HO are given by Eqs. (13.84-13.95), (13.261-13.270), and (13.440- 13.448). The polar intersection angle θ' is given by Eq. (13.261) where rn = r2spZ = 0.91771α0 is the radius of the C2sp3 shell. Substitution of Eqs. (13.499-13.500) into Eq. (13.261) gives 0' = 84.54° (13.505)
Then, the angle θC2spiH0 the radial vector of the C2sp3 HO makes with the internuclear axis is θC2sp,H0 = 180° -84.54° = 95.46° (13.506) as shown in Figure 14. The angle ΘH^UO between the internuclear axis and the point of intersection of each H2 -type ellipsoidal MO with the C2sp3 radial vector given by Eqs. (13.442-13.443), (13.503), and (13.506) is
ΘHIMO = 46.30° (13.507)
Then, the distance dHiM0 along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals given by Eqs. (13.445), (13.499), and (13.507) is dHiM0 = 1.13305α0 = 5.99585 X IQT11 m (13.508)
The distance d 3 along the internuclear axis from the origin of the C atom to the point of intersection of the orbitals given by Eqs. (13.447), (13.500), and (13.508) is dc2spΗo = °-O8739flo = 4.62472 X 10~12 m (13.509)
As shown in Eq. (13.494), the uniform charge-density in the C2sp3 HO is increased by a factor of 0.25 and the H -atom density is decreased by a factor of 0.25 for by each C -H bond. Using the orbital composition of CH2 (Eq. (13.494)), the radii of
CIs = 0.17113α0 (Eq. (10.51)) and C2sp3 = 0.9177 Ia0 (Eq. (10.424)) shells, and the parameters of the CH2 MO given by Eqs. (13.3-13.4), (13.499-13.501), and (13.503-13.509), the charge-density of the CH2 MO comprising the linear combination of two C- H -bond MOs is shown in Figure 16. Each C-H-bond MO comprises a H2 -type ellipsoidal MO and a C2sp3 HO having the dimensional diagram shown in Figure 14.
ENERGIES OF CH2
The energies of CH2 are two times those of CH and are given by the substitution of the semiprincipal axes (Eqs. (13.499-13.500) and (13.503)) into the energy equations Eq. (13.495) and (Eqs. (13.449-13.452)) that are multiplied by two:
Figure imgf000198_0001
(13.514) where E7, (CH2) is given by Eq. (13.495) which is reiteratively matched to Eq. (13.496) within five-significant-figure round-off error.
VIBRATION OF CH2
The vibrational energy levels of CH2 may be solved as two equivalent coupled harmonic oscillators by developing the Lagrangian, the differential equation of motion, and the eigenvalue solutions [2] wherein the spring constants are derived from the central forces as given in the Vibration of Ηydrogen-Type Molecular Ions section and the Vibration of Ηydrogen-Type Molecules section.
THE DOPPLER ENERGY TERMS OF 12CH. 2
The reentrant oscillation of hybridized orbitals in the transition state is not coupled. Therefore, the equations of the radiation reaction force of dihydrogen and dideuterium carbide are the same as those of the corresponding hydrogen carbide radicals with the substitution of the dihydrogen and dideuterium carbide parameters. Using Eqs. (11.136) and (13.140-13.142), the angular frequency of the reentrant oscillation in the transition state is
Figure imgf000198_0002
where b is given by Eq. (13.503). The kinetic energy, Eκ , is given by Planck's equation
(Eq. (11.127)):
Eκ = hω = £2.52077 X l O16 radls ^ 16.59214 eV (13.516)
In Eq. (11.181), substitution of Ex (H2) (Eqs. (11.212) and (13.75)), the maximum total energy of each H2 -type MO3 for Ehv , the mass of the electron, me, for M , and the kinetic energy given by Eq. (13.516) for Eκ gives the Doppler energy of the electrons of each of the two bonds for the reentrant orbit:
Figure imgf000199_0001
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of CH2 due to the reentrant orbit of each bond in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by Eq. (13.517) and EKvib , the average kinetic energy of vibration which is 1/2 of the vibrational energy of each C -H bond. Using ωe given by Eq. (13.458) for EKvjb of the transition state having two independent bonds, E 'osc ( 12CH2 ) per bond is
E'osc (nCH2) = ED +EKvib = ED +U Jt (13.518)
E 'osύ (12CH2) = -0.25493 eF + -(0.35532 eV) = -0.07727 eV (13.519)
Given that the vibration and reentrant oscillation is for two C-H bonds, Eosc ( 12CH2 ) , is:
Figure imgf000199_0002
= 2[ -0.25493 eF+-(0.35532 eF) ] (13.520)
= -0.15454 eV TOTAL AND BOND ENERGIES OF 12CH2
Eτ+osc (12CH2), the total energy of the 12CH2 radical including the Doppler term, is given by
the sum of Eτ (CH2 ) (Eq. (13.497)) and Eosc ( 12CH2 ) given by Eq. (13.520):
ETH>sc {CH2) = Ve +T + Vm +Vp + E(c,2sp3) + E04UCH2)
(13.521) = ET (CH2) + E0SC ( 12CH2)
= -49.66493 eV-2\ 0.25493 eV--hJ—
From Eqs. (13.518-13.522), the total energy of 12CH2 is Eτ+osc ( 12CH2 ) = -49.66493 e V + Eosc ( 12CH2 )
= -49.66493 eV -li 0.25493 eF —(0.35532 eF) j (13.523)
= -49.81948 e V
where ω given by Eq. (13.458) was used for the term.
Figure imgf000200_0002
12CH2 has the same electronic configuration as 14iVH . The dissociation of the bond of the dihydrogen carbide radical forms a free hydrogen atom with one unpaired electron and a C2.sp3 HO with three unpaired electrons as shown in Eq. (13.422) wherein the magnetic moments cannot all cancel. Thus, the bond dissociation of 12CH2 gives rise to 12CH with the same electronic configuration as N as given by Eq. (10.134). The N configuration is more stable than H as shown in Eqs. (10.141-10.143). The lowering of the energy of the reactants decreases the bond energy. The total energy of carbon is reduced by the energy in the field of the two magnetic dipoles given by Eq. (7.46) and Eq. (13.424):
(13.524)
Figure imgf000200_0003
The CH2 bond dissociation energy, ED (12CH2) , is given by the sum of the total energies of
the CH radical and the hydrogen atom minus the sum of Eτ+osc (12CH2) and E(magnetic) :
ED (12CH2) = E(12CH) + E(H)- ET,0SC (l2CH2)-E(magmtic) (13.525)
where Ex(12CH) is given by the sum of the energies of the C2sp3 HO, E(c,2.sp3) given by
Eq. (13.428), ED (H) given by Eq. (13.154), and the negative of the bond energy of 12CH given by Eq. (13.489):
E(12CH) = -13.59844 eF-14.63489 eF-3.47 eV = -31.70333 eV (13.526)
Thus, the 12CH2 bond dissociation energy, ED (12CH2) , given by Eqs. (13.154), and (13.523- 13.526) is
ED ( 12CH2 ) = - (31.70333 e V + 13.59844 e V) - (ET+OSC ( 12CH2 ) + E {magnetic)) = -45.30177 eF-(-49.81948 eF + 0.14803 eF)
= 4.36968 eV (13.527) The experimental 12CH2 bond dissociation energy is [39]
ED (12CH2) = 4.33064 eV (13.528)
BOND ANGLE OF 12CH2
The CH2 MO comprises a linear combination of two C-H -bond MOs. Each C-H -bond MO comprises the superposition of a H2 -type ellipsoidal MO and the C2sp3 HO with a relative charge density of 0.75 to 1.25; otherwise, the C2sp3 shell is unchanged. A bond is also possible between the two H atoms of the C-H bonds. Such H-H bonding would decrease the C-H bond strength since electron density would be shifted from the C - H bonds to the H-H bond. Thus, the bond angle between the two C-H bonds is determined by the condition that the total energy of the H2 -type ellipsoidal MO between the terminal H atoms of the C-H bonds is zero. From Eqs. (11.79) and (13.228), the distance from the origin to each focus of the H-H ellipsoidal MO is
Figure imgf000201_0001
The internuclear distance from Eq. (13.229) is
The length of the semiminor axis of the prolate spheroidal H-H MO b = c is given by Eq. (13.62).
The bond angle of CH2 is derived by using the orbital composition and an energy matching factor as in the case with NH2 and NH3.. Since the two H2 -type ellipsoidal MOs comprise 75% of the H electron density of H2 and the energy of each H2 -type ellipsoidal
MO is matched to that of the C2sp3 HO; the component energies and the total energy E7, of the H-H bond are given by Eqs. (13.67-13.73) except that Ve , T , and V1n are corrected for the hybridization-energy-matching factor of 0.91771 given by Eq. (13.430). Substitution of Eq. (13.529) into Eq. (13.233) with the hybridization factor gives
Figure imgf000202_0002
(13.531)
From the energy relationship given by Eq. (13.531) and the relationship between the axes given by Eqs. (13.529-13.530) and (13.62-13.63), the dimensions of the H-H MO can be solved.
The most convenient way to solve Eq. (13.531) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is a = 5.1500α0 = 2.7253 X 10"10 m (13.532)
Substitution of Eq. (13.532) into Eq. (13.529) gives c' = 1.6047 O0 = 8.4916 X 10"11 m (13.533)
The internuclear distance given by multiplying Eq. (13.533) by two is
2c' = 3.2O94cto = l.6983 X 10"10 m (13.534)
Substitution of Eqs. (13.532-13.533) into Eq. (13.62) gives b = c = 4.8936an = 2.5896 X 10"lu m (13.535) Substitution of Eqs. (13.532-13.533) into Eq. (13.63) gives e = 0.3116 (13.536)
Using, 2c'H_H (Eq. (13.534)), the distance between the two H atoms when the total energy of the corresponding MO is zero (Eq. (13.531)), and 2c\_H (Eq. (13.501)), the internuclear distance of each C - H bond, the corresponding bond angle can be determined from the law of cosines. Using, Eq. (13.242), the bond angle θ between the C - H bonds is
θ = cos -1 = 100.22° (13.537)
Figure imgf000203_0001
The experimental angle between the C- H bonds is [38]
0 = 102.4° (13.538) The results of the determination of bond parameters of CH2 are given in Table 13.1.
The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
METHYL RADICAL (CH3)
The methyl radical CH3 is formed by the reaction of a hydrogen atom with a dihydrogen carbide radical:
CH2 + H -> CH3 (13.539)
CH3 can be solved using the same principles as those used to solve and NH3 with the exception that the carbon 2s and 2p shells hybridize to form a single 2spl shell as an energy minimum. Three diatomic molecular orbitals (MOs) developed in the Nature of the Chemical Bond of Ηydrogen-Type Molecules and Molecular Ions section serve as basis functions in a linear combination with three carbon 2sp3 hybridized orbitals (ΗOs) to form the MO of CH3. The solution is very similar to that of CH2 except that there are three CH bonds in CH3. FORCE BALANCE OF CH3
CH3 comprises three chemical bonds between carbon and hydrogen atoms. Each C-H bond comprises two spin-paired electrons with one from an initially unpaired electron of the carbon atom and the other from the hydrogen atom. Each H -atom electron forms a H2 -type ellipsoidal MO with an unpaired C -atom electrons. However, such a bond is not possible with the outer two C electrons in their ground state since the resulting H2 -type ellipsoidal MO would have a shorter internuclear distance than the radius of the carbon 2p shell which is not energetically stable, and only two electrons are unpaired. Thus, when bonding the carbon 2s and 2p shells hybridize to form a single 2sp* shell as an energy minimum. The electron configuration and the energy, E\ C,2spl\ , of the C2sp3 shell is given by Eqs. (13.422), and (13.428), respectively.
For each C-H bond, a C2sp3 electron combines with the His electron to form a molecular orbital. The proton of the H atom and the nucleus of the C atom are along each internuclear axis and serve as the foci. As in the case of H2 , each of the three C-H -bond MOs is a prolate spheroid with the exception that the ellipsoidal MO surface cannot extend into C2spl HO for distances shorter than the radius of the C2sp3 shell since it is energetically unfavorable. Thus, each MO surface comprises a prolate spheroid at the H proton that is continuous with the C2sp3 shell at the C atom whose nucleus serves as the other focus. The radius and the energy of the C2sp3 shell are unchanged with bond formation. The central paramagnetic force due to spin of each C -H bond is provided by the spin-pairing force of the CH3 MO that has the symmetry of an « orbital that superimposes with the C2sp3 orbitals such that the corresponding angular momenta are unchanged.
The energies in the CH3 MO involve only each C2.yp3 and each His electron with the formation of each C -H bond. The sum of the energies of the H2 -type ellipsoidal MOs is matched to that of the C2sp3 shell. As in the cases with of OH , H2O , NH , NH2 , NH3 , CH , and CH2 the linear combination of each H2 -type ellipsoidal MO with the C2sp3 HO must involve a 25% contribution from the H2 -type ellipsoidal MO to the C2,sp3 HO in order to match potential, kinetic, and orbital energy relationships. Thus, the CH3 MO must comprise three C-H bonds with each comprising 75% of a H2 -type ellipsoidal MO and a
C2sp3 HO:
3[l C2sp3 + 0.75 H2 Mθ] → CH3 MO (13.540)
The force balance of the CH3 MO is determined by the boundary conditions that arise from the linear combination of orbitals according to Eq. (13.540) and the energy matching condition between the hydrogen and C2sp3 HO components of the MO.
The force constant k ' to determine the ellipsoidal parameter c ' of the each H2 -type- ellipsoidal-MO component of the CH3 MO in terms of the central force of the foci is given by Eq. (13.59). The distance from the origin of each C -H -bond MO to each focus c' is given by Eq. (13.60). The internuclear distance is given by Eq. (13.61). The length of the semiminor axis of the prolate spheroidal C -H -bond MO b = c is given by Eq. (13.62). The eccentricity, e , is given by Eq. (13.63). The solution of the semimajor axis a then allows for the solution of the other axes of each prolate spheroid and eccentricity of each C -H-bond MO. Since each of the three prolate spheroidal C-H -bond MOs comprises a H2 -type- ellipsoidal MO that transitions to the C2sp3 HO, the energy E(c,2sp3) in Eq. (13.428) adds to that of the three corresponding H2 -type ellipsoidal MOs to give the total energy of the CH3 MO. From the energy equation and the relationship between the axes, the dimensions of the CH3 MO are solved.
The energy components of Ve , Vp, T , and V1n are three times those of CH corresponding to the three C-H bonds. Since the each prolate spheroidal H2 -type MO transitions to the C2sp3 HO and the energy of the C2sp3 shell must remain constant and equal to the E(c,2sρ3) given by Eq. (13.428), the total energy Eτ (CH3) of the CH3 MO is given by the sum of the energies of the orbitals corresponding to the composition of the linear combination of the C2sp3 HO and the three H2 -type ellipsoidal MOs that forms the CH3
MO as given by Eq. (13.540). Using Eq. (13.431), Eτ (CH3) is given by ET (CH3) = ET + E(C,2sp3)
Figure imgf000206_0001
The three hydrogen atoms and the hybridized carbon atom can achieve an energy minimum as a linear combination of three H2 -type ellipsoidal MOs each having the proton and the carbon nucleus as the foci. Hybridization gives rise to the C2sp3 HO-shell Coulombic energy ECoulomb (c,2sp3) given by Eq. (13.435). To meet the equipotential condition of the union of the H2 -type-ellipsoidal-MO and the C2sp3 HO, the electron energies in Eqs. (13.431), (13.495), and (13.541) were normalized by the ratio of 14.82575 eV , the magnitude of ECoulomb [c,2spl) given by Eq. (13.425), and 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243). The factor given by Eq. (13.430) normalized the energies to match that of the Coulombic energy alone to meet the energy matching condition of each C-H -bond MO under the influence of the proton and the C nucleus. Each C -H -bond MO comprises the same C2sp3 shell having its energy normalized to that of the Coulombic energy between the electron and a charge of +e at the carbon focus of the CH3 MO. Thus, the energy of the CH3 MO is also given by the sum of that of the three H2 -type ellipsoidal MOs given by Eq. (11.212) minus two times the Coulombic energy, ECoulomb (H) = -13.605804 eV , of the two redundant +e 's of the linear combination:
ET (3H2 -2H) = -
Figure imgf000206_0002
= 3(-31.63536831 eF)-2(-13.605804 eF) = -67.69450 eF (13.542) E7 (CH3) given by Eq. (13.541) is set equal to three times the energy of the H2-type ellipsoidal MO minus two times the Coulombic energy of H given by Eq. (13.542):
89 eV = -67.69450 eV
Figure imgf000206_0003
(13.543) From the energy relationship given by Eq. (13.543) and the relationship between the axes given by Eqs. (13.60-13.63), the dimensions of the CH3 MO can be solved. Substitution of Eq. (13.60) into Eq. (13.543) gives
Figure imgf000207_0001
The most convenient way to solve Eq. (13.544) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is α = 1.62893fl0 = 8.61990 X lO"11 m (13.545) Substitution of Eq. (13.545) into Eq. (13.60) gives c' = 1.04209α0 = 5.51450 X 10"11 m (13.546) The internuclear distance given by multiplying Eq. (13.546) by two is
2c' = 2.08418α0 = 1.10290 X 10"10 m (13.547) The experimental bond distance is [38]
2c' = 1.079 X lO"10 m (13.548) Substitution of Eqs. (13.545-13.546) into Eq. (13.62) gives b = c = 1.25198a, = 6.62518 X 10"n m (13.549)
Substitution of Eqs. (13.545-13.546) into Eq. (13.63) gives e = 0.63974 (13.550) The nucleus of the H atom and the nucleus of the C atom comprise the foci of each H2 - type ellipsoidal MO. The parameters of the point of intersection of each H2 -type ellipsoidal MO and the C2sp3 HO are given by Eqs. (13.84-13.95), (13.261-13.270), and (13.434- 13.442). The polar intersection angle θ' is given by Eq. (13.261) where rn = r2ψ3 = 0.91771α0 is the radius of the C2sp3 shell. Substitution of Eqs. (13.545-13.546) into Eq. (13.261) gives 0' = 85.65° (13.551) Then, the angle θ Clsp 3'HO the radial vector of the C2sp3 HO makes with the internuclear axis is θ ClSp1HO = 180°-85.65° = 94.35° (13.552) as shown in Figure 14. The angle ΘHiU0 between the internuclear axis and the point of intersection of each H2 -type ellipsoidal MO with the C2sp3 radial vector given by Eqs. (13.442-13.443), (13.549), and (13.552) is
ΘH%MO = 46.96° (13.553) Then, the distance dH uo along the internuclear axis from the origin of H2 -type ellipsoidal
MO to the point of intersection of the orbitals given by Eqs. (13.445), (13.545), and (13.553) is d HτMo = l.H 172α0 = 5.88295 X 1(T11 m (13.554)
The distance dC2s 3ff0 along the internuclear axis from the origin of the C atom to the point of intersection of the orbitals given by Eqs. (13.447), (13.546), and (13.554) is dm C 2ιSp ι Mm(J = 0.06963a, = 3.68457 X 10"12 m (13.555)
As shown in Eq. (13.540), the uniform charge-density in the C2sp3 HO is increased by a factor of 0.25 and the H -atom density is decreased by a factor of 0.25 for by each C -H bond. Using the orbital composition of CH3 (Eq. (13.540)), the radii of CIJ = 0.17113α0 (Eq. (10.51)) and C2sp3 = 0.9177 Ia0 (Eq. (10.424)) shells, and the parameters of the CH3 MO given by Eqs. (13.3-13.4), (13.545-13.547), and (13.549-13.555), the charge-density of the CH3 MO comprising the linear combination of three C-H -bond MOs is shown in Figure 17. Each C-H-bond MO comprises a H2 -type ellipsoidal MO and a C2sp3 HO having the dimensional diagram shown in Figure 14.
ENERGIES OF CH3
The energies of CH3 are three times those of CH and are given by the substitution of the semiprincipal axes (Eqs. (13.545-13.546) and (13.549)) into the energy equations Eq. (13.541) and (Eqs. (13.449-13.452)) that are multiplied by three:
Ve -108.94944 eV (13.556)
Figure imgf000208_0001
Vp 39.16883 eV (13.557)
Figure imgf000208_0002
T eV (13.558)
Figure imgf000209_0001
Figure imgf000209_0002
EJCH3) eF = -67.69444 eK
Figure imgf000209_0003
(13.560) where Eτ (CH3) is given by Eq. (13.541) which is reiteratively matched to Eq. (13.542) within five-significant-figure-round-off-error.
viBRATiON OF CH3
The vibrational energy levels of CH3 may be solved as three equivalent coupled harmonic oscillators by developing the Lagrangian, the differential equation of motion, and the eigenvalue solutions [2] wherein the spring constants are derived from the central forces as given in the Vibration of Ηydrogen-Type Molecular Ions section and the Vibration of Ηydrogen-Type Molecules section.
THE DOPPLER ENERGY TERMS OF 12CH
The reentrant oscillation of hybridized orbitals in the transition state is not coupled. Therefore, the equations of the radiation reaction force of methyl radical are the same as those of the corresponding hydrogen carbide radicals with the substitution of the methyl radical parameters. Using Eqs. (11.136) and (13.140-13.142), the angular frequency of the reentrant oscillation in the transition state is
Q.15e2 4πεnb3 ω = \ ?— = 2.55577 X 1016 rad/s (13.561) me where b is given by Eq. (13.549). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)):
Eκ = hω = h2.55577X 1016 rad/s = 16.82249 eV (13.562) In Eq. (11.181), substitution of E7- (H2) (Eqs. (11.212) and (13.75)), the maximum total energy of each H2 -type MO, for E1n, , the mass of the electron, me , for M , and the kinetic energy given by Eq. (13.562) for Eκ gives the Doppler energy of the electrons of each of the three bonds for the reentrant orbit:
(13.563)
Figure imgf000210_0001
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of CH3 due to the reentrant orbit of each bond in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by Eq. (13.563) and EKvώ , the average kinetic energy of vibration which is 1/2 of the vibrational energy of each C -H bond. Using ωe given by Eq. (13.458) for EKvώ of the transition state having three independent bonds, E 'osc ( 12CH3 ) per bond is
(13.564)
Figure imgf000210_0002
E'osc (12CH3 ) = -0.25670 eF+-(0.35532 eV) = -0.07904 eV (13.565)
Given that the vibration and reentrant oscillation is for three C -H bonds, Eosc ( 12CH3 j , is:
(13.566)
Figure imgf000210_0003
= -0.23711 eV
TOTAL AND BOND ENERGIES OF 12CH3
Eτ+Osc (12CH3) , the total energy of the 12CH3 radical including the Doppler term, is given by the sum of E7. (CH3) (Eq. (13.543)) and Eosc (12CH3) given by Eq. (13.566):
Eτ,oΛCH3) = Ve +T + Vm +Vp + E(C,2sp3) + EOSc (UCH,)
— (13.567)
= £r(cH3)+E412cH3)
Figure imgf000211_0001
From Eqs. (13.564-13.568), the total energy of 12CH3 is ET+OSC ( 12CH3 ) = -67.69450 eV + Eosc ( 12CH3 )
= -67.69450 eVs( 0.25670 <?F--(0.35532 eF) (13.569)
= -67.93160 eV
where ω given by Eq. (13.458) was used for the term.
Figure imgf000211_0002
5 The CH3 bond dissociation energy, ED ( 12CH3 ) , is given by the sum of the total
energies of the CH2 radical and the hydrogen atom minus Eτ+osc ( 12CH3 j :
ED ( 12CH3 ) = E( 12CH2)+ E(H) - ET+OSC (12CH3) (13.570)
where E7-(12CH2) is given by the sum of the energies of the C2sp3 HO, E(C,2.sp3) given by Eq. (13.428), 2E0 (H) given by Eq. (13.154), and the negative of the bond energies of 10 12CH given by Eq. (13.489) and 12CH2 given by Eq. (13.528):
E(12CH2) = 2(-13.59844 eF)-14.63489 eF-3.47 eK-4.33064 eF = -49.63241 eF (13.571)
Thus, the 12CH3 bond dissociation energy, ED (12CH3) , given by Εqs. (13.154), and (13.569- 13.571) is
En (12CH3) = -(-49.63241 eV- 13.59844 eV)-ET+0SC (12CH3)
15 = -63.23085eF-(-67.93160 eF) (13.572)
= 4.70075 eV
The experimental 12CH3 bond dissociation energy is [40]
ED ( 12CH3) = 4.72444 e V (13.573) BOND ANGLE OF 12CH3
Using, 2c \_H (Eq. (13.534)), the distance between the two H atoms when the total energy of the corresponding MO is zero (Eq. (13.531)), and 2c'c_H , the internuclear distance of each C-H bond (Eq. (13.547)), the corresponding bond angle can be determined from the law of cosines. Using Eq. (13.537), the bond angle θ between the C-H bonds is
Figure imgf000212_0001
= cos"1 (-0.18560) (13.574)
= 100.70°
The CH3 radical has a pyramidal structure with the carbon atom along the z-axis at the apex and the hydrogen atoms at the base in the xy-plane. The distance dorigin_H from the origin to the nucleus of a hydrogen atom given by Eqs. (13.534) and (13.412) is
<,„,„_„ = 1.85293α0 (13.575)
The height along the z-axis of the pyramid from the origin to C nucleus dhejght given by Eqs. (13.414), (13.547), and (13.575) is
«/^ = 0.95418^ (13.576) The angle θv of each C-H bond from the z-axis given by Eqs. (13.416), (13.575), and (13.576) is θv = 62.75° (13.577)
The CH3 MO shown in Figure 17 was rendered using these parameters.
The results of the determination of bond parameters of CH3 are given in Table 13.1. The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
METHANE MOLECULE (CH4) The methane molecule CH4 is formed by the reaction of a hydrogen atom with a methyl radical:
CH3 + H -» cH 4 (13.578) CH4 can be solved using the same principles as those used to solve and CH3 wherein the carbon 2s and 2p shells hybridize to form a single 2sp3 shell as an energy minimum. Four diatomic molecular orbitals (MOs) developed in the Nature of the Chemical Bond of
Ηydrogen-Type Molecules and Molecular Ions section serve as basis functions in a linear combination with four carbon 2sp3 hybridized orbitals (ΗOs) to form the MO of CH4 . The solution is very similar to that of CH3 except that there are four CH bonds in CH4 , Methane is the simplest hydrocarbon that can be solved using the results for CH3. From the solution of CH2 as well as CH3 , more complex hydrocarbons can be solved using these radical as basis elements with bonding between the C2sp3 hybridized carbons.
FORCE BALANCE OF CH4
CH4 comprises four chemical bonds between carbon and hydrogen atoms. Each
C-H bond comprises two spin-paired electrons with one from an initially unpaired electron of the carbon atom and the other from the hydrogen atom. Each H -atom electron forms a H2 -type ellipsoidal MO with an unpaired C -atom electrons. However, such a bond is not possible with the outer two C electrons in their ground state since the resulting H2 -type ellipsoidal MO would have a shorter internuclear distance than the radius of the carbon 2p shell which is not energetically stable, and only two electrons are unpaired. Thus, when bonding the carbon 2s and 2p shells hybridize to form a single 2sp3 shell as an energy minimum. The electron configuration and the energy, E(C,2sp3) , of the C2sp3 shell is given by Eqs. (13.422), and (13.428), respectively.
For each C-H bond, a C2sp3 electron combines with the HIs electron to form a molecular orbital. The proton of the H atom and the nucleus of the C atom are along each internuclear axis and serve as the foci. As in the case of H2 , each of the four C-H -bond MOs is a prolate spheroid with the exception that the ellipsoidal MO surface cannot extend into the C2sp3 HO for distances shorter than the radius of the C2sp3 shell since it is energetically unfavorable. Thus, each MO surface comprises a prolate spheroid at the H proton that can be solve as being continuous with the C2sp3 shell at the C atom whose nucleus serves as the other focus. The radius and the energy of the C2sp3 shell are unchanged with bond formation. The central paramagnetic force due to spin of each C- H bond is provided by the spin-pairing force of the CH4 MO that has the symmetry of an s orbital that superimposes with the C2sp3 orbitals such that the corresponding angular momenta are unchanged. The energies in the CH4 MO involve only each C2sp3 and each His electron with the formation of each C-H bond. The sum of the energies of the H2 -type ellipsoidal MOs is matched to that of the C2sp3 shell. As in the cases with of OH , H2O , NH , NH2 , NH3 , CH , CH2 , and CH3 the CH4 , the CH4 MO must comprise four C- H bonds with each having 75% of a H2 -type ellipsoidal MO and a C2,sp3 HO in a linear combination in order to match potential, kinetic, and orbital energy relationships:
4[l C2sp3 + 0.75 H2 MO\~ ~> CH^ MO (13.579)
The force balance of the CH4 MO is determined by the boundary conditions that arise from the linear combination of orbitals according to Eq. (13.579) and the energy matching condition between the hydrogen and C2sp3 HO components of the MO.
The force constant k{ to determine the ellipsoidal parameter c' of the each H2-type- ellipsoidal-MO component of the CH4 MO in terms of the central force of the foci is given by Eq. (13.59). The distance from the origin of each C- H-bond MO to each focus c' is given by Eq. (13.60). The intemuclear distance is given by Eq. (13.61). The length of the semiminor axis of the prolate spheroidal C-H -bond MO b = c is given by Eq. (13.62). The eccentricity, e , is given by Eq. (13.63). The solution of the semimajor axis a then allows for the solution of the other axes of each prolate spheroid and eccentricity of each C- H -bond MO. Since each of the four prolate spheroidal C -H-bond MOs comprises a H2-type- ellipsoidal MO that transitions to the C2spl HO, the energy E (c, 2sps ) in Eq. (13.428) adds to that of the four corresponding H2 -type ellipsoidal MOs to give the total energy of the CH4 MO. From the energy equation and the relationship between the axes, the dimensions of the CH4 MO are solved. The energy components of Ve > Vp, T, and V111 are four times those of CH corresponding to the four C-H bonds. Since the each prolate spheroidal H2 -type MO transitions to the C2sp3 HO and the energy of the C2sp3 shell must remain constant and equal to the E(c,2sp3) given by Eq. (13.428), the total energy Eτ (CH4) of the CH4 MO is given by the sum of the energies of the orbitals corresponding to the composition of the linear combination of the C2sp3 HO and the four H2 -type ellipsoidal MOs that forms the CH4
MO as given by Eq. (13.579). Using Eq. (13.431), E7 (CH4) is given by
ET (CHA) = ET +E(C,2sp3)
(13.580)
Figure imgf000215_0001
The four hydrogen atoms and the hybridized carbon atom can achieve an energy minimum as a linear combination of four H2 -type ellipsoidal MOs each having the proton and the carbon nucleus as the foci. Hybridization gives rise to the C2sp3 HO-shell Coulombic energy ^coulomb (C,2sp3) given by Eq. (13.435). To meet the equipotential condition of the union of the H2 -type-ellipsoidal-MO and the C2sp3 HO, the electron energies in Eqs. (13.431),
(13.495), (13.541), and (13.580) were normalized by the ratio of 14.82575 eV , the magnitude of ECoulomb (C,2sp3) given by Eq. (13.425), and 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243). The factor given by Eq. (13.430) normalized the energies to match that of the Coulombic energy alone to meet the energy matching condition of each C -H -bond MO under the influence of the proton and the C nucleus. Each C -H -bond MO comprises the same C2sp3 shell having its energy normalized to that of the Coulombic energy between the electron and a charge of +e at the carbon focus of the CH4 MO. Thus, the energy of the CH4 MO is also given by the sum of that of the four H2 -type ellipsoidal MOs given by Eq. (11.212) minus three times the Coulombic energy, ECmlomb (H) = -13.605804 eV , of the three redundant +e 's of the linear combination:
Figure imgf000216_0001
= 4(-31.63536831 eF)-3(-13.605804 eF) (13.581) = -85.72406 eV
E1. (CH4) given by Eq. (13.580) is set equal to four times the energy of the H2 -type ellipsoidal MO minus three times the Coulombic energy of H given by Eq. (13.581):
ET (CH4) = - -85.72406 eV
Figure imgf000216_0002
(13.582)
From the energy relationship given by Eq. (13.582) and the relationship between the axes given by Eqs. (13.60-13.63), the dimensions of the CH4 MO can be solved. Substitution of Eq. (13.60) into Eq. (13.543) gives
Figure imgf000216_0003
The most convenient way to solve Eq. (13.583) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is α = l.6234Oa0 = 8.59066 X 10"11 m (13.584) Substitution of Eq. (13.584) into Eq. (13.60) gives c' = l.O4O32βo = 5.50514 JST 10"11 m (13.585) The internuclear distance given by multiplying Eq. (13.585) by two is
2c' = 2.08064α0 = 1.10103 X 10"10 m (13.586) The experimental bond distance is [41]
2c' = 1.087 X 10~10 m (13.587) Substitution of Eqs. (13.584-13.585) into Eq. (13.62) gives b = c = 1.24626αn = 6.59492 X 10 ,"-πu m (13.588)
Substitution of Eqs. (13.584-13.585) into Eq. (13.63) gives e = 0.64083 (13.589) The nucleus of the H atom and the nucleus of the C atom comprise the foci of each H2 - type ellipsoidal MO. The parameters of the point of intersection of each H2 -type ellipsoidal MO and the C2sp3 HO in the absence of the other three are given by Eqs. (13.84-13.95),
(13.261-13.270), (13.434-13.442), and (13.551-13.555). The polar intersection angle θ' is given by Eq. (13.261) where rn = r2^ = 0.9177Ia0 is the radius of the C2sp3 shell.
Substitution of Eqs. (13.584-13.585) into Eq. (13.261) gives 0' = 86.20° (13.590)
Then, the angle θQ2s ,HQ the radial vector of the C2sp3 HO makes with the internuclear axis is
0 CiJHO = 180° -86.20° = 93.80° (13.591) as shown in Figure 14. The angle ΘH MO between the internuclear axis and the point of intersection of each H2 -type ellipsoidal MO with the C2sp3 radial vector given by Eqs. (13.442-13.443), (13.588), and (13.591) is
ΘHIMO = 47.29° (13.592)
Then, the distance dHiM0 along the internuclear axis from the origin of H2 -type ellipsoidal
MO to the point of intersection of the orbitals given by Eqs. (13.445), (13.584), and (13.592) is dHlMO = 1-1012Ia0 = 5.82734 Z 10~n m (13.593)
The distance dQ2s 3/rø along the internuclear axis from the origin of the C atom to the point of intersection of the orbitals given by Eqs. (13.447), (13.585), and (13.593) is d^ JH = 0.06089α0 = 3.22208 JST 10'12 m (13.594)
The H2 -type ellipsoidal MOs do not actually directly contact the C2sp3 HO. As discussed in the Force Balance of H2O section, with the addition of the fourth C-H bond, the H2 -type ellipsoidal MOs may linearly combine to form a continuous two-dimensional surface of equipotential equivalent to that of the MOs if they did contact the C2sp3 HO. However, Eqs. (13.579-13.580) must hold based on conservation of momentum and the potential, kinetic, and total energy relationships. In order that there is current continuity given the constraints of Eqs. (13.579-13.580), the existence of a self-contained, continuous- current, linear-combination of the H2 -type ellipsoidal MOs requires that electrons are divisible between the combination H2 -type MO and the C2sp3 HO. This is not possible. Thus, at the points of intersection of the H2 -type MOs of methane, symmetry, electron indivisibility, current continuity, and conservation of energy and angular momentum require that the current between the C2sp3 shell and points of mutual contact is projected onto and flows along the radial vector to the surface of C2sp3 shell. This current designated the bisector current (BC) meets the C2sp3 surface and does not travel to distances shorter than its radius. Moreover, an energy minimum is obtained when the H -atom charge-density of each C -H-bond MO is decreased by a factor of 0.25 with a corresponding 0.25 increase in that of the three other C-H -bond MOs. In this case, the angular momentum components of the transferred current mutually cancel. The geometry of the equivalent bonds is tetrahedral. The symmetry point group is Td . This geometry is equivalent to the indistinguishable bonds positioned uniformly on a spherical surface or also at the apexes of a cube. The predicted angle θ between the C -H bonds is
0 = 109.5° (13.595)
The experimental bond angle is [41]
0 = 109.5° (13.596) The polar angle φ at which the H2 -type ellipsoidal MOs intersect is given by the bisector of the angle θ between the C-H bonds: φ = 109:5 = 54 75o (13.597)
With the carbon nucleus defined as the origin and one of the C-H bonds defined as the positive x-axis, the polar-coordinate angle of the intersection occurs at ^' = 54.75o + 180o = 234.57° (13.598)
The polar radius r,at this angle is given by Eqs. (13.84-13.85):
Figure imgf000218_0001
Substitution of Eqs. (13.584-13.585) and (13.589) into Eq. (13.599) gives η = 1.52223α0 = 8.05530 X 10"11 m (13.600) Using the orbital composition of CH4 (Eq. (13.579)), the radii of CIs = 0.17113α0
(Eq. (10.51)) and C2sp3 = 0.9177kr0 (Eq. (10.424)) shells, and the parameters of the CH4 MO given by Eqs. (13.3-13.4), (13.584-13.586), and (13.588-13.600), the charge-density of the CH4 MO comprising the linear combination of four C-H-bond MOs is shown in Figure 18. Each C-H-bond MO having the dimensional diagram shown in Figure 14 comprises a H2 -type ellipsoidal MO and a C2sp3 HO according to Eq. (13.579). But, based on the Td symmetry of the H2 -type MOs, the charge is distributed 1:1 between the H2 -type MOs and the C2sp3 shell.
ENERGIES OF CH4
The energies of CH4 are four times those of CH and are given by the substitution of the semiprincipal axes (Eqs. (13.584-13.585) and (13.588)) into the energy equations Eq. (13.580) and (Eqs. (13.449-13.452)) that are multiplied by four:
Fe = 4(0.91771) ~2f In α + , ~ h% = -145.86691 eV (13.601)
SπεosJ a2 - b2 a-yja2 -b2
Vp (13.602)
Figure imgf000219_0001
Figure imgf000219_0002
(13.605) where E7 (CH4) is given by Eq. (13.580) which is reiteratively matched to Eq. (13.581) within five-significant-figure round-off error.
VIBRATION OF CH4 The vibrational energy levels of CH4 may be solved as four equivalent coupled harmonic oscillators by developing the Lagrangian, the differential equation of motion, and the eigenvalue solutions [2] wherein the spring constants are derived from the central forces as given in the Vibration of Ηydrogen-Type Molecular Ions section and the Vibration of Hydro gen-Type Molecules section. THE DOPPLER ENERGY TERMS OF 12CH4
The reentrant oscillation of hybridized orbitals in the transition state is not coupled. Therefore, the equations of the radiation reaction force of methane are the same as those of OH, CH , CH2 , and CH3 with the substitution of the methane parameters. Using Eqs. (11.136) and (13.140-13.142), the angular frequency of the reentrant oscillation in the transition state is
ω 1016 radls (13.606)
Figure imgf000220_0001
where b is given by Eq. (13.588). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)): Eκ = hω = £2.57338 X IO16 rad/s = 16.9384I eF (13.607)
In Eq. (11.181), substitution of E7 (H1) (Eqs. (11.212) and (13.75)), the maximum total energy of each H2 -type MO, for E1n, , the mass of the electron, me , for M , and the kinetic energy given by Eq. (13.607) for Eκ gives the Doppler energy of the electrons of each of the four bonds for the reentrant orbit:
(13.608)
Figure imgf000220_0002
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of CH4 due to the reentrant orbit of each bond in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, E030 , is given by the sum of the corresponding energies, ED given by Eq. (13.608) and EKvib , the average kinetic energy of vibration which is 1/2 of the vibrational energy of each C-H bond. Using ωe given by Eq. (13.458) for EKvlb of the transition state having four independent bonds, E 'røc ( 12CH4 ) per bond is
(13.609)
Figure imgf000220_0003
E'osc (12CH4) = -0.25758 eF+-(0.35532 eV) = -0.07992 eV (13.610) The reentrant orbit for the binding of a hydrogen atom to a CH3 radical involves four C-H bonds. Since the vibration and reentrant oscillation is along four bonds, Eosc for 12CH4 , E,,,(nCHt), is:
Figure imgf000221_0001
= 4[ -0.25758 eF+-(0.35532 eV)) (13.611)
= -0.31967 eV
TOTAL AND BOND ENERGIES OF 12CH4
Eτ+osc ( 12CH4 ) , the total energy of the 12CH4 radical including the Doppler term, is given by
the sum of E7. (CH4) (Eq. (13.582)) and Eosc ( 12CH4 ) given by Eq. (13.611):
Eτ+osc (CH4) = Ve +T + Vm +Vp + E(C,2sp3) + Eosc (12CH4)
(13.612)
= Er (cH4)+ E(KC (12cH4)
Figure imgf000221_0002
= -85.72406 eV-4 0.25758 eV--h /—
(13.613) From Eqs. (13.609-13.613), the total energy of 12CH4 is
Eτ+osc ( 12CH4 ) = -85.72406 e V + Eosc ( 12CH4 )
= -85.72406 eF-4[ 0.25758 eK--(0.35532 eF) j (13.614) = -86.04373 eV
where ω given by Eq. (13.458) was used for the h i I —k term.
V μ The CH4 bond dissociation energy, E0 (12CH4) , is given by the sum of the total
energies of the CH3 radical and the hydrogen atom minus Eτ+osc ( 12CH4 ) :
JE/) (12cH4) = £(12cH3)+E(H)-JEr+OiC (12cH4) (13.615)
where JS71(12CH3) is given by the sum of the energies of the C2sp% HO, E(c,2sp3") given by Eq. (13.428), 2>ED (H) given by Eq. (13.154), and the negative of the bond energies of 12CH given by Eq. (13.489), 12CH2 given by Eq. (13.528), and 12CH3 given by Eq. (13.573):
1^-3.47 eV - 4.33064 eV - 4.72444 eV )
Thus, the 12CH4 bond dissociation energy, ED (12CH4) , given by Eqs. (13.154), and (13.614- 13.616) is
E0 (12CH4) = -(67.95529 eV + 13.59844 e V) -Eτ+osc (12CH4) = -81.55373 eV -(-86.04373 eV) (13.617)
= 4.4900 eK
The experimental 12CH4 bond dissociation energy is [40]
E0 (12CH4) = 4.48464 e V (13.618)
The results of the determination of bond parameters of CH4 are given in Table 13.1. The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
NITROGEN MOLECULE The nitrogen molecule can be formed by the reaction of two nitrogen atoms:
N + N → N2 (13.619)
The bond in the nitrogen molecule comprises a H2 -type molecular orbital (MO) with two paired electrons. The force balance equation and radius rη of the Ip shell of N is derived in the Seven-Electron Atoms section. With the formation of the H2 -type MO by the contribution of a 2p electron from each N atom, a diamagnetic force arises between the remaining 2p electrons and the H2 -type MO. This force from each JV causes the H2 -type
MO to move to greater principal axes than would result with the Coulombic force alone. But, the integer increase of the central field and the resulting increased Coulombic as well as magnetic central forces on the remaining 2p electrons of each JV decrease the radius of the corresponding shell such that the energy minimum is achieved that is lower than that of the reactant atoms. The resulting electron configuration of JV2 is Ls1 2 Is2 ^s \ 2$l 1*v\ 2plσi 2 where the subscript designates the JV atom, 1 or 2, σ designates the H2 -type MO, and the orbital arrangement is
(13.620)
Figure imgf000223_0001
Nitrogen is predicted to be diamagnetic in agreement with observations [42].
FORCE BALANCE OF THE 2p SHELL OF THE NITROGEN ATOMS OF THE NITROGEN MOLECULE
For each JV atom, force balance for the outermost 2p electron of JV2 (electron 6) is achieved between the centrifugal force and the Coulombic and magnetic forces that arise due to interactions between electron 6 and the other 2ρ -shell as well as the 2s -shell electrons due to spin and orbital angular momentum. The forces used are derived in the Seven- Electron Atoms section. The central Coulomb force on the outer-most 2ρ shell electron of JV2 (electron 6) due to the nucleus and the inner five electrons is given by Eq. (10.70) with the appropriate charge and radius:
Figure imgf000224_0001
for r > r5. The 2/? shell possess an external electric field given by Eq. (10.92) for r > r6.
The energy is minimized with conservation of angular momentum. This condition is met when the diamagnetic force, Fdiamagmlic , of Eq. (10.82) due to the p -orbital contribution is the same as that of the reactant nitrogen atoms given by Eq. (10.136) with r6 replacing rη :
Figure imgf000224_0002
And, Fmag 2 corresponding to the conserved orbital angular momentum of the three orbitals is given by Eq. (10.89):
Figure imgf000224_0003
The electric field external to the 2p shell given by Eq. (10.92) for r > r6 gives rise to a second diamagnetic force, Fdlωnaglielic 2 , given by Eq. (10.93). Fdiamagnetjc 2 due to the binding of the p-orbital electron having an electric field of +1 outside of its radius is:
F diamagnetic 2 (13.624)
Figure imgf000224_0004
In addition, the contribution of a 2p electron from each N atom in the formation of the σ MO gives rise to a paramagnetic force on the remaining two 2p electrons that pair. The force, Vmag 3 > follows from Eq. (10.11) wherein the two radii are equal to r6 and the direction is positive, central:
Figure imgf000224_0005
energy. This AO spin-pairing force reduces the radius directly to reduce the energy, and it can also cancel the contribution of the corresponding electron to Fdiamagneljc to further reduce the energy.
The radius of the 2p shell is calculated by equating the outward centrifugal force to the sum of the electric (Eq. (13.621)) and diamagnetic (Eqs. (13.622) and (13.624)), and paramagnetic (Eqs. (13.623) and (13.625)) forces as follows:
Figure imgf000225_0001
1 .
Substitution of v6 = (Eq. (1.56)) and .? = - into Eq. (13.626) gives: mer6 Z
Figure imgf000225_0002
(13.627) The quadratic equation corresponding to Eq. (13.627) is
Figure imgf000225_0003
(13.628)
The solution of Eq. (13.628) using the quadratic formula is:
Figure imgf000225_0004
r3 i« units of aQ The positive root of Eq. (13.629) must be taken in order that rβ > 0 . Substitution of
-^- = 0.69385 (Eq. (10.62) with Z = 7 ) into Eq. (13.629) gives an r6 = 0.78402α0 (13.630) ENERGIES OF THE 2p SHELL OF THE NITROGEN ATOMS OF THE
NITROGEN MOLECULE
The central forces on the 2p shell of each N are increased with the formation of the σ MO, which reduces the shell's radius and increases its total energy. The Coulombic energy terms of the total energy of the two N atoms at the new radius are calculated and added to the energy of the σ MO to give the total energy of N2 . Then, the bond energy is determined from the total N2 energy.
The radius r7 of each nitrogen atom before bonding is given by Eq. (10.142): r7 = 0.93084α0 (13.631) Using the initial radius r7 of each N atom and the final radius r6 of the NIp shell of N2 (Eq. (13.630)) and by considering that the central Coulombic field decreases by an integer for each successive electron of the shell, the sum ET (N2,2p) of the Coulombic energy change of the N2p electrons of both atoms is determined using Eq. (10.102):
(13.632)
Figure imgf000226_0002
FORCE BALANCE OF THE σ MO OF THE NITROGEN MOLECULE
The 2p shell gives rise to two diamagnetic forces on the σ MO. As given for the hydrogen molecule in the Hydrogen-Type Molecules section, the σ MO comprises two electrons, σ electron 1 and σ electron 2, that are bound at ξ = 0 as a equipotential prolate spheroidal MO by the central Coulombic field due to the nitrogen atoms at the foci and the spin pairing force on σ electron 2 due to σ electron 1 that initially has smaller semiprincipal axes. The spin- pairing force given in Eq. (11.200) is equal to one half the centrifugal force of the two electrons. The spin-pairing electron of the σ MO is also repelled by the remaining 2p electrons of each N according to Lenz law, and the force is based on the total number of these electrons ne that interact with the binding σ -MO electron. This diamagnetic force
Figure imgf000226_0001
*s °f the same form as me molecular spin-pairing force but in the opposite direction. The force follows from the derivations of Eqs. (10.219) and (11.200) which gives:
Figure imgf000227_0001
In addition, there is a relativistically corrected Lorentzian force H?dlamagnelicM02 on the pairing electron of the σ MO that follows from Eqs. (7.15) and (11.200):
XdiamagriettcMOl (13.634) where L is the magnitude of the angular momentum of each N atom at a focus that is the source of the diamagnetism at the σ -MO.
The force balance equation for the σ -MO of the nitrogen molecule given by Eq. (11.200) and Eqs. (13.633-13.634) with ne = 2 and \∑\ = h is
Figure imgf000227_0003
^+^)^τττD = --^-τD (13.637)
Zj 2mea2b2 Sπsoab2
Figure imgf000227_0004
Substitution of Z = 7 into Eq. (13.638) gives α = 2.14286αo = 1.13395 X 10~10 m (13.639) Substitution of Eq. (13.639) into Eq. (11.79) is c' = 1.0351Oa0 = 5.47750 ΛT 10"" m (13.640)
The internuclear distance given by multiplying Eq. (13.640) by two is
2c' = 2.0702Oa0 = 1.09550 JST 10"10 m (13.641)
The experimental bond distance from Ref. [28] and Ref. [43] is 2c' = 1.09769 X lO"10 m (13.642)
2c' = 1.094X 10-10 m (13.643)
Substitution of Eqs. (13.639-13.640) into Eq. (11.80) is b = c = 1.87628α0 = 9.92882 X \QTn m (13.644)
Substitution of Eqs. (13.639-13.640) into Eq. (11.67) is e = 0.48305 (13.645) Using the electron configuration of N2 (Eq. (13.620)), the radii of the NIs = 0.14605α0 (Eq.
(10.51)), _V2J = 0.69385O0 (Eq. (10.62)), and NIp = 0.78402α0 (Eq. (13.630)) shells and the parameters of the σ MO of N1 given by Eqs. (13.3-13.4), (13.639-13.641), and (13.644- 13.645), the dimensional diagram and charge-density of the N2 MO are shown in Figures 19 and 20, respectively.
Despite the predictions of standard quantum mechanics that preclude the imaging of a molecular orbital, the full three-dimensional structure of the outer molecular orbital of N2 has been recently tomographically reconstructed [44]. The charge-density surface observed is consistent with that shown in Figure 20. This result constitutes direct evidence that electrons are not point-particle probability waves that have no form until they are "collapsed to a point" by measurement. Rather they are physical, two-dimensional equipotential charge density surfaces.
SUM OF THE ENERGIES OF THE σ MO AND THE AOs OF THE NITROGENMOLECULE
The energies of the N2 σ MO are given by the substitution of the semiprincipal axes (Eqs. (13.639-13.640) and (13.644)) into the energy equations (Eqs. (11.207-11.212)) of H2 :
Figure imgf000228_0001
ET = Ve +T + Vm +Vp (13.650)
Substitution of Eqs. (11.79) and (13.646-13.649) into Eq. (13.650) gives
Figure imgf000228_0002
where Eτ (N2,σ) is the total energy of the σ MO of JV2 . The sum, Ex (N2) , of
ET (N2,2p), the 2p (AO) contribution given by Eq. (13.632), and E7, (JV2, σ) , the σ MO contribution given by Eq. (13.651) is:
Eτ{N2) = Eτ (N2,2p) + Eτ (N2,σ)
= -27.37174 eV-11.32906 eV (13.652)
= -38.70080 eF
VIBRATION OF N2
The vibrational energy levels of JV2 may be solved by determining the Morse potential curve from the energy relationships for the transition from two JV atoms whose parameters are given by Eqs. (10.134-10.143) to the two JV atoms whose parameter r6 is given by Eq. (13.630) and the σ MO whose parameters are given by Eqs. (13.639-13.641) and (13.644- 13.645). As shown in the Vibration of Hydrogen-type Molecular Ions section, the harmonic oscillator potential energy function can be expanded about the internuclear distance and expressed as a Maclaurin series corresponding to a Morse potential after Karplus and Porter (K&P) [15] and after Eq. (11.134). Treating the Maclaurin series terms as anharmonic perturbation terms of the harmonic states, the energy corrections can be found by perturbation methods.
THE DOPPLER ENERGY TERMS OF THE NITROGEN MOLECULE
The equations of the radiation reaction force of nitrogen are the same as those of H2 with the substitution of the nitrogen parameters. Using Eqs. (11.231-11.233), the angular frequency of the reentrant oscillation in the transition state is
o) = 1.31794X 1016 rod Is (13.653)
Figure imgf000229_0001
where a is given by Eq. (13.639). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)): Eκ = hω = M.31794 X 1016 radls = 8.67490 eV (13.654) In Eq. (11.181), substitution of E7 (N2 ) for Elw , the mass of the electron, me , for M , and the kinetic energy given by Eq. (13.654) for Eκ gives the Doppler energy of the electrons of the reentrant orbit:
Figure imgf000230_0001
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of the N2 MO due to the reentrant orbit in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by Eq. (13.655) and EKvώ, the average kinetic energy of vibration which is 1/2 of the vibrational energy. Using the experimental N2 ωe of 2358.57 cm'1 (0.29243 e V) [28] for EKvjb of the transition state, E0SC(N2) is
Ko (N2) = ED+EKvib = ED +U - (13.656)
Ksc(N2) = -0.22550 eV +-(0.29243 eV) = -0.07929 eV (13.657)
TOTALANDBONDENERGIESOFTHENITROGENMOLECULE
ET+0SC(N2), the total energy of N2 including the Doppler term, is given by the sum of E7(N2) (Eq. (13.652)) and E030(N2) given by Eq. (13.657):
Eτ+0SC(N2) = K+T + Vm+Vp+Eτ(N2,2p) + E0ΛN2)
= Eτ(N2,σ) + Eτ(N2,2p)+ E0Sc(N2) (13.658)
= E7(N2) + E0SC(N2) iw
Figure imgf000231_0001
From Eqs. (13.656-13.659), the total energy of the N2 MO is ET,osc (N2) = -3S.700Z0 eV + Eosc (N2)
= -38.70080 eV- 0.22550 eV +-(0.29243 eV) (13.660) = -38.78009 eV
where the experimental ω was used for the h — term.
The N2 bond dissociation energy, E D (N2) , is given by the difference in the total energies of the two N atoms and Eτ+osc (N2) :
ED (N2) = 2E{N)-ET+OSC (N2) (13.661) where the energy of a nitrogen atom is [6]
E(N) = -14.53414 eV (13.662) Thus, the N2 bond dissociation energy, ED (N2) , given by Eqs. (13.660-13.662) is
ED [N2) = -2(14.53414 eV)-ET+0SC (N2)
= -29.06828 eV- (-38.78009 eV) (13.663)
= 9.71181 eV The experimental N2 bond dissociation energy from Ref. [43] and Ref. [45] is
ED (N2) = 9.756 e V (13.664)
E D (N2) = 9.764 e V (13.665) The results of the determination of bond parameters of N2 are given in Table 13.1. The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
OXYGEN MOLECULE The oxygen molecule can be formed by the reaction of two oxygen atoms:
O+ O -> O2 (13.666)
The bond in the oxygen molecule comprises a H2 -type molecular orbital (MO) with two paired electrons. The force balance equation and radius rs of the 2p shell of O is derived in the Eight-Electron Atoms section. With the formation of the H2 -type MO by the contribution of a 2p electron from each O atom, a diamagnetic force arises between the remaining 2p electrons and the H2 -type MO. This force from each O causes the H2 -type
MO to move to greater principal axes than would result with the Coulombic force alone. But, the integer increase of the central field and the resulting increased Coulombic as well as magnetic central forces on the remaining 2p electrons of each O decrease the radius of the corresponding shell such that the energy minimum is achieved that is lower than that of the reactant atoms. The resulting electron configuration of O2 is \s\\s\2s\2s\2p\2p\σ\2 where the subscript designates the O atom, 1 or 2, σ designates the H2 -type MO, and the orbital arrangement is
(13.667)
Figure imgf000232_0001
Oxygen is predicted to be paramagnetic in agreement with observations [42].
FORCE BALANCE OF THE 2p SHELL OF THE OXYGEN ATOMS OF THE OXYGENMOLECULE For each O atom, force balance for the outermost 2p electron of O2 (electron 7) is achieved between the centrifugal force and the Coulombic and magnetic forces that arise due to interactions between electron 7 and the other 2p -shell as well as the 2s -shell electrons due to spin and orbital angular momentum. The forces used are derived in the Eight-Electron Atoms section. The central Coulomb force on the outer-most 2p shell electron of O2 (electron 7) due to the nucleus and the inner six electrons is given by Eq. (10.70) with the appropriate charge and radius:
Figure imgf000233_0001
for r > r6. The 2p shell possess an external electric field given by Eq. (10.92) for r > r7.
The energy is minimized with conservation of angular momentum. This condition is met when the magnetic forces are the same as those of the reactant oxygen atoms with r7 replacing rs . The diamagnetic force, Fdiamagnetlc , of Eq. (10.82) due to the p -orbital contributions is given by Eq. (10.156):
VdiamasneUc
Figure imgf000233_0002
And, ¥mag 2 corresponding to the conserved spin and orbital angular momentum given by Eq. (10.157) is
Figure imgf000233_0003
The electric field external to the 2p shell given by Eq. (10.92) for r > rη gives rise to a second diamagnetic force, Fdiamaglielic 2 , given by Eq. (10.93). Fdiamagtielic 2 due to the binding of the p-orbital electron having an electric field of +1 outside of its radius is :
diamagnetic 2
Figure imgf000233_0004
Z-6 1~ + I)I, (13.671)
The radius of the 2p shell is calculated by equating the outward centrifugal force to the sum of the electric (Eq. (13.688)) and diamagnetic (Eqs. (13.669) and (13.671)), and paramagnetic (Eq. (13.670)) forces as follows:
Figure imgf000234_0001
% 1
Substitution of V7 = (Eq. (1.56)) and s = - into Eq. (13.672) gives: merη
Figure imgf000234_0002
(13.673)
The quadratic equation corresponding to Eq. (13.673) is
Figure imgf000234_0003
(13.674)
The solution of Eq. (13.674) using the quadratic formula is:
Figure imgf000234_0004
r3 in units of aQ The positive root of Eq. (13.675) must be taken in order that rη > 0. Substitution of
-^- = 0.59020 (Eq. (10.62) with Z = S) into Eq. (13.675) gives an r7 = 0.91088α0 (13.676)
ENERGIES OF THE 2p SHELL OF THE OXYGEN ATOMS OF THE OXYGEN MOLECULE
The central forces on the 2p shell of each O are increased with the formation of the σ MO5 which reduces the shell's radius and increases its total energy. The Coulombic energy terms of the total energy of the two O atoms at the new radius are calculated and added to the energy of the σ MO to give the total energy of O2. Then, the bond energy is determined from the total O2 energy.
The radius r8 of each oxygen atom before bonding is given by Eq. (10.162): r8 = a0 (13.677)
Using the initial radius r& of each O atom and the final radius r7 of the O2p shell of O2 (Eq. (13.676)) and by considering that the central Coulombic field decreases by an integer for each successive electron of the shell, the sum Eτ {θ2,1p) of the Coulombic energy change of the 02 p electrons of both atoms is determined using Eq. (10.102):
Figure imgf000235_0001
= -2(13.60580 eF)(0.09784)(2 + 3 + 4) (13.678) = -23.96074 eV
FORCE BALANCE OF THE σ MO OF THE OXYGEN MOLECULE
The force balance equation for the σ-MO of the oxygen molecule given by Eq. (11.200) and
Eqs. (13.633-13.634) with ne = 2 and \L\ = J-H is
Figure imgf000235_0002
n2 D = - _£>_ I+I-I D (13.680) mea 2 b1.2 $πε0ab2 2 Z 2rn aΨ
Figure imgf000235_0003
Figure imgf000236_0001
Substitution of Z = 8 into Eq. (13.682) gives a = 2.60825α0 = 1.38023 X 10~10 m (13.683)
Substitution of Eq. (13.683) into Eq. (11.79) is c' = 1.14198α0 = 6.04312Z 10-π m (13.684)
The internuclear distance given by multiplying Eq. (13.684) by two is
2c' = 2.28397fl0 = 1.20862 X 10~10 m (13.685)
The experimental bond distance is [28]
2c' = 1.20752 JST 10"10 m (13.686) Substitution of Eqs. (13.683-13.684) into Eq. (11.80) is ft = C = 2.34496Λ0 = 1.24090 JSf 10~10 m (13.687)
Substitution of Eqs. (13.683-13.684) into Eq. (11.67) is e = 0.43783 (13.688)
Using the electron configuration of O2 (Eq. (13.667)), the radii of the Ols = 0Λ2739a0 (Eq. (10.51)), O2s = 0.59020α0 (Eq. (10.62)), and O2p = 0.91088α0 (Eq. (13.676)) shells and the parameters of the σ MO of O2 given by Eqs. (13.3-13.4), (13.683-13.685), and (13.687- 13.688), the dimensional diagram and charge-density of the O2 MO are shown in Figures 21 and 22, respectively.
SUM OF THE ENERGIES OF THE σ MO AND THE AOs OF THE OXYGEN MOLECULE
The energies of the O2 σ MO are given by the substitution of the semiprincipal axes (Eqs. (13.683-13.684) and (13.687)) into the energy equations (Eqs. (11.207-11.212)) of H2 :
Figure imgf000236_0002
Figure imgf000237_0001
ET = Ve +T + Vm +Vp (13.693) Substitution of Eqs. (11.79) and (13.689-13.692) into Eq. (13.693) gives
Eτ (θ2,σ) = -8.31814 eV (13.694)
Figure imgf000237_0002
where Eτ (O2, σ) is the total energy of the σ MO of O2. The sum, Eτ (O2) , of Eτ (O2,2p) , the 2p AO contribution given by Eq. (13.678), and E1. (O2, σ), the σ MO contribution given by Eq. (13.694) is:
ET (O2) = ET (O2,2P) + Eτ (θ2,σ)
= -23.96074 eV ^8.31814 eF (13.695)
= -32.27888 eV
VIBRATION OF O2
The vibrational energy levels of O2 may be solved by determining the Morse potential curve from the energy relationships for the transition from two O atoms whose parameters are given by Eqs. (10.154-10.163) to the two O atoms whose parameter rη is given by Eq. (13.676) and the σ MO whose parameters are given by Eqs. (13.683-13.685) and (13.687- 13.688). As shown in the Vibration of Hydrogen-type Molecular Ions section, the harmonic oscillator potential energy function can be expanded about the internuclear distance and expressed as a Maclaurin series corresponding to a Morse potential after Karplus and Porter (K&P) [15] and after Eq. (11.134). Treating the Maclaurin series terms as anharmonic perturbation terms of the harmonic states, the energy corrections can be found by perturbation methods. THE DOPPLER ENERGY TERMS OF THE OXYGEN MOLECULE
The equations of the radiation reaction force of oxygen are the same as those of H2 with the substitution of the oxygen parameters. Using Eqs. (11.231-11.233), the angular frequency of the reentrant oscillation in the transition state is
Figure imgf000238_0001
where a is given by Eq. (13.683). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)):
Eκ = hω = h9.8l432X 1016 rod Is = 6.45996 eV (13.697)
In Eq. (11.181), substitution of E1, (O2 ) for E1n, , the mass of the electron, me , for M , and the kinetic energy given by Eq. (13.697) for Eκ gives the Doppler energy of the electrons of the reentrant orbit:
ED Si eV (13.698)
Figure imgf000238_0002
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of the O2 MO due to the reentrant orbit in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by Eq. (13.698) and EKvib , the average kinetic energy of vibration which is 1/2 of the vibrational energy. Using the experimental O2 ωe of 1580.19 cm"1 (0.19592 eV) [28] for EKvib of the transition state, E0110 [O2) is
KAO2) = ED +EKvib = ED +U Jt (13.699)
E0SC [02) = -QΛ62?>l eV+-(QΛ9592 eV) = -Q.m35 eV (13.700)
TOTAL AND BOND ENERGIES OF THE OXYGEN MOLECULE
Eτ+osc [θ2) , the total energy of O2 including the Doppler term, is given by the sum of E7 (O2) (Eq. (13.695)) and E050 (O2) given by Eq. (13.700): Eτ+oAo 2)=K+τ+vm+vp+ετ(o2,2p)+εosc(o2)
= Eτ(θ2,σ) + Eτ(θ2,2p) + Eosc(θ2) (13.701) = ET(O2) + EOSC(O2)
Figure imgf000239_0001
= -32.27888 eV - 0.16231
Figure imgf000239_0002
From Eqs. (13.699-13.702), the total energy of the O2 MO is Er+0SC (O2) = -32.27888 eV + E0SC (O2)
= -32.27888 eK-0.16231 eK+-(0.19592 eF) (13.703) = -32.34323 e V
where the experimental ωe was used for the h I— term.
The O2 bond dissociation energy, E0(O2), is given by the difference in the total energies of the two O atoms and Eτ+osc (O2):
ED (O2) = 2E(O)-ET+OSC (O2) (13.704) where the energy of an oxygen atom is [6] E(O) = -13.61806 eV (13.705)
Thus, the O2 bond dissociation energy, ED (O2), given by Eqs. (13.703-13.705) is
ED(O2) = -2(l3.6U06eV)-ET+osc(O2)
= -27.23612 eF-(-32.34323 eV) (13.706)
= 5.10711 eV
The experimental O2 bond dissociation energy from Ref. [46] and Ref. [47] is
E0(O2) = 5.11665 eV (13.707) E0 (O2) = 5.116696 eV (13.708) The results of the determination of bond parameters of O2 are given in Table 13.1. The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
FLUORINE MOLECULE The fluorine molecule can be formed by the reaction of two fluorine atoms:
F + F → F2 (13.709)
The bond in the fluorine molecule comprises a H2 -type molecular orbital (MO) with two paired electrons. The force balance equation and radius r9 of the 2ρ shell of F is derived in the Nine-Electron Atoms section. With the formation of the H2 -type MO by the contribution of a 2p electron from each F atom, a diamagnetic force arises between the remaining 2p electrons and the H2 -type MO. This force from each F causes the H2 -type MO to move to greater principal axes than would result with the Coulombic force alone. But, the integer increase of the central field and the resulting increased Coulombic as well as magnetic central forces on the remaining 2p electrons of each F decrease the radius of the corresponding shell such that the energy minimum is achieved that is lower than that of the reactant atoms. The resulting electron configuration of F2 is \$\\s\2s\2s\2p{2p\σl2 where the subscript designates the F atom, 1 or 2, σ designates the H2 -type MO5 and the orbital arrangement is
(13.710)
Figure imgf000240_0001
Fluorine is predicted to be diamagnetic in agreement with observations [42].
FORCE BALANCE OF THE 2p SHELL OF THE FLUORINE ATOMS OF THEFLUORINEMOLECULE
For each F atom, force balance for the outermost 2p electron of F2 (electron 8) is achieved between the centrifugal force and the Coulombic and magnetic forces that arise due to interactions between electron 8 and the other 2p -shell as well as the 2s -shell electrons due to spin and orbital angular momentum. The forces used are derived in the Nine-Electron Atoms section. The central Coulomb force on the outer-most 2p shell electron of F2
(electron 8) due to the nucleus and the inner seven electrons is given by Eq. (10.70) with the appropriate charge and radius:
_ (Z -7)g2 H Q 7 I n tele - —/ J — *r (13.711)
for r > rη . The 2p shell possess an external electric field given by Eq. (10.92) for r > rs . The energy is minimized with conservation of angular momentum. This condition is met when the diamagnetic force, FΛamagnetιc , of Eq. (10.82) due to the p -orbital contributions is the same as that of the reactant fluorine atoms given by Eq. (10.176) with rs replacing r9 :
Vdιamagnenc (13-712)
Figure imgf000241_0001
Thus, ΕΛamagmtιc due to the two filled 2p orbitals per F atom is twice that of N2 given by Eq. (13.622) having one filled 2p orbital per N atom. Fmag 2 corresponding to the conserved spin and orbital angular momentum is also the same as that of the reactant fluorine atoms given by Eq. (10.177) and that of N2 given by Eq. (13.623) where the outer radius of the 2 p shell of the F atoms of F2 is r8 . i 2h2 )
F-* 2 = 7 Z mers — r3 VΦ + l)ir (13-713)
The electric field external to the 2p shell given by Eq. (10.92) for r > rs gives rise to a second diamagnetic force, ΕdιamagneUc 2, given by Eq. (10.93). Fdιamagnetιc 2 due to the binding of the p-orbital electron having an electric field of +1 outside of its radius is :
Figure imgf000242_0001
In addition, the contribution of a 2/» electron from each F atom in the formation of the σ MO gives rise to a paramagnetic force on the remaining paired 2p electrons. The force ¥mag 3 is given by Eq. (13.625) wherein the radius is r8 : h2 mag 3 ]S(s + l)ir (13.715)
4mers
The radius of the 2p shell is calculated by equating the outward centrifugal force to the sum of the electric (Eq. (13.711)) and diamagnetic (Eqs. (13.712) and (13.714)), and paramagnetic (Eqs. (13.713) and (13.715)) forces as follows:
Figure imgf000242_0002
Substitution of vs = — (Eq. (1.56)) and s = - into Eq. (13.716) gives: mer%
Figure imgf000242_0003
(13.717)
The quadratic equation corresponding to Eq. (13.717) is
Figure imgf000242_0004
(13.718)
The solution of Eq. (13.718) using the quadratic formula is:
Figure imgf000243_0001
r3 in units of aQ The positive root of Eq. (13.719) must be taken in order that rs > 0. Substitution of
-2- = 0.51382 (Eq. (10.62) with Z = 9) into Eq. (13.719) gives
Ctn r8 = 0.73318α0 (13.720)
ENERGIES OF THE 2p SHELL OF THE FLUORINE ATOMS OF THE
FLUORINEMOLECULE
The central forces on the 2p shell of each F are increased with the formation of the σ MO, which reduces the shell's radius and increases its total energy. The Coulombic energy terms of the total energy of the two F atoms at the new radius are calculated and added to the energy of the σ MO to give the total energy of F2. Then, the bond energy is determined from the total F2 energy.
The radius r9 of each fluorine atom before bonding is given by Eq. (10.182): r9 = 0.78069α0 (13.721) Using the initial radius r9 of each F atom and the final radius r8 of the FIp shell of F2
(Eq. (13.720)) and by considering that the central Coulombic field decreases by an integer for each successive electron of the shell, the sum Eτ (F2, 2/?) of the Coulombic energy change of the F2p electrons of both atoms is determined using Eq. (10.102):
Figure imgf000243_0002
= -2(13.60580 eF)(0.0830l)(2 + 3 + 4 + 5) (13.722) = -31.62353 eV FORCE BALANCE OF THE σ MO OF THE FLUORINE MOLECULE
The relativistic diamagnetic force FdiamagljellcM02 of F2 is one half that of N2 due to the two versus one filled 2p orbitals per atom at the focus. The force balance equation for the σ - MO of the fluorine molecule is given by Eq. (11.200) and Eqs. (13.633-13.634) with the correction of 1/2 due the two 2p orbitals per F after Eqs. (10.2-10.11)5 ne - 2 , and \∑\ = h :
Figure imgf000244_0001
Substitution of Z = 9 into Eq. (13.726) gives
« = 3.55556αo = 1.88152X 10"10 m (13.727)
Substitution of Eq. (13.727) into Eq. (11.79) is e' = 1.33333α0 = 7.05569 X 10"11 m (13.728) The internuclear distance given by multiplying Eq. (13.728) by two is
2c' = 2.66667o0 = 1.41114 JST 10~10 m (13.729)
The experimental bond distance is [28]
2c' = 1.41193 X 10~10 m (13.730)
Substitution of Eqs. (13.727-13.728) into Eq. (11.80) is b = c = 3.29609a0 = 1.74421 X lO'10 m (13.731)
Substitution of Eqs. (13.727-13.728) into Eq. (11.67) is e = 0.37500 (13.732)
Using the electron configuration of F2 (Eq. (13.710)), the radii of the FIs = 0.11297α0 (Eq. (10.51)), F2s = 0.513S2a0 (Eq. (10.62)), and 7^ = 0.73318^ (Eq. (13.720)) shells and the parameters of the σ MO of F2 given by Eqs. (13.3-13.4), (13.727-13.728), and (13.731- 13.732), the dimensional diagram and charge-density of the F2 MO are shown in Figures 23 and 24, respectively. SUM OF THE ENERGIES OF THDE σ MO AND THE AOs OF THE FLUORINE MOLECULE The energies of the F2 σ MO are given by the substitution of the semiprincipal axes (Eqs. (13.683-13.684) and (13.687)) into the energy equations (Eqs. (11.207-11.212)) of H2 :
(13.733)
Figure imgf000245_0001
Figure imgf000245_0002
ET = Ve +T + Vm +Vp (13.737)
Substitution of Eqs. (11.79) and (13.733-13.736) into Eq. (13.737) gives
Figure imgf000245_0003
where Eτ [F2,σ) is the total energy of the σ MO of F2. The sum, E7 [F2), of E7, (F2, 2p) , the 2 p AO contribution given by Eq. (13.722), and Eτ [F2, σ), the σ MO contribution given by Eq. (13.738) is:
Eτ [F2) = Eτ (F2,2p) + Eτ (F2,σ)
= -31.62353 eF-4.75562 eV (13.739)
= -36.37915 eV
VIBRATION OF F2 The vibrational energy levels of F2 may be solved by determining the Morse potential curve from the energy relationships for the transition from two F atoms whose parameters are given by Eqs. (10.174-10.183) to the two F atoms whose parameter r8 is given by Eq. (13.720) and the σ MO whose parameters are given by Eqs. (13.727-13.729) and (13.731-
13.732). As shown in the Vibration of Hydrogen-type Molecular Ions section, the harmonic oscillator potential energy function can be expanded about the internuclear distance and expressed as a Maclaurin series corresponding to a Morse potential after Karplus and Porter (K&P) [15] and after Eq. (11.134). Treating the Maclaurin series terms as anharmonic perturbation terms of the harmonic states, the energy corrections can be found by perturbation methods.
THE DOPPLER ENERGY TERMS OF THE FLUORINE MOLECULE The equations of the radiation reaction force of fluorine are the same as those of H2 with the substitution of the fluorine parameters. Using Eqs. (11.231-11.233), the angular frequency of the reentrant oscillation in the transition state is
11 n01155 rod Is (13.740) where a is given by Eq. (13.727). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)):
Eκ = hω = h6Λ6629X \015 rod Is = 4.05876 eV (13.741)
In Eq. (11.181), substitution of E1. [F2) for Ehv , the mass of the electron, me, for M , and the kinetic energy given by Eq. (13.741) for Eκ gives the Doppler energy of the electrons of the reentrant orbit:
Figure imgf000246_0002
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of the F2 MO due to the reentrant orbit in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by Eq. (13.742) and EKvΛ , the average kinetic energy of vibration which is 1/2 of the vibrational energy. Using the experimental F2 ωe of 916.64 cm"1 (0.1136S eF) [28] for EKvφ of the transition state, E0SC (F2) is
Figure imgf000247_0001
Eosc(F2) = -0Λ4499 eV +-(0.11365 eV) = -0.08817 eV (13.744)
TOTALANDBOND ENERGIES OF THEFLUORINEMOLECULE Eτ+osc [F2 ) , the total energy of F2 including the Doppler term, is given by the sum of E7(F2) (Eq. (13.739)) and E0SC(F2) given by Eq. (13.744):
Eτ+0SC(F2) = Ve+T + Vm+Vp+Eτ(F2,2p) + E0ΛF2)
= Eτ[F2,σ) + Eτ(F2,2p) + E0Sc(E2) (13-745)
= E7(F2) + E0SC(F2)
Figure imgf000247_0002
From Eqs. (13.743-13.746), the total energy of the F2 MO is ET+OSC(E2) = -36.37915 eV + E0SC(F2)
= -36.37915 eV -0.14499 eV + -(0.\\365 eV) (13.747)
= -36.46732 eV
where the experimental ω was used for the term.
Figure imgf000247_0003
The F2 bond dissociation energy, ED (F2 ) , is given by the difference in the total energies of the two F atoms and Eτ÷osc (F2):
ED (F2) = 2E(F)-ET+OSC (F2) (13.748) where the energy of a fluorine atom is [6] E(F) = -17.42282 e V (13.749)
Thus, the F2 bond dissociation energy, E0 (F2), given by Eqs. (13.747-13.749) is
E0 [F2) = -2(17.42282 eV)-ET+0SC [F2)
= -34.84564 eV -(-36.46732 eV) (13.750)
= 1.62168 eV The experimental F2 bond dissociation energy is [48] E0 (F2) = 1.606 eV (13.751)
The results of the determination of bond parameters of F2 are given in Table 13.1. The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
CHLORINE MOLECULE
The chlorine molecule can be formed by the reaction of two chlorine atoms:
Cl + Cl → Cl2 (13.752)
The chlorine molecule can be solved by using the hybridization approach used to solve the methane series CHn=l 23 4 . In the methane series, the 2s and 2p shells of carbon hybridize to form a single 2spz shell to achieve an energy minimum, and in a likewise manner, the 3s and 3p shells of chlorine hybridize to form a single 3sp3 shell which forms the bonding orbital of Cl2.
FORCE BALANCE OF CZ2
CZ2 has two spin-paired electrons in a chemical bond between the chlorine atoms. The CZ2 molecular orbital (MO) is determined by considering properties of the binding atoms and the boundary constraints. The prolate spheroidal H2 MO developed in the Nature of the Chemical Bond of Hydrogen-Type Molecules section satisfies the boundary constraints; thus, each Cl atom could contribute a 3p electron to form a σ MO (H2 -type ellipsoidal MO) as in the case of N2 , O2, and F2. However, such a bond is not possible with the outer Cl electrons in their ground state since the resulting 3p shells of chlorine atoms would overlap which is not energetically stable. Thus, when bonding, the chlorine 3s and 3p shells hybridize to form a single 3sp3 shell to achieve an energy minimum.
The Cl electron configuration given in the Seventeen-Electron Atoms section is Is22s22p63s23p5 , and the orbital arrangement is
(13.753)
Figure imgf000249_0001
corresponding to the ground state 2P3°/2. The radius rιη of the 3p shell given by Eq. (10.363) is r17 = 1.05158α0 (13.754) ,
The energy of the chlorine 3p shell is the negative of the ionization energy of the chlorine atom given by Eq. (10.364). Experimentally, the energy is [6] E(3p shell) = -EQonization; Cl) = -12.96764 eV (13.755)
The Cl3s atomic orbital (AO) combines with the Cl3p AOs to form a single 3sp3 hybridized orbital (HO) with the orbital arrangement
2sp3 state (13.756)
Figure imgf000249_0002
where the quantum numbers (£,me) are below each electron. The total energy of the state is given by the sum over the seven electrons. Using only the largest-force terms of the outer most and next inner shell, the calculated energies for the chlorine atom and the ions: Cl , Cl+ , Cl2+ , Cl3+ , α4+ , C/5+ and Cl6+ are given in Eqs. (10.363-10.364), (10.353-10.354), (10.331-10.332), (10.309-10.310), (10.288-10.289), (10.255-10.256), and (10.235-10.236), respectively. The sum Eτ ( Cl, 3sp3) of the experimental energies of Cl and these ions is [6]
, 3s _ ("12.96764 eV + 23.814 eV + 39.61 eV + 53.4652 eVΛ r V ' Sp f ~ ^+67.8 eF + 97.03 eF + l 14.1958 eV J (13.757)
= 408.88264 eV The spin and orbital-angular-momentum interactions cancel such that the energy of the
ET (Cl,3sp3\ is purely Coulombic. By considering that the central field decreases by an integer for each successive electron of the shell, the radius r3s 3 of the Cl3sp3 shell may be calculated from the Coulombic energy using Eq. (10.102):
Figure imgf000250_0001
where Z = 17 . Using Eqs. (10.102) and (13.758), the Coulombic energy ECouhmb (ci,3sp3) of the outer electron of the Cl3sp3 shell is
= -14.60295 eV (13.759)
Figure imgf000250_0002
The calculated energy of the C2sp3 shell of 14.63489 eV given by Eq. (13.428), and nitrogen's calculated energy of 14.61664 e V given by Eq. (10.143) is a close match with ECoulomb (Cl,3sp3 ) .
The unpaired Cl3$p3 electron from each of two chlorine atoms combine to form a molecular orbital. The nuclei of the Cl atoms are along the internuclear axis and serve as the foci. Due to symmetry, the other Cl electrons are equivalent to point charges at the origin. (See Eqs. (19-38) of Appendix IV.) Thus, the energies in the Cl MO involve only the two Cl3sp3 electrons. The forces are determined by these energies.
As in the case of H2 , the MO is a prolate spheroid with the exception that the ellipsoidal MO surface cannot extend into Cl3sp3 HO for distances shorter than the radius of the Cl3sp3 shell of each atom. Thus, the MO surface comprises a partial prolate spheroid in between the nuclei and is continuous with the Cl3sp3 shell at each Cl atom. The energy of the H2 -type ellipsoidal MO is matched to that of the Cl3sp3 shell. As in the case with OH, NH , and CH (where the latter also demonstrates sp3 hybridization) the linear combination of the H2 -type ellipsoidal MO with each Cl3sp3 HO must involve a 25% contribution from the H2 -type ellipsoidal MO to the Cl3sp3 HO in order to match potential, kinetic, and orbital energy relationships. Thus, the Cl2 MO must comprise two Cl3sp3 HOs and 75% of a H2- type ellipsoidal MO divided between the two Cl3sp3 HOs:
2 Cl3sp3 + 0.75 H2 MO -> Cl2 MO (13.760) The force balance of the Cl2 MO is determined by the boundary conditions that arise from the lineai' combination of orbitals according to Eq. (13.760) and the energy matching condition between the H2 -type-elHpsoidal-MO and C/3^p3-HO components of the MO. As in the case with OH (Eq. (13.57)), NH (Eq. (13.247)), and CH (Eq. 13.429)), the H2- type ellipsoidal MO comprises 75% of the CZ2 MO; so, the electron charge density in Eq. (11.65) is given by -0.75e . Since the chlorine atoms of CZ2 are hybridized and the k parameter is different from unity in order to meet the boundary constraints, both k and k ' must comprise the corresponding hybridization factors. (In contrast, the chlorine atom of a C -Cl bond of an alkyl chloride is not hybridized, and only k' must comprise the corresponding hybridization factor.) The force constant k' to determine the ellipsoidal parameter c1 in terms of the central force of the foci is given by Eq. (13.59), except that k' is divided by two since the H2 -type-ellipsoidal-MO is physically divided between two Cl3sps
ΗOs. In addition, the energy matching at both Cl3sp3 ΗOs further requires that k' be corrected the hybridization factor given by Eq. (13.762). Thus, k' of the H2-type- ellipsoidal-MO component of the CZ2 MO is
Figure imgf000251_0001
The distance from the origin to each focus c1 is given by Eq. (13.60). The internuclear distance is given by Eq. (13.61). The length of the semiminor axis of the prolate spheroidal Cl - Cl -bond b = c is given by Eq. (13.62). The eccentricity, e , is given by Eq. (13.63). The solution of the semimajor axis a then allows for the solution of the other axes of each prolate spheroid and eccentricity of the Cl2 MO. Since the CZ2MO comprises a H2- type-ellipsoidal MO that transitions to the Cl3sp3 ΗOs at each end of the molecule, the energy E(ci,3sp3) in Eq. (13.759) adds to that of the H2 -type ellipsoidal MO to give the total energy of the Cl2 MO. From the energy equation and the relationship between the axes, the dimensions of the Cl2 MO are solved.
The energy components of Ve , Vp , T , and Vn, are those of H2 (Eqs. (11.207-11.211)) except that they are corrected for electron hybridization. Hybridization gives rise to the CBsp3 HO-shell Coulombic energy ECoιιlomb (cl,3sp3) given by Eq. (13.759). To meet the equipotential condition of the union of the H2 -type-ellipsoidal-MO with each Cl3sp3 HO5 the electron energies are normalized by the ratio of 14.60295 eV , the magnitude of Ecouio,nb (Ch3sp3) given by Eq. (13.759), and 13.605804 eV , the magnitude of the
Coulombic energy between the electron and proton of H given by Eq. (1.243). This normalizes the energies to match that of the Coulombic energy alone to meet the energy matching condition of the Cl2 MO under the influence of the two Cl3sp3 HOs bridged by the H2 -type-ellipsoidal MO. The hybridization energy factor Ccn 3 is
H 3 762Ϊ
Figure imgf000252_0001
The total energy E7. (Cl2) of the CZ2 MO is given by the sum of the energies of the orbitals, the H2 -type ellipsoidal MO and the two Cl3sp3 HOs, that form the hybridized Cl2 MO. Eτ (Cl2) follows from by Eq. (13.74) for OH5 but the energy of the Cl3sp3 HO given by Eq. (13.759) is substituted for the energy of O and the H2 -type-ellipsoidal-MO energies are those of H2 (Eqs. (11.207-11.212)) multiplied by the electron hybridization factor rather than by the factor of 0.75 :
ET {Cl2) = ET +ECoulomb (Cl,3sp3)
Figure imgf000252_0002
To match the boundary condition that the total energy of the entire the H2 -type ellipsoidal MO is given by Eqs. (11.212) and (13.75), E1 (Cl2) given by Eq. (13.763) is set equal to Eq. (13.75):
-31.63537 eV
Figure imgf000252_0003
(13.764)
From the energy relationship given by Eq. (13.764) and the relationship between the axes given by Eqs. (13.60-13.63), the dimensions of the CZ2 MO can be solved. Substitution of Eqs. (13.60) and (13.761) into Eq. (13.764) gives
Figure imgf000253_0001
(13.765)
The most convenient way to solve Eq. (13.765) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is a = 2.4650Oa0 = 1.30442 X 10"10 m (13.766)
Substitution of Eq. (13.766) into Eq. (13.60) gives c' = 1.87817α0 = 9.93887 X 10~n m (13.767) The internuclear distance given by multiplying Eq. (13.767) by two is
2c' = 3.75635αn = 1.98777 X 10 i-lO m (13.768) The experimental bond distance is [28]
2c' = 1.988 X lO'10 m (13.769)
Substitution of Eqs. (13.766-13.767) into Eq. (13.62) gives b = c = l.59646a0 = 8.44810 X 10"n m (13.770)
Substitution of Eqs. (13.766-13.767) into Eq. (13.63) gives e = 0.76194 (13.771)
The Cl nuclei comprise the foci of the H2 -type ellipsoidal MO. The parameters of the point of intersection of the Tf2 -type ellipsoidal MO and the Cl3sp3 HO are given by Eqs. (13.84- 13.95) and (13.261-13.270). The polar intersection angle θ' is given by Eq. (13.261) where rn =r3sp, = 0.93172o0 is the radius of the CBsp3 shell. Substitution of Eqs. (13.766-13.767) into Eq. (13.261) gives
0' = 81.72° (13.772)
Then, the angle θ .. ,„ the radial vector of the Cl3sp3 HO makes with the internuclear axis
IS θa^Ho =180°-81.72° = 98.28° (13.773) as shown in Figure 25. The distance from the point of intersection of the orbitals to the internuclear axis must be the same for both component orbitals. Thus, the angle ωt - ΘH uo between the internuclear axis and the point of intersection of the H2 -type ellipsoidal MO with the CBsp3 radial vector obeys the following relationship:
>V ∞θatfm = 0-93172β° sin*W*o = ^in^0 (13.774) such that ^7755
Figure imgf000254_0001
with the use of Eq. (13.773). Substitution of Eq. (13.770) into Eq. (13.775) gives θHiMO = 35.28° (13.776)
Then, the distance dH^U0 along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by d H,Mo = a cos θHiM0 (13.777)
Substitution of Eqs. (13.766) and (13.776) into Eq. (13.777) gives d HxMo = 2.01235α0 =1.06489 X 10~10 m (13.778)
The distance dr._ 3 along the internuclear axis from the origin of each Cl atom to the point of intersection of the orbitals is given by da3ψ3m = dHiM0 -S (13.779)
Substitution of Eqs. (13.768) and (13.778) into Eq. (13.779) gives dmspi[10 = 0.13417α0 = 7.10022 X 10~12 m (13.780)
As shown in Eq. (13.760), a factor of 0.25 of the charge-density of the H2 -type ellipsoidal MO is distributed on each Cl3sp3 HO. Using the orbital composition of CZ2 (Eq. (13.760)), the radii of the ClIs = 0.05932α0 (Eq. (10.51)), Cl2s = 0.25344α0 (Eq. (10.62)), C/2/? = 0.31190α0 (Eq. (10.212)), and CBsp* = 0.93172α0 (Eq. (13.758)) shells, and the parameters of the CZ2 MO given by Eqs. (13.3-13.4), (13.766-13.768), and (13.770-13.771), the dimensional diagram and charge-density of the CZ2 MO comprising the linear combination of the H2-type ellipsoidal MO and two Cl3sp3 HOs according to Eq. (13.760) are shown in Figures 25 and 26, respectively. ENERGIES OF Cl2
The energies of Cl2 are given by the substitution of the semiprincipal axes (Eqs. (13.766- 13.767) and (13.770)) into the energy equations, (Eq. (13.763) and Eqs. (11.207-11.211) of H2) that are corrected for electron hybridization using Eq. ( 13.762) :
Ve -27.02007 eV (13.781)
Figure imgf000255_0001
Figure imgf000255_0002
E7, (35CZ2) = - eV = -31.63849 eF
Figure imgf000255_0003
(13.785) where E7, [Cl2) is given by Eq. (13.763) which is reiteratively matched to Eq. (13.75) within five-significant-figure round-off error.
VIBRATION AND ROTATION OF Cl2
In Cl2 , the division of the H2 -type ellipsoidal MO between the two Cl3sp3 HOs and the hybridization must be considered in determining the vibrational parameters. One approach is to use Eq. (13.761) for the force constant and r3 3 given by Eq. (13.758) for the distance parameter of the central force in Eq. (11.213) since the H2 -type ellipsoidal MO is energy matched to the Cl3sp3 HOs. With the substitution of the Cl2 parameters in Eqs. (11.213- 11.217), the angular frequency of the oscillation is
Figure imgf000256_0001
= 1.01438X 1014 rad/s where c' is given by Eq. (13.767), and the reduced mass of Cl2 is given by:
Figure imgf000256_0003
where /røp is the proton mass. Thus, during bond formation, the perturbation of the orbit determined by an inverse-squared force results in simple harmonic oscillatory motion of the orbit, and the corresponding frequency, ω{θ) , for 35CZ2 given by Eqs. (11.136), (11.148), and (13.786) is
1014 radiansls (13.788)
Figure imgf000256_0002
where the reduced nuclear mass of 35 CZ2 is given by Eq.(13.787) and the spring constant, A: (0) , given by Eqs. (11.136) and (13.786) is k (0) = 30\.19 Nm~l (13.789)
The 35CZ2 transition-state vibrational energy, Evib (θ) or ωe , given by Planck's equation (Eq.
(11.127)) is:
Evib (0) = ωe = hω = M.0U38 X 1014 radls = 0.06677 eV = 53%.52 cm'1 (13.790) ωe , from the experimental curve fit of the vibrational energies of 35Cl2 is [28]
0^ 559.7 CW"1 (13.791)
Using Eqs. (13.112-13.118) with Evjb (θ) given by Eq. (13.790) and D0 given by Eq. (13.807), the 35Cl2 ϋ = l -> ϋ = 0 vibrational energy, Evib (l) is
Evib (1) = 0.0659 eV (531.70 cm'1) (13.792) The experimental vibrational energy of 35Cl2 using ωe and ωexe [28] according to K&P [15] is Evib (l) = 0.0664 eV (535.55 cm"1) (13.793)
Using Eq. (13.113) with Evib (l) given by Eq. (13.792) and D0 given by Eq. (13.807), the anharmonic perturbation term, ωQx0 , of 35CZ2 is β)oxo = 3.41 cm"1 (13.794) The experimental anharmonic perturbation term, ωQxQ , of 35CZ2 [28] is
O)0X0 = 2.68 cm"1 (13.795)
The vibrational energies of successive states are given by Eqs. (13.790), (13.112), and (13.794).
Using Eqs. (13.133-13.134) and the internuclear distance, r = 2c' , and reduced mass of 35CZ2 given by Eqs. (13.768) and (13.787), respectively, the corresponding Be is
Be = 0.2420 cm'1 (13.796)
The experimental Be rotational parameter of 35Cl2 is [28]
5^ 0.2440 Cm"1 (13.797)
THE DOPPLER ENERGY TERMS OF Cl2
The equations of the radiation reaction force of the symmetrical Cl2 MO are the given by Eqs. (11.231-11.233) with the substitution of the Cl2 parameters and the substitution of the force factor of Eq. (13.761). The angular frequency of the reentrant oscillation in the transition state is
Figure imgf000257_0001
where a is given by Eq. (13.766). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)):
Eκ = hω = hi.6.31418 X 1015 radls = 4.15610 eV (13.799)
In Eq. (11.181), substitution of the total energy of Cl2, Ex (Cl2) , (Eq. (13.764)) for E1n, , the mass of the electron, me, for M , and the kinetic energy given by Eq. (13.799) for Eκ gives the Doppler energy of the electrons for the reentrant orbit:
Figure imgf000258_0001
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of Cl2 due to the reentrant orbit in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by
Eq. (13.800) and EKvιb , the average kinetic energy of vibration which is 1/2 of the vibrational energy of Cl2. Using the experimental 35CZ2 ωe of 559.7 cm"1 (0.06939 eF) [28] for E '1Kvib of the transition state, Egsc (35ClA is
(13.801)
Eosc (35CZ2) = -0.12759 eV +-(0.06939 eV) = -0.09289 eV (13.802)
TOTAL AND BOND ENERGIES OF CZ2
Eτ+osc i^sCl2^ , the total energy of the 35Cl2 radical including the Doppler term, is given by the
sum of Eτ (Cl2) (Eq. (13.764)) and Eosc (35Cl2) given by Eq. (13.802):
ET+OSC [ 35Cl2) = Ve +T + Vm + Vp + ECmhmb (Cl,3sp3) + Eosc (35Cl2)
(13.803) = ET (Cl2) + Eosc (35Cl2)
Figure imgf000258_0002
= -31.63537
Figure imgf000258_0003
(13.804)
From Eqs. (13.801-13.804), the total energy of 35Cl2 is ET+0SC (35Cl2) = -3l.63537 eV + E050(35Cl2)
= -31.63537 eF-0.12759 eF+-(0.06939 eF) (13.805)
= -31.72826 e V
where the experimental ωe (Eq. (13.791)) was used for the h i — term.
The Cl2 bond dissociation energy, ED (35Cl2 ) , is given by the difference between the
total energies of the two Cl3sps HOs and £r+osc ( 35CZ2 ) :
ED (35O2) = 2ECoulomb (Cl,3sp3 )-ET,OSC (35CZ2) (13.806)
ECoulomb (Cl,3sp3) is . given by Eq. (13.759); thus, the 35Cl2 bond dissociation energy,
E0 (35Cl2) , given by Eqs. (13.759) and (13.805-13.806) is
ED (35Cl2) = -2(14.60295 eV)- Eτ,osc (35Cl2)
= -29.20590 eV -(-31.72826 eV) (13.807)
= 2.52236 eV
The experimental 35Cl2 bond dissociation energy is [49] ED (35Cl2) = 2.51412 eV (13.808)
The results of the determination of bond parameters of Cl2 are given in Table 13.1.
The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
CARBON NITRIDE RADICAL
The carbon nitride radical can be formed by the reaction of carbon and nitrogen atoms:
C + N → CN (13.809)
The bond in carbon nitride radical comprises a H2 -type molecular orbital (MO) with two paired electrons. The force balance equations and radii, r6 and rη , of the 2p shell of C and N are derived in the Six-Electron Atoms section and Seven-Electron Atoms section, respectively. With the formation of the H2 -type MO by the contribution of a 2p electron from each of the C and N atoms, a diamagnetic force arises between the remaining 2p electrons of each atom and the H2 -type MO. This force from each atom causes the H2 -type MO to move to greater principal axes than would result with the Coulombic force alone. But, the integer increase of the central field and the resulting increased Coulombic as well as magnetic central forces on the remaining 2p electrons of each atom decrease the radii of the corresponding shells such that the energy minimum is achieved that is lower than that of the reactant atoms. The resulting electron configuration of CN is
Cls2N\s2C2s2N2s2C2ptN2p2σC t N where σ designates the H2 -type MO5 and the orbital arrangement is
(13.810)
Figure imgf000260_0001
The carbon nitride radical is predicted to be weakly paramagnetic .
FORCE BALANCE OF THE 2p SHELL OF THE CARBON ATOM OF THE
CARBONNITRIDERADICAL For the C atom, force balance for the outermost 2p electron of CN (electron 5) is achieved between the centrifugal force and the Coulombic and magnetic forces that arise due to interactions between electron 5 and the 2s -shell electrons due to spin and orbital angular momentum. The forces used are derived in the Six-Electron Atoms section. The central Coulomb force on the outer-most 2p shell electron of CN (electron 5) due to the nucleus and the inner four electrons is given by Eq. (10.70) with the appropriate charge and radius:
(13.811)
Figure imgf000260_0002
for r > r4. The 2p shell possess an external electric field given by Eq. (10.92) for r > r5.
The single unpaired carbon 2p electron gives rise to a diamagnetic force on the σ - MO as given by Eqs. (13.835-13.839). The corresponding Newtonian reaction force cancels
F d,iamagnetic , of Eq. (10.82). The energy is minimized with conservation of angular momentum. This condition is met when
^,« = 0 (13.812)
And, Fmag 2 corresponding to the maximum orbital angular momentum of the three 2p orbitals given by Eq. (10.89) is
mag 2 /φ + l)ir (13.813)
Figure imgf000261_0001
The electric field external to the 2p shell given by Eq. (10.92) for r > r5 gives rise to a second diamagnetic force, Vdiamagnelic 2, given by Eq. (10.93). Fdiamagnelic 2 due to the binding of the p-orbital electron having an electric field of +1 outside of its radius is :
diamagnetic 2 (13.814)
Figure imgf000261_0002
The radius of the 2p shell is calculated by equating the outward centrifugal force to the sum of the electric (Eq. (13.811)) and diamagnetic (Eqs. (13.812) and (13.814)), and paramagnetic (Eq. (13.813)) forces as follows:
Figure imgf000261_0003
Substitution of v5 = (Eq. (1.56)) and s = — into Eq. (13.815) gives: mer5 2
Figure imgf000261_0004
The quadratic equation corresponding to Eq. (13.816) is
Figure imgf000261_0005
(13.817)
The solution of Eq. (13.817) using the quadratic formula is:
Figure imgf000262_0001
r3 in units of a0 The positive root of Eq. (13.818) must be taken in order that r5 > 0 . Substitution of
-^- = 0.84317 (Eq. (10.62) with Z = 6) into Eq. (13.818) gives an r5 = 0.88084α0 (13.819)
FORCE BALANCE OF THE 2p SHELL OF THE NITROGEN ATOM OF
THE CARBON NITRIDE RADICAL For the N atom, force balance for the outermost 2p electron of CN (electron 6) is achieved between the centrifugal force and the Coulombic and magnetic forces that arise due to interactions between electron 6 and the other 2p -shell as well as the 2s -shell electrons due to spin and orbital angular momentum. The forces used are derived in the Seven-Electron Atoms section. The central Coulomb force on the outer-most 2p shell electron of CN (electron 6) due to the nucleus and the inner five electrons is given by Eq. (10.70) with the appropriate charge and radius:
_ (Z-5)e2 *«fe - ~ 2 l *r (13.820)
AnSnT6 for r > r5 , The 2p shell possess an external electric field given by Eq. (10.92) for r > r6.
The forces to determine the radius of the N2p shell of N in CN are the same as those of N in TV2 except that in CN there is a contribution from the Newtonian reaction force that arises from the single unpaired carbon 2 p electron. The energy is minimized with conservation of ang °ular momentum. This condition is met when F d .i.amagnetic of JV in CiV is canceled by the σ-MO -reaction force. Eq. (13.622) becomes
Frf,»,;c = 0 (13.821)
And, Fmag 2 corresponding to the conserved orbital angular momentum of the three orbitals given by Eq. (10.89) is
Figure imgf000263_0001
mas2 Z msfo
The electric field external to the 2p shell given by Eq. (10.92) for r > r6 gives rise to a second diamagnetic force, Fdiamagnetic 2 , given by Eq. (10.93). Fdlamagnetic 2 due to the binding of the p-orbital electron having an electric field of +1 outside of its radius is
Figure imgf000263_0002
The JV forces Fele , Fmag 2 , Fdiamagmtic 2 , and Fmag 3 of CN axe the same as those of TV2 given by Eqs. (13.621) and (13.623-13.624), respectively. In both cases, the contribution of a 2p electron from the JV atom in the formation of the σ MO gives rise to a paramagnetic force on the remaining two 2p electrons that pair. Thus, the force, Fmαg 3 of CN , given by Eq. (13.625) is
(13.824)
Figure imgf000263_0005
The radius of the 2p shell is calculated by equating the outward centrifugal force to the sum of the electric (Eq. (13.820)) and diamagnetic (Eqs. (13.821) and (13.823)), and paramagnetic (Eqs. (13.822) and (13.824)) forces as follows:
Figure imgf000263_0003
Substitution of v* = (Eq. (1.56)) and s = — into Eq. (13.626) gives: m,r6
Figure imgf000263_0004
(13.826) The quadratic equation corresponding to Eq. (13.826) is
Figure imgf000264_0001
(13.827)
The solution of Eq. (13.827) using the quadratic formula is:
Figure imgf000264_0002
r3 in units of a0 The positive root of Eq. (13.828) must be taken in order that r6 > 0 . Substitution of
-^- = 0.69385 (Eq. (10.62) with Z = 7) into Eq. (13.828) gives
Ctn r6 = 0.76366α0 (13.829)
ENERGIES OF TKDE Ip SHELLS OF THE CARBON AND NITROGEN
ATOMS OF THE CARBON NITRIDE RADICAL
The central forces on the 2p shell of the C and N atoms are increased with the formation of the σ MO which reduces each shell's radius and increases its total energy. The Coulombic energy terms of the total energy of the C and N atoms at the new radii are calculated and added to the energy of the σ MO to give the total energy of CN . Then, the bond energy is determined from the total CN energy.
The radius r6 of the carbon atom before bonding is given by Eq. (10.122):
Figure imgf000264_0003
Using the initial radius r6 of the C atom and the final radius rs of the C2p shell of CN (Eq. (13.819)) and by considering that the central Coulombic field decreases by an integer for each successive electron of the shell, the sum ET (CN,C2p) , of the Coulombic energy change of the C2p electron is determined using Eq. (10.102):
ET {CN,C2p) = -8.33948 eV
Figure imgf000265_0001
(13.831)
The radius r7 of the nitrogen atom before bonding is given by Eq. (10.142): r7 = 0.93084α0 (13.832)
Using the initial radius r7 of the N atom and the final radius r6 of the NIp shell of CN
(Eq. (13.829)) and by considering that the central Coulombic field decreases by an integer for each successive electron of the shell, the sum ET (CN,N2p) of the Coulombic energy change of the N2p electron is determined using Eq. (10.102):
Figure imgf000265_0003
= -(13.60580 eF)(0.23518)(2 + 3) (13.833)
= -15.99929 e V
FORCE BALANCE OF THE σ MO OF THE CARBON NITRIDE RADICAL The diamagnetic force VdUmagmticMOl for the σ-MO of the CTV molecule due to the two paired electrons in the N2p shell given by Eq. (13.633) with ne = 2 is:
^dicmagneticMOl = ~ Ϊ7T DH (13.834)
/.mea b
The force ^dlamagneticM02 is given by Eq. (13.634) except that the force is the summed over the individual diamagnetic-force terms due to each component of angular momentum L1 acting on the electrons of the σ -MO from each atom having a nucleus of charge Zy at one of the foci of the σ -MO:
(13.835)
Figure imgf000265_0002
Using Eqs. (11.200), (13.633-13.634), and (13.834-13.835), the force balance for the σ -MO of the carbon nitride radical comprising carbon with charge Z1 = O and |-^| = ^ and I and J-ZL31 = % is
Figure imgf000266_0001
Figure imgf000266_0002
(13.839)
Figure imgf000266_0003
Substitution of Z1 = 6 and Z2 = 7 into Eq. (13.839) gives
« = 2.45386α0 = 1.29853 X 1(T10 w (13.840) Substitution of Eq. (13.840) into Eq. (11.79) is c' = 1.10767α0 = 5.86153 X 1(T11 m (13.841) The internuclear distance given by multiplying Eq. (13.841) by two is
2c' = 2.21534αo = i:i7231 X lO"10 m (13.842) The experimental bond distance from Ref. [28] is
2c' =1.17181 X lO"10 m (13.843)
Substitution of Eqs. (13.840-13.841) into Eq. (11.80) is i = c = 2.18964αo = 1.15871 X lO"10 m (13.844)
Substitution of Eqs. (13.840-13.841) into Eq. (11.67) is e = 0.45140 (13.845)
Using the electron configuration of CN (Eq. (13.810), the radii of the Cl^ = 0.17113α0 (Eq.
(10.51)), C2s = 0.84317α0 (Eq. (10.62)), CIp = 0.88084α0 (Eq. (13.819)), Ms = 0.14605α0 (Eq. (10.51)), N2s = 0.69385α0 (Eq. (10.62)), and N2p = 0.76366aQ (Eq (13.829)) shells and the parameters of the σ MO of CN given by Eqs. (13.3-13.4), (13.840-13.842), and (13.844-13.845), the dimensional diagram and charge-density of the CiV MO are shown in Figures 27 and 28, respectively.
SUM OF THE ENERGIES OF THE σ MO AND THE AOs OF THE CARBON NITRIDE RADICAL
The energies of the CN σ MO are given by the substitution of the semiprincipal axes (Eqs. (13.840-13.841) and (13.844)) into the energy equations (Eqs. (11.207-11.212)) of H2 :
Figure imgf000267_0001
ET = Ve +T + Vm +Vp (13.850)
Substitution of Eqs. (11.79) and (13.846-13.849) into Eq. (13.850) gives
Eτ {CN,σ) -9.18273 eV (13.851)
Figure imgf000267_0002
where Eτ (CN,σ) is the total energy of the σ MO of CN . The sum, ET (CN) , of JEr (CN,C2p) , the C2p AO contribution given by Eq. (13.831), ET (CN,N2p), the N2p AO contribution given by Eq. (13.833), and Eτ (CN,σ) , the σ MO contribution given by
Eq. (13.851) is:
Eτ [CN) = E7 (CN, C2p) + Eτ (CN, N2p) +Eτ (N2,σ)
= -8.33948 eF-15.99929 eF-9.18273 eV (13.852) = -33.52149 eV VIBRATION OF CN
The vibrational energy levels of CN may be solved by determining the Morse potential curve from the energy relationships for the transition from a C atom and N atom whose parameters are given by Eqs. (10.115-10.123) and (10.134-10.143), respectively, to a C atom whose parameter r5 is given by Eq. (10.819), a N atom whose parameter r6 is given by Eq.
(13.829), and the σ MO whose parameters are given by Eqs. (13.840-13.842) and (13.844- 13.845). As shown in the Vibration of Hydrogen-type Molecular Ions section, the harmonic oscillator potential energy function can be expanded about the internuclear distance and expressed as a Maclaurin series corresponding to a Morse potential after Karplus and Porter (K&P) [15] and after Eq. (11.134). Treating the Maclaurin series terms as anharmonic perturbation terms of the harmonic states, the energy corrections can be found by perturbation methods.
THE DOPPLER ENERGY TERMS OF THE CARBON NITRIDE RADICAL The equations of the radiation reaction force of CTV are the same as those of H2 with the substitution of the CN parameters. Using Eqs. (11.231-11.233), the angular frequency of the reentrant oscillation in the transition state is
I 1Λ01166 radls (13.853)
Figure imgf000268_0001
where a is given by Eq. (13.840). The kinetic energy, Eκ, is given by Planck's equation (Eq. (11.127)):
Eκ = hω = M.07550X 1016 radls = 7.07912 eV (13.854)
In Eq. (11.181), substitution of ET [CN) for Ehv , the mass of the electron, me, for M , and the kinetic energy given by Eq. (13.854) for Eκ gives the Doppler energy of the electrons of the reentrant orbit:
ED s E,W eV (13.855)
Figure imgf000268_0002
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of the CN MO due to the reentrant orbit in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by Eq. (13.855) and EKvib , the average kinetic energy of vibration which is 1/2 of the vibrational energy. Using the experimental CN ωe of 2068.59 cm'1 (0.25647 eV) [28] for EKvjb of the transition state, EOSC [CN) is
Figure imgf000269_0001
Eosc (CN) = -0.17684 eV + -(0.25647 eV) = -0.04860 eV (13.857)
J*
TOTAL AND BOND ENERGIES OF THE CARBON NITRIDE RADICAL
ET+0SC [CN) , the total energy of CN including the Doppler term, is given by the sum of Eτ (CN) (Eq. (13.852)) and Eosc (CN) given by Eq. (13.857):
Eτ÷0SC (CN) = Ve +T + Vm + Vp +Eτ (CN,C2p) + Eτ (CN,N2p) + E0SC (CN)
= Eτ (CN, σ)+Eτ (CN, C2p) + Eτ (CN, N2p) + Eosc (CN) (13.858)
= ET (CN) + EOSC (CN)
Figure imgf000269_0002
From Eqs. (13.856-13.859), the total energy of the CN MO is ^W (CN) = -33.52149 eV + Eosc (CN)
= -33.52149 eV -0, Vi '684 eV >-(0.25647 eV) (13.860) = -33.56970 eV where the experimental ω was used for the h. \ — term.
The CN bond dissociation energy, ED (CN) , is given by the difference between the sum of the energies of the C and N atoms and Eτ+osc (CN) :
ED (CN) = E(C) + E(N) - ET+0SC (CN) (13.861) where the energy of a carbon atom is [6]
E(C) = -U.2603O eV (13.862) and the energy of a nitrogen atom is [6]
E(N) = -14.53414 eV (13.863)
Thus, the CN bond dissociation energy, ED (CN) , given by Eqs. (13.860-13.863) is
ED (CN) = -(11.26030 eV + 14.53414 eV)-ET+osc (CN) = -25.79444 eV - (-33.5691O eV) (13.864)
= 7.77526 eV The experimental CN bond dissociation energy is [50]
ED29S (CN) = 7.773I eV (13.865)
The results of the determination of bond parameters of CiV are given in Table 13.1. The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
CARBON MONOXIDE MOLECULE
The carbon monoxide molecule can be formed by the reaction of carbon and oxygen atoms: C + O → CO (13.866)
The bond in the carbon monoxide molecule comprises a double bond, a H2 -type molecular orbital (MO) with four paired electrons. The force balance equation and radius r6 of the 2p shell of C is derived in the Six-Electron Atoms section. The force balance equation and radius rs of the 2p shell of O is derived in the Eight-Electron Atoms section. With the formation of the H2 -type MO by the contribution of two 2p electrons from each of the C and O atoms, a diamagnetic force arises between the remaining outer shell atomic electrons, the 2s electrons of C and the 2p electrons of O , and the H2 -type MO. This force from C and O causes the H2 -type MO to move to greater principal axes than would result with the
Coulombic force alone. But, the factor of two increase of the central field and the resulting increased Coulombic as well as magnetic central forces on the remaining O2p electrons decrease the radius of the corresponding shell such that the energy minimum is achieved that is lower than that of the reactant atoms. The resulting electron configuration of CO is CIs20Is2CIs202s202 p2 σc 4 p where σ designates the H2 -type MO, and the orbital arrangement is σ state
Figure imgf000271_0001
2p state
s state
(13.867)
s state
Figure imgf000271_0003
Figure imgf000271_0002
Carbon monoxide is predicted to be diamagnetic in agreement with observations [42].
FORCE BALANCE OF THE 2p SHELL OF THE OXYGEN ATOM OF THE
CARBONMONOXIDEMOLECULE
For the O atom, force balance for the outermost 2p electron of CO (electron 6) is achieved between the centrifugal force and the Coulombic and magnetic forces that arise due to interactions between electron 6 and the other 2p electron as well as the 2s -shell electrons due to spin and orbital angular momentum. The forces used are derived in the Eight-Electron Atoms section. The central Coulomb force on the outer-most 2p shell electron of CO (electron 6) due to the nucleus and the inner five electrons is given by Eq. (10.70) with the appropriate charge and radius:
^ =^^-K (13-868) for r > r5. The 2p shell possess a +2 external electric field given by Eq. (10.92) for r > r6.
The energy is minimized with conservation of angular momentum. This condition is met when the diamagnetic force, Hάamagnellc , of Eq. (10.82) due to the p -orbital contribution is given by:
F diamagnetic (13.869)
Figure imgf000272_0001
And, Fmag 2 corresponding to the conserved spin and orbital angular momentum given by Eq. (10.157) is
mag 2 (13.870)
Z mer6 r3
The electric field external to the 2p shell given by Eq. (10.92) for r > r6 gives rise to a second diamagnetic force, ¥ώamagnehc 2 , given by Eq. (10.93). ¥ώamagmlιc 2 due to the binding of the p-orbital electron having an electric field of +2 outside of its radius is :
F diamagnetic 2 (13.871)
Figure imgf000272_0002
In addition, the contribution of two 2p electrons in the formation of the σ molecular orbital (MO) gives rise to a paramagnetic force on the remaining paired 2p electrons. The force Fmag 3 is given by Eq. (13.625) wherein the radius is r6 :
F n2 mag 3 (13.872)
Arn ej',6
The radius of the 2p shell is calculated by equating the outward centrifugal force to the sum of the electric (Eq. (13.868)) and diamagnetic (Eqs. (13.869) and (13.871)), and paramagnetic (Eqs. (13.870) and (13.872)) forces as follows:
Figure imgf000272_0003
Substitution of v6 = ■ n (Eq. (1.56)) and s = - into Eq. (13.873) gives: mer6
Figure imgf000272_0004
(13.874)
The quadratic equation corresponding to Eq. (13.874) is
Figure imgf000273_0001
The solution of Eq. (13.875) using the quadratic formula is:
Figure imgf000273_0002
r3 in units of a0 The positive root of Eq. (13.876) must be taken in order that r6 > 0. Substitution of
-^- = 0.59020 (Eq. (10.62) with Z = 8) into Eq. (13.876) gives an r6 = 0.68835α0 (13.877)
ENERGIES OF THE 2s AND 2P SHELLS OF THE CARBON ATOM AND THE 2p SHELL OF THE OXYGEN ATOM OF THE CARBON MONOXIDE MOLECULE
With the formation of the H2 -type MO by the contribution of two 2p electrons from the C atom, the remaining outer-shell atomic electrons comprise the 2s electrons, which are unchanged by bonding with oxygen. However, the total energy of the CO molecule, which is subtracted from the sum of the energies of the carbon and oxygen atoms to determine the bond energy, is increased by the ionization energies of C+ and O+ given by Eqs. (10.113- 10.114) and (10.152-10.153), respectively. Experimentally, the energies are [6]
EQonization; C+) = 24.38332 eV (13.878) EQonization; O+) = 35.11730 eV (13.879)
In addition, the central forces on the 2p shell of the O atom are increased with the formation of the σ MO, which reduces the shell's radius and increases its total energy. The
Coulombic energy terms of the total energy of the O atom at the new radius are calculated and added to the ionization energies of C+ and O+ , and the energy of the σ MO to give the total energy of CO . Then, the bond energy is determined from the total CO energy.
The radius r8 of the oxygen atom before bonding is given by Eq. (10.162): rs = a0 (13.880)
Using the initial radius rs of the O atom and the final radius r6 of the O2p shell (Eq. (13.877)) and by considering that the central Coulombic field decreases by an integer for each successive electron of the shell, the sum Eτ (O,2p) of the Coulombic energy change of the O2p electrons of the O atom is determined using Eq. (10.102):
Figure imgf000274_0001
= -(13.60580 eF)(0.45275)(3 + 4) (13.881)
= -43.11996 eV
FORCE BALANCE OF THE σ MO OF THE CARBON MONOXIDE MOLECULE
The force balance can be considered due to a second pair of two electrons binding to a molecular ion having +2e at each focus and a first bound pair. Then, the forces are the same as those of a molecule ion having +e at each focus. The diamagnetic force FdiamagneUcMOl for the σ -MO of the CO molecule due to the two paired electrons in each of the C2s and 02 p shells is given by Eq. (13.633) with ne = 2 :
^diamagmI,cMOl = -^—ψ:Dh (13.882)
The force FaamagnelicM02 is given by Eqs. (13.634) and (13.835) as the sum of the contributions due to carbon with Z = Zx and oxygen with Z = Z2. VdiamagneljcM01 for CO with Z, = h is
vdiamagneticMo2 <13-883)
Figure imgf000274_0002
The force balance equation for the σ -MO of the carbon monoxide molecule given by Eqs.
(11.200), (13.633-13.634), and (13.882-13.883) is
(.3.8S4)
Figure imgf000275_0003
mea2b2
Figure imgf000275_0001
("-BBS)
Figure imgf000275_0002
Substitution of Z1 = 6 and Z2 = 8 into Eq. (13.887) gives α = 2.29167α0 = 1.21270 X 10"10 m (13.888)
Substitution of Eq. (13.888) into Eq. (11.79) is c' = l.O7O44αo = 5.66450 X lO"11 m (13.889)
The internuclear distance given by multiplying Eq. (13.889) by two is
2c' = 2.14087tf0 = 1.13290 X 10"10 m (13.890)
The experimental bond distance is [28]
2c' = 1.12823 X 10"10 m (13.891) Substitution of Eqs. (13.888-13.889) into Eq. (11.80) is ό = c = 2.0263Oa0 = 1.07227 X lO'10 m (13.892)
Substitution of Eqs. (13.888-13.889) into Eq. (11.67) is e = 0.46710 (13.893)
Using the electron configuration of CO (Eq. (13.867)), the radii of the CIs = 0.17113α0 (Eq. (10.51)), C2s = 0.84317α0 (Eq. (10.62)), Ob = 0.12739α0 (Eq. (10.51)), O2s = 0.59020α0 (Eq. (10.62)), and O2;? = 0.68835α0 (Eq. (13.877)) shells and the parameters of the σ MO of CO given by Eqs. (13.3-13.4), (13.888-13.890), and (13.892-13.893), the dimensional diagram and charge-density of the CO MO are shown in Figures 29 and 30, respectively. SUM OF THE ENERGIES OF THE σ MO AND THE AOs OF THE CARBON MONOXIDE MOLECULE
The energies of the CO σ MO are given by the substitution of the semiprincipal axes (Eqs. (13.888-13.889) and (13.892)) into the energy equations (Eqs. (11.207-11.212)) of H2 except that the terms based on charge are multiplied by four and the kinetic energy term is multiplied by two due to the σ -MO double bond with two pairs of paired electrons:
Figure imgf000276_0001
(13.896)
Figure imgf000276_0002
Figure imgf000276_0003
F = V +T+V +V (13.898)
Substitution of Eqs. (11.79) and (13.894-13.897) into Eq. (13.898) gives
Figure imgf000276_0004
where Eτ (CO, σ) is the total energy of the σ MO of CO . The total energy of CO5 Eτ [Cθ) , is given by the sum of E(ionization; C+) , the energy of the second electron of carbon (Eq. (13.878)) donated to the double bond, E(ionization; O+) , the energy of the second electron of oxygen (Eq. (13.879)) donated to the double bond, Eτ (θ,2p) , the O2p AO contribution due to the decrease in radius with bond formation (Eq. (13.881)), and Ej. (CO, σ) , the σ MO contribution given by Eq. (13.899):
Eτ (CO) = Eiionization; C+ ) + EQomzation; 0+) + Eτ (0,2p) + Eτ (C0,σ)
= 24.38332 eF + 35.11730 eF-43.11996 eF-52.13425 eF (13.900)
= -35.75359 eV VIBRATION OF CO
The vibrational energy levels of CO may be solved by determining the Morse potential curve from the energy relationships for the transition from a C atom and O atom whose parameters are given by Eqs. (10.115-10.123) and (10.154-10.163), respectively, to a C atom whose parameter r4 is given by Eq. (10.61), an O atom whose parameter r6 is given by Eq.
(13.877), and the σ MO whose parameters are given by Eqs. (13.888-13.890) and (13.892- 13.893). As shown in the Vibration of Hydrogen-type Molecular Ions section, the harmonic oscillator potential energy function can be expanded about the internuclear distance and expressed as a Maclaurin series corresponding to a Morse potential after Karplus and Porter (K&P) [15] and after Eq. (11.134). Treating the Maclaurin series terms as anharmonic perturbation terms of the harmonic states, the energy corrections can be found by perturbation methods.
THE DOPPLER ENERGY TERMS OF THE CARBON MONOXIDE MOLECULE
The equations of the radiation reaction force of carbon monoxide are the same as those of H2 with the substitution of the CO parameters except that there is a factor of four increase in the central force in Eq. (11.231) due to the double bond. Using Eqs. (11.231-11.233), the angular frequency of the reentrant oscillation in the transition state is
2.38335 X l O16 ra<//s (13.901)
Figure imgf000277_0001
where a is given by Eq. (13.888). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)):
Eκ = Λfl) = 82.38335 X lO16 rod Is = 15.68762 eV (13.902)
In Eq. (11.181), substitution of E7. [CO) for E1n, , the mass of the electron, me, for M , and the kinetic energy given by Eq. (13.902) for Eκ gives the Doppler energy of the electrons of the reentrant orbit:
I.
Figure imgf000277_0002
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of the CO MO due to the reentrant orbit in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by Eq. (13.903) and EKvtb , the average kinetic energy of vibration which is 1/2 of the vibrational energy. Using the experimental CO ωe of 2169.81 cm"1 (0.26902 eV) [28] for EKvib of the transition state, E\SC [CO) per bond is
E'osc [CO) = ED +EKvib = ED +U ' (13.904)
E \sc [CO) = -0.28016 eV +-(0.26902 eV) = -0.14564 eV (13.905)
Since the σ MO bond is a double bond with twice a many electrons as a single bond, E \sc [CO) is multiplied by two to give
E0Sc [CO) = -0,29129 eV (13.906)
TOTAL AND BOND ENERGIES OF THE CARBON MONOXIDE MOLECULE
Eτ+osc (CO) , the total energy of CO including the Doppler term, is given by the sum. of
E1. (CO) (Eq. (13.900)) and EOSC (CO) given by Eq. (13.906):
(13"907)
Figure imgf000278_0001
= E1 (CO) + Eosc (CO) Er+0SC (CO)
Figure imgf000279_0001
= -35.75359 eF-2(θ.28O16
Figure imgf000279_0002
From Eqs. (13.906-13.908), the total energy of the CO MO is
ET+OSC (CO) = -35.75359 eV + Eosc (CO)
= -35.75359 eV + (-029ϊ29 eV) (13.909)
= -36.04488 eF
where the experimental oe was used for the h i f —it" term.
The CO bond dissociation energy, ED (CO) , is given by the difference between the sum of the energies of the C and O atoms and Eτ+osc (CO) :
ED (CO) = E(C) + E(O)-ET+0SC (CO) (13.910) where the energy of a carbon atom is [6]
E(C) = -11.2603O eK (13.911) and the energy of an oxygen atom is [6]
E(O) = -13.61806 eV (13.912)
Thus, the CO bond dissociation energy, ED (CO) , given by Eqs. (13.909-13.912) is
ED (CO) = -(11.26030 eF + 13.61806 eV)-ET+0SC (CO)
= -24.87836 eV - (-36.04488 eV) (13.913)
= 11.16652 eV
The experimental CO bond dissociation energy is [49] ED29S (CO) = I 1.15696 e V . (13.914) The results of the determination of bond parameters of CO are given in Table 13.1. The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
NITRIC OXIDE RADICAL The nitric oxide radical can be formed by the reaction of nitrogen and oxygen atoms:
N + O → NO (13.915)
The bond in the nitric oxide radical comprises a double bond, a H2 -type molecular orbital (MO) with four paired electrons. The force balance equation and radius r7 of the 2p shell of N is derived in the Seven-Electron Atoms section. The force balance equation and radius r% of the 2p shell of O is derived in the Eight-Electron Atoms section. With the formation of the H2 -type MO by the contribution of two 2p electrons from each of the N and O atoms, a diamagnetic force arises between the remaining outer shell atomic electrons, the 2s and 2 p electrons of N and O, and the H1 -type MO. This force from N and O causes the H2- type MO to move to greater principal axes than would result with the Coulombic force alone. But, the factor of two increase of the central field and the resulting increased Coulombic as well as magnetic central forces on the remaining N and O electrons decrease the radii of the corresponding shells such that the energy minimum is achieved that is lower than that of the reactant atoms. The resulting electron configuration of NO is
Nls2Ols2N2s2O2s2N2p1O2p2σN 4 fi where σ designates the H2 -type MO5 and the orbital arrangement is
(13.916)
Figure imgf000281_0003
Nitric oxide is predicted to be weakly paramagnetic in agreement with observations [42].
FORCE BALANCE OF THE 2p SHELL OF THE NITROGEN ATOM OF THE NITRIC OXIDE RADICAL
For the N atom, force balance for the outermost 2p electron of NO (electron 5) is achieved between the centrifugal force and the Coulombic and magnetic forces that arise due to interactions between electron 5 and the 2s -shell electrons due to spin and orbital angular momentum. The forces used are derived in the Seven-Electron Atoms section. The central Coulomb force on the outer-most 2p shell electron of NO (electron 5) due to the nucleus and the inner four electrons is given by Eq. (10.70) with the appropriate charge and radius:
F*
Figure imgf000281_0001
for r > r4. The 2p shell possess a +2 external electric field given by Eq. (10.92) for r > r5.
The energy is minimized with conservation of angular momentum. This condition is met when the magnetic forces of N in NO are the same as those of N in the nitrogen molecule with r5 replacing r6 and with an increase of the central field by an integer. The diamagnetic force, FΛamagmlιc , of Eq. (10.82) due to the p -orbital contribution is given by Eq. (13.622) with r5 replacing r6 :
Figure imgf000281_0002
And, Fmag 2 corresponding to the conserved orbital angular momentum of the three orbitals is also the same as that of N2 given by Eq. (13.623) with r5 replacing r6 :
Figure imgf000282_0001
The electric field external to the 2p shell given by Eq. (10.92) for r > rs gives rise to a second diamagnetic force, Fdlamagmtlc 2, given by Eq. (10.93). FΛβBMpiββc 2 due to the binding of the p-orbital electron having an electric field of +2 outside of its radius follows from Eq. (13.624):
F d.iamagneltc 2 (13.920)
Figure imgf000282_0002
In addition to the N forces Fdβ, ¥diamagnetic , Fmag 2, and Vdιamagnelic 2 of NO being the same as N2 given by Eqs. (13.621-13.624), respectively, Fefø, Fmαg 2, and Vdiamagnetic 2 are also the same as those of CN (Eqs. (13.820) and (13.822-13.823)). In the N2 and CN cases, the contribution of a 2p electron from the N atom in the formation of the σ MO gives rise to an additional paramagnetic force on the remaining two 2p electrons that pair. However, the force, Fmag 3 , is absent in NO since the single outer electron is unpaired.
The radius of the 2 p shell is calculated by equating the outward centrifugal force to the sum of the electric (Eq. (13.917)) and diamagnetic (Eqs. (13.918) and (13.920)), and paramagnetic (Eq. (13.919)) forces as follows:
Figure imgf000282_0003
h 1
Substitution of v< = (Eq. (1.56)) and s = - into Eq. (13.921) gives: mer5 2
Figure imgf000282_0004
(13.922)
The quadratic equation corresponding to Eq. (13.922) is
Figure imgf000283_0001
(13.923)
The solution of Eq. (13.923) using the quadratic formula is:
Figure imgf000283_0002
r3 in units of aQ The positive root of Eq. (13.924) must be taken in order that r5 > 0 . Substitution of
'3 _ = 0.69385 (Eq. (10.62) with Z = I) into Eq. (13.924) gives an r5 = 0.74841α0 (13.925)
FORCE BALANCE OF THE 2p SHELL OF THE OXYGEN ATOM OF THE NITRIC OXIDE RADICAL
For the O atom, force balance for the outermost 2p electron of NO (electron 6) is achieved between the centrifugal force and the Coulombic and magnetic forces that arise due to interactions between electron 6 and the other 2p electron as well as the 2s -shell electrons due to spin and orbital angular momentum. The forces used are derived in the Eight-Electron Atoms section. The central Coulomb force on the outer-most 2p shell electron of NO (electron 6) due to the nucleus and the inner five electrons is given by Eq. (10.70) with the appropriate charge and radius:
_ (Z-5)e2
S l enllen — — ; 2 τ—l r (13.926)
4πε0r6 for r > r5. The 2p shell possess an external electric field of +2 given by Eq. (10.92) for r > r6. The energy is minimized with conservation of angular momentum. This condition is met when the diamagnetϊc force, ¥dlamagllellc , of Eq. (10.82) due to the p -orbital contribution is given by:
F d.iamagnetic (13.927)
Figure imgf000284_0001
And, Fmag 2 corresponding to the conserved spin and orbital angular momentum given by Eqs. (10.157) and (13.670) is
Figure imgf000284_0002
The electric field external to the 2p shell given by Eq. (10.92) for r > r6 gives rise to a second diamagnetic force, Vdiamagnelic 2 , given by Eq. (10.93). Ydiamagmlic 2 due to the binding of the p-orbital electron having an electric field of +2 outside of its radius is :
Figure imgf000284_0003
In addition, the contribution of two 2p electrons in the formation of the σ MO gives rise to a paramagnetic force on the remaining paired 2p electrons. The force Fmαg 3 is given by Eq. (13.625) wherein the radius is r6 :
Figure imgf000284_0004
The radius of the 2p shell is calculated by equating the outward centrifugal force to the sum of the electric (Eq. (13.926)) and diamagnetic (Eqs. (13.927) and (13.929)), and paramagnetic (Eqs. (13.928) and (13.930)) forces as follows:
Figure imgf000284_0005
Substitution of vΛ = (Eq. (1.56)) and s = - into Eq. (13.931) gives: m.r6
Figure imgf000284_0006
(13.932)
The quadratic equation corresponding to Eq. (13.932) is
Figure imgf000285_0001
The solution of Eq. (13.933) using the quadratic formula is:
Figure imgf000285_0002
r3 in units of aQ The positive root of Eq. (13.934) must be taken in order that r6 > 0 . Substitution of
-^- = 0.59020 (Eq. (10.62) with Z = 8) into Eq. (13.934) gives an r, = 0.7046Oan (13.935)
ENERGIES OF THE 2P SHELLS OF THE NITROGEN ATOM AND
OXYGEN ATOM OF THE NITRIC OXIDE RADICAL
With the formation of the H2 -type MO by the contribution of two 2p electrons from each of the N and O atoms, the total energy of the NO molecule, which is subtracted from the sum of the energies of the nitrogen and oxygen atoms to determine the bond energy, is increased by the ionization energies of N+ and O+ given by Eqs. (10.132-10.133) and (10.152- 10.153), respectively. Experimentally, the energies are [6]
E(ionization; N+) = 29.6013 eV (13.936)
E(ionization; O+) = 35.11730 eV (13.937)
In addition, the central forces on the 2p shells of the JV and O atoms are increased with the formation of the σ MO which reduces each shell's radius and increases its total energy. The Coulombic energy terms of the total energy of the N and O atoms at the new radii are calculated and added to the ionization energies of N+ and O+ , and the energy of the σ MO to give the total energy of NO . Then, the bond energy is determined from the total NO energy.
The radius rη of the nitrogen atom before bonding is given by Eq. (10.142): r7 = 0.93084α0 (13.938)
Using the initial radius rη of the N atom and the final radius r5 of the N2p shell (Eq. (13.925)) and by considering that the central Coulombic field decreases by an integer for each successive electron of the shell, the sum E1, (N, 2p) of the Coulombic energy change of the N2p electrons of the N atom is determined using Eq. (10.102):
(13.939)
Figure imgf000286_0001
= -10.68853 eV The radius r& of the oxygen atom before bonding is given by Eq. (10.162): r& = α0 (13.940) Using the initial radius rg of the O atom and the final radius r6 of the 02 p shell (Eq. (13.935)) and by considering that the central Coulombic field decreases by an integer for each successive electron of the shell, the sum E1, (O,2p) of the Coulombic energy change of the O2p electrons of the O atom is determined using Eq. (10.102):
(13.941)
Figure imgf000286_0002
= -39.92918 eV
FORCE BALANCE OF THE σ MO OF THE NITRIC OXIDE RADICAL
The force balance can be considered due to a second pair of two electrons binding to a molecular ion having +2e at each focus and a first bound pair. Then, the forces are the same as those of a molecule ion having +e at each focus. The diamagnetic force ^ώamagneUcMO\ f°r the σ -MO of the NO molecule due to the two paired electrons in the O2p shell is given by Eq. (13.633) with ne = 2 : h2
F L, diamagnelicMOl η 2 L2 -D 1^L* ξ (13.942)
^diamagmticMoi of the nitric oxide radical comprising nitrogen with charge Z1 = 7 and JX1I = ^
and = % is given by the corresponding sum of the
Figure imgf000287_0001
contributions. Using Eq. (13.835), F diamagneticMO 2 for NO is
diamagneticMOI (13.943)
Figure imgf000287_0002
The general force balance equation for the σ -MO of the nitric oxide radical given by Eqs. (11.200), (13.633-13.634), and (13.942-13.943) is the same as that of CN (Eq. (13.836)):
Figure imgf000287_0003
Substitution of Z1 = 7 and Z2 = 8 into Eq. (13.947) gives α = 2.39i58α0 = 1.26557X 10-10 m (13.948) Substitution of Eq. (13.948) into Eq. (11.79) is c' = 1.09352α0 = 5.78666 X 10~π m (13.949) The internuclear distance given by multiplying Eq. (13.949) by two is 2c' = 2.18704α0 = 1.15733 X 1(T10 m (13.950)
The experimental bond distance is [28]
2c' = 1.15077X 10-10 m (13.951)
Substitution of Eqs. (13.948-13.949) into Eq. (11.80) is b = c = 2Λ2693ao = 1.12552 X lO-10 m (13.952)
Substitution of Eqs. (13.948-13.949) into Eq. (11.67) is e = 0.45724 (13.953)
Using the electron configuration of NO (Eq. (13.916)), the radii of the NIs = 0.14605α0 (Eq.
(10.51)), N2s = 0.69385α0 (Eq. (10.62)), N2p = 0.7484Ia0 (Eq. (13.925)), Ols = 0.12739α0 (Eq. (10.51)), O2s = 0.5902Oa0 (Eq. (10.62)), and O2p = 0.7046Oa0 (Eq. (13.935)) shells and the parameters of the σ MO of NO given by Eqs. (13.3-13.4), (13.948-13.950), and (13.952-13.953), the dimensional diagram and charge-density of the NO MO are shown in Figures 31 and 32, respectively.
SUM OF THE ENERGIES OF THE σ MO AND THE AOs OF THE NITRIC OXIDE RADICAL
The energies of the NO σ MO are given by the substitution of the semiprincipal axes (Eqs. (13.948-13.949) and (13.952)) into the energy equations (Eqs. (11.207-11.212)) of H2 except that the terms based on charge are multiplied by four and the kinetic energy term is multiplied by two due to the σ -MO double bond with two pairs of paired electrons:
Ve -98.30623 eV (13.954)
Vp (13.955)
Figure imgf000288_0001
10.27631 eV (13.956)
Figure imgf000288_0002
Vn, eV (13.957)
Figure imgf000288_0003
ET = Ve +T + Vm +Vp (13.958)
Substitution of Eqs. (11.79) and (13.954-13.957) into Eq. (13.958) gives
Figure imgf000289_0001
where E7 [NO, σ) is the total energy of the σ MO of NO. The total energy of NO, E7 [NO) , is given by the sum of E (ionization; N+) , the energy of the second electron of nitrogen (Eq. (13.936)) donated to the double bond, E (ionization; O+) , the energy of the second electron of oxygen (Eq. (13.937)) donated to the double bond, E1. (N, 2 p) , the N2p AO contribution due to the decrease in radius with bond formation (Eq. (13.939)), ET (O,2p) , the Olp AO contribution due to the decrease in radius with bond formation
(Eq. (13.941)), and Eτ (N0,σ), the σ MO contribution given by Eq. (13.959):
. I E (ionization; N+) + E(ionization; O+) ^ ' ~ [+ET (N, 2p) + Eτ (0, 2p) + Eτ (NO, σ)^
Figure imgf000289_0002
VIBRATION OF NO
The vibrational energy levels of NO may be solved by determining the Morse potential curve from the energy relationships for the transition from a N atom and O atom whose parameters are given by Eqs. (10.134-10.143) and (10.154-10.163), respectively, to a N atom whose parameter r5 is given by Eq. (13.925), an O atom whose parameter r6 is given by Eq. (13.935), and the σ MO whose parameters are given by Eqs. (13.948-13.950) and (13.952.-13.953). As shown in the Vibration of Hydrogen-type Molecular Ions section, the harmonic oscillator potential energy function can be expanded about the internuclear distance and expressed as a Maclaurin series corresponding to a Morse potential after Karplus and Porter (K&P) [15] and after Eq. (11.134). Treating the Maclaurin series terms as anharmonic perturbation terms of the harmonic states, the energy corrections can be found by perturbation methods. THE DOPPLER ENERGY TERMS OF THE NITRIC OXIDE RADICAL
The equations of the radiation reaction force of nitric oxide are the same as those of H2 with the substitution of the NO parameters except that there is a factor of four increase in the central force in Eq. (11.231) due to the double bond. Using Eqs. (11.231-11.233) and (13.901), the angular frequency of the reentrant oscillation in the transition state is
Figure imgf000290_0001
where a is given by Eq. (13.948). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)):
Eκ = hω = h2.23557X 1016 radls = 14.71493 eV (13.962) In Eq. (11.181), substitution of E7. [NO) for Ehv , the mass of the electron, me, for M , and the kinetic energy given by Eq. (13.962) for Eκ gives the Doppler energy of the electrons of the reentrant orbit:
Figure imgf000290_0002
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of the NO MO due to the reentrant orbit in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by Eq. (13.963) and EKvιb , the average kinetic energy of vibration which is 1/2 of the vibrational energy. Using the experimental NO ωe of 1904.20 cm"1 (0.23609 eV) [28] for EKvώ of the transition state, E \sc [NO) per bond is
E\sc [NO) = ED + EKvιb = ED +U £ (13.964)
E\sc [NO) = -0.26134 eF+-(0.23609 eV) = -0.14329 eV (13.965)
Since the σ MO bond is a double bond with twice a many electrons as a single bond, E \sc [NO) is multiplied by two to give E0^ (NO) = -0.28658 eF (13.966) TOTAL AND BOND ENERGIES OF THE NITRIC OXIDE RADICAL ET+OSC (NO), the total energy of NO including the Doppler term, is given by the sum of Ex (NO) (Eq. (13.960)) and Eosc (Nθ) given by Eq. (13.966):
O+)"
Figure imgf000291_0001
Ετ (NO, σ) + E(ionization; N*) + E (ionization; O+Y
(13.967)
+ET (N, 2p) + Eτ (0, 2p) + Eosc (NO) = E7 (NO) + Eosc (NO)
Figure imgf000291_0002
From Eqs. (13.966-13.968), the total energy of the NO MO is
ET*. (NO) = -34.43653 eV + Eosc (NO) = -34.43653 eV + (-0.28658) (13.969) = -34.72312 eV
where the experimental ω was used for the h /— term.
The NO bond dissociation energy, ED (NO), is given by the difference between the sum of the energies of the N and O atoms and Eτ+osc (NO) :
ED (NO) ^ E(N) + E(O)- Eτ+osc (NO) (13.970) where the energy of a nitrogen atom is [6] E(N) = -14.53414 eV (13.871) and the energy of an oxygen atom is [6]
E(O) = -13.61806 e V (13.972)
Thus, the NO bond dissociation energy, ED (NO), given by Eqs. (13.969-13.972) is
ED (NO) = -(14.53414 eF + 13.61806 eV)-ET^0SC (NO) = -28.15220 67-(-34.72312 6F) (13.973)
= 6.57092 eV The experimental NO bond dissociation energy is [49]
ED29S (NO) = 6.5353 eV (13.974)
The results of the determination of bond parameters of NO are given in Table 13.1. The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
Table 13.1. The calculated and experimental bond parameters of H3 , D3 , OH , OD , H2O, D2O, NH , ND, NH2 , ND2, NH3 , ND3 , CH , CD, CH2 , CH3, CH4 , N2 , O2 , F2 , Cl2, CN , CO, and NO.
Figure imgf000292_0001
Figure imgf000293_0001
Figure imgf000294_0001
Figure imgf000295_0001
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45. R. Η. Christian, R. E. Duff, F. L. Yarger, "Equation of state of gases by shock wave measurements. II. The dissociation energy of nitrogen", J. Chem. Phys., Vol. 23, No. 11, (1955), pp. 2045-2049. 46. K. M. Ervin, I. Anusiewicz, P. Skurski, J. Simons, W. C. Lineberger, The only stable state of O2 is the X 2Tl g ground state and it (still!) has an adiabatic electron detachment energy of 0.45 eV, J. Phys. Chem. A, Vol. 107, (2003), pp. 8521-8529. 47. P. C. Cosby, D. L. Huestis, On the dissociation energy of O2 and the energy of the
Oj b 4 Σ~ state, J. Chem. Phys., Vol. 97, No. 9, (1992), pp. 6108-6112.
48. J. Yang, Y. Hao, J. Li, C. Zhou, Y. Mo, A combined zero electronic kinetic energy spectroscopy and ion-pair dissociation imaging study of the F2 +(X2Et^) structure, J. Chem. Phys., Vol. 122, No. 13, (2005), 134308-1-134308-7.
49. D. R. Lide, CRC Handbook of Chemistry and Physics, 79th Edition, CRC Press, Boca Raton, Florida, (1998-9), pp. 9-51 to 9-57.
50. D. R. Lide, CRC Handbook of Chemistry and Physics, 86th Edition, Taylor & Francis, Boca Raton, Florida, (2005-6), pp. 9-54 to 9-62.
Section III
MORE POLYATOMIC MOLECULES AND HYDROCARBONS
Additional polyatomic molecules can be solved using the same principles as those used to solve hydrogen molecular ions and molecules wherein the hydrogen molecular orbitals (MOs) and hydrogen atomic orbitals serve as basis functions for the MOs. The MO must (1) be a solution of Laplace's equation to give a equipotential energy surface, (2) correspond to an orbital solution of the Newtonian equation of motion in an inverse-radius-squared central field having a constant total energy, (3) be stable to radiation, and (4) conserve the electron angular momentum of h . Energy of the MO must be matched to that of the outermost atomic orbital of a bonding heteroatom in the case where a minimum energy is achieved with a direct bond to the AO. Alternatively, the MO is continuous with the AO containing paired electrons that do not particpate in the bond. Rather, they only provide a means for the energy matched MO to form a continuous equipotential energy surface, hi the case that an independent MO is formed, the AO force balance causes the remaining electrons to be at lower energy and a smaller radius. In another case, the atomic orbital may hybridize in order to achieve a bond at an energy minimum, and the sharing of electrons between two or more such orbitals to form a MO permits the participating hybridized orbitals to decrease in energy through a decrease in the radius of one or more of the participating orbitals. Representative cases were solved. Specifically, the results of the determination of bond parameters of carbon dioxide (CO2), nitrogen dioxide (NO2), ethane (CH3CH3), ethylene (CH1CH2), acetylene (CHCH), benzene (C6H6), propane (C3H8), butane (C4H10), pentane (C5H12 ), hexane (C6H14), heptane (C7H16), octane (C8H18), nonane (C9H20), decane (C10H22), undecane (CnH24), dodecane (C12H26), and octadecane (C18H38 ) are given in Table 14.1. The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
CARBON DIOXIDE MOLECULE The carbon dioxide molecule can be formed by the reaction of carbon monoxide and an oxygen atom: CO + O ^ CO2 (14.1)
Each equivalent bond in the carbon dioxide molecule comprises a double bond that is energy- matched to the filled C2s orbital. Each such bond comprises 75% of a H2 -type MO with four paired electrons as a basis set such that three electrons can be assigned to each C = O bond. Thus, the two C2p electrons combine with the four O2p electrons, two from each O, as a linear combination to form the two C = O bonds of CO2. The force balance equation and radius rs of the 2p shell of O is derived in the Eight-Electron Atoms section. With the formation of the H2 -type MOs by the contribution of two 2p electrons from each of the two O atoms, a factor of two increase of the central field on the remaining O2p electrons arises. The resulting increased Coulombic as well as magnetic central forces decrease the radii of the O2p shells such that the energy minimum is achieved that is lower than that of the reactant atoms. The resulting electron configuration of CO2 is C\s1Ox\s202\s2C2s2Oϊ2s1022s2Oι2p2022p2σ0 6 i C Oι where the subscripts designate the O atom, 1 or 2, σ designates the H2 -type MO, and the orbital arrangement is
(14.2)
Figure imgf000301_0001
Carbon dioxide is predicted to be diamagnetic in agreement with observations [I].
FORCE BALANCE OF THE 2p SHELL OF THE OXYGEN ATOM OF THE CARBONDIOXIDEMOLECULE For each O atom, force balance for the outermost 2p electron of CO2 (electron 6) is achieved between the centrifugal force and the Coulombic and magnetic forces that arise due to interactions between electron 6 and the other 2p electrons as well as the 2s -shell electrons due to spin and orbital angular momentum. The forces used are derived in the Eight-Electron Atoms section. The central Coulomb force on the outer-most 2p shell electron of CO (electron 6) due to the nucleus and the inner five electrons is given by Eq, (10.70) with the appropriate charge and radius:
Figure imgf000302_0001
for r > r5. The 2p shell possess a +2 external electric field given by Eq. (10.92) for r > r6. The energy is minimized with conservation of angular momentum. This condition is met when the diamagnetic force, Fd/omogBeήc , of Eq. (10.82) due to the p -orbital contribution is given by:
diamagnetic (14.4)
Figure imgf000302_0002
where 5 = 1/2 . And, ~Fmag 2 corresponding to the conserved spin and orbital angular momentum given by Eq. (10.157) is
Figure imgf000302_0003
The electric field external to the 2p shell given by Eq. (10.92) for r > r6 gives rise to a second diamagnetic force, FΛanagnetw 2 , given by Eq. (10.93). Fώamagnetιc 2 due to the binding of the p-orbital electron having an electric field of +2 outside of its radius is :
F ώ, amagnetic 2 (14.6)
Figure imgf000302_0004
The radius of the 2p shell is calculated by equating the outward centrifugal force to the sum of the electric (Eq. (14.3)) and diamagnetic (Eqs. (14.4) and (14.6)), and paramagnetic (Eq. (14.5)) forces as follows:
Figure imgf000302_0005
(14.7)
Substitution of v6 = (Eq. (1.56)) and s = - into Eq. (14.7) gives: mer6
Figure imgf000303_0001
The quadratic equation corresponding to Eq, (14.8) is
Figure imgf000303_0002
(14.9) The solution of Eq. (14.9) using the quadratic formula is:
Figure imgf000303_0003
r3 m units of a0 The positive root of Eq. (14.10) must be taken in order that r6 > 0 . Substitution of
-^- = 0.59020 (Eq. (10.62) with Z = 8) into Eq. (14.10) gives an r6 = 0.74776α0 (14.11)
ENERGIES OF THE 2s AND 2p SHELLS OF THE CARBON ATOM AND THE 2p SHELL OF THE OXYGEN ATOMS OF THE CARBON DIOXIDE MOLECULE
Consider the determination of the total energy of CO2 from the reaction of a carbon atom with two oxygen atoms. With the formation of the Tf2 -type MO by the contribution of two 2p electrons from the C atom, the remaining outer-shell atomic electrons comprise the 2,y electrons which are unchanged by bonding with two oxygen atoms. However, the total energy of the CO2 molecule, which is subtracted from the sum of the energies of the oxygen atom and carbon monoxide molecule to determine the O - CO bond energy, is increased by the ionization energies of C , C+ , O, and 2O+ given by Eqs. (14.12-14.15), respectively.
Experimentally, the energies are [2]
Eiionization; C) = W .26030 e V (14.12)
E(ionization; C+) = 24.38332 eV (14.13) EQonization; O) = 13.61806 eV (14.14)
EQonization; O+) = 35.11730 eV (14.15)
In addition, the central forces on the 2p shell of the O atom are increased with the formation of the σ MO which reduces the shell's radius and increases its total energy. The Coulombic energy terms of the total energy of each O atom at the new radius are calculated and added to the ionization energies of C , C+ , O, and 2O+ , and the energy of the σ MO to give the total energy of CO2. Then, the bond energy is determined from the total CO2 energy.
The radius rs of each oxygen atom before bonding is given by Eq. (10.162): rs = a0 (14.16) Using the initial radius rs of each O atom and the final radius r6 of the 02 p shell (Eq.
(14.11)) and by considering that the central Coulombic field decreases by an integer for each successive electron of the shell, the sum ET {O,2p) of the Coulombic energy change of the
O2p electrons of each O atom is determined using Eq. (10.102):
Figure imgf000304_0001
= -(13.60580 eK)(0.33733)(3 + 4) (14.17)
= -32.12759 eV
FORCE BALANCE OF THE σ MO OF THE CARBON DIOXIDE
MOLECULE
As in the case of H2 , the σ MO is a prolate spheroid with the exception that the ellipsoidal
MO surface cannot extend into the C atom for distances shorter than the radius of the C2,s shell; nor, can it extend into the O atom for distances shorter than the radius of the O2p shell. Thus, the MO surface of each C = O bond comprises a prolate spheroid that bridges and is continuous with the 2s and 2p shells of the O and C atoms whose nuclei serve as the foci. The energy of each prolate spheroid is matched to that of the C2.? and O2p shells. As in the case of previous examples of energy-matched MOs such as OH and NH , the
C = O -bond MO must comprise 75% of a H2 -type ellipsoidal MO in order to match potential, kinetic, and orbital energy relationships. However, the paired electrons of the C2s and 02 p shells are not involved in bonding. Rather, the AOs permit a continuous surface comprising the two C = O -bond MOs having six paired electrons, two from each of the C and the two O atoms:
2(0.75 H2 MO) → CO2 MO (14.18)
The force balance of the CO2 MO is determined by the boundary conditions that arise from the linear combination of orbitals according to Eq. (14.18) and the energy matching condition between the carbon and oxygen components of the MO.
Similar to the OH and H2O cases given by Eqs. (13.57) and (13.162), the H2-type ellipsoidal MO comprises 75% of the CO2 MO; so, the electron charge density in Eq. (11.65) is given by -0.75e . Thus, k} of the each H2 -type-ellipsoidal-MO component of the CO2
MO is given by Eq. (13.59). The distance from the origin of each C = O -bond MO to each focus c' is given by Eq. (13.60). The internuclear distance is given by Eq. (13.61). The length of the semiminor axis of the prolate spheroidal C = O -bond MO b = c given by Eq. (13.62). The eccentricity, e , is given by Eq. (13.63). Then, the solution of the semimajor axis a allows for the solution of the other axes of the prolate spheroidal and eccentricity of the CO2 MO.
The energy components of Ve , Vp, T , V1n , and E7. of the CO2 σ MO are the same as those of OH given by Eqs. (13.67-13.73), except that the terms based on charge are multiplied by four and the kinetic energy term is multiplied by two due to each σ -MO double bond:
Figure imgf000305_0001
where E7. (C = O, σ) is the total energy of each C = O σ MO of CO2. The total energy of a
H2 -type ellipsoidal MO is given by Εqs. (11.212) and (13.75). A minimum energy is obtained when each double bond of the σ MO of CO2 comprises the energy equivalent of four H2 -type ellipsoidal MOs. For each C = O bond to match the energy of the C2.s orbital, the ionization energy of C and C+ (Eqs. (14.12-14.13)) must be added for each bond of the double bond. Thus, the total energy of each C = O -bond MOs is
Figure imgf000306_0001
= 2(2(-31.63536831 eF) + 11.26030 eF + 24.38332 eK) (14.20)
= -55.25423 eV
Eτ (C = O5 σ) given by Eq. (14.19) is set equal to Eq. (14.20):
Figure imgf000306_0002
From the energy relationship given by Eq. (14.21) and the relationship between the axes given by Eqs. (13.60-13.63), the dimensions of the CO2 MO can be solved. Substitution of Eq. (13.60) into Eq. (14.21) gives
Eτ (C = O,σ) = e55.25423 eF (14.22)
Figure imgf000306_0003
The most convenient way to solve Eq. (14.22) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is α = 1.80703α0 = 9.56239 X 10~π m (14.23) Substitution of Eq. (14.23) into Eq. (13.60) is c' = 1.09758α0 = 5.80815 X 10"11 m (14.24) The internuclear distance given by multiplying Eq. (14.24) by two is 2c' = 2.19516αn = 1.16163 X 10"10 m (14.25)
The experimental bond distance is [3]
2c' = 1.1600X 10-10 m (14.26)
Substitution of Eqs. (14.23-14.24) into Eq. (13.62) is b = c = l.4355Oa0 = 7.59636 X lO"11 m (14.27)
Substitution of Eqs. (14.23-14.24) into Eq. (13.63) is e = 0.60740 (14.28)
The C and O nuclei comprise the foci of each H2 -type ellipsoidal MO defined as O = C = O . Consider the left-hand C = O bond of the two equivalent bonds in the absence of the right-hand bond. The parameters of the point of intersection of the H2 -type ellipsoidal
MO and the C2s AO are given by Eqs. (13.84-13.95) and (13.261-13.270). The polar intersection angle θ' is given by Eq. (13.261) where rn = r4 = 0.84317α0 is the radius of the
C2s shell. Substitution of Eqs. (14.23-14.24) into Eq. (13.261) gives 0' = 54.53° (14.29)
Then, the angle θC2sA0 the radial vector of the C2s AO makes with the intemuclear axis is
Θ C2SΛO = 180°-54.53° = l 25.47° (14.30) as shown in Figure 33. The distance from the point of intersection of the orbitals to the intemuclear axis must be the same for both component orbitals. Thus, the angle cot = ΘH Mo between the intemuclear axis and the point of intersection of the H2 -type ellipsoidal MO with the CIs radial vector obeys the following relationship: r 4 s™ec2sAo = 0.84317«0 ύnθC2sA0 = bύnθHiM0 (14.31) such that
V1^A)
Figure imgf000307_0001
with the use of Eq. (14.30). Substitution of Eq. (14.27) into Eq. (14.32) gives
6^o = 28.58° (14.33)
Then, the distance dH u0 along the intemuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by du2Mo = a∞sθHJM) (14.34) Substitution of Eqs. (14.23) and (14.33) into Eq. (14.34) gives dHiM0 = 1.58687β0 = 8.39737 X Wn m (14.35)
The distance dC2sAO along the intemuclear axis from the origin of the C atom to the point of intersection of the orbitals is given by dC2sAO = ^H2MO - 0 ' (14-36) Substitution of Eqs. (14.24) and (14.35) into Eq. (14.36) gives dC2sΛ0 = 0.48929α0 = 2.58922 X lO"11 m (14.37)
The C and O nuclei comprise the foci of each H2 -type ellipsoidal MO defined as
O - C = O . Consider the right-hand C - O bond of the two equivalent bonds. The parameters of the point of intersection of the H2 -type ellipsoidal MO and the O2p AO are given by Eqs. (13.84-13.95) and (13.261-13.270). The polar intersection angle 0' is given by
Eq. (13.261) where rn = r6 = 0.7 '477 '6a0 is the radius of the O2p shell. Substitution of Eqs. (14.23-14.24) into Eq. (13.261) gives
0' = 30.18° (14.38) Then, the angle θ02pA0 the radial vector of the 02 p AO makes with the internuclear axis is θ02pA0 =180°-30.18° = 149.82° (14.39) ' as shown in Figure 33. The distance from the point of intersection of the orbitals to the internuclear axis must be the same for both component orbitals. Thus, the angle ωt = θHiMO between the internuclear axis and the point of intersection of the H2 -type ellipsoidal MO with the 02 p radial vector obeys the following relationship: r6 sinθO2pAO = 0.74776a0 smθO2pAO = bsmθHiMO (14.40) such that
Figure imgf000308_0001
with the use of Eq. (14.39). Substitution of Eq. (14.27) into Eq. (14.41) gives θHiM0 = 15.18° (14.42)
Then, the distance dH u0 along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by dH2MO = a C0S ΘH,MO (1 4-43)
Substitution of Eqs. (14.23) and (14.42) into Eq. (14.43) gives dH,uo = l-74396θo = 9.22862 X 10~n m (14.44)
The distance d02pA0 along the internuclear axis from the origin of each O atom to the point of intersection of the orbitals is given by d2pAo = dH2MO -c' (14.45)
Substitution of Eqs. (14.24) and (14.44) into Eq. (14.45) gives do2PΛo = 0.64637α0 = 3.42047 X 10"n m (14.46)
As shown in Eq. (14.18), each C = O bond comprises a factor of 0.75 of the charge- density of double that of the Tf2 -type ellipsoidal MO. Using the electron configuration of
CO2 (Eq. (14.2)), the radii of the Cl1S = O-Hl Da0 (Eq. (1Q.51)), C2.? = 0.84317a0 (Eq. (10.62)), OL? = 0.12739α0 (Eq. (10.51)), 02s = 0.5902Oa0 (Eq. (10.62)), and O2p = 0.74776α0 (Eq. (14.11)) shells and the parameters of the σ MO of CO2 given by Eqs. (13.3-13.4), (14.23-14.25), and (14.27-14.28), the dimensional diagram and charge-density of the CO2 MO are shown in Figures 33 and 34, respectively.
SUM OF THE ENERGIES OF THE σ MO AND THE AOs OF THE CARBON DIOXIDE MOLECULE
The energies of the CO2 σ MO are given by the substitution of the semiprincipal axes (Eqs.
(14.23-14.24) and (14.27)) into the energy equations of OH (Eqs. (13.67-13.73)), except that the terms based on charge are multiplied by four and the kinetic energy term is multiplied by two due to each σ -MO double bond:
V. -104.83940 er (14.47)
Figure imgf000309_0001
Figure imgf000309_0002
(14.50)
Figure imgf000309_0003
ET = Ve +T + Vm +Vp (14.51)
Substitution of Eqs. (13.60) and (14.47-14.50) into Eq. (14.51) gives
Eτ [C eV
Figure imgf000309_0004
(14.52) where E1. (C = O,σ) is the total energy of each C = O σ MO of CO2 given by Eq. (14.19) which is reiteratively matched to Eq. (14.20) within five-signifϊcant-figure round off error.
The total energy of CO2, E7 [CO2) , is given by the sum of E (ionization; C) and
EQonization; C+) , the sum of the energies of the first and second electrons of carbon (Eqs. (14.12-14.13)) donated to each double bond, the sum of EQonization; O) and two times EQonization; O+) , the energies of the first and second electrons of oxygen (Eqs. (14.14- 14.15)) donated to the double bonds, two times E1. (θ,2p) , the Q2p AO contribution due to the decrease in radius with the formation of each bond (Eq. (14.17)), and two times Eτ (C = O,σ) , the σ MO contribution given by Eq. (14.22):
Figure imgf000310_0001
VIBRATION OF CO2
The vibrational energy levels of CO2 may be solved by determining the Morse potential curve from the energy relationships for the transition from a C atom and two O atoms whose parameters are given by Eqs. (10.115-10.123) and (10.154-10.163), respectively, to a C atom whose parameter r4 is given by Eq. (10.61), two O atoms whose parameter r6 is given by Eq.
(14.11), and the σ CO2 MO whose parameters are given by Eqs. (14.23-14.25) and (14.27-
14.28). As shown in the Vibration of Hydrogen-type Molecular Ions section, the harmonic oscillator potential energy function can be expanded about the internuclear distance and expressed as a Maclaurin series corresponding to a Morse potential after Karplus and Porter (K&P) [4] and after Eq. (11.134). Treating the Maclaurin series terms as anharmonic perturbation terms of the harmonic states, the energy corrections can be found by perturbation methods.
THE DOPPLER ENERGY TERMS OF THE CARBON DIOXIDE MOLECULE
The equations of the radiation reaction force of carbon dioxide are the same as those of OH with the substitution of the CO2 parameters except that there is a factor of four increase in the central force in Eq. (13.140) due to the double bond. Using Eqs. (13.140-13.142), the angular frequency of the reentrant oscillation in the transition state is
4Λ6331 X 1016 rad/ s (14.54)
Figure imgf000311_0001
where b is given by Eq. (14.27). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)):
Eκ = hω = h4Λ633l X 1016 radls = 27.40365 eV (14.55)
In Eq. (11.181), substitution of E7. (CO2)12 for E1n, , the mass of the electron, me, for M , and the kinetic energy given by Eq. (14.55) for Eκ gives the Doppler energy of the electrons of the reentrant orbit:
(R56)
Figure imgf000311_0002
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The transition state comprises O — CO, oxygen binding to CO . Vibration of the linear XYZ-molecular transition state corresponds to U3 [5] with the maximum kinetic energy localized to the nascent C- O bond. In this case, the kinetic energy of the nuclei is the maximum for this bond. Thus, EKvib is the vibrational energy. The decrease in the energy of the CO2 MO due to the reentrant orbit in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by Eq. (14.56) and EKvib , the vibrational energy. Using the experimental CO2 Evib (υ3) of 2349 cm"1 (0.29124 e F) [6] for EKvib of the transition state, Eosc (CO2) is
Eosc {CO2) = ED +EKvib = ED +Evib (14.57)
Ksc [COi) = -0.28619 eV + Q29UA eV = 0.00505 eV (14.58) TOTAL AND BOND ENERGIES OF THE CARBON DIOXIDE MOLECULE
Eτ+osc [CO2) , the total energy of CO2 including the Doppler term, is given by the sum of
Er (CO2) (Eq. (14.53)) and Eosc (CO2) given by Eq. (14.58):
(14.59)
Figure imgf000312_0002
Figure imgf000312_0001
= -55.26841 eV - 0.28619 eV + Evώ
(14.60) From Eqs. (14.57-14.60), the total energy of the CO2 MO is
Eτ÷osc (CO2) = -55.25476 eV + Eosc (CO2)
= -55.25476 eV + 0.00505 eV (14.61) = -55.26336 eV where the experimental Evιb was used. As in the case of the dissociation of the bond of the hydroxyl radical, an oxygen atom is formed with dissociation of CO2. O has two unpaired electrons as shown in Eq. (13.55) which interact to stabilize the atom as shown by Eq. (10.161-10.162). The lowering of the energy of the reactants decreases the bond energy. Thus, the total energy of oxygen is reduced by the energy in the field of the two magnetic dipoles given by Eq. (7.46) and Eq.
(13.101):
(14.62)
Figure imgf000313_0001
The CO2 bond dissociation energy, ED (CO2) , is given by the sum of the energies of the CO and the O atom minus the sum of Eτ+osc (CO2) and E(magnetic) :
ED (CO2 ) = E(CO) + E(O)- (E{magnetic) + Eτ+osc (CO2 )) (14.63)
The energy of an oxygen atom is given by Eq. (14.14) and E7(CO) is given by the sum of the experimental energies of C (Eq. (14.12)), O (Eq. (14.14)), and the negative of the bond energy of CO (Eq. (13.914)): E(CO) = -U.26030 eV -13.618060 eV -U.15696 eV = -36.03532 eV (14.64)
The energy of O is given by the negative of the corresponding ionization energy given in Eq. (4.14). Thus, the CO2 bond dissociation energy, E0 (CO2) , given by the Eqs. (4.14) and (14.61-14.64) is
ED (CO2) = -(36.03532 eF + 13.618060 eV)-(E(magnetic) + Eτ÷0SC (CO2)) 1 = -49.65338 eV-(0ΛU4l eV -55.26336 eV) (14.65)
= 5.49557 eV The experimental CO2 bond dissociation energy is [7]
^298 (CO2) = 5.516 eV (14.66)
The results of the determination of bond parameters of CO2 are given in Table 14.1. The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
NITROGEN DIOXIDE MOLECULE
The nitrogen dioxide molecule can be formed by the reaction of nitric oxide and an oxygen atom: N0 + 0 → N02 (14.67)
The bonding in the nitrogen dioxide molecule comprises two double bonds, each a H2 -type
MO with four paired electrons wherein the central N atom is shared by both bonds such that six electrons can be assigned to the two N = O bonds. Thus, two N2p electrons combine with the four 02 p electrons, two from each O, as a linear combination to form the two overlapping N = O bonds of NO2. The force balance equation and radius rη of the 2p shell of N is derived in the Seven-Electron Atoms section. The force balance equation and radius r& of the 2p shell of O is derived in the Eight-Electron Atoms section. With the formation of each of the two H2 -type MOs by the contribution of two 2p electrons each from the N and O atoms, a diamagnetic force arises between the remaining outer shell atomic electrons, the 2s and 2p electrons of N and O, and the H2 -type MO. This force from N and O causes the H2 -type MO to move to greater principal axes than would result with the
Coulombic force alone. But, the factor of two increase of the central field and the resulting increased Coulombic as well as magnetic central forces on the remaining N and O electrons decrease the radii of the corresponding shells such that the energy minimum is achieved that is lower than that of the reactant atoms. The resulting electron configuration of NO2 is
NIs2O1 Is2O2 ls2N2s2Ol 2s2 O22s2N2p10l 2p2O22p2σO 6 i N Oι where the subscripts designate the O atom, 1 or 2, σ designates the H2 -type MO, and the orbital arrangement is
(14.68)
Figure imgf000314_0001
Nitrogen dioxide is predicted to be weakly paramagnetic in agreement with observations [I]. FORCE BALANCE OF THE 2p SHELL OF THE NITROGEN ATOM OF
NITROGEN DIOXIDE
For the N atom, force balance for the outermost 2p electron of NO2 (electron 5) is achieved between the centrifugal force and the Coulombic and magnetic forces that arise due to interactions between electron 5 and the 2s -shell electrons due to spin and orbital angular momentum. The forces used are derived in the Seven-Electron Atoms section. The central Coulomb force on the outer-most 2p shell electron of NO (electron 5) due to the nucleus and the inner four electrons is given by Eq. (10.70) with the appropriate charge and radius:
Figure imgf000315_0001
for r > rA . The 2p shell possess a +2 external electric field given by Eq. (10.92) for r > r5. The energy is minimized with conservation of angular momentum. This condition is met when the magnetic forces of N in NO2 are the same as those of N in NO . They are also the same as those of N in the nitrogen molecule with rs replacing r6 and with an increase of the central field by an integer. The diamagnetic force, Vdjamagnelic , of Eq. (10.82) due to the p - orbital contribution is given by Eq. (13.918):
diamasneiic (14.70)
Figure imgf000315_0002
And, Fmag 2 corresponding to the conserved orbital angular momentum of the three orbitals is also the same as that of NO2 given by Eq. (13.919):
Figure imgf000315_0003
The electric field external to the 2p shell given by Eq. (10.92) for r > r5 gives rise to a second diamagnetic force, Fdiamagnetic 2, given by Eq. (10.93). ¥diamagιietlc 2 due to the binding of the p-orbital electron having an electric field of +2 outside of its radius is given by Eq. (13.920):
Figure imgf000315_0004
The radius of the 2p shell is calculated by equating the outward centrifugal force to the sum of the electric (Eq. (14.69)) and diamagnetic (Eqs. (14.70) and (14.72)), and paramagnetic (Eq. (14.71)) forces as follows:
Figure imgf000316_0001
Substitution of V5 = (Eq. (1.56)) and s = - into Eq. (14.73) gives: mer5
Figure imgf000316_0002
The quadratic equation corresponding to Eq. (14.74) is
Figure imgf000316_0005
Figure imgf000316_0003
(14.75)
The solution of Eq. (14.75) using the quadratic formula is:
Figure imgf000316_0004
r3 in units of αQ The positive root of Eq. (14.76) must be taken in order that r5 > 0 . Substitution of
-l- = 0.69385 (Eq. (10.62) with Z = 7 ) into Eq. (14.76) gives αn rs = 0.7484Ia0 (14.77)
FORCE BALANCE OF THE 2P SHELL OF EACH OXYGEN ATOM OF NITROGEN DIOXIDE For each O atom, force balance for the outermost 2p electron of NO2 (electron 6) is achieved between the centrifugal force and the Coulombic and magnetic forces that arise due to interactions between electron 6 and the other 2p electron as well as the 2s -shell electrons due to spin and orbital angular momentum. The forces used are derived in the Eight-Electron Atoms section. The central Coulomb force on the outer-most 2ρ shell electron of NO2
(electron 6) due to the nucleus and the inner five electrons is given by Eq. (10.70) with the appropriate charge and radius:
Figure imgf000317_0001
for r > r5 m The 2p shell possess an external electric field of +2 given by Eq. (10.92) for r > r6. The energy is minimized with conservation of angular momentum. This condition is met when the magnetic forces of O in NO2 are the same as those of O in NO . The diamagnetic force, Εdtamagnettc , of Eq. (10.82) due to the p -orbital contribution given by Eq. (13.927) is
Figure imgf000317_0002
And, Fmag 2 corresponding to the conserved spin and orbital angular momentum given by Eq. (13.928) is
F * '-m_„,a„g„ i,T = — (14.80)
Figure imgf000317_0003
The electric field external to the 2p shell given by Eq. (10.92) for r > r6 gives rise to a second diamagnetic force, Εdiamagnetic 2 , given by Eq. (10.93). FdtomogHe//c 2 due to the binding of the p-orbital electron having an electric field of +2 outside of its radius given by Eq. (13.929) is
diamagnetic 2 (14.81)
Figure imgf000317_0004
In addition, the contribution of two 2p electrons in the formation of the σ MO gives rise to a paramagnetic force on the remaining paired 2p electrons. The force Fmag 3 is given by Eq. (13.930) is n2 (14.82)
Amer6 The radius of the 2p shell is calculated by equating the outward centrifugal force to the sum of the electric (Eq. (14.78)) and diamagnetic (Eqs. (14.79) and (14.81)), and paramagnetic (Eqs. (14.80) and (14.82)) forces as follows:
Figure imgf000318_0001
Substitution of v6 = (Eq. (1.56)) and s = — into Eq. (14.83) gives: mtr6
Figure imgf000318_0002
(14.84)
The quadratic equation corresponding to Eq. (14.84) is
Figure imgf000318_0003
(14.85)
The solution of Eq. (14.85) using the quadratic formula is:
Figure imgf000318_0004
r3 in units of a0 The positive root of Eq. (14.86) must be taken in order that r6 > 0 . Substitution of
'3 _ = 0.59020 (Eq. (10.62) with Z = 8) into Eq. (14.86) gives an r6 = 0.7046Oa0 (14.87)
ENERGIES OF THE 2P SHELLS OF THE NITROGEN ATOM AND OXYGEN ATOMS OF NITROGEN DIOXIDE
Consider the determination of the total energy of NO2 from the reaction of a nitrogen atom with two oxygen atoms. With the formation of each H2 -type MO by the contribution of two 2p electrons from each of the N and the two O atoms, the total energy of the NO2 molecule, which is subtracted from the sum of the energies of the nitrogen and oxygen atoms to determine the bond energy, is increased by the ionization energies of N , N+ , O, and 2O+ given by Eqs. (14.88-14.91), respectively. Experimentally, the energies are [2]
EQonization; N) = 14.53414 eV (14.88)
Eiionization; N+) = 29.6013 eV (14.89) Eiionization; O) = 13.61806 eV (14.90)
E(ionization; O+) = 35.11730 eV (14.91)
In addition, the central forces on the 2p shells of the N and O atoms are increased with the formation of the σ MOs which reduces each shell's radius and increases its total energy. The change per bond is the same as that of NO since the final radii given by Eq. (14.77) and (14.87) are the same for NO and NO2. The Coulombic energy terms of the total energy of the N and O atoms at the new radii are calculated and added to the ionization energies of N , N+ , O , and 2O+ , and the energy of the σ MOs to give the total energy of
NO2. Then, the bond energy is determined from the total NO2 energy.
The radius r7 of the nitrogen atom before bonding is given by Eq. (10.142): r7 = 0.93084α0 (14.92)
Using the initial radius r7 of the N atom and the final radius r5 of the NIp shell (Eq. (14.77)) and by considering that the central Coulombic field decreases by an integer for each successive electron of the shell, the sum ET (N,2p) of the Coulombic energy change of the N2p electrons of the N atom is determined using Eq. (10.102):
Figure imgf000319_0001
= -(13.60580 eF)(0.26186)(3) (14.93)
= -10.68853 eV The radius r8 of the oxygen atom before bonding is given by Eq. (10.162): r8 = a0 (14.94) Using the initial radius r& of the O atom and the final radius r6 of the O2p shell (Eq.
(14.87)) and by considering that the central Coulombic field decreases by an integer for each successive electron of the shell, the sum Eτ (θ,2p) of the Coulombic energy change of the O2p electrons of the O atom is determined using Eq. (10.102):
Figure imgf000320_0001
= (13.60580 e F) (0.41925a"1) (3 + 4) (14.95)
= -39.92918 eV
FORCE BALANCE OF THE σ MO OF NITROGEN DIOXIDE
The force balance can be considered due to a second pair of two electrons binding to a molecular ion having +2e at each focus and a first bound pair. Then, the forces are the same as those of a molecule ion having +e at each focus. The diamagnetic force VdiamaglieticMOΪ for each σ -MO of the NO2 molecule due to the two paired electrons in the O2p shell is given by Eq. (13.633) with ne = 2 : h2
* diamagneticMOl = η 2τ 2 ^lξ (14.9θj
This is also the corresponding force of NO given by Eq. (13.942). ^diamagnetiCMO2 0^ tne nitrogen dioxide molecule comprising nitrogen with charge Z1 = 7 and |li| = fø and
[3 \L2\ = A-TI and the two oxygen atoms, each with Z2 = 8 and |Z3| = /Ϊ is given by the
corresponding sum of the contributions. Using Eq. (13.835), I \iamasmMcMoi f°r NO2 is
Figure imgf000320_0002
This is also the corresponding force of NO given by Eq. (13.943) except the term due to oxygen is twice that of NO due to the two oxygen atoms of NO2. The general force balance equation for the σ -MO of the nitrogen dioxide molecule given by Eqs. (11.200), and (14.97-
2
14.98) is also the same as that of CiV (Eq. (14.836)) except for the doubling of the — term
Z2 due to the two oxygen atoms:
Figure imgf000321_0001
Substitution of Z1 = 7 and Z2 = 8 into Eq. (14.101) gives a = 2.5165San = 1.33171 X 10 i-lO m (14.102)
Substitution of Eq. (14.102) into Eq. (11.79) is c' = 1.12173αn = 5.93596 X 10 ι-π /W (14.103)
The internuclear distance given by multiplying Eq. (14.103) by two is
2c' = 2.24347αo = 1.18719 JST 10"10 m (14.104) The experimental bond distance is [3]
2c' = 1.193 X W10 m (14.105) Substitution of Eqs. (14.102-14.103) into Eq. (11.80) is b = c = 2.25275αn = 1.19210 JT 10"10 m (14.106) Substitution of Eqs. (14.102-14.103) into Eq. (11.67) is e = 0.44574 (14.107)
The bonding in the nitrogen dioxide molecule comprises two double bonds, each a
H2 -type MO with four paired electrons wherein the central N atom is shared by both bonds such that six electrons can be assigned to the two N = O bonds. Thus, two N2p electrons combine with the four O2p electrons, two from each O , as a linear combination to form the two overlapping N = O bonds of NO2. Using the electron configuration of NO2 (Eq. (14.68)), the radii of the NLy = 0.14605α0 (Eq. (10.51)), N2s = 0.69385α0 (Eq. (10.62)),
N2/? = 0.74841α0 (Eq. (14.77)), OL? = 0.12739a0 (Eq. (10.51)), O2s = 0.59020α0 (Eq. (10.62)), and 02^ = 0.70460^ (Eq. (14.87)) shells and the parameters of the σ MOs of NO2 given by Eqs. (13.3-13.4), (14.102-14.104), and (14.106-14.107), the dimensional diagram and charge-density of the NO2 MO are shown in Figures 35 and 36, respectively.
SUM OF THE ENERGIES OF THE σ MOs AND THE AOs OF NITROGEN DIOXIDE
The energies of each NO2 σ MO are the same as those of NO (Eqs. (13.954-13.958)). They are given by the substitution of the semiprincipal axes (Eqs. (14.102-14.103) and (14.106)) into the energy equations (Eqs. (11.207-11.212)) of H2 except that the terms based on charge are multiplied by four and the kinetic energy term is multiplied by two due to the σ -MO double bond with two pairs of paired electrons:
Figure imgf000322_0001
Vp = 22 = 48.51704 eV (14.109)
8πεnΛja2 -b2
9.24176 eV (14.110)
Figure imgf000322_0002
Figure imgf000322_0003
ET = Ve +T + Vm +Vp (14.112)
Substitution of Eqs. (11.79) and (14.108-14.111) into Eq. (14.112) gives
Er (N = -44.51329 e V (14.113)
Figure imgf000322_0004
where E1, (N = O, σ) is the total energy of each σ MO of NO2. The total energy of NO2 ,
E7 [NO2) , is given by the sum of E(ionization; N) and E(ionization; N+) , the sum of the energies of the first and second electrons of nitrogen (Εqs. (14.88-14.89)) donated to each double bond, the sum of E (ionization; O) and two times Eiionization; O+) , the energies of the first and second electrons of oxygen (Eqs. (14.90-14.91)) donated to the double bonds, ET (N,2p) , the N2p AO contribution due to the decrease in radius with the formation of each bond (Eq. (14.93)), two times Eτ (θ,2p) , the O2p AO contribution due to the decrease in radius with the formation of each bond (Eq. (14.95)), and two times Eτ (N = O,σ) , the σ MO contribution given by Eq. (14.113):
Figure imgf000323_0001
VIBRATION OF NO2 The vibrational energy levels of NO2 may be solved by determining the Morse potential curve from the energy relationships for the transition from a N atom and two O atoms whose parameters are given by Eqs. (10.134-10.143) and (10.154-10.163), respectively, to a N atom whose parameter r5 is given by Eq. (14.77), two O atoms whose parameter r6 is given by Eq. (14.87), and the σ MOs whose parameters are given by Eqs. (14.102-14.104) and (14.106.-14.107). As shown in the Vibration of Hydrogen-type Molecular Ions section, the harmonic oscillator potential energy function can be expanded about the internuclear distance and expressed as a Maclaurin series corresponding to a Morse potential after Karplus and Porter (K&P) [4] and after Eq. (11.134). Treating the Maclaurin series terms as anharmonic perturbation terms of the harmonic states, the energy corrections can be found by perturbation methods. THE DOPPLER ENERGY TERMS OF NITROGEN DIOXIDE The equations of the radiation reaction force of nitrogen dioxide are the same as those of NO with the substitution of the NO2 parameters. Using Eq. (13.961), the angular frequency of the reentrant oscillation in the transition state is
2.07110 X lO16 radls (14.115)
Figure imgf000324_0001
where a is given by Eq. (14.102). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)):
Eκ = hω = h2.(m 10 X lO16 radls = 13.63231 eV (14.116) In Eq. (11.181), substitution of E1. (NO2)/ 2 for Ehv , the mass of the electron, me , for M , and the kinetic energy given by Eq. (14.116) for Eκ gives the Doppler energy of the electrons of the reentrant orbit:
Figure imgf000324_0002
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The transition state comprises O — NO, oxygen binding to NO . As in the case of CO2 bond formation, vibration in the transition state corresponds to υ3 [5] with the maximum kinetic energy localized to the nascent N-O bond. In this case, the kinetic energy of the nuclei is the maximum for this bond. Thus, EKvtb is the vibrational energy. The decrease in the energy of the NO2 MO due to the reentrant orbit in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, E010 , is given by the sum of the corresponding energies, ED given by Eq. (14.117) and
EKvib , the vibrational energy. Using the experimental NO2 Evjb (υ3) of 1618 cm'1 (0.20061 eV) [6] for EKvib of the transition state, E0n (NO2) is
Eosc (NO2 ) = ED + Emb = ED + Evib (14.118) IOie (NO2) = -0.18840 eF + 0.20061 eF = 0.01221 eK (14.119) TOTAL AND BOND ENERGIES OF NITROGEN DIOXIDE
Eτ+osc [NO2) , the total energy of NO2 including the Doppler term, is given by the sum of ET (NO2) (Eq. (14.114)) and Eosc (NO2) given by Eq. (14.119):
(14.120)
Figure imgf000325_0001
= ET (NO2) + EOSC (NO2)
Figure imgf000325_0002
= -51.58536 eF - 0.18840 eV + E, vib
From Eqs. (14.119-14.121), the total energy of the NO2 MO is
^+o. (^O2) = -51.58536 eF + EOiC (NO2)
= -51.58536 βF + 0.01221 eF (14.122)
= -51.57315 eV where the experimental Evιb was used.
As in the case of the dissociation of the bond of the hydroxyl radical, an oxygen atom is formed with dissociation of NO2. O has two unpaired electrons as shown in Eq. (13.55) which interact to stabilize the atom as shown by Eq. (10.161-10.162). The lowering of the energy of the reactants decreases the bond energy. Thus, the total energy of oxygen is reduced by the energy in the field of the two magnetic dipoles given by Eq. (7.46) and Eq.
(13.101):
Eimagnetic) = 2π^ff = tøψL = O.\ 1441 eV (14.123) mea0 a0
The NO2 bond dissociation energy, E0 (NO2) , is given by the sum of the energies of the NO and the O atom minus the sum of Eτ+osc (NO2) and Eimagnetic) :
ED (NO2 ) = E (NO) + E(O)- (E(magnetic) + Eτ+Osc (NO2 )) (14.124)
The energy of an oxygen atom is given by the negative of Eq. (14.90), and E7-(NO) is given by the sum of the experimental energies of N (negative of Eq. (14.88)), O, and the negative of the bond energy of NO (Eq. (13.974)): EiNO) = -14.53414 eV -13.618060 eV- 6.53529 eV = -34.68749 eV (14.125)
Thus, the NO2 bond dissociation energy, E0 (NO2) , given by Eqs. (4.90) and (14.112- 14.125) is
E0 (NO2) = - (34.68749 e V + 13.618060 e V) - (Eimagnetic) + Eτ+osc (NO2 ))
= -48.30555 eF-(0.11441 eF-51.57315 eV) (14.126)
= 3.15319 eV The experimental NO2 bond dissociation energy is [7] E0298 (NO2) = 3.16I eF (14.127)
BOND ANGLE OF NO2
The NO2 MO comprises a linear combination of two N = O -bond MOs. A bond is also possible between the two O atoms of the N = O bonds. Such O = O bonding would decrease the N = O bond strength since electron density would be shifted from the N = O bonds to the O = O bond. Thus, the bond angle between the two N = O bonds is determined by the condition that the total energy of the H2 -type ellipsoidal MO between the terminal O atoms of the N = O bonds is zero. From Eqs. (11.79) and (13.228), the distance from the origin to each focus of the O = O ellipsoidal MO is
'■
Figure imgf000326_0001
The internuclear distance from Eq. (13.229) is
Figure imgf000327_0001
The length of the semiminor axis of the prolate spheroidal O = O MO h = c is given by Eq. (13.167).
The component energies and the total energy E1, of the O = O bond are given by the energy equations (Eqs. (11.207-11.212), (11.213-11.217), and (11.239)) of H1 except that the terms based on charge are multiplied by four and the kinetic energy term is multiplied by two due to the O = O double bond with two pairs of paired electrons. Substitution of Eq. (14.128) into Eqs. (11.207-11.212) gives
Figure imgf000327_0002
From the energy relationship given by Eq. (14.130) and the relationship between the axes given by Eqs. (14.128-14.129) and (13.167-14.168), the dimensions of the O = O MO can be solved.
The most convenient way to solve Eq. (14.130) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is α = 8.3360αn = 4.4112X 10 -10 m (14.131)
Substitution of Eq. (14.131) into Eq. (14.128) gives c' = 2.0416αn =1.0804X 10 ,-10 m (14.132)
The internuclear distance given by multiplying Eq. (14.132) by two is
2c' = 4.083Ia0 = 2.1607 X lO"10 m (14.133) Substitution of Eqs. (14.131-14.132) into Eq. (14.167) gives b = c = 8.082Ian = 4.2769 X lO"10 m (14.134)
Substitution of Eqs. (14.131-14.132) into Eq. (14.168) gives e = 0.2449 (14.135) From, 2c'c=c (Eq. (14.133)), the distance between the two O atoms when the total energy of the corresponding MO is zero (Eq. (14.130)), and
Figure imgf000327_0003
(Eq. (14.104)), the internuclear distance of each N = O bond, the corresponding bond angle can be determined from the law of cosines. Using, Eqs. (13.240-13.242), the bond angle θ between the N = O bonds is
Figure imgf000328_0001
= cos"1 (-0.6562) (14.136)
= 131.012° The experimental angle between the N = O bonds is [3]
0 = 134.1° (14.137)
The results of the determination of bond parameters of NO2 are given in Table 14.1.
The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
ETHANE MOLECULE (CH3CH,)
The ethane molecule CH3CH3 is formed by the reaction of two methyl radicals:
CH3 + CH3 -> CH3CH3 (14.138) CH3CH3 can be solved using the same principles as those used to solve CH3 , wherein the 2s and 2ρ shells of each C hybridize to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between two C2sp3 hybridized orbitals (HOs) to form a molecular orbital (MO) permits each participating hybridized orbital to decrease in radius and energy. First, two sets of three H atomic orbitals (AOs) combine with two sets of three carbon 2sp3 HOs to form two methyl groups comprising a linear combination of six diatomic H2 -type MOs developed in the Nature of the Chemical Bond of Hydrogen-Type Molecules and Molecular Ions section. Then, the two CH3 groups bond by forming a H2 -type MO between the remaining C2sp3 HO on each carbon.
FORCE BALANCE OF THE C-C -BOND MO OF ETHANE
CH3CH3 comprises a chemical bond between two CH3 radicals wherein each methyl radical comprises three chemical bonds between carbon and hydrogen atoms. The solution of the parameters of CH3 is given in the Methyl Radical (CH3) section. Each C-H bond of CH3 having two spin-paired electrons, one from an initially unpaired electron of the carbon atom and the other from the hydrogen atom, comprises the linear combination of 75% H2 - type ellipsoidal MO and 25% C2sp3 HO. The proton of the H atom and the nucleus of the C atom are along each internuclear axis and serve as the foci. As in the case of H2 , each of the three C-H -bond MOs is a prolate spheroid with the exception that the ellipsoidal MO surface cannot extend into C2sp3 HO for distances shorter than the radius of the C2sp3 shell since it is energetically unfavorable. Thus, each MO surface comprises a prolate spheroid at the H proton that is continuous with the C2sp3 shell at the C atom whose nucleus serves as the other focus. The electron configuration and the energy, E(c,2sp3) , of the C2sp3 shell is given by Eqs. (13.422) and (13.428), respectively. The central paramagnetic force due to spin of each C-H bond is provided by the spin-pairing force of the CH3 MO that has the symmetry of an i orbital that superimposes with the C2sp3 orbitals such that the corresponding angular momenta are unchanged.
Two CH3 radicals bond to form CH3CH3 by forming a MO between the two remaining C2,yp3-ΗO electrons of the two carbon atoms. However, in this case, the sharing of electrons between two C2sp3 HOs to form a molecular orbital (MO) comprising two spin- paired electrons permits each C2sp3 HO to decrease in radius and energy.
As in the case of the C-H bonds, the C-C -bond MO is a prolate-spheroidal-MO surface that cannot extend into C2sp3 HO for distances shorter than the radius of the C2sp3 shell of each atom. Thus, the MO surface comprises a partial prolate spheroid in between the carbon nuclei and is continuous with the C2sp3 shell at each C atom. The energy of the H2 - type ellipsoidal MO is matched to that of the C2sp3 shell. As in the case of previous examples of energy-matched MOs such as those of OH , NH , CH , and the C = O -bond MO of CO2 , the C - C -bond MO of ethane must comprise 75% of a H2 -type ellipsoidal MO in order to match potential, kinetic, and orbital energy relationships. Thus, the C - C -bond MO must comprise two C2ψ3ΗOs and 75% of a H2 -type ellipsoidal MO divided between the two C2sp3 ΗOs:
2 C2sp3 + 0.75 H2 MO → C -C -bond MO (14.139) The linear combination of the H2 -type ellipsoidal MO with each C2sp3 HO further comprises an excess 25% charge-density contribution from each C2sp3 HO to the C-C - bond MO to achieve an energy minimum. The force balance of the C- C -bond MO is determined by the boundary conditions that arise from the linear combination of orbitals according to Eq. (14.139) and the energy matching condition between the C2sp3-HO components of the MO.
Similarly, the energies of each CH3 MO involve each C2sp3 and each HIs electron with the formation of each C-H bond. The sum of the energies of the H2 -type ellipsoidal MOs is matched to that of the C2sp3 shell. This energy is determined by the considering the effect of the donation of 25% electron density from the two C2sp3 ΗOs to the C -C -bond
MO. The 2sp3 hybridized orbital arrangement given by Eq. (13.422) is
2sp3 state (14.140)
Figure imgf000330_0001
where the quantum numbers (£,me) are below each electron. The total energy of the state is given by the sum over the four electrons. The sum E7. (c, 2sp3 ) of calculated energies of C , C+ , C2+, and C3+ from Eqs. (10.123), (10.113-10.114), (10.68), and (10.48), respectively, is
E7. (C,2sp3 ) = 64.3921 eF + 48.3125 eF + 24.2762 eF + 11.27671 eV
= 148.25751 eV which agrees well with the sum of 148.02532 eV from the experimental [2] values. Consider the case of the C2sp3 HO of each methyl radical. The orbital-angular-momentum interactions cancel such that the energy of the Eτ (c,2sp3) is purely Coulombic. By considering that the central field decreases by an integer for each successive electron of the shell, the radius r2s 3 of the C2sp3 shell may be calculated from the Coulombic energy using Eq. (10.102): (14.142)
Figure imgf000330_0002
where Z = 6 for carbon. Using Eqs. (10.102) and (14.142), the Coulombic energy
^coubmb (C,2.ζp3) of the outer electron of the C2sp3 shell is
Figure imgf000331_0001
During hybridization, one of the spin-paired 2s electrons is promoted to Cl sp3 shell as an unpaired electron. The energy for the promotion is the magnetic energy given by Eq. (13.152) at the initial radius of the 2s electrons. From Eq. (10.62) with Z = 6 , the radius r3 of C2s shell is r3 = 0.84317α0 (14.144)
Using Eqs. (13.152) and (14.144), the impairing energy is
E(magnetic) 0,19086 βF (14.145)
Figure imgf000331_0002
Using Eqs. (14.143) and (14.145), the energy E(c,2sp3) of the outer electron of the C2sp3 shell is
Figure imgf000331_0003
= -14.82575 eV + 0.19086 eV (14.146)
= -14.63489 eV
Next, consider the formation of the C-C -bond MO of ethane from two methyl radicals, each having a C2sp3 electron with an energy given by Eq. (14.146). The total energy of the state is given by the sum over the four electrons. The sum Eτ (Cetlmιe,2sp3) of calculated energies of C2sp3, C+ , C2+, and C3+ from Eqs. (10.123), (10.113-10.114), (10.68), and (10.48), respectively, is
Eτ [Cetham,2sp3) = -(βA392\ eF + 48.3125 eF + 24.2762 eV + E(c,2sp3))
= -(64.3921 eF + 48.3125 eF + 24.2762 eF + 14.63489 eF)(14.147) = -151.61569 eV where E[c,2sp3^ is the sum of the energy of C , -11.27671 eV , and the hybridization energy. The orbital-angular-momentum interactions also cancel such that the energy of the Eτ (Cetham , 2sp3 ) is purely Coulombic.
The sharing of electrons between two C2sp3 HOs to form a. C-C -bond MO permits each participating hybridized orbital to decrease in radius and energy. In order to further satisfy the potential, kinetic, and orbital energy relationships, each C2sp% HO donates an excess of 25% of its electron density to the C-C -bond MO to form an energy minimum. By considering this electron redistribution in the ethane molecule as well as the fact that the central field decreases by an integer for each successive electron of the shell, the radius r et)mm2s 3 0^ ^ C2sp3 s^e^ 0^ ethane may be calculated from the Coulombic energy using Eq. (10.102):
(14.148)
Figure imgf000332_0001
= 0.87495α0
Using Eqs. (10.102) and (14.148), the Coulombic energy ECoulomb [Ceiham,2sp3) of the outer electron of the C2sp3 shell is
Figure imgf000332_0002
= -15.55033 eV
During hybridization, one of the spin-paired 2* electrons is promoted to C2spl shell as an unpaired electron. The energy for the promotion is the magnetic energy given by Eq. (13.152). Using Eqs. (14.145) and (14.149), the energy ∑(Cethme,2sp3) of the outer electron of the C2spz shell is
E(Cethamasp*) =
Figure imgf000332_0003
= -15.55033 eF + 0.19086 eF (14.150)
= -15.35946 eV
Thus, ET(C-C,2sp3) , the energy change of each C2sp3 shell with the formation of the C -C -bond MO is given by the difference between Eq. (14.146) and Eq. (14.150):
ET (C-C,2sp3) = E(Celhans>2sp3)-E(C,2sp3)
= -15.35946 eV -(-14.63489 eV) (14.151)
= -0.72457 e V The H2 -type ellipsoidal MO comprises 75% of the C-C -bond MO shared between
two C2sp3 ΗOs corresponding to the electron charge density in Eq. (11.65) of — '- . But,
the additional 25% charge-density contribution to the C-C -bond MO causes the electron
charge density in Eq. (11.65) to be is given by — = -0.5e . Thus, the force constant k' to
determine the ellipsoidal parameter c ' in terms of the central force of the foci given by Eq. (11.65) is
V -&&- (14.152)
4πε0
The distance from the origin to each focus c' is given by substitution of Eq. (14.152) into Eq. (13.60). Thus, the distance from the origin of the C -C -bond MO to each focus c' is given by
Figure imgf000333_0001
The internuclear distance from Eq. (14.153) is
2c' = 2^ (14.154)
The length of the semiminor axis of the prolate spheroidal C-C -bond MO b = c is given by Eq. (13.62). The eccentricity, e , is given by Eq. (13.63). The solution of the semimajor axis a then allows for the solution of the other axes of each prolate spheroid and eccentricity of the C-C-bond MO. Since the C- C -bond MO comprises a H2 -type-ellipsoidal MO that transitions to the Cethane2sp3 HO of each carbon, the energy E(Celhane,2sp3} in Eq. (14.150) adds to that of the H2 -type ellipsoidal MO to give the total energy of the C-C -bond MO. From the energy equation and the relationship between the axes, the dimensions of the
C-C -bond MO are solved. Similarly, E(Cellmm,2sp3) is added to the energy of the H2- type ellipsoidal MO of each C-H bond of the methyl groups to give its total energy. From the energy equation and the relationship between the axes, the dimensions of the equivalent C-H -bond MOs of the methyl groups in ethane are solved. The general equations for the energy components of Ve , Vp, T , Vn , and E7, of the
C-C -bond MO are the same as those of the CH MO as well as each C-H -bond MO of the methyl groups except that energy of the Celhme2sp3 HO is used. Since the prolate spheroidal H2 -type MO transitions to the Celhmιe2sp3 HO of each carbon and the energy of the Cetham2sp3 shell must remain constant and equal to the E{Celham,2sp^ given by Eq.
(14.150), the total energy Eτ (C-C,σ) of the σ component of the C -C -bond MO is given by the sum of the energies of the orbitals corresponding to the composition of the linear combination of the Cethane2spi HO and the H2 -type ellipsoidal MO that forms the σ component of the C -C -bond MO as given by Eq. (14.139) with the electron charge redistribution. Using Eqs. (13.431) and (14.150), Eτ (C-C,σ) is given by
Eτ {C-C,σ) = Eτ +E(Cethans,2sp3)
Sπεoc '
Figure imgf000334_0001
To match the boundary condition that the total energy of the entire the H2 -type ellipsoidal MO is given by Eqs. (11.212) and (13.75), Eτ (C -C,σ) given by Eq. (14.155) is set equal to Eq. (13.75):
Figure imgf000334_0002
= -31.63536831 eV
From the energy relationship given by Eq. (14.156) and the relationship between the axes given by Eqs. (14.153-14.154) and (13.62-13.63), the dimensions of the C-C -bond MO can be solved.
Substitution of Eq. (14.153) into Eq. (14.156) gives
%πεQJaa0
Figure imgf000334_0003
The most convenient way to solve Eq. (14.157) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is α = 2.1O725αo = 1.1151 I X lO'10 m (14.158)
Substitution of Eq. (14.158) into Eq. (14.153) gives c' = 1.45164α0 = 7.68173 X 10~n m (14.159)
The internuclear distance given by multiplying Eq. (14.159) by two is
2c' = 2.90327«0 = 1.53635 X 10"10 m (14.160) The experimental bond distance is [3] 2c' = 1.5351 X 1(T10 m (14.161)
Substitution of Eqs. (14.158-14.159) into Eq. (13.62) gives b = c = l.52750a0 = 8.08317X 10-u m (14.162)
Substitution of Eqs. (14.158-14.159) into Eq. (13.63) gives e = 0.68888 (14.163)
The nucleus of the C atoms comprise the foci of the H2 -type ellipsoidal MO. The parameters of the point of intersection of the H2 -type ellipsoidal MO and the Celham2sp3 HO are given by Eqs. (13.84-13.95) and (13.261-13.270). The polar intersection angle θ' is given by Eq. (13.261) where rn = rΛhm2^ = 0.87495α0 is the radius of the Celhane2sp3 shell. Substitution of Eqs. (14.158-14.159) into Eq. (13.261) gives
#' = 67.33° (14.164)
Then, the angle θ t- '^ethane ^SP 3 "O the radial vector of the C2sp3 HO makes with the internuclear axis is θ c c i 3 = 180°-67.33° = 112.67° (14.165) as shown in Figure 37.
Consider the right-hand intersection point. The distance from the point of intersection of the orbitals to the internuclear axis must be the same for both component orbitals. Thus, the angle cot = θc_c H M0 between the internuclear axis and the point of intersection of the H2 - type ellipsoidal MO with the Celham2sp3 radial vector obeys the following relationship: r eth.am ,lsp ,* sin0 C_-C-elhme ,2sp ,3 H„O = 0.87495a °, sinø, C,-C „ellmι2sp 3Ηθ = Z>sin6L C- rCe,hm, HH1MMΩO ( V14.166) / such that θ - dn-' °-87495α° Shi θc-c^»o ,in-x 0-87495α0 sinl 12.67°
with the use of Eq. (14.166). Substitution of Eq. (14.162) into Eq. (14.167) gives Θ C-C^H,MO = 31-91° (14.168)
Then, the distance dc_c H M0 along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by dr L ~ rL ethane ,H H1MMnO = acosθr C- rC elUam ,H H 2MUDO ( \14.169) J
Substitution of Eqs. (14.158) and (14.168) into Eq. (14.169) gives dc-cclhm,,HlMo = l-78885α0 = 9.46617 X 1(T11 m (14.170)
The distance <f 3 along the internuclear axis from the origin of the C atom to the point of intersection of the orbitals is given by dc-ccllmc 2sP>Ho = dc-ccllmt,H>Mo - *' (14.171) Substitution of Eqs. (14.159) and (14.170) into Eq. (14.171) gives dc-cΛ∞^no = °'33721α0 = 1.78444 X 10"11 m (14.172)
FORCE BALANCE OF THE CH3 MOs OF ETHANE
Each of the two equivalent CH3 MOs must comprise three C -H bonds with each comprising 75% of a H2 -type ellipsoidal MO and a C2sp3 HO as given by Eq. (13.540):
3[l C2sp3 +0.75 H2 MO] → CH3 MO (14.173)
The force balance of the CH3 MO is determined by the boundary conditions that arise from the linear combination of orbitals according to Eq. (13.540) and the energy matching condition between the hydrogen and C2spz HO components of the MO.
; The force constant k ' to determine the ellipsoidal parameter c ' of the each H2 -type- ellipsoidal-MO component of the CH3 MO in terms of the central force of the foci is given by Eq. (13.59). The distance from the origin of each C-H-bond MO to each focus c' is given by Eq. (13.60). The internuclear distance is given by Eq. (13.61). The length of the semiminor axis of the prolate spheroidal C-H-bond MO b = c is given by Eq. (13.62). The eccentricity, e , is given by Eq. (13.63). The solution of the semimajor axis a then allows for the solution of the other axes of each prolate spheroid and eccentricity of each C-H -bond MO. Since each of the three prolate spheroidal C- H-bond MOs comprises an H2-type- ellipsoidal MO that transitions to the Cellmm2sρ3 HO of ethane, the energy E(Cethane,2sp3j of Eq. (14.150) adds to that of the three corresponding H2 -type ellipsoidal MOs to give the total energy of the CH3 MO. From the energy equation and the relationship between the axes, the dimensions of the CH3 MO are solved. The energy components of Ve , Vp, T , and V1n are the same as those of methyl radical, three times those of CH corresponding to the three C-H bonds except that energy of the Celhane2sp3 HO is used. Since the each prolate spheroidal H2 -type MO transitions to the
Cellmm2spl HO and the energy of the CethaHe2sp3 shell must remain constant and equal to the E(Cellmne,2sp3 ) given by Eq. (14.150), the total energy E7^ (CH3) of the CH3 MO is given by the sum of the energies of the orbitals corresponding to the composition of the linear combination of the Celhane2sp3 HO and the three H2 -type ellipsoidal MOs that forms the
CH3 MO as given by Eq. (13.540). Using Eq. (13.431), Ew (CH3) is given by
^,w (CH3) =
Figure imgf000337_0001
E1, (CH3) given by Eq. (14.174) is set equal to three times the energy of the H2 -type ellipsoidal MO minus two times the Coulombic energy of H given by Eq. (13.542): 3e2
E7 [CH3) = - (0.91771) [ 2-!-SL kfi±iLi -15.35946 eV = -67.69450 eV
8πεϋc ' V \ 2 a ) a-c'
(14.175)
From the energy relationship given by Eq. (14.175) and the relationship between the axes given by Eqs. (13.60-13.63), the dimensions of the CH3 MO can be solved. Substitution of Eq. (13.60) into Eq. (14.175) gives
e52.33505 (14.176)
Figure imgf000337_0002
The most convenient way to solve Eq. (14.176) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is fl = 1.64469α0 = 8.70331 X 10"n m (14.177) Substitution of Eq. (14.177) into Eq. (14.60) gives c' = l.O4712αo = 5.5411 I X lO"11 m (14.178) The internuclear distance given by multiplying Eq. (14.178) by two is
2c' = 2.09424α0 = 1.10822 X 10"10 m (14.179) The experimental bond distance is [3] 2c' = 1.0940 X 10~10 m (14.180)
Substitution of Eqs. (14.177-14.178) into Eq. (14.62) gives δ = c = 1.26828αo = 6.71145 X 10~u m (14.181)
Substitution of Eqs. (14.177-14.178) into Eq. (14.63) gives e = 0.63667 (14.182)
The nucleus of the H atom and the nucleus of the C atom comprise the foci of each
H2 -type ellipsoidal MO. The parameters of the point of intersection of the H2 -type ellipsoidal MO and the Cethane2sp3 HO are given by Eqs. (13.84-13.95) and (13.261-13.270). The polar intersection angle θ' is given by Eq. (13.261) where rn ^ T 3 = 0.87495α0 is
the radius of the Celhane2sp3 shell. Substitution of Eqs. (14.177-14.178) into Eq. (13.261) gives θ ' = 79.34° (14.183)
Then, the angle θn „ . -, the radial vector of the C2sp3 HO makes with the internuclear
C ~" ethane lSP "U axis is 9 C-H1^HO = 180°-79.34° = 100.66° (14.184) as shown in Figure 38.
The distance from the point of intersection of the orbitals to the internuclear axis must be the same for both component orbitals. Thus, the angle ωt = ΘC_H e M M0 between the internuclear axis and the point of intersection of the H2 -type ellipsoidal MO with the Cethme2spl radial vector obeys the following relationship: r . , sinβ> Ju = 0.87495a, sin6> „ ,. = δsin0r ff H MO (14.185) such that
θc H H MO
Figure imgf000338_0001
with the use of Eq. (14.184). Substitution of Eq. (14.181) into Eq. (14.186) gives
ΘC-H^,H2MO = 42.68° (14.187)
Then, the distance dc_H H^o along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by dr H H Un ~ a C0^ θr H jy Un (14.188) Substitution of Eqs. (14.177) and (14.187) into Eq. (14.188) gives dc-HaUmc,Hτuo = 1-2090Ia0 = 6.39780 X 10"u m (14.189)
The distance dn . along the internuclear axis from the origin of the C atom to the point of intersection of the orbitals is given by dc-HCOm^Hθ - dc-HΛm^uo -C (14.190) Substitution of Eqs. (14.178) and (14.189) into Eq. (14.190) gives dc-Hlll^Ho = °-1618H = 8.56687X 10"12 m (14.191)
BOND ANGLE OF THE CH3 GROUPS Each CH3 MO comprises a linear combination of three C -H-bond MOs. Each C-H- bond MO comprises the superposition of a H2 -type ellipsoidal MO and the Cethane2sp3 HO.
A bond is also possible between the two H atoms of the C-H bonds. Such H-H bonding would decrease the C-H bond strength since electron density would be shifted from the
C-H bonds to the H-H bond. Thus, the bond angle between the two C-H bonds is determined by the condition that the total energy of the H2 -type ellipsoidal MO between the terminal H atoms of the C-H bonds is zero. From Eqs. (11.79) and (13.228), the distance from the origin to each focus of the H-H ellipsoidal MO is
Figure imgf000339_0001
The internuclear distance from Eq. (13.229) is
2c1 = 2 P^- (14.193)
The length of the semiminor axis of the prolate spheroidal H-H MO b = c is given by Eq. (14.62).
The bond angle of the CH3 groups of ethane is derived by using the orbital composition and an energy matching factor as in the case with the CH3 radical. Since the two H2 -type ellipsoidal MOs initially comprise 75% of the H electron density of H2 and the energy of each H2 -type ellipsoidal MO is matched to that of the Cetham2sp3 HO, the component energies and the total energy Eτ of the H-H bond are given by Eqs. (13.67- 13.73) except that Ve , T , and Vm are corrected for the hybridization-energy-matching factor of 0.87495. Hybridization with 25% electron donation to the C-C -bond gives rise to the
Cetham2sp% HO-shell Coulombic energy ECoulomb (Celhane,2sp") given by Eq. (14.149). The corresponding normalization factor for determining the zero of the total H-H bond energy is given by the ratio of 15.55033 eV , the magnitude of ECoulomb (Celhane,2sp3} given by Eq. (14.149), and 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243). The hybridization energy factor C,,AαMeC2- 3/rø is
Figure imgf000340_0002
Substitution of Eq. (14.152) into Eq. (13.233) with the hybridization factor of 0.87495 gives
Figure imgf000340_0001
(14.195)
From the energy relationship given by Eq. (14.195) and the relationship between the axes given by Eqs. (14.192-14.193) and (14.62-14.63), the dimensions of the H-H MO can be solved. The most convenient way to solve Eq. (14.195) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is α = 5.7000αn = 3.0163 X 10 ,-10 m (14.196)
Substitution of Eq. (14.196) into Eq. (14.192) gives d = 1.6882Λ0 = 8.9335 X 10"11 m (14.197) The internuclear distance given by multiplying Eq. (14.197) by two is
2c' = 3.3764α0 = 1.7867 Jf 10"10 m (14.198) Substitution of Eqs. (14.196-14.197) into Eq. (14.62) gives δ = c = 5.4443αn = 2.8810 X lO"10 m (14.199) Substitution of Eqs. (14.196-14.197) into Eq. (14.63) gives e = 0.2962 (14.200)
From, 2c'H_H (Eq. (14.198)), the distance between the two H atoms when the total energy of the corresponding MO is zero (Eq. (14.195)), and 2c'c_H (Eq. (14.179)), the internuclear distance of each C-H bond, the corresponding bond angle can be determined from the law of cosines. Using, Eq. (13.242), the bond angle θ between the C- H bonds is
Figure imgf000341_0001
The experimental angle between the C- H bonds is [8]
0 = 107.4° (14.202) The CH3 radical has a pyramidal structure with the carbon atom along the z-axis at the apex and the hydrogen atoms at the base in the xy-plane. The distance dorigin^H from the origin to the nucleus of a hydrogen atom given by Eqs. (14.198) and (13.412) is
<U,w, = l-94936α0 (14.203)
The height along the z-axis of the pyramid from the origin to C nucleus dheight given by Eqs. (13.414), (14.179), and (14.203) is
4^ = 0.76540^ (14.204)
The angle θv of each C -H bond from the z-axis given by Eqs. (13.416), (14.203), and (14.204) is θv = 68.563° (14.205) The C -C bond is along the z-axis. Thus, the bond angle ΘC_C_H between the internuclear axis of the C-C bond and a H atom of the methyl groups is given by θc_c_H = 180-θv (14.206)
Substitution of Eq. (14.205) into Eq. (14.206) gives θC-c-H = 111-44° (14.207) The experimental angle between the C-C-H bonds is [3] θ c-c-H = 111.17° (14.208)
The CH2CH1 MO shown in Figure 39 was rendered using these parameters. A minimum energy is obtained with a staggered configuration consistent with observations [3]. The charge-density in the C - C -bond MO is increased by a factor of 0.25 with the formation of the Cellmne2sp3 HOs each having a smaller radius. Using the orbital composition of the CH3 groups (Eq. (14.173)) and the C - C -bond MO (Eq. (14.139), the radii of CLy = O.17113α0 (Eq. (10.51)) and Cclhane2sp3 = 0.87495α0 (Eq. (14.148)) shells, and the parameters of the C - C -bond (Eqs. (13.3-13.4), (14.158-14.160), and (14.162-14.172)), the parameters of the C -H-bond MOs (Eqs. (13.3-13.4), (14.177-14.179), and (14.181- 14.191)), and the bond-angle parameters (Eqs. (14.195-14.208)), the charge-density of the CH3CH3 MO comprising the linear combination of two sets of three C - H-bond MOs and a C - C -bond MO bridging the two methyl groups is shown in Figure 39. Each C - H -bond MO comprises a H2 -type ellipsoidal MO and a Cethme2sp3 HO having the dimensional diagram shown in. Figure 38. The C - C -bond MO comprises a H2 -type ellipsoidal MO bridging two Celhane2sp3 ΗOs having the dimensional diagram shown in Figure 37. ENERGIES OF THE CH3 GROUPS The energies of each CH3 group of ethane are given by the substitution of the semiprincipal axes (Eqs. (14.177-14.178) and (14.181)) into the energy equations of the methyl radical (Eqs. (13.556-13.560)), with the exception that E(Cethαne,2sp3 ) replaces E(c,2sp3) in Eq. (13.560):
Figure imgf000342_0001
3e2 V P = = 38.98068 eV (14.210)
8πεjα2 -b2
(14.211)
V (14.212)
Figure imgf000342_0002
Eτ ( CH, ) = — -67.69451 eV
Figure imgf000342_0003
(14.213) where E7, (CH3) is given by Eq. (14.174) which is reiteratively matched to Eq. (13.542) within five-significant-figure round off error. VIBRATION OF THE 12CH3 GROUPS
The vibrational energy levels of CH3 in ethane may be solved as three equivalent coupled harmonic oscillators by developing the Lagrangian, the differential equation of motion, and the eigenvalue solutions [9] wherein the spring constants are derived from the central forces as given in the Vibration of Ηydrogen-Type Molecular Ions section and the Vibration of Ηydrogen-Type Molecules section.
THE DOPPLERENERGY TERMS OF TEE 12CH3 GROUPS The equations of the radiation reaction force of the methyl groups in ethane are the same as those of the methyl radical with the substitution of the methyl-group parameters. Using Eq. (13.561), the angular frequency of the reentrant oscillation in the transition state is
= 2.50664 X 1016 radls (14.214)
Figure imgf000343_0001
where b is given by Eq. (14.181). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)):
Eκ = hω = h2.50664 X 1016 mdls = 16.49915 eV (14.215)
In Eq. (11.181), substitution of E7- (H2) (Εqs. (11.212) and (13.75)), the maximum total energy of each H2 -type MO, for Ehv , the mass of the electron, me , for M , and the kinetic energy given by Eq. (14.215) for Eκ gives the Doppler energy of the electrons of each of the three bonds for the reentrant orbit:
ED - Eh (14.216)
Figure imgf000343_0002
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of CH3 due to the reentrant orbit of each bond in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by Eq. (14.216) and EKvjb , the average kinetic energy of vibration which is 1/2 of the vibrational energy of each C-H bond. Using ωe given by Eq. (13.458) for EKvib of the transition state having three independent bonds, E \lham osc ( 12CH3 ) per bond is
(14.2.17)
Figure imgf000344_0001
E 'en™, osc (12CH3) = -0.25422 eF +-(0.35532 eV) = -0.07656 eV (14.218)
Given that the vibration and reentrant oscillation is for three C-H bonds, Eetham osc ( 12CH3 ) , is:
(14.219)
Figure imgf000344_0002
= -0.22967 e V
TOTAL AND DIFFERENCE ENERGIES OF THE 12CH3 GROUPS EeihaneT+osc ( 12^^3 ) » tne total energy of each 12CH3 group including the Doppler term, is given by the sum of E^ (CH3) (Eq. (14.213)) and Eetham BSC ( 12CH3 ) given by Eq. (14.219):
Khamτ+oΛCHi) = V e +τ + Vm +Vp + E{Cethamasp*) + Eethan^
(14.220)
= ET«hm (CH3 ) + EeΛane osc ( "^3 )
Figure imgf000344_0003
From Eqs. (14.217-14.221), the total energy of each 12CH3 is ^Wr+0,. ( 12C
(14.222)
Figure imgf000345_0003
where ω given by Eq. (13.458) was used for the term.
Figure imgf000345_0001
The total energy for each methyl radical given by Eq. (13.569) is Eradicalτ+Osc ( 12CH3 ) = -67.69450 e V + Eradical osc ( 12CH3 ) (14.223)
Figure imgf000345_0004
= -67.93160 eF The difference in energy between the methyl groups and the methyl radical ΛEr+oκ, ( 12CH3 ) is given by two times the difference between Eqs. (14.222) and (14.223): ΔET±O∞ ( 12CH3 J = 2 {EethaneT+0SC ( 12CH3 J - Eradicalτ÷osc ( 12CH3 JJ
= 2(-67.92417 eF-(-67.93160 eF)) (14.224)
= 0.01487 eV
SUM OF THE ENERGIES OF THE C-C σ MO AND THE HOs OF ETHANE
The energy components of Ve , Vp , T , Vm , and E7, of the C-C -bond MO are the same as those of the CH MO as well as each C -H -bond MO of the methyl groups except that energy of the Cetham 2$p* HO is used. The energies of each C -C -bond MO are given by the substitution of the semiprincipal axes (Eqs. (14.158-14.159) and (14.162)) into the energy equations of the CH MO (Eqs. (13.449-13.453)), with the exception that E(Cethane,2spl)
replaces E (C,2SP 3 ) in Eq. (13.453):
Figure imgf000345_0002
Vn, -3.45250 eV (14.228)
Figure imgf000346_0001
Er [C-Ca) eK = -31.63535 eF
Figure imgf000346_0002
(14.229) where E7, (C-C,σ) is the total energy of the C-C σ MO given by Eq. (14.155) which is reiteratively matched to Eq. (13.75) within five-significant-figure round off error.
The total energy of the C-C -bond MO, Eτ [C-C], is given by the sum of two times
Eτ (C-C,2sp*j , the energy change of each C2sp3 shell due to the decrease in radius with the formation of the C -C -bond MO (Eq. (14.151)), and Eτ (C-C,σ) , the σ MO contribution given by Eq. (14.156): Eτ (C-C) = 2Eτ (C -C,2sp3) + Eτ (C-C,σ)
Figure imgf000346_0003
= 2(-0.72457 eF) + (-31.63537 eF)
= -33.08452 eV
VIBRATION OF ETHANE
The vibrational energy levels of CH3CH3 may be solved as two sets of three equivalent coupled harmonic oscillators with a bridging harmonic oscillator by developing the Lagrangian, the differential equation of motion, and the eigenvalue solutions [9] wherein the spring constants are derived from the central forces as given in the Vibration of Hydrogen- Type Molecular Ions section and the Vibration of Hydrogen-Type Molecules section.
THE DOPPLER ENERGY TERMS OF THE C-C -BOND MO OF ETHANE The equations of the radiation reaction force of the symmetrical C-C -bond MO are given by Eqs. (11.231-11.233), except the force-constant factor is 0.5 based on the force constant £' of Eq. (14.152), and the C-C-bond MO parameters are used. The angular frequency of the reentrant oscillation in the transition state is
Figure imgf000347_0001
ω = Δ Aπε°a* = 9.55643 X 1015 radls (14.231) e where a is given by Eq. (14.158). The kinetic energy, Eκ , is given by Planck's .equation (Eq. (11.127)):
Eκ = hω = £9.55643 X 1015 radls = 6.29021 eV (14.232) In Eq. (11.181), substitution of ET (C-C) (Eq. (14.230)) for Ehv , the mass of the electron, me , for M , and the kinetic energy given by Eq. (14.232) for Eκ gives the Doppler energy of the electrons of each of the three bonds for the reentrant orbit:
ED s - -33.08450 (14.233)
Figure imgf000347_0002
Figure imgf000347_0003
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of the C-C- bond MO due to the reentrant orbit of the bond in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by Eq. (14.233) and EKvιb , the average kinetic energy of vibration which is 1/2 of the vibrational energy of the C -C bond. Using the experimental C-C Evώ (υ3) of 993 cm'1 (0.12312 eV) [10] for EKvιb of the transition state,
E0SC (C-C,σ) is
(14.234)
Figure imgf000347_0004
E0SC (C - C, σ) = -0Λ64\6 eV +-(0.12312 eV) = -0.10260 eV (14.235)
TOTAL ENERGIES OF THE C-C-BONDMO OF ETHANE
Eτ+osc (C-C), the total energy of the C-C -bond MO including the Doppler term, is given by the sum of E7. (C - C) (Eq. (14.230)) and Eosc (C - C, σ) given by Eq. (14.235): Eτ,Oi.c(C-C) = Ve+T + Vm + Vp + E(Cethane,2sp3) + 2Eτ(C-C,2sp3) + Eosc(C-C,σ) = ET(C-C,σ) + 2ET(C-C,2sp3) + EBSC(C-C,σ) = Eτ(C-C) + E0SC(C-C,σ)
(14.236)
Figure imgf000348_0001
Figure imgf000348_0003
(14.237) From Eqs. (14.234-14.237), the total energy of the C-C -bond MO is ET+OSC (C - C) = -31.63537 eV + 2ET (c - C, 2sp3 ) + Eosc (C - C, σ)
= -31.63537 eF + 2(-0.72457eF)-0.16416eF + -(0.12312eF) (14.238)
= -33.18712 eV
where the experimental E tb was used for the term.
Figure imgf000348_0002
BONDENERGY OF THE C-C BOND OFETHANE The dissociation energy of the C-C bond of CH3CH3, E0(H3C-CH3), is given by two times E(c,2sp3) (Eq. (14.146)), the initial energy of the C2sp3 HO of each CH3 radical
that bond with a single C-C bond, minus the sum of AET+0SC (12CH3) (Eq. (14.224)), the energy change going from the methyl radicals to the methyl groups of ethane, and Eτ+osc (C-C) (Eq. (14.238)). Thus, the dissociation energy of the C-C bond of CH3CH3 , is
ED(H3C-CH3) = 2(E(C,2sp3))-(AET+0SC(nCH3) + ET+0SC{C-C))
= 2(-14.63489 eF)-(θ.O1487 eF-33.18712 eV) (14239)
= 2(-14.63489eF)-(33.17225 eV) = 3.90247 eV The experimental dissociation energy of the C-C bond of CH3CH3 is [6]
ED (H3C-CH3) = 3.89690 eV (14.240)
The results of the determination of bond parameters of CH3CH3 are given in Table
14.1. The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
ETHYLENE MOLECULE (CH2CH2)
The ethylene molecule CH2CH2 is formed by the reaction of two dihydrogen carbide radicals:
CH2 + CH2 -» CH2CH2 (14.241)
CH2CH2 can be solved using the same principles as those used to solve the methane series CHn=1 23 4 , wherein the 2s and 2p shells of each C hybridize to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between two C2sp3 hybridized orbitals (ΗOs) to form a molecular orbital (MO) permits each participating hybridized orbital to decrease in radius and energy. First, two sets of two H atomic orbitals (AOs) combine with two sets of two carbon 2sp3 ΗOs to form two dihydrogen carbide groups comprising a linear combination of four diatomic H2 -type MOs developed in the Nature of the Chemical Bond of
Ηydrogen-Type Molecules and Molecular Ions section. Then, the two CH2 groups bond by forming a H2 -type MO between the remaining two C2sp3 ΗOs on each carbon atom.
FORCEBALANCE OF THE C=C-BOND MO OF ETHYLENE
CH2CH2 comprises a chemical bond between two CH2 radicals wherein each radical comprises two chemical bonds between carbon and hydrogen atoms. The solution of the parameters of CH2 is given in the Dihydrogen Carbide (CH2 ) section. Each C-H bond of
CH2 having two spin-paired electrons, one from an initially unpaired electron of the carbon atom and the other from the hydrogen atom, comprises the linear combination of 75% H2 - type ellipsoidal MO and 25% C2sρ3 HO. The proton of the H atom and the nucleus of the C atom are along each internuclear axis and serve as the foci. As in the case of H2 , each of the two C -H -bond MOs is a prolate spheroid with the exception, that the ellipsoidal MO surface cannot extend into C2sp3 HO for distances shorter than the radius of the C2sp3 shell since it is energetically unfavorable. Thus, each MO surface comprises a prolate spheroid at the H proton that is continuous with the C2sp3 shell at the C atom whose nucleus serves as the other focus. The electron configuration and the energy, E(c,2sp3) , of the C2sp3 shell is given by Eqs. (13.422) and (13.428), respectively. The central paramagnetic force due to spin of each C- H bond is provided by the spin-pairing force of the CH2 MO that has the symmetry of an s orbital that superimposes with the C2sp3 orbitals such that the corresponding angular momenta are unchanged. Two CH2 radicals bond to form CH2CH2 by forming a MO between the two pairs of remaining C2sp3 -HO electrons of the two carbon atoms. However, in this case, the sharing of electrons between four C2sp3 HOs to form a molecular orbital (MO) comprising four spin-paired electrons permits each C2sp3 HO to decrease in radius and energy.
As in the case of the C -H bonds, the C = C -bond MO is a prolate-spheroidal-MO surface that cannot extend into C2sp3 HO for distances shorter than the radius of the C25p3 shell of each atom. Thus, the MO surface comprises a partial prolate spheroid in between the carbon nuclei and is continuous with the C2sp3 shell at each C atom. The energy of the H2 - type ellipsoidal MO is matched to that of the C2sp3 shell. As in the case of previous examples of energy-matched MOs such as those of OH , NH , CH , the C = O -bond MO of CO2 , and the C - C -bond MO of CH3CH3 , the C = C -bond MO of ethylene must comprise
75% of a H2 -type ellipsoidal MO in order to match potential, kinetic, and orbital energy relationships. Thus, the C = C -bond MO must comprise a linear combination of two MOs wherein each comprises two C2,sp3ΗOs and 75% of a H2 -type ellipsoidal MO divided between the C2sp3 ΗOs:
2(2 C2sp3 +0.75 H2 MO) → C = C -bond MO (14.242)
The linear combination of each H2 -type ellipsoidal MO with each C2sp3 HO further comprises an excess 25% charge-density contribution from each C2sp3 HO to the C = C - bond MO to achieve an energy minimum. The force balance of the C = C -bond MO is determined by the boundary conditions that arise from the linear combination of orbitals according to Eq. (14.242) and the energy matching condition between the C2sp3-HO components of the MO. Similarly, the energies of each CH2 MO involve each C2spz and each HIs electron with the formation of each C-H bond. The sum of the energies of the H2 -type ellipsoidal MOs is matched to that of the C2sp3 shell. This energy is determined by the considering the effect of the donation of 25% electron density from the two pairs of C2sp3 ΗOs to the C = C -bond MO with the formation of the Cethykne2sp3 ΗOs each having a smaller radius. The 2sp3 hybridized orbital arrangement is given by Eq. (14.140). The sum Eτ (c ,2sp3) of calculated energies of C , C+ , C2+, and C3+ is given by Eq. (14.141). The radius r2s 3 of the
C2sp3 shell is given by Eq. (14.142). The Coulombic energy ECoulomb (C,2sp3) and the
energy E(c,2sp3^ of the outer electron of the C2sp3 shell are given by Eqs. (14.143) and
(14.146), respectively. Next, consider the formation of the C = C -bond MO of ethylene from two CH2 radicals, each having a C2sp3 electron with an energy given by Eq. (14.146). The total energy of the state is given by the sum over the four electrons. The sum Eτ [Cethylene i2sp3jθϊ calculated energies of C2sp3 , C+ , C2+, and C3+ from Eqs. (10.123), (10.113-10.114), (10.68), and (10.48), respectively, is
ET (Celhylene,2sp3 ) = -(64.3921 eV + 48.3125 eV + 24.2762 eV + E(C,2sp3)) = -(64.3921 eV + 48.3125 eV + 24.2762 eV + 14.63489 eV)
= -151.61569 eV (14.243) where E[c,2spl) (Eq. (14.146)) is the sum of the energy of C , -11.27671 eV , and the hybridization energy. The orbital-angular-momentum interactions also cancel such that the energy of the ET (Cethy!ene,2sp3} is purely Coulombic. The sharing of electrons between two pairs of C2sp3 ΗOs to form a C = C -bond MO permits each participating hybridized orbital to decrease in radius and energy, hi order to further satisfy the potential, kinetic, and orbital energy relationships, each participating C2sp3 HO donates an excess of 25% per bond of its electron density to the C = C -bond MO to form an energy minimum. By considering this electron redistribution in the ethylene molecule as well as the fact that the central field decreases by an integer for each successive electron of the shell, the radius r , , of the C2sp3 shell of ethylene may be calculated from the Coulombic energy using Eq. (10.102):
r . , , 3 0.85252α0 eihyiemisp
Figure imgf000352_0001
(14.244) where Z = 6 for carbon. Using Eqs. (10.102) and (14.244), the Coulombic energy
Ecouiomb (C e,hyiene > 2sP3) of the outer electron of the C2sp3 shell is
(14-245)
Figure imgf000352_0002
During hybridization, one of the spin-paired 2s electrons is promoted to C2sρ3 shell as an unpaired electron. The energy for the promotion is the magnetic energy given by Eq. (13.152). Using Eqs. (14.145) and (14.245), the energy E[Ce!liylene,2sp3) of the outer electron of the C2.sp3 shell is
Figure imgf000352_0003
= -15.95955 eV + 0.19086 eV (14.246)
= -15.76868 eV
Thus, ET (c = C,2sp3} , the energy change of each C2sp3 shell with the formation of the C = C -bond MO is given by the difference between Eq. (14.146) and Eq. (14.246): ET (c = C,2sp3) = E(Celhy!em,2Sp3 )-E(C,2sp3)
= -15.76868 eF-(-14.63489 eV) (14.247)
= -1.13380 eV As in the case of Cl2, each H2 -type ellipsoidal MO comprises 75% of the C = C -bond MO shared between two C2sp3 HOs corresponding to the electron charge density in Eq. (11.65)
of — '- . But, the additional 25% charge-density contribution to each bond of the C = C -
— Θ bond MO causes the electron charge density in Eq. (11.65) to be is given by — = -0.5e . The corresponding force constant k' is given by Eq. (14.152). In addition, the energy matching at both C2spz HOs further requires that k' be corrected by the hybridization factor given by Eq. (13.430). Thus, the force constant k' to determine the ellipsoidal parameter c' in terms of the central force of the foci (Eq. (11.65)) is given by
t._C C_lSp ,6H..O.M≥l = 0.9177lM≥ (14.248)
4πεn 4πεn
The distance from the origin to each focus c' is given by substitution of Eq. (14.248) into Eq. (13.60). Thus, the distance from the origin of the component of the double C = C -bond MO to each focus c ' is given by
Figure imgf000353_0001
The internuclear distance from Eq. (14.249) is
Figure imgf000353_0002
The length of the semiminor axis of the prolate spheroidal C = C -bond MO b = c is given by Eq. (13.62). The eccentricity, e , is given by Eq. (13.63). The solution of the semimaj or axis a then allows for the solution of the other axes of each prolate spheroid and eccentricity of the C = C -bond MO. From the energy equation and the relationship between the axes, the dimensions of the C = C -bond MO are solved.
The general equations for the energy components of Ve , Vp , T , V1n , and Eτ of the
C = C -bond MO are the same as those of the CH MO except that energy of the Cethyhm2sp3 HO is used and the double-bond nature is considered. In the case of a single bond, the prolate spheroidal H2 -type MO transitions to the Celhylem2sp2 HO of each carbon, and the energy of the Cethylene2sp3 shell must remain constant and equal to the E(Celhyleιie,2sp3) given by Eq.
(14.246). Thus, the energy E(Cethylene,2sp2) in Eq. (14.246) adds to that of the energies of the corresponding H2 -type ellipsoidal MO. The second bond of the double C = C -bond MO also transitions to the Celhylene2sp3 HO of each C . The energy of a second H2 -type ellipsoidal MO adds to the first energy component, and the two bonds achieve an energy minimum as a linear combination of the two H2 -type ellipsoidal MOs each having the carbon nuclei as the foci. Each C -C -bond MO comprises the same Cethylene2sp3 HO shells of constant energy given by Eq. (14.246). As in the case of the water, NH2 , and ammonia molecules given by Eqs. (13.180), (13.320), and (13.372), respectively, the energy of the redundant shell is subtracted from the total energy of the linear combination of the σ MO. Thus, the total energy Eτ (C = C,σ) of the σ component of the C = C -bond MO is given by the sum of the energies of the two bonds each comprising the linear combination of the Cethy!ene2sp3 HO and the H2-ty$Q ellipsoidal MO as given by Eq. (14.242) wherein the E1. terms add positively, the E[Cethylene,2sp%\ terms cancel, and the energy matching condition between the components is provided by Eq. (14.248). Using Eqs. (13.431) and (14.246), Eτ {C = C,σ) is given by
E7 (C = C,σ) = Eτ +E(Celhylene,2sp3)-E(Cethyhne,2sp3)
Figure imgf000354_0001
The total energy term of the double C = C -bond MO is given by the sum of the two H2 -type ellipsoidal MOs given by Eq. (11.212). To match this boundary condition, Eτ {C = C,σ) given by Eq. (14.251) is set equal to two times Eq. (13.75):
Eτ (C (14.252)
Figure imgf000354_0002
From the energy relationship given by Eq. (14.252) and the relationship between the axes given by Eqs. (14.249-14.250) and (13.62-13.63), the dimensions of the C - C -bond MO can be solved.
Substitution of Eq. (14.249) into Eq. (14.252) gives
Figure imgf000354_0003
The most convenient way to solve Eq. (14.253) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is a = 1.47228α0 = 7.79098 X 10~" m (14.254)
Substitution of Eq. (14.254) into Eq. (14.249) gives c' = 1.2666Ia0 = 6.70259 X 10~u m (14.255) The internuclear distance given by multiplying Eq. (14.255) by two is 2c' = 2.5332Ia0 = 1.34052 X 10~10 m (14.256)
The experimental bond distance is [3]
2c' = 1.339 X lO"10 m (14.257)
Substitution of Eqs. (14.254-14.255) into Eq. (13.62) gives b = c = 0.75055tf0 = 3.97173 X l(Tn m (14.258)
Substitution of Eqs. (14.252-14.255) into Eq. (13.63) gives e = 0.86030 (14.259)
The nucleus of the C atoms comprise the foci of the H2 -type ellipsoidal MO. The parameters of the point of intersection of the H2 -type ellipsoidal MO and the Cethylem2spl HO are given by Eqs. (13.84-13.95) and (13.261-13.270). The polar intersection angle θ' is given by Eq. (13.261) where rn = r thylem2sp3 = 0.85252α0 is the radius of the Cethylem2sps shell.
Substitution of Eqs. (14.254-14.255) into Eq. (13.261) gives θ' = 129.84° (14.260)
Then, the angle #„ „ „ . „ the radial vector of the C2sp3 HO makes with the internuclear axis is
^ . , 3ϋ =180°-129.84° = 50.16° (14.261) as shown in Figure 40.
Consider the right-hand intersection point. The distance from the point of intersection of the orbitals to the internuclear axis must be the same for both component orbitals. Thus, the angle ωt = θr r „ un between the internuclear axis and the point of intersection of the H, - type ellipsoidal MO with each Cethylene2sp3 radial vector obeys the following relationship: r . , , s sin0_ _ , ^n = 0.85252α0 sin<9 „ , itm = bsmθc_c H MO (14.262) such that
Figure imgf000355_0001
with the use of Eq. (14.261). Substitution of Eq. (14.258) into Eq. (14.263) gives θc c H MO = 60.70° (14.264)
Then, the distance dc=c H MO along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by dC=Cellφm,H2MO = a COS θC=Cφlem,H2MO (14.265)
Substitution of Eqs. (14.254) and (14.264) into Eq. (14.265) gives dc=c!l¥smM2Mo = 0.7204Oa0 = 3.81221 X 10"11 m (14.266)
The distance d , along the internuclear axis from the origin of the C atom to the point of intersection of the orbitals is given by dC~Celhylιm2sp>HO = C'~ dC=CΛyUm,H2M0 (14.267)
Substitution of Eqs. (14.255) and (14.266) into Eq. (14.267) gives dc-celhykn^Ho = °-5462(H = 2-89038 X 10"" m (14.268)
FORCE BALANCE OF THE CH2 MOs OF ETHYLENE
Each of the two equivalent CH2 MOs must comprise two C-H bonds with each comprising 75% of a H2 -type ellipsoidal MO and a C2sp3 HO as given by Eq. (13.494):
2[l C2sp3 + 0.75 H2 MO] → CH2 MO (14.269)
The force balance of the CH2 MO is determined by the boundary conditions that arise from the linear combination of orbitals according to Eq. (13.494) and the energy matching condition between the hydrogen and C2sp3 HO components of the MO.
The force constant k' to determine the ellipsoidal parameter c' of the each H2-type- ellipsoidal-MO component of the CH2 MO in terms of the central force of the foci is given by Eq. (13.59). The distance from the origin of each C-H -bond MO to each focus c' is given by Eq. (13.60). The internuclear distance is given by Eq. (13.61). The length of the semiminor axis of the prolate spheroidal C-H-bond MO b = c is given by Eq. (13.62). The eccentricity, e , is given by Eq. (13.63). The solution of the semimajor axis a then allows for the solution of the other axes of each prolate spheroid and eccentricity of each C-H -bond MO. From the energy equation and the relationship between the axes, the dimensions of the CH2 MO are solved.
Consider the formation of the double C = C -bond MO of ethylene from two CH2 radicals, each having a C2jp3 shell with an energy given by Eq. (14.146). The energy components of F6 , Vp, _T, Vn, , and Eτ are the same as those of the dihydrogen carbide radical, two times those of CH corresponding to the two C-H bonds, except that two times Eτ (C = C,2sp*) is subtracted from E1. (CH2) of Eq. (13.495). The subtraction of the energy change of the C2sp3 shells with the formation of the C - C -bond MO matches the energy of the C-H-bond MOs to the decrease in the energy of the C2sp3 ΗOs. Using Eqs. (13.495) and (14.247), E^jCH2) is given by
E^JCH 2) = Eτ + E(C,2sp3)-2Eτ (c = C,2sps)
Figure imgf000357_0001
ETι (CH2) given by Eq. (14.270) is set equal to two times the energy of the H2 -type ellipsoidal MO minus the Coulombic energy of H given by Eq. (13.496):
Figure imgf000357_0002
(14.271)
From the energy relationship given by Eq. (14.271) and the relationship between the axes given by Eqs. (13.60-13.63), the dimensions of the CH2 MO can be solved.
Substitution of Eq. (13.60) into Eq. (14.271) gives
Figure imgf000357_0003
The most convenient way to solve Eq. (14.272) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is a = 1.56946αo = 8.30521 X 10"11 m (14.273) Substitution of Eq. (14.273) into Eq. (13.60) gives c' = 1.02289α0 = 5.41290 X 10"n m (14.274)
The internuclear distance given by multiplying Eq. (14.274) by two is
2c' = 2.04578α0 = 1.08258 X 10~10 m (14.275) The experimental bond distance is [3] 2c' = 1.087 X 1(T10 m (14.276)
Substitution of Eqs. (14.273-14.274) into Eq. (14.62) gives δ = c = 1.19033α0 = 6.29897 X 10~" m (14.277)
Substitution of Eqs. (14.273-14.274) into Eq. (14.63) gives β = 0.65175 (14.278)
The nucleus of the H atom and the nucleus of the C atom comprise the foci of each
H2 -type ellipsoidal MO. The parameters of the point of intersection of the H2 -type ellipsoidal MO and the Cethy!ens2sp3 HO are given by Eqs. (13.84-13.95) and (13.261-13.270). The polar intersection angle θ' is given by Eq. (13.261) where rn = reihytøie2sp3 = 0.85252α0 is
the radius of the Cellhylem2sp3 shell. Substitution of Eqs. (14.273-14.274) into Eq. (13.261) gives
0' = 84.81° (14.279)
Then, the angle Qn „ „ %a. the radial vector of the C2sp3 HO makes with the internuclear
axis is θ B , 3 = 180°-84.81° = 95.19° ■ (14.280) as shown in Figure 41.
The distance from the point of intersection of the orbitals to the internuclear axis must be the same for both component orbitals. Thus, the angle ωt = θc-He,hy,m,H2Mo between the internuclear axis and the point of intersection of the H2 -type ellipsoidal MO with the Celhykne2sp3 radial vector obeys the following relationship: r 3 sin6> „ ,„„ = 0.85252α0 sin6> „ , 3nn = hsinθr H H MO (14.281) such that
Figure imgf000358_0001
with the use of Eq. (14.280). Substitution of Eq. (14.277) into Eq. (14.282) gives
QC C~H H ethylene ' H H2MM0O = 45.50° ( V14.283) '
Then, the distance dc_Hah kmfliU0 along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by dc-He:l≠em,H2Mo = a∞sθc_Hcι¥smιH2M0 (14.284)
Substitution of Eqs. (14.273) and (14.283) into Eq. (14.284) gives dc-Hel¥me,H2Mo = l-10002α0 = 5.82107 X 1(T11 m (14.285)
The distance d . along the intemuclear axis from the origin of the C atom to the point of intersection of the orbitals is given by d C-Hellψkm 2^H0 = dC-Hs,h≠m,H2M0 ~ C (14.286)
Substitution of Eqs. (14.274) and (14.285) into Eq. (14.286) gives dc-HMy,^Ho = °-O7713*o = 4.08171 X 10-12 m (14.287)
BOND ANGLE OF THE CH2 GROUPS
Each CH2 MO comprises a linear combination of two C -H -bond MOs. Each C- H -bond MO comprises the superposition of a H2 -type ellipsoidal MO and the Ceth)>lme2sp3 HO. A bond is also possible between the two H atoms of the C-H bonds. Such H-H bonding would decrease the C-H bond strength since electron density would be shifted from the C-H bonds to the H-H bond. Thus, the bond angle between the two C-H bonds is determined by the condition that the total energy of the H2 -type ellipsoidal MO between the terminal H atoms of the C-H bonds is zero. From Eqs. (11.79) and (13.228), the distance from the origin to each focus of the H-H ellipsoidal MO is
Figure imgf000359_0001
The intemuclear distance from Eq. (13.229) is
2c1 = 2 A^- (14.289)
The length of the semiminor axis of the prolate spheroidal H-H MO b = c is given by Eq. (14.62).
The bond angle of the CH2 groups of ethane is derived by using the orbital composition and an energy matching factor as in the case with the dihydrogen carbide radical and the CH3 groups of ethane. Since the two H2 -type ellipsoidal MOs initially comprise
75% of the H electron density of H2 and the energy of each H2 -type ellipsoidal MO is matched to that of the Celhyhm2sp3 HO, the component energies and the total energy Eτ of the H - H bond are given by Eqs. (13.67-13.73) except that Ve , T , and V1n are corrected for the hybridization-energy-matching factor of 0.85252 . Hybridization with 25% electron donation to the C = C -bond gives rise to the Celhykm2sp2 HO-shell Coulombic energy
Ecoubmb (C e,hyiene> 2sP3 ) §iven bv Ecl- (14-245). The corresponding normalization factor for determining the zero of the total H -H bond energy is given by the ratio of 15.95955 eV , the magnitude of ECoulomb (Cellψlem,2sp3 ) given by Eq. (14.245), and 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H given by Eq. ( v1.243) J. The hy Jbridization energ byJ factor C elh1y!,eneC2sp 3 3HO is
(14.290)
Figure imgf000360_0002
Substitution of Eq. (14.290) into Eq. (13.233) or Eq. (14.195) with the hybridization factor of 0.85252 gives
Figure imgf000360_0001
(14.291)
From the energy relationship given by Eq. (14.291) and the relationship between the axes given by Eqs. (14.192-14.193) and (14.62-14.63), the dimensions of the H-H MO can be solved.
The most convenient way to solve Eq. (14.291) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is α = 6.040Oa0 = 3.1962 X lO"10 m (14.292) Substitution of Eq. (14.292) into Eq. (14.288) gives c' = 1.7378a0 = 9.1961 X 10~u m (14.293) The internuclear distance given by multiplying Eq. (14.293) by two is 2c' = 3.4756α0 = 1.8392 X 1(T10 m (14.294)
Substitution of Eqs. (14.292-14.293) into Eq. (14.62) gives
6 = c = 5.7846«0 = 3.061 I Z 1(T10 m (14.295)
Substitution of Eqs. (14.292-14.293) into Eq. (14.63) gives e = 0.2877 (14.296)
From, 2c'H_H (Eq. (14.294)), the distance between the two H atoms when the total energy of the corresponding MO is zero (Eq. (14.291)), and 2c'c_H (Eq. (14.275)), the internuclear distance of each C-H bond, the corresponding bond angle can be determined from the law of cosines. Using, Eq. (13.242), the bond angle ΘHCH between the C -H bonds is
Figure imgf000361_0001
The experimental angle between the C-H bonds is [11]
ΘHCH =116.6° (14.298)
The C - C bond is along the z-axis. Thus, based on the symmetry of the equivalent bonds, the bond angle ΘC=C_H between the internuclear axis of the C = C bond and a H atom of the
CH2 groups is given by
Figure imgf000361_0002
Substitution of Eq. (14.298) into Eq. (14.299) gives
ΘC+C_H = 121.85° (14.300) The experimental angle between the C = C-H bonds is [11]
61 C=C-H = 121.7° (14.301) and [3]
Figure imgf000361_0003
The C = C bond and H atoms of ethylene line in a plane, and rotation about the C = C is not possible due to conservation of angular momentum in the two sets of spin-paired electrons of the double bond. The CH2CH2 MO shown in Figure 42 was rendered using these parameters.
The charge-density in the C = C -bond MO is increased by a factor of 0.25 per bond with the formation of the Cethylem2sp3 ΗOs each having a smaller radius. Using the orbital composition of the CH2 groups (Eq. (14.269)) and the C = C -bond MO (Eq. (14.242), the radii of CIs = 0.17113α0 (Eq. (10.51)) and Celhylem2sp3 = 0.85252α0 (Eq. (14.244)) shells, and the parameters of the C = C -bond (Eqs. (13.3-13.4), (14.254-14.256), and (14.258-14.268)), the parameters of the C- H-bond MOs (Eqs. (13.3-13.4), (14.273-14.275), and (14.277- 14.287)), and the bond-angle parameters (Eqs. (14.297-14.302)), the charge-density of the
CH2CH2 MO comprising the linear combination of two sets of two C - H -bond MOs and a
C = C -bond MO bridging the two CH2 groups is shown in Figure 42. Each C -H -bond
MO comprises a H2 -type ellipsoidal MO and a Celhylene2sp3 HO having the dimensional diagram shown in Figure 41. The C = C -bond MO comprises a H2 -type ellipsoidal MO bridging two Celhyhne2sp3 ΗOs having the dimensional diagram shown in Figure 40.
ENERGIES OF THE CH2 GROUPS
The energies of each CH2 group of ethylene are given by the substitution of the semiprincipal axes (Eqs. (14.273-14.274) and (14.277)) into the energy equations of dihydrogen carbide (Eqs. (13.510-13.514)), with the exception that two times E1. (c = C,2sp3) (Eq. (14.247)) is subtracted from E7 (CH2) in Eq. (13.514):
Ve eV (14.303)
Figure imgf000362_0001
(14.304)
Figure imgf000362_0005
(14.305)
Figure imgf000362_0004
-12.1073O eF (14.306)
Figure imgf000362_0002
Figure imgf000362_0003
(14.307) where ETtΛ≠m (CH2) is given by Eq. (14.270) which is reiteratively matched to Eq. (13.496) within five-significant-figure round off error.
VIBRATION OF THE 12CH2 GROUPS The vibrational energy levels of CH2 in ethylene may be solved as two equivalent coupled harmonic oscillators by developing the Lagrangian, the differential equation of motion, and the eigenvalue solutions [9] wherein the spring constants are derived from the central forces as given in the Vibration of Hydrogen-Type Molecular Ions section and the Vibration of Hydrogen-Type Molecules section.
THE DOPPLER ENERGY TERMS OF THE 12CH2 GROUPS
The equations of the radiation reaction force of the CH2 groups in ethylene are the same as those of the dihydrogen carbide radical with the substitution of the CH2 -group parameters. Using Eq. (13.515), the angular frequency of the reentrant oscillation in the transition state is
(14.308)
Figure imgf000363_0002
where b is given by Eq. (14.277). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)):
Eκ = hω = £2.75685 X 1016 rad Is = 18.14605 eV (14.309)
In Eq. (11.181), substitution of ET (H2) (Eqs. (11.212) and (13.75)), the maximum total energy of each H2 -type MO, for E1n, , the mass of the electron, me , for M , and the kinetic energy given by Eq. (14.309) for Eκ gives the Doppler energy of the electrons of each of the two bonds for the reentrant orbit:
Figure imgf000363_0001
hi addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of CH2 due to the reentrant orbit of each bond in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by Eq. (14.310) and EKvlb , the average kinetic energy of vibration which is 1/2 of the vibrational energy of each C-H bond. Using ωe given by Eq. (13.458) for EKvjb of the transition state having two independent bonds, E 'elhykne osc (12CH2) per bond is
(14.311)
Figure imgf000364_0001
E \thylem osc (12CH2) = -0.26660 eV + 1(0.35532 eV) = -0.08894 eV (14.312)
Given that the vibration and reentrant oscillation is for two C-H bonds, Eethylene osc ( 12CH2 ) , is:
(14.313)
Figure imgf000364_0004
= -0.17788 eV
TOTAL AND DIFFERENCE ENERGIES OF THE 12CH, GROUPS
''ethyhneT+osc (12CH2) , the total energy of each 12CH2 group including the Doppler term, is
given by the sum of E^ (CH2) (Eq. (14.307)) and Eethylem osc ( 12CH2 ) given by Eq. (14.313):
■JethylemT+osc (CH2)
(14.314)
Figure imgf000364_0002
Figure imgf000364_0003
From Eqs. (14.313-14.315), the total energy of each 12CH2 is
E^T+OSC ( 12CH2 ) = -49.66493 e V + Eelh≠em osc ( 12CH2 )
= -49.66493 eV~li 0.26660 eV --(0.35532 eV)) (14.316)
= -49.84282 eV
where ω given by Eq. (13.458) was used for the term.
Figure imgf000365_0001
The total energy for each dihydrogen carbide radical given by Eq. (13.523) is EradicalT+0SC ( 12CH2 ) = -49.66493 eV + Eradicalosc ( 12CH2 )
Figure imgf000365_0002
= -49.81948 eV The difference in energy between the CH2 groups and the dihydrogen carbide radical
AET+OSC ( 12CH2 ) is given by two times the difference between Eqs. (14.316) and (14.317):
Figure imgf000365_0003
( "^^2 ))
= 2(-49.84282 eF-(-49.81948 eF)) (14.318)
= -0.04667 eV
SUM OF THE ENERGIES OF THE C = C σ MO AND THE HOs OF ETHYLENE
The energy components of Ve , Vp , T , V1n , and E1, of the C = C -bond MO are the same as those of the CH MO except that each term is multiplied by two corresponding to the double bond and the energy term corresponding to the Cethy!em2sp3 ΗOs in the equation for E7. is zero. The energies of each C = C -bond MO are given by the substitution of the semiprincipal axes (Eqs. (14.254-14.255) and (14.258)) into two times the energy equations of the CH MO (Eqs. (13.449-13.453)), with the exception that zero replaces E(c,2ψ3) in Eq. (13.453):
Ve -102.08992 eV (14.319)
Figure imgf000365_0004
e2 V = 2 . = 21.48386 eV (14.320)
P %πεja2 -b2 eV (14.321)
Figure imgf000366_0001
Figure imgf000366_0002
where E7. (C - C,σ) is the total energy of the C = C σ MO given by Eq. (14.251) which is reiteratively matched to two times Eq. (13.75) within five-significant-figure round off error. The total energy of the C = C -bond MO, E1. [C = C) , is given by the sum of two times E1. (C = C, 2sp3 ) , the energy change of each C2sp3 shell due to the decrease in radius with the formation of the C = C -bond MO (Eq. (14.247)), and Eτ (C = C,σ), the σ MO contribution given by Eq. (14.252):
Eτ (C = C) = 2Eτ (c = C,2sp3 ) + Eτ (C = C,σ)
Figure imgf000366_0003
= 2(-l.13380 eV) + (-63.27074 eV) = -65.53833 eV
VIBRATION OF ETHYLENE
The vibrational energy levels of CH2CH2 may be solved as two sets of two equivalent coupled harmonic oscillators with a bridging harmonic oscillator by developing the Lagrangian, the differential equation of motion, and the eigenvalue solutions [9] wherein the spring constants are derived from the central forces as given in the Vibration of Hydrogen- Type Molecular Ions section and the Vibration of Hydrogen-Type Molecules section.
THE DOPPLER ENERGY TERMS OF THE C = C -BOND MO OF ETHYLENE
The equations of the radiation reaction force of the C = C -bond MO are given by Eq. (13.142), except the force-constant factor is (0.93172)0.5 based on the force constant k' of Eq. (14.248), and the C = C -bond MO parameters are used. The angular frequency of the reentrant oscillation in the transition state is
Figure imgf000367_0001
where b is given by Eq. (14.258). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)):
Eκ = hω = h4.30680 X l O16 rad Is = 28.34813 eV (14.326)
In Eq. (11.181), substitution of Eτ (C = C)/ 2 (Eq. (14.324)) for E1n, , the mass of the electron, me , for M , and the kinetic energy given by Eq. (14.326) for Eκ gives the Doppler energy of the electrons of each of the two bonds for the reentrant orbit:
En . ,Q34517 eV (14.327)
Figure imgf000367_0002
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of the C = C - bond MO due to the reentrant orbit of the bond in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by Eq. (14.327) and EKvjb , the average kinetic energy of vibration which is 1/2 of the vibrational energy of the C = C bond. Using the experimental C = C Evib (υ3) of 1443.5 cm"1 (0.17897 eV) [12] for EKvib of the transition state having two bonds, E 'osc (C = C, σ) per bond is
E L (C = C,σ) = ED + EKvlb = ED +U P- (14.328) z \ μ
E'osc (C = C,σ) = -0.34517 eV +-(QΛ1$91 eV) = -0.25568 eV (14.329)
Given that the vibration and reentrant oscillation is for two C-C bonds of the C = C double bond, Eelhy!ene osc (C = C, σ) , is:
Figure imgf000367_0003
(14.330) TOTALENERGIES OF THE C=C-BONDMO OF ETHYLENE
Eτ+osc [C = C) , the total energy of the C = C -bond MO including the Doppler term, is given by the sum of Eτ [C = C) (Eq. (14.324)) and Eeth≠em osc (C = C5 σ) given by Eq. (14.330):
Eτ+BSC (C = C) = Ve +T + Vm + Vp +2Eτ (c = C,2sp3) + Eelhylem osc (C = C,σ)
= Eτ {C = C,σ) + 2Er (c = C,2sp3) + Eellvlem osc (C = C,σ) (14.331)
= Eτ (C = C) + Eethylene osc (C = C,σ)
Figure imgf000368_0001
From Eqs. (14.330-14.332), the total energy of the C = C -bond MO is
^r+<we (C = C) = -63.27074 eF + 2Er (c = C,2^3) + Ee//^ ∞c (C = C,σ)
= -63.27074 eV + 2(-ϊ.13380 e7)-2| 0.34517 eF-l(0.17897 eF) = -66.04969 e V
(14.333)
where the experimental E ib was used for the
Figure imgf000368_0002
BOND ENERGYOF THE C=C BOND OF ETHYLENE
The dissociation energy of the C = C bond of CH2CH2 , ED (H2C = CH2), is given by four times E(C,2^3) (Eq. (14.146)), the initial energy of each C2sp" HO of each CH2 radical
that forms the double C = C bond, minus the sum of ΔET+OSC ( 12CH2 ) (Eq. (14.318)), the energy change going from the dihydrogen carbide radicals to the CH2 groups of ethylene, and ET+OSC (C = C) (Eq. (14.333)). Thus, the dissociation energy of the C = C bond of CH2CH2 , is C))
Figure imgf000369_0001
= 4(-14.63489 eV)- (-0.04667 eV -66.04969 eV) (14 334)
= 4(-14.63489 eV)- (-66.09636 eV) = 7.55681 eV
The experimental dissociation energy of the C = C bond of CH2CH2 is [7]
ED (H2C -CH2) = 7.5969 eV (14.335)
The results of the determination of bond parameters of CH2CH2 are given in Table 14.1. The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
ACETYLENE MOLECULE (CHCH) The acetylene molecule CHCH is formed by the reaction of two hydrogen carbide radicals:
CH +CH → CHCH (14.336)
CHCH can be solved using the same principles as those used to solve the methane series CHn=1 234 as well as ethane, wherein the 2s and 2p shells of each C hybridize to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between two C2sp3 hybridized orbitals (HOs) to form a molecular orbital (MO) permits each participating hybridized orbital to decrease in radius and energy. First, two sets of one H atomic orbital (AO) combine with two sets of one carbon 2sp3 HO to form two hydrogen carbide groups comprising a linear combination of two diatomic H2 -type MOs developed in the Nature of the Chemical Bond of Hydrogen-Type Molecules and Molecular Ions section. Then, the two CH groups bond by forming a H2 -type MO between the remaining three C2sp3 HOs on each carbon atom.
FORCE BALANCE OF THE C ≡ C -BOND MO OF ACETYLENE
CHCH comprises a chemical bond between two CH radicals wherein each radical comprises a chemical bond between a carbon and a hydrogen atom. The solution of the parameters of CH is given in the Hydrogen Carbide (CH ) section. The C -H bond of CH having two spin-paired electrons, one from an initially unpaired electron of the carbon atom and the other from the hydrogen atom, comprises the linear combination of 75% H2 -type ellipsoidal MO and 25% C2sp3 HO. The proton of the H atom and the nucleus of the C atom are along each internuclear axis and serve as the foci. As in the case of H2 , the C-H- bond MOs is a prolate spheroid with the exception that the ellipsoidal MO surface cannot extend into C2sp3 HO for distances shorter than the radius of the C2sp3 shell since it is energetically unfavorable. Thus, the MO surface comprises a prolate spheroid at the H proton that is continuous with the C2sp3 shell at the C atom whose nucleus serves as the other focus. The electron configuration and the energy, E(C,2sp3} , of the C2sp3 shell is given by Eqs. (13.422) and (13.428), respectively. The central paramagnetic force due to spin of the C-H bond is provided by the spin-pairing force of the CH MO that has the symmetry of an s orbital that superimposes with the C2sp3 orbitals such that the corresponding angular momenta are unchanged.
Two CH radicals bond to form CHCH by forming a MO between the two pairs of three remaining C2sp3 -HO electrons of the two carbon atoms. However, in this case, the sharing of electrons between two C2sp3 HOs to form a MO comprising six spin-paired electrons permits each C2sp3 HO to decrease in radius and energy.
As in the case of the C-H bonds, the C ≡ C -bond MO is a prolate-spheroidal-MO surface that cannot extend into C2sp3 HO for distances shorter than the radius of the C2sp3 shell of each atom. Thus, the MO surface comprises a partial prolate spheroid in between the carbon nuclei and is continuous with the C2sp3 shell at each C atom. The energy of the H2- type ellipsoidal MO is matched to that of the C2sp3 shell. As in the case of previous examples of energy-matched MOs such as those of OH , NH , CH , the C = O -bond MO of CO2 , the C - C -bond MO of CH3CH3 , and the C = C -bond MO of CH2CH2 , the C ≡ C - bond MO of acetylene must comprise 75% of a H2 -type ellipsoidal MO in order to match potential, kinetic, and orbital energy relationships. Thus, the C ≡ C -bond MO must comprise a linear combination of three MOs wherein each comprises two C2^p3HOs and 75% of a H2 -type ellipsoidal MO divided between the C2sp3 HOs:
3{2C2sp3 +0.75 H2 Mθ) → C ≡ C -bond MO (14.337) The linear combination of each H2 -type ellipsoidal MO with each C2sp3 HO further comprises an excess 25% charge-density contribution from each C2sp3 HO to the C = C - bond MO to achieve an energy minimum. The force balance of the C ≡ C -bond MO is determined by the boundary conditions that arise from the linear combination of orbitals according to Eq. (14.337) and the energy matching condition between the C2sp3-UO components of the MO.
Similarly, the energies of each CH MO involve each C2sp3 and each His electron with the formation of each C-H bond. The sum of the energies of the H2 -type ellipsoidal
MOs is matched to that of the C2sp3 shell. This energy is determined by the considering the effect of the donation of 25% electron density from the three pairs of C2sp3 HOs to the
C ≡ C -bond MO with the formation of the Cacetylene2sp3 HOs each having a smaller radius.
The 2sp3 hybridized orbital arrangement is given by Eq. (14.140). The sum E1. (C,2sp3 ) of calculated energies of C, C+ , C2+, and C3+ is given by Eq. (14.141). The radius r^ of the
C2sp3 shell is given by Eq. (14.142). The Coulombic energy ECoulomb (C,2sp3} and the
energy E(c,2sp3) of the outer electron of the C2sp3 shell are given by Eqs. (14.143) and
(14.146), respectively.
Next, consider the formation of the C ≡ C -bond MO of acetylene from two CH radicals, each having a C2sp3 electron with an energy given by Eq. (14.146). The total energy of the state is given by the sum over the four electrons. The sum Eτ [Cacetylene,2sp3)oϊ calculated energies of C2sp3 , C+ , C2+, and C3+ from Eqs. (10.123), (10.113-10.114), (10.68), and (10.48), respectively, is
Eτ {Cace≠em,2spl) = -{^^92\ eF + 48.3125 eF + 24.2762 eV + E(C,2sp3))
= -(64.3921 eF + 48.3125 eF + 24.2762 eF + 14.63489 eV) = -151.61569 eV (14.338) where E[c,2sp3) (Eq. (14.146)) is the sum of the energy of C , -11.2767I eF , and the hybridization energy. The orbital-angular-momentum interactions also cancel such that the energy of the Eτ (Cace≠em,2sp3} is purely Coulombic. The sharing of electrons between three pairs of C2sp3 HOs to form a C ≡ C -bond
MO permits each participating hybridized orbital to decrease in radius and energy. In order to further satisfy the potential, kinetic, and orbital energy relationships, each participating C2sp 3 HO donates an excess of 25% of its electron density to the C = C -bond MO to form an energy minimum. By considering this electron redistribution in the acetylene molecule as well as the fact that the central field decreases by an integer for each successive electron of the shell, ' the radius r acelyl ,ene „2sp 3 i of the C2sp "3 shell of acety Jlene mayJ be calculated from the
Coulombic energy using Eq. (10.102):
(14.339)
Figure imgf000372_0001
= 0.83008α0 where Z = 6 for carbon. Using Eqs. (10.102) and (14.339), the Coulombic energy Ecouiomb {c aceφne^P2) of the outer electron of the C2sp3 shell is
(14.340)
Figure imgf000372_0003
During hybridization, one of the spin-paired 2s electrons is promoted to C2spl shell as an unpaired electron. The energy for the promotion is the magnetic energy given by Eq. (13.152). Using Eqs. (14.145) and (14.340), the energy E(Cacetylene,2spl) of the outer electron of the C2sp3 shell is '
_16 39089 e V + 0.19086 eV = -16.20002 eV
Figure imgf000372_0002
(14.341) Thus, Eτ (c ≡ C,2$p3 } , the energy change of each C2sp3 shell with the formation of the C ≡ C -bond MO is given by the difference between Eq. (14.146) and Eq. (14.341):
E7 (C ≡ C,2sp') = E(Cace¥ene,2sp3yE(c,2sp3)
= -16.20002 eV -(-14.63489 eV) (14.342)
= -1.56513 eV As in the case of CZ25 each H2 -type ellipsoidal MO comprises 75% of the C ≡ C -bond MO shared between two C2sp3 HOs corresponding to the electron charge density in Eq. (11.65)
of — '- . But, the additional 25% charge-density contribution to each bond of the C = C -
bond MO causes the electron charge density in Eq. (11.65) to be is given by — = -0.5e . The
corresponding force constant k ' to determine the ellipsoidal parameter c ' in terms of the central force of the foci (Eq. (11.65)) is given by Eq. (14.152). The distance from the origin to each focus c' is given by Eq. (14.153). The internuclear distance is given by Eq. (14.154). The length of the semiminor axis of the prolate spheroidal C ≡ C -bond MO b = c is given by Eq. (13.62). The eccentricity, e , is given by Eq. (13.63). The solution of the semimajor axis a then allows for the solution of the other axes of each prolate spheroid and eccentricity of the C = C -bond MO. From the energy equation and the relationship between the axes, the dimensions of the C ≡ C -bond MO are solved.
The general equations for the energy components of Ve , Vp, T , V1n , and E1, of the
C ≡ C -bond MO are the same as those of the CH MO except that energy of the Cacelylene2sp3 HO is used and the triple-bond nature is considered. In the case of a single bond, the prolate spheroidal H2 -type MO transitions to the Cacetylem2sp3 HO of each carbon, and the energy of the Caceφm2sp3 shell must remain constant and equal to the E{Cacejylene,2sp3} given by Eq.
(14.391). Thus, the energy E(Cacelylene,2sp3 ) in Eq. (14.391) adds to that of the energies of the corresponding H2 -type ellipsoidal MO. The second and third bonds of the triple C ≡ C - bond MO also transition to each Cacetylene2sp3 HO of each C . The energy of a second and a third H2 -type ellipsoidal MO adds to the first energy component, and the three bonds achieve an energy minimum as a linear combination of the three H2 -type ellipsoidal MOs each having the carbon nuclei as the foci. Each C -C -bond MO comprises the same Cacetylem2sp3 HO shells of constant energy given by Eq. (14.391). As in the case of the water, NH2 , ammonia, and ethylene molecules given by Eqs. (13.180), (13.320), (13.372), and (14.251), respectively, the energy of the redundant shell is subtracted from the total energy of the linear combination of the σ MO. Thus, the total energy Eτ (C ≡ C,σ) of the σ component of the C ≡ C -bond MO is given by the sum of the energies of the three bonds each comprising the linear combination of the Cacely!ene2sp3 HO and the H2 -type ellipsoidal MO as given by Eq.
(14.337) wherein the Eτ terms add positively and the E(Cacelylene>2sp3 } term is positive due to the sum over a negative and two positive terms. Using Eqs. (13.431) and (14.341), Eτ (C ≡ C, σ) is given by
Eτ (C - C, σ) = Eτ + E (Cacelylem , 2sp3 ) - E (Cacelykm , 2sp3 ) - E (c→ene , 2sp3 )
Figure imgf000374_0001
(14.343)
The total energy term of the double C ≡ C -bond MO is given by the sum of the three H2 - type ellipsoidal MOs given by Eq. (11.212). To match this boundary condition, E1. (C s C, σ) given by Eq. (14.343) is set equal to three times Eq. (13.75):
Figure imgf000374_0002
(14.344)
From the energy relationship given by Eq. (14.344) and the relationship between the axes given by Eqs. (14.153-14.154) and (13.62-13.63), the dimensions of the C ≡ C -bond MO can be solved. Substitution of Eq. (14.153) into Eq. (14.344) gives
Figure imgf000374_0003
The most convenient way to solve Eq. (14.345) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is α = 1.28714a0 = 6.81122 JST lO"11 m (14.346) Substitution of Eq. (14.346) into Eq. (14.153) gives c' = 1.13452α0 = 6.00362 X 10~n m (14.347) The internuclear distance given by multiplying Eq. (14.347) by two is
2c' = 2.26904α0 = 1.20072 X 10~10 m (14.348) The experimental bond distance is [3] 2c' = 1.203 X lO"10 m (14.349) Substitution of Eqs. (14.346-14.347) into Eq. (13.62) gives b = c = 0.60793α0 = 3.21704 X l(Tn m (14.350) Substitution of Eqs. (14.346-14.347) into Eq. (13.63) gives e = 0.88143 (14.351) The nucleus of the C atoms comprise the foci of the H2 -type ellipsoidal MO. The parameters of the point of intersection of the H2 -type ellipsoidal MO and the Cacetylem2spz HO are given by Eqs. (13.84-13.95) and (13.261-13.270). The polar intersection angle 0' is given by Eq. (13.261) where rn = r→mtW = 0.83008«0 is the radius of the Cacetylem2sp3 shell. Substitution of Eqs. (14.346-14.347) into Eq. (13.261) gives 0' = 137.91° (14.352)
Then, the angle θ 3 the radial vector of the C2sp3 HO makes with the internuclear axis is θn n , 3^ = 180° -137.91° = 42.09° (14.353) as shown in Figure 43.
Consider the right-hand intersection point. The distance from the point of intersection of the orbitals to the internuclear axis must be the same for both component orbitals. Thus, the angle ωt - θc≡c t HiM0 between the internuclear axis and the point of intersection of the
H2 -type ellipsoidal MO with each Cacelylene2sp3 radial vector obeys the following relationship: r , ., 3 sin6> , ,_ = 0.83008αo sin0_ _ , !on = bshxθr_c „ ^ (14.354) such that
Figure imgf000375_0001
with the use of Eq. (14.353). Substitution of Eq. (14.350) into Eq. (14.355) gives θc_c H Mn = 66.24° (14.356)
Then, the distance dCmC^ ^^MO along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by dr C- =r(- acetylene >" H2M Mn(-> = a cos&r C-=rL acetylene ,H n2MunO ( V14.357) /
Substitution of Eqs. (14.346) and (14.356) into Eq. (14.357) gives dc*c→im,H2Mo = 0.51853α0 = 2.74396X 10'11 m (14.358)
The distance dn r , ,„. along the intemuclear axis from the origin of the C atom to the point of intersection of the orbitals is given by dc*c→m itfHo = c '" ^<W,,« ,/fϊMo (14.359) Substitution of Eqs. (14.347) and (14.358) into Eq. (14.359) gives dc*c→en^o = 0.61599α0 = 3.25966X 10"11 m (14.360)
FORCE BALANCE OF THE CH MOs OF ACETYLENE
The C -Tf bond of each of the two equivalent CH MOs must comprise 75% of a H2 -type ellipsoidal MO and a C2sp3 HO as given by Eq. (13.429):
1 C2sp3 + 0.75 H2 MO → CH MO (14.361)
The force balance of the CH MO is determined by the boundary conditions that arise from the linear combination of orbitals according to Eq. (13.429) and the energy matching condition between the hydrogen and C2sp3 HO components of the MO.
The force constant k ' to determine the ellipsoidal parameter c ' of the each H2 -type- ellipsoidal-MO component of the CH MO in terms of the central force of the foci is given by Eq. (13.59). The distance from the origin of each C-H-bond MO to each focus c' is given by Eq. (13.60). The intemuclear distance is given by Eq. (13.61). The length of the semiminor axis of the prolate spheroidal C-H-bond MO b = c is given by Eq. (13.62). The eccentricity, e , is given by Eq. (13.63). The solution of the semimajor axis α then allows for the solution of the other axes of each prolate spheroid and eccentricity of each C-H -bond MO. From the energy equation and the relationship between the axes, the dimensions of the CH MO are solved.
Consider the formation of the triple C ≡ C -bond MO of acetylene from two CH radicals, each having a C2sp3 shell with an energy given by Eq. (14.146). The energy components of Ve , Vp, T , V1n , and E7, are the same as those of the hydrogen carbide radical, except that two times E7, (c ≡ C, 2sp3 ) is subtracted from ET (CH) of Eq. (13.495). The subtraction of the energy change of the C2sp3 shells with the formation of the C ≡ C -bond MO matches the energy of the C- H -bond MOs to the decrease in the energy of the C2sp3
HOs. Using Eqs. (13.495) and (14.342), Eτ (CH) is given by
Eτ→m (CH) = Eτ +E(casp3)-2Eτ(c ≡C,2sp3)
Figure imgf000377_0001
Eτ (CH) given by Eq. (14.362) is set equal to the energy of the H2 -type ellipsoidal MO given by Eq. (13.75):
Figure imgf000377_0002
(14.363)
From the energy relationship given by Eq. (14.363) and the relationship between the axes given by Eqs. (13.60-13.63), the dimensions of the CH MO can be solved. Substitution of Eq. (13.60) into Eq. (14.363) gives
Figure imgf000377_0003
The most convenient way to solve Eq. (14.364) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is α = 1.48719α0 = 7.86987X 10"n m (14.365) Substitution of Eq. (14.365) into Eq. (14.60) gives c' = 0.99572α0 = 5.26913 X 10"11 m (14.366) The internuclear distance given by multiplying Eq. (14.366) by two is
2c' = 1.99144α0 = 1.05383 X 10~10 m (14.367) The experimental bond distance is [3] 2c' = 1.060X 10-10 w (14.368)
Substitution of Eqs. (14.365-14.366) into Eq. (14.62) gives ό = c = 1.10466α0 = 5.84561 X 10"n m (14.369)
Substitution of Eqs. (14.365-14.366) into Eq. (14.63) gives e = 0.66953 (14.370) The nucleus of the H atom and the nucleus of the C atom comprise the foci of each
H2 -type ellipsoidal MO. The parameters of the point of intersection of the H2 -type ellipsoidal MO and the Cacely!βm2spι HO are given by Eqs. (13.84-13.95) and (13.261-
13.270). The polar intersection angle 0' is given by Eq. (13.261) where
^ r » = r acet iemisp 3 = 0-83008α0 is the radius of the Cacetylene2sp3 shell. Substitution of Eqs.
(14.365-14.366) into Eq. (13.261) gives
0' = 90.99° (14.371)
Then, the angle 0 3tjn the radial vector of the C2sp3 HO makes with the internuclear axis is 0 6> , 3j = 180o-90.99° = 89.01° (14.372) as shown in Figure 43. The distance from the point of intersection of the orbitals to the internuclear axis must be the same for both component orbitals. Thus, the angle ωt =÷ θr „ „ un between the internuclear axis and the point of intersection of the H7 -type ellipsoidal MO with the Cacelylene2sp3 radial vector obeys the following relationship: 5 r acelyl ,ene2 „sp 3 3 sin6> C-H →cm „2sp i* 'HonO = 0.83008αo ° sin<9 C-H a →em .2sp i i HanO = bsaiθr C-H Haa¥sm,H „τ^M„0 (\14.373) J such that
Figure imgf000378_0001
with the use of Eq. (14.372). Substitution of Eq. (14.369) into Eq. (14.374) gives
^^,^0 = 48.71° (14.375) 0 Then, the distance dc_H HiM0 along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by dp C-H H<mt>h∞iH MiMiunO = ^cosθr C-Hffaxφm,H M'φUInO ( V14.376) /
Substitution of Eqs. (14.365) and (14.375) into Eq. (14.376) gives dc-H→em,H,uo = 0.98145α0 = 5.19359X 10"u m (14.377) 5 The distance dr 3 along the internuclear axis from the origin of the C atom to the point of intersection of the orbitals is given by d C-H→melrfHO = C '~ dC-H→tm,H2M0 (14.378)
Substitution of Eqs. (14.366) and (14.377) into Eq. (14.378) gives VH. = °-O1427αo = 7-55329X 10-13 m (14.379)
With the C ≡ C double bond along one axis, the minimum energy is obtained with the C-H-bond MO at a maximum separation. Thus, the bond angle ΘCΞC_H between the internuclear axis of the C ≡ C bond and the H atom of the CH groups is θCΞC_H = \m° (14.380)
The experimental angle between the C ≡ C -H bonds is [6]
ΘCSC-H = ^° (14.381)
The CHCH MO shown in Figure 44 was rendered using these parameters.
The charge-density in the C ≡ C -bond MO is increased by a factor of 0.25 per bond with the formation of the Cacely,ene2sp3 HOs each having a smaller radius. Using the orbital composition of the CH groups (Eq. (14.361)) and the C ≡ C -bond MO (Eq. (14.337), the radii of Ck = 0.17113α0 (Eq. (10.51)) and Cacetylem2sp3 = 0.83008α0 (Eq. (14.339)) shells, and the parameters of the C ≡ C -bond (Eqs. (13.3-13.4), (14.346-14.348), and (14.350- 14.360)), the parameters of the C-H-bond MOs (Eqs. (13.3-13.4), (14.365-14.367), and (14.369-14.379)), and the bond-angle parameter (Eqs. (14.380-14.381)), the charge-density of the CHCH MO comprising the linear combination of two C-H -bond MOs and a C ≡ C - bond MO bridging the two CH groups is shown in Figure 44. Each C-H-bond MO comprises a H2 -type ellipsoidal MO and a Cαcelylem2sp3 HO having the dimensional diagram shown in Figure 43. The C s C -bond MO comprises a H2 -type ellipsoidal MO bridging two Cαcetylene2sp3 HOs having the dimensional diagram also shown in Figure 43.
ENERGIES OF THE CH GROUPS
The energies of each CH group of acetylene are given by the substitution of the semiprincipal axes (Eqs. (14.365-14.366) and (14.369)) into the energy equations of hydrogen carbide (Eqs. (13.510-13.514)), with the exception that two times Eτ (c ≡ C,2sp3) (Eq. (14.342)) is subtracted from Eτ (CH) in Eq. (13.514):
Ve = (0.91771) , In 7 = -40.62396 eV (14.382)
V = . =13.66428 eV (14.383)
8πεja2 -h2
Figure imgf000380_0001
(14.386) where Er^ ^ (CH") is given by Eq. (14.362) which is reiteratively matched to Eq. (13.75) within five-significant-figure round off error.
VIBRATION OF THE 12CH GROUPS
The vibrational energy levels of CH in acetylene may be solved using the methods given in the Vibration and Rotation of CH section.
THE DOPPLERENERGY TERMS OF THE 12CH GROUPS The equations of the radiation reaction force of the CH groups in acetylene are the same as those of the hydrogen carbide radical with the substitution of the CH -group parameters. Using Eq. (13.477), the angular frequency of the reentrant oscillation in the transition state is
3.08370X 1016 rαJ/5 (14.387)
Figure imgf000380_0002
where b is given by Eq. (14.369). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)):
Eκ = hω = /13.08370 X 1016 rad Is = 20.29747 e V (14.388) In Eq. (11.181), substitution of E2- (H2) (Εqs. (11.212) and (13.75)), the maximum total energy of each H2 -type MO, for Ehv , the mass of the electron, me , for M , and the kinetic energy given by Eq. (14.388) for Eκ gives the Doppler energy of the electrons for the reentrant orbit:
Figure imgf000380_0003
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of CH due to the reentrant orbit of each bond in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by Eq. (14.389) and EKvlb , the average kinetic energy of vibration which is 1/2 of the vibrational energy of each C-H bond. Using ωe given by Eq. (13.458) for EKvib ofthe transition state, Eacetykm osc (12CH) is
Eacelylem osc ( 12CH) (14.390)
Figure imgf000381_0001
E→em osc {nCH) = -0.28197 eV +^(0.35532 eV) = -0.10430 eV (14.391)
TOTAL AND DIFFERENCE ENERGIES OF THE 12CH GROUPS
E ■a1 ceiyieneT+osc ( ^CH) , the total energy of each 12CH group including the Doppler term, is given by the sum of £WeJCH) (Eq. (14.386)) and Eacetylene osc (12CH) given by Eq. (14.391):
'Ve +T + Vm +Vp + E(C,2sp*)
~* acetyleneT+osc (CH) = -2ET (C * C, 2sp3 ) + E→em osc ( 12CH) (14.392)
Figure imgf000381_0002
From Eqs. (14.391-14.393), the total energy of each 12CH is E→emT,0SC ( 12CH) = -31.63537 eV + Eacelyleικ osc ( 12CH)
= -31.63537 eV-( 0.28197 eF-i(0.35532 eV) J (14.394)
= -31.73967 e V
where ωe given by Eq. (13.458) was used for the h \ I—k term.
\ μ
The total energy for each hydrogen carbide radical given by Eq. (13.485) is
Eradica!T+0SC {12CH) = -31.63537 e V + Eradicalosc (12CH)
= -31.63537 eV- 0.24966 eV+ -(035532 eV) (14.395)
= -31.70737 eF The difference in energy between the CH groups and the hydrogen carbide radical
ΔET+OSC ( 12CH) is given by two times the difference between Eqs. (14.394) and (14.395):
Figure imgf000382_0001
( 12CH) - Emdjcalτ+Osc ( 12CH))
= 2(-31.73967 eF-(-31.70737 eF)) (14.396)
= -0.06460 eF
SUM OF THE ENERGIES OF THE C = C σ MO AND THE HOs OF ACETYLENE
The energy components of Ve , Vp, T , V1n , and E1. of the C = C -bond MO are the same as those of the CH MO except that each term is multiplied by three corresponding to the triple bond and the energy term corresponding to the Cacetylme2spl ΗOs in the equation for E1. is positive. The energies of each C = C -bond MO are given by the substitution of the semiprincipal axes (Eqs. (14.346-14.347) and (14.350)) into three times the energy equations of the CH MO (Eqs. (13.449-13.453)), with the exception that E(c,2sp3) in Eq. (13.453) is positive and given by Eq. (14.341):
Figure imgf000382_0002
Figure imgf000383_0001
Eτ [C ≡ C,σ) eV = -94.90616 eV
Figure imgf000383_0002
(14.401) where E1. (C ≡ C,σ) is the total energy of the C ≡ C σ MO given by Eq. (14.343) which is reiteratively matched to three times Eq. (13.75) within five-significant-figure round off error. The total energy of the C ≡ C -bond MO, E1. (C = C) , is given by the sum of two times E1. (C ≡ C,2sp3) , the energy change of each C2sp3 shell due to the decrease in radius with the formation of the C ≡ C -bond MO (Eq. (14.342)), and E1 (C ≡ C, σ), the σ MO contribution given by Eq. (14.344):
E7 [C = C) = 2ET (C ≡ C,2sp3) + Eτ (C & C,σ)
Figure imgf000383_0003
= 2(-1.56513 eF) + (-94.90610 eF) = -98.03637 eV
(14.402)
VIBRATION OF ACETYLENE The vibrational energy levels of CHCH may be solved as two equivalent coupled harmonic oscillators with a bridging harmonic oscillator by developing the Lagrangian, the differential equation of motion, and the eigenvalue solutions [9] wherein the spring constants are derived from the central forces as given in the Vibration of Hydrogen-Type Molecular Ions section and the Vibration of Hydrogen-Type Molecules section. THE DOPPLER ENERGY TERMS OF THE C ≡ C -BOND MO OF
ACETYLENE
The equations of the radiation reaction force of the C ≡ C -bond MO are given by Eq. (14.231), except that the C ≡ C -bond MO parameters are used. The angular frequency of the reentrant oscillation in the transition state is
ω = 2.00186 X 1016 rod I s (14.403)
Figure imgf000384_0001
where a is given by Eq. (14.346). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)):
Eκ = hω = h2.00l86X 1016 rad/s = 13.17659 eV (14.404) In Eq. (11.181), substitution of E1, (C ≡ C)/3 (Eq. (14.402)) for E1n, , the mass of the electron, me, for M , and the kinetic energy given by Eq. (14.404) for Eκ gives the Doppler energy of the electrons of each of the three bonds for the reentrant orbit:
ED s * (14.405)
Figure imgf000384_0002
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of the C ≡ C - bond MO due to the reentrant orbit of the bond in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, E0110 , is given by the sum of the corresponding energies, ED given by Eq. (14.405) and EKvlb , the average kinetic energy of vibration which is 1/2 of the vibrational energy of the C ≡ C bond. Using the experimental C ≡ C Evib (υ3) of 3374 cm'1 (0.41833 eV) [6] for EKvib of the transition state having three bonds, E\sc (C ≡ C,σ) per bond is
E \sc (C m C,σ) = ED + EKvtb = ED +U fc (14.406)
E ' osc (C ≡ C, σ) = -0.23468 eV+ -(0.41833 eV) = -0.02551 eV (14.407)
Given that the vibration and reentrant oscillation is for three C-C bonds of the C ≡ C triple bond, Eacetylme osc (C ≡ C,σ) , is: (14.408)
Figure imgf000385_0001
TOTAL ENERGIES OF THE C≡C -BOND MO OF ACETYLENE
Eτ+oso [C-C), the total energy of the C≡C -bond MO including the Doppler term, is given by the sum of Eτ [C ≡ C) (Eq. (14.402)) and Eacelylem osc (C ≡ C, σ) given by Eq. (14.408):
Figure imgf000385_0002
= ET(C≡ C, σ) + 2ET (C ≡ C, 2sp3 ) + E→em osc (C ≡ C5 σ) (14.409) = Eτ(C≡C) + E→eneosc(C≡C,σ)
Figure imgf000385_0003
From Eqs. (14.408-14.410), the total energy of the C≡C -bond MO is Eτ+osc(C≡
Figure imgf000385_0004
= -98.11291 eV
(14.411)
where the experimental E , was used for the term.
Figure imgf000385_0005
BONDENERGY OF THE C≡C BOND OF ACETYLENE
As in the case of 12CH2 and 14NH5 the dissociation of the C ≡ C bond forms three unpaired electrons per central atom wherein the magnetic moments cannot all cancel. The energy per atom E(magnetic) is given by Eq. (13.524). Thus, the dissociation energy of the C ≡ C bond of CHCH , ED (HC ≡ CH) , is given by six times E(c,2sp3 ) (Eq. (14.146)), the initial energy of each C2sp3 HO of each CH radical that forms the triple C ≡ C bond, minus the sum of ΔET+OSC (12CH) (Eq. (14.396)), the energy change going from the hydrogen carbide radicals to the CH groups of acetylene, Er+oso (C ≡ C) (Eq. (14.411)), and two times E(magnetic) given by Eq. (13.524). Thus, the dissociation energy of the C ≡ C bond of CHC H, is
ED (HC ≡ CH) = 6 (E (C52sp3 )) - (ΔEr+OiC ( 12CH) + Eτ+osc (C ≡ C) + 2E(magnetic))
= 6(-14.63489 eF)-(-0.06460 eF-98.11291 eF + 0.29606 eF) (14 412) = 6(-14.63489 eF)-(-97.88145 eV) = 10.07212 eV The experimental dissociation energy of the C ≡ C bond of CHCH is [7]
ED (HC ≡ CH) = 10.0014 eV (14.413)
The results of the determination of bond parameters of CHCH are given in Table 14Jl . The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
BΕNZΕNΕ MOLECULE (C6H6) The benzene molecule C6H6 is formed by the reaction of three ethylene molecules:
3CH2CH2 → C6H6 + 3H2 . (14.414)
C6H6 can be solved using the same principles as those used to solve ethylene wherein the 2s and 2p shells of each C hybridize to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between two C2sp3 hybridized orbitals (ΗOs) to form a molecular orbital (MO) permits each participating hybridized orbital to decrease in radius and energy. Each 2sp3 HO of each carbon atom initially has four unpaired electrons. Thus, the 6 H atomic orbitals (AOs) of benzene contribute six electrons and the six sp3 -hybridized carbon atoms contribute twenty-four electrons to form six C-H bonds and six C = C bonds. Each
C -H bond has two paired electrons with one donated from the H AO and the other from the C2sp3 HO. Each C = C bond comprises a linear combination of a factor of 0.75 of four paired electrons (three electrons) from two sets of two C2spi HOs of the participating carbon atoms. Each C -H and each C = C bond comprises a linear combination of one and two diatomic H2 -type MOs developed in the Nature of the Chemical Bond of Hydrogen-Type Molecules and Molecular Ions section, respectively.
FORCE BALANCE OF THE C = C -BOND MO OF BENZENE C6H6 can be considered a linear combination of three ethylene molecules wherein a
C -H bond of each CTf2 group of H2C = CH2 is replaced by a C = C bond to form a six- member ring of carbon atoms. The solution of the ethylene molecule is given in the Ethylene Molecule (CH2CH2) section. Before forming ethylene groups, the 2sp3 hybridized orbital arrangement of each carbon atom is given by Eq. (14.140). The sum Eτ (c,2$p3 } of calculated energies of C, C+ , C2+, and C3+ is given by Eq. (14.141). The radius r^ of the
C2sp3 shell is given by Eq. (14.142). The Coulombic energy ECoulomb (C,2sp3) and the
energy E(C,2sp3^) of the outer electron of the C2sp3 shell are given by Eqs. (14.143) and (14.146), respectively. Two C-T2 radicals bond to form CH2CH2 by forming a MO between the two pairs of remaining C2sp3 -HO electrons of the two carbon atoms. However, in this case, the sharing of electrons between four C2sp3 HOs to form a MO comprising four spin- paired electrons permits each C2sp3 HO to decrease in radius and energy. The C = C -bond MO is a prolate-spheroidal-MO surface that cannot extend into C2sp3 HO for distances shorter than the radius of the C2sp3 shell of each atom. Thus, the MO surface comprises a partial prolate spheroid in between the carbon nuclei and is continuous with the C2,sp3 shell at each C atom. The energy of the H2 -type ellipsoidal MO is matched to that of each C2sp3 shell. As in the case of previous examples of energy-matched MOs such as those of OH , NH , CH , the C = O -bond MO of CO2 , and the C - C -bond MO of CH1CH1 , the C = C - bond MO of ethylene must comprise 75% of a H2 -type ellipsoidal MO in order to match potential, kinetic, and orbital energy relationships. Thus, the C = C -bond MO must comprise a linear combination of two MOs wherein each comprises two C2sp3ΗOs and 75% of a H2- type ellipsoidal MO divided between the C2sp3 HOs:
2(2 C2sp3 +0.75 H2 MO) -» C = C -bond MO (14.415)
The linear combination of each H2 -type ellipsoidal MO with each C2sp3 HO further comprises an excess 25% charge-density contribution from each C2sp3 HO to the C = C - bond MO to achieve an energy minimum. The force balance of the C = C -bond MO is determined by the boundary conditions that arise from the linear combination of orbitals according to Eq. (14.415) and the energy matching condition between the C2sp3-RO components of the MO.
The sharing of electrons between two pairs of Clsp3 HOs to form a C = C -bond MO permits each participating hybridized orbital to decrease in radius and energy. The sum ET (Cethylem,2sp3) of calculated energies of C2sp3 , C+ , C2+, and C3+ is given by Eq. (14.243). In order to further satisfy the potential, kinetic, and orbital energy relationships,
' each participating C2sp3 HO donates an excess of 25% of its electron density to the C = C - bond MO to form an energy minimum. By considering this electron redistribution in the ethylene molecule as well as the fact that the central field decreases by an integer for each successive electron of the shell, ' the radius r eth .yl ,ene „lsp 3 of the C2sp * 3 shell of ethy Jlene calculated from the Coulombic energy is given by Eq. (14.244). The Coulombic energy Ecouiomb (Cdhyum'Zψ3) of me outer electron of the C2sp3 shell is given by Eq. (14.245). The energy E(Cethylem,2sp3) of the outer electron of the C2sp3 shell is given by Eq. (14.246).
Eτ {C = C, 2sp3 ) (Eq. (14.247), the energy change of each C2sp3 shell with the formation of
the C = C -bond MO is given by the difference between E [Cethyhm , 2sρ3 ) and E [C, 2sρ3 ) . Consider the case where three sets of C = C -bond MOs form bonds between the two carbon atoms of each molecule to form a six-member ring such that the six resulting bonds comprise eighteen paired electrons. Each bond comprises a linear combination of two MOs wherein each comprises two C2sp3Η.Os and 75% of a H2 -type ellipsoidal MO divided between the C2sp3 HOs:
Figure imgf000389_0001
The linear combination of each H2 -type ellipsoidal MO with each C2sp3 HO further comprises an excess 25% charge-density contribution per bond from each C2sp3 HO to the C = C -bond MO to achieve an energy minimum. Thus, the dimensional parameters of each bond C = C -bond are determined using the same equations as those used to determine the same parameters of the C = C -bond MO of ethylene (Eqs. (14.242-14.268)) while matching the boundary conditions of the structure of benzene. The energies of each C = C bond of benzene are also determined using the same equations, as those of ethylene with the parameters of benzene. The result is that the energies are essentially given as 0.75 times the energies of the C = C -bond MO of ethylene (Eqs. (14.251-14.253) and (14.319-14.333).
The derivation of the dimensional parameters of benzene follows the same procedure as the determination of those of ethylene. As in the case of ethylene, each Tf2 -type ellipsoidal MO comprises 75% of the C = C -bond MO shared between two C2sp3 HOs
corresponding to the electron charge density in Eq. (11.65) of — : . But, the additional
Z*
25% charge-density contribution to each bond of the C = C -bond MO causes the electron
— Q charge density in Eq. (11.65) to be is given by — = -0.5e . The corresponding force constant
ky is given by Eq. (14.152). In addition, the energy matching at all six C2sp3 HOs further requires that k' be corrected by a hybridization factor (Eq. (13.430)) as in the case of ethylene, expect that the constraint that the bonds connect a six-member ring of C = C bonds of benzene rather two C2sp3 HOs of ethylene decreases the hybridization factor of benzene compared to that of ethylene (Eq. (14.248)).
Since the energy of each H2 -type ellipsoidal MO is matched to that of all the continuously connected Cbenzem2sp3 HOs, the hybridization-energy-matching factor is 0.85252 . Hybridization with 25% electron donation to each C = C -bond gives rise to the Cbemene2sp3 HO-shell Coulombic energy ECmιlomb (cbemene,2sp3) given by Eq. (14.245). The corresponding hybridization factor is given by the ratio of 15.95955 eV , the magnitude of ECBu1omb {Cbemem,2sp2) given by Eq. (14.245), and 13.605804 eV , the magnitude of the
Coulombic energy between the electron and proton of H given by Eq. (1.243). The hybridization energy factor C4-^ Vjrø is
Figure imgf000390_0004
Thus, the force constant k' to determine the ellipsoidal parameter c' in terms of the central force of the foci (Eq. (11.65)) is given by
beneneClsp HO ^ ' 4^
The distance from the origin to each focus c' is given by substitution of Eq. (14.418) into Eq. (13.60). Thus, the distance from the origin of the component of the double C - C -bond MO to each focus c1 is given by
Figure imgf000390_0001
The internuclear distance from Eq. (14.419) is
Figure imgf000390_0002
The length of the semiminor axis of the prolate spheroidal C = C -bond MO b = c is given by Eq. (13.62). The eccentricity, e , is given by Eq. (13.63). The solution of the semimajor axis a then allows for the solution of the other axes of each prolate spheroid and eccentricity of the C = C -bond MO. From the energy equation and the relationship between the axes, the dimensions of the C = C -bond MO are solved.
The general equations for the energy components of Ve , Vp, T , V1n , and Eτ of the C = C -bond MO of benzene are the same as those of the CH2CH2 MO except that energy of the Cbeι1zene2sp3 HO is used and the hybridization factor is given by Eq. (14.417). Using Eqs. (14.251) and (14.417), Eτ (C = C,σ) is given by
ET [C = C,a) = ET +E(Cbemene,2S/)-E(Cbenzem,2sp3)
Figure imgf000390_0003
The total energy term of the double C - C -bond MO is given by the sum of the two H2 -type ellipsoidal MOs given by Eq. (11.212). To match this boundary condition, E1, (C = C, σ) given by Eq. (14.421) is set equal to two times Eq. (13.75):
Eτ (C = C,σ) = -63.27074 e V (14.422)
Figure imgf000391_0001
From the energy relationship given by Eq. (14.422) and the relationship between the axes given by Eqs. (14.419-14.420) and (13.62-13.63), the dimensions of the C = C -bond MO can be solved.
Substitution of Eq. (14.419) into Eq. (14.422) gives
e63.27074 (14.423)
Figure imgf000391_0002
The most convenient way to solve Eq. (14.423) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is α = 1.47348Λ0 = 7.79733 X 10~u m (14.424)
Substitution of Eq. (14.424) into Eq. (14.4129) gives c' = 1.31468α0 = 6.95699 X 10~n m (14.425) The internuclear distance given by multiplying Eq. (14.425) by two is
2c' = 2.62936α0 = 1.39140 X 10"10 m (14.426)
The experimental bond distance is [3]
2c' = 1.339X 10-10 w (14.427)
Substitution of Eqs. (14.424-14.425) into Eq. (13.62) gives h = c = 0.6654Oa0 = 3.52116 X 10"11 m (14.428)
Substitution of Eqs. (14.424-14.425) into Eq. (13.63) gives e = 0.89223 (14.429)
The nucleus of the C atoms comprise the foci of the H2 -type ellipsoidal MO. The parameters of the point of intersection of the H2 -type ellipsoidal MO and the Cbenzene2sp3 HO are given by Eqs. (13.84-13.95) and (13.261-13.270). Each benzene carbon atom contributes
(0.75)(-l.13380 eV) = -0.85035 eV (Eqs. (14.483) and (14.493)) to each of the two C = C - bond MOs and (0.5)(-l.13380 eV) = -0.56690 eV (Eq. (14.467)) to the corresponding C -H -bond MO. The energy contribution due to the charge donation at each carbon superimposes linearly. The radius of rbmgm^ = 0.79597α0 is calculated using Eq. (14.518) using the total energy donation to each bond with which it is participates in bonding. The polar intersection angle θx is given by Eq. (13.261) where r = r , = 0.79597αn is the
radius of the Cbemem2sp3 shell. Substitution of Eqs. (14.424-14.425) into Eq. (13.261) gives
0' = 134.24° (14.430)
Then, the angle θ C-^benam^Sp Z HzinU the radial vector of the C2sp3 HO mak ^ es with the internuclear axis is
0 . = 180°-134.24° = 45.76° (14.431) as shown in Figure 45.
Thus, the 12CH4 bond dissociation energy, ED (12CH4) , given by Eqs. (13.154), and (13.614-
13.616) is
ED (12CH4) = -(67.95529 eF +13.59844 eF)-Er+røc (12CH4)
= -81.55373 eF-(-86.04373 eV) (13.617)
= 4.4900 eF
The experimental 12CH4 bond dissociation energy is [40] £o (12CH4) = 4.48464 e V (13.618)
Consider the right-hand intersection point. The distance from the point of intersection of the orbitals to the internuclear axis must be the same for both component orbitals. Thus, the angle ωt = 6>C=Q tH MO between the internuclear axis and the point of intersection of the H2- type ellipsoidal MO with each Cbemem2sp3 radial vector obeys the following relationship: r benzemlsp 3 3 sin6> C=Cbtmem 2 nsp.z HO = 0.79597αo ° sin<9 C=CUmeM ,2sp z 3HυOn = hsmθc C_-cCb!mtm , HH2MM0O ( \14.432) J such that
Figure imgf000392_0001
with the use of Eq. (14.431). Substitution of Eq. (14.428) into Eq. (14.433) gives
6C-C^H1MO = 58.98° (14.434) Then, the distance dc=Cb^mΑM0 along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by dc-Cbm^uo = a cos θc=ctm,H2Mθ (14.435)
Substitution of Eqs. (14.424) and (14.434) into Eq. (14.435) gives dc=cbmΛMo = 0.75935α0 = 4.01829 X lO"11 m (14.436)
The distance d „ „ , n along the internuclear axis from the origin of the C atom to the point of intersection of the orbitals is given by dc-cbmis^o = c -dc=cbm,H2Mo (14-437)
Substitution of Eqs. (14.425) and (14.436) into Eq. (14.437) gives dc c . ,m = 0.55533α0 = 2.93870 X 10"n m (14.438)
FORCE BALANCE OF THE CH MOs OF BENZENE
Benzene can also be considered as comprising chemical bonds between six CH radicals wherein each radical comprises a chemical bond between carbon and hydrogen atoms. The solution of the parameters of CH is given in the Hydrogen Carbide (CH) section. Each C- H bond of CH having two spin-paired electrons, one from an initially unpaired electron of the carbon atom and the other from the hydrogen atom, comprises the linear combination of 75% H2 -type ellipsoidal MO and 25% C2sp3 HO as given by Eq. (13.439):
1 C2sp3 + 0.75 H2 MO → CH MO (14.439)
The proton of the H atom and the nucleus of the C atom are along each internuclear axis and serve as the foci. As in the case of H2, the C-H-bond MO is a prolate spheroid with the exception that the ellipsoidal MO surface cannot extend into C2sp3 HO for distances shorter than the radius of the C2sp3 shell since it is energetically unfavorable. Thus, each MO surface comprises a prolate spheroid at the H proton that is continuous with the C2sp3 shell at the C atom whose nucleus serves as the other focus.
The force balance of the CH MO is determined by the boundary conditions that arise from the linear combination of orbitals according to Eq. (14.439) and the energy matching condition between the hydrogen and C2sp3 HO components of the MO. The force constant k' to determine the ellipsoidal parameter c' of the each H2 -type-ellipsoidal-MO component of the CH MO in terms of the central force of the foci is given by Eq. (13.59). The distance from the origin of each C -H -bond MO to each focus c' is given by Eq. (13.60). The internuclear distance is given by Eq. (13.61). The length of the semiminor axis of the prolate spheroidal C-H -bond MO b = c is given by Eq. (13.62). The eccentricity, e , is given by Eq. (13.63). The solution of the semimajor axis a then allows for the solution of the other axes of each prolate spheroid and eccentricity of each C- H -bond MO. From the energy equation and the relationship between the axes, the dimensions of the CH MO are solved.
Consider the formation of the double C = C -bond MOs of benzene wherein ethylene formed from two CH2 radicals, each having a C2sp3 shell with an energy given by Eq. (14.146), serves as a basis element. The energy components of F8 , Vp, T , V1n , and E7, are the same as those of the hydrogen carbide radical, except that E7, ( C = C, 2sp3 ) is subtracted from Eτ (CH) of Eq. (13.495). As in the case of the CH2 groups of ethylene (Eq. (14.270)), the subtraction of the energy change of the C2 sp3 shell per H with the formation of the C = C -bond MO matches the energy of each C -H -bond MO to the decrease in the energy of the corresponding C2sp3 HO. Using Eqs. (13.431) and (14.247), E7, m (CH) is given by
E7, (CH) = ET +E(C,2sp3)-ET (C = C, 2sp3 )
Figure imgf000394_0001
Eτ (CH) given by Eq. (14.440) is set equal to the energy of the H2 -type ellipsoidal MO given by Eq. (13.75):
Figure imgf000394_0002
(14.441) From the energy relationship given by Eq. (14.441) and the relationship between the axes given by Eqs. (13.60-13.63), the dimensions of the CH MO can be solved. Substitution of Eq. (13.60) into Eq. (14.441) gives
Figure imgf000394_0003
The most convenient way to solve Eq. (14.442) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is α = 1.6006Ia0 = 8.47006 X 10"11 m (14.443)
Substitution of Eq. (14.443) into Eq. (14.60) gives c' = 1.03299α0 = 5.46636 X 10~n m (14.444)
The internuclear distance given by multiplying Eq. (14.444) by two is
2c' = 2.06598_70 =1.09327 X 10"10 m (14.445)
The experimental bond distance is [3]
2c' = 1.101 JST l(r10 m (14.446) Substitution of Eqs. (14.443-14.444) into Eq. (14.62) gives έ = c = 1.22265«0 = 6.47000 X 10'11 m (14.447)
Substitution of Eqs. (14.443-14.444) into Eq. (14.63) gives e = 0.64537 (14.448)
The nucleus of the H atom and the nucleus of the C atom comprise the foci of each H2 -type ellipsoidal MO. The parameters of the point of intersection of the H1 -type ellipsoidal MO and the Cbenzene2sp3 HO are given by Eqs. (13.84-13.95) and (13.261-13.270). The polar intersection angle θ' is given by Eq. (13.261) where rn = rbeιaene2sp3 = 0.79597α0 is
the radius of the Cbemem2sp3 shell. Substitution of Eqs. (14.443-14.444) into Eq. (13.261) gives #' = 74.42° (14.449)
Then, the angle θ _H 2s 3/rø the radial vector of the C2sp3 HO makes with the internuclear axis is θ r C-H »benzem ,2sp 3 HmO =180o-74.42o = 105.58° (14.450) as shown in Figure 46.
The distance from the point of intersection of the orbitals to the internuclear axis must be the same for both component orbitals. Thus, the angle ωt = θc_H HiMO between the internuclear axis and the point of intersection of the H2 -type ellipsoidal MO with the Cbemme2sp3 radial vector obeys the following relationship: r „ 3 sin<9 , . = 0.79597a, sinø „ , = &sin6> „ H MO (14.451) such that
θ -
Figure imgf000396_0001
with the use of Eq. (14.450). Substitution of Eq. (14.447) into Eq. (14.452) gives
QC-H^H1MO = 38.84° (14.453) Then, the distance dc_H H M0 along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by dc~Hbm ,H2Mo = <>∞sθc_Hbm HiMO (14.454)
Substitution of Eqs. (14.443) and (14.453) into Eq. (14.454) gives dc-Hbm,HlMo = 1.24678.% = 6.59767 X 10"11 m (14.455) The distance dn „ „ ,„„ along the internuclear axis from the origin of the C atom to the point of intersection of the orbitals is given by dc-Hbm2Ψ>Ho = dc-HbmΛMo ~cl (14-456)
Substitution of Eqs. (14.444) and (14.455) into Eq. (14.456) gives dc-Hbιm^Ho = 0.21379α0 = 1.13131 X 10'11 m (14.457) The basis set of benzene, the ethylene molecule, is planar with bond angles of approximately 120° (Eqs. (14.298-14.302)). To form a closed ring of equivalent planar bonds, the C = C bonds of benzene form a planar hexagon. The bond angle θc=c=c between the internuclear axis of any two adjacent C = C bonds is
0c=c=c = 120° (14.458) The bond angle ΘC=C_H between the internuclear axis of each C = C bond and the corresponding H atom of each CH group is
Oc-c-H = 120° (14.459)
The experimental angle between the C = C = C bonds is [13-15]
0c=c=c = 120° (14.460) The experimental angle between the C = C-H bonds is [13-15]
6^ =120° (14.461)
The C6H6 MO shown in Figure 47 was rendered using these parameters.
The charge-density in the C = C -bond MO is increased by a factor of 0.25 per bond with the formation of the Cbenzene2sp3 HOs each having a smaller radius. Using the orbital composition of the CH groups (Eq. (14.439)) and the C = C -bond MO (Eq. (14.416), the radii of Cb = 0.17113α0 (Eq. (10.51)) and C benzem2sp3 = 0.79597 aQ (Eq. (14.520)) shells, and the parameters of the C = C -bond (Eqs. (13.3-13.4), (14.424-14.426), and (14.428-
14.438)), the parameters of the C -H-bond MOs (Eqs. (13.3-13.4), (14.443-14.445), and (14.447-14.457)), and the bond-angle parameters (Eqs. (14.458-14.459)), the charge-density of the C6H6 MO comprising the linear combination of six sets of C -H -bond MOs with
3e bridging Q=C -bond MOs is shown in Figure 47. Each C -H -bond MO comprises a H2 - type ellipsoidal MO and a Cbenzem2sp3 HO having the dimensional diagram shown in Figure 46. The C = C -bond MO comprises a H2 -type ellipsoidal MO bridging two sets of two Cbemme 2sp3 ΗOs having the dimensional diagram shown in Figure 45.
ENERGIES OF THE CH GROUPS
The energies of each CH group of benzene are given by the substitution of the semiprincipal axes (Eqs. (14.443-14.444) and (14.447)) into the energy equations of hydrogen carbide (Eqs. (13.449-13.453)), with the exception that Eτ (C = C,2sp3) (Eq. (14.247)) is subtracted from Ej. (CH) in Eq. (13.453):
Figure imgf000397_0001
Vm -5.79470 eF (14.465)
Figure imgf000397_0002
Eτ = -31.63539 eV
Figure imgf000397_0003
(14.466) where ETht (CH) is given by Eq. (14.440) which is reiteratively matched to Eq. (13.75) within five-significant-figure round off error. The total energy of the C-H-bond MO, ETbm (C-H) , is given by the sum of
0.5ET (c = C,2sp3) , the energy change of each C2sp3 shell per single bond due to the decrease in radius with the formation of the corresponding C = C -bond MO (Eq. (14.247)), and EThaaaa [CH), the σ MO contribution given by Eq. (14.441):
\^ (C-tf) = (0.5)2^ (c = C, V) + ^ (CH)
Figure imgf000398_0001
= (0.5)(-1.13379 eF) + (-31.63537 e7)
= -32.20226 e V
VIBRATION OF THE 12CH GROUPS
The vibrational energy levels of CH in benzene may be solved using the methods given in the Vibration and Rotation of CH section.
THE DOPPLER ENERGY TERMS OF THE 12CH GROUPS The equations of the radiation reaction force of the CH groups in benzene are the same as those of the hydrogen carbide radical with the substitution of the CH -group parameters. Using Eq. (13.477), the angular frequency of the reentrant oscillation in the transition state is
ω = = 2.64826 X lO16 rod I s (14.468)
Figure imgf000398_0002
where b is given by Eq. (14.447). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)):
Eκ = hω = h2.64826X 1016 radls = 17.43132 eV (14.469)
In Eq. (11.181), substitution of E7 (H2) (Eqs. (11.212) and (13.75)), the maximum total energy of each H2 -type MO, for E1n, , the mass of the electron, me , for M , and the kinetic energy given by Eq. (14.469) for Eκ gives the Doppler energy of the electrons for the reentrant orbit: ED eF (14.470)
Figure imgf000399_0001
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of CH due to the reentrant orbit of each bond in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by Eq. (14.470) and EKvtb , the average kinetic energy of vibration which is 1/2 of the vibrational energy of the C-H bond. Using ωs given by Eq. (13.458) for EKvjb of the transition, Ebenzem osc (12CH) per bond is
Ebenzene osc {12CH) (14.471)
Figure imgf000399_0002
Ebemene osc (12CH) = -0.26130 eV +^-(0.35532 eV) = -0.08364 eV (14.472)
TOTAL AND BOND ENERGIES OF THE 12CH GROUPS
EbenzemT+0SC (12CH) , the total energy of each 12CH group including the Doppler term, is given
by the sum of E3^ (C - H) (Eq. (14.467)) and Ebemem osc ( 12CH) given by Eq. (14.472):
-'benzeneT-i-osc (CH)
Figure imgf000399_0003
Figure imgf000399_0004
(14.474) From Εqs. (14.472-14.474), the total energy of each 12CH is EbmT+osc ( 12CH) = -32.20226 eV + Ebm osc ( 12CH2 )
= -32.20226 eV - j 0.26130 eF--(0.35532 eV) J (14.475)
= -32.28590 eF
where ωe given by Eq. (13.458) was used for the term.
Figure imgf000400_0002
As in the case of 12CH2 , WNH , and acetylene, the dissociation of the C = C bonds forms three unpaired electrons per central atom wherein the magnetic moments cannot all cancel. The energy per atom E(magnetic) is given by Eq. (13.524). Thus, the bond dissociation energy of each CH group of the linear combination to form benzene,
EDbm (12CH) , is given by the sum of the total energies of the C2sp3 HO and the hydrogen
atom minus the sum of EbemeneT+osc (12CH) and E(jnagnetic) given by Eq. (13,524):
^ ( l2CH) = E (C' 2sP3 ) + EW ~{E^,Kr+0Sc ( 12CH) + E(magnetic)) (14.476) E(C,2.yp3) is given by Eq. (13.428), ED (H) is given by Eq. (13.154), and E(magnetic) is
given by Eq. (13.524). Thus, EDb∞ (12CH) given by Eqs. (13.154), (13.428), (13.524),
(14.475), and (14.476) is
EDb_XuCH) = -(Um489 eV + ϊ3S9844 eV)-(EbememT+^^
= -28.23333 eV - (-32.28590 eV + 0.14803 eV) (14.477)
= 3.90454 eV
SUM OF THE ENERGIES OF THE C = C σ MO ELEMENT ANX) THE HOs OF BENZENE
The energy components of F6 , Vp, T , V1n , and E7. of the C = C -bond MO of benzene are the same as those of the CH2CH2 MO except that the hybridization factor is given by Eq.
(14.417). The energies of each C = C -bond MO are given by the substitution of the semiprincipal axes (Eqs. (14.424-14.425) and (14.428)) into energy equations of the CH2CH2 MO (Eqs. (14.319-14.323)), with the exception that the hybridization factor is 0.85252 (Eq. (14.417)):
Ve eV (14.478)
Figure imgf000400_0001
Figure imgf000401_0001
(14.480)
(14.481)
eV (14.482)
Figure imgf000401_0002
where 2?r (C = C,σ) is the total energy of the C = C σ MO given by Eq. (14.421) which is reiteratively matched to two times Eq. (13.75) within five-significant-figure round off error. The total energy of the C = C -bond MO, E7. [C = C) , is given by the sum of two times ET (C = C, 2sp* ) , the energy change of each C2spi shell due to the decrease in radius with the formation of the C = C -bond MO (Eq. (14.247)), and Er (C = C,σ) , the σ MO contribution given by Eq. (14.422):
ET {C = C) = 2ET (C = C,2sp3) + ET (C = C,σ)
Figure imgf000401_0003
= 2(-1.13380 eF) + (-63.27074 eF) = -65.53833 eF which is the same E7. (C = C,σ) of ethylene given by Eq. (14.324).
VIBRATION OF BENZENE The C = C vibrational energy levels of C6H6 may be solved as six sets of equivalent coupled harmonic oscillators where each C is a further coupled to the corresponding C-H oscillator by developing the Lagrangian, the differential equation of motion, and the eigenvalue solutions [9] wherein the spring constants are derived from the central forces as given in the Vibration of Ηydrogen-Type Molecular Ions section and the Vibration of Ηydrogen-Type Molecules section.
THE DOPPLER ENERGY TERMS OF THE C = C -BOND MO ELEMENT OF BENZENE
The equations of the radiation reaction force of the C = C -bond MO of benzene are given by Eq. (13.142), except the force-constant factor is (0.85252)0.5 based on the force constant k' of Eq. (14.418), and the C = C -bond MO parameters are used. The angular frequency of the reentrant oscillation in the transition state is
1016 radls (14.484)
Figure imgf000402_0001
where b is given by Eq. (14.428). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)):
Eκ = hω = H4.97272X 1016 radls = 32.73133 eV (14.485) In Eq. (11.181), substitution of Ex (H2) (Eqs. (11.212) and (13.75)), the maximum total energy of each H2 -type MO, for Ehv , the mass of the electron, me , for M , and the kinetic energy given by Eq. (14.485) for Eκ gives the Doppler energy of the electrons for the reentrant orbit:
ED -0.35806 eV (14.486)
Figure imgf000402_0002
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of the C = C - bond MO due to the reentrant orbit of the bond in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by Eq. (14.486) and EKvjb , the average kinetic energy of vibration which is 1/2 of the vibrational energy of the C = C bond. Using the experimental
C = C Evib (υl6) of 1584.8 cm"1 (0.19649 eV) [16] for EKvjb of the transition state having two bonds, E 'osc (C = C, σ) per bond is
E (14.487)
Figure imgf000402_0003
E'osc (C = C,σ) = -0.35806 eF+-(0.19649 eV) = -0.25982 eV (14.488) Given that the vibration and reentrant oscillation is for two C-C bonds of each C = C double bond, Ebemene QSC (C = C, σ) , is:
Figure imgf000403_0005
(14.489)
TOTAL ENERGIES OF THE C = C -BOND MO ELEMENT OF BENZENE
Eτ+osc (C = C), the total energy of the C = C -bond MO of benzene including the Doppler term, is given by the sum of Ex (C = C) (Eq. (14.483)) and Ebemem osc (C = C, σ) given by Eq. (14.489):
Eτ+osc(C = C) = Ve+T + Vm+Vp+2Eτ(C = C,2sp3) + Ebenzeneosc(C = C,σ) =Eτ(C = C,σ) + 2Eτ(C = C,2Sp3) + Ebmosc(C = C,σ) (14.490)
= Eτ(C = C) + Ebenzeneosc(C = C,σ)
y
Figure imgf000403_0001
(14.491)
= -65.53833 eV -2\ 0.35806
Figure imgf000403_0002
From Eqs. (14.489-14.491), the total energy of the C = C -bond MO is Eτ+osc (C = C) = -63.2707 '4 eF + 2Er (C = C, 2ψ3 ) + Eirø oiC (C = C, σ)
Figure imgf000403_0003
= -66.05796 eF
(14.492)
where the experimental E b was used for the
Figure imgf000403_0004
TOTAL BOND DISSOCIATION ENERGY OF BENZENE
Ethylene serves as a basis element for the C = C bonding of benzene wherein each of the six
C-C bonds of benzene comprises (0.75) (4) = 3 electrons according to Eq. (14,416). The
total energy of the bonds of the eighteen electrons of the C-C bonds of benzene,
ET \C6H6,C=c) , is given by (6) (0.75) times Eτ+osc (C = C) (Eq. (14.492)), the total energy of the C = C -bond MO of benzene including the Doppler term, minus eighteen times E(c,2sp3) (Eq. (14.146)), the initial energy of each C2sp3 HO of each C that forms the
3e double C = C bonds. Thus, the total energy of the six C = C bonds of benzene is = C)-nE(C,2Spi)
Figure imgf000404_0001
= (6) (0.75) (-66.05796 eF)-18(-14.63489 eV) (14.493)
= -297.26081 eV - (-263.42798 eV) = -33.83284 eV Each of the C - H bonds of benzene comprises two electrons according to Eq. (14.439).
From the energy of each C-H bond, -EDbm (12CH) (Eq. (14.477)), the total energy of the twelve electrons of the six C - H bonds of benzene, E7. (C6H6 , C - H) , is given by
ET (C6H6,C -H) = (6)(~EDbm (12CH)) = 6 (-3.90454 eF) = -23.42724 eV (14.494)
The total bond dissociation energy of benzene, ED (C6H6) , is given by the negative sum of
ET \ C6H6,C=C \ (Eq. (14.493)) and ET (C6H6,C-H) (Eq. (14.494)):
Figure imgf000404_0002
= -((-33.83284 eV) + (-23.42724 eV)) (14.495)
= 57.2601 eV The experimental total bond dissociation energy of benzene, E7. (C6H6) , is given by the negative difference between the enthalpy of its formation (ΔH/ (benzene (gas))) and the sum of the enthalpy of the formation of the gaseous carbons (ΔHf (C(gas))) and hydrogen (AH f (H(gas))) atoms. The heats of formation are [17-18]
AH7 (benzene (gas)) = 82.9 Ul mole (0.8592 eV I molecule) (14.496) AHf (C(gas)) = 716.68 kj/mole (7.42774 eV/ molecule) (14.497)
AH f (H (gas)) = 217.998 U I mole (2.259353 eV I molecule) (14.498)
Thus, the total bond dissociation energy of benzene, ED (C6H6), is
ED (C6H6)- Eτ (C6H6) = ~(AHf (benzene(gas)y(6AHf (C(gas)) + 6AHf (H (gas))))
= -(0.8592 eF-6(7.42774 eF + 2.259353 eV))
= 57.26 eV (14.499) where E1. (C6H6) is the total energy of the bonds. The results of the determination of bond parameters of C6H6 are given in Table 14.1. The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
CONTINUOUS-CHAIN ALKANES (CnH2n+2, « = 3,4,5...oo)
The continuous chain alkanes, CnH2n+2, are the homologous series comprising terminal methyl groups at each end of the chain with » - 2 methylene ( CH2 ) groups in between:
CH3 (CH2 )M_2 CH3 (14.500) CnH2n+2 can be solved using the same principles as those used to solve ethane and ethylene wherein the 2s and 2p shells of each C hybridize to form a single 2s£>3 shell as an energy minimum, and the sharing of electrons between two C2^p3 hybridized orbitals (ΗOs) to form a molecular orbital (MO) permits each participating hybridized orbital to decrease in radius and energy. Three H AOs combine with three carbon 2spz ΗOs and two H AOs combine with two carbon 2sps ΗOs to form each methyl and methylene group, respectively, where each bond comprises a H2 -type MO developed in the Nature of the Chemical Bond of Ηydrogen-Type Molecules and Molecular Ions section. The CH3 and CH2 groups bond by forming H2 -type MOs between the remaining C2sp3 ΗOs on the carbons such that each carbon forms four bonds involving its four C2sp" ΗOs. FORCE BALANCE OF THE C-C-BOND MOs OF CONTINUOUS-CHAIN
ALKANES
CnH2n+2 comprises a chemical bond between two terminal CH3 radicals and n - 2 CH2 radicals wherein each methyl and methylene radical comprises three and two chemical bonds, respectively, between carbon and hydrogen atoms. The solution of the parameters of CH3 is given in the Methyl Radical (CH3) section. The solution of the parameters of CH2 is given in the Dihydrogen Carbide Radical ( CH2 ) section and follows the same procedure.
Each C- H bond having two spin-paired electrons, one from an initially unpaired electron of the carbon atom and the other from the hydrogen atom, comprises the linear combination of 75% H2 -type ellipsoidal MO and 25% C2sp3 HO as given by Eq. (13.429):
1 C2sp3 + 0.75 H2 MO → C -H MO (14.501)
The proton of the H atom and the nucleus of the C atom are along each internuclear axis and serve as the foci. As in the case of H2 , each of the C-H -bond MOs is a prolate spheroid with the exception that the ellipsoidal MO surface cannot extend into C2sp3 HO for distances shorter than the radius of the C2sp3 shell since it is energetically unfavorable. Thus, each MO surface comprises a prolate spheroid at the H proton that is continuous with the C2sp3 shell at the C atom whose nucleus serves as the other focus. The electron configuration and the energy, E(c,2sp3) , of the C2sp3 shell is given by Eqs. (13.422) and
(13.428), respectively. The central paramagnetic force due to spin of each C-H bond is provided by the spin-pairing force of the CH3 or CH2 MO that has the symmetry of an s orbital that superimposes with the C2sp3 orbitals such that the corresponding angular momenta are unchanged. The energies of each CH3 and CH2 MO involve each C2sp3 and each HIs electron with the formation of each C-H bond. The sum of the energies of the
H2 -type ellipsoidal MOs is matched to that of the C2sp3 shell. The force balance of the
C -H -bond MO is determined by the boundary conditions that arise from the linear combination of orbitals according to Eq. (14.139) and the energy matching condition between the C2sp3-ΗO components of the MO. The CH3 and CH2 groups form C-C bonds comprising H2 -type MOs between the remaining C2sp3 ΗOs on the carbons such that each carbon forms four bonds involving its four C2sp3 ΗOs. The sharing of electrons between any two C2sp3 ΗOs to form a molecular orbital (MO) comprising two spin-paired electrons permits each C2sp3 HO to decrease in radius and energy. As in the case of the C -H bonds, each C -C -bond MO is a prolate-spheroidal-MO surface that cannot extend into C2sp3 HO for distances shorter than the radius of the C2sp3 shell of each atom. Thus, the MO surface comprises a partial prolate spheroid in between the carbon nuclei and is continuous with the C2sp3 shell at each C atom. The energy of the H2 -type ellipsoidal MO is matched to that of the C2sp3 shell. As in the case of previous examples of energy-matched MOs such as the C-C -bond MO of ethane, each C -C -bond MO of CnH2n+2 must comprise 75% of a H2 -type ellipsoidal MO in order to match potential, kinetic, and orbital energy relationships. Thus, the C -C -bond MO must comprise two C2sp3 ΗOs and 75% of a H2 -type ellipsoidal MO divided between the two C2sp3 ΗOs:
2 C2sp3 + 0.75 H2 M0 → C-C- bond MO (14.502)
The linear combination of the H2 -type ellipsoidal MO with each C2sp3 HO further comprises an excess 25% charge-density contribution from each C2$p3 HO to the C -C - bond MO to achieve an energy minimum. The force balance of the C-C -bond MO is determined by the boundary conditions that arise from the linear combination of orbitals according to Eq. (14.502) and the energy matching condition between the C2sp3 -HO components of the MO. Before bonding, the 2sp3 hybridized orbital arrangement of each carbon atom is given by Eq. (14.140). The sum Eτ (c, 2s/?3 ) of calculated energies of C , C+ , C2+, and C3+ is given by Eq. (14.141). The radius r2sp, of the Clsp3 shell is given by Eq. (14.142). The Coulombic
energy ECoulomb [c,2sp3 \ and the energy E( C, 2sp3\ of the outer electron of the C2,sp3 shell are given by Eqs. (14.143) and (14.146), respectively.
The formation of each C-C bond of CnH2n+2 further requires that the energy of all H2 -type prolate spheroidal MOs (σ MOs) be matched at all C2sp3 HOs since they are continuous throughout the molecule. Thus, the energy of each C2sp3 HO must be a linear combination of that of the CH3 and CH2 groups that serve as basis elements. Each CH3 forms one C-C bond, and each CH2 group forms two. Thus, the energy of each C2sp3 HO of each CH3 and CH2 group alone is given by that in ethane and ethylene, respectively.
The parameters of ethane and ethylene are given by Eqs. (14.147-14.151) and (14.244- 14.247), respectively. The alkane parameters can be determined by first reviewing those of ethane and ethylene.
With the formation of the C-C -bond MO of ethane from two methyl radicals, each having a C2sp3 electron with an energy given by Eq. (14.146), the total energy of the state is given by the sum over the four electrons. The sum E7, (Celham,2sp3 ) of calculated energies of C2sp\ C+ , C2+, and C3+ given by Eq. (14.147), is
ET (Cetham,2sp3) = -(64.3921 eF + 48.3125 eF + 24.2762 eV + E(C,2sp3 ))
= -(64.3921 eF + 48.3125 eF + 24.2762 eF + 14.63489 eF)(14.503) = -151.61569 eV where E(c,2sp3) is the sum of the energy of C , -11.2767I eF , and the hybridization energy. The orbital-angular-momentum interactions also cancel such that the energy of the Eτ (c ethane > 2SP" ) is P^y Coulombic.
The sharing of electrons between two C2^p3 ΗOs to form a C - C -bond MO permits each participating hybridized orbital to decrease in radius and energy. In order to further satisfy the potential, kinetic, and orbital energy relationships, each C2sp3 HO donates an excess of 25% of its electron density to the C-C -bond MO to form an energy minimum. By considering this electron redistribution in the ethane molecule as well as the fact that the central field decreases by an integer for each successive electron of the shell, the radius r ethane2s 3 0^ ^e C2,yp3 shell of ethane may be calculated from the Coulombic energy using
Eq. (10.102):
Figure imgf000408_0001
(14.504) Using Eqs. (10.102) and (14.504), the Coulombic energy ECoulomb [Cethane,2sp3 ) of the outer electron of the Clsp shell is
—e —e
Ecouiomb {Cethane,2sp*) -15.55033 eV (14.505)
Figure imgf000409_0001
During hybridization, one of the spin-paired 2s electrons is promoted to C2sp3 shell as an unpaired electron. The energy for the promotion is the magnetic energy given by Eq.
(14.145). Using Eqs. (14.145) and (14.505), the energy E(CβΛam,2sp3) of the outer electron of the C2,sp3 shell is
E(Celhane,2sp3) = -15.35946 eF
Figure imgf000409_0002
(14.506) Thus, ET (C-C,2sp3 ) , the energy change of each C2sp3 shell with the formation of the C-C -bond MO is given by the difference between Eq. (14.146) and Eq. (14.506):
Eτ (C-C,2ψ3 ) = E(Cetham,2sp3)-E(C,2sp3) = -15.35946 eF-(-14.63489 eF) = -0.72457 eF
(14.507)
Next, consider the formation of the C = C -bond MO of ethylene from two CH2 radicals, each having a C2,sp3 electron with an energy given by Eq. (14.146). The sum
Eτ {Cejhylem>2$p3}oϊ calculated energies of C2sp3 , C+ , C2+, and C3+ is given by Eq.
(14.147). The sharing of electrons between two pairs of C2,sp3 HOs to form a C = C -bond MO permits each participating HO to decrease in radius and energy. In order to further satisfy the potential, kinetic, and orbital energy relationships, each participating C2,sp3 HO donates an excess of 25% of its electron density to the C = C -bond MO to form an energy minimum. By considering this electron redistribution in the ethylene molecule as well as the fact that the central field decreases by an integer for each successive electron of the shell, the radius reth km2s 3 of the C2sp3 shell of ethylene may be calculated from the Coulombic energy using Eqs. (10.102) and (14.147): (14.508)
Figure imgf000410_0001
= 0.85252α0 where Z = 6 for carbon. Using Eqs. (10.102) and (14.508), the Coulombic energy Ecouiomb fcethyhm^P*) of the outer electron of the C2sp3 shell is
Figure imgf000410_0002
During hybridization, one of the spin-paired 2s electrons is promoted to C2spl shell as an unpaired electron. The energy for the promotion is the magnetic energy given by Eq. (14.145). Using Eqs. (14.145) and (14.509), the energy E(Cethylem,2sp3) of the outer electron of the Clsp1 shell is
Figure imgf000410_0003
(14.510)
Thus, ET (c = C,2sp3} , the energy change of each C2sp3 shell with the formation of the C = C -bond MO is given by the difference between Eq. (14.146) and Eq. (14.510):
ET (c = C,2sp3) = E(Celhylem,2sp3)-E(c,2sp3)
= -15.76868 eV - (-14.63489 eV) (14.511)
= -1.13380 eV
To meet the energy matching condition for all σ MOs at all C2sp* HOs, the energy E(Calkane,2sρ3) of the outer electron of the C2,sp3 shell of each alkane carbon atom must be
the average of E(Celham,2sp3) (Eq. (14.506)) and E(Cethy!ene,2sp3) (Eq. (14.510)):
(M 512)
Figure imgf000410_0004
= -15.56407 eV And, E3^ (C - C, 2sp3 ) , the energy change of each C2sp3 shell with the formation of each
C-C -bond MO, must be the average of ET (C -C,2sp3) (Eq. (14.507)) and
ET (C = C,2sp3) (Eq. (14.511)):
E1
(14.513)
Figure imgf000411_0001
= -0.92918 eV Using Eq. (10.102), the radius ralkane2s 3 of the C2sp3 shell of each carbon atom of CnH2n+2 may be calculated from the Coulombic energy using the initial energy ECoulomb (c,2sp") = -U.82575 eV (Eq. (14.143)) and Ex^ (C- C, 2sp3 ) Eq. (14.513)), the energy change of each C2sp3 shell with the formation of each C -C -bond MO. Consider the case of a methyl carbon which donates E7, (C-C,2sp3) Eq. (14.513)) to a single C-C bond:
(14.514)
Figure imgf000411_0004
= 0.86359α0
Using Eqs. (10.102) and (14.514), the Coulombic energy ECoulomb (Calkane,2sp3) of the outer electron of the C2sp3 shell is
ECoulomb (14.515)
Figure imgf000411_0002
During hybridization, one of the spin-paired 25 electrons is promoted to C2sp3 shell as an unpaired electron. The energy for the promotion is the magnetic energy given by Eq. (14.145). Using Eqs. (14.145) and (14.515), the energy E(Calkam,2sp3) of the outer electron of the C2sp3 shell is
-15.56407 eF
Figure imgf000411_0003
(14.516)
Thus, ETΛm (C - C, 2sp3 ) , the energy change of each C2sp3 shell with the formation of each C -C -bond MO is given by the difference between Eq. (14.146) and Eq. (14.516):
ETaikam (C-C,2sp3) = E(Calkane,2sp3)- E(c,2Spi) = -\5.56401 eV -(-14.634S9 eV) = -0.9291S eV (14.517) which agrees with Eq. (14.513).
The energy contribution due to the charge donation at each carbon superimposes linearly. In general, the radius rmgl2s 3 of the C2sp3 HO of a carbon atom of a group of a
given molecule is calculated using Eq. (14.514) by considering , the total
Figure imgf000412_0002
energy donation to each bond with which it participates in bonding. The general equation for the radius is given by
Figure imgf000412_0001
The C2sρ3 HO of each methyl group of an alkane contributes -0.92918 eV to the corresponding single C - C bond; thus, the corresponding C2sp3 HO radius is given by Eq. (14.514). The Clsp3 HO of each methylene group of CnH2n+2 contributes -0.92918 eV to each of the two corresponding C-C bond MOs. Thus, the radius of each methylene group of an alkane is given by
(14.519)
Figure imgf000412_0003
= 0.81549α0
As in the case with ethane, the H2 -type ellipsoidal MO comprises 75% of the C-C - bond MO shared between two C2sp3 HOs corresponding to the electron charge density in
Eq. (11.65) of — '- . But, the additional 25% charge-density contribution to the C-C - Q bond MO causes the electron charge density in Eq. (11.65) to be is given by — = -0.5e . Thus, the force constant k' to determine the ellipsoidal parameter c1 in terms of the central force of the foci is given by Eq. (14.152). The distance from the origin of the C-C -bond MO to each focus c' is given by Eq. (14.153). The internuclear distance from is given by Eq. (14.154). The length of the semiminor axis of the prolate spheroidal C-C -bond MO b = c is given by Eq. (13.62). The eccentricity, e , is given by Eq. (13.63). The solution of the semimajor axis a then allows for the solution of the other axes of each prolate spheroid and eccentricity of the C-C -bond MO. Since the C - C -bond MO comprises a H2 -type- ellipsoidal MO that transitions to the Cdkam2sp3 HO of each carbon, the energy E(C aihme> 2sP3 ) "* E(l- (14.512) adds to that of the H2 -type ellipsoidal MO to give the total energy of the C-C -bond MO. From the energy equation and the relationship between the axes, the dimensions of the C-C-bond MO are solved. Similarly, E(Calkam,2sp3} is added to the energy of the H2 -type ellipsoidal MO of each C-H bond of the methyl and methylene groups to give their total energy. From the energy equation and the relationship between the axes, the dimensions of the equivalent C-H -bond MOs of the methyl and methylene groups in the alkane are solved.
The general equations for the energy components of Ve , Vp, T , V1n , and Eτ of each
C-C-bond MO are the same as those of the CH" MO except that energy of the Ca!kane2sp3 HO is used. The energy components at each carbon atom superimpose linearly and may be treated independently. Since each prolate spheroidal H2 -type MO transitions to the Calkane2sp3 HO of each corresponding carbon of the bond and the energy of the Calkane2sp3 shell treated independently must remain constant and equal to the E(Calkam,2$p3 j given by
Eq. (14.512), the total energy ϋ^/tae (C - C, σ) ofthe σ component of each C-C-bond MO is given by the sum of the energies of the orbitals corresponding to the composition of the linear combination of the Calkam2sp3 HO and the H2 -type ellipsoidal MO that forms the σ component of the C-C-bond MO as given by Eq. (14.502) with the electron charge redistribution. The total number of C-C bonds in CnH2n+2 is n-ϊ . Using Eqs. (13.431) and (14.512), E7. (C-C, σ) ofthe n-\ bonds is given by ^*. (C-C,σ) = {n -\)(ET + E(Calkam,2sp>))
Figure imgf000414_0001
To match the boundary condition that the total energy of each H2 -type ellipsoidal MO is given by Eqs. (11.212) and (13.75), E7, (C-C,σ) given by Eq. (14.520) is set equal to (« -l) times Eq. (13.75):
Eτ (C-C,σ) =
Figure imgf000414_0002
(14.521)
From the energy relationship given by Eq. (14.521) and the relationship between the axes given by Eqs. (14.153-14.154) and (13.62-13.63), the dimensions of the C-C -bond MO can be solved. Substitution of Eq. (14.153) into Eq. (14.521) gives
Figure imgf000414_0003
The most convenient way to solve Eq. (14.522) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is β = 2.12499α0 = 1.12450 JT 10~10 m (14.523) Substitution of Eq. (14.523) into Eq. (14.155) gives c' = 1.45774α0 = 7.71400 X 10~π m (14.524) The internuclear distance given by multiplying Eq. (14.524) by two is
2c' = 2.91547α0 = 1.54280 JT 10""10 m (14.525) The experimental C-C bond distance of propane is [3] 2c' = 1.532X 10-10 m (14.526)
The experimental C-C bond distance of butane is [3]
2c' = 1.531 X 10 ,-10 m (14.527) Substitution of Eqs. (14.523-14.524) into Eq. (13.62) gives δ = c = 1.54616α0 = 8.18192 X 10"11 m (14.528) Substitution of Eqs. (14.523-14.524) into Eq. (13.63) gives e = 0.68600 (14.529)
The nucleus of the C atoms comprise the foci of each H2 -type ellipsoidal MO. The parameters of the point of intersection of the H2-IyPe ellipsoidal MO and the Calkane2sp3 HO are given by Eqs. (13.84-13.95) and (13.261-13.270). The polar intersection angle Θx is given by Eq. (13.261) where for methylene bonds rn = rΛ(meV = rmtl≠m2^ =0.81549α0 is the
radius of the Calkam2sp3 shell given by Eq. (14.519). Substitution of Eqs. (14.523-14.524) into Eq. (13.261) gives
#' = 56.41° (14.530)
Then, the angle θ . the radial vector of the C2sp3 HO makes with the internuclear axis is
Θ C-CΛ^HO = 180°- 56.41° = 123.59° (14.531) as shown in Figure 48.
Consider the right-hand intersection point. The distance from the point of intersection of the orbitals to the internuclear axis must be the same for both component orbitals. Thus, the angle ωt = θc_c & H^o between the internuclear axis and the point of intersection of the H2 - type ellipsoidal MO with the Calkam2sp3 radial vector obeys the following relationship: r n „ 3 sin0 ,„ = 0.81549a, sinø „ , = -7sin0r r „ Mn (14.532) such that
Figure imgf000415_0001
with the use of Eq. (14.531). Substitution of Eq. (14.528) into Eq.. (14.533) gives θr r H Mn = 26S6o (14.534)
Then, the distance d,,^ H M0 along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by dr r ff m = flcos^ r u Mn (14.535)
Substitution of Eqs. (14.523) and (14.534) into Eq. (14.535) gives
JC-C^H1MO = 1.90890θo = 1.01015 * lO""10 m (14.536) The distance dr n 3 along the internuclear axis from the origin of the C atom to the point of intersection of the orbitals is given by dc-cAm^m = dc-cΛaι,^Mθ -C (14.537)
Substitution of Eqs. (14.524) and (14.536) into Eq. (14.537) gives dc c . o = 0.45117α0 = 2.38748 X 10-u w (14.538)
FORCE BALANCE OF THE CH3 MOs OF CONTINUOUS-CHAIN
ALKANES
Each of the two CH3 MOs must comprise three equivalent C-H bonds with each comprising 75% of a H2 -type ellipsoidal MO and a C2sp3 HO as given by Eq. (13.540):
3[l C2sp3 +0.75 H2 Mθ] -> CH3 MO (14.539)
The force balance of the CH3 MO is determined by the boundary conditions that arise from the linear combination of orbitals according to Eq. (14.539) and the energy matching condition between the hydrogen and C2spz HO components of the MO.
The force constant k' to determine the ellipsoidal parameter c' of the each H2-type- ellipsoidal-MO component of the CH3 MO in terms of the central force of the foci is given by Eq. (13.59). The distance from the origin of each C- H-bond MO to each focus c' is given by Eq. (13.60). The internuclear distance is given by Eq. (13.61). The length of the semiminor axis of the prolate spheroidal C-H-bond MO h = c is given by Eq. (13.62). The eccentricity, e , is given by Eq. (13.63). The solution of the semimajor axis a then allows for the solution of the other axes of each prolate spheroid and eccentricity of each C-H -bond MO. Since each of the three prolate spheroidal C - H-bond MOs comprises an H2-type- ellipsoidal MO that transitions to the Calkam2sp3 HO of CnH2n+2, the energy E(CMam,2sρ3) of Eq. (14.512) adds to that of the three corresponding H2 -type ellipsoidal MOs to give the total energy of the CH3 MO. From the energy equation and the relationship between the axes, the dimensions of the CH3 MO are solved. The energy components of Ve , Vp, T, and V1n are the same as those of methyl radical, three times those of CH corresponding to the three C-H bonds except that energy of the Calkam2spi HO is used. Since the each prolate spheroidal H2 -type MO transitions to the Cα/toιe2sp3 HO and the energy of the Calkam2sp3 shell must remain constant and equal to the E(Calkane,2sp3) given by Eq. (14.512), the total energy E7,^ (CH3) of the CH3 MO is given by the sum of the energies of the orbitals corresponding to the composition of the linear combination of the Calkane2sp3 HO and the three H2 -type ellipsoidal MOs that forms the
CH3 MO as given by Eq. (14.539). Using Eq. (13.431) or Eq. (13.541), Ew (CH3) is given by
Eτ,JCH3) = Eτ + E(Calkane,2sp3)
Figure imgf000417_0001
E1. (CH3) given by Eq. (14.540) is set equal to three times the energy of the H2 -type ellipsoidal MO minus two times the Coulombic energy of H given by Eq. (13.542):
Figure imgf000417_0002
(14.541) From the energy relationship given by Eq. (14.541) and the relationship between the axes given by Eqs. (13.60-13.63), the dimensions of the CH3 MO can be solved. Substitution of Eq. (13.60) into Eq. (14.541) gives
Figure imgf000417_0003
The most convenient way to solve Eq. (14.542) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is a = 1.6492Oa0 = 8.72720 X 10'11 m (14.543)
Substitution of Eq. (14.543) into Eq. (14.60) gives c' =1.04856a0 = 5.54872 X \QTn m (14.544) The internuclear distance given by multiplying Eq. (14.544) by two is
2c' = 2.0971 Ia0 = 1.10974 X 10"10 m (14.545) The experimental C-H bond distance of propane is [3]
2c' = 1.107 X l 0"10 W (14.546) Substitution of Eqs. (14.543-14.544) into Eq. (14.62) gives
6 = c = 1.27295α0 = 6.73616 JST 1(T1' m (14.547) Substitution of Eqs. (14.543-14.544) into Eq. (14.63) gives e = 0.63580 (14.548)
The nucleus of the H atom and the nucleus of the C atom comprise the foci of each
H2 -type ellipsoidal MO. The parameters of the point of intersection of the H2 -type ellipsoidal MO and the Calkane2sp3 HO are given by Eqs. (13.84-13.95) and (13.261-13.270). The polar intersection angle θ' is given by Eq. (13.261) where rn =ralhme2s i = 0.86359#0 is
the radius of the Calkane2sp3 shell. Substitution of Eqs. (14.543-14.544) into Eq. (13.261) gives
6»' -= 77.49° (14.549)
Then, the angle 6> 3 the radial vector of the C2sp3 HO makes with the internuclear
C~HMmc 2sP H0 axis is
On , = 180° -77.49° = 102.51° (14.550) as shown in Figure 49.
The distance from the point of intersection of the orbitals to the internuclear axis must be the same for both component orbitals. Thus, the angle ωt = θc_H & H^uo between the internuclear axis and the point of intersection of the H2 -type ellipsoidal MO with the
Calkam 2sp3 radial vector obeys the following relationship: , r al.k.ane „2sp 3 s sin6> C-HalUm ,lsp .^HO = 0.86359α00 sin<9 C-H „Mam „2sp 3 zHunO = bsinθc C-H „01^3H „2M MnO ( V14.551) / such that
Figure imgf000418_0001
with the use of Eq. (14.550). Substitution of Eq. (14.547) into Eq. (14.552) gives θc-HΛm,H2Mo = 41.48° (14.553) Then, the distance dc_H^MiHiM0 along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by dc-H^t,H2M0 = acosθc_HaikamtBiM0 (14.554)
Substitution of Eqs. (14.543) and (14.553) into Eq. (14.554) gives dc-HΛamAMo = l-23564α0 = 6.53871 X 10"n m (14.555)
The distance <tf 3 along the internuclear axis from the origin of the C atom to the point of intersection of the orbitals is given by d C-Halkam2sp>HO = dC-Halkam,H2MO ~ C > (14.556)
Substitution of Eqs. (14.544) and (14.555) into Eq. (14.556) gives d u „ ian = 0.18708α0 = 9.89999 X 10"12 rø (14.557)
BOND ANGLE OF THE CH3 AND CH2 GROUPS
Each CH3 MO comprises a linear combination of three C -H -bond MOs. Each C -H- bond MO comprises the superposition of a H2 -type ellipsoidal MO and the Calkane2sp3 HO. A bond is also possible between the two H atoms of the C-H bonds. Such H- H bonding would decrease the C -H bond strength since electron density would be shifted from the C-H bonds to the H-H bond. Thus, the bond angle between the two C- H bonds is determined by the condition that the total energy of the H2 -type ellipsoidal MO between the terminal H atoms of the C-H bonds is zero. From Eqs. (11.79) and (13.228), the distance from the origin to each focus of the H-H ellipsoidal MO is
Figure imgf000419_0001
The internuclear distance from Eq. (14.558) is
(14.559)
Figure imgf000419_0002
The length of the semiminor axis of the prolate spheroidal H-H MO b - c is given by Eq. (14.62).
The bond angle of the CH3 groups of CnH2n+2 is derived by using the orbital composition and an energy matching factor as in the case with the CH3 radical. Since each pair of H2 -type ellipsoidal MOs initially comprise 75% of the H electron density of H2 and the energy of each H2 -type ellipsoidal MO is matched to that of the Calkam2spl HO5 the component energies and the total energy Eτ of the H- H bond are given by Eqs. (13.67- 13.73) except that Ve , T , and V1n are corrected for the hybridization-energy-matching factor of 0.86359. Hybridization with 25% electron donation to the C -C -bond gives rise to the Calkane2sp3 HO-shell Coulombic energy ECoulomb [Calkam, 2sp3 ) given by Eq. (14.515). The corresponding normalization factor for determining the zero of the total H -H bond energy is given by the ratio of 15.75493 eV , the magnitude of ECouiomb {C aikane> 2sP3) given bv Ecl- (14.515), and 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243). The hybridization energy factor CalkamC2s ,HQ is
Figure imgf000420_0002
Substitution of Eq. (14.558) into Eq. (13.233) with the hybridization factor of 0.86359 gives
Figure imgf000420_0001
(14.561) From the energy relationship given by Eq. (14.561) and the relationship between the axes given by Eqs. (14.558-14.559) and (14.62-14.63), the dimensions of the H -H MO can be solved.
The most convenient way to solve Eq. (14.561) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is α = 5.8660ø0 =3.1042 X 10'lϋ m (14.562) Substitution of Eq. (14.562) into Eq. (14.558) gives c' = 1.7126a0 = 9.0627 X 10"11 m (14.563) The internuclear distance given by multiplying Eq. (14.563) by two is
Id = 3A252a0 = 1.8125 X 1(T10 m (14.564)
Substitution of Eqs. (14.562-14.563) into Eq. (14.62) gives έ = c = 5.61O4αo = 2.9689 X lO"10 m (14.565) Substitution of Eqs. (14.562-14.563) into Eq. (14.63) gives e = 0.2920 (14.566)
Using 2c'H_H (Eq. (14.564)), the distance between the two H atoms when the total energy of the corresponding MO is zero (Eq. (14.561)), and 2cx c_H , the internuclear distance of each C-H bond, the corresponding bond angle can be determined from the law of cosines. Since the internuclear distance of each C- H bond of CH3 (Eq. (14.545)) and
CH2 (Eq. (14.597)) are sufficiently equivalent, the bond angle determined with either is within experimental error of being the same. Using, Eqs. (13.242), (14.545), and (14.564), the bond angle θ between the C-H bonds is
2(2.09711)2 -(3.4252) 2 Λ θ = cos"1 = cos"1 (-0.33383) = 109.50° (14.567)
2(2.09711) The experimental angle between the C-H bonds is [19]
0 = 109.3° (14.568)
The CH3 radical has a pyramidal structure with the carbon atom along the z-axis at the apex and the hydrogen atoms at the base in the xy-plane. The distance dorJgin_H from the origin to the nucleus of a hydrogen atom given by Eqs. (14.564) and (13.412) is dorigln_H = \.9115AaQ (14.569)
The height along the z-axis of the pyramid from the origin to C nucleus dhejghl given by Eqs.
(13.414), (14.545), and (14.569) is dheιghι = 0.6980Oa0 (14.570)
The angle θv of each C-H bond from the z-axis given by Eqs. (13.416), (14.569), and (14.570) is θv = 70.56° (14.571)
The C-C bond is along the z-axis. Thus, the bond angle ΘC_C_H between the internuclear axis of the C-C bond and a H atom of the methyl groups is given by θc_c_H = 180-θv (14.572) Substitution of Eq. (14.571) into Eq. (14.572) gives
Θ C~C-H = 109.44° (14.573)
The experimental angle between the C-C-H bonds is [19]
ΘC-C-H = 109.3° (14.574) The CnH2n+2 MOs shown in Figures 50-60 were rendered using these parameters. A minimum energy is obtained with a staggered configuration consistent with observations [3].
ENERGIES OF THE CH3 GROUPS
The energies of each CH3 group of CnH2n+2 are given by the substitution of the semiprincipal axes, (Eqs. (14.543-14.544) and (14.547)) into the energy equations of methyl radical (Eqs. (13.556-13.560)), with the exception that E(Calkane,2sp3) (Eq. (14.514))
replaces E[C,2sp3) in Eq. (13.560):
eV (14.575)
Figure imgf000422_0001
Figure imgf000422_0002
(14.577)
(14.578)
Figure imgf000422_0003
Figure imgf000422_0004
where E1, (CHΛ is given by Eq. (14.540) which is reiteratively matched to Eq. (13.542) within five-significant-figure round off error.
VIBRATION OF THE 12CH3 GROUPS
The vibrational energy levels of the C-H bonds of CH3 in CnH2n+2 may be solved as three equivalent coupled harmonic oscillators by developing the Lagrangian, the differential equation of motion, and the eigenvalue solutions [9] wherein the spring constants are derived from the central forces as given in the Vibration of Hydrogen-Type Molecular Ions section and the Vibration of Hydrogen-Type Molecules section.
THE DOPPLER ENERGY TERMS OF THE 12CH3 GROUPS The equations of the radiation reaction force of the methyl groups in CnH2n+2 are the same as those of the methyl radical with the substitution of the methyl-group parameters. Using Eq. (13.561), the angular frequency of the reentrant oscillation in the transition state is
2.49286 X 1016 rad I s (14.580)
Figure imgf000423_0001
where b is given by Eq. (14.547). The kinetic energy, Eκ, is given by Planck's equation (Eq. (11.127)):
Eκ = hω = h2Λ9286 X 1016 rad Is =16.40846 eV (14.581)
In Eq. (11.181), substitution of ET (H2) (Eqs. (11.212) and (13.75)), the maximum total energy of each H2 -type MO, for Ehv , the mass of the electron, me , for M , and the kinetic energy given by Eq. (14.581) for Eκ gives the Doppler energy of the electrons of each of the three bonds for the reentrant orbit:
Figure imgf000423_0002
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of CH3 due to the reentrant orbit of each bond in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by Eq. (14.582) and EKvιb > the average kinetic energy of vibration which is 1/2 of the vibrational energy of each C-H bond. Using ωe given by Eq. (13.458) for EKvιb of the transition state having three independent bonds, E \lhme osc ( 12CH71 ) per bond is
(14.583)
Figure imgf000423_0003
E\lkam osc (12CH3) = -0.25352 eF + -(0.35532 eF) = -0.07586 eV (14.584) Given that the vibration and reentrant oscillation is for three C-H bonds, Kikam 0∞ (12CH3), is:
(14.585)
Figure imgf000424_0001
= -0.22757 e V
TOTAL BOND ENERGIES OF THE 12CH3 GROUPS
EaikaneT+osc ( 12CH3 ) , the total energy of each 12CH3 group including the Doppler term, is given
by the sum of E7^ (CH3) (Eq. (14.579)) and Ealkane osc ( 12CH3 ) given by Eq. (14.585):
EΛ^^ (CH,) = V. +T+Vu +Vp ^E(c^,7^) + Ε^ aκ {uCHι )
(14.586)
= ^jCH3 ) + £α/faπ_ (12CH3)
Figure imgf000424_0002
(14.587) From Eqs. (14.585-14.587), the total energy of each 12CH3 is
(14.588)
Figure imgf000424_0003
Figure imgf000424_0004
The total CH3 bond dissociation energy, EDalhm (12CH3) is given by the sum of the
initial C2sp3 HO energy, E(c,2spz) (Eq. (14.146)), and three times the energy of the
hydrogen atom, ED (H) (Eq. (13.154)), minus EaltøeT+osc ( 12CH3 ) (Eq. (14.588)):
E**- ( 12rø 3 ) = E(C,2sp3 ) + 3E(H) - EalkamT+mc ( 12CH3 ) (14.589) Thus, the total 12CH3 bond dissociation energy, ED (12CH3) is
E°*- (12CH3) = -(14.63489 ^ + 3(13.59844 eV)) -(EalkaneT+0SC (12CH2))
= -55.43021 eV -(-67.92207 eV) . (14.590)
= 12.49186 eV
FORCE BALANCE OF THE CH2 MOs OF CONTINUOUS-CHAIN
ALKANES Each of the CH2 MOs must comprise two equivalent C -H bonds with each comprising
75% of a H2 -type ellipsoidal MO and a C2sp3 HO as given by Eq. (13.494):
2[l C2sp3 + 0.75 H2 MO] → CH2 MO (14.591)
The force balance of each CH2 MO is determined by the boundary conditions that arise from the linear combination of orbitals according to Eq. (14.591) and the energy matching condition between the hydrogen and C2sp3 HO components of the MO.
The force constant k' to determine the ellipsoidal parameter cx of the each H2-type- ellipsoidal-MO component of the CH2 MO in terms of the central force of the foci is given by Eq. (13.59). The distance from the origin of each C-H-bond MO to each focus c' is given by Eq. (13.60). The internuclear distance is given by Eq. (13.61). The length of the semiminor axis of the prolate spheroidal C-H -bond MO b — c is given by Eq. (13.62). The eccentricity, e , is given by Eq. (13.63). The solution of the semimajor axis a then allows for the solution of the other axes of each prolate spheroid and eccentricity of each C-H -bond MO. Since each of the two prolate spheroidal C -H -bond MOs comprises an H2 -type- ellipsoidal MO that transitions to the Calkane2$p3 HO of CnH2n+2, the energy E(Calkane,2sp3) of Eq. (14.512) adds to that of the two corresponding H2 -type ellipsoidal MOs to give the total energy of the CH2 MO. From the energy equation and the relationship between the axes, the dimensions of the CH2 MO are solved.
The energy components of Ve , Vp , T , and V1n are the same as those of dihydrogen carbide radical, two times those of CH corresponding to the two C -H bonds except that energy of the Calkane2sp3 HO is used. Since the each prolate spheroidal H2 -type MO transitions to the Calkam2sp3 HO and the energy of the Calkane2sp3 shell treated independently must remain constant and equal to the E(Calkane,2sp3) given by Eq. (14.512), the total energy ETik (CH2) of the CH2 MO is given by the sum of the energies of the orbitals corresponding to the composition of the linear combination of the Calkane2sp3 HO and the two H2 -type ellipsoidal MOs that forms the CH2 MO as given by Eq. (14.591). Using Eq. (13.431) or Eq. (13.495), E7^ (CH2) is given by
ETΛJCH2) = Eτ +E(Ca!kane,2sP 3)
2e2 (14.592)
(0.9177l)f2--^lln^£!-lj-15.56407 eF
%πεQc '
E1, (CH2 ) given by Eq. (14.592) is set equal to two times the energy of the H2 -type ellipsoidal MO minus the Coulombic energy of H given by Eq. (13.496):
ET (CH2) = - -49.66493 eV
Figure imgf000426_0001
(14.593)
From the energy relationship given by Eq. (14.593) and the relationship between the axes given by Eqs. (13.60-13.63), the dimensions of the CH2 MO can be solved. Substitution of Eq. (13.60) into Eq. (14.593) gives
Figure imgf000426_0002
The most convenient way to solve Eq. (14.594) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures is α = 1.67122αo = 8.84370 X 10~n m (14.595) Substitution of Eq. (14.595) into Eq. (14.60) gives c' = 1.05553β0 = 5.58563 X 1(TU m (14.596)
The internuclear distance given by multiplying Eq. (14.596) by two is
2c' = 2.111O6tfo = 1.11713 X l(T10 m (14.597) The experimental C-H bond distance of butane is [3]
2c' = 1.117 X lO"10 m (14.598)
Substitution of Eqs. (14.595-14.596) into Eq. (14.62) gives fc = c = 1.29569β0 = 6.85652 X 10"u m (14.599)
Substitution of Eqs. (14.595-14.596) into Eq. (14.63) gives e = 0.63159 (14.600)
The nucleus of the H atom and the nucleus of the C atom comprise the foci of each
H2 -type ellipsoidal MO. The parameters of the point of intersection of the Tf2 -type ellipsoidal MO and the Calkane2sp3 HO are given by Eqs. (13.84-13.95) and (13.261-13.270). The polar intersection angle θy is given by Eq. (13.261) where rn = ^ lem2spi = 0.81549α0 is
the radius of the CβMH)Λm2spi shell (Eq. (14.521)). Substitution of Eqs. (14.595-14.596) into
Eq. (13.261) gives
0' = 68.47° (14.601)
Then, the angle θ 3 an the radial vector of the C2spi HO makes with the internuclear axis is θr w , , = 180°-68.47o = 111.53° (14.602) as shown in Figure 49. The distance from the point of intersection of the orbitals to the internuclear axis must be the same for both component orbitals. Thus, the angle ωt - ^c-H lka ,H2MO between the internuclear axis and the point of intersection of the H2 -type ellipsoidal MO with the Calkam2sp3 radial vector obeys the following relationship: r n , sin0> , 3£ = 0.81549αo sin# „ „ 3w = 6sin0r „ H UΩ (14.603) such that
Figure imgf000427_0001
with the use of Eq. (14.602). Substitution of Eq. (14.599) into Eq. (14.604) gives
ΘC-H^MO = 35.84° (14.605) Then, the distance dc_H^m>H2M0 along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by dr H W Mrt = βcos ^ „ „ , „ (14.606)
Substitution of Eqs. (14.595) and (14.605) into Eq. (14.606) gives dc-H^jw = I 35486αo = 7.16963 X 10"" m (14.607)
The distance d „ „ , n along the internuclear axis from the origin of the C atom to the point of intersection of the orbitals is given by d C-HΛm,2sP>HO = dC-HΛaM,HlM0 ~ * (14.608)
Substitution of Eqs. (14.596) and (14.605) into Eq. (14.608) gives dr H , . = 0.29933αQ = 1.58400 X 10-" m (14.609)
The charge-density in each C-C -bond MO is increased by a factor of 0.25 with the formation of the Calkam2spl HOs each having a smaller radius. Using the orbital composition of the C-C -bond MOs (Eq. (14.504), CH3 groups (Eq. (14.539)), and the CH2 groups (Eq.
(14.591)), the radii of CIs = 0.17113α0 (Eq. (10.51)), Calkam2sp3 = 0.86359α0 (Eq. (14.514)), and Cglkam2sp3 = Cmethylene2sp3 = 0.81549α0 (Eq. (14.521)) shells, the parameters of the
C-C -bonds (Eqs. (13.3-13.4), (14.523-14.525), and (14.528-14.538)), the parameters of the C-H-bond MOs of the CH3 groups (Eqs. (13.3-13.4), (14.544-14.545), and (14.547-
14.557)), the parameters of the C- H-bond MOs of the CH2 groups (Eqs. (13.3-13.4),
(14.595-14.597), and (14.599-14.609)), and the bond-angle parameters (Eqs. (14.562- 14.574)), the charge-density of the CnH2n+2 MO comprising the linear combination 2n + 2
C-H-bond MOs and n-l C- C -bond MOs, each bridging one or more methyl or methylene groups is shown for representative cases where data was available [17-18]. Propane, butane, pentane, hexane, heptane, octane, nonane, decane, undecane, dodecane, and octadecane are shown in Figures 50-60, respectively. Each C-H-bond MO comprises a H2 -type ellipsoidal MO and a Cal]ωm2spi HO having the dimensional diagram shown in Figure 48. Each C -C-bond MO comprises a H2 -type ellipsoidal MO bridging two Calkane2sp3 ΗOs having the dimensional diagram shown in Figure 49. ENERGIES OF THE CH2 GROUPS
The energies of each CH2 group of CnH2n+2 are given by the substitution of the semiprincipal axes (Eqs. (14.595-14.596) and (14.599)) into the energy equations of dihydrogen carbide radical (Eqs. (13.510-13.514)), with the exception that E(Calkam,2sp3}
(Eq. (14.512)) replaces E{C,2sp*) in Eq. (13.514):
Ve (14.610)
Figure imgf000429_0001
Figure imgf000429_0002
(14.613)
V = -49.66493 eV
Figure imgf000429_0003
(14.614) where E7^ (CH2) is given by Eq. (14.592) which is reiteratively matched to Eq. (13.496) within five-significant-figure round off error.
VIBRATION OF THE 12CH2 GROUPS
The vibrational energy levels of the C -H bonds of CH2 in CnH2n+2 may be solved as two equivalent coupled harmonic oscillators by developing the Lagrangian, the differential equation of motion, and the eigenvalue solutions [9] wherein the spring constants are derived from the central forces as given in the Vibration of Ηydrogen-Type Molecular Ions section and the Vibration of Ηydrogen-Type Molecules section.
THE DOPPLER ENERGY TERMS OF THE 12CH2 GROUPS
The equations of the radiation reaction force of the methylene groups in CnH2n+2 are the same as those of the dihydrogen carbide radical with the substitution of the methylene-group parameters. Using Eq. (13.515), the angular frequency of the reentrant oscillation in the transition state is
Figure imgf000430_0001
where b is given by Eq. (14.599). The kinetic energy, Eκ , is given by Planck's equation (Eq. (11.127)):
Eκ = hω = h2A215\ X 1016 radls = 15.97831 eV (14.616)
In Eq. (11.181), substitution of E7 (H2) (Eqs. (11.212) and (13.75)), the maximum total energy of each H2 -type MO, for Ehv , the mass of the electron, me , for M , and the kinetic energy given by Eq. (14.616) for Eκ gives the Doppler energy of the electrons of each of the three bonds for the reentrant orbit:
En a , & . -31.63537 eV. H15*7"1 ^ - -0.25017 eV
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of CH2 due to the reentrant orbit of each bond in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by Eq. (14.617) and EKvlb , the average kinetic energy of vibration which is 1/2 of the vibrational energy of each C-H bond. Using ωe given by Eq. (13.458) for EKvib of the transition state having two independent bonds, E \lkam osc ( 12CH2 ) per bond is
E 'alkane o4nCH2) (14.618)
Figure imgf000430_0002
E } alkam osc (12CH2) = -0.25017 eV + -(0.35532 eV) = -0.07251 eV (14.619)
Given that the vibration and reentrant oscillation is for two C-H bonds, Ealkam osc ( 12CH2 ) , is:
W « (12CH2) = 2(ED +U β- j = 2^-0.25017 βF+|(0.35532 eV)j = -0.14502 eV (14.620)
TOTAL BOND ENERGIES OF THE 12CH2 GROUPS
EaikaneT+osc ( 12CH2 ) , the total energy of each 12CH2 group including the Doppler term, is given
by the sum of E^ (CH2) (Eq. (14.614)) and Ealkane osc ( 12CH2 ) given by Eq. (14.620):
EalkaneT+osc (CH 2) = K +T + Vm +Vp +E(Calkane,2sp') + Ealkane osc (l2CH2)
(14.621)
= ^w (CH 2)+εalkane osc ( 12CH2)
Figure imgf000431_0001
(14.622) From Eqs. (14.620-14.622), the total energy of each 12CH2 is
EalkaneT+osc ( 12CH2 ) = -49.66493 eV + Ealkam osc ( 12CH2 )
= -49.66493 eV-li 0.25017 eV --(0.35532 eV) j (14.623)
= -49.80996 eV
where ω given by Eq. (13.458) was used for the term.
Figure imgf000431_0002
The derivation of the total CH2 bond dissociation energy, ED ( 12CH2 ) follows
from that of the bond dissociation energy of dihydrogen carbide radical, ED ( 12CH2 ) , given
by Eqs. (13.524-13.527). E0^ (12CH2) is given by the sum of the initial C2.yp3 HO energy,
E[C,2sps) (Eq. (14.146)), and two times the energy of the hydrogen atom, E(H) (Eq.
(13.154)), minus the sum of EalkaneT+osc ( 12CH2 ) (Eq. (14.623)) and E(magnetic) (Eq. 13.524)): EDΛmc (12CH2) = E(c,2sp') + 2E(H)~EalkaneT+0SC (nCH2)-E{magnetic) (14.624)
Thus, the total 12CH2 bond dissociation energy, ED κ ( 12CH2 ) is
Figure imgf000432_0001
= -41.83177 eF-(-49.80996 eF + 0.14803 eF) = 7.83016 eV
(14.625)
SUM OF TEDE ENERGIES OF THE C-C σ MOs AND THE HOs OF CONTINUOUS-CHAIN ALKANES The energy components of Ve , Vp, T , V1n , and E7, of the C-C -bond MOs are the same as those of the CH MO except that energy of the Cdkane2sp3 HO is used. The energies of each C- C -bond MO are given by the substitution of the semiprincipal axes (Eqs. (14.523- 14.524) and (14.528)) into the energy equations of the CH MO (Eqs. (13.449-13.453)), with the exception that E(Callcane,2spz) (Eq. (14.512)) replaces E(c,2sp3) in Eq. (13.453). The total number of C - C bonds of CnH2n+2 is n - 1. Thus, the energies of the n - 1 bonds is given by
Figure imgf000432_0002
= -(«-1)31.63537 eV (14.630) where Eτ (C - C, σ) is the total energy of the C - C σ MOs given by Eq. (14.520) which is reiteratively matched to Eq. (13.75) within five-significant-figure round off error. Since there are two carbon atoms per bond, the number of C-C bonds is n -1 , and the energy change of each C2sp3 shell due to the decrease in radius with the formation of each C -C -bond MO is ETΛM (C -C,2sp3) (Eq. (14.517)), the total energy of the C-C -
bond MOs, E7^ (C- C), is given by the sum of 2(n-l)ETβm (C-C,2sp3) and ETΛam (C - C, σ) , the σ MO contribution given by Eq. (14.630):
E ra,JC-C) = 2(n-l)ETaiJC-C,2sPη + ETΛJC-C,σ)
Figure imgf000433_0001
= («-l)(2(-0.92918 eF) + (-31.63537 eF))
= -(«-1)33.49373 eV
VIBRATION OF CONTINUOUS-CHAIN ALKANES
The vibrational energy levels of the C-C bonds of CnH2n+2 may be solved as n - 1 sets of coupled carbon harmonic oscillators wherein each carbon is further coupled to two or three equivalent H harmonic oscillators by developing the Lagrangian, the differential equation of motion, and the eigenvalue solutions [9] wherein the spring constants are derived from the central forces as given in the Vibration of Hydrogen-Type Molecular Ions section and the Vibration of Hydrogen-Type Molecules section.
THE DOPPLER ENERGY TERMS OF THE C-C-BOND MOs OF CONTINUOUS-CHAIN ALKANES
The equations of the radiation reaction force of each symmetrical C-C -bond MO are given by Eqs. (11.231-11.233), except the force-constant factor is 0.5 based on the force constant k' of Eq. (14.152), and the C-C-bond MO parameters are used. The angular frequency of the reentrant oscillation in the transition state is
O mdls (14.632)
Figure imgf000433_0002
where a is given by Eq. (14.523). The kinetic energy, Ex , is given by Planck's equation
(Eq. (11.127)):
Eκ = hω = £9.43699X 1015 radls = 6.21159 eV (14.633)
In Eq. (11.181), substitution of E7^ (C-C) (Eq. (14.631)) with n = 2 for E1n, , the mass of the electron, me, for M , and the kinetic energy given by Eq. (14.633) for Eκ gives the Doppler energy of the electrons of each of the bonds for the reentrant orbit:
S. (14.634)
Figure imgf000434_0001
In addition to the electrons, the nuclei also undergo simple harmonic oscillation in the transition state at their corresponding frequency. The decrease in the energy of each C-C - bond MO due to the reentrant orbit of the bond in the transition state corresponding to simple harmonic oscillation of the electrons and nuclei, Eosc , is given by the sum of the corresponding energies, ED given by Eq. (14.634) and EKvώ , the average kinetic energy of vibration which is 1/2 of the vibrational energy of each C-C bond. Using the ethane experimental C-C Evιb (υ3) of 993 cm"1 (0.12312 eV) [10] for EKvib of the transition state having n-\ independent bonds, E'all!am osc {C-C,σ) per bond is
(14.635)
Figure imgf000434_0003
E'alkane osc (C-C,σ) = -0Λ6515 eV +-(0.12312 eV) = -0.10359 eV (14.636)
Given that the vibration and reentrant oscillation is for n - 1 C-C bonds, Ealkane osc (C — C,σ) , is:
Figure imgf000434_0002
= (rc-l)[ -0.16515 eF + -(0.12312 eF) J (14.637)
= -(n-l)0.10359 eF TOTAL ENERGIES OF THE C-C-BOND MOs OF CONTINUOUS-CHAIN
ALKANES
EaikamT+osc [C-C) , the total energy of the n-\ bonds of the C- C -bond MOs including the . Doppler term, is given by the sum of E7.^ (C-C) (Eq. (14.631)) and Ealkam osc (C-C,σ) given by Eq. (14.637):
Figure imgf000435_0001
= ^- (C-C>σ) + 2(n-l)Eτ^ (C-C,2ψ3) + ffl/fe,ne røc (C-C,σ) = E T^ (C-C) + Ealkam osc {C-C,a)
(14.638)
Figure imgf000435_0002
From Eqs. (14.637-14.639), the total energy of the n - 1 bonds of the C - C -bond MOs is
Figure imgf000435_0003
= -(«-1)33.59732 eF
(14.640)
where the experimental E .. was used for the term.
Figure imgf000435_0004
TOTAL BOND ENERGY OF THE C-C BONDS OF CONTINUOUS- CHAINALKANES
Since there are two carbon atoms per bond and the number of C - C bonds is n - 1 , the total bond energy of the C-C bonds of CnH2n+2, ED (C-C)n_χ , is given by 2(n-\)E[C,2sp%)
5 minus EalkaneT+0SC (C-C) (Eq. (14.640)) where E(c,2sp3) (Eq. (14.146)) is the initial energy of each C2sp3 HO of the CuT3 and CH2 groups that bond to the C -C bonds. Thus, the total dissociation energy of the C-C bonds of C11H2n+2 , is
E0 (C -C)n^1 = 2(n-l)(E{C,2sp3))-{Ealah,er+0 JC -C))
= 2 (w - 1) (-14.63489 eV)-(n-l) (-33.59732 eV) ^ ,,14,.0,4,1,)
= (« -1)(2 (-14.63489 eF)-(-33.59732 eF))
= («-0(4.32754 eV)
0 TOTAL ENERGY OF CONTINUOUS-CHAIN ALKANES
E0 (CnH2n+2) , the total bond dissociation energy of CnH2n+2, is given as the sum of the energy components due to the two methyl groups, n- 2 methylene groups, and n -\ C -C bonds where each energy component is given by Eqs. (14.590), (14.625), and (14.641), respectively. Thus, the total bond dissociation energy of CnH2n+2 is 5 ED (cji^) = ED (c-c)^ +2ED^ (aCH3)+{n--2)E0^ (acH2) {UM2)
= («-0(4.32754 eF) + 2(l2.49186 eF) + («-2)(7.83016 eV) The experimental total bond dissociation energy of CnH2n+2, E0^ (CnH2n+2) , is given
by the negative difference between the enthalpy of its formation (ΔHf (CnH2n+2 (gas)) ) and the sum of the enthalpy of the formation of the reactant gaseous carbons (AHf (C(gas))) and hydrogen (AH7 (H (gas))) atoms:
o EOa? (CnH2n+2 ) = - {Δff, (CnH2n+2 (gas)) - [nAHf (C (gas)) + (2n + 2)AHf (H(gas)J\)
= - {AHf (CnH2n+2 (gas)) -[«7.42774 eV+{2n + 2) 2.259353 eV~]} (14.643) where the heats of formation atomic carbon and hydrogen gas are given by [17-18]
ΔHf (C (gas)) = 716.68 kJlmole (7.42774 eV I molecule) (14.644)
AH f (H (gas)) = 217.998 kJlmole (2.259353 eV I molecule) (14.645) Using the corresponding experimental AH f (CnH2n+2 (gas)) [18], E0 (CnH2n+2) was determined from propane, butane, pentane, hexane, heptane, octane, nonane, decane, undecane, dodecane, and octadecane in the corresponding sections, and the results of the determination of the total energies are given in Table 14.1. The calculated results are based on first principles and given in closed-form, exact equations containing fundamental constants only. The agreement between the experimental and calculated results is excellent.
Using the results for CnH2n+2 and the functional groups as basis sets that are linearly combined, the exact solution for the dimensional parameters, charge density functions, and energies of all molecules can be obtained. For example, one or more of the hydrogen atoms of the solution for CnH2n+2 can be substituted with one or more of the previously solved functional groups or derivative functional groups to give a desired molecule. The solution is given by energy matching each group to CnH2n+2. Substitution of one or more H 's of
CnH2n+2 with functional groups from the list of CH3 , other CnH2n+2 groups, H2C = CH2 , HC = CH , F , Cl , O, OH, NH , NH2 , CN , NO, NO2, CO, CO2, and C6H6 give the solutions of branched alkanes, alkenes, and alkynes, alkyl halides, ethers, alcohols, amides, amines, nitriles, alkyl nitrosos, alkyl nitrates, aldehydes, ketones, carbolylic acids, esters, and substituted aromatics.
PROPANE (C3H8) Using Eq. (14.642) with n - 3 , the total bond dissociation energy of C3H8 is
ED (C2H%) = ED (C -C)2 + 2ED^CH3) + ED^CH2)
= (2)(4.32754 eF) + (2)(l2.49186 eF) + (l)(7.83016 eV) (14.646) = 41.46896 eV
Using Eq. (14.643), the experimental total bond dissociation energy of C3H8, ED (C3H8) , given by the negative difference between the enthalpy of its formation (AH f (C3H8 (gas) = -1.0758 eV) ) [18] and the sum of the enthalpy of the formation of the gaseous carbons ( ΔH/ (C (gas)) ) and hydrogen ( AH f (H(gas)) ) atoms is
Eoa? (C3^s) = -{ΔH, (C3H8 (gas))-[3AHf (C(gas))+ ZAHf (H(gas))]}
= -{-1.0758 eV- [(3) 7.42774 eV + (8) 2.259353 eV]} (14.647)
= 41.434 eV The charge-density of the C3H8 molecular orbital (MO) comprising a linear combination of two methyl groups and one methylene group is shown in Figure 50.
BUTANE (C4H10) Using Eq. (14.642) with n = 4 , the total bond dissociation energy of C4H10 is
ED (C4H10) = ED (c-c\ + 2Eo^e (12cH3)+ 2^_ (12CH2)
= (3)(432154 eV) + (2)(l2A9186 eV) + (2)(7.830l6 eV) (14.648)
= 53.62666 e V
Using Eq. (14.643), the experimental total bond dissociation energy of C4H10 , EDm (C4H10) , given by the negative difference between the enthalpy of its formation (AH f (C4H10 (gas) = -1.3028 eV) ) [18] and the sum of the enthalpy of the formation of the gaseous carbons (AH f (C(gas))) and hydrogen (AH7 (H(gas))) atoms is
EDexp (c4Hlo) = -{ΔH/ (c4H10 (gα.))-[4ΔH/ (c(gα,))+i0ΔH/ (H(gα5))]}
= -{-1.3028 eV- [(4) 7.42774 eV + (l 0)2.259353 eV]} = 53.61 eF (14.649) The charge-density of the C4H10 molecular orbital (MO) comprising a linear combination of two methyl and two methylene groups is shown in Figure 51.
PENTANE (C5H12) .
Using Eq. (14.642) with n - 5 , the total bond dissociation energy of C5H12 is
ED (C5H12) = ED (C-C)4 +2£o_ (12cH3)+3ED_ (12CH2)
= (4)(4.32754 eF)+(2)(l2.49186 eF) + (3)(7.83016 eV) (14.650)
= 65.78436 eV
Using Eq. (14.643), the experimental total bond dissociation energy of C5H12 , ED<χ (C5H12), given by the negative difference between the enthalpy of its formation (AHf (C5H12 (gas) = -1.5225 eV)) [18] and the sum of the enthalpy of the formation of the gaseous carbons (AH f (C(gas))) and hydrogen (AH f (H(gas))) atoms is EDa? [C5H12) = ~{Wf (C5Hn (gas))-[5AHf (C(gas))+l2AHf (H{gas))]}
= -{-1.5225 eK-[(5)7.42774 eF + (l2)2.259353 eF]} (14.651)
= 65.77 eV
The charge-density of the C5H11 molecular orbital (MO) comprising a linear combination of two methyl and three methylene groups is shown in Figure 52.
HEXANE (C6H14)
Using Eq. (14.642) with n = 6 , the total bond dissociation energy of C6H14 is
ED (C6H14 ) = E0 (C-C)5 + 2ED^s ( 12CH3 ) + 4EDΛ_ ( 12CH2 )
= (5)(4.32754 eF) + (2)(l2.49186 eF) + (4)(7.83016 eF) (14.652)
= 77.94206 e V
Using Eq. (14.643), the experimental total bond dissociation energy of C6H14 , ED (C6H14) , given by the negative difference between the enthalpy of its formation ( AHf (C6H14 (gas) = ~l .7298 e V) ) [18] and the sum of the enthalpy of the formation of the gaseous carbons (ΔHf (C(gas))) and hydrogen (ΔH7 (H (gas))) atoms is
EDcxp (C6H14) = -{ΔH, (C6H14 (gas))-[6AHf (C(gas)) + l4AHf (H(gas))]} = -{-1.7298 eV- [(6)7.42774 eV + (14)2.259353 eV^ = 77.93 eV (14.653) The charge-density of the C6H14 molecular orbital (MO) comprising a linear combination of two methyl and four methylene groups is shown in Figure 53.
HEPTANE (C7H16)
Using Eq. (14.642) with n = 7 , the total bond dissociation energy of C7H16 is
E0 (C7H16) = E0 (c-c)6 +2EθΛmc (12cH3)+5ED_ (12CH2)
= (6)(4.32754 eF) + (2)(l2.49186 eF) + (5)(7.83016 eV) (14.654)
= 90.09976 eV Using Eq. (14.643), the experimental total bond dissociation energy of C7H16 , ED∞ (C7H16) , given by the negative difference between the enthalpy of its formation (AHf (C7H16 (gas) = -1.9443 eV)) [18] and the sum of the enthalpy of the formation of the gaseous carbons (AHf (C(gas))) and hydrogen (ΔH/ (H(gαs))) atoms is
EDnv (C7H16) = -{Δff, (C7H16 (gas))-[lAHf (C (gas)) +16 AH f (H(gasj)]}
= -{-1.9443 eV - [(?) 7.42774 <?F + (16)2.259353 e^]} = 90.09 eV (14.655) The charge-density of the C7H16 MO comprising a linear combination of two methyl and five methylene groups is shown in Figure 54.
OCTANE (C8H18)
Using Eq. (14.642) with n = 8 , the total bond dissociation energy of C8H18 is
ED (CsHιs) = ED (C -C)η +2ED^CH3) + 6EDaiJi2CH2) = (7)(4.32754 eF) + (2)(l2.49186 eF) + (6)(7.83016 eF) (14.656)
= 102.25746 eV
Using Eq. (14.643), the experimental total bond dissociation energy of C8H18, ED^ (C8H18) , given by the negative difference between the enthalpy of its formation (AH f (C8H18 (gas) = -2.1609 eV)) [18] and the sum of the enthalpy of the formation of the gaseous carbons (AHf (C(gas))) and hydrogen (AH f (H(gas))) atoms is
E^ (C8H18 ) = - [AHf (C8H18 (gas)) - [8ΔH, (C (gas)) + 1 SAHf (H (gas))]} = -{-2.1609 eF-[(8)7.42774 eV + (18)2.259353 eV]}
= 102.25 eV (14.657)
The charge-density of the C8H18 MO comprising a linear combination of two methyl and six methylene groups is shown in Figure 55.
NONANE (C9H20)
Using Eq. (14.642) with n = 9 , the total bond dissociation energy of C9H20 is ED(C^M) = ED{C-C)t +2E^ (*CH3) + 7E^ (»CH2)
= (8)(4.32754 eF) + (2)(l2.49186 eF) + (7)(7.83016 er) (14.658)
= 114.41516 eV
Using Eq. (14.643), the experimental total bond dissociation energy of C9H20, EDa (C9H20), given by the negative difference between the enthalpy of its formation (ΔH7 (C9H20 (gas) = -2.3651 eV)) [18] and the sum of the enthalpy of the formation of the gaseous carbons ( ΔH ' f (C (gas)) ) and hydrogen ( ΔH f (H (gas)) ) atoms is
^A, (C&o ) = - (AH7 (C9H20 (gas)) - [9 AH, (C(gas)) + 20AHf (H (gas))]} = -{-2.3651 eV- [(9)7.42774 eV + (20)2.259353 eV]} = 114.40 eF (14.659) The charge-density of the C9H20 MO comprising a linear combination of two methyl and seven methylene groups is shown in Figure 56.
DECANE (C10H22)
Using Eq. (14.642) with n - 10 , the total bond dissociation energy of C10H22 is
ED (C10H22 ) = E0 (C -C)9 + 2EDΛ_ ( 12CH3 ) + *EDφ_ ( 12CH2 )
= (9)(4.32754 <?r)+(2)(l2.49186 eF) + (8)(7.83016 eF) (14.660) = 126.57286 eV Using Eq. (14.643), the experimental total bond dissociation energy of C10H22, ED (C10H22) , given by the negative difference between the enthalpy of its formation (AH f (C10H22 (gas) = -2.5858 eV)) [18] and the sum of the enthalpy of the formation of the gaseous carbons (ΔH f (C(gas))) and hydrogen (AH f (H(gas))) atoms is
^ (C10H22) = -{ΔH, (C10H22 (gaS))-[l0AHf (C(gas)) + 22AHf (H(gas))]}
= -{-2.5858 eF-[(lθ)7.42774 eF + (22)2.259353 eF]} = 126.57 eV (14.661) The charge-density of the C10H22 molecular orbital (MO) comprising a linear combination of two methyl and eight methylene groups is shown in Figure 57. UNDECANE (C11H24)
Using Eq. (14.642) with n - 11 , the total bond dissociation energy of C11H24 is
ED (C11H24) = ED (C -C)10 + 2^ {»CH3) + 9ED_ (12CH2)
= (l0)(4.32754 eF) + (2)(l2.49186 eF)+(9)(7.83016 eF) (14.662) = 138.73056 eV Using Eq. (14.643), the experimental total bond dissociation energy of C11H24 , EDa (C11H24) , given by the negative difference between the enthalpy of its formation (AH f (C11H24 (gas) = -2.8066 eV) ) [18] and the sum of the enthalpy of the formation of the gaseous carbons (AH f (C(gas))) and hydrogen (AH f (H (gas))) atoms is
^κp (c1A) = -{ΔH/ (cuH24(gα5))-[iiΔH/ (c(g^))+24ΔH/(H(g^))]}
= -{-2.8066 eV- [(11)7.42774 eV + (24) 2.259353 eV]} = 138.736 eV (14.663)
The charge-density of the CnH24 MO comprising a linear combination of two methyl and nine methylene groups is shown in Figure 58.
DODECANE (C12H26)
Using Eq. (14.642) with n - Yl , the total bond dissociation energy of C12H26 is
ED (CnH26) = ED (c-c)n +2ED^ (12cH3)+ioE,fl;w (12CH2)
= (l l)(4.32754 eF) + (2)(l2.49186 eF) + (l0)(7.83016 eF) (14.664) = 150.88826 eV Using Eq. (14.643), the experimental total bond dissociation energy of C12H26,
ED (C12H26), given by the negative difference between the enthalpy of its formation (ΔHr (C12H26 (gas) = -2.9994 eV)) [18] and the sum of the enthalpy of the formation of the gaseous carbons (AH f (C(gas))) and hydrogen (AH f (H(gas))) atoms is EDap (CnH26 ) = -{ΔHf (C12H26 (gas)) - [l2Δff, (C (gas)) + 26ΔH , (H (gas))]}
= -{-2.9994 eV- [(12)7.42774 eF + (26)2.259353 eF]} (14.665)
= 150.88 eV
The charge-density of the C12H26 MO comprising a linear combination of two methyl and ten methylene groups is shown in Figure 59.
OCTADECANE (C18H38)
Using Eq. (14.642) with n - 18 , the total bond dissociation energy of C18H38 is
ED (C18H38) = ED (C-C\η +2EDΛm (12CH3) + 16£D_ (12CH2)
= (17)(4.32754 eF) + (2)(l2.49186 eF) + (l6)(7.83016 eV) (14.666) = 223.83446 eV
Using Eq. (14.643), the experimental total bond dissociation energy of C18H38 , ED (C18H38) , given by the negative difference between the enthalpy of its formation ( AH f (C18H38 (gas) = -4.2970 e V) ) [18] and the sum of the enthalpy of the formation of the gaseous carbons ( /SJH f (C (gas)) ) and hydrogen ( ΔHf (H (gas))) atoms is
EDav (CΑ) = -{ΔΗ7 (C18H38 (gas))-[\SAHf (C(gas)) + 3SAHf (H(gas))]} = -{-4.2970 eV- [(18) 7.42774 eF + (38)2.259353 eV]} = 223.85 eV (14.667)
The charge-density of the C18H38 molecular orbital (MO) comprising a linear combination of two methyl and sixteen methylene groups is shown in Figure 60.
Table 14.1. The calculated and experimental bond parameters of CO2, NO2, CH3CH3 , CH2CH2 , CHCH , benzene, propane, butane, pentane, hexane, heptane, octane, nonane, decane, undecane, dodecane, and octadecane.
Figure imgf000443_0001
Figure imgf000444_0001
Figure imgf000445_0001
References for Section III
1. D. R. Lide, CRC Handbook of Chemistry and Physics, 79th Edition, CRC Press, Boca
Raton, Florida, (1998-9), pp. 4-130 to 4-135. 2. D. R. Lide, CRC Handbook of Chemistry and Physics, 79th Edition, CRC Press, Boca Raton, Florida, (1998-9), p. 10-175.
3. D. R. Lide, CRC Handbook of Chemistry and Physics, 79th Edition, CRC Press, Boca Raton, Florida, (1998-9), pp. 9-15 to 9-41.
4. M. Karplus, R. N. Porter, Atoms and Molecules an Introduction for Students of Physical Chemistry, The Benjamin/Cummings Publishing Company, Menlo Park, California,
(1970), pp. 447-484.
5. G. Ηerzberg, Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules, Krieger Publishing Company, Malabar, FL, (1945), p. 174.
6. D. R. Lide, CRC Handbook of Chemistry and Physics, 79th Edition, CRC Press, Boca Raton, Florida, (1998-9), pp. 9-76 to 9-79.
7. D. R. Lide, CRC Handbook of Chemistry and Physics, 79th Edition, CRC Press, Boca Raton, Florida, (1998-9), pp. 9-63 to 9-69.
8. R. L. DeKock, Η. B. Gray, Chemical Structure and Bonding, The Benjamin/Cummings Publishing Company, Menlo Park, CA, (1980), p. 162. 9. G. R. Fowles, Analytical Mechanics, Third Edition, Holt, Rinehart, and Winston, New York, (1977), pp. 251-305. 10. G. Herzberg, Molecular Spectra and Molecular Structure II. Infrared and Raman
Specti-a of Polyatomic Molecules, Van Nostrand Reinhold Company, New York, New- York, (1945), p. 344.
11. R. L. DeKock, H. B. Gray, Chemical Structure and Bonding, The Benjamin/Cummings Publishing Company, Menlo Park, CA, (1980), p. 179.
12. G. Herzberg, Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand Reinhold Company, New York, New York, (1945), p. 326.
13. W. I. F. David, R. M. Ibberson, G. A. Jeffrey, J. R. Ruble, "The structure analysis of deuterated benzene and deuterated nitromethane by pulsed-neutron powder diffraction: a comparison with single crystal neutron analysis", Physica B (1992), 180 & 181, pp. 597- 600..
14. G. A. Jeffrey, J. R. Ruble, R. K. McMullan, J. A. Pople, "The crystal structure of deuterated benzene," Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 414, No. 1846, (Nov. 9, 1987), pp. 47-57.
15. H. B. Burgi, S. C. Capelli, "Getting more out of crystal-structure analyses," Helvetica Chimica Acta, Vol. 86, (2003), pp. 1625-1640.
16. G. Herzberg, Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand Reinhold Company, New York, New York, (1945), pp. 362-369.
17. D. R. Lide, CRC Handbook of Chemistry and Physics, 79th Edition, CRC Press, Boca Raton, Florida, (1998-9), pp. 9-63.
18. D. R. Lide, CRC Handbook of Chemistry and Physics, 79th Edition, CRC Press, Boca Raton, Florida, (1998-9), pp. 5-1 to 5-60. 19. R. J. Fessenden, J. S. Fessenden, Organic Chemistry, Willard Grant Press. Boston, Massachusetts, (1979), pp. 46-48. Section IV
ORGANIC MOLECULAR FUNCTIONAL GROUPS
AND MOLECULES
Organic molecules comprising an arbitrary number of atoms can be solved using the similar principles and procedures as those used to solve alkanes of arbitrary length. Alkanes can be considered to be comprised of the functional groups of CH3 , CH2, and C - C . These groups with the corresponding geometrical parameters and energies can be added as a linear sum to give the solution of any straight chain alkane as shown in the Continuous-Chain Alkanes section. Similarly, the geometrical parameters and energies of all functional groups such as alkanes, branched alkanes, alkenes, branched alkenes, alkynes, alkyl fluorides, alkyl chlorides, alkyl bromides, alkyl iodides, alkene halides, primary alcohols, secondary alcohols, tertiary alcohols, ethers, primary amines, secondary amines, tertiary amines, aldehydes, ketones, carboxylic acids, carboxylic esters, amides, N-alkyl amides, N,N-dialkyl amides, urea, acid halides, acid anhydrides, nitriles, thiols, sulfides, disulfides, sulfoxides, sulfones, sulfites, sulfates, nitro alkanes, nitrites, nitrates, conjugated polyenes, aromatics, heterocyclic aromatics, substituted aromatics, and others can be solved. The functional-group solutions can be made into a linear superposition and sum, respectively, to give the solution of any organic molecule. The solutions of the functional groups can be conveniently obtained by using generalized forms of the geometrical and energy equations. The equations and sections that are referenced by not contained in this text refer to those sections and equations of the book by R. L. Mills entitled, "The Grand Unified Theory of Classical Quantum Mechanics", June 2006 Edition, Cadmus Professional Communications-Science Press Division, Ephrata, PA, ISBN 0963517171, Library of Congress Control Number 2005936834; posted at http://www.blacklightpower.com/bookdownload.shtml which is incorporated in its entirety by reference.
Consider the case wherein at least two atomic orbital hybridize as a linear combination of electrons at the same energy in order to achieve a bond at an energy minimum, and the sharing of electrons between two or more such orbitals to form a MO permits the participating hybridized orbitals to decrease in energy through a decrease in the radius of one or more of the participating orbitals. The force generalized constant ft' of a H2 -type ellipsoidal MO due to the equivalent of two point charges of at the foci is given by:
Figure imgf000448_0001
where C1 is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the molecule or molecular ion which is 0.75 (Eq. (13.59)) in the case of H bonding to a central atom and 0.5 (Eq. (14.152)) otherwise, and C2 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of the chemical bond. From Eqs. (13.58-13.63), the distance from the origin of the MO to each focus c1 is given by:
Figure imgf000448_0002
The internuclear distance is
Figure imgf000448_0003
The length of the semiminor axis of the prolate spheroidal MO b = c is given by
Figure imgf000448_0004
And, the eccentricity, e , is c'
(15.5) a
From Eqs. (11.207-11.212), the potential energy of the two electrons in the central field of the nuclei at the foci is
V e = (15.6)
Figure imgf000448_0005
The potential energy of the two nuclei is
Figure imgf000448_0006
The kinetic energy of the electrons is
Figure imgf000448_0007
And, the energy, Vm , of the magnetic force between the electrons is
Figure imgf000449_0001
The total energy of the H2 -type prolate spheroidal MO, Eτ (ff.jwσ), is given by the sum of the energy terms:
ET (H2rn)= V + T + Vm + Vp (15.10)
Figure imgf000449_0002
where nx is the number of equivalent bonds of the MO and applies in the case of functional groups. In the case of independent MOs not in contact with the bonding atoms, the terms based on charge are multiplied by cB0 , the bond-order factor. It is 1 for a single bond, 4 for an independent double bond as in the case of the CO2 and NO2 molecules, and 9 for an independent triplet bond. Then, the kinetic energy term is multiplied by c\o which is 1 for a single bond, 2 for a double bond, and 9/2 for a triple bond. C1 is the fraction of the H2 -type ellipsoidal MO basis function of an MO which is 0.75 (Eqs. (13.67-13.73)) in the case of H bonding to an unhybridized central atom and 1 otherwise, and c2 is the factor that results in an equipotential energy match of the participating the MO and the at least two atomic orbitals of the chemical bond. Specifically, to meet the equipotential condition and energy matching conditions for the union of the H2 -type-ellipsoidal-MO and the ΗOs or AOs of the bonding atoms, the factor c2 of a H2 -type ellipsoidaL MO may given by (i) one, (ii) the ratio of the
Coulombic or valence energy of the AO or HO of at least one atom of the bond and 13.605804 eV , the Coulombic energy between the electron and proton of H , (iii) the ratio of the valence energy of the AO or HO of one atom and the Coulombic energy of another, (iv) the ratio of the valence energies of the AOs or HOs of two atoms, (v) the ratio of two c2 factors corresponding to any of cases (ii)-(iv), and (vi) the product of two different c2 factors corresponding to any of the cases (i)-(v). Specific examples of the factor c2 of a H2 -type ellipsoidal MO given in previous sections are 0.936127 , the ratio of the ionization energy of JV 14.53414 eV and 13.605804 eV , the Coulombic energy between the electron and proton of H ; 0.91771 , the ratio of 14.82575 eV ,' -£ C„oul,omb, ( Vc,2sp3\ / and 13.605804 eV ;
0.87495 , the ratio of 15.55033 eV , -ECmhmb (Cetoe,2ψ3), and 13.605804 eV ;
0.85252 , the ratio of 15.95955 eV , -ECoφmb (Celhy!ene,2sp*y and 13.605804 eV ;
0.85252 , the ratio of 15.95955 eV , -ECoulomb(Cbcnzene,2sp3y and 13.605804 eV , and
0.86359 , the ratio of 15.55033 eV , -ECWowA(Cfl/toie,2sp3), and 13.605804 eV .
In the generalization of the hybridization of at least two atomic-orbital shells to form a shell of hybrid orbitals, the hybridized shell comprises a linear combination of the electrons of the atomic-orbital shells. The radius of the hybridized shell is calculated from the total
Coulombic energy equation by considering that the central field decreases by an integer for each successive electron of the shell and that the total energy of the shell is equal to the total
Coulombic energy of the initial AO electrons. The total energy Eτ \atom,msp3Λ (m is the integer of the valence shell) of the AO electrons and the hybridized shell is given by the sum of energies of successive ions of the atom over the n electrons comprising total electrons of the at least one AO shell.
(15.12)
Figure imgf000450_0001
where /Pn is the mth ionization energy (positive) of the atom. The radius r^ 3 of the hybridized shell is given by: (15Λ3)
Figure imgf000450_0002
Then, the Coulombic energy ECouhmb [atom,msp3 j of the outer electron of the atom msp3 shell is given by
( y15.14) J
Figure imgf000450_0004
In the case that during hybridization at least one of the spin-paired AO electrons is unpaired in the hybridized orbital (HO), the energy change for the promotion to the unpaired state is the magnetic energy E (magnetic) at the initial radius r of the AO electron: (15.15)
Figure imgf000450_0003
Then, the energy E\atom,msp3 J of the outer electron of the atom msp3 shell is given by the
sum of ECoulmb (atom,msp3j and E(magnetic) :
Figure imgf000451_0001
Consider next that the at least two atomic orbitals hybridize as a linear combination of electrons at the same energy in order to achieve a bond at an energy minimum with another atomic orbital or hybridized orbital. As a further generalization of the basis of the stability of the MO, the sharing of electrons between two or more such hybridized orbitals to form a MO permits the participating hybridized orbitals to decrease in energy through a decrease in the radius of one or more of the participating orbitals. In this case, the total energy of the hybridized orbitals is given by the sum of E\atom,msp3 J and the next energies of successive ions of the atom over the n electrons comprising the total electrons of the at least two initial AO shells. Here, E\atom,msp3 J is the sum of the first ionization energy of the atom and the
hybridization energy. An example of E [atom, msp3 J fox E{C,2sp3 J is given in Eq. (14.503) where the sum of the negative of the first ionization energy of C, -11.27671 eV , plus the hybridization energy to form the Clsp3 shell given by Eq. (14.146) is
E(C525p3)= -14.63489 eV .
Thus, the sharing of electrons between two atom msp3 HOs to form an atom-atom- bond MO permits each participating hybridized orbital to decrease in radius and energy. In order to further satisfy the potential, kinetic, and orbital energy relationships, each atom msp3 HO donates an excess of 25% per bond of its electron density to the atom-atom-bond MO to form an energy minimum wherein the atom-atom bond comprises one of a single, double, or triple bond. In each case, the radius of the hybridized shell is calculated from the Coulombic energy equation by considering that the central field decreases by an integer for each successive electron of the shell and the total energy of the shell is equal to the total Coulombic energy of the initial AO electrons plus the hybridization energy. The total energy
Eτ (mol.atom,msp3 j {m is the integer of the valence shell) of the HO electrons is given by the sum of energies of successive ions of the atom over the n electrons comprising total electrons of the at least one initial AO shell and the hybridization energy: (15.17)
Figure imgf000452_0004
where ZPn is the røth ionization energy (positive) of the atom and the sum of -/P1 plus the hybridization energy is E(atom,msp3\ Thus, the radius r , of the hybridized shell is given by:
(15.18)
Figure imgf000452_0001
where s = 1,2,3 for a single, double, and triple bond, respectively. The Coulombic energy
^couiomb ynol.atom,msp3) of the outer electron of the atom msp3 shell is given by
(15.19)
Figure imgf000452_0003
hi the case that during hybridization at least one of the spin-paired AO electrons is unpaired in the hybridized orbital (HO), the energy change for the promotion to the unpaired state is the magnetic energy E(magnetic) at the initial radius r of the AO electron given by Eq.
(15.15). Then, the energy E[mol.atom,msp3 J of the outer electron of the atom msp3 shell is
given by the sum of ECoulomb \mol.atom,msp3 J and E(magnetic) :
Figure imgf000452_0002
Eτ (atom — atom,msp3 J, the energy change of each atom msp3 shell with the formation of the
atom-atom-bond MO is given by the difference between E\mol.atom,msp3 Λ and
E\atom,msp3 J:
E7, {atom - atom, msp3 J= E (moLatom, msp3 V E [atom, msp3 J (15.21)
As examples from prior sections, ECoI,/omό (mol.atom,msp3 J is one of:
E Coulomb ^ ethylene^ y E Coulnnb ψ ethane>2SP )> E Coulomb ^} acetylene'1^ } mά
^- 'Coulomb Λ ~ atlume>^SP )>
Ecou,omb (atom,msp3) is one of ECoulomb (c,2sp3) and ECoulomb (c/,3ψ3); E(mol.atom,msp3) is one of E(Cellφm,2sp3), E(c ,w2ψ3),
E(Cacelylene,2sp3)E(Ca!kane,2sp3),
E (atom, msp3 ) is one of and E (c, 2sp3 ) and E (CI, 3sp3 );
Eτ (atom- atom,msp3^ is one of E(C - C,2.ψ3)> E(C = C ,2Sp3^, and. E(C ≡ C,2sp3J, atom msp3 is one of C2sp3 , O3sp3
Eτ \atom- atom(sΛmsp3 ) is Eτ (C - C, 2sp3 J and ET (atom- atom(s2\msp3 J is
£r (c = C,2sp3), and r msp , is one of r C2sp 3 , r ethaneisp 3 , r ethylenelsp . , r acetylenelsp , , r alkane2sp 3 , and r CI3sp , .
In the case of the C2sp3 HO, the initial parameters (Eqs. (14.142-14.146)) are
Figure imgf000453_0001
= -14.82575 eV + 0.19086 eF (15.25)
= -14.63489 eV In Eq. (15.18),
(15.26)
Figure imgf000453_0002
Eqs. (14.147) and (15.17) give
£r (molatom,msp3y= E7 (CeΛow,2ψ3)= -151.61569 eF (15.27)
Using Eqs. (15.18-15.28), the final values of r^ , , ECoa/on)6 (C2sp3 ), and E(C2sp3), and the
resulting β j c - c,C2stf \ °f the MO due to charge donation from the HO to the MO where BO
C - C refers to the bond order of the carbon-carbon bond for different values of the parameter 5 are given in Table 15.1.
Table 15.1. The final values of /^ 3 , ECguhmb (C2sp*y and E(C2sp3^ and the resulting
BO E \ C °C,C2sp 31 °f me MO due to charge donation from the HO to the MO where C - C
) refers to the bond order of the carbon-carbon bond.
Figure imgf000454_0003
In another generalized case of the basis of forming a minimum-energy bond with the constraint that it must meet the energy matching condition for all MOs at all HOs or AOs, the energy Eψιol.atom,msp3 J of the outer electron of the atom msp3 shell of each bonding atom
must be the average of E [mohatom, msp3 J for two different values of s :
Figure imgf000454_0001
hi this case, Eτ (atom - atom, msp3 \ the energy change of each atom msp3 shell with the formation of each atom-atom-bond MO, is average for two different values of s :
Figure imgf000454_0002
Consider an aromatic molecule such as benzene given in the Benzene Molecule section. Each C = C double bond comprises a linear combination of a factor of 0.75 of four paired electrons (three electrons) from two sets of two C2sp3 HOs of the participating carbon atoms. Each C - H bond of CH having two spin-paired electrons, one from an initially unpaired electron of the carbon atom and the other from the hydrogen atom, comprises the linear combination of 75% H2 -type ellipsoidal MO and 25% C2sp3 HO as given by Eq.
(13.439). However, Eτ \atom- atom,msp3 >\ of the C - iϊ-bond MO is given by
0.5E7, (c = C,2sp3) (Eq. (14.247)) corresponding to one half of a double bond that matches the condition for a single-bond order for C - H that is lowered in energy due to the aromatic character of the bond.
A further general possibility is that a minimum-energy bond is achieved with satisfaction of the potential, kinetic, and orbital energy relationships by the formation of an MO comprising an allowed multiple of a linear combination of H2 -type ellipsoidal MOs and corresponding HOs or AOs that contribute a corresponding allowed multiple (e.g. 0.5, 0.75, 1) of the bond order given in Table 15.1. For example, the alkane MO given in the
Continuous-Chain Alkanes section comprises a linear combination of factors of 0.5 of a single bond and 0.5 of a double bond.
Consider a first MO and its HOs comprising a linear combination of bond orders and a second MO that shares a HO with the first. In addition to the mutual HO, the second MO comprises another AO or HO having a single bond order or a mixed bond order. Then, in order for the two MOs to be energy matched, the bond order of the second MO and its HOs or its HO and AO is a linear combination of the terms corresponding to the bond order of the mutual HO and the bond order of the independent HO or AO. Then, in general,
E1, (atom- atom,msp3 \ the energy change of each atom msp3 shell with the formation of each atom-atom-bond MO, is a weighted linear sum for different values of s that matches the energy of the bonded MOs, HOs, and AOs:
N
Eτ [atom - atom, msp3 J= ]T cs Eτ [atom - atom ($n \ msp3 J (15.30)
H=I where c s n is the multiple of the BO of s "M . The radius r msp , of the atom ms -p^ 3 shell of each
bonding atom is given by the Coulombic energy using the initial energy ECoulomb [atom,msp3 j
and Eτ [atom- atom, msp3 \ the energy change of each atom msp3 shell with the formation of each atom-atom-bond MO:
Figure imgf000455_0001
where ^cOM/om* (C2^3)= ~14-825751 eK - The Coulombic energy ECoulomb (mol.atom,msp3) of the outer electron of the atom msp3 shell is given by Eq. (15.19). In the case that during hybridization, at least one of the spin-paired AO electrons is unpaired in the hybridized orbital (HO), the energy change for the promotion to the unpaired state is the magnetic energy E(magnetic) (Eq. (15.15)) at the initial radius r of the AO electron. Then, the energy
E (mol. atom, msp3 J of the outer electron of the atom msp3 shell is given by the sum of
ECmlomb the
Figure imgf000456_0001
energy change of each atom msp3 shell with the formation of the atom-atom-bond MO is given by the difference between E\mol.atom,msp3 Λ and E\atom,msp3 J given by Eq.
(15.21). Using Eq. (15.23) for ECouhmb (c,2sρ3λ in Eq. (15.31), the single bond order energies given by Eqs. (15.18-15.27) and shown in Table 15.1, and the linear combination energies (Eqs. (15.28-15.30)), the parameters of linear combinations of bond orders and linear combinations of mixed bond orders are given in Table 15.2.
Table 15.2. The final values of rC2^ , ECoulomb (<22sp3 ), and E(θ2sp3^ and the resulting
E i CB- C,C2sp3 1 of the MO comprising a linear combination of H2 -type ellipsoidal MOs and
corresponding ΗOs of single or mixed bond order where cSn is the multiple of the bond order
parameter Eτ (atom - atom (sn \ msp3 J given in Table 15.1.
Figure imgf000457_0001
Consider next the radius of the AO or HO due to the contribution of charge to more than one bond. The energy contribution due to the charge donation at each atom such as carbon superimposes linearly. In general, the radius r molisp , of the C2sp3 HO of a carbon
atom of a given molecule is calculated using Eq. (14.514) by considering V E mo! \ VM, O,2sp% \ s the total energy donation to each bond with which it participates in bonding. The general equation for the radius is given by
Figure imgf000458_0003
The Coulombic energy ECoulomb \mol.atom,msp3 J of the outer electron of the atom msp3 shell is given by Eq. (15.19). In the case that during hybridization, at least one of the spin-paired
AO electrons is unpaired in the hybridized orbital (HO), the energy change for the promotion to the unpaired state is the magnetic energy E(magnetic) (Eq. (15.15)) at the initial radius r of the AO electron. Then, the energy E\mol.atom,msp3 \ of the outer electron of the
atom msp3 shell is given by the sum of ECoulomb\mol.atom,msp3 λ and E(magnetic) (Eq. (15.20)).
For example, the C2sp3 HO of each methyl group of an alkane contributes -0.92918 eV (Eq. (14.513)) to the corresponding single C - C bond; thus, the corresponding C2sp3 HO radius is given by Eq. (14.514). The Clsp3 HO of each methylene group of CnH2n+2 contributes -0.92918 eV to each of the two corresponding C- C bond
MOs. Thus, the radius (Eq. (15.32)), the Coulombic energy (Eq. (15.19)), and the energy (Eq. (15.20)) of each alkane methylene group are
(15.33)
Figure imgf000458_0001
Figure imgf000458_0002
In the determination of the parameters of functional groups, heteroatoms bonding to
C2sp3 HOs to form MOs are energy matched to the C2sp3 HOs. Thus, the radius and the energy parameters of a bonding heteroatom are given by the same equations as those for C2sρ3 HOs. Using Eqs. (15.15), (15.19-15.20), (15.24), and (15.32) in a generalized fashion,
the final values of the radius of the HO or AO, fAlom UO AO> ECoΦmb Vtol-atom>msP j> m&
E (Cmol 2sp3 J are calculated using \MO, 2sp3 \ the total energy donation to each bond
Figure imgf000459_0001
with which an atom participates in bonding corresponding to the values of β ^ _ c,C2sp3 1 °f
the MO due to charge donation from the AO or HO to the MO given in Tables 15.1 and 15.2.
Table 15.3.A. The final values of rAtom HO AO, ECmlomb (molatom,msp3), and E(Cmol2spl) f BO \ calculated using the values of #J Cc -- C c,,CC22sspp33 \\ given in Tables 15.1 and 15.2
Figure imgf000460_0001
Figure imgf000461_0001
Table 15.3.B. The final values of rAtomHOAO, ECoulomb(mol.atom,msp3), and E(CIIIO!2SP 3)
( BO calculated for heterocyclic groups using the values of & f . C-C,C2sp3 given in Tables 15.1
and 15.2.
H
Figure imgf000463_0001
Figure imgf000464_0001
The energy of the MO is matched to each of the participating outermost atomic or hybridized orbitals of the bonding atoms wherein the energy match includes the energy contribution due to the AO or HO's donation of charge to the MO. The force constant k' (Eq. (15.1)) is used to determine the ellipsoidal parameter c1 (Eq. (15.2)) of the each H2- type-ellipsoidal-MO in terms of the central force of the foci. Then, c' is substituted into the energy equation (from Eq. (15.11))) which is set equal to nx times the total energy of H2 where nλ is the number of equivalent bonds of the MO and the energy of H2,
-31.63536831 eV , Eq. (11.212) is the minimum energy possible for a prolate spheroidal MO. From the energy equation and the relationship between the axes, the dimensions of the
MO are solved. The energy equation has the semimajor axis a as it only parameter. The solution of the semimajor axis a then allows for the solution of the other axes of each prolate spheroid and eccentricity of each MO (Eqs. (15.3-15.5)). The parameter solutions then allow for the component and total energies of the MO to be determined. The total energy, ET (H2MO), is given by the sum of the energy terms (Eqs. (15.6-
15.11)) plus ET (AOI HO):
E7 (H1M0)= Ve + T + Vm + Vp + Eτ (AO I HO) (15.36)
Figure imgf000465_0001
where nλ is the number of equivalent bonds of the MO, c, is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the group, c2 is the factor that results in an equipotential energy match of the participating at least two atomic orbitals of each chemical bond, and Eτ CiO I Hθ) is the total energy comprising the difference of the energy
E(A0 I HO) of at least one atomic or hybrid orbital to which the MO is energy matched and
any energy component AEH MO (AO I Hθ) due to the AO or HO' s charge donation to the MO.
ET (A0 I H0)= E(AO / HO)~ AE^M0 (AO I H0) - (15.38) s specific examples given in previous sections, E7 (AO I HO) is one from the group of
E7 (AO I Hθ)= E(p2p shell)= ~E(ionization; O) = -13.6181 eV ;
E7 (AO I HO)= E(NIp shell)= -E(ionization; N) = -14.53414 eV ;
Eτ (AO I HO)= E(c,2sp*y -14.63489 eV ;
Eτ (AO I HO)= ECoulomb (α,3ψ3)= -14.60295 eV ;
E7 (AO I Hθ)= E(ionization; C) + E(ionization; C+) ;
Eτ (AO I HOY E(Celhane,2sp3)= -15.35946 eV ;
Eτ (AO I HO)= +E(Cethyle)ie,2sp3y E(Celhylene,2sP>),
Eτ (AO I Hθ)= E((J,2sp3y 2ET (C = C,25p3)= -14.63489 eV - (-2.26758 eV); ET(AO/ Hθy E(c^,2tfyE(c^,2&yE(£^,2tfy 1620002 eV;
E7 (AO I HOy E(c,2sp*y 2E7 (c ≡ C,2ψ3)= -14.63489 eV - (-3.13026 eV~);
E7 (AO I HOy E(Cbm,2sp3y E(Cbe>jzem,2sP 3);
E7 (AO I HO)= E(c,2sp3y E1, (C = C,2^3)= -14.63489 eV - (-1.13379 eV), and
E7 (AO I HO)= E(Calkam,2sp3y- 15.56407 eV .
To solve the bond parameters and energies, c' (Eq.
Figure imgf000466_0001
(15.2)) is substituted into E7 (H^MO) to give
Figure imgf000466_0002
The total energy is set equal to Eφasis energies) which in the most general case is given by the sum of a first integer W1 times the total energy of H2 minus a second integer n2 times the total energy of H, minus a third integer «3 times the valence energy of E (AO) (e.g.
E(N)= -14.53414 eV) where the first integer can be 1,2,3... , and each of the second and third integers can be 0, 1, 2, 3.... E(basis energies) = n, (-31.63536831 eV)- n2 (-13.605804 eV)- K3E(AO) (15.40)
In the case that the MO bonds two atoms other than hydrogen, Eihasis energies) is nx times the total energy of H2 where nχ is the number of equivalent bonds of the MO and the energy of H2, -31.6353683I eF , Eq. (11.212) is the minimum energy possible for a prolate spheroidal MO: E(basis energies) = n, (-31.63536831 eV) (15.41)
ET (H2MO), is set equal to Eibasis energies) , and the semimajor axis a is solved. Thus, the semimajor axis a is solved from the equation of the form:
Figure imgf000467_0001
The distance from the origin of the H2 -type-ellipsoidal-MO to each focus c1 , the internuclear distance 2c1, and the length of the semiminor axis of the prolate spheroidal H2 -type MO b = c are solved from the semimajor axis a using Eqs. (15.2-15.4). Then, the component energies are given by Eqs. (15.6-15.9) and (15.39).
The total energy of the MO of the functional group, Eτ (MO), is the sum of the total energy of the components comprising the energy contribution of the MO formed between the participating atoms and Eτ(atom- atom,msp3.Aθ\ the change in the energy of the AOs or
ΗOs upon forming the bond. From Eqs. (15.39-15.40), Eτ (MO) is
E7. (MO)= E(basis energies) + Eτ (atom - atom, msp3.AOj (15.43)
During bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energy Eosc is the sum of the Doppler, ED , and average vibrational kinetic energies, EKvjb :
Figure imgf000468_0001
where nx is the number of equivalent bonds of the MO , k is the spring constant of the equivalent harmonic oscillator, and μ is the reduced mass. The angular frequency of the reentrant oscillation in the transition state corresponding to ED is determined by the force between the central field and the electrons in the transition state. The force and its derivative are given by
Figure imgf000468_0005
(l5-45) and
Figure imgf000468_0002
such that the angular frequency of the oscillation in the transition state is given by
Figure imgf000468_0003
where R is the semimajor axis a or the semiminor axis b depending on the eccentricity of the bond that is most representative of the oscillation in the transition state, cBO is the bond- order factor which is 1 for a single bond and when the MO comprises nx equivalent single bonds as in the case of functional groups. cB0 is 4 for an independent double bond as in the case of the CO2 and NO2 molecules and 9 for an independent triplet bond. C10 is the fraction of the H2 -type ellipsoidal MO basis function of the oscillatory transition state of a chemical bond of the group, and C20 is the factor that results in an equipotential energy match of the participating at least two atomic orbitals of the transition state of the chemical bond. Typically, C10 = C1 and C2o = C2. The kinetic energy, Eκ , corresponding to E~ D is given by Planck's equation for functional groups:
Figure imgf000468_0004
The Doppler energy of the electrons of the reentrant orbit is
Figure imgf000469_0001
Egsc given by the sum of ED and EKvlb is
Figure imgf000469_0002
Eήv of a group having ^1 bonds is given by Eτ (MO)! nx such that
Figure imgf000469_0003
Eτ+osc (ρwup) is given by the sum of E7, (MO) (Eq. (15.42)) and Eosc (Eq. (15.51)):
Eτ+osc &°"ph Eτ (MO)+ Ksι:
Figure imgf000469_0004
(15.52) The total energy of the functional group E7, (group) is the sum of the total energy of the components comprising the energy contribution of the MO formed between the participating atoms, E(basis energies) , the change in the energy of the AOs or HOs upon forming the bond (E7, (atom- atom,msp3.AOj), the energy of oscillation in the transition state, and the change in magnetic energy with bond formation, Emag . From Eq. (15.52), the total energy of the group Eτ (ρm»P) is
Figure imgf000470_0001
The change in magnetic energy E mag which arises due to the formation of unpaired electrons in the corresponding fragments relative to the bonded group is given by
2πμ0e % %πμoμB
E mag = c „ • ■ = c. (15.54) m2r3 where r3 is the radius of the atom that reacts to form the bond and C3 is the number of electron pairs.
(15.55)
Figure imgf000470_0002
The total bond energy of the group ED (ρmup) is the negative difference of the total energy of the group (Eq. (15.55)) and the total energy of the starting species given by the sum of
Figure imgf000470_0003
AO I HO /) and
Figure imgf000470_0004
AO I HO J): ED (15.56)
Figure imgf000471_0001
In the case of organic molecules, the atoms of the functional groups are energy matched to the C2sp3 HO such that
E(A0/ HO)= -14.63489 eV (15.57) For examples of Ema from previous sections:
(15.58)
Figure imgf000471_0002
Figure imgf000471_0003
In the general case of the solution of an organic functional group, the geometric bond parameters are solved from the semimajor axis and the relationships between the parameters by first using Eq. (15.42) to arrive at a . Then, the remaining parameters are determined using Eqs. (15.1-15.5). Next, the energies are given by Eqs. (15.52-15.59). To meet the equipotential condition for the union of the Tf2 -type-ellipsoidal-MO and the HO or AO of the atom of a functional group, the factor c2 of a H2 -type ellipsoidal MO in principal Eqs. (15.42) and (15.52) may given by
(i) one: C2 = I (15.61)
(ii) the ratio that is less than one of 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243), and the magnitude of the Coulombic energy of the participating AO or HO of the atom, ECmlomb \MO.atom,msp3Λ given
by Eqs. (15.19) and (15.31-15.32). For |-5CθHtøwό (MO.atom,msp3]> 13.605804 eV :
(15.62)
Figure imgf000472_0004
Figure imgf000472_0001
(iii) the ratio that is less than one of 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243), and the magnitude of the valence energy, Eiyalence), of the participating AO or HO of the atom where E(valence) is the ionization energy or E\MO.atom,msp3 Λ given by Eqs. (15.20) and (15.31-15.32). For
I E(valence)\ > 13.605804 eV :
Figure imgf000472_0002
For \E(valence)\ <13.605804 eV :
(15 65)
Figure imgf000472_0003
(iv) the ratio that is less than one of the magnitude of the Coulombic energy of the participating AO or HO of a first atom, ECoulonib {MO.atom,msp3 j given by Eqs. (15.19) and
(15.31-15.32), and the magnitude of the valence energy, Eiyalence) , of the participating AO or HO of a second atom to which the first is energy matched where Eiyalence) is the ionization energy or E[MO.atom,msp3λ given by Eqs. (15.20) and (15.31-15.32). For
Ecouiomb (MO.ato?n,msp3y Eiyalence) :
Figure imgf000473_0001
Ecouionb ψθ.atom,msp J
For Ecouiomb (MO. atom, msp3 ]< E(yalence) :
Figure imgf000473_0002
(v) the ratio that is less than one of the magnitude of the valence-level energies, En(valence) , of the AO or HO of the nth participating atom of two that are energy matched where Eiyalence) is the ionization energy or E [MO. atom, msp3 J given by Eqs. (15.20) and (15.31-15.32):
C2 = E^aknc^ (15.68)
E2(yalence)
(vi) the factor that is the ratio of the hybridization factor C1 \\ \ of the valence AO or
HO of a first atom and the hybridization factor c2 [2j of the valence AO or HO of a second
atom to which the first is energy matched where c2 (nj is given by Eqs. (15.62-15.68);
alternatively C1 is the hybridization factor C1 (C\ of the valence AOs or HOs a first pair of
atoms and the hybridization factor c2 {2j of the valence AO or HO a third atom or second pair to which the first two are energy matched:
C2 v) (15 69)
2 c2 (2) { }
(vii) the factor that is the product of the hybridization factor c2 π. J of the valence AO
or HO of a first atom and the hybridization factor c2 \2 Λ of the valence AO or HO of a second
atom to which the first is energy matched where c2 (nλ is given by Eqs. (15.62-15.69); alternatively C2 is the hybridization factor c2 (C\ of the valence AOs or HOs a first pair of
atoms and the hybridization factor α, (2) of the valence AO or HO a third atom or second pair to which the first two are energy matched:
C2 = c2 (l>2 (2) (15.70) The hybridization factor c2 corresponds to the force constant k (Eqs. (11.65) and (13.58)). In the case that the valence or Coulombic energy of the AO or HO is less than 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243), then C2 corresponding to Jfc1 (Eq. (15.1)) is given by Eqs. (15.62-15.70).
Specific examples of the factors c2 and C2 of a H2 -type ellipsoidal MO of Eq. (15.51) given in following sections are
Figure imgf000474_0001
Figure imgf000475_0001
Figure imgf000476_0002
MO INTERCEPT ANGLES AND DISTANCES
Consider the general case of Eqs. (13.84-13.95) wherein the nucleus of a B atom and the nucleus of a A atom comprise the foci of each H2 -type ellipsoidal MO of an A - B bond.
The parameters of the point of intersection of each H2 -type ellipsoidal MO and the y4-atom AO are determined from the polar equation of the ellipse:
(15.71)
Figure imgf000476_0003
. V '
The radius of the A shell is rA , and the polar radial coordinate of the ellipse and the radius of the A shell are equal at the point of intersection such that
(15.72)
Figure imgf000476_0004
The polar angle θ' at the intersection point is given by
Figure imgf000476_0001
Then, the angle ΘA AO the radial vector of the A AO makes with the internuclear axis is 0^0 = 180°- 0' (15.74)
The distance from the point of intersection of the orbitals to the internuclear axis must be the same for both component orbitals such that the angle ωt = ΘH MQ between the internuclear
axis and the point of intersection of each H2 -type ellipsoidal MO with the A radial vector obeys the following relationship: rJκθA AO ÷ bήκθHiUO (15.75) such that
(15.76)
Figure imgf000476_0005
The distance dH M0 along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by dH2Mo = a cos θH1Mo (15.77)
The distance dA Λo along the internuclear axis from the origin of the A atom to the point of intersection of the orbitals is given by
BOND ANGLES
Further consider an ACB MO comprising a linear combination of C - A -bond and C - B - bond MOs where C is the general central atom. A bond is also possible between the A and B atoms of the C - A and C - B bonds. Such A - B bonding would decrease the C - A and C - B bond strengths since electron density would be shifted from the latter bonds to the former bond. Thus, the ZACB bond angle is determined by the condition that the total energy of the H2 -type ellipsoidal MO between the terminal A and B atoms is zero. The force constant k' of a H2 -type ellipsoidal MO due to the equivalent of two point charges of at the foci is given by:
Figure imgf000477_0001
where C1 is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the molecule which is 0.75 (Eq. (13.59)) for a terminal A- H {A is H or other atom) and 1 otherwise and C2 is the factor that results in an equipotential energy match of the participating at least two atomic orbitals of the chemical bond and is equal to the corresponding factor of Eqs. (15.42) and (15.52). The distance from the origin of the MO to each focus c' of the A- B ellipsoidal MO is given by:
Figure imgf000477_0002
The internuclear distance is
(15.81)
Figure imgf000477_0003
The length of the semiminor axis of the prolate spheroidal A- B MO b = c is given by Eq.
(15.4).
The component energies and the total energy, E1, (H2MO), of the A - B bond are given by the energy equations (Eqs. (11.207-11.212), (11.213-11.217), and (11.239)) of H2 except that the terms based on charge are multiplied by cB0 , the bond-order factor which is 1 for a single bond and when the MO comprises nx equivalent single bonds as in the case of functional groups. cB0 is 4 for an independent double bond as in the case of the CO2 and
NO2 molecules. The kinetic energy term is multiplied by c'B0 which is 1 for a single bond, 2 for a double bond, and 9/2 for a triple bond. The electron energy terms are multiplied by C1 , the fraction of the H2 -type ellipsoidal MO basis function of a terminal chemical bond which is 0.75 (Eq. (13.233)) for a terminal A - H {A is H or other atom) and 1 otherwise. The electron energy terms are further multiplied by c2' , the hybridization or energy-matching factor that results in an equipotential energy match of the participating at least two atomic orbitals of each terminal bond. Furthermore, when A - B comprises atoms other than H, E1, {atom - atom, msp3.AOJ, the energy component due to the AO or HO ' s charge donation to the terminal MO, is added to the other energy terms to give E1. (H1Mo):
Figure imgf000478_0001
(15.82)
The radiation reaction force in the case of the vibration of A - B in the transition state corresponds to the Doppler energy, ED , given by Eq. (11.181) that is dependent on the motion of the electrons and the nuclei. The total energy that includes the radiation reaction of the A - B MO is given by the sum of ET (H2MO) (Eq. (15.82)) and Eosc given Eqs. (11.213-
11.220), (11.231-11.236), and (11.239-11.240). Thus, the total energy E1. (A- B) of the A - B MO including the Doppler term is
Figure imgf000479_0001
(15.83) where C10 is the fraction of the H2 -type ellipsoidal MO basis function of the oscillatory transition state of the A - B bond which is 0.75 (Eq. (13.233)) in the case of H bonding to a central atom and 1 otherwise, C2o is the factor that results in an equipotential energy match of the participating at least two atomic orbitals of the transition state of the chemical bond, and
TYl T)I μ = — — ^- is the reduced mass of the nuclei given by Eq. (11.154). To match the boundary Tn1 + W2 condition that the total energy of the A - B ellipsoidal MO is zero, E7, (A - Bj given by Eq. (15.83) is set equal to zero. Substitution of Eq. (15.81) into Eq. (15.83) gives
Figure imgf000479_0002
(15.84)
The vibrational energy-term of Eq. (15.84) is determined by the forces between the central field and the electrons and those between the nuclei (Eqs. (11.141-11.145)). The electron- central-field force and its derivative are given by
Figure imgf000479_0003
and c c'e
/'(α)= 2c (15.86)
BO 4πεna3
The nuclear repulsion force and its derivative are given by
Figure imgf000480_0001
and
Figure imgf000480_0002
such that the angular frequency of the oscillation is given by
Figure imgf000480_0003
Since both terms of Eo∞ = ED + EKvjb are small due to the large values of a and c\ to very good approximation, a convenient form of Eq. (15.84) which is evaluated to determine the bond angles of functional groups is given by
Figure imgf000480_0004
From the energy relationship given by Eq. (15.90) and the relationship between the axes given by Eqs. (15.2-15.5), the dimensions of the A- B MO can be solved. The most convenient way to solve Eq. (15.90) is by the reiterative technique using a computer. A factor c2 of a given atom in the determination of c2' for calculating the zero of the total A - B bond energy is typically given by Eqs. (15.62-15.65). In the case of a H- H terminal bond of an alkyl or alkenyl group, c2' is typically the ratio of c2 of Eq. (15.62) for the H- H bond which is one and c2 of the carbon of the corresponding C- H bond:
Figure imgf000481_0007
In the case of the determination of the bond angle of the ACH MO comprising a linear combination of C - J -bond and C - H -bond MOs where A and C are general, C is the central atom, and c2 for an atom is given by Eqs. (15.62-15.70), c2 of the A - H terminal bond is typically the ratio of c2 of the A atom for the A - H terminal bond and c2 of the C atom of the corresponding C - H bond:
Figure imgf000481_0001
In the case of the determination of the bond angle of the COH" MO of an alcohol comprising a linear combination of C - O -bond and O- H -bond MOs where C , O , and H are carbon, oxygen, and hydrogen, respectively, c2' of the C - H terminal bond is typically 0.91771 since the oxygen and hydrogen atoms are at the Coulomb potential of a proton and an electron (Eqs. (1.236) and (10.162), respectively) that is energy matched to the C2sp3 HO.
In the determination of the hybridization factor c2' of Eq. (15.90) from Eqs. (15.62-
15.70), the Coulombic energy, ECoulomb or the energy,
Figure imgf000481_0003
Figure imgf000481_0002
the radius r Λ-B AorBsp ,3 of the A or B AO or HO of the heteroatom of the A- B terminal bond
MO such as the C2sp3 HO of a terminal C - C bond is calculated using Eq. (15.32) by considering the total energy donation to each bond with which it
Figure imgf000481_0004
participates in bonding as it forms the terminal bond. The Coulombic energy
Figure imgf000481_0005
of the outer electron of the atom msp3 shell is given by Eq. (15.19). In the case that during hybridization, at least one of the spin-paired AO electrons is unpaired in the hybridized orbital (HO), the energy change for the promotion to the unpaired state is the magnetic energy E(magnetic) (Eq. (15.15)) at the initial radius r of the AO electron.
Then, the energy E{MO.atom,msp3 J of the outer electron of the atom msp3 shell is given by
the sum of ECoulomb E(magmtic) (Eq. (15.20)).
Figure imgf000481_0006
In the specific case of the terminal bonding of two carbon atoms, the c2 factor of each carbon given by Eq. (15.62) is determined using the Coulombic energy Ecouiomb (c - C C2sp3^ of the outer electron of the C2sp3 shell given by Eq. (15.19) with the radius rc_cc2^ of each C2sp3 HO of the terminal C-C bond calculated using Eq. (15.32)
by considering ∑ET the total energy donation to each bond with which it
Figure imgf000482_0001
participates in bonding as it forms the terminal bond including the contribution of the methylene energy, 0.92918 eV (Eq. (14.513)), corresponding to the terminal C-C bond. The corresponding
Figure imgf000482_0002
Eq. (15.90) is
ET(C-C C2sp3)= -1.85836 eV .
In the case that the terminal atoms are carbon or other heteroatoms, the terminal bond comprises a linear combination of the HOs or AOs; thus, c2' is the average of the hybridization factors of the participating atoms corresponding to the normalized linear sum:
c2' = - (c2' (atom l)+ c2' (atom 2)) (15.93)
In the exemplary cases of C-C , O- O, and N-N where C is carbon:
Figure imgf000482_0003
In the exemplary cases of C-N , C-O, and C-S ,
Figure imgf000482_0004
where C is carbon and c2 (c to B^ is the hybridization factor of Eqs. (15.52) and (15.84) that matches the energy of the atom B to that of the atom C in the group. For these cases, the corresponding Eτ\atom- atom,msp~ '.AOJ term in Eq. (15.90) depends on the hybridization and bond order of the terminal atoms in the molecule, but typical values matching those used in the determination of the bond energies (Eq. (15.56)) are ET(C-O -1.65376 eV ;
Figure imgf000482_0005
ET(C-N C2sp\N2p)= -1.44915 eV ; Eτ(c-S C2sp\S2p)= -0.72457 eV ;
Eτ(θ-O O2p.O2p)= -1.44915 eV ; Eτ(θ-O O2p.O2p)= -1.65376 eV ; ET (N - N N2p.N2py -1.44915 eV , ET (N - 0 N2p.O2p)= -1.44915 eV ;
Ex (F - F F2p.F2p)= -1.44915 eV ; Eτ (CI - Cl Cl3p.CI3p)= -0.92918 eV ;
E1. (Br - Br BrAp.Br<\p)= -0.92918 eV ; E7 (i - 1 I5p.I5p)= -0.36229 eV ;
Eτ (c - F C2sp\F2p)= -1.85836 eV ; Eτ (c- Cl C2sp\CBp)= -0.92918 eV ;
Eτ (c - Br C2sp\Br 4p)= -0.72457 eV ; Eτ (c - I C2sp315p)= -0.36228 eV , and Eτ (O - Cl O2p.CBp)= -0.92918 eV .
In the case that the terminal bond is X - X where X is a halogen atom, C1 is one, and c2' is the average (Eq. (15.93)) of the hybridization factors of the participating halogen atoms given by Eqs. (15.62-15.63) where ECouhmb (MO.atom,msp3 J is determined using Eq.
(15.32) and ECoulomb (MO.atom,msp^= 13.605804 eV for X = I. The factor C1 of Eq.
(15.90) is one for all halogen atoms. The factor C2 of fluorine is one since it is the only halogen wherein the ionization energy is greater than that 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243). For each of the other halogens, Cl, Br , and /, C2 is the hybridization factor of Eq. (15.52) given by Eq. (15.70) with c2 (l) being that of the halogen given by Eq. (15.68) that matches the
valence energy of X ( E1 (valence)) to that of the C2sp3 HO (E2 (valence)= -14.63489 eV ,
Eq. (15.25)) and to the hybridization of C2,sp3 HO (c2 (2)= 0.91771 , Eq. (13.430)). Eτ (atom- atom, msp" .AO 1J of Eq. (15.90) is the maximum for the participating atoms which is -1.44915 eV , -0.929IS eF 5 -0.92918 eV , and -0.33582 eV for F, Cl, Br , and /, respectively.
Consider the case that the terminal bond is C - X where C is a carbon atom and X is a halogen atom. The factors C1 and C1 of Eq. (15.90) are one for all halogen atoms. For
X = F , c2' is the average (Eq. (15.95)) of the hybridization factors of the participating carbon and F atoms where C2 for carbon is given by Eq. (15.62) and c2 for fluorine matched to carbon is given by Eq. (15.70) with c2 (JJ for the fluorine atom given by Eq. (15.68) that
matches the valence energy of F (Eχ (valence}= -17.42282 eV ) to that of the C2sp3 HO (E2 (valence)= -14.63489 eV, Eq. (15.25)) and to the hybridization of C2sp3 HO
(c2 (2)= 0.91771 , Eq. (13.430)). The factor C2 of fluorine is one since it is the only halogen wherein the ionization energy is greater than that 13.605804 eV , the magnitude of the
Coulombic energy between the electron and proton of H given by Eq. (1.243). For each of the other halogens, Cl, Br , and I, c\ is the hybridization factor of the participating carbon atom since the halogen atom is energy matched to the carbon atom. C2 of the terminal-atom bond matches that used to determine the energies of the corresponding C - X -bond MO. Then, C2 is the hybridization factor of Eq. (15.52) given by Eq. (15.70) with c2 (l) for the
halogen atom given by Eq. (15.68) that matches the valence energy of X {Eλ (valence \) to
that of the C2sp3 HO ( E2 (valence)= -14.63489 eV , Eq. (15.25)) and to the hybridization of
C2sp3 HO (c2 (2)= 0.91771 , Eq. (13.430)). Eτ (atom- atom,msp\Aθ) of Eq. (15.90) is the maximum for the participating atoms which is -1.85836 eV , -0.92918 eV , -0.72457 eV , and -0.33582 eV for F , Cl , Br , and /, respectively.
Consider the case that the terminal bond is H - X corresponding to the angle of the atoms HCX where C is a carbon atom and X is a halogen atom. The factors C1 and C1 of
Eq. (15.90) are 0.75 for all halogen atoms. For X = F , c2' is given by Eq. (15.69) with c2 of the participating carbon and F atoms given by Eq. (15.62) and Eq. (15.65), respectively. The factor C2 of fluorine is one since it is the only halogen wherein the ionization energy is greater than that 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243). For each of the other halogens, Cl , Br , and /, c2 is also given by Eq. (15.69) with c2 of the participating carbon given by Eq. (15.62) and c2 of the participating X atom given by c2 = 0.91771 (Eq. (13.430)) since the X atom is energy matched to the Clsp^ HO. hi these cases, C2 is given by Eq. (15.65) for the corresponding atom X where C2 matches the energy of the atom X to that of H . Using the distance between the two atoms A and B of the general molecular group ACB when the total energy of the corresponding A - B MO is zero, the corresponding bond angle can be determined from the law of cosines:
S1 2 + S2 2 - Is^cosineθ = S3 2 (15.96) With S1 = 2cc' _A , the internuclear distance of the C- A bond, s2 = 2cc'_B , the internuclear distance of each C - B bond, and s3 = 2c'A_B , the internuclear distance of the two terminal atoms, the bond angle G^08 between the C - A and C - B bonds is given by
Figure imgf000485_0001
(15>98)
Figure imgf000485_0002
Consider the exemplary structure CbCa(Oa)Ob wherein Ca is bound to C4, O0 , and O0. In the general case that the three bonds are coplanar and two of the angles are known, say θλ and Q2 , then the third θ3 can be determined geometrically:
#3 = 360 - θx - θ2 (15.99) In the general case that two of the three coplanar bonds are equivalent and one of the angles is known, say ^1 , then the second and third can be determined geometrically. (15,00)
Figure imgf000485_0004
ANGLES AND DISTANCES FOR AN MO THAT FORMS AN ISOSCELES TRIANGLE
In the general case where the group comprises three A - B bonds having B as the central atom at the apex of a pyramidal structure formed by the three bonds with the A atoms at the base in the xy-plane. The C3v axis centered on B is defined as the vertical or z-axis, and any two A - B bonds form an isosceles triangle. Then, the angle of the bonds and the distances from and along the z-axis are determined from the geometrical relationships given by Eqs. (13.412-13.416): the distance d ori .gin— B D from the orig °in to the nucleus of a terminal B atom is g ^iven by *
(15.101)
Figure imgf000485_0005
the height along the z-axis from the origin to the A nucleus dhejghl is given by
(15.102)
Figure imgf000485_0003
the angle θv of each A- B bond from the z-axis is given by
Figure imgf000486_0001
Consider the case where the central atom B is further bound to a fourth atom C and the B- C bond is along the z-axis. Then, the bond ΘMBC given by Eq. (14.206) is θZABC = nQ - θv (15.104)
DIHEDRAL ANGLE
Consider the plane defined by a general ACA MO comprising a linear combination of two
C- A -bond MOs where C is the central atom. The dihedral angle ΘZBCIACA between the ACA -plane and a line defined by a third bond with C , specifically that corresponding to a
C - B -bond MO, is calculated from the bond angle 0^01 and the distances between the A ,
B , and C atoms. The distance dχ along the bisector of θ^^ from C to the internuclear- distance line between A and A , 2cf A_A , is given by
dx = 2c'c_Λcos-^- (15.105)
where 2c y C_A is the internuclear distance between A and C . The atoms A , A , and 5 define the base of a pyramid. Then, the pyramidal angle ΘZABA can be solved from the internuclear distances between A and A , 2c 'A_A , and between A and 5, 2c '^ 5 , using the law of cosines (Eq. (15.98)):
Figure imgf000486_0002
Then, the distance d2 along the bisector of Q/ΛBA from B to the internuclear-distance line 2c'A_A , is given by
Figure imgf000486_0003
The lengths dχ , dz , and 2c'c_B define a triangle wherein the angle between dχ and the internuclear distance between B and C, 2c'C β 5 is the dihedral angle ^ZBCMω that can be solved using the law of cosines (Eq. (15.98)): (15.108)
Figure imgf000487_0001
SOLUTION OF GEOMETRICAL AND ENERGY PARAMETERS OF MAJOR FUNCTIONAL GROUPS AND CORRESPONDING ORGANIC MOLECULES
The exemplary molecules given in the following sections were solved using the solutions of organic chemical functional groups as basis elements wherein the structures and energies where linearly added to achieve the molecular solutions. Each functional group can be treated as a building block to form any desired molecular solution from the corresponding linear combination. Each functional group element was solved using the atomic orbital and hybrid orbital spherical orbitsphere solutions bridged by molecular orbitals comprised of the H2 - type prolate spheroidal solution given in the Nature of the Chemical Bond of Hydrogen-Type Molecules section. The energy of each MO was matched at the HO or AO by matching the hybridization and total energy of the MO to the AOs and HOs. The energy Emag (e.g. given
by Eq. (15.58)) for a C2sp3 HO and Eq.(15.59) for an O2p AO) was subtracted for each set of unpaired electrons created by bond breakage.
The bond energy is not equal to the component energy of each bond as it exists in the molecule; although, they are close. The total energy of each group is its contribution to the total energy of the molecule as a whole. The determination of the bond energies for the creation of the separate parts must take into account the energy of the formation of any radicals and any redistribution of charge density within the pieces and the corresponding energy change with bond cleavage. Also, the vibrational energy in the transition state is dependent on the other groups that are bound to a given functional group. This will effect the functional-group energy. But, because the variations in the energy based on the balance of the molecular composition are typically of the order of a few hundreds of electron volts at most, they were neglected.
The energy of each functional-group MO bonding to a given carbon HO is independently matched to the HO by subtracting the contribution to the change in the energy of the HO from the total MO energy given by the sum of the MO contributions and E(C,2SP^= -14.63489 eV (Eq. (13.428)). The intercept angles are determined from Eqs. (15.71-15.78) using the final radius of the HO of each atom. The final carbon-atom radius is determined using Eqs. (15.32) wherein the sum of the energy contributions of each atom to all the MOs in which it participates in bonding is determined. This final radius is used in Eqs. (15.19) and (15.20) to calculate the final valence energy of the HO of each atom at the corresponding final radius. The radius of any bonding heteroatom that contributes to a MO is calculated in the same manner, and the energy of its outermost shell is matched to that of the MO by the hybridization factor between the carbon-HO energy and the energy of the heteroatomic shell. The donation of electron density to the AOs and HOs reduces the energy. The donation of the electron density to the MO's at each AO or HO is that which causes the resulting energy to be divided equally between the participating AOs or HOs to achieve energy matching.
The molecular solutions can be used to design synthetic pathways and predict product yields based on equilibrium constants calculated from the heats of formation. New stable compositions of matter can be predicted as well as the structures of combinatorial chemistry reactions. Further important pharmaceutical applications include the ability to graphically or computationally render the structures of drugs that permit the identification of the biologically active parts of the molecules to be identified from the common spatial charge-density functions of a series of active molecules. Drugs can be designed according to geometrical parameters and bonding interactions with the data of the structure of the active site of the drug.
To calculate conformations, folding, and physical properties, the exact solutions of the charge distributions in any given molecule are used to calculate the fields, and from the fields, the interactions between groups of the same molecule or between groups on different molecules are calculated wherein the interactions are distance and relative orientation dependent. The fields and interactions can be determined using a finite-element-analysis approach of Maxwell's equations. AROMATIC AND HETEROCYCLIC COMPOUMDS
Aromatic and heterocyclic molecules comprise at least one of an aromatic or a cyclic conjugated alkene functional group. The latter was described in the Cyclic and Conjugated
Alkenes section. The aromatic bond is uniquely stable and requires the sharing of the electrons of multiple H2 -type MOs. The results of the derivation of the parameters of the benzene molecule given in the Benzene Molecule (C6H6) section can be generalized to any aromatic function group(s) of aromatic and heterocyclic compounds.
C6H6 can be considered a linear combination of three ethylene molecules wherein a
C - H bond of each CH2 group of H2C = CH2 is replaced by a C = C bond to form a six- member ring of carbon atoms. The solution of the ethylene molecule is given in the Ethylene
Molecule (CH. CH,) section. The radius r , (0.85252αn) of the C2sp3 shell of ethylene calculated from the Coulombic energy is given by Eq. (14.244). The Coulombic energy Ecouiomb
Figure imgf000489_0001
(-15.95955 eV ) of the outer electron of the C2,sp3 shell is given by Eq.
(14.245). The energy E (Celhylem,2sp3λ (-15.76868 eV ) of the outer electron of the C2sp3 shell is given by Eq. (14.246). Eτ (c = C,2ap3) (-1.13380 eV ) (Eq. (14.247), the energy change of each C2sp3 shell with the formation of the C = C -bond MO is given by the difference between
E\Ceth lene,2sp3 J and E\C,2sp3 Y C6H6 can be solved using the same principles as those used to solve ethylene wherein the 2s and 2p shells of each C hybridize to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between two C2sp3 hybridized orbitals (ΗOs) to form a molecular orbital (MO) permits each participating hybridized orbital to decrease in radius and energy. Each 2sp3 HO of each carbon atom initially has four unpaired electrons. Thus, the 6 H atomic orbitals (AOs) of benzene contribute six electrons and the six sp3 -hybridized carbon atoms contribute twenty-four electrons to form six C- H bonds and six
C = C bonds. Each C- H bond has two paired electrons with one donated from the H AO and the other from the C2sp3 HO. Each C = C bond comprises a linear combination of a factor of 0.75 of four paired electrons (three electrons) from two sets of two C2,sp3 HOs of the participating carbon atoms. Each C - H and each C = C bond comprises a linear combination of one and two diatomic H2 -type MOs developed in the Nature of the Chemical Bond of Ηydrogen-Type Molecules and Molecular Ions section, respectively. Consider the case where three sets of C = C -bond MOs form bonds between the two carbon atoms of each molecule to form a six-member ring such that the six resulting bonds comprise eighteen paired electrons. Each bond comprises a linear combination of two MOs wherein each comprises two C2sρ3 ΗOs and 75% of a H2 -type ellipsoidal MO divided between the C2sp3 ΗOs:
Figure imgf000489_0002
The linear combination of each H2 -type ellipsoidal MO with each C2sp3 HO further comprises an excess 25% charge-density contribution per bond from each C2sp3 HO to the C = C -bond MO to achieve an energy i minimum. Thus, the dimensional parameters of each bond C = C- bond are determined using Eqs. (15.42) and (15.1-15.5) in a form that are the same equations as those used to determine the same parameters of the C = C -bond MO of ethylene (Eqs. (14.242- 14.268)) while matching the boundary conditions of the structure of benzene.
Hybridization with 25% electron donation to each C = C -bond gives rise to the
C irø V HO-shell Coulombic energy ECmhmb (cbm,2sp3) given by Eq. (14.245). To meet the equipotential condition of the union of the six C2sp3 HOs, C2 and C2 of Eq. (15.42) for the
3e aromatic C=C -bond MO is given by Eq. (15.62) as the ratio of 15.95955 eV , the magnitude of E couhmb ^ umem^P^) (Ecl- (14-245))> md 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H (Eq. (1.243)):
Q (benzeneC2sp3HO)= c, (benzeneC2sp3Hθ)= 13-605804 eV = 0.85252 (15.143) 2 V ' 2 V ' 15.95955 eV
The energies of each C-C bond of benzene are also determined using the same equations as those of ethylene (Eqs. (14.251-14.253) and (14.319-14.333) with the parameters of
3e benzene. Ethylene serves as a basis element for the C=C bonding of benzene wherein each of the six C=C bonds of benzene comprises (θ.75)(4)= 3 electrons according to Eq. (15.142).
3e
The total energy of the bonds of the eighteen electrons of the C=C bonds of benzene,
Eτ (c6H6,C=cJ, is given by (6)(θ.75) times Er+røc(c = c) (Eq. (14.492)), the total energy of the C 3=eC -bond MO of benzene including the Doppler term, minus eighteen times E f[C,2sp3 \)
3e
(Eq. (14.146)), the initial energy of each C2sp3 HO of each C that forms the C=C bonds of
3e bond order two. Thus, the total energy of the six C=C bonds of benzene with three electron per aromatic bond given by Eq. (14.493) is
(15.144)
Figure imgf000490_0001
= -33.83284 eV The results of benzene can be generalized to the class of aromatic and heterocyclic compounds.
E1n, of an aromatic bond is given by E1, (H2) (Eqs. (11.212) and (14.486)), the maximum total energy of each H2 -type MO such that
(15.145)
Figure imgf000490_0002
The factor of 0.75 corresponding to the three electrons per aromatic bond of bond order two given in the Benzene Molecule (C6H6) section modifies Eqs. (15.52-15.56). Multiplication of the total energy given by Eq. (15.55) by fχ - 0.75 with the substitution of Eq. (15.145) gives the total energy of the aromatic bond:
Figure imgf000491_0001
The total bond energy of the aromatic group ED (ρmuP) is the negative difference of the total energy of the group (Eq. (15.146)) and the total energy of the starting species given by the sum
Figure imgf000491_0002
AO I Hθ\ J
Figure imgf000491_0003
Since there are three electrons per aromatic bond, c4 is three times the number of aromatic bonds.
Benzene can also be considered as comprising chemical bonds between six CH radicals wherein each radical comprises a chemical bond between carbon and hydrogen atoms. The solution of the parameters of CH is given in the Hydrogen Carbide (CH) section. Those of the benzene are given in the Benzene Molecule (C6H6 ) section. The energy components of F , V , T , Vm , and E1, are the same as those of the hydrogen carbide radical, except that Eτ (c -1.13379 eV (Eq. (14.247)) is subtracted from Er (CH) of Eq. (13.495) to
Figure imgf000491_0004
match the energy of each C- H -bond MO to the decrease in the energy of the corresponding C2sp3 HO. In the corresponding generalization of the aromatic CH group, the geometrical parameters are determined using Eq. (15.42) and Eqs. (15.1-15.5) with .
Figure imgf000491_0005
The total energy of the benzene C - H -bond MO, Eτ [C - H] , given by Eq. (14.467) is the sum of 0.5E7, [C = C,2sp3 X the energy change of each C2,sp3 shell per single bond due to
3e the decrease in radius with the formation of the corresponding C=C -bond MO (Eq. (14.247)), and E (CH ), the σ MO contribution given by Eq. (14.441). In the corresponding generalization of the aromatic CH group, the energy parameters are determined using Eqs. (15.146-15.147) with Z1 = 1 and Eτ . Thus, the energy
Figure imgf000491_0006
contribution to the single aromatic CH bond is one half that of the C=C double bond
3e contribution. This matches the energies of the CH and C=C aromatic groups, conserves the electron number with the equivalent charge density as that of 5 = lin Eqs. (15.18-15.21), and further gives a minimum energy for the molecule. Breakage of the aromatic C=C bonds to give CH groups creates unpaired electrons in these fragments that corresponds to C3 = I in Eq.
(15.56) with Emag given by Eq. (15.58).
Each of the C - H bonds of benzene comprises two electrons according to Eq. (14.439). From the energy of each C - H bond, -En (12CH ) (Eq. (14.477)), the total energy of the benzene \ S twelve electrons of the six C - H bonds of benzene, ET (C6H6,C - H\ given by Eq. (14.494) is
Eτ (C6H6,C - H)= (^EDb (12CH))= 6(-3.90454 eV)= -23.42724 eV (15.148)
The total bond dissociation energy of benzene, ED [C6H6X given by Eq. (14.495) is the negative
sum (Eq. (14.493)) and E7, (C6H6,C - H) (Eq. (14.494)):
Figure imgf000492_0001
Figure imgf000492_0002
= -((-33.83284 e V )+ (-23.42724 eF)) (15.149)
= 57.2601 eV Using the parameters given in Tables 15.214 and 15.216 in the general equations (Eqs. (15.42), (15.1-15.5), and (15.146-15.147)) reproduces the results for benzene given in the Benzene Molecule (C6H6 ) section as shown in Tables 15.214 and 15.216.
The symbols of the functional groups of aromatics and hertocyclics are given in Table 15.213. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11), (15.17-15.56), and (15.146-15.147)) parameters of aromatics and hertocyclics are given in Tables 15.214, 15.215, and 15.216, respectively. The total energy of benzene given in Table 15.217 was calculated as the sum over the integer multiple of each En φrøap) of Table 15.216 corresponding to functional-group composition of the molecule. The bond angle parameters of benzene determined using Eqs. (15.79-15.108) are given in Table 15.218.
Figure imgf000492_0003
Figure imgf000493_0001
Figure imgf000493_0002
Figure imgf000494_0001
Figure imgf000495_0001
Figure imgf000495_0002
Figure imgf000495_0003
Figure imgf000496_0001
Figure imgf000496_0002
Figure imgf000497_0001
Figure imgf000498_0001
Figure imgf000499_0001
Figure imgf000500_0001
Figure imgf000501_0001
Figure imgf000502_0001
Figure imgf000503_0001
Figure imgf000504_0001
Figure imgf000505_0001
Figure imgf000506_0001
Figure imgf000507_0001
Figure imgf000508_0001
REFERENCES FOR SECTION IV
1. D. R. Lide, CRC Handbook of Chemistry and Physics, 86th Edition, CRC Press, Taylor Sc
Francis, Boca Raton, (2005-6), pp. 9-19 to 9-45. 2. G. Herzberg, Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand Reinhold Company, New York, New York, (1945), p. 344. 3. D. R. Lide, CRC Handbook of Chemistry and Physics, 86th Edition, CRC Press, Taylor &
Francis, Boca Raton, (2005-6), pp. 9-63; 9-64 to 9-76. 4. R. J. Fessenden, J. S. Fessenden, Organic Chemistry, Willard Grant Press. Boston, Massachusetts, (1979), p. 320.
5. cyclohexane at http ://webbook.nist. gov/.
6. G. Herzberg, Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand Reinhold Company, New York, New York, (1945), p. 326.
7. b 1 2-methyl- 1 -propene at http ://webbook.nist. gov/.
8. al 2-methyl- 1 -propene at http://webbook.nist.gov/.
9. 2-butyne at http ://webbook.nist. gov/.
10. trifluoromethane at http://webbook.nist.gov/. 11. fluoroethane at http://webbook.nist. gov/.
12. G. Herzberg, Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Poly atomic Molecules, Krieger Publishing Company, Malabar, FL, (1991), p. 195.
13. chloromethane at http://webbook.nist.gov/.
14. methyl bromide at http ://webbook.nist. gov/. 15. tetrabromomethane at http ://webbook.nist. gov/.
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25. D. Lin-Vien. N. B. Colthup, W. G. Fateley, J. G. Grasselli, The Handbook of Infrared and Raman Frequencies of Organic Molecules, Academic Press, Inc., Harcourt Brace Jovanovich, Boston, (1991), p. 482.
26. acetaldehyde at http ://webbook.nist. gov/. 27. acetone at http ://webbook.nist. go v/.
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30. formic acid at http://webbook.nist. gov/.
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32. methyl formate at http://webbook.nist.gov/.
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Section Vl Software Program
The present invention relates to a system of computing and rendering the nature of at least one specie selected from a group of diatomic molecules having at least one atom that is other than hydrogen, polyatomic molecules, molecular ions, polyatomic molecular ions, or molecular radicals, or any functional group therein, comprising physical, Maxwellian solutions of charge, mass, and current density functions of said specie, said system comprising: processing means for processing physical, Maxwellian equations representing charge, mass, and current density functions of said specie; and an output device in communication with the processing means for displaying said physical, Maxwellian solutions of charge, mass, and current density functions of said specie.
In one embodiment, for example, the system comprises five components: (1) the graphical user interface (GUI); (2) the routine for parsing between an input chemical structure or name and taking the input and activating a routine to call up the parts of the molecule (functional groups), which are used for determining the energies and structure to be rendered; (3) the functional-group data base that has an organization of the theoretical solutions; (4) the rendering engine, which calculates and enables manipulations of the image, such as a three- dimensional model in response to commands, as well as responds to commands for data parameters corresponding to the image such as bond energies and charge distribution and geometrical parameters; and (5) data transfer system for inputting numerical data into or out of the computational components and storage components of the main system. The system further comprises spreadsheets with solutions of the bond parameters with output in any standard spreadsheet format. The system also comprises a data-handling program to transfer data from the spreadsheets into the main program.
The output may be, for example, at least one of graphical, simulation, text, and numerical data. The output may be the calculation of at least one of: (1) a bond distance between two atoms; (2) a bond angle between three of the atoms; (3) a bond energy between two atoms; (4) orbital intercept distances and angles; and (5) charge-density functions of atomic, hybridized, and molecular orbitals, wherein the bond distance, bond angle, and bond energy are calculated from physical solutions of the charge, mass, and current density functions of atoms and atomic ions, which solutions are derived from Maxwell's equations using a constraint that a bound electron(s) does not radiate under acceleration. In other embodiments, the charge, current, energy, and geometrical parameters are output to be inputs to other programs that may be used in further applications. For example, the data of heats of formation may be input to another program to be used to predict stability (existence of compounds) equilibrium constants and to predict synthetic pathways. That is, a novel composition of matter may be discovered by calculating at least one of a bond distance between two of the atoms, a bond angle between three of the atoms, and a bond energy between two of the atoms, orbital intercept distances and angles, charge-density functions of atomic, hybridized, and molecular orbitals, the bond distance, bond angle, and bond energy being calculated from physical solutions of the charge, mass, and current density functions of atoms and atomic ions, which solutions are derived from Maxwell's equations using a constraint that a bound electron(s) does not radiate under acceleration. The charge and current density functions may be used to predict the electric and magnetic fields of the species to determine other properties due to the interaction of the fields between species. These fields and the predictions of field interactions may be computed using Maxwell's equations. In one embodiment, finite-element analysis is used to predict or calculate the interaction and resulting properties, such as the freezing point, boiling point, density, viscosity, and refractive index. Furthermore, the output data can be used to give thermodynamic, spectroscopic, and other properties, aid in drug design and other applications with or without direct visualization. Furthermore, the data can be input into other programs of the system, which calculate thermodynamic and other properties, or performs a simulation, such as a chemical reaction or molecular dynamics.
The output data may be used to predict a composition of matter comprising a plurality of atoms, the improvement comprising a novel property or use discovered by calculation of at least one of a bond distance between two of the atoms, a bond angle between three of the atoms, and a bond energy between two of the atoms, orbital intercept distances and angles, charge-density functions of atomic, hybridized, and molecular orbitals, the bond distance, bond angle, and bond energy being calculated from physical solutions of the charge, mass, and current density functions of atoms and atomic ions, which solutions are derived from Maxwell's equations using a constraint that a bound electron(s) does not radiate under acceleration. The novel property, for example, may be a new pharmaceutical use, or stability at room temperature of a novel arrangement of atoms or ions.
In one embodiment, the output device of the system is a display that displays at least one of visual or graphical media. The display may be at least one of static or dynamic. At least one of vibration, rotation, and translation may be displayed. The displayed information may be used for at least one of modeling reactivity, predicting physical properties, and aiding in drug and material design. The output device may be a monitor, video projector, printer, or one-, two- or three-dimensional rendering device. The displayed information may be used to model other molecules and provides utility to anticipate their reactivity and physical properties. Additionally, data may be output and used in the same and additional applications as the rendered models and representations of the calculated physical solutions. The processing means of the system may be a general-purpose computer. The general-purpose computer may comprise a central processing unit (CPU)5 one or more specialized processors, system memory, a mass storage device such as a magnetic disk, an optical disk, or other storage device, an input means. The input means may comprise a serial port, USB port, microphone input, camera input, keyboard or mouse. The processing means comprises a special purpose computer or other hardware system. The system may comprise computer program products such as computer readable medium having embodied therein program code means. The computer readable media may be any available media which can be accessed by a general purpose or special purpose computer. The computer readable media may comprise, for example, at least one of RAM, ROM, EPROM, CD ROM, DVD or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium that can embody the desired program code means and which can be accessed by a general purpose or special purpose computer. The program code means may comprise executable instructions and data, which cause a general purpose computer or special purpose computer to perform a certain function of a group of functions. Commercial examples of suitable program language includes, for example, C++, C, JAVA, FORTRAN, Python and Assembly Languages, programmed with an algorithm based on the physical solutions, and the computer may be a PC, mainframe, supercomputer, or cluster of computers. Commercial examples of suitable programs include, for example, APIs like OpenGL, DirectX, FOX GUI toolkit, and Qt. This program may be developed to run on at least one of operating systems like Windows XP, Windows 2000, Windows Vista, MAC OS, MAC OS X, Linux, Unix, Mx and other Unix-type operating systems.
Millsian software is designed to render 3-D models of molecules, molecular ions, molecular radicals, functional groups thereof, and related structure and property information and produce useful data output and application of the parameters of these species, wherein the nature of their bound electrons and chemical bonds are solved using Dr. Randell L. Mills' Classical Quantum Mechanics theory described in the reference: R. L. Mills, "The Grand Unified Theory of Classical Quantum Mechanics", June 2006 Edition, Cadmus Professional Communications-Science Press Division, Ephrata, PA, ISBN 0963517171, Library of
Congress Control Number 2005936834; posted at http://www.blacklightpower.com/bookdownload.shtml, which is incorporated by this reference in its entirety. Figure 76 provides a flow chart diagram, which is an example of a software system that can be utilized for this purpose, which example is not intended to limit the scope of the disclosed inventions. The main parts of this exemplary software system illustrated in Figure 76 will now be further explained:
Start: A user can start the program by running an executable program file. That might be done, for example, by double clicking the program icon on a Windows-based operating system, or typing the name of the executable file on the command line and pressing the 'Enter' key on a Linux or Unix operating system. The program initially starts by reading data files located in specific directories. The names of those directories and locations are fixed according to the type and format of the data files. Data files: There are two types of data files used in the software system: functional group data files and molecule data files. Functional group data files contain information about various functional groups. Functional groups are the basic bonding elements or units that each typically comprise an atom, or at least two atoms bound together as found within a molecule (e.g. -Cl , C=C, C=O, CH3). Functional groups typically dictate or define properties and structure of the molecule. Similar functional groups in different molecules typically react in similar ways when subjected to a particular set of reaction conditions. Molecule data files contain information about molecules, molecular ions, and molecular radicals. These data files are processed according to their file formats.
File formats: Millsian software employs two kinds of file formats for storing information about the structure, energies and names of molecules and functional groups: raw- data format and hierarchical format. In a raw-data file, all information is stored as is, below the header describing the type of information. As shown in the Table below, for example, the names of the molecule or functional group are listed below the #NAMES header. The names and positions of the atoms are listed below the #ATOMS header, and so on.
Figure imgf000518_0001
Most of the functional groups files and some simple molecule files are stored in raw data format. The other file format used with the Millsian software system, the hierarchical file format, represents the information in a graph style, in which nodes are connected to other nodes through links. This format is designed to construct molecules by attaching different functional groups in a desired manner. Using this file format, the user can construct complex molecules built from solved functional groups.
For example, as shown in the Table below, the pentane data file includes a #GROUPS header below which is list all of the functional groups that form a part of the pentane molecule. Under the #GROUP_LINKS header is information about how these functional groups are connected to each other to construct pentane.
Figure imgf000519_0001
Processing data files: As further shown in the flow diagram of Figure 76 for the exemplary Millsian software system, the program first processes the functional group data file and constructs the functional-group objects, which are complete 3-D representations of the functional groups and their related information supplied in the corresponding file. These objects are then ready to be visually displayed through use of a molecule viewer. Next, the program processes the molecule data file. If the molecule date file is in raw-data format, then the program makes a molecule object directly from it. If the file is in hierarchical format, then the program calculates geometric parameters from listed functional groups. The program stores all functional-group objects and molecule objects using internal data structures.
Visualization/ User Interactions: As shown in Figures 77 and 78, the molecule viewer displays the functional-group objects and molecule objects and provides basic interaction capabilities with the displayed objects, such as rotating, scaling, and moving the objects. The molecule viewer also provides other visualization options, such as viewing molecules in wire frame mode, viewing coordinate axes, and changing of the transparency and lighting. The user, for example, can also select parts of a molecule for visualization, like a nucleus, atomic orbital, molecular orbital, or bond axis. The viewer also includes a drop down information window, which provides, for example,' related information about molecules, such as bond angles, component functional groups, and total heat of formation. In one embodiment, a user can create new molecules by joining functional groups. For this purpose, the user can select a first functional group. Next, the user can select an open bond from the functional group where the user desires to attach another group. Next, the user can select another or the same functional group, followed by selecting an open bond from the other group. The user can join the two selected functional groups at the selected open bonds by clicking on 'Join Groups'. This method of joining functional groups at open bonds can be repeated to form the desired molecule.
SMILES input: The Simplified Molecular Input Line Entry Specification or SMILES is a specification for unambiguously describing the structure of chemical molecules using short
ASCII strings. Through the user interface, a user can enter SMILES to construct molecules, provided the new molecule comprises functional groups that are in the database. Once a
SMILES is entered, a parser reads in and breaks it down into component functional groups.
The software system then attaches the component functional groups to create the new molecule object, which can then be viewed using the molecule viewer.
While the claimed invention has been described in detail and with reference to specific embodiments thereof, it will be apparent to one of ordinary skill in the art that various changes and modifications can be made to the claimed invention without departing from the spirit and scope thereof.
CONTINUOUS-CHAIN ALKANES ( CnH2n+2, n = 3,4,5...°° )
The continuous-chain alkanes, CnH2n+2 , are the homologous series comprising terminal methyl groups at each end of the chain with n - 2 methylene ( CH2 ) groups in between:
CH3 (CH2)^ CH3 (15.109) CnH2n+2 can be solved using the same principles as those used to solve ethane and ethylene wherein the 2s and 2p shells of each C hybridize to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between two C2sp3 hybridized orbitals (ΗOs) to form a molecular orbital (MO) permits each participating hybridized orbital to decrease in radius and energy. Three H AOs combine with three carbon 2sp3 ΗOs and two H AOs combine with two carbon 2sp3 ΗOs to form each methyl and methylene group, respectively, where each bond comprises a H2 -type MO developed in the Nature of the Chemical Bond of Hydrogen- Type Molecules and Molecular Ions section. The CH3 and CH2 groups bond by forming H2 -type MOs between the remaining C2sp3 ΗOs on the carbons such that each carbon forms four bonds involving its four C2sp3 ΗOs. For the alkyl C- C group, Eτ ( atom— atom,msp3. AO) is -1.85836 eV where both energy contributions are given by Eq. (14.513). It is based on the energy match between the C2sp3 ΗOs of the chain comprising methylene groups and terminal methyl groups.
The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of straight-chain alkanes are given in Tables 15.4, 15.5, and 15.6, respectively. The total energy of each straight-chain alkane given in Table 15.7 was calculated as the sum over the integer multiple of each ED (GWUP) of Table 15.6 corresponding to functional-group composition of the molecule. The bond angle parameters of straight-chain alkanes determined using Eqs. (15.79-15.108) are given in Table 15.8. In this angle table and those given in subsequent sections when c2' is given as the ratio of two values of c2 designated to Atom 1 and Atom 2 and corresponding to ECm!hmhlc of Atom 1 and Atom 2,
Figure imgf000521_0001
Figure imgf000522_0001
Figure imgf000522_0002
Figure imgf000523_0001
Figure imgf000524_0002
Figure imgf000524_0001
BRANCHED ALKANES (CnH2n+2, « = 3,4,5...«»)
The branched-chain alkanes, C11H2n+2 , comprise at least two terminal methyl groups ( CH3 ) at each end of the chain, and may comprise methylene (CH2), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C- C bonds can be identified. The n-alkane C - C bond is the same as that of straight-chain alkanes. In addition, the C - C bonds within isopropyl ((CH3) CH) and t-butyl ((CH3 J3 C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C - C bonds comprise functional groups. The branched-alkane groups are solved using the same principles as those used to solve the methyl and methylene functional groups wherein the 2s and 2p AOs of each
C hybridize to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between two C2sp3 ΗOs to form a MO permits each participating hybridized orbital to decrease in radius and energy. Eτ[atom- atom,mspl .AOj of each C- C-bond MO in Eq. (15.52) due to the charge donation from the C atoms to the MO is -1.85836 eV or -1.44915 eV based on the energy match between the C2sp3 ΗOs corresponding to the energy contributions equivalent to those of methylene, -0.92918 eV (Eq. (14.513), or methyl, -0.72457 e V (Eq. (14.151)), groups, respectively.
The symbols of the functional groups of branched-chain alkanes are given in Table 15.9. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of branched-chain alkanes are given in Tables 15.10, 15.11, and 15.12, respectively. The total energy of each branched-chain alkane given in Table 15.13 was calculated as the sum over the integer multiple of each Eυ[amup) of Table 15.12 corresponding to functional-group composition of the molecule. The bond angle parameters of branched-chain alkanes determined using Eqs. (15.79-15.108) are given in Table 15.14.
Figure imgf000526_0001
Figure imgf000527_0001
Figure imgf000527_0002
Figure imgf000528_0001
Figure imgf000529_0001
Figure imgf000530_0001
ALKENES ( CnH2n, n = 3,4,5...oo )
The straight and branched-chain alkenes, CnH2n , comprise at least one carbon-carbon double bond comprising a functional group that is solved equivalently to the double bond of ethylene. The double bond may be bound to one, two, three, or four carbon single bonds that substitute for the hydrogen atoms of ethylene. Based on the condition of energy matching of the orbital, any magnetic energy due to unpaired electrons in the constituent fragments, and differences in oscillation in the transition state, three distinct functional groups can be identified: C vinyl single bond to -C(C) = C , C vinyl single bond to -C(H) = C , and C vinyl single bond to -C(C) = CH2 . In addition, CH2 of the -C = CH2 moiety is an alkene functional group.
The alkyl portion of the alkene may comprise at least two terminal methyl groups (CH3 ) at each end of the chain, and may comprise methylene (CH2), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C - C bond is the same as that of straight-chain alkanes. In addition, the C- C bonds within isopropyl ((CH3) CH) and t-butyl ([CH3) C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C - C bonds comprise functional groups. The branched-chain-alkane groups in alkenes are equivalent to those in branched-chain alkanes. The solution of the functional groups comprises the hybridization of the 2s and 2p AOs of each C to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between two C2sp3 ΗOs to form a MO permits each participating hybridized orbital to decrease in radius and energy. E1. ( atom- atom, msp1. AO) of the C = C - bond MO in Eq. (15.52) due to the charge donation from the C atoms to the MO is equivalent to that of ethylene, -2.26759 eV , given by Eq. (14.247). E1 [atom - atom,msp3.A(ή of each C- C-bond MO in Eq. (15.52) is -1.85836 eV or -1.44915 eV based on the energy match between the C2sp3 ΗOs corresponding to the energy contributions equivalent to those of methylene, -0.92918 eV (Eq. (14.513), or methyl, -0.72457 eF (Eq. (14.151)), groups, respectively.
The symbols of the functional groups of alkenes are given in Table 15.15. The geometrical (Eqs. (15.1-15.5) and (15.41)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of alkenes are given in Tables 15.16, 15.17, and 15.18, respectively. The total energy of each alkene given in Table 15.19 was calculated as the sum over the integer multiple of each ED (cmψ) of Table 15.18 corresponding to functional- group composition of the molecule. For each set of unpaired electrons created by bond breakage , the C2sp3 HO magnetic energy Emag that is subtracted from the weighted sum of the
ED {Group) (eV) values based on composition is given by Eq. (15.58). The bond angle parameters of alkenes determined using Eqs. (15.79-15.108) are given in Table 15.20.
Figure imgf000533_0001
Figure imgf000534_0001
Figure imgf000535_0001
Figure imgf000536_0001
Figure imgf000537_0001
ALKYNES ( C H2M_2, n = 3,4,5...-)
The straight and branched-chain alkynes, CnH2n_2 , have at least one carbon-carbon triple bond comprising a functional group that is solved equivalently to the triple bond of acetylene. The triple bond may be bound to one or two carbon single bonds that substitute for the hydrogen atoms of acetylene. Based on the energy matching of the mutually bound C , these C - C -bond MOs are defined as primary and secondary C - C functional groups, respectively, that are unique to alkynes. In addition, the corresponding terminal CH of a primary alkyne comprises a functional group that is solved equivalently to the methylyne group of acetylene as given in the Acetylene Molecule section. The alkyl portion of the alkyne may comprise at least two terminal methyl groups ( CH3 ) at each end of the chain, and may comprise methylene (CH2 ), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C - C bonds within isopropyl ((CH3) CH) and t-butyl ((CH3 ) C ) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C - C bonds comprise functional groups. The branched-chain-alkane groups in alkynes are equivalent to those in branched-chain alkanes.
The solution of the functional groups comprises the hybridization of the 2s and 2p AOs of each C to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between two C2sp3 ΗOs to form a MO permits each participating hybridized orbital to decrease in radius and energy. Eτ iatom- atom,msp3.AOJ of the C ≡ C-bond MO in Eq. (15.52) due to the charge donation from the C atoms to the MO is equivalent to that of acetylene, -3.13026 eV , given by Eq. (14.342). E7. [atom- atom,msp3 \Aθ] of each -alkyl-bond MO in Eq. (15.52) is -1.85836 eV or -1.44915 eV based on the energy match between the C2sp3 ΗOs corresponding to the energy contributions equivalent to those of methylene, -0.92918 eV (Eq. (14.513), or methyl, -0.72457 eV (Eq. (14.151)), groups, respectively. For the C- C groups each, comprising a C single bond to C ≡ C , E.J atom- atom, msp3. AOj is
-0.72457 eV based on the energy match between the C2sp3 ΗOs for the mutually bound C of the single and triple bonds. The parameter ω of each group is matched for oscillation in the transition state based on the group being primary or secondary. The symbols of the functional groups of alkynes are given in Table 15.21. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of alkynes are given in Tables 15.22, 15.23, and 15.24, respectively. The total energy of each alkyne given in Table 15.25 is calculated as the sum over the integer multiple of each ED (c,muP) of Table 15.24 corresponding to functional- group composition of the molecule. The bond angle parameters of alkynes determined using Eqs. (15.79-15.108) are given in Table 15.26. Each C of the C ≡ C group can further bond with only one atom, and the bond is linear as a minimum of energy as in the case of acetylene.
Figure imgf000540_0001
Figure imgf000541_0001
Figure imgf000542_0001
Figure imgf000543_0001
ALKYL FLUORIDES ( CnH2tι+2_mFm, n = 1,2,3,4,5...°° m = 1,2,3...«» )
The branched-chain alkyl fluorides, C) E2n^2_n Fm , may comprise at least two terminal methyl groups (CH3 ) at each end of the chain, and may comprise methylene (CH2), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds wherein at least one H is replaced by a fluorine. The C- F bond comprises a functional group for each case of F replacing a H of methane in the series H4_mC— Fm, m = 1,2,3,4, and F replacing a
H of an alkane. The methyl, methylene, methylyne functional groups are equivalent to those of branched-chain alkanes. Six types of C- C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C- C bonds within isopropyl ((CH3)2 CH) and t-butyl ((CH3) C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C- C bonds comprise functional groups that are equivalent to those of branched-chain alkanes.
The solution of the C- F functional groups comprises the hybridization of the 2s and 2p AOs of each C to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between the C2sp3 HO and the F AO to form a molecular orbital (MO) permits each participating orbital to decrease in radius and energy. In alkyl fluorides, the C2sp3 HO has a hybridization factor of 0.91771 (Eq. (13.430)) with a corresponding energy of E[c,2spή = -14.63489 eV (Eq. (15.25)), and the F AO has an energy of
E[F) = -17.42282 eV . To meet the equipotential condition of the union of the C - F H2- type-ellipsoidal-MO with these orbitals, the hybridization factor c2 of Eq. (15.52) for the C - F -bond MO given by Eqs. (15.68) and (15.70) is
= 0.77087 (15.110)
Figure imgf000544_0001
E1. (atom- atom,msp3.Aθ) of the C - F -bond MO in Eq. (15.52) based on the charge donation from F to the MO is determined by the linear combination that results in a energy that is a minimum which does not exceed the energy of the AO of the F atom to which it is energy matched.
The symbols of the functional groups of branched-chain alkyl fluorides are given in Table 15.27. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of branched-chain alkyl fluorides are given in Tables 15.28, 15.29, and 15.30, respectively. The total energy of each branched-chain alkyl fluoride given in Table 15.31 was calculated as the sum over the integer multiple of each ED (omtφ) of Table 15.30 corresponding to functional-group composition of the molecule. For each set of unpaired electrons created by bond breakage, the C2sp3 HO magnetic energy Ema that is subtracted from the weighted sum of the ED (βmιφ) (eV) values based on composition is given by Eq. (15.58). In the case of trifluoromethane, Emg is positive since the term due to the fluorine atoms cancels that of the CH group. The C- C bonds to the CHF group (one H bond to C) were each treated as an iso C- C bond. The C- C bonds to the CF group (no H bonds to C) were each treated as a tert-butyl C - C . Ema was subtracted for each t-butyl group. The bond angle parameters of branched-chain alkyl fluorides determined using Eqs. (15.61-15.70), (15.79-15.108) and (15.110) are given in Table 15.32.
Figure imgf000545_0001
Figure imgf000546_0001
Figure imgf000546_0002
Figure imgf000547_0001
Figure imgf000548_0001
ALKYL CHLORIDES ( CnH2n+2^C/,,, « = 1,2,3,4,5...«. «1 = 1,2,3...«»)
The branched-chain alkyl chlorides, CHH2n+2_mC/m , may comprise at least two terminal methyl groups (CH3 ) at each end of the chain, and may comprise methylene (CH2), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds wherein at least one H is replaced by a chlorine. The C - Cl bond comprises a functional group for each case of Cl replacing a H of methane for the series H^1nC - Cl1n, m = 1,2,3, with the
C - Cl bond of CCl^ comprising another functional group due to the limitation of the minimum energy of Cl matched to that of the C2sp3 HO. In addition, the C - Cl bond due to Cl replacing a H of an alkane is a function group. The methyl, methylene, methylyne functional groups are equivalent to those of branched-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C - C bond is the same as that of straight-chain alkanes. In addition, the C - C bonds within isopropyl ((CH3) CH) and t-butyl ((CH3J3 C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C- C bonds comprise functional groups that are equivalent to those of branched-chain alkanes. The solution of the C - Cl functional groups comprises the hybridization of the 2s and
2 p AOs of each C to form a single 2s/?3 shell as an energy minimum, and the sharing of electrons between the C2sp3 HO and the Cl AO to form a MO permits each participating orbital to decrease in radius and energy. In alkyl chlorides, the energy of chorine is less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). Thus, c2 in Eq. (15.52) is one, and the energy matching condition is determined by the C2 parameter. Then, C2,sp3 HO has a hybridization factor of 0.91771 (Eq. (13.430)) with a corresponding energy of £(c,2ap3) = -14.63489 e-V (Eq. (15.25)), and the Cl AO has an energy of E(Cl) = -12.96764 eV . To meet the equipotential condition of the union of the C- Cl H1 - type-ellipsoidal-MO with these orbitals, the hybridization factor C2 of Eq. (15.52) for the C - C/ -bond MO given by Eqs. (15.68) and (15.70) is
C2 (C2sp3HO to Cl) = ^Cl\, c2 (C2sp3Hθ) = ~n-96164 eV (θ.9177l) = 0.81317 (15.110) Λ ' E{c,2sp") n ' -14.63489 eV K J
The valence energy of the carbon 2p is -11.2603 eV and that of the Cl AO is -12.96764 eV . The energy difference is more than that of 2U1JC - C^Sp3 J given by Eq. (14.151) for a single bond. Thus, Er( atom- atom, msp\ Aθ) of the C- C/ -bond MO in Eq. (15.52) due to the charge donation from the C and CZ atoms to the MO is -1.44915 eV based on the energy match between the C2sp3 HO and the Cl AO corresponding to the energy contributions equivalent to those of methyl groups, -0.72457 eV (Eq. (14.151)). The symbols of the functional groups of branched-chain alkyl chlorides are given in
Table 15.33. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of branched-chain alkyl chlorides are given in Tables 15.34, 15.35, and 15.36, respectively. The total energy of each branched-chain alkyl chloride given in Table 15.37 was calculated as the sum over the integer multiple of each ED {c,mΨ) of Table 15.36 corresponding to functional-group composition of the molecule. For each set of unpaired electrons created by bond breakage, the C2sp3 HO magnetic energy Emg that was subtracted from the weighted sum of the ED(amuP) ieV) values based on composition is given by Eq. (15.58). The C - C bonds to the CHCl group (one H bond to C) were each treated as an iso C - C bond. The C- C bonds to the CCl group (no H bonds to C) were each treated as a tert-butyl C - C . Ema ι was subtracted for each t-butyl group. The bond angle parameters of branched-chain alkyl chlorides determined using Eqs. (15.61-15.70), (15.79-
15.108) and (15.111) are given in Table 15.38.
Figure imgf000550_0001
Figure imgf000551_0001
Figure imgf000551_0002
Figure imgf000552_0001
Figure imgf000553_0001
Figure imgf000554_0001
ALKYL BROMIDES (CnH2n+2_m.5rm, « = 1,2,3,4,5...°° m = 1,2,3...°° )
The branched-chain alkyl bromides, CnHZn+2_mBrm, may comprise at least two terminal methyl groups (CH3) at each end of the chain, and may comprise methylene (CH2), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds wherein at least one H is replaced by a bromine. The C - Br bond comprises a functional group for each case of Br replacing a H of methane for the series HA_JC— Brm, m- 1,2,3, with the
C - Br bond of CBr4 comprising another functional group due to the limitation of the minimum energy of Br matched to that of the C2^p3 HO. In addition, the C - Br bond due to Br replacing a H of an alkane is a function group. The methyl, methylene, methylyne functional groups are equivalent to those of branched-chain alkanes. Six types of C- C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C- C bonds within isopropyl ((CH3) CH) and t-butyl ((CH3) C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C- C bonds comprise functional groups that are equivalent to those of branched-chain alkanes. The solution of the C - Br functional groups comprises the hybridization of the 2s and
2p shells of each C to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between the C2,yp3 hybridized orbital (HO) and the Br AO to form a molecular orbital (MO) permits each participating orbital to decrease in radius and energy. In alkyl bromides, the energy of bromine is less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). Thus, c2 in Eq. (15.52) is one, and the energy matching condition is determined by the C2 parameter. Then, the C2sp3 HO has a hybridization factor of 0.91771 (Eq. (13.430)) with a corresponding energy of E(c,2sp3) = -14.63489 eV (Eq. (15.25)), and the Br AO has an energy of £(£r) = -11.81381 eV . To meet the equipotential condition of the union of the C - Br H2-type-ellipsoidal-MO with these orbitals, the hybridization factor C2 of Eq. (15.52) for the C - .Sr -bond MO given by Eqs. (15.68) and (15.70) is
C2 (C2sp3HO (15.112) n y
Figure imgf000555_0001
J The valence energy of the carbon 2p is -11.2603 eK and that of the Br AO is -11.81381 eV . The energy difference is less than that of EAC - C^spΛ given by Eq. (14.151) for a single
bond. Thus, Eτ [atom- atom,msp2 \A(ή of the alkyl C - Br -bond MO in Eq. (15.52) due to the charge donation from the C and Br atoms to the MO is -0.92918 eV (Eq. (14.513) based on the maximum single-bond-energy contribution of the Clsp1 HO. Er I atom - atom,msp3. AOj of the series CBr1nH^111 m = 1,2,3 is equivalent to those of methyl groups, -0.72457 eV (Eq. (14.151)). For CBr4, Eτ [atom- atom,mspi .Aθ) of the C - Br -bond MO in Eq. (15.52) due to the charge donation from the C and Br atoms to the MO is -0.36229 eV (Eqs. (15.18-15.20 and Eq, (15.29) with a linear combination of s = 1 , Eτ {atom- atom,msp3.AOj = -0.72457 eV
and .E7, 1 atom - atom,msp3.AOj = 0) based on the maximum charge density on the C2.jp3 HO.
The symbols of the functional groups of branched-chain alkyl bromides are given in Table 15.39. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of branched-chain alkyl bromides are given in Tables 15.40, 15.41, and 15.42, respectively. The total energy of each branched-chain alkyl bromide given in Table 15.43 was calculated as the sum over the integer multiple of each E D (aroup) of Table 15.42 corresponding to functional-group composition of the molecule. For each set of unpaired electrons created by bond breakage, the C2sp3 HO magnetic energy E that was subtracted from the weighted sum of the ED(a,»uP) (eV) values based on composition is given by Eq. (15.58). The C- C bonds to the CHBr group (one H bond to C) were each treated as an iso C - C bond. The C - C bonds to the CBr group (no H bonds to C) were each treated as a tert-butyl C- C . E is subtracted for each t-butyl group. In the case of 2,3- dibromo-2-methylbutane, Emg is positive since the terms due to the two bromine atoms cancel that of the t-butyl and CH groups. The bond angle parameters of branched-chain alkyl bromides determined using Eqs. (15.61-15.70), (15.79-15.108) and (15.112) are given in Table 15.44.
Figure imgf000557_0001
Figure imgf000558_0001
Figure imgf000558_0002
Figure imgf000559_0001
Figure imgf000560_0001
ALKYL IODIDES (C H2n+2.,, /,„, n = 1,2,3, 4,5...°° w = 1,2,3...«.)
The branched-chain alkyl iodides, CHH2n+2_mIm , may comprise at least two terminal methyl groups (CH3 ) at each end of the chain, and may comprise methylene (CH2 ), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds wherein at least one H is replaced by an iodine atom. The C- I bond comprises a functional group for
/ replacing a H of methane (CH3/) or for / replacing a H of an alkane corresponding to the series CnH2n+2_m/m . The C- I bond of each of CH2Z2 and CHI3 comprise separate functional groups due to the limitation of the minimum energy of I matched to that of the C2sp3 HO. The methyl, methylene, methylyne functional groups are equivalent to those of branched-chain alkanes. Six types of C- C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C - C bonds within isopropyl ([CH3J CH) and t-butyl ((CH3) C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t- butyl C- C bonds comprise functional groups that are equivalent to those of branched-chain alkanes. The solution of the C- I functional groups comprises the hybridization of the 2s and
2p AOs of each C to form a single 2sp* shell as an energy minimum, and the sharing of electrons between the C2sp3 HO and the / AO to form a MO permits each participating orbital to decrease in radius and energy. In alkyl iodides, the energy of iodine is less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). Thus, c2 in Eq. (15.52) is one, and the energy matching condition is determined by the C2 parameter. Then, the C2spi HO has a hybridization factor of 0.91771 (Eq. (13.430)) with a corresponding energy of E(c,2sp3) = -14.63489 eV (Eq. (15.25)). The / AO has an energy of E(I) = -10.45126 eV . To meet the equipotential condition of the union of the C - 1 H2 -type-ellipsoidal-MO with these orbitals, the hybridization factor C2 of Eq. (15.51) for the C - I -bond MO given by Eqs. (15.68) and (15.70) is
C2 (Clsp'HO to I) = ,£(/) C2 (C2Sp3Hθ) = ~10-45126 eV (0.9177l) = 0.65537 (15.113) 2 V ' E(c,2sp3) V ' -14.63489 eV y ' V '
The valence energy of the carbon 2p is -11.2603 eV and that of the / AO is -10.45126 eV . The energy difference is positive. Thus, based on the maximum charge density on the C2sp3 HO Eτ [atom- atom,msp% ΛO) of the C - /-bond MO in Eq. (15.52) due to the charge donation from the C and / atoms to the MO is -0.36229 eV (Eqs. (15.18-15.20 and Eq. (15.29) with a linear combination of s ~ l , Er( atom- atom,msp*. Aθ) = -0.72457 eV and
Eτ [atom - atom,msp3.Aθ) = 0) for methyl and alkyl iodides, -0.18114 eV for diiodomethane, and 0 for CHZ3 .
The symbols of the functional groups of branched-chain alkyl iodides are given in Table 15.45. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of branched-chain alkyl iodides are given in Tables 15.46, 15.47, and 15.48, respectively. The total energy of each branched-chain alkyl iodide given in Table 15.49 was calculated as the sum over the integer multiple of each ED (cwuP) of Table 15.48 corresponding to functional-group composition of the molecule. For each set of unpaired electrons created by bond breakage, the C2sp* HO magnetic energy Ema that was subtracted from the weighted sum of the ED [cm,P) (eV) values based on composition is given by Eq. (15.58). The C- C bonds to the CHI group (one H bond to C ) were each treated as an iso C- C bond. The C - C bonds to the CI group (no H bonds to C) were each treated as a tert-butyl C - C. Emg is subtracted for each t-butyl group. The bond angle parameters of branched-chain alkyl iodides determined using Eqs. (15.61-15.70), (15.79-15.108) and (15.113) are given in Table 15.50.
Figure imgf000562_0001
Figure imgf000563_0001
Figure imgf000564_0001
Figure imgf000565_0001
Figure imgf000566_0001
ALKENYL HALIDES (CnH2)i_mXm, n = 3,4,5...∞ m = 1,2,3...°°)
The branched-chain alkenyl halides, CnH2n+2_nXm with X = F, Cl, Br, I , may comprise alkyl and alkenyl functional groups wherein at least one H is replaced by a halogen atom. In the case that a halogen atom replaces an alkyl H , the C- X bond comprises the alkyl-halogen functional groups given in their respective sections. The alkenyl halogen C- X bond comprises a separate functional group for each case of X bonding to the C = C -bond functional group given in the Alkenes section. In addition the CH group of the moiety XCH = C comprises a functional group unique to alkenyl halides. The straight and branched-chain alkenes, CnH2n , comprise at least one carbon-carbon double bond comprising a functional group that is solved equivalently to the double bond of ethylene. The double bond may be bound to one, two, three, or four carbon single bonds that substitute for the hydrogen atoms of ethylene. The three distinct functional groups given in the Alkenes section are C vinyl single bond to -C(C) = C , C vinyl single bond to -C(H) = C , and C vinyl single bond to -C(C) = CH2 .
In addition, CH2 of the -C = CH2 moiety is also an alkene functional group solved in the Alkenes section.
Consider the case where X= Cl substitutes for a carbon single bond or a hydrogen atom. Based on the condition of energy matching of the orbital, any magnetic energy due to unpaired electrons in the constituent fragments, and differences in oscillation in the transition state, two distinct C - Cl functional groups can be identified: Cl vinyl single bond to -C(C) = C and Cl vinyl single bond to -C(H) = C . The alkenyl-halide CH group is equivalent to that solved in the Hydrogen Carbide ( CH) section except that AE H M0 [AO I HO) = -1.13379 eV in order to energy match to the C- Cl and C = C bonds.
The alkyl portion of the alkenyl halide may comprise at least two terminal methyl groups (CH3 ) at each end of the chain, and may comprise methylene (CH2), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C- C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C- C bonds within isopropyl ((CH.) CH) and t-butyl ((CH3) C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C- C bonds comprise functional groups. The branched-chain-alkane groups in alkene halides are equivalent to those in branched-chain alkanes. Eτ \ atom- atom,msp' .AOj of the C = C -bond MO in Eq. (15.52) due to the charge donation from the C atoms to the MO is equivalent to that of ethylene, -2.26759 eV , given by Eq. (14.247). Eτ (atom - atom,msp2.Aθ) of each C- C -bond MO in Eq. (15.52) i s
-1.85836 eV or -1.44915 eV based on the energy match between the C2sp3 HOs corresponding to the energy contributions equivalent to those of methylene, -0.92918 eV (Eq. (14.513), or methyl, -0.72457 eV (Eq. (14.151)), groups, respectively.
The solution of each C - X functional group comprises the hybridization of the 2s etna
2 p AOs of the C atom to form- a single 2sp3 shell as an energy minimum, and the sharing of electrons between the C2sp2 HO and the X AO to form a MO permits each participating orbital to decrease in radius and energy. The alkenyl C - X -bond functional groups comprise single bonds and are equivalent to those of the corresponding alkyl halides except that the halogen AO and the C - X -bond MO are each energy matched to the alkene C2spi HO. In alkenyl halides with X = Cl, Br, or I , the energy of the halogen atom is less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). Thus, c2 in Eq. (15.52) is one, and the energy matching condition is determined by the C, parameter. For example, the hybridization factor C2 of Eq. (15.52) for the alkenyl C - Cl -bond MO given by Eq. (15.111) is C2 [C2sp3HO to Cl) = 0.81317.
Eτ [atom~
Figure imgf000568_0001
of the alkenyl C - Cl -bond MO in Eq. (15.52) due to the charge donation from the C and Cl atoms to the MO is -0.72457 eV for the Cl vinyl single bond to -C(H) = C C- Cl group and -0.92918 eV for the Cl vinyl single bond to
-C(C) = C C - Cl group. It is based on the energy match between the Cl atom and the C2sp3 HO of an unsubstituted vinyl group and a substituted vinyl group given by Eqs. (14.151) and (14.513), respectively.
The symbols of the functional groups of branched-chain alkenyl chlorides are given in Table 15.51. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of branched-chain alkenyl chlorides are given in Tables 15.52, 15.53, and 15.54, respectively. The total energy of each branched- chain alkenyl chloride given in Table 15.55 was calculated as the sum over the integer multiple of each ED (υmuP) of Table 15.54 corresponding to functional-group composition of the molecule. The bond angle parameters of branched-chain alkenyl chlorides determined using Eqs. (15.61- 15.70), (15.79-15.108) and (15.111) are given in Table 15.56.
Figure imgf000569_0001
Figure imgf000570_0001
Figure imgf000571_0001
Figure imgf000572_0001
Figure imgf000573_0001
ALCOHOLS (C H2n+2O,,, « = l,2,3,4,5...oo )
The alkyl alcohols, CnH2n+2O1n , comprise an OH functional group and two types of
C- O functional groups, one for methyl alcohol and the other for general alkyl alcohols. The alkyl portion of the alkyl alcohol may comprise at least two terminal methyl groups (CH3) at each end of the chain, and may comprise methylene (CH2), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C- C bonds within isopropyl ((CH3) CH) and t-butyl ((CH3) C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C- C bonds comprise functional groups. The branched-chain-alkane groups in alcohols are equivalent to those in branched-chain alkanes.
The OH functional group was solved in the Ηydroxyl Radical (OH ) section. Each C -O group is solved by hybridizing the 2s and 2p AOs of the C atom to form a single 2sp3. shell as an energy minimum, and the sharing of electrons between the Cisp" HO and the O AO to form a MO permits each participating orbital to decrease in radius and energy. In alkyl alcohols, the C2sp3 HO has a hybridization factor of 0.91771 (Eq. (13.430)) with a corresponding energy of ε(c,2sp3} = -14.63489 eV (Eq. (15.25)) and the O AO has an energy of -E(OJ = -13.61806 eV . To meet the equipotential condition of the union of the C - O H2 - type-ellipsoidal-MO with these orbitals, the hybridization factor C2 of Eq. (15.52) for the C - O -bond MO given by Eqs. (15.68) and (15.70) is c = 0.85395 (15.114)
Figure imgf000574_0001
Eγiatom - atom,msp3.AOj of the C- 0-bond MO in Eq. (15.52) due to the charge donation from the C and O atoms to the MO is -1.65376 eV for the CH3 - OH C- O group. It is based on the energy match between the OH .group and the C2sp3 HO of a methyl group and is given by the linear combination of -0.92918 eV (Eq. (14.513)) and -0.72457 eV (Eq . (14.151)), respectively. For the alkyl C- O group, Er (atom- atom,msp3.Aθ) is
—1.85836 eV . It is based on the energy match between the O AO and the C2sp3 HO of a methylene group where both energy contributions are given by Eq. (14.513). The symbols of the functional groups of branched-chain alkyl alcohols are given in
Table 15.57. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of alkyl alcohols are given in Tables
15.58, 15.59, and 15.60, respectively. The total energy of each alkyl alcohol given in Table 15.61 was calculated as the sum over the integer multiple of each ED{c,nmP) of Table 15.60 corresponding to functional-group composition of the molecule. The bond angle parameters of alkyl alcohols determined using Eqs. (15.79-15.108) are given in Table 15.62.
Figure imgf000576_0001
Figure imgf000577_0001
Figure imgf000578_0001
Figure imgf000579_0001
Figure imgf000580_0001
ETHERS (C H2n+2O ,, » = 2,3,4,5...« )
The alkyl ethers, CnH2n+2On , comprise two types of C - O functional groups, one for methyl or t-butyl groups corresponding to the C and the other for general alkyl groups. The alkyl portion of the alkyl ether may comprise at least two terminal methyl groups (CH3 ) at each end of the chain, and may comprise methylene (CH2), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C- C bonds within isopropyl ((CH3) CH) and t-butyl ((CH3J3C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C- C bonds comprise functional groups. The branched-chain-alkane groups in ethers are equivalent to those in branched-chain alkanes.
Each C- O group is solved by hybridizing the 2s and 2p AOs of the C atom to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between the C2sp3 HO and the O AO to form a MO permits each participating orbital to decrease in radius and energy.
In alkyl ethers, the C2sp3 HO has a hybridization factor of 0.91771 (Eq. (13.430)) and an energy of E{c,2sp3) = -14.63489 eV (Eq. (15.25)) and the O AO has an energy of E\θ) - -13.61806 eV . To meet the equipotential condition of the union of the C - O H2- type-ellipsoidal-MO with these orbitals, the hybridization factor c2 of Eq. (15.52) for the C- O-bond MO given by Eq. (15.1 13) is c2(C2sp3HO to O) - 0.85395 . E1, (atom- atom,msp3.AOj of the C- O-bond MO in Eq. (15.52) due to the charge donation from the C and O atoms to the MO is -1.44915 eV for the CH3 - O - and (CH3)3 C- O - C- O groups. It is based on the energy match between the O AO, initially at the Coulomb potential of a proton and an electron (Eqs. (1.236) and (10.162), respectively), and the C2sp3 H O of a methyl group as given by Eq. ( 14. 15 1 ). For the alkyl C- O group, Er(atom— atom,msp3.Aθ) is -1.65376 eV . It is based on the energy match between the O
AO and the C2sp3 HO of a methylene group and is given by the linear combination of -0.72457 eV (Eq. (14.151)) and -0.92918 eV (Eq. (14.513)), respectively. The symbols of the functional groups of branched-chain alkyl ethers are given in Table
15.63. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of alkyl ethers are given in Tables
15.64, 15.65, and 15.66, respectively. The total energy of each alkyl ether given in Table 15.67 was calculated as the sum over the integer multiple of each ED {G>ΏUP) of Table 15.66 corresponding to functional-group composition of the molecule. The bond angle parameters of alkyl ethers determined using Eqs. (15.79-15.108) are given in Table 15.68.
Figure imgf000583_0001
Figure imgf000584_0001
Figure imgf000585_0001
Figure imgf000586_0001
PRIMARY AMINES ( C H2n+2+mNm, « = 1,2,3, 4,5...«»)
The primary amines, CnH2n+2+mNm, comprise an NH2 functional group and a C - N functional group. The alkyl portion of the primary amine may comprise at least two terminal methyl groups (CH3 ) at each end of the chain, and may comprise methylene (CH2 ), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C - C bonds within isopropyl ((CH3J CH) and t-butyl
((CH3) C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C - C bonds comprise functional groups. The branched-chain-alkane groups in primary amines are equivalent to those in branched-chain alkanes.
The primary amino (NH2 ) functional group was solved using the procedure given in the
Dihydrogen Nitride ( NH2) section. Using the results of Eqs. (13.245-13.368), the primary amino parameters in Eq. (15.52) are n, = 2 , C1 = 0.75 , C2 = 0.93613 (Eqs. (13.248-13.249)), C1n = 1.5 , and c, = 0.75. In primary amines, the C2sp3 HO of the C- NH2 -bond MO has an energy of E(c,2sp3) = -15.35946 eV (Eq. (15.18) with s = l and Eqs. (15.19-15.20)) and the N AO has an energy of £(N) = -14.53414 eV . To meet the equipotential condition of the union of the N- H H2 -type-ellipsoidal-MO with the C2sp* HO, the hybridization factor c2 of Eq. (15.52) for the N- H-bond MO given by Eq. (15.68) is (!5.I 15)
Figure imgf000587_0001
The C - N group is solved by hybridizing the 2s and 2p AOs of the C atom to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between the C2.sp3 HO and the N AO to form a MO permits each participating orbital to decrease in radius and energy. In primary amines, the C2spi HO has a hybridization factor of 0.91771 (Eq. (13.430)) with a corresponding energy of E(c,2sp3} = -14.63489 eV (Eq. (15.25)), and the N AO has an energy of £(N) = -14.53414 eV . To meet the equipotential condition of the union of the C- N H2 -type-ellipsoidal-MO with these orbitals, the hybridization factor C2 of Eq. (15.52) for the C - N -bond MO given by Eqs. (15.68) and (15.70) is (15.116)
Figure imgf000588_0001
Er(atom- atom,msp3.AC)) of the C- N -bond MO in Eq. (15.52) due to the charge donation from the C and N atoms to the MO is -1.44915 eV . It is based on the energy match between the N of the NH2 group and the C2sp\ HO corresponding to the energy contributions to the single bond that are equivalent to those of methyl groups, -0.72457 eV (Eq. (14.151)), where the N- H bonds are also energy matched to the C- N bond.
The symbols of the functional groups of branched-chain primary amines are given in Table 15.69. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of primary amines are given in Tables
15.70, 15.71, and 15.72, respectively. The total energy of each primary amine given in Table
15.73 was calculated as the sum over the integer multiple of each ED [GWUP) of Table 15.72 corresponding to functional-group composition of the molecule. The bond angle parameters of primary amines determined using Eqs. (15.79-15.108) are given in Table 15.74.
Figure imgf000588_0002
Figure imgf000589_0001
Figure imgf000589_0002
Figure imgf000590_0001
Figure imgf000590_0002
Figure imgf000591_0001
SECONDARY AMINES ( CnH2n^n N1n, n = 2,3,4,5...- )
The secondary amines, CnH2n+2^n N m , comprise an NH functional group and two types of C- N functional groups, one for the methyl group corresponding to the C of C- N and the other for general alkyl secondary amines. The alkyl portion of the secondary amine may comprise at least two terminal methyl groups (CH3 ) at each end of the chain, and may comprise methylene (CH2), and methylyne (CH) functional groups as well as C bound by carbon- carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C- C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C - C bonds within isopropyl ((CH3) CH) and t-butyl ((CH3] C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C- C bonds comprise functional groups. The branched-chain-alkane groups in secondary amines are equivalent to those in branched-chain alkanes.
The secondary amino ( NH ) functional group was solved using the procedure given in the Hydrogen Nitride (NH) section. Using the results of Eqs. (13.245-13.316), the secondary amino parameters in Eq. (15.52) are n, = 1 , C1 = 0.75 , C2 = 0.93613 (Eqs. (13.248-13.249)),
C1n = 0.75 , and c, = 0.75. In secondary amines, the C2sp* HO of the C- NH -bond MO has an energy of E[c,2sp3) = -15.56407 eV (Eqs. (14.514-14.516)); Eq. (15.29) with s = l and s = 2 , Eq. (15.31), and Eqs. (15.19-15.20)) and the N AO has an energy of i?(N) = -14.53414 eV . To meet the equipotential condition of the union of the N- H H2- type-ellipsoidal-MO with the C2.ψ3 HO, the hybridization factor c2 of Eq. (15.52) for the N- H -bond MO given by Eq. (15.68) is
C2 [H tO TN) = ^^ 3, = -1453414 ey = 0.93383 (15.117)
2 V ; E(c,2sp3) -15.56407 eV
The C - N group is solved by hybridizing the 2s and 2p AOs of the C atom to form a single
2sp3 shell as an energy minimum, and the sharing of electrons between the C2.sp3 HO and the N AO to form a MO permits each participating orbital to decrease in radius and energy. In secondary amines, the C2sp* HO has a hybridization factor of 0.91771 (Eq. (13.430)) with a corresponding energy of ε(c,2sp3} = -14.63489 eV (Eq. (15.25)), and the N AO has an energy of EyNj - -14.53414 eV . To meet the equipotential condition of the union of the C - N H2 -type-ellipsoidal-MO with these orbitals, the hybridization factor C2 of Eq. (15.52) for the C - N -bond MO given by Eq. (15.116) is c2 (C2sp3 HO to N) = 0.91140 .
As given in the Continuous-Chain Alkanes (CnH2n+2, n - 3, 4,5...∞) section, each methylene group forms two single bonds, and the energy of each C2sp3 HO of each CH2 group alone is given by that in ethylene, -1.13379 eV (Eq. (14.511)). In secondary amines, the N of the NH group also binds to two C2sp3 ΗOs and the corresponding Eτi atom - atom, msp3.AOj of each C- N -bond MO in Eq. (15.52) due to the charge donation from the C and N atoms to the MO is -1.13379 eV . It is based on the energy match between the N of the NH group to the two C2sp3 ΗOS corresponding to the energy contributions to each of the two single bonds that are equivalent to those of independent methylene groups, -1.13379 eV (Eq. (14.51 1)), where the N- H bond is also energy matched to the C - N bonds. E1. ( atom - atom, msp3.Aθ) of the C- N -bond MO in Eq. (15.52) due to the charge donation from the C and N atoms to the MO is -1.13379 eV . It is based on the energy match between the N of the NH group to two C2sp3 ΗOs corresponding to the energy contributions to the single bond that are equivalent to those of methyl groups, -0.72457 eV (Eq. (14.151)), where the N - H bonds are also energy matched to the C- N bond.
The symbols of the functional groups of branched-chain secondary amines are given in Table 15.75. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of secondary amines are given in Tables 15.76, 15.77, and 15.78, respectively. As in the case of NH2 (Eq. (13.339)), C1n = 2C1 rather than C1n = C1 in Eq. (15.52) for the C - N bond. The total energy of each secondary amine given in Table 15.79 was calculated as the sum over the integer multiple of each ED (GW«P) of Table 15.78 corresponding to functional-group composition of the molecule. The bond angle parameters of secondary amines determined using Eqs. (15.79-15.108) are given in Table 15.80.
Figure imgf000594_0001
Figure imgf000595_0001
Figure imgf000595_0002
Figure imgf000596_0001
Figure imgf000596_0002
Figure imgf000597_0001
TERTIARY AMINES ( CnH2n+3N, n = 3,4,5...°° )
The tertiary amines, CnH2n+3N , have three C- N bonds to methyl or alkyl groups wherein C - N comprises a functional group. The alkyl portion of the tertiary amine may comprise at least two terminal methyl groups ( CH3 ) at each end of the chain, and may comprise methylene (CH2), and methylyne (CH) functional groups as well as C bound by carbon- carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C- C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C- C bonds within isopropyl
((CH3) CH) and t-butyl ((CH3J3C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C- C bonds comprise functional groups. The branched-chain-alkane groups in tertiary amines are equivalent to those in branched-chain alkanes.
The C - N group is solved by hybridizing the 2s and 2p AOs of the C atom to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between the C2sp3 HO and the N AO to form a MO permits each participating orbital to decrease in radius and energy. In tertiary amines, the C2sp3 HO has a hybridization factor of 0.91771 (Eq. (13.430)) with a corresponding energy of E(c,2sp3) = -14.63489 eV (Eq. (15.25)), and the N AO has an energy of E( N) ~ -14.53414 eV . To meet the equipotential condition of the union of the C- N H2 -type-ellipsoidal-MQ with these orbitals, the hybridization factor e2 of Eq. (15.52) for the C- N -bond MO given by Eq. (15.116) is c2 (C2sp3 HO to N) = 0.91140. As given in the Continuous-Chain Alkanes (CnH2n+2, n - 3,4,5...∞) section, the energy of each C2sp3 HO must be a linear combination of that of the CH3 and CH2 groups that serve as basis elements. Each CH3 forms one C-C bond, and each CH2 group forms two. Thus, the energy of each C2sp3 HO of each CH3 and CH2 group alone is given by that in ethane,
-0.72457 eV (Eq. (14.151)), and ethylene, -1.13379 eV (Eq. (14.511)), respectively. In order to match the energy of the component ΗOs and MOs for the entire molecule, the energy
E.r [C- C,2sp3 ) given as a linear combination of these basis elements is -0.92918 eV (Eq.
(14.513)). In tertiary amines, the N binds to three C2spl ΗOs and the corresponding E1. (atom - atom, msp3 Λθ\ of each C- N -bond MO in Eq. (15.52) due to the charge donation from the C and N atoms to the MO is -0.92918 eV . It comprises a linear combination of the energy for a primary amine, -0.72457 eV and a secondary amine, -1.13379 eV .
The symbols of the functional groups of branched-chain tertiary amines are given in
Table 15.81. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of tertiary amines are given in Tables
15.82, 15.83, and 15.84, respectively. The total energy of each tertiary amine given in Table
15.85 was calculated as the sum over the integer multiple of each Eυ{cmuP) of Table 15.84 corresponding to functional-group composition of the molecule. The bond angle parameters of tertiary amines determined using Eqs. (15.79-15.108) are given in Table 15.86.
Figure imgf000600_0001
Figure imgf000601_0001
Figure imgf000602_0001
Figure imgf000602_0002
C
Figure imgf000603_0001
ALDEHYDES (CnH1nO, w = 1,2,3,4,5...°°)
The alkyl aldehydes, CnH2nO, each have a HC = O moiety that comprises a C = O functional group and a CH functional group. The single bond of carbon to the carbonyl carbon atom, C - C(O)H , is a functional group, In addition to the C = O functional group, formaldehyde comprises a CH2 functional group. The alkyl portion of the alkyl aldehyde may comprise at least two terminal methyl groups (CH3 ) at each end of the chain, and may comprise methylene (CH2), and methylyne (CH) functional groups as well as C bound by carbon- carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C- C bonds within isopropyl
((CH3J CH) and t-butyl ((CH3) C ) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C- C bonds comprise functional groups. The branched-chain-alkane groups in aldehydes are equivalent to those in branched-chain alkanes.
The CH functional group was solved in the Hydrogen Carbide (CH) section except that Ema is not subtracted since unpaired electrons are not created with fragmentation of the CH functional group of aldehydes. The CH2 functional group of formaldehyde is solved in the Dihydrogen Carbide (CH2) section except that the energy of each C - H MO is matched to the initial energy of the C2sp3 HO (Eq. (15.25)). The C = O and C - C(O)H groups are solved by hybridizing the 2s and 2p AOs of each C atom to form a single 2sp2 shell as an energy minimum, and the sharing of electrons between the C2sp3 HO and the O AO or between two C2sp3 HOs3 respectively, to form a MO permits each participating orbital to decrease in radius and energy. In alkyl aldehydes, the C2sp3 HO has a hybridization factor of 0.91771 (Eq. (13.430)) with a corresponding energy of £(c,2-sp3) = -14.63489 eV (Eq. (15.25)) and the O AO has an energy of E(O) = -13.61806 eV . To meet the equipotential condition of the union of the C = O H2 -type-ellipsoidal-MO with these orbitals, the hybridization factor c2 of Eq. (15.52) for the C = 0-bond MO given by Eq. (15.114) is c2(C2sp3HO to θ) = 0.85395. The unpaired electrons created by bond breakage of the double C = O bond requires that two times the O2p AO magnetic energy Emag (Eq. (15.60)) be subtracted from the total energy to give ED{a-ouP) (eV) for C = O .
Eτ ( atom - atom, msp* .AO) of the C = C7 -bond MO in Eq. (15.52) due to the charge donation from the C and O atoms to the MO is -2.69893 eV which is an energy minimum for the double bond between the pair of C2sp3 HO electrons of the C atom and the pair of AO electrons of the O atom. It is given as a linear combination of the energy contributions corresponding to a double bond, -1.13379 eV (Eq. (14.247)), and a triple bond, -1.56513 eV
(Eq. (14.342)). The triple bond contribution includes the Clsp^ HO electron of the C - H bond in addition to the pair involved directly in the double bond with O . Eτ ( atom- atom,mspi. AOJ of the C - C(O)H group is equivalent to that of an alkane, -1.85836 eV , where both energy contributions are given by Eq. (14.513). It is based on the energy match between the C2sp3 HOs of the aldehyde. In order to match energy between the groups bonded to the C = O , electron-density is shared. Due to the interaction in the transition state between the groups based on the sharing, C)o = 2C1 rather than C1n = C1 in Eq. (15.52) for the C - C(O)H bond.
The symbols of the functional groups of alkyl aldehydes are given in Table 15.87. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of alkyl aldehydes are given in Tables 15.88, 15.89, and 15.90, respectively. The total energy of each alkyl aldehyde given in Table 15.91 was calculated as the sum over the integer multiple of each En [cm«P) of Table 15.90 corresponding to functional-group composition of the molecule. The bond angle parameters of alkyl aldehydes determined using Eqs. (15.79-15.108) are given in Table 15.92.
Figure imgf000606_0001
Figure imgf000607_0001
Figure imgf000607_0002
Figure imgf000608_0001
Figure imgf000608_0002
Figure imgf000609_0001
KETONES ( CnH2nO, « = l,2,3,4,5...o=)
The alkyl ketones, CnH2nO, each have a C = O moiety that comprises a functional group. Each of the two single bonds of carbon to the carbonyl carbon atom, C - C(O) , is also a functional group. The alkyl portion of the alkyl ketone may comprise at least two terminal methyl groups (CH3 ) at each end of the chain, and may comprise methylene (CH2 ), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C- C bonds can be identified. The n-alkane C - C bond is the same as that of straight-chain alkanes. In addition, the C- C bonds within isopropyl ((CH3] CH) and t-butyl ((CH3)3 C) groups and the isopropyl to isopropyl, isopropyl to t-butyl5 and t-butyl to t-butyl
C - C bonds comprise functional groups. The branched-chain-alkane groups in ketones are equivalent to those in branched-chain alkanes.
The C = O and C - C[O) groups are solved by hybridizing the 2s and 2p AOs of each
C atom to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between the C2sp3 HO and the O AO or between two C2sp3 HOs, respectively, to form a MO permits each participating orbital to decrease in radius and energy. In alkyl ketones, the C2sp3 HO has a hybridization factor of 0.91771 (Eq. (13.430)) with a corresponding energy of E(C,2$P3) = -14.63489 eV (Eq. (15.25)) and the O AO has an energy of
E\ θ\ = -13.61806 eV . To meet the equipotential condition of the union of the C = O H2- type-ellipsoidal-MO with these orbitals, the hybridization factor c2 of Eq. (15.52) for the C = O-bond MO given by Eq. (15.114) is C2 [C2sp3HO to θ) = 0.85395 . The unpaired electrons created by bond breakage of the double C - O bond requires that two times the O2p AO magnetic energy E (Eq. (15.60)) be subtracted from the total energy to give ED (am,p) (eV) for C = O . As in the case with aldehydes, Eτ ( atom- atom, msp* .AO ) of the C = O-bond MO in
Eq. (15.52) due to the charge donation from the C and O atoms to the MO is -2.69893 eV which is an energy minimum for the double bond between the pair of C2sp3 HO electrons of the C atom and the pair of AO electrons of the O atom. It is given as a linear combination of the energy contributions corresponding to a double bond, -1.13379 eV (Eq. (14.247)), and a triple bond, -1.56513 eV (Eq. (14.342)). The triple bond contribution includes the C2sp3 HO electron of the C - C(O) bond in addition to the pair involved directly in the double bond with
O . Consequently, E1, (atom- atom, msp3 Λθ) of the C - C(O) -bond MO is -1.44915 βV , corresponding to the energy contributions of the two C2spl HOs to the single bond that are equivalent to those of methyl groups, -0.72457 eV (Eq. (14.151)). Since there are two C - C(O) bonds in ketones versus one in aldehydes, Clo = C1 in Eq. (15.52) for each C - C(O) ketone bond.
The symbols of the functional groups of alkyl ketones are given in Table 15.93. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.55)) parameters of alkyl ketones are given in Tables 15.94, 15.95, and 15.96, respectively. The total energy of each alkyl ketone given in Table 15.97 was calculated as the sum over the integer multiple of each ED{amuP) of Table 15.96 corresponding to functional-group composition of the molecule. For each set of unpaired electrons created by bond breakage , the C2sp3 HO magnetic energy Ema that is subtracted from the weighted sum of the ED (GWUP) (eV) values based on composition is given by Eq. (15.58). The bond angle parameters of alkyl ketones determined using Eqs. (15.79-15.108) are given in Table 15.98.
Figure imgf000611_0001
Figure imgf000612_0001
Figure imgf000612_0002
Figure imgf000613_0001
Figure imgf000614_0001
CARBOXYLIC ACIDS (CnH2nO2, « = l,2,3,4,5...oo)
The alkyl carboxylic acids, CnH2nO2 , comprise a C = O functional group, and the single bond of carbon to the carbonyl carbon atom, C - C(O) , is also a functional group. Formic acid has a HC = O moiety that comprises a more stable C = O functional group and a CH functional group. All carboxylic acids further comprise a C - OH moiety that comprises C - O and OH functional groups. The alkyl portion of the alkyl carboxylic acid may comprise at least two terminal methyl groups (CH3 ) at each end of the chain, and may comprise methylene (CH2), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C - C bond is the same as that of straight-chain alkanes. In addition, the C - C bonds within isopropyl ([CH2) CH) and t-butyl {{CHΛ C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t- butyl C - C bonds comprise functional groups. The branched-chain-alkane groups in carboxylic acids are equivalent to those in branched-chain alkanes.
The CH functional group was solved in the Hydrogen Carbide ( CH) section except that the energy of the C - H MO is matched to the carbon-atom contribution to AEH UQ [AOIHO] and
Er(αtom- αtom,msp3.Aθ) of the C- O group. The alkyl carboxylic acid C = O and
C - C(O) groups are equivalent to those given in the Aldehydes section except that EKvιh is that of a carboxylic acid. The formic acid C = O group is solved equivalently to that of the alkyl carboxylic acid group, except that ΔEH m {AOIHO) and Er ( atom- atom, msp3.Aθ) correspond to a 25% increase in the donation of charge density from the orbitals of the atoms to the C = O MO due to the presence of a H bound to the carbonyl carbon. Also, EKvlh is that corresponding to formic acid. The C - O and OH groups are equivalent to those of alkyl alcohols given in the corresponding section except that the energy of the C - O MO is matched to that of the C = O group and EKvιh is that of a carboxylic acid. AEH M0 {AOIHO) of the C - O group is equal to
Eγ l atom - atom, msp3.AOj of the alkyl C = O group in order to match the energies of the corresponding MOs.
As in the case with aldehydes and ketones, Eτ \ atom- atom,mspl .AOj of the C = O- bond MO in Eq. (15.52) of alky carboxylic acids due to the charge donation from the C and O atoms to the MO is -2.69893 eV which is an energy minimum for the double bond between the pair of C2sp2 HO electrons of the C atom and the pair of AO electrons of the carbonyl O atom. It is given as a linear combination of the energy contributions corresponding to a double bond, -1.13379 eV (Eq. (14.247)), and a triple bond, -1.56513 eV (Eq. (14.342)). The triple bond contribution includes the energy match of the carbonyl C2sp3 HO electron with the O of the C - 0-bond MO in addition to the pair involved directly in the double bond with the carbonyl O .
Eτ\ atom- atom,msp3.AO) of the formic acid C - O-bond MO in Eq. (15.52) due to the charge donation from the C and O atoms to the MO is -3.58557 eV . This is also an energy minimum for the double bond between the pair of C2sp3 HO electrons of the C atom and the pair of AO electrons of the carbonyl O atom. It is given as a linear combination of the energy contributions corresponding to a triple bond, -1.56513 eV (Eq. (14.342)), and a quadruple bond, -2.02043 eV (Eqs. (15.18-15.21) with S = A )) where the bond order components are increased by an integer over that of alkyl carboxylic acids due to the presence of a H bound to the carbonyl carbon.
E1. f atom— atom,msp3.AOj of the carboxylic acid C — C(O) group is equivalent to that of alkanes and aldehydes, -1.85836 eV , where both energy contributions are given by Eq. (14.513). It is based on the energy match between the C2sp3 ΗOs of the carboxylic acid. As in the case of aldehydes, C, β = 2C1 in Eq. (15.52).
Er[atom- atom,msp* .AOj of the carboxylic acid C - O group is equivalent to that of alkyl alcohols, -1.85836 eV . It is based on the energy match between the O AO and the C2sp* HO of a methylene group (the maximum hybridization for a single bond) where both energy contributions are given by Eq. (14.513). E.A atom - atom, msp1.AOj of the C- O group matches that of the C - C(O) group.
The symbols of the functional groups of alkyl carboxylic acids are given in Table 15.99. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of alkyl carboxylic acids are given in Tables 15.100, 15.101, and 15.102, respectively. The total energy of each alkyl carboxylic acid given in Table 15.103 was calculated as the sum over the integer multiple of each Eυ (aιmφ) of Table 15.102 corresponding to functional-group composition of the molecule. For each set of unpaired electrons created by bond breakage , the C2sp3 HO magnetic energy E that is subtracted from the weighted sum of the E0 (o™/>) (eV) values based on composition is given by Eq.
(15.58). The bond angle parameters of alkyl carboxylic acids determined using Eqs. (15.79- 15.108) are given in Table 15.104.
Figure imgf000617_0001
Figure imgf000618_0001
Figure imgf000618_0002
Figure imgf000619_0001
Figure imgf000620_0001
CARBOXYLIC ACID ESTERS (CnH2nO2, n = l,2,3,4,5...oo )
The alkyl carboxylic acid esters, CnH2nO2 , comprise a C = O functional group, and the single bond of carbon to the carbonyl carbon atom, C - C(O) , is also a functional group. Formic acid ester has a HC = O moiety that comprises a more stable C = O functional group and a CH functional group. AU carboxylic acid esters further comprise a COR moiety that comprises a C - O functional group and three types of O - R functional groups, one for R comprising methyl, one for R comprising an alkyl ester group of a formate, and one for R comprising an alkyl ester group of an alkyl carboxylate. The alkyl portion of the alkyl carboxylic acid ester may comprise at least two terminal methyl groups ( CH3 ) at each end of the chain, and may comprise methylene (CH2 ), and methylyne ( CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C - C bond is the same as that of straight-chain alkanes. In addition, the C - C bonds within isopropyl ((CH3) CH) and t-butyl ((CH3) C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C- C bonds comprise functional groups. The branched-chain-alkane groups in carboxylic acid esters are equivalent to those in branched-chain alkanes. • i
The CH functional group is equivalent to that of formic acid. The alkyl carboxylic acid ester C = O and C - C(O) groups are equivalent to those given in the Carboxylic Acids section. The formic acid ester C = O group is equivalent to that given in the Carboxylic Acids section except that EKvιh is that corresponding to a formic acid ester. The C- O group is equivalent to that given in the Carboxylic Acids section except that the parameters corresponding to oscillation of the bond in the transition state, ED (eV) and EKvιh , are those of a carboxylic acid ester. As in the case with the alkyl ethers, each 0 - C group is solved by hybridizing the 2s and 2p AOs of the C atom to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between the C2sp3 HO and the O AO to form a MO permits each participating orbital to decrease in radius and energy. To meet the equipotential condition of the union of the O - C H2 -type-ellipsoidal-MO with other orbitals of the molecule, the hybridization factor c2 ofEq. (15.51) for the O - C -bond MO given by Eq. (15.114) is c2 (C2sp3HO to θ) = 0.85395 . E7 [atom~ atom,msp3.AOJ (Eq. (15.52)) of (1) the C = O group of alky carboxylic acid esters, (2) the C = O group of formic acid esters, (3) the alkyl carboxylic acid ester C - C(O) group, and (4) the carboxylic acid ester C- O group are equivalent to those of the corresponding carboxylic acids. The values given in the Carboxylic Acids section are -2.69893 eV , -3.58557 eV , -1.85836 eV , and -1.85836 eV , respectively.
Eτ ( atom- atom, msp3. AOj of the C- O group matches that of the C - C(O) group. Also, as in the case of aldehydes, C)o = 2C1 in Eq. (15.52) for the C - C(O) group.
Eτ(atom- atom,msp3.Aθ) of the O - C -bond MO in Eq. (15.52) due to the charge donation from the C and O atoms to the MO is -1.13379 eV for the 0 - CH3 group of formate and alkyl carboxylates, -1.44915 eV for the O - R group of alkyl carboxylates, and -1.85836 eV for the O - R group of alkyl formates, where R is an alkyl group. Each is based on the energy match between the O AO, initially at the Coulomb potential of a proton and an electron (Eqs. (1.236) and (10.162), respectively), the C2sp* HO of the methyl or alkyl ester group, and the carbonyl carbon. The increasing energy contributions to the single bond correspond to the increasing hybridization of linear combinations of increasing bond order. The energy contributions corresponding to one half of a double bond and those of the methyl-methyl and methylene-methylene bonds are -1.13379 eV (Eq. (14.247)), two times -0.72457 e V (Eq. (14.151)), and two times -0.92918 eV (Eq. (14.513)), respectively.
The symbols of the functional groups of alkyl carboxylic acid esters are given in Table 15.105. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of alkyl carboxylic acid esters are given in Tables 15.106, 15.107, and 15.108, respectively. The total energy of each alkyl carboxylic acid ester given in Table 15.109 was calculated as the sum over the integer multiple of each ED (GWUP) of Table 15.108 corresponding to functional-group composition of the molecule. For each set of unpaired electrons created by bond breakage , the C2sp3 HO magnetic energy Emg that is subtracted from the weighted sum of the ED[Gm,,P) (eV) values based on composition is given by Eq. (15.58). The bond angle parameters of alkyl carboxylic acid esters determined using Eqs. (15.79-15.108) are given in Table 15.110.
Figure imgf000623_0001
Figure imgf000624_0001
Figure imgf000625_0001
Figure imgf000626_0001
Figure imgf000627_0001
Figure imgf000628_0001
Figure imgf000629_0001
AMIDES (c H2n+1No, * = 1,2,3,4,5...«.)
The alkyl amides, CnH2n+1NO , comprise a C = O functional group, and the single bond of carbon to the carbonyl carbon atom, C - C(O) , is also a functional group. Formamide has a HC = O moiety that comprises a more stable C - O functional group and a CH functional group that is equivalent to that of the CH (i) of aldehydes given in the corresponding section. It is also equivalent to that of the iso- CH group of branched-chain-alkyl portion of the alkyl amide except that Emg (Eq. (15.58)) is not subtracted from ED (am,P) . All amides further comprise a C - NH2 moiety that comprises a NH2 functional group and two types of C - N functional groups, one for formamide and the other for alkyl amides ( RC(O)NH2 where R is alkyl). The alkyl portion of the alkyl amide may comprise at least two terminal methyl groups ( CH3 ) at each end of the chain, and may comprise methylene (CH2 ), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C - C bonds within isopropyl ((CH3) CH) and t-butyl ((CH3) C ) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C- C bonds comprise functional groups. The branched-chain-alkane groups in amides are equivalent to those in branched-chain alkanes.
The NH2 functional group was solved in the Dihydrogen Nitride (NH2) section except that the energy of the N - H MO is matched to the nitrogen-atom contribution to ΔEH M0 [AOIHO)
and E7, (atom — atom, msp^. AO) of the C- N group. Both alkyl amide C = O groups and the C - C(O) group are equivalent to those given in the Carboxylic Acid Esters section except that EKvjh of the C - C(O) group is matched to that of an amide. The C - N groups are equivalent to those of alkyl amines given in the corresponding section except that the energy of the C - N MO is matched to that of the C = O group and EKvjh is that of a amide. AE H MO [AOIHO) of the
C - N group is equal to Eτ ( atom- atom,msp3. AO ) of the alkyl C = O and C - N groups in order to match the energies of the corresponding MOs.
As in the case of primary amines, each C - N group is solved by hybridizing the 2s and
2p AOs of the C atom to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between the C2sp3 HO and the N AO to form a MO permits each participating orbital to decrease in radius and energy. To meet the equipotential condition of the union of the C- N H2 -type-ellipsoidal-MO with other orbitals of the molecule, the hybridization factor C2 ofEq. (15.52) for the C- N-bond MO given by Eq. (15.114) is C2 (C2sp3HO to N) = 0.91140 . Eτ\ atom- atom,msp3.AOj (Eq. (15.52)) of the C - O group of alky amides and the
C = O group of formamide are equivalent to those of the corresponding carboxylic acids and esters. The values given in the Carboxylic Acids section are -2.69893 eV and -3.58557 eV , respectively. ,
Eτ [atom- atom,msp3.AO) of the amide C - C(O) group is the same as alkanes, aldehydes, carboxylic acids, and carboxylic acid esters, -1.85836 eV , where both energy contributions are given by Eq. (14.513). Also, as in the case of aldehydes, C1n = 2Cλ in Eq.
(15.52).
In order to match energy throughout the chain of the amide molecule,
Eτ\ atom- atom,msp3.AOj of the C- N -bond MO in Eq. (15.52) due to the charge donation from the C and N atoms to the MO is -1.65376 eV . It is based on the energy match between the C2sp3 HO of the carbonyl and the primary amino group NH2 . It is given by the linear combination of -0.92918 eV (Eq. (14.513)) which matches the contiguous C - C(O) or HC(O) group and -0.72457 eV (Eq. (14.151)), the contribution of a primary amino group given in the Primary Amines section. The symbols of the functional groups of alkyl amides are given in Table 15.111. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of alkyl amides are given in Tables 15.112, 15.113, and 15.114, respectively. The total energy of each alkyl amide given in Table 15.115 was calculated as the sum over the integer multiple of each ED [c,u>uP) of Table 15.114 corresponding to functional-group composition of the molecule. The bond angle parameters of alkyl amides determined using Eqs. (15.79-15.108) are given in Table 15.116.
Figure imgf000632_0001
Figure imgf000633_0001
Figure imgf000634_0001
Figure imgf000635_0001
Figure imgf000636_0001
N-ALKYL AND N3N-DIALKYL-AMIDES ( CnH2n+1NO, n = 2,3,4,5...- )
The N-alkyl and N,N-dialkyl amides, CnH2n^NO , comprise a C = O functional group, and the single bond of carbon to the carbonyl carbon atom, C - C(O) , is also a functional group. Formamide has a HC = O moiety that comprises a more stable C = O functional group and a CH functional group that is equivalent to that of the iso- CH group of branched-chain-alkyl portion of the N-alkyl or N,N-dialkyl amide. All amides further comprise a C - N(R^R2 moiety that comprises two types of C -N functional groups, one for formamide and the other for alkyl amides (RC(O)N[R1JR2 where R is alkyl). The N or N,N-dialkyl moiety comprises three additional groups depending on the alkyl substitution of the nitrogen. In the case of a single methyl or alkyl substitution, the NH - C bond and NH are functional groups, and the
N- C bond of a di-substituted nitrogen is the third.
The alkyl portion of the Ν-alkyl or Ν,Ν-dialkyl amide may comprise at least two terminal methyl groups (CH3) at each end of the chain, and may comprise methylene (CH2 ), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes.
Six types of C- C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C- C bonds within isopropyl ([CH3J CH) and t-butyl ([CH3) C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl
C- C bonds comprise functional groups. The branched-chain-alkane groups in N-alkyl or N,N-dialkyl amides are equivalent to those in branched-chain alkanes.
The NH functional group was solved in the Hydrogen Nitride ( NH) section except that the energy of the N - H MO is matched to the nitrogen-atom contribution toAEH MO [AO/HO) and
E.A atom- atom,msp*. AOj of the C- N group. The C- C(O) group, both Ν-alkyl or Ν,Ν- dialkyl amide C = O groups, and both C- N groups are equivalent to those given in the Amides section.
As in the case of primary amines, each N - C group is solved by hybridizing the 2s and 2p AOs of the C atom to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between the C2sp3 HO and the N AO to form a MO permits each participating orbital to decrease in radius and energy. To meet the equipotential condition of the union of the N - C H2 -type-ellipsoidal-MO with other orbitals of the molecule, the hybridization factor C2 ofEq. (15.52) for the N - C -bond MO given by Eq. (15.114) is C2 (C2sp3HO to N) = 0.91140 . Eτ (atom- atom,msp3.Ao) of the Ν-substituted amide C - C(O) group is the same as alkanes, aldehydes, carboxylic acids, carboxylic acid esters, and amides, -1.85836 eV , where both energy contributions are given by Eq. (14.513). Also, as in the case of aldehydes, C10 = 2C1 in Eq. (15.52).
Er ( atom- atom, msp3.AOj (Eq. (15.52)) of the C = O group of Ν-substituted alky amides and the C = O group of Ν-substituted formamide are equivalent to those of the corresponding carboxylic acids, carboxylic esters, and amides. The values given in the Carboxylic Acids section are -2.69893 eV and -3.58557 eV , respectively.
E1. (atom— atom,m$p3.AO] of both C- N functional groups are the same as those of
the corresponding groups of amides, -1.65376 eV . Er[atom- atom,msp3.AOj of the singly- substituted NH - C -bond MO in Eq. (15.52) due to the charge donation from the N and C atoms to the MO is -0.92918 eV . It is equivalent to that of tertiary amines and matches the energy of the NH- C group to that of the C - N group wherein Eτ (atom- atom,msp3.AOj of the latter is a linear combination of -0.92918 eV (Eq. (14.513)) and -0.72457 e V (Eq. (14.151)).
Figure imgf000638_0001
atom,msp3.AOj of the doubly-substituted N - C -bond MO i s
-0.72457 eV . It is equivalent to that of the contribution of each atom of a primary amine and also matches the energy of the N - C group to that of the C - N group by matching one of the components of E7 I atom— atom,msp3 '.AOj of the latter.
The symbols of the functional groups of Ν-alkyl and Ν,Ν-dialkyl amides are given in Table 15.117. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.1 1) and (15.17-15.56)) parameters of N-alkyl and N,N-dialkyl amides are given in Tables 15.118, 15.119, and 15.120, respectively. The total energy of each N-alkyl or N,N-dialkyl amide given in Table 15.121 was calculated as the sum over the integer multiple of each ED {am,Ψ) of Table 15.120 corresponding to functional-group composition of the molecule. The bond angle parameters of N-alkyl and N,N-dialkyl amides determined using Eqs. (15.79-15.108) are given in Table 15.122.
Figure imgf000639_0001
Figure imgf000640_0001
Figure imgf000641_0001
Figure imgf000642_0001
Figure imgf000643_0001
Figure imgf000644_0001
UREA (CH4N2C)
Urea, CH4N2O , comprises a C = O functional group and two C - NH2 moieties that each comprise a NH2 functional group and a C - N functional group. The C = O group is equivalent to that given for formamide in the Amides section except that the energy terms due to oscillation in the transition state are matched to that of urea. The NH2 and C - N functional groups are also equivalent to those given in the Amides section. Eτ(atom- atom,msp3.AO)
(Eq. (15.52)) of the C = O and C - N groups are equivalent to those of formamide. The values given in the Amides section are -3.58557 eV , and -1.65376 eV , respectively.
The symbols of the functional groups of urea are given in Table 15.123. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs.
(15.6-15.11) and (15.17-15.56)) parameters of urea are given in Tables 15.124, 15.125, and
15.126, respectively. The total energy of urea given in Table 15.127 was calculated as the sum over the integer multiple of each ED {βmιp) of Table 15.126 corresponding to functional-group composition of the molecule. The bond angle parameters of urea determined using Eqs. (15.79- 15.108) are given in Table 15.128.
Figure imgf000645_0001
Figure imgf000646_0001
Figure imgf000647_0001
Figure imgf000647_0002
Figure imgf000648_0001
CARBOXYLIC ACID HALIDES ( CnH2n-1QJ, X= F,Cl,Br,I; « = 1,2,3,4,5.»°°)
The alkyl carboxylic acid halides, CnH2n^OX , comprise a C = O functional group, and the single bond of carbon to the carbonyl carbon atom, C - C(O) , is also a functional group. All carboxylic acid halides further comprise a C - X functional group where Z is a halogen atom. The alkyl portion of the alkyl carboxylic acid halide may comprise at least two terminal methyl groups (CH3 ) at each end of the chain, and may comprise methylene (CH2 ), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C - C bonds within isopropyl ((CH3 ) CH) and t-butyl
((CH3J C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl
C - C bonds comprise functional groups. The branched-chain-alkane groups in carboxylic acids are equivalent to those in branched-chain alkanes.
The alkyl carboxylic acid halide C = O and C — C(O) groups are equivalent to those given in the Aldehydes section and the Ketones section, respectively. The values of
EΛαtom — αtom,mspι .Aθ\ given in these sections are — 2.69893 e V and — 1.44915 eV , respectively.
As in the case of alkyl halides, each (O)C- X group is solved by hybridizing the 2s and 2 p AOs of the C atom to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between the C2sp3 HO and the X AO to form a MO permits each participating orbital to decrease in radius and energy. For example, to meet the equipotential condition of the union of the (O)C- Cl H2 -type-ellipsoidal-MO with other orbitals of the molecule, the hybridization factor C2 of Eq. (15.52) for the (O)C- Cl -bond MO given by Eq. (15.111) is C2 (C2sp3HO to Cl) - 0.81317 . The solution is equivalent to that of the alkyl chloride bond except that the energy parameters corresponding to oscillation in the transition state are matched to those of a carboxylic acid chloride.
As in the case with the C - Cl group of alkyl chlorides, E.A atom- atom,msp3.AOj of the (O)C- Cl -bond MO in Eq. (15.52) of alky carboxylic acid chlorides due to the charge donation from the C and Cl atoms to the MO is -1.44915 eV where both energy contributions are given by Eq. (14.511). This matches the energy of the C - C(O) functional group with that of the {O)C- Cl group within the carboxylic acid chloride molecule.
The symbols of the functional groups of alkyl carboxylic acid chlorides are given in Table 15.129. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of alkyl carboxylic acid chlorides are given in Tables 15.130, 15.131, and 15.132, respectively. The total energy of each alkyl carboxylic acid chloride given in Table 15.133 was calculated as the sum over the integer multiple of each E0 (βmup) of Table 15.132 corresponding to functional-group composition of the molecule. The bond angle parameters of alkyl carboxylic acid chlorides determined using Eqs. (15.79-15.108) are given in Table 15.134.
Figure imgf000651_0001
Figure imgf000652_0001
Figure imgf000653_0001
Figure imgf000654_0001
CARBOXYLIC ACID ANHYDRIDES ( CnH2n_2O3, n = 2,3,4,5...oo )
The alkyl carboxylic acid anhydrides, CπH2n_2C>3 , have two (θ)C - O moieties that each comprise C - O and C- O functional groups. The single bond of carbon to the carbonyl carbon atom, C — C(O) , is also a functional group. The alkyl portion of the alkyl carboxylic acid anhydride may comprise at least two terminal methyl groups (CH3 ) at each end of the chain, and may comprise methylene (CH2), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C- C bonds within isopropyl ((CH3) CH) and t-butyl ((CH3) C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C- C bonds comprise functional groups. The branched-chain-alkane groups in carboxylic acid anhydrides are equivalent to those in branched- chain alkanes.
The alkyl carboxylic acid anhydride C = O and C - C(O) groups are equivalent to those given in the Carboxylic Acid Esters section and the Ketones section, respectively. The values of
Eri atom - atom,msp3. AOJ given in these sections are -2.69893 eV and -1.44915 eV , respectively. The C- O group is also equivalent to that given in the Carboxylic Acid Esters section except that E1. [atom— atom,msp3.AO \ is equivalent to that of an alkyl ether as given in the corresponding section and the energy terms due to oscillation in the transition state are matched to that of a carboxylic acid anhydride.
For the C- O group, E7. (atom- atom,msp3.AOj is -1.65376 eV . It is based on the energy match between the O AO and the C2sp3 HO of each C - C(O) group and is given by the linear combination of -0.72457 eV (Eq. (14.151)) and -0.92918 eV (Eq. (14.513)), respectively. This matches -0.72457 eV , the energy contribution of each of the C2.sp3 HOs to each C - C(O) functional group, with that of the corresponding energy component of the C- O group and gives a minimum energy within the carboxylic acid anhydride molecule.
The symbols of the functional groups of alkyl carboxylic acid anhydrides are given in Table 15.135. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of alkyl carboxylic acid anhydrides are given in Tables 15.136, 15.137, and 15.138, respectively. The total energy of each alkyl carboxylic acid anhydride given in Table 15.139 was calculated as the sum over the integer multiple of each ED{cmφ) of Table 15.138 corresponding to functional-group composition of the molecule. The bond angle parameters of alkyl carboxylic acid anhydrides determined using Eqs. (15.79-15.108) are given in Table 15.140.
Figure imgf000657_0001
Figure imgf000658_0001
Figure imgf000659_0001
Figure imgf000660_0001
NITRILES ( CnH2n^N, n = 2,3,4,5...°- )
The nitriles, CnH2n^N , comprise a C ≡ N functional group, and the single bond of carbon to the nitrile carbon atom, C - CN , is also a functional group. The alkyl portion of the nitrile may comprise at least two terminal methyl groups ( CH3 ) at each end of the chain, and may comprise methylene (CH2 ), and methylyne ( CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C - C bonds within isopropyl ([CH3J CH) and t-butyl ((CH3J C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C - C bonds comprise functional groups. The branched-chain- alkane groups in nitriles are equivalent to those in branched-chain alkanes.
The nitrile C ≡ N is solved equivalently to acetylene as given in the Acetylene Molecule section except that the energy for AEH MO {AO/HO) is two times that given in Eq. (14.343),
16.20002 eV , in order to match the N AOs to that of the nitrile C2sp3 HO having a bond order of three. E.A atom- atom^msp1 , AOJ of the C = N functional group is -1.56513 eV (Eq.
(14.342)) corresponding to the third-order bonded C2sp* HO.
The C - CN functional group is equivalent to that of an alkyl C- C group given in the Continuous-Chain Alkanes section except that E1. [H1Mo) and EKvlh are those corresponding to a nitrile. As given in the Continuous-Chain Alkanes section, E.A atom - atom, msp3. AOJ of the alkyl C- C group is -1.85836 eV where both energy contributions are given by Eq. (14.513). It is based on energy matching within the nitrile. It corresponds to the maximum-magnitude energy contributions of a single-bonded and a third-order bonded C2sp3 HO.
The symbols of the functional groups of nitriles are given in Table 15.141. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of nitriles are given in Tables 15.142, 15.143, and 15.144, respectively.1 The total energy of each nitrile given in Table 15.139 was calculated as the sum over the integer multiple of each ED (c;,mφ) of Table 15.144 corresponding to functional- group composition of the molecule. For each set of unpaired electrons created by bond breakage, the C2spi HO magnetic energy Ema that is subtracted from the weighted sum of the ED[cmup) (eV) values based on composition is given by Eq. (15.58). The bond angle parameters of nitriles determined using Eqs. (15.79-15.108) are given in Table 15.146. The C of the C = N group can further bond with only one atom, and the bond is linear as a minimum of energy as in the case of acetylene and alkynes.
Figure imgf000664_0001
Figure imgf000665_0001
Figure imgf000666_0001
THIOLS (C ,, » = 1,2,3,4.5..*»)
The alkyl thiols, CnH2n+2S1n , comprise a SH functional group and a C- S functional group. The alkyl portion of the alkyl thiol may comprise at least two terminal methyl groups (CH3) at each end of the chain, and may comprise methylene (CH2), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C- C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C- C bonds within isopropyl ((CH3) CH) and t-butyl ((CH3)^ C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C- C bonds comprise functional groups. The branched-chain-alkane groups in thiols are equivalent to those in branched-chain alkanes.
The parameters of the SH functional group is solved using Eq. (15.41). As in the case of the C- H bonds of CHn n = 1,2,3, the S- H -bond MO is a partial prolate spheroid in between the sulfur and hydrogen nuclei and is continuous with the S3p shell. The energy of the H2 -type ellipsoidal MO is matched to that of the S3p shell and comprises 75% of a H2 -type ellipsoidal MO in order to match potential, kinetic, and orbital energy relationships. Since the energy of S , E{S) = -10.36001 eV , is less that that of H, the linear combination of the H2- type ellipsoidal MO with the S3p shell further comprises an excess 50% charge-density donation from H to the S3p shell of the S- H -bond MO to achieve an energy minimum. The initial total energy of the shell is given by the sum over the four 3p electrons. From Eq.
(15.12), the sum Er (s,3p)ofthe energies of S, S+ , S2+, and S3+ [38] is
ET(S,3p) = 10.36001 eF + 23.33788 eF + 34.79 eF + 47.222 eV .
= 115.70989 eV
By considering that the central field decreases by an integer for each successive electron of the shell, the radius r. of the S3p shell may be calculated from the Coulombic energy using Eqs. (15.13) and (15.118):
(15.119)
Figure imgf000667_0001
where Z = 16 for sulfur. Using Eqs. (15.14) and (15.119), the Coulombic energy EOm!ιmh(S,3p) of the outer electron of the S3p shell is (15.120)
Figure imgf000668_0002
V
The sharing of the electrons between the S and H atoms permits the formation an
S- H -bond MO that is lowered more in energy than the participating S3p orbital which consequently increases in energy. By considering the 50% electron redistribution in the S- H group as well as the fact that the central field decreases by an integer for each successive electron of the shell, the radius rs_H3 of the S3p shell may be calculated from the Coulombic energy using Eq. (15.18)
(15.121)
Figure imgf000668_0001
where the $ = -1 in Eq. (15.18) due to the charge donation from H to S . Using Eqs. (15.19) and (15.121), the Coulombic energy ECm/mώySs_H,3p) of the outer electron of the S3p shell is
(15.122)
Figure imgf000668_0003
Thus, ET [S- H,3pj, the energy change of each S3p shell with the formation of the S- H- bond MO is given by the difference between Eq. (15.120) and Eq. (15.122): Er(S- H,3p) = E(Ss_H,3p)- E(s,3p)
= -11.01999 eF -(-l 1.57099 eF) (15.123)
= 0.55100 eF Then, in Eq. (15.42):
E1 (AO / HO) = E(s)- Er (S- H,3p)
= -10.36001 eF- 0.55100 eF (15.124)
= -10.91101 eV
And, in Eq. (15.56),
E1. (atom - atom,msp\AO) = 0.55100 eV (15.125) Due to the charge donation from H to S , c, = 1 in both Eqs. (15.42) and (15.56). As in the case of the C- H -bond MO, C, = 0.75 based on the orbital composition. In alkyl thiols, the energy of sulfur is less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). Thus, C2 in Eq. (15.52) is also one, and the energy matching condition is determined by the C2 parameter. Using the energy of S , E(S) = -10.36001 eV in Eq. (15.65), the hybridization factor C2 of Eq. (15.52) for the S - H -bond MO is
(15.126)
Figure imgf000669_0001
Since the energy of S is matched to the Coulombic energy between the electron and proton of
H , 4HM) > E««,a, U "'*>) = E(HM) = -13.60580 eV , Emιlιa/ (s Aoi HO) = E(H) = -13.59844 eV , and Emgg is that corresponding to E(H(O0)) given by
Eq. (15.58). ED[am,φ) for hydrogen sulfide is equivalent to that of the SH functional group, and the ED (ttιωφ) (eV) for dihydrogen sulfide follows the same derivation as that for the SH functional group except that the parameters correspond to n( = 2 rather than n, = 1 in Eqs.
(15.42) and (15.56). Furthermore, with the energy of S matched to the Coulombic energy between the electron and proton of H , the energy of the C - S -bond MO is the sum of the component energies of the H2 -type ellipsoidal MO given in Eq. (15.42) with E{AO/HO) = 0 and
E1 (AO / HO) = AEN ^j (AO / HO) . Then, the solution of the C - S functional group
comprises the hybridization of the 2s and 2p AOs of C to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between the C2sp3 HO and the S AO to form a MO permits each participating orbital to decrease in radius and energy. Since the energy of sulfur is less than the Coulombic energy between the electron and proton of H given by Eq, (1.243), c, in Eq. (15.52) is one, and the energy matching condition is determined by the C2 parameter. Then, C2,sρ3 HO has a hybridization factor of 0.91771 (Eq. (13.430)) with a corresponding energy of E(c,2sp*) = -14.63489 eV (Eq. (15.25)), and the S AO has an energy of E(s) = -10.36001 eV . To meet the equipotential condition of the union of the C - S H2 -type-ellipsoidal-MO with these orbitals, the hybridization factor C2 of Eq. (15.51) for the C - S -bond MO given by Eqs. (15.68) and (15.70) is
C2 (C2sp3HO to S) = ,EW ^ c2 (C2sp3Hθ) = ~1036m *V (0.91771) = 0.64965 (15.127) 2 { F ' E(c,2sp ) V ' -14.63489 eV X '
Since the sulfur is energy matched to E(H(CI0 M in the 5- H -bond MO,
Er ( atom - atom, msp3.AOj of the C - S -bond MO in Eq. (15.52) due to the charge donation from the C and S atoms to the MO is -0.72457 eV corresponding to the energy contribution equivalent to that of a methyl group (Eq. (14.151)).
The symbols of the functional groups of branched-chain alkyl thiols are given in Table
15.147. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.1 1) and (15.17-15.56)) parameters of alkyl thiols are given in Tables
15.148, 15.149, and 15.150, respectively. The total energy of each alkyl thiol given in Table 15.151 was calculated as the sum over the integer multiple of each ED[GmuP) of Table 15.150 corresponding to functional-group composition of the molecule. For each set of unpaired electrons created by bond breakage, the C2sp3 HO magnetic energy Ema that is subtracted from the weighted sum of the ED(am«P) (eV) values based on composition is given by Eq.
(15.58). The C - C bonds to the HCSH group (one H bond to C ) were each treated as an iso C- C bond. The C- C bonds to the CSH group (no H bonds to C) were each treated as a tert-butyl C - C . Emg was subtracted for each t-butyl group. The bond angle parameters of alkyl thiols determined using Eqs. (15.79-15.108) are given in Table 15.152.
Figure imgf000671_0001
Figure imgf000672_0001
Figure imgf000673_0001
Figure imgf000674_0002
Figure imgf000674_0001
SULFIDES ( CnH2n+2S111, H = 2,3,4,5...OO )
The alkyl sulfides, CnH2n+2Sn , comprise two types of C - S functional groups, one for t- butyl groups corresponding to the C and the other for the remaining general alkyl groups including methyl. The alkyl portion of the alkyl sulfide may comprise at least two terminal methyl groups (CH3 ) at each end of the chain, and may comprise methylene (CH2), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C- C bonds within isopropyl ((CH3) CH) and t-butyl ([CH3) C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl
C- C bonds comprise functional groups. The branched-chain-alkane groups in sulfides are equivalent to those in branched-chain alkanes.
Each C - S group is solved by hybridizing the 2s and 2p AOs of the C atom to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between the C2sp3 HO and the S AO to form a MO permits each participating orbital to decrease in radius and energy. Since the energy of sulfur is less than the Coulombic energy between the electron and proton of H given by Eq. (1.243), c2 in Eq. (15.52) is one, and the energy matching condition is determined by the C2 parameter. As in the case of thiols, C2 of Eq. (15.52) for the C - S -bond MO given by Eq. (15.127) is C2 (C2sp3HO to s) = 0.64965 . The C - S group of alkyl sulfides is equivalent to that of thiols where
Er (atom- atom,msp3.Aθ) is -0.72457 eV (Eq. (14.151)). The t-butyl- C - S group is also equivalent to that of thiols except that the energy parameters corresponding to the oscillation in the transition state are matched to those of the t-butyl group.
The symbols of the functional groups of branched-chain alkyl sulfides are given in Table 15.153. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of alkyl sulfides are given in Tables 15.154, 15.155, and 15.156, respectively. Consider that the C - S bond is along the x axis in the xy-plane. The S nucleus is at the focus +c and the C nucleus is at the focus -c. The elliptic angle β' is taken as counterclockwise from the x-axis for S and as clockwise from the -x-axis for C . The total energy of each alkyl sulfide given in Table 15.157 was calculated as the sum over the integer multiple of each ED (amuP) of Table 15.156 corresponding to functional-group composition of the molecule. Em given by Eq. (15.58) was subtracted for each t-butyl group.
The bond angle parameters of alkyl sulfides determined using Eqs. (15.79-15.108) are given in Table 15.158.
Figure imgf000677_0001
Figure imgf000678_0001
Figure imgf000679_0001
Figure imgf000680_0001
Figure imgf000681_0001
DISULFIDES ( C H2n+2S2n,, « = 2,3,4,5...oo)
The alkyl disulfides, CnH2n+2S2n , comprise C - S and S - S functional groups. The alkyl portion of the alkyl disulfide may comprise at least two terminal methyl groups ( CH3 ) at each end of the chain, and may comprise methylene (CH2 ), and methylyne ( CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C - C bonds within isopropyl ([CH3) CH) and t-butyl ((CH3) C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C - C bonds comprise functional groups. The branched-chain-alkane groups in disulfides are equivalent to those in branched-chain alkanes.
Each C - S group is equivalent to that of general alkyl sulfides given in the corresponding section. As in the case of thiols and sulifϊdes, C2 of Eq. (15.52) for the C- S - bond MO given by Eq. ( 15.127) is C2 (C2sp3HO to s) = 0.64965 and Eτ [atom- atom,msp\AO) is -0.72457 eV (Eq. (14.151)).
The S - S group is solved as an H, -type-ellipsoidal-MO that is energy matched to the energy of sulfur, E(S) = -10.36001 eV , such that E(AO I HO) = -10.36001 eV in Eq. (15.42) with E1, [AO I HO) = E[AO I HO) . The S - S -bond MO is further energy matched to the C2sp3 HO of the C - S -bond MO. C2 of Eq. (15.52) for the 5- S-bond MO given by Eq. (15.127) is also C2 (C2sp3HO to s) = 0.64965 . In order to match Er[atom- atom,msp2 ' .Aθ)
of the C - S group (-0.72457 e V (Eq. (14.151))), Eτ {atom- atom,τnsp3.Aθ) of the S - S - bond MO is determined using a linear combination of the AOs corresponding to -0.72457 eV and 0 eV in Eq. (15.29), Eq. (15.31), and Eqs. (15.19-15.20). The result corresponding to bond order 1/21 in Table 15.2 is E1, (atom- atom,msp3.Aθ) = -0.36229 eV . The symbols of the functional groups of branched-chain alkyl disulfides are given in
Table 15.159. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of alkyl disulfides are given in Tables 15.160, 15.161, and 15.162, respectively. The total energy of each alkyl disulfide given in Table 15.163 was calculated as the sum over the integer multiple of each ED(cm<P) of Table 15.162 corresponding to functional-group composition of the molecule. E given by Eq.
(15.58) was subtracted for each t-butyl group. The bond angle parameters of alkyl disulfides determined using Eqs. (15.79-15.108) are given in Table 15.164.
Figure imgf000684_0001
Figure imgf000685_0001
Figure imgf000686_0001
Figure imgf000687_0001
SULFOXIDES ( C H2n+2 (£6>)m , « = 2,3,4,5...~ )
The alkyl sulfoxides, CnH2n+2 (SO) , comprise a C-SO- C moiety that comprises
C -S and 50 functional groups. The alkyl portion of the alkyl sulfoxide may comprise at least two terminal methyl groups (CH3) at each end of the chain, and may comprise methylene (CH2), and methylyne ( CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C- C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C- C bonds within isopropyl ((CH3) CH) and t-butyl ((CH3J3C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t- butyl C- C bonds comprise functional groups. The branched-chain-alkane groups in sulfoxides are equivalent to those in branched-chain alkanes.
The electron configuration of oxygen is l$22s22p* , and the orbital arrangement given by Eq. (10.154) has two unpaired electrons corresponding to the ground state 3P2. The 50 functional group comprises a double bond between the two unpaired electrons of O . The sulfur atom is energy matched to the C2spi HO. In alkyl sulfoxides, the C2spi HO has a hybridization factor of 0.91771 (Eq. (13.430)) with a corresponding energy of E(c,2spή = -14.63489 eV (Eq. (15.25)), and the S AO has an initial energy of
E(S) = -10.36001 eV [38]. To meet the equipotential condition of the union of the 5 = O H2- type-ellipsoidal-MO with these orbitals, the hybridization factor c2 of Eq. (15.52) for the S = O -bond MO given by Eqs. (15.68) and (15.70) is
C2(O to S3sp3 to C2sp3Hθ) = -γ-!-c2(C2sp3Hθ)
_ -13.61806 eV -10.3600I eV = 1.20632
The 5 atom also forms a single bond with each of the C2.sp3 HOs of the two C- S groups. The formation of these bonds is permitted by the hybridization of the four electrons of the S3p shell to give the orbital arrangement:
3sp3 state (15.129)
Figure imgf000688_0001
where the quantum numbers (i,me) are below each electron. The 3s shell remains unchanged. Then, the Coulombic energy ECoulomh(S,3sp3 \ of the outer electron of the S3sp3 shell given by Eq. (15.118) with r^ = 1.17585α0 (Eq. (15.119)) is -11.57099 eV . Using Eq. (15.16) with the
radius of the sulfur atom rI6 = 1.32010α0 given by Eq. (10.341), the energy E(S3sp3) of the 5 outer electron of the S3sp3 shell is given by the sum of ECmhmh(S3sp3) and E{magnetic) :
(m30)
Figure imgf000689_0001
= -11.57099 eV + 0.04973 = -11.52126 eV
Then, the hybridization energy Ehyhrjώzal. [S3spΛ of the S3sp3 HO is
= -11.52126 eV- 10.3600I eF (15.131)
= -1.16125 eV
The SO group is matched to the C -S group with which it shares the common 10 hybridized 5 atom. Consequently, Eh hrldr.alj [S3sp3\ is subtracted from ET \ Groupj in the determination of ED\Groupj (Eq. 15.56)). Furthermore, the energy of the S = O -bond MO is the sum of the component energies of the H2-type ellipsoidal MO given in Eq. (15.42) with the energy matched to the final energy of the hybridized S atom such that E(AO I HO) = E(S3SP 3 )=- 11.52126 eV a n d
15 USHi^(A0 / H0)= Elvi^(S3spi) = -lΛ6125 eV . T h e n ,
Er(AO / Hθ) = E(S) = -\0.36QQ\ eV . Also, Ef [atom- atom, msp\Aθ) of the S = O bond is zero since there are no bonds with a C2sp3 HO.
The C - S group is solved as an energy minimum by hybridizing the 2s and 2p AOs of the C atom to form a single 2sp3 shell and by hybridizing the four S3p electrons to form a
20 53^p3 shell, and the sharing of electrons between the C2^p3 HO and the 6"3^p3 HO to form a
MO permits each participating orbital to decrease in radius and energy. Using the Coulombic energy of the S3sp3 shell, ECoιιhmh(s3sp3) given by Eq. (15.120) in Eq. (15.63), the S3sp3 -shell
hybridization factor, c2 (53,Sp3 J , is
0-85045 (15.132)
Figure imgf000690_0001
As in the case of thiols, sulfides, and disulfides, the energy of sulfur is less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). Thus, C1 and c2 are equal to one in Eq. (15.52), and the energy matching condition is determined by the C2 parameter. In alkyl sulfoxides, the C2sp3 HO has a hybridization factor of 0.91771 (Eq. (13.430)) with a corresponding energy of E[c,2sp3) = -14.63489 eV (Eq. (15.25)) and the. S3sp3 HO has an energy of E(S3sp3)=- 11.52126 eV (Eq. (15.130)). To meet the equipotential condition of the union of the C -S H2 -type-ellipsoidal-MO with these orbitals, the hybridization factor C2 of Eq. (15.52) for the C - S -bond MO given by Eqs. (15.68) and (15.70) is
C2 (c2sp3HO to S3sp3) = } Ψ !s C2(S3sp3) = ~1 L52126 gV (θ,85O45) = 0.66951 (15.133) n ' E(c,2sp ) -14.63489 eV K '
As in the case of thiols, sulfides, and disulfides, with the energy of S matched to the Coulombic energy between the electron and proton of H , the energy of the C - S -bond MO is the sum of the component energies of the H2 -type ellipsoidal MO given in Eq. (15.42) with
E(AO/HO) = 0 and Er(AO I Hθ) = AEH MO(AO I HO) . For sulfoxides ,
AEH MO (AO / Hθ) = -0.72457 eV . Fu rth er eq ui v al en tl y ,
E7 [atom - atom,msp\Aθ) = -0.72457 eV (Eq. (14.151)). The symbols of the functional groups of branched-chain alkyl sulfoxides are given in
Table 15.165. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of alkyl sulfoxides are given in Tables 15.166, 15.167, and 15.168, respectively. Consider that the C - Sbond is along the x axis in the xy-plane. The S nucleus is at the focus +c and the C nucleus is at the focus -c. The elliptic angle θ' is taken as counterclockwise from the x-axis for S and as clockwise from the -x-axis for C . The total energy of each alkyl sulfoxide given in Table 15.169 was calculated as the sum over the integer multiple of each ED(GmuP) of Table 15.168 corresponding to functional- group composition of the molecule. The bond angle parameters of alkyl sulfoxides determined using Eqs. (15.79-15.108) are given in Table 15.170.
Figure imgf000691_0001
Figure imgf000692_0001
Figure imgf000692_0002
Figure imgf000693_0001
Figure imgf000694_0001
SULFOXIDES ( CnH2n+2 (sO)m , n = 2,3,4,5...oo )
The allcyl sulfoxides, CnH2n+2 (so) , comprise a C- SO - C moiety that comprises
C - S and SO functional groups. The alkyl portion of the alkyl sulfoxide may comprise at least two terminal methyl groups (CH3 ) at each end of the chain, and may comprise methylene (CH2), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C- C bonds within isopropyl ((CH3) CH) and t-butyl ((CH3J C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t- butyl C - C bonds comprise functional groups. The branched-chain-alkane groups in sulfoxides are equivalent to those in branched-chain alkanes.
The electron configuration of oxygen is ls22s22pA , and the orbital arrangement given by Eq. (10.154) has two unpaired electrons corresponding to the ground state 3P2. The SO functional group comprises a double bond between the two unpaired electrons of O . The sulfur atom is energy matched to the C2spl HO. In alkyl sulfoxides, the C2sp3 HO has a hybridization factor of 0.91771 (Eq. (13.430)) with a corresponding energy of ε{c,2sp^ = -14.63489 eV (Eq. (15.25)), and the S AO has an initial energy of
-5(S) = -10.3600I eF [38]. To meet the equipotential condition of the union of the S = O H2- type-ellipsoidal-MO with these orbitals, the hybridization factor c2 of Eq. (15.52) for the S - C-bond MO given by Eqs. (15.68) and (15.70) is
C1(O to S3sp3 to C2sp3Hθ)
Figure imgf000695_0001
= -lM1806£y
-10.36001 eV K '
= 1.20632
The S atom also forms a single bond with each of the C2sp3 ΗOs of the two C - S groups. The formation of these bonds is permitted by the hybridization of the four electrons of the S3p shell to give the orbital arrangement:
(15.129)
Figure imgf000695_0002
where the quantum numbers {i,me) are below each electron. The 3s shell remains unchanged.
Then, the Coulombic energy ECgιιhmh[s,3sp3) of the outer electron of the S3sp3 shell given by Eq. (15.118) with r^ = 1.17585α0 (Eq. (15.119)) is -11.57099 eV . Using Eq. (15.16) with the
radius of the sulfur atom r16 = 1.32010α0 given by Eq. (10.341), the energy E[S3sp3) of the outer electron of the S3sp3 shell is given by the sum of E(.mιhmh[S3sp3) and E(magnetic) :
Figure imgf000696_0001
, , , .... .. = -11,57099 eV + 0.04973 , .
= -11.52126 eV
Then, the hybridization energy Eh≠rjdlzaljm [s3sp3) of the 53^3 HO is
V**, (5V) = E(S3SP 3)- E[S)
= -11.52126 eF- 10.36001 eF (15.131)
= -1.16125 eK The 50 group is matched to the C - S group with which it shares the common hybridized S atom. Consequently, Ehyhr.d.zaljm\ S3sp3\ is subtracted from ET [Groupj in the determination of ED\Group) (Eq. 15.56)). Furthermore, the energy of the S = O -bond MO is the sum of the component energies of the H2-type ellipsoidal MO given in Eq. (15.42) with the energy matched to the final energy of the hybridized S atom such that E(AO / HO) = E(S3sp3)=- \ 1.52126 eV a n d MHi^(A0 / H0) = Elφ^(S3spι) = -lΛ6125 eV . T h e n ,
E1. [AO I HO) = E(S) = -10.3600I eF . Also, E7. [atom- atom, msp\Aθ) of the S= O bond is zero since there are no bonds with a C2sp3 HO.
The C - S group is solved as an energy minimum by hybridizing the 2s and 2p AOs of the C atom to form a single 2sp3 shell and by hybridizing the four £3/? electrons to form a S3sp3 shell, and the sharing of electrons between the C2sp3 HO and the 53^3 HO to form a
MO permits each participating orbital to decrease in radius and energy. Using the Coulombic energy of the S3sp3 shell, ^^(s^3) given by Eq. (15.120) in Eq. (15.63), the S3sp3 -shell hybridization factor, c2 lS3$p3), is
(15.132)
Figure imgf000697_0001
As in the case of thiols, sulfides, and disulfides, the energy of sulfur is less than the Coulonibic energy between the electron and proton of H given by Eq. (1.243). Thus, cχ and C2 are equal to one in Eq. (15.52), and the energy matching condition is determined by the C2 parameter. In alkyl sulfoxides, the C2sp3 HO has a hybridization factor of 0.91771 (Eq. (13.430)) with a corresponding energy of E(c,2sp2) = -14.63489 eV (Eq. (15.25)) and the
S3sp3 HO has an energy of E(s3spή= - 11.52126 eV (Eq. (15.130)). To meet the equipotential condition of the union of the C -S H2 -type-ellipsoidal-MO with these orbitals, the hybridization factor C2 of Eq. (15.52) for the C - S -bond MO given by Eqs. (15.68) and (15.70) is
C2 (C2sp3HO to S3sp3) = / Ψ L c2 (S3$p3) = ~π-52126 gF (0,85045) = 0.66951 (15.133) Λ y ' E(c,2sp3) Λ ' -14.63489 eV V '
As in the case of thiols, sulfides, and disulfides, with the energy of S matched to the Coulombic energy between the electron and proton of H , the energy of the C- S -bond MO is the sum of the component energies of the H2 -type ellipsoidal MO given in Eq. (15.42) with
E{AOIHO) = Q and E7(AO I HO) = Δ£H UO (AO I HO) . For sulfoxides,
AEH M0 (AO I HO) = -0.72457 eV . Fu rth er eq u i v al en tl y ,
E1. (atom - atom, msp3.AO) = -0.72457 e V (Eq. (14.151)). The symbols of the functional groups of branched-chain alkyl sulfoxides are given in
Table 15.165. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of alkyl sulfoxides are given in Tables 15.166, 15.167, and 15.168, respectively. Consider that the C - S bond is along the x axis in the xy-plane. The 5 nucleus is at the focus +c and the C nucleus is at the focus -c. The elliptic angle Θ' is taken as counterclockwise from the x-axis for S and as clockwise from the -x-axis for C . The total energy of each alkyl sulfoxide given in Table 15.169 was calculated as the sum over the integer multiple of each E0 (omup) of Table 15.168 corresponding to functional- group composition of the molecule. The bond angle parameters of alkyl sulfoxides determined using Eqs. (15.79-15.108) are given in Table 15.170.
Figure imgf000698_0001
Figure imgf000699_0001
Figure imgf000699_0002
Figure imgf000700_0001
Figure imgf000701_0001
DIMETHYL SULFOXIDE DIHEDRAL ANGLE
The dihedral angle θ^0/csc between the plane defined by the CSC MO comprising a linear combination of two S- C -bond MOs and a line defined by the S = O -bond MO where S is the central atom is calculated using the results given in Table 15.170 and Eqs. (15.105-15.108). The distance d] along the bisector of θzcsc from S to the internuclear-distance line between C and
C , 2c'c,_c , is given by
(15.134)
Figure imgf000702_0003
where 2c's._c is the internuclear distance between S and C . The atoms C , C , and O define the base of a pyramid. Then, the pyramidal angle θzcoc can be solved from the internuclear distances between C and C , 2c 'r r , and between C and O, 2c 'c_0 , using the law of cosines (Eq. (15.106)):
Figure imgf000702_0004
Figure imgf000702_0001
= 60.27°
Then, the distance d. along the bisector of θ,coc from O to the internuclear-distance line 2c'(,_c , is given by
(15.136)
Figure imgf000702_0002
The lengths ds , d2 , and 2c's=0 define a triangle wherein the angle between rf, and the internuclear distance between O and S , 2c\=0 , is the dihedral angle θ^^n.^, that can be solved using the law of cosines (Eq. (15.108)):
(15.137)
Figure imgf000702_0005
= 115.74C The experimental [1] dihedral angle Θ/i!=0/csc is
- 115 5° <15-138)
SULFITES ( C H2n+2 (SOj)11 , n = 2,3,4,5...oo )
The alkyl sulfites, C11H2n+2 (SO3) , comprise a C - O - SO - O - C moiety that comprises two types C- O functional groups, one for methyl and one for alkyl, and O- S and SO functional groups. The alkyl portion of the alkyl sulfite may comprise at least two terminal methyl groups (CH3 ) at each end of the chain, and may comprise methylene (CH2), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C- C bonds within isopropyl ((CH3) CH) and t-butyl ((CH3) C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl
C- C bonds comprise functional groups. The branched-chain-alkane groups in sulfites are equivalent to those in branched-chain alkanes.
The SO functional group is equivalent to that of sulfoxides with
Eτ ( atom - atom, msp3.AOj = O as given in the Sulfoxides section. The methyl and alkyl C- O functional groups having Eτ ( atom- atom, msp3. AO) = -1.44915 eV and
EJatom- atom,msp3.AO) = -1.65376 eV , respectively, are equivalent to the corresponding ether groups given in the Ethers section except for the energy terms corresponding to oscillation of the bond in the transition state.
The electron configuration of oxygen is Is22s22p4 , and the orbital arrangement given by Eq.. (10.154) has two unpaired electrons corresponding to the ground state 3P2. The SO functional group comprises a double bond between the S atom and the two unpaired electrons of O . The S atom also forms single bonds with two additional oxygen atoms that are each further bound to methyl or alkyl groups. The first bond-order bonding in the O- S groups is between the sulfur atom and a O2p AO of each oxygen of the two bonds. The formation of these four bonds with the sulfur atom is permitted by the hybridization of the four electrons of the S3p shell to give the orbital arrangement given by Eq. (15.129). Then, the Coulombic energy ECmι/mώ[S,3sp3) of the outer electron of the S3sp3 shell given by Eq. (15.120) with r3φ, = 1.17585α0 (Eq. (15.119)) is -11.57099 eV . Using Eq. (15.16) with the radius of the sulfur atom r]6 = 1.3201Oa0 given by Eq. (10.341), the energy E\ S3sp2 J of the outer electron of the S3sp3 shell given by the sum of ECouh AS3sp3) and E{magnetic) is
£(S3^3) = -11.52126 eF(Eq. (15.130)).
Thus, the O- S group is solved as an energy minimum by hybridizing the four S3p electrons to form a S3sp3 shell, and the sharing of electrons between the O2p AO and the
S3sp3 HO to form a MO permits each participating orbital to decrease in radius and energy. As in the case of thiols, sulfides, disulfides, and sulfoxides, the energy of sulfur is less than the
Coulombic energy between the electron and proton of H given by Eq. (1.243). Thus, c, and c2 are equal to one in Eq. (15.52), and the energy matching condition is determined by the C2 parameter. Each C2sp3 HO has a hybridization factor of 0.91771 (Eq. (13.430)) with a corresponding energy of E{c,2sp3) = -14.63489 eV (Eq. (15.25)), and the S HO has an energy of E\S3sp3 J = -11.52126 eV . To meet the equipotential condition of the union of the O- S H2 -type-ellipsoidal-MO with these orbitals with the oxygen that further bonds to a
C2sp3 HO, the hybridization factor C2 of Eq. (15.52) for the O - S -bond MO given by Eqs. (15.68) and (15.70) is
Figure imgf000705_0001
. -1152126 «V( )
-13.61806 eVy '
= 0.77641
As in the case of thiols, sulfides, disulfides, and sulfoxides, with the energy of S matched to the Coulombic energy between the electron and proton of H , the energy of the O- S -bond MO is the sum of the component energies of the H2 -type ellipsoidal MO given in Eq. (15.42) with E(A0 / Hθ) = 0 and ET[AO I H0) = AEH MO (AO I HO) . For sulfites, ΔEHIM0 (A0 / H0) = -0.92918 eV a n d e q u i v a l e n t l y ,
E1 ( atom- alom,msp3. AOj = -0.92918 eV (Eq. (14.513)) due to the maximum energy match with the oxygen AO as in the case with carboxylic acid esters. The symbols of the functional groups of branched-chain alkyl sulfites are given in Table
15.177. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of alkyl sulfites are given in Tables
15.178, 15.179, and 15.180, respectively. The total energy of each alkyl sulfite given in Table 15.175 was calculated as the sum over the integer multiple of each ED [GWUP) of Table 15.180 corresponding to functional-group composition of the molecule. The bond angle parameters of alkyl sulfites determined using Eqs. (15.79-15.108) are given in Table 15.182.
Figure imgf000706_0001
Figure imgf000707_0001
Figure imgf000708_0001
Figure imgf000709_0001
Figure imgf000710_0001
SULFATES (CHH2n+2 [S04)m , « = 2,3,4,5...°o )
The alkyl sulfates, CnH2n+2 (SO4) , comprise a C - O - SO2 - O - C moiety that comprises two types C - O functional groups, one for methyl and one for alkyl, and O - S and SO2 functional groups. The alkyl portion of the alkyl sulfate may comprise at least two terminal methyl groups (CH3) at each end of the chain, and may comprise methylene (CH2), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C- C bonds within isopropyl ([CH3J CH) and t-butyl ((CH3] C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl
C- C bonds comprise functional groups. The branched-chain-alkane groups in sulfates are equivalent to those in branched-chain alkanes.
The methyl and alkyl C- O functional groups having
E1, [atom - atom,mψι \Aθ} = -1.44915 eV and E1 [atom - atom^sp* .AO) = -1.65376 eV , respectively, are equivalent to the corresponding groups given in the Sulfites section. The O -S functional group having E1, (atom- atom,msp3.AOj = -0.92918 e^ is equivalent to that given in the Sulfites section. The SO2 functional group is equivalent to that of sulfones with Er\ atom - atom,msp3.AO) — 0 as given in the Sulfones section.
The symbols of the functional groups of branched-chain alkyl sulfates are given in Table 15.183. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of alkyl sulfates are given in Tables
15.184, 15.185, and 15.186, respectively. The total energy of each alkyl sulfate given in Table
15.187 was calculated as the sum over the integer multiple of each Eυ[owup) of Table 15.186 corresponding to functional-group composition of the molecule. The bond angle parameters of alkyl sulfates determined using Eqs. (15.79-15.108) are given in Table 15.188.
Figure imgf000712_0001
Figure imgf000713_0001
Figure imgf000714_0001
Figure imgf000715_0001
NiTROALKANES (c H2H+2_m (No2)m, « == 1,2,3,4,5...-)
The nitroalkanes, Cn H2n+ 2_m (NO2) , comprise a NO2 functional group and Ά C - N functional group. The alkyl portion of the nitroalkane may comprise at least two terminal methyl groups (CH3 ) at each end of the chain, and may comprise methylene (CH2), and methylyne (CH) functional groups as well as C bound by carbon-carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C - C bond is the same as that of straight-chain alkanes. In addition, the C- C bonds within isopropyl ((CH3J CH) and t-butyl
((CHJ C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C- C bonds comprise functional groups. The branched-chain-alkane groups in nitroalkanes are equivalent to those in branched-chain alkanes.
The electron configuration of oxygen is \s22s22pA , and the orbital arrangement given by
Eq. (10.154) has two unpaired electrons corresponding to the ground state 1P1 . The electron configuration of nitrogen is \s22s22pz , and the orbital arrangement given by Eq. (10.134) has three unpaired electrons corresponding to the ground state 4^2. The bonding in the nitro ( NO2) functional group is similar to that in the SO2 group given previously. It also has similarities to the bonding in the carbonyl functional group. In the NO2 group, the two unpaired electrons of the O atoms form a MO with two unpaired electrons of the nitrogen atom such that the MO comprises a linear combination of two bonds, each of bond order two involving the nitrogen AOs and oxygen AOs of both oxygen atoms. The nitrogen atom is then energy matched to the C2sp3 HO. In nitroalkanes, the C2sp' HO has a hybridization factor of
0.91771 (Eq. (13.430)) with a corresponding energy of ε(c,2spή = -14.63489 eV (Eq. (15.25)), the N AO has an energy of E(N) = -14.53414 eV , and the O AO has an energy of £■(0) = -13.61806 eV [38]. To meet the equipotential condition of the union of the N - O H2 -type-ellipsoidal-MO with these orbitals, the hybridization factor C2 of Eq. (15.52) for the N = O -bond MO given by Eqs. (15.68) and (15.70) is
Figure imgf000716_0001
Since there are two O atoms in a linear combination that comprises the bonding of the NO2 g σroup L , the unp L aired electrons of each O cancel each others effect such that E mag is not subtracted from the total energy of NO2. Additionally,
Eτ(atom- atom,msp\Aθ) - -3J\613 eV = A{~0.929\^ eV) (Eq. (14.513)) is the maximum given the bonding involves four electrons comprising two bonds, each having a bond order of one.
The C- N group is equivalent to that of primary amines except that the energies corresponding to vibration in the transition state are matched to a nitroalkane and AEN^0 (AO I HO) = -0.72457 eV for nitroalkane and AE^0 (AO I Hθ) = -1.44915 eV for
primary amines. Whereas, E7, (atom - atom,msp2,Aθ) = -1.44915 eV for both functional groups. . This condition matches the energy of the C - N group with the NO2 having M^0 (AO / HO) = O . , .
The symbols of the functional groups of branched-chain nitroalkanes are given in Table
15.189. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of nitroalkanes are given in Tables
15.190, 15.191, and 15.192, respectively. The total energy of each nitroalkane given in Table 15.193 was calculated as the sum over the integer multiple of each ED {amuP) of Table 15.192 corresponding to functional-group composition of the molecule. Ema given by Eq. (15.58) was subtracted for each t-butyl group. The bond angle parameters of nitroalkanes determined using Eqs. (15.79-15.108) are given in Table 15.194.
Figure imgf000717_0001
Figure imgf000718_0001
Figure imgf000718_0002
Figure imgf000719_0001
Figure imgf000719_0002
Figure imgf000720_0001
ALKYL NITRITES ( CnH2n+2_m (NO2)m , « = l,2,3,4,5...°o)
The alkyl nitrites, CnH2n+2_m (NO2)m ., comprise a RC - O - NO moiety that comprises
C- O, O - N , and NO functional groups. The alkyl portion of the alkyl nitrite may comprise at least two terminal methyl groups (CH3 ) at each end of the chain, and may comprise methylene (CH2 ), and methylyne (CH) functional groups as well as C bound by carbon- carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C - C bond is the same as that of straight-chain alkanes. In addition, the C- C bonds within isopropyl ((CH3] CH) and t-butyl ([CH3) C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C- C bonds comprise functional groups. The branched-chain-alkane groups in alkyl nitrites are equivalent to those in branched-chain alkanes.
The electron configuration of oxygen is Is22s22p4 , and the orbital arrangement given by
Eq. (10.154) has two unpaired electrons corresponding to the ground state 3P2. The electron configuration of nitrogen is \s12s22pi, and the orbital arrangement given by Eq. (10.134) has three unpaired electrons corresponding to the ground state <S3/2. The bonding in the nitro ( NO ) functional group is similar to that in the SO group given previously. It also has similarities to the bonding in the carbonyl functional group. In the NO group, the two unpaired electrons of the O atom form a MO with two unpaired electrons of the nitrogen atom such that the MO comprises a double bond. The nitrogen atom is then energy matched to the O- N functional group that is further energy matched to the C2sp3 HO of the C- O functional group. To meet the equipotential condition of the union of the N = O H2 -type-ellipsoidal-MO with other orbitals of the molecule, the hybridization factor c2 of Eq. (15.51) for the N = O -bond MO given by Eq. (15.140) is c2(θ to N2p to C2sp3Hθ) = 0.85987 .
As in the case of the carbonyl group, two unpaired O electrons result upon bond breakage of the N = O bond which requires that two times Ema of oxygen (Eq. (15.59)) be
subtracted from the total energy of NO . Additionally, Eτ ( atom - atom,m$p3. AO) and AEH^MO(AO / Hθ) are equal to -0.92918 eV (Eq. (14.513)) which matches the energy of the
N = O bond with the contiguous O- N bond and matches the energy contribution of an oxygen atom. The O - N functional group comprise a single-bond, H2 -type-ellipsoidal-MO between the remaining unpaired nitrogen electron and an unpaired electron of the second oxygen atom which further forms a single bond with the C2spi HO of the C- O functional group. In alkyl nitrites, the hybridization factor c2 of Eq. (15.52) for the C - O -bond MO given by Eq. (15.114) is c2 (C2sp3HO to θ) = 0.85395 . The hybridization factor C2 of Eq. (15.52) for a C - N -bond MO given by Eq. (15.116) is c2 {C2sp3HO to N) = 0.91140 . Thus, the hybridization factor c2 of Eq. (15.52) for O - N that bridges the C - O and N = O bonds given by Eq. (15.69) is
(15.141)
Figure imgf000722_0001
Er ( atom- atom,mspi\Aθ\ = -0.92918 eV in order to match the energy of the NO group and Ei AO I Hθ) = -15.35946 eV in order to match the C - O functional group.
The C- O functional group is equivalent to that of an ether as given in the corresponding section except that Er ( atom- atom, msp3. AOj and AEH MO[AO I HOj are both
-0.72457 eV which matches the energy contribution of an independent C2sp3 HO (Eq . (14.151)). Also, the energy terms corresponding to the oscillation of the bond in the transition state are matched to a nitrite.
The symbols of the functional groups of branched-chain alkyl nitrites are given in Table
15.195. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of alkyl nitrites are given in Tables 15.196, 15.197, and 15.198, respectively. The total energy of each alkyl nitrite given in Table
15.199 was calculated as the sum over the integer multiple of each ED (υnmP) of Table 15.198 corresponding to functional-group composition of the molecule. The bond angle parameters of alkyl nitrites determined using Eqs. (15.79-15.108) are given in Table 15.200.
Figure imgf000723_0001
Figure imgf000724_0001
Figure imgf000725_0001
Figure imgf000726_0001
ALKYL NITRITES (C1H2n+2JNO2I1 , » - 1.2.3,4,5...-)
The alkyl nitrites, Cn H2n+2_m (NO2)* , comprise a RC - O - NO moiety that comprises
C - O, O- N , and NO functional groups. The alkyl portion of the alkyl nitrite may comprise at least two terminal methyl groups ( CH3 ) at each end of the chain, and may comprise methylene (CH2 ), and methylyne (CH) functional groups as well as C bound by carbon- carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C - C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C- C bonds within isopropyl ((CH3J2 CH) and t-butyl ((CH^ C) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C- C bonds comprise functional groups. The branched-chain-alkane groups in alkyl nitrites are equivalent to those in branched-chain alkanes.
The electron configuration of oxygen is Is2Is2Ip4 , and the orbital arrangement given by
Eq. (10.154) has two unpaired electrons corresponding to the ground state 3P2. The electron configuration of nitrogen is \s22s22p3 , and the orbital arrangement given by Eq. (10.134) has three unpaired electrons corresponding to the ground state 4S^2. The bonding in the nitro ( NO ) functional group is similar to that in the SO group given previously. It also has similarities to the bonding in the carbonyl functional group. In the NO group, the two unpaired electrons of the O atom form a MO with two unpaired electrons of the nitrogen atom such that the MO comprises a double bond. The nitrogen atom is then energy matched to the O — N functional group that is further energy matched to the C2sp3 HO of the C- O functional group. To meet the equipotential condition of the union of the N = O H2 -type-ellipsoidal-MO with other orbitals of the molecule, the hybridization factor c2 of Eq. (15.51) for the N = 0-bond MO given by Eq. (15.140) is C2 (θ to N2p to C2sp3Hθ) = 0.85987 .
As in the case of the carbonyl group, two unpaired O electrons result upon bond breakage of the N = O bond which requires that two times Emα of oxygen (Eq. (15.59)) be
subtracted from the total energy of NO . Additionally, E1, ( atom - atom,msp3. AOj and AEH MO (AO I HO) are equal to -0.92918 eF (Eq. (14.513)) which matches the energy of the
N = O bond with the contiguous O- N bond and matches the energy contribution of an oxygen atom. The O - N functional group comprise a single-bond, H2 -type-ellipsoidal-M O between the remaining unpaired nitrogen electron and an unpaired electron of the second oxygen atom which further forms a single bond with the C2sp3 HO of the C- O functional group. In alkyl nitrites, the hybridization factor C2 of Eq. (15.52) for the C - O -bond MO given by Eq. (15.114) is c2 (C2sp3HO to θ) = 0.85395. The hybridization factor c2 of Eq. (15.52) for a C - N -bond MO given by Eq. (15.116) is c2 (C2sp3 HO to N) = 0.91140 . Thus, the hybridization factor C2 of Eq. (15.52) for O - N that bridges the C - O and N = O bonds given by Eq. (15.69) is
. . c2 (C2sp3HO to N) 0 91140
C2 (N2p to O2p) = -^ ^- f = = 1.06727 (15.141)
; c2 (C2sp3HO to θ) 0.85395 E1. i atom — atom,mspl .AO J = -0.92918 eV in order to match the energy of the NO group and E[^AO I HOJ = -15.35946 eV in order to match the C - O functional group.
The C - O functional group is equivalent to that of an ether as given in the corresponding section except that Eτ ( atom- atom, msp3.Aθ) and AEH MO (AO I HOJ are both
-0.72457 e V which matches the energy contribution of an independent C2sp3 HO (Eq. (14.151)). Also, the energy terms corresponding to the oscillation of the bond in the transition state are matched to a nitrite.
The symbols of the functional groups of branched-chain alkyl nitrites are given in Table
15.195. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of alkyl nitrites are given in Tables 15.196, 15.197, and 15.198, respectively. The total energy of each alkyl nitrite given in Table
15.199 was calculated as the sum over the integer multiple of each ED (υmuP) of Table 15.198 corresponding to functional-group composition of the molecule. The bond angle parameters of alkyl nitrites determined using Eqs. (15.79-15.108) are given in Table 15.200.
Figure imgf000729_0001
Figure imgf000730_0001
Figure imgf000731_0001
Figure imgf000731_0002
Figure imgf000732_0001
ALKYL NITRATES (CnH2n+2JNO2I, » = U3Λ5...-)
The alkyl nitrates, Cn H2n+2 (NO3) , comprise a RC - O - NO2 moiety that comprises C- O, O- N , and NO2 functional groups. The alkyl portion of the alkyl nitrate may comprise at least two terminal methyl groups ( CH3 ) at each end of the chain, and may comprise methylene (CH2 ), and methylyne (CH) functional groups as well as C bound by carbon- carbon single bonds. The methyl and methylene functional groups are equivalent to those of straight-chain alkanes. Six types of C- C bonds can be identified. The n-alkane C- C bond is the same as that of straight-chain alkanes. In addition, the C - C bonds within isopropyl ((C1H3J CH) and t-butyl ((CH3J C ) groups and the isopropyl to isopropyl, isopropyl to t-butyl, and t-butyl to t-butyl C - C bonds comprise functional groups. The branched-chain-alkane groups in alkyl nitrates are equivalent to those in branched-chain alkanes.
The NO2 functional group is equivalent to that of nitro alkanes with the exception that
ΔEH MO (AO I HO) as well as Eτ [atom- atom,msp3.Aθ) is equal to -3.71673 eV in order to match the group energy to that of the contiguous O- N bond. Furthermore, the O- N group with E7, (atom - atom, msp" .Aθ\ = -0.92918 eV is equivalent to that of nitrites as given in the corresponding section.
The C - O functional group is equivalent to that of an ether as given in the corresponding section except that E.A atom- atom,msp3. Aθ) and AEH m yAO I Hθ\ are both
-0.92918 eV which matches the energy contribution of an independent C2sp3 HO (Eq . (14.513)). Also, the energy terms corresponding to the oscillation of the bond in the transition state are matched to a nitrate.
The symbols of the functional groups of branched-chain alkyl nitrates are given in Table
15.201. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of alkyl nitrates are given in Tables
15.202, 15.203, and 15.204, respectively. The total energy of each alkyl nitrate given in Table 15.205 was calculated as the sum over the integer multiple of each ED {βιmφ) of Table 15.204 corresponding to functional-group composition of the molecule. The bond angle parameters of alkyl nitrates determined using Eqs. (15.79-15.108) are given in Table 15.206.
Figure imgf000734_0001
Figure imgf000735_0001
Figure imgf000736_0001
Figure imgf000736_0002
Figure imgf000737_0001
CYCLIC AND CONJUGATED ALKENES
( C&n*2-τm-ic> n = 3,4,5...oo, m = 1,2,3..., c = 0 or 1 )
The cyclic and conjugated alkenes are represented by the general formula
^ιΑn+2-zm-2c' » = 3,4,5...∞, m = 1,2,3..., c = 0 or 1 where /w is the number of double bonds , and c = 0 for a straight-chain alkene and c = 1 for a cyclic alkene. They have at least one carbon-carbon double bond comprising a functional group that is solved equivalently to the double bond of ethylene. Consider the cyclic and conjugated alkenes 1,3-butadiene, 1,3- pentadiene, 1,4-pentadiene, 1,3-cyclopentadiene, and cyclopentene. Based on the condition of energy matching of the orbital, any magnetic energy due to unpaired electrons in the constituent fragments, and differences in oscillation in the transition state, five distinct C - C functional groups can be identified as given in Table 15.208. The designation of the structure of the groups are shown in Figures 6 IA-E. In addition, CH2 of any -C = CH2 moiety is an conjugated alkene functional group. The alkyl portion of the cyclic or conjugated alkene may comprise at least one terminal methyl group (CH3 ), and may comprise methylene (CH2 ), and methylyne ( CH ) functional groups that are equivalent to those of branched-chain alkanes.
The solution of the functional groups comprises the hybridization of the 2s and 2p
AOs of each C to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between two C2sp3 ΗOs to form a MO permits each participating hybridized orbital to decrease in radius and energy. The C- C groups are solved in the same manner as those of the branched-chain alkanes given in the corresponding section. For example, the cyclopentene
C1 - Ch group is equivalent to the n-C - C alkane group. Many of the corresponding energies of the molecules of this class are similar, and they can be related to one another based on the structure. For example, cyclopentadiene is formed by ring closure of 1,3-pentadiene with the elimination of H from the terminal methyl and methylene groups. Thus, the energy of each of the corresponding carbon-carbon bonds in cyclopentadiene is the same as that in 1,3-pentadiene except that the difference between the energies of the 1,3-pentadiene Cc - Cd and the cyclopentadiene Ca - Ch groups is the magnetic energy (Eq. (15.58)) which is subtracted from the Cn - CΛ total bond energy according to Eqs. (13.524-13.527) due to the formation of a CH group from the methylene group. Eτ [atom- atom,mspl ,Aθ} of the C = C-bond MO in Eq. (15.52) due to the charge donation from the C atoms to the MO is equivalent to that of ethylene, -2.26759 eV , given by Eq. (14.247). Eτ [atom - atom,msp\Aθ) of each C - C-bond MO in Eq. (15.52) i s
-2.26759 eV or -1.85836 eV based on the energy match between the C2sp3 HOs corresponding to the energy contributions equivalent to those of alkene, -1.13379 eV (Eq. (14.247)), or methylene, -0.92918 eV (Eq. (14.513)), groups, respectively, that are contiguous with the C - C-bond carbons. In the former case, the total energy of the C - C bond MO is matched to that of the alkane energy in the determination of the bond length. The charge density of 0.5e must be donated to the C- C bond in order to match the energy of the adjacent flanking double bonds. This further lowers the total energy of the C - C -bond MO and increases the C - C bond energy. This additional lowering of the C - C-bond energy by additional charge donation over that of an alkane bond due to adjacent double bonds is called conjugation.
The symbols of the functional groups of cyclic and conjugated alkenes are given in Table 15.207. The structures of 1,3 -butadiene, 1,3-pentadiene, 1,4-pentadiene, 1,3-cyclopentadiene, and cyclopentene are shown in Figures 61A-E, respectively. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of cyclic and conjugated alkenes are given in Tables 15.208, 15.209, and 15.210, respectively. The total energy of each cyclic or conjugated alkenes given in Table 15.21 1 was calculated as the sum over the integer multiple of each ED(amuP) of Table 15.210 corresponding to functional-group composition of the molecule. For each set of unpaired electrons created by bond breakage , the C2.yp3 HO magnetic energy Ema t that is subtracted from the weighted sum of the ED [c,ιv,ψ) (eV) values based on composition is given by Eq. (15.58). The bond angle parameters of cyclic and conjugated alkenes determined using Eqs. (15.79-15.108) are given in Table 15.212.
Figure imgf000740_0001
Figure imgf000740_0002
Figure imgf000741_0001
Figure imgf000742_0001
Figure imgf000742_0002
Figure imgf000743_0001
AROMATIC AND HETEROCYCLIC COMPOUNDS
Aromatic and heterocyclic molecules comprise at least one of an aromatic or a cyclic conjugated alkene functional group. The latter was described in the Cyclic and Conjugated
Alkenes section. The aromatic bond is uniquely stable and requires the sharing of the electrons of multiple H2 -type MOs. The results of the derivation of the parameters of the benzene molecule given in the Benzene Molecule (C6H6 ) section can be generalized to any aromatic function group(s) of aromatic and heterocyclic compounds.
CH, can be considered a linear combination of three ethylene molecules wherein a
C - H bond of each CH2 group of H2C = CH2 is replaced by a C = C bond to form a six- member ring of carbon atoms. The solution of the ethylene molecule is given in the Ethylene
Molecule (CH0CH. ) section. The radius r , , (0.85252αn ) of the C2spl shell of ethylene calculated from the Coulombic energy is given by Eq. (14.244). The Coulombic energy ECoulmΛ {Cdhyhnι!,2spΛ (-15.95955 eV ) of the outer electron of the C2sp3 shell is given by Eq.
(14.245). The energy j?(c,%fe∞,2,5p3) (-15.76868 eV ) of the outer electron of the C2sp3 shell
is given by Eq. (14.246). E1. (c = C,2sp3) (-1.13380 eV ) (Eq. (14.247), the energy change of each C2spi shell with the formation of the C = C -bond MO is given by the difference between
EyCelh lme,2sp3) and E\ C,2spΛ . C6H6 can be solved using the same principles as those used
to solve ethylene wherein the 2s and 2p shells of each C hybridize to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between two C2sp3 hybridized orbitals (ΗOs) to form a molecular orbital (MO) permits each participating hybridized orbital to decrease in radius and energy. Each 2sp3 HO of each carbon atom initially has four unpaired electrons. Thus, the 6 H atomic orbitals (AOs) of benzene contribute six electrons and the six sp3 -hybridized carbon atoms contribute twenty-four electrons to form six C - H bonds and six
C = C bonds. Each C- H bond has two paired electrons with one donated from the H AO and the other from the C2sp3 HO. Each C = C bond comprises a linear combination of a factor of 0.75 of four paired electrons (three electrons) from two sets of two C2sp3 HOs of the participating carbon atoms. Each C - H and each C = C bond comprises a linear combination of one and two diatomic H2 -type MOs developed in the Nature of the Chemical Bond of Ηydrogen-Type Molecules and Molecular Ions section, respectively.
Consider the case where three sets of C = C -bond MOs form bonds between the two carbon atoms of each molecule to form a six-member ring such that the six resulting bonds comprise eighteen paired electrons. Each bond comprises a linear combination of two MOs wherein each comprises two C2.sp3 ΗOs and 75% of a H2 -type ellipsoidal MO divided between the C2sp3 ΗOs:
Figure imgf000745_0001
The linear combination of each H2 -type ellipsoidal MO with each C2spi HO further comprises an excess 25% charge-density contribution per bond from each C2sp3 HO to the C = C -bond
MO to achieve an energy minimum. Thus, the dimensional parameters of each bond C = C - bond are determined using Eqs. (15.42) and (15.1-15.5) in a form that are the same equations as those used to determine the same parameters of the C = C -bond MO of ethylene (Eqs. (14.242- 14.268)) while matching the boundary conditions of the structure of benzene.
Hybridization with 25% electron donation to each C = C -bond gives rise to the
C^V HO-shell Coulombic energy EVmιhnh{Chcm,2Sp3) given by Eq. (14.245). To meet the equipotential condition of the union of the six C2sp3 HOs, c2 and C2 of Eq. (15.42) for the
3e aromatic C= C -bond MO is given by Eq. (15.62) as the ratio of 15.95955 eV , the magnitude of Ecm,iomb {C henzem>2sP3) (E<1- (14-245))> and 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H (Eq. (1.243)): (15.143)
Figure imgf000745_0002
3β
The energies of each C=C bond of benzene are also determined using the same equations as those of ethylene (Eqs. (14.251-14.253) and (14.319-14.333) with the parameters of
3c benzene. Ethylene serves as a basis element for the C=C bonding of benzene wherein each of
3c the six C=C bonds of benzene comprises (θ.75)(4) = 3 electrons according to Eq. (15.142). 3e
The total energy of the bonds of the eighteen electrons of the C = C bonds of benzene,
E1, [C6H6, C=C) , is given by (δ)(θ.75) times -5r+βB.(c = c) (Eq. (14.492)), the total energy of
3« . . the C= C -bond MO of benzene including the Doppler term, minus eighteen times E[C, 2sp3)
(Eq. (14.146)), the initial energy of each C2sp3 HO of each C that forms the C=C bonds of
3e bond order two. Thus, the total energy of the six C=C bonds of benzene with three electron per aromatic bond given by Eq. (14.493) is
Figure imgf000746_0001
= (6)(0.75)(-66.05796 eV) ~ lδ(-14.63489 eV) (15.144)
= -297.26081 e F - (-263.42798 eV) = -33.83284 eV The results of benzene can be generalized to the class of aromatic and heterocyclic compounds.
Ehv of an aromatic bond is given by Er ( H2 ] (Eqs. (11.212) and (14.486)), the maximum total energy of each H2 -type MO such that
(15.145)
Figure imgf000746_0002
The factor of 0.75 corresponding to the three electrons per aromatic bond of bond order two given in the Benzene Molecule (C6H6 ) section modifies Eqs. (15.52-15.56). Multiplication of the total energy given by Eq. (15.55) by fχ = 0.75 with the substitution of Eq. (15.145) gives the total energy of the aromatic bond:
(15.146)
Figure imgf000746_0003
The total bond energy of the aromatic group E0 (cmup) is the negative difference of the total energy of the group (Eq. (15.146)) and the total energy of the starting species given by the sum of <A« fo ΛO I HO) and c5Em. M (c5 AO/ Hθ) :
Figure imgf000747_0001
Since there are three electrons per aromatic bond, c, is three times the number of aromatic bonds.
Benzene can also be considered as comprising chemical bonds between six CH radicals wherein each radical comprises a chemical bond between carbon and hydrogen atoms. The solution of the parameters of CH is given in the Hydrogen Carbide (CH) section. Those of the benzene are given in the Benzene Molecule ( C6H6 ) section. The energy components of Ve , V ,
T , Vm , and Eτ are the same as those of the hydrogen carbide radical, except that ET(c = C,2sp3) = -1.13379 eV (Eq. (14.247)) is subtracted from £r (CH) of Eq. (13.495) to match the energy of each C— H -bond MO to the decrease in the energy of the corresponding C2sp3 HO. In the corresponding generalization of the aromatic CH group, the geometrical parameters are determined using Eq. ( 15.42) and Eqs. (15.1 -15.5) with E7. [atom- atom,msp\Λθ) = -1.13379 eV .
The total energy of the benzene C - H -bond MO, E1. (C - H) , given by Eq. (14.467)
is the sum of Q.5ET [ C = C,2sp3 J , the energy change of each C2sp3 shell per single bond due to
3e the decrease in radius with the formation of the corresponding C=C -bond MO (Eq. (14.247)), and E1. (CH) , the σ MO contribution given by Eq. (14.441). In the corresponding generalization of the aromatic CH group, the energy parameters are determined using Eqs.
-1.13379 eV
(15.146-15.147) with ./J = 1 and E1. (atom - atom,msp\Aθ) = ■ . Thus, the energy 3a contribution to the single aromatic CH bond is one half that of the C=C double bond contribution. This matches the energies of the CH and C=C aromatic groups, conserves the electron number with the equivalent charge density as that of s = lin Eqs. (15.18-15.21), and
3s further gives a minimum energy for the molecule. Breakage of the aromatic C=C bonds to give CH groups creates unpaired electrons in these fragments that corresponds to c3 = 1 in Eq.
(15.56) with Emag given by Eq. (15.58).
Each of the C - H bonds of benzene comprises two electrons according to Eq. (14.439). From the energy of each C - H bond, -En ( 12CH) (Eq. (14.477)), the total energy of the
twelve electrons of the six C - H bonds of benzene, E7, (C6H6, C - H) , given by Eq. (14.494) is
ET (C6H6,C - H) = (6)(-EDi ( 12CH)] = 6(-3.90454 eF) = -23.42724 eV (15.148)
The total bond dissociation energy of benzene, E0 [C6H6 ) , given by Eq. (14.495) is the negative
sum of £r C6H6, C=C I (Eq. (14.493)) and Er(C6H6,C - H) (Eq. (14.494)):
Figure imgf000748_0001
= -((-33.83284 eV) + (-23.42724 eV)) (15.149)
= 57.2601 eV Using the parameters given in Tables 15.214 and 15.216 in the general equations (Eqs. (15.42), (15.1-15.5), and (15.146-15.147)) reproduces the results for benzene given in the Benzene Molecule (C6H6 ) section as shown in Tables 15.214 and 15.216.
The symbols of the functional groups of aromatics and hertocyclics are given in Table 15.213. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11), (15.17-15.56), and (15.146-15.147)) parameters of aromatics and hertocyclics are given in Tables 15.214, 15.215, and 15.216, respectively. The total energy of benzene given in Table 15.217 was calculated as the sum over the integer multiple of each ED (ϋm,φ) of Table 15.216 corresponding to functional-group composition of the molecule. The bond angle parameters of benzene determined using Eqs. (15.79-15.108) are given in Table 15.218.
Figure imgf000749_0001
Figure imgf000750_0001
Figure imgf000751_0001
NAPHTHALENE
Naphthalene has the formula C10H8 and comprises a planar molecule with two aromatic rings that share a common C - C group. In order to be aromatic, the total number of bonding electrons must be a multiple of 3 since the number of electrons of the aromatic bond is (θ.75j(4j = 3 as shown in the Benzene section. In the case of naphthalene, the peripheral 10 carbons form the aromatic MO with the center bridged by a C- C single bond. Then, 30 electrons of the 48 available form aromatic bonds, two electrons form the bridging C- C single bond, and 16 electrons form the eight C - H single bonds. The energies of the aromatic carbons are given by the same equations as those of benzene (Eqs. (15.42), (15.1-15.5), and (15.146-15.147)), except that there are 10 in naphthalene versus six in benzene. Since there are three electrons per aromatic bond, c, is three times ten, the number of aromatic bonds. Similarly, the aromatic C - H group of naphthalene is equivalent to that of benzene.
To meet the equipotential condition of the union of the ten C2sp3 ΗOs bridged by the C - C single bond, the parameters C1 , C2 , and C2n of Eq. (15.42) are one for the C- C group, C]o and C1 are 0.5, and C1 given by Eq. (15.142) is c2 (C2sp3Hθ) = 0.85252. Otherwise, the solutions of the C- C bond parameters are equivalent to those of the replaced C -H groups with E(AO I HO) = -14.63489 eV and LEH UQ {ΛO I HO) - -\.13379 eV in Eq. (15.41 ).
Similarly, the energy parameters are determined using Eqs. (15.52-15.56) with n / 3 A rA -1-13379 eV E1, 1 atom - atom, msp .AO 1J = .
The symbols of the functional groups of naphthalene are given in Table 15.219. The corresponding designation of the structure is shown in Figure 62. The geometrical (Eqs. (15.1- 15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11), (15.17-15.56), and (15.146-15.147)) parameters of naphthalene are given in Tables 15.220, 15.221, and 15.222, respectively. The total energy of naphthalene given in Table 15.223 was calculated as the sum over the integer multiple of each Eυ (amuP) of Table 15.222 corresponding to functional-group composition of the molecule. The bond angle parameters of naphthalene determined using Eqs. (15.79-15.108) are given in Table 15.224.
Figure imgf000753_0001
Figure imgf000754_0001
Figure imgf000755_0001
TOLUENE
Toluene has the formula C7H8 and comprises the benzene molecule with one hydrogen atom replaced by a methyl group corresponding to a CH3 functional group and a C - C
functional group. The aromatic C=C and C- H functional groups are equivalent to those of benzene given in Aromatic and Heterocyclic Compounds section. The CH3 functional group is the same as that of continuous and branched-chain alkanes given in the corresponding sections.
The bond between the methyl and aromatic ring comprises a C - C functional group that is are solved using the same principles as those used to solve the alkane functional groups wherein the 2s and 2p AOs of each C hybridize to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between two C2sp3 ΗOs to form a MO permits each participating hybridized orbital to decrease in radius and energy. To match energies within the MO that bridges methyl and aromatic carbons, EyAO I Hθ\ and AEH MO \AO I HO} in Eq.
-1 13379 eV (15.41) are -15.35946 eF (Eq. (14.155)) and — : , respectively.
To meet the equipotential condition of the union of the aromatic and methyl C2sp3 ΗOs of the C- C single bond, the parameters C1 , C2 , and C20 of Eq. (15.42) are one for the C- C group, CJo and C, are 0.5, and C1 given by Eq. (13.430) is c2 (C2sp3Hθ) = 0.91771. To match the energies of the functional groups, Er (αtom - αtom,mspz '.AOJ of the C - C -bond MO in Eq. (15.52) due to the charge donation from the C atoms to the MO is -1.13379 eV which is the same energy per C2sp3 HO as that of the replaced C - H group. The symbols of the functional groups of toluene are given in Table 15.225. The corresponding designation of the structure is shown in Figure 63. The geometrical (Eqs. (15.1- 15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11), (15.17-15.56), and (15.146-15.147)) parameters of toluene are given in Tables 15.226, 15.227, and 15.228, respectively. The total energy of toluene given in Table 15.229 was calculated as the sum over the integer multiple of each ED {αo«P) of Table 15.228 corresponding to functional-group composition of the molecule. The bond angle parameters of toluene determined using Eqs. (15.79-15.108) are given in Table 15.230.
Figure imgf000757_0001
Figure imgf000758_0001
Figure imgf000759_0001
CHLOROBENZENES
Chlorobenzenes have the formula C6H6_mClm and comprise the benzene molecule with at least one hydrogen atom replaced by a chlorine atom corresponding to a C - Cl functional group. The aromatic C=C and C-H functional groups are equivalent to those of benzene given in Aromatic and Heterocyclic Compounds section.
The small differences between energies of ortho, meta, and para-dichlorobenzene is due to differences in the energies of vibration in the transition state that contribute to E .
Two types of C - Cl functional groups can be identified based on symmetry that determine the parameter R in Eq. (15.48). One corresponds to the special case of 1,3,5 substitution and the other corresponds to other cases of single or multiple substitutions of Cl for H . P- dichlorobenzene is representative of the bonding with R = a . 1,2,3-trichlorbenzene is the particular case wherein is R = b. Also, beyond the binding of three chlorides Ema is subtracted for each additional Cl due to the formation of an unpaired electrons on each C- Cl bond. The bond between the chlorine and aromatic ring comprises two C - Cl functional groups that are solved using the same principles as those used to solve the alkyl chloride functional groups as given in the corresponding section wherein the 2s and Ip AOs of each
C hybridize to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between the C2sp3 HO and Cl AO to form a MO permits each participating hybridized orbital to decrease in radius and energy. As in the case of alkyl chlorides, C1 of Eq. (15.52) for each C- Cl -bond MO is one, and the energy matching condition is determined by the C2 parameter given by Eq. (15.111) which is C1 (C2sp3HO to C/) = 0.81317. To match energies within the MO that bridges the chlorine AO and aromatic carbon C2sp3 HO, E{A0 I HO) and MH UO (AO / HO) in Eq. (15.42) are -14.63489 eV and -2.99216 eV , respectively. The latter matches twice that of the replaced C- H-bond MO plus EΛ atom- atom,msp3.AOy To match the energies of the functional groups,
Er ( atom- atom,msp3. AOj of the C - Cl -bond MO in Eq. (15.53) due to the charge donation from the C and Cl atoms to the MO is -0.72457 eV (Eq. (14.151)). The symbols of the functional groups of chlorobenzenes are given in Table 15.231. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11), (15.17-15.56), and (15.146-15.147)) parameters of chlorobenzenes are given in Tables 15.232, 15.233, and 15.234, respectively. The total energy of each chlorobenzene given in Table 15.235 was calculated as the sum over the integer multiple of each ED(cwup) of Table 15.234 corresponding to functional-group composition of the molecule. For each set of unpaired electrons created by bond breakage , the C2sp3 HO magnetic energy Ema r that is subtracted from the weighted sum of the ED[Gmφ) (eV) values based on composition is given by Eq. (15.58). The bond angle parameters of chlorobenzenes determined using Eqs. (15.79-15.108) are given in Table 15.236.
Figure imgf000761_0001
Figure imgf000762_0001
Figure imgf000763_0001
Figure imgf000764_0001
PHENOL
Phenol has the formula C6H6O and comprises the benzene molecule with one hydrogen atom replaced by a hydroxyl corresponding to an OH functional group and a
C- O functional group. The aromatic C=C and C-H functional groups are equivalent to those of benzene given in Aromatic and Heterocyclic Compounds section. The OH functional group is the same as that of alcohols given in the corresponding section.
The bond between the hydroxyl and aromatic ring comprises a C - O functional group that is are solved using the same principles as those used to solve the alcohol functional groups wherein the 2s and 2p AOs of each C hybridize to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between the C2sp3 HO and O AO to form a MO permits each participating hybridized orbital to decrease in radius and energy. In aryl alcohols, the aromatic C2sp3 HO has a hybridization factor of 0.85252 (Eq.
(15.143)) with an initial energy of E(c,2sp3) = -14.63489 eV (Eq. (15.25)) and the O AO has an energy of E[θ\ = -13.61806 eV . To meet the equipotential condition of the union of the C - O H2 -type-ellipsoidal-MO with these orbitals, the hybridization factor c2 of Eq. (15.52) for the C- O-bond MO given by Eqs. (15.68) and (15.70) is
Figure imgf000765_0001
■ = -13*1806 eV -14.63489 eV = 0.79329
Eτ [atom- atom,msp\Aθ) of the C- O-bond MO in Eq. (15.52) due to the charge donation from the C and O atoms to the MO is -1.49608 eV . It is based on the energy match between the OH group and the C2spz HO of an aryl group and is given by the linear combination of -0.92918 eV (Eq. (14.513)) and -1.13379 eV (Eq. (14.247)), respectively.
The symbols of the functional groups of phenol are given in Table 15.237. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs.
(15.6-15.11), (15.17-15.56), and (15.146-15.147)) parameters of phenol are given in Tables 15.238, 15.239, and 15.240, respectively. The total energy of phenol given in Table 15.241 was calculated as the sum over the integer multiple of each ED{am,φ) of Table 15.240 corresponding to functional-group composition of the molecule. The bond angle parameters of phenol determined using Eqs. (15.79-15.108) are given in Table 15.242.
Figure imgf000766_0001
Figure imgf000767_0001
Figure imgf000768_0001
Figure imgf000769_0001
ANILNE
Aniline and methyl aniline have the formula C6H7N and C7H9N , respectively. They comprise the benzene and toluene molecules with one hydrogen atom replaced by an amino group corresponding to an NH2 functional group and a C - N functional group. The
3c aromatic C = C and C- H functional groups are equivalent to those of benzene given in Aromatic and Heterocyclic Compounds section. The C- C and CH3 functional groups of methyl anilines are equivalent to those of toluene given in the corresponding section.
The aryl amino ( NH2 ) functional group was solved using the procedure given in the
Dihydrogen Nitride (NH2) section. Using the results of Eqs. (13.245-13.368), the aryl amino parameters in Eq. (15.51) are «, = 2 , C1 = 0.75 , C2 = 0.93613 (Eqs. (13.248-13.249)), Ch) = 1.5 , and c, = 0.75. In the determination of the hybridization factor C2 of Eq. (15.52) for the N- H-bond MO of aryl amines, the C2.ψ3 HO of the C - NH2 -bond MO has an energy of E[c,2spz} = -15.76868 eV (Eq. (15.18) corresponding to _? = 2 in Eqs. (15.18- 15.20), and the N AO has an energy of E[N) = -14.53414 eV . To meet the equipotential condition of the union of the N - H H2 -type-ellipsoidal-MO with the C2sp* HO, the hybridization factor c given by Eq. (15.68) is
= 0.92171 (15.151)
Figure imgf000770_0001
The bond between the amino and aromatic ring comprises a C - N functional group that is the. same as that of 2° amines (methylene) except that the energies corresponding to oscillation in the transition state are those of aniline. The group is solved using the same principles as those used to solve the primary and secondary-amine functional groups wherein the 2s and Ip AOs of each C hybridize to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between the C2sps HO and N AO to form a MO permits each participating hybridized orbital to decrease in radius and energy. The hybridization is determined in a similar manner to that of the C - O group of phenol. In anilines, the aromatic C2sp3 HO has a hybridization factor of 0.85252 (Eq. (15.143)) with an initial energy of
Figure imgf000770_0002
-14.63489 eV (Eq. (15.25)) and the N AO has an energy of E(N) = -14.53414 eV . To meet the equipotential condition of the union of the C- O
H2 -type-ellipsoidal-MO with these orbitals, the hybridization factor C1 of Eq. (15.51) for the C- O -bond MO given by Eqs. (15.68) and (15.70) is
C2 (arylC2sp3HO to N) =
Figure imgf000771_0001
1453414 eV (0.85252) <1..I 52)
-14.63489 eV = 0.84665 Er { atom ~ atom,msp3. AO) of the C - N -bond MO in Eq. (15.52) due to the charge donation from the C and N atoms to the MO is -1.13379 eV (Eq. (14.247)). It is based on the energy match between the NH2 group and the C2sp3 HO of the aryl group and is twice that of the aryl C - H group that it replaces. The symbols of the functional groups of aniline and methyl-substituted anilines are given in Table 15.243. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.1 1), (15.17-15.56), and (15.146-15.147)) parameters of aniline and methyl-substituted anilines are given in Tables 15.244, 15.245, and 15.246, respectively. The total energy of each aniline and methyl-substituted aniline given in Table 15.247 was calculated as the sum over the integer multiple of each E0 (<j,mtp) of Table
15.246 corresponding to functional-group composition of the molecule. The bond angle parameters of aniline and methyl-substituted anilines determined using Eqs. (15.79-15.108) are given in Table 15.248.
Figure imgf000771_0002
Figure imgf000772_0001
Figure imgf000773_0001
Figure imgf000774_0001
ARYL NITRO COMPOUNDS
Aryl nitro compounds have a hydrogen of an aryl group replaced by a nitro corresponding to an NO2 functional group and a C - N functional group. Examples include nitrobenzene, nitrophenol, and nitroanilne with formulas C6H5NO2 , C6H5NO3 , and
3« C6H6N2O2 , respectively. The aromatic C=C and C - H functional groups are equivalent to those of benzene given in Aromatic and Heterocyclic Compounds section. The OH and C- O functional groups of nitrophenols are the same as those of phenol given in the corresponding section. The NH2 and C - N functional groups of nitroanilines are the same as those of aniline given in the corresponding section. The differences between the total bond energies of the nitroanilines given in Table 15.252 are due to differences in the Eoκ term.
For simplicity and since the differences are small, the E09, terms for nitroanilines were taken as the same.
The NO2 group is the same as that given in the Νitroalkanes section. The bond between the nitro and aromatic ring comprises a C - N functional group that is the same as that of nitroalkanes given in the corresponding section except that
Eτ[αtom- αtom,msp\Aθ) is -0.72457 eV , one half of that of the C - N -bond MO of nitroalkanes and equivalent to that of methyl (Eq. (14.151)) in order to maintain the independence and aromaticity of the benzene functional group. In addition, the energy terms due to oscillation in the transition state correspond to those of an aryl nitro compound. The symbols of the functional groups of aryl nitro compounds are given in Table
15.249. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11), (15.17-15.56), and (15.146-15.147)) parameters of aryl nitro compounds are given in Tables 15.250, 15.251, and 15.252, respectively. The total energy of each aryl nitro compound given in Table 15.253 was calculated as the sum over the integer multiple of each ED(αmup) of Table 15.252 corresponding to functional-group composition of the molecule. For each set of unpaired electrons created by bond breakage , the C2sp3 HO magnetic energy Emα that is subtracted from the weighted sum of the ED{umuP) (eV) values based on composition is given by Eq. (15.58). The bond angle parameters of aryl nitro compounds determined using Eqs. (15.79-15.108) are given in Table 15.254.
Figure imgf000776_0001
Figure imgf000777_0001
Figure imgf000778_0001
Figure imgf000779_0001
BENZOIC ACID COMPOUNDS
Benzoic acid compounds have a hydrogen of an aryl group replaced by a carboxylic acid group corresponding to an C- C(O)-OH moiety that comprises C = O and OH functional groups that are the same as those of carboxylic acids given in the corresponding section. The single bond of aryl carbon to the carbonyl carbon atom, C - C{0) , is also a functional group. This group is also equivalent to the same group of carboxylic acids except that AEH MO (AO I HO) in Eq. (15.42) and Eτ[atom- atom,msp3 '.AO) in Eq. (15.52) are
-1.13379 eV both -1.29147 eV which is a linear combination of — '■ ,
E1J atom- atom,msp3.AO) of the C- H group that the C - C(O) group replaces, and that of an independent C2sp3 HO, -0.72457 eV (Eq. (14.151)).
Examples include benzoic acid, chlorobenzoic acid, and aniline carboxylic acid with formulas C1H6O2 , C1H5O2Cl , and C7H7NC2, respectively. The aromatic C=C and C- H functional groups are equivalent to those of benzene given in Aromatic and Heterocyclic Compounds section. The NH2 and C- N functional groups of aniline carboxylic acids are the same as those of aniline given in the corresponding section. The C- Cl functional group of 2-chlorobenzoic acids corresponding to meta substitution is equivalent to that of chlorobenzene given in the corresponding section. The C- Cl functional group of 3 or 4- chlorobenzoic acids corresponding to ortho and para substitution is also equivalent to that of chlorobenzene, except that AEH MO[AO / Hθ) in Eq. ( 15.42) and
Eτ[atom- atom,msp\Aθ) in Eq. (15.52) are both -0.92918 eV (Eq. (14.513)) since each of these positions can form a resonance structure with the carboxylic acid group which is permissive of greater charge donation from the C2,sp3 HO.
The symbols of the functional groups of benzoic acid compounds are given in Table 15.255. The corresponding designations of benzoic acid is shown in Figure 64. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11), (15.17-15.56), and (15.146-15.147)) parameters of benzoic acid compounds are given in Tables 15.256, 15.257, and 15.258, respectively. The total energy of each benzoic acid compound given in Table 15.259 was calculated as the sum over the integer multiple of each ED (aroΨ) of Table 15.258 corresponding to functional-group composition of the molecule. The bond angle parameters of benzoic acid compounds determined using Eqs. (15.79-15.108) are given in Table 15.260.
Figure imgf000781_0001
Figure imgf000782_0001
O
Figure imgf000783_0001
Figure imgf000784_0001
Figure imgf000785_0002
Figure imgf000785_0001
ANISOLE
Anisole has the formula C7H SO and comprises the phenol molecule with the hydroxyl hydrogen atom replaced by the moiety -O - CH3 to form an ether comprising aromatic and methyl functional groups as well as two types of C - O functional groups, one for aryl carbon to oxygen and one for methyl carbon to oxygen. The aromatic C=C and C- H functional groups are equivalent to those of benzene given in Aromatic and Heterocyclic Compounds section. The CH3 and methyl C - O functional groups are the same as those of the corresponding ether groups given in the corresponding section.
The C- O functional group comprising the bond between the ether oxygen and aromatic ring is equivalent to that of the methyl ether C - O functional group except that
AEH UO (AO I HO) in Eq. (15.42) and E7. (atom- atom,msp\Aθ) in Eq. (15.52) are both
-1.13379 eV (Eq. (14.247)). Eτ( atom— atom,msp3.AO) is based on the energy match between the OCH3 group and the C2.sp3 HO of the aryl group and is twice that of the aryl
C - H group that it replaces. The symbols of the functional groups of anisole are given in Table 15.261. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11), (15.17-15.56), and (15.146-15.147)) parameters of anisole are given in Tables 15.262, 15.263, and 15.264, respectively. The total energy of anisole given in Table 15.265 was calculated as the sum over the integer multiple of each ED [βwiφ) of Table 15.264 corresponding to functional-group composition of the molecule. The bond angle parameters of anisole determined using Eqs. (15.79-15.108) are given in Table 15.266.
Figure imgf000786_0001
Figure imgf000787_0001
Figure imgf000787_0002
Figure imgf000788_0001
Figure imgf000789_0001
PYRROLE
Pyrrole having the formula C4H5N comprises the conjugated alkene 1,3-butdiene that forms a cyclic structure by terminal-atom bonding to a NH functional group. The two symmetrical carbon-to-nitrogen bonds comprise the C- N- C functional group. The 1,3- butdiene moiety comprises C - C , C = C , and CH functional groups. The C - C and C = C groups are equivalent to the corresponding groups of 1,3-butdiene given in the Cyclic and Conjugated Alkenes section except that the energies terms of the corresponding to oscillation in the transition state match pyrrole. Furthermore, the conjugated double bonds have the same bonding as in 1,3-butdiene except that the hybridization terms C2 of the C - C and C = C groups and C2 and C20 of the C = C group in Eqs. (15.42) and (15.52) become t h a t o f b e n z e n e g i v e n b y E q . ( 1 5 . 1 4 3 ) ,
( C2 (benzeneC2sp3Hθ) = c2 (benzeneC2sp3 Hθ) = 0.85252 ), in the cyclic pyrrole MO which has aromatic character. The bonding in pyrrole, furan, and thiophene are the same except for the energy match to the corresponding heteroatoms. The hybridization permits double-bond character in the carbon-heteroatom bonding.
The NH group is solved equivalently to that of a secondary amine as given in the corresponding section except that the hybridization term C2 is that of the amino group of aniline in order provide double-bond character to match the group to the other orbitals of the molecule. Similarly, the CH functional group is equivalent to that of 1,3-butdiene, except that AEH U0 (AO I H0) = -2.26758 eV (Eq. (14.247)) in Eq. (15.42) in order to provide matching double-bond character.
The solution of the C- N- C functional group comprises the hybridization of the 2s and 2p AOs of each C to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between two C2sp3 ΗOs and the nitrogen atom to form a MO permits each participating hybridized orbital to decrease in radius and energy. Thus, the C - N - C - bond MO comprising a linear combination of two single bonds is solved in the same manner as a double bond with «, = 2 in Eqs. (15.42) and (15.52). The hybridization factor c2 [arylC2sp3HO to N) = 0.84665 (Eq. (15.152)) matches the double-bond character of the C2sp3 ΗOs to the N atom of the NH group, and C2 and C20 in Eqs. (15.42) and (15.52) become that of benzene given by Eq. (15.143), C2 (benzeneC2sp3Hθ) = 0.85252. Furthermore, AE^M0 [AO I HO) in Eq. (15.42) and Eτ [atom - atom,msp2 \AOJ in Eq.
(15.52) are both -0.92918 eV (Eq. (14.513)) per atom corresponding to -3.71673 eV in total. This is the maximum energy for a single bond and corresponds to methylene character as given in the Continuous-Chain Alkanes section.
The symbols of the functional groups of pyrrole are given in Table 15.267. The structure of pyrrole is shown in Figure 65. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of pyrrole are given in Tables 15.268, 15.269, and 15.270, respectively. The total energy of pyrrole given in Table 15.271 was calculated as the sum over the integer multiple of each En[GmUp) of Table 15.270 corresponding to functional- group composition of the molecule.
The bond angle parameters of pyrrole determined using Eqs. (15.79-15.108) are given in Table 15.272.
Figure imgf000791_0001
Figure imgf000792_0001
Figure imgf000793_0001
Figure imgf000794_0001
Figure imgf000795_0002
Figure imgf000795_0001
FURAN
Furan having the formula C4H4O comprises the conjugated alkene 1,3-butdiene that forms a cyclic structure by terminal-atom bonding to an oxygen atom. The two symmetrical carbon-to-oxygen bonds comprise the C - O - C functional group. The 1,3-butdiene moiety comprises C - C , C = C , and CH functional groups. The CH, C - C , and C = C groups are equivalent to the corresponding groups of pyrrole given in the corresponding section.
The C - O- C functional group of furan is solved in a similar manner as that of the C - N - C group of pyrrole. The solution of the C - O- C functional group comprises the hybridization of the 2s and 2p AOs of each C to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between two C2sp3 ΗOs and the oxygen atom to form a MO permits each participating hybridized orbital to decrease in radius and energy. Thus, the C - O - C -bond MO comprising a linear combination of two single bonds is solved in the same manner as a double bond with n, = 2 in Eqs. (15.42) and (15.52). The hybridization factor c2[arylC2sp3HO to θ) = 0.79329 (Eq. (15.150)) matches the double- bond character of the C2spi ΗOs to the O atom, and C2 and C20 in Eqs. (15.42) and (15.52) become that of benzene given by Eq. (15.143), C2 [benzenedsp3 Hθ) = 0.85252.
Furthermore, Er[atom- atom, msp3.AOj^ in Eq. (15.52) is -0.92918 eV (Eq. (14.513)) per atom corresponding to -3.71673 e V in total.
The symbols of the functional groups of furan are given in Table 15.273. The structure of furan is shown in Figure 66. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of furan are given in Tables 15.274, 15.275, and 15.276, respectively. The total energy of furan given in Table 15.277 was calculated as the sum over the integer multiple of each Eυ(ϋmuP) of Table 15.276 corresponding to functional-group composition of the molecule. The bond angle parameters of furan determined using Eqs. (15.79-15.108) are given in Table 15.278.
Figure imgf000797_0001
Figure imgf000797_0002
Figure imgf000798_0001
Figure imgf000799_0001
THIOPHENE
Thiophene having the formula C4H4S comprises the conjugated alkene 1,3-butdiene that forms a cyclic structure by terminal-atom bonding to an oxygen atom. The two symmetrical carbon-to-oxygen bonds comprise the C- S- C functional group. The 1,3-butdiene moiety comprises C - C , C = C , and CH functional groups. The CH , C - C , and C = C groups are equivalent to the corresponding groups of pyrrole and furan given in the corresponding sections.
The C- S- C functional group of thiophene is solved in a similar manner as that of the
C- N- C group of pyrrole and the C- O- C group of furan. The solution of the C- S - C functional group comprises the hybridization of the 2s and 2p AOs of each C to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between two C2sp3 HOs and the oxygen atom to form a MO permits each participating hybridized orbital to decrease in radius and energy. Thus, the C - £ - C -bond MO comprising a linear combination of two single bonds is solved in the same manner as a double bond with «, = 2 in Eqs. (15.42) and (15.52).
In thiophene, the energy of sulfur is less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). Thus, c2 in Eq. (15.52) is
C2 [benzeneC2sp3HOj = 0.85252 to match the double-bond character of the C2sp* HOs, and the energy matching condition is further determined by the C2 parameter. Using the energy of S , E(S) = -10.36001 eV in Eq. (15.68) and the C2sp3 HO energy of E(c,2sp^ = -15.76868 eV (Eq. (15.18) corresponding to s = 2 in Eqs. (15.18-15.20), the hybridization factor C2 of Eq. (15.52) for the C - S - C -bond MO is
C2 (S3p to aryl-type C2sp3Hθ) = y '3j ;λ = ~10-36001 eV _ Q.65700 (15.153) n F y ^ F ' E(c,2sp3) -15.76868 eV
C10 is also given by Eq. (15.153). Furthermore, ΔEH (AO / HO) of the C- S- C -
bond MO in Eq. (15.42) and Er[atom- atom,mspι.AO) in Eq. (15.52) are both -0.72457 eV per atom corresponding to -2.89830 eV in total. The energy contribution equivalent to that of a methyl group (Eq. (14.151)) and that of the C-S -bond MO of thiols given in the corresponding section matches the energy of the sulfur atom to the C2spz HOs.
The symbols of the functional groups of thiophene are given in Table 15.279. The structure of thiophene is shown in Figure 67. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of thiophene are given in Tables 15.280, 15.281, and 15.282, respectively. The total energy of thiophene given in Table 15.283 was calculated as the sum over the integer multiple of each ED [Group) of Table 15.282 corresponding to functional-group composition of the molecule. The bond angle parameters of thiophene determined using Eqs. (15.79-15.108) are given in Table 15.284.
Figure imgf000801_0001
Figure imgf000802_0001
Figure imgf000802_0002
OO O
Figure imgf000803_0001
Figure imgf000804_0001
IMIDAZOLE
Imidazole having the formula C3H4N2 comprises a conjugated system that is equivalent to pyrrole with one of the conjugated CH groups replaced by a nitrogen atom. The CH, NH , and C = C groups are equivalent to the corresponding groups of pyrrole, furan, and thiophene where present. In addition, the nitrogen substitution creates a C - N = C moiety comprising C - N and N = C functional groups. The C - N bonding is the same as that of a tertiary amine except that the hybridization term C2 in Eqs. (15.42) and
(15.52) is that of the amino group of aniline, c2 (arylC2sp3HO to N) = 0.84665 (Eq.
(15.152)). The hybridization factor provides double-bond character to match the group to the other orbitals of the molecule. AEH MO(AO / HO) in Eq. ( 15.42) and
Eτ [atom - atom ,msp3 \AO] in Eq. (15.52) are both -0.92918 eV (Eq. (14.513)). This matches the energy of the group to that of the contiguous N - C group wherein AEH MO (AO I HO) in Eq. (15.42) and E7 [atom- atom, msp\Aθ) in Eq. (15.52) are both
-0.92918 eV (Eq. (14.513)) per atom of the double bond with aromatic character as in the case of the prior heterocyclic compounds. As in the prior cases of pyrrole, furan, and thiophene, «, = 2 and C2 and C20 are the same as C2 (benzeneC2sp3Hθ) = 0.85252 (Eq. (15.143)) in Eqs. (15.42) and (15.52). To match the energy of the nitrogen to the C2sp3 HO, c2 of the N = C -bond MO is also given by Eq. (15.152). These parameters also provide an energy match to the C - N - C group. As in the case of pyrrole, the C- N- C -bond MO comprising a linear combination of two single bonds is solved in the same manner as a double bond with «, = 2 in Eqs.
(15.42) and (15.52). The hybridization factor c2 [arylC2sp3 HO to N) = 0.84665 (Eq.
(15.152)) matches the double-bond character of the C2sp3 HOs to the N atom of the NH group, and C2 and C20 in Eqs. (15.42) and (15.52) become that of benzene given by Eq. (15.143), C2 (benzeneC2spiHθ) = 0.85252. Furthermore, AEH m [AO / Hθ) in Eq. (15.42)
and Eτ [atom- atom,msp\Aθ) in Eq. (15.52) are both -0.92918 eV (Eq. (14.513)) per atom corresponding to -3.71673 eV in total. The symbols of the functional groups of imidazole are given in Table 15.285. The structure of imidazole is shown in Figure 68. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of imidazole are given in Tables 15.286, 15.287, and 15.288, respectively. The total energy of imidazole given in Table 15.289 was calculated as the sum over the integer multiple of each ED (omup) of Table 15.288 corresponding to functional-group composition of the molecule. The bond angle parameters of imidazole determined using Eqs. (15.79-15.108) are given in Table 15.290.
Figure imgf000806_0001
Figure imgf000807_0001
Figure imgf000807_0002
Figure imgf000808_0001
Figure imgf000809_0001
PYRIDINE
Pyridine has the formula C5H5N and comprises the benzene molecule with one CH
3e group replaced by a nitrogen atom which gives rise to a C=N functional group. The
3e aromatic C = C and C- H functional groups are equivalent to those of benzene given in the Aromatic and Heterocyclic Compounds section with the aromaticity maintained by the
3e electrons from nitrogen in the C=N group which is also aromatic.
As in the case of the aromatic carbons of benzene, each pyridine C2sp3 HO initially has four unpaired electrons. Each C - H bond has two paired electrons with one donated from the H AO and the other from the C2sp3 HO. In pyridine the three N2p electrons are
3c 3e donated to the aromatic bond. Thus, as in the case of the C-C group, each C=N bond comprises a linear combination of a factor of 0.75 of four paired electrons (three electrons) from the C2sp3 HO and the N2p AO of the participating carbon and nitrogen atoms, respectively.
3e
The solution of the C=N functional group comprises the hybridization of the 2s and 2 p AOs of each C to form a single 2sp3 shell as an energy minimum, and the sharing of electrons between the C2sp3 HO and the nitrogen atom to form a MO permits each
3e participating hybridized orbital to decrease in radius and energy. The C=N -bond MO is solved as a double bond with «, = 2 in Eqs. (15.42) and (15.147). The hybridization factor c2 (C2sp3HO to N) = 0.91140 (Eq. (15.116)) matches the double-bond character of the C2sp3 HO to the N atom, and C2 and C20 in Eqs. (15.42) and (15.147) are also given by Eq.
(15.1 16) in order to match the nitrogen to the aromatic C2,sp3 HO such that
AEH m(AO / HO) = O in Eq. (15.42). Furthermore, Er {atom - atom,msp3.Aθ) of the
3«
C=N -bond MO in Eq. (15.147) due to the charge donation from the C and N atoms to the MO is -1.44915 eV corresponding to an energy contribution from each atom that is equivalent to that of an independent methyl group, -0.72457 e V (Eq. (14.151)). The contributions are also the same as those for a primary amine group as given in the corresponding section. As in the case of benzene, the aromatic E1. [cmiφ) and E0 (ϋm<φ) are given by Eqs. (15.146) and (15.147), respectively, with fλ - 0.75. The breakage of the CNC bonds results in three unpaired electrons on the N atom. Thus, the corresponding E given by Eq. (15.60) was normalized for the two bonds per atom and for fχ - 0.75 and was
subtracted from the total energy of the C=N -bond MO in Eq. (15.147). The pyridine vibrational energies are similar to those of benzene [6OJ; thus, the value for benzene was used.
The symbols of the functional groups of pyridine are given in Table 15.291. The corresponding designation of the structure is shown in Figure 69. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11), (15.17-15.56), and (15.146-15.147)) parameters of pyridine are given in Tables 15.292, 15.293, and 15.294, respectively. The total energy of pyridine given in Table 15.295 was calculated as the sum over the integer multiple of each ED {a,oup) of Table 15.294 corresponding to functional-group composition of the molecule. The bond angle parameters of pyridine determined using Eqs. (15.79-15.108) are given in Table 15.296.
Figure imgf000811_0001
Figure imgf000812_0001
Figure imgf000813_0001
Figure imgf000814_0001
PYRIMIDINE
Pyrimidine has the formula C4H4N2 and comprises the pyridine molecule with one
3« additional CH group replaced by a nitrogen atom which gives rise to a second C=N functional group that is equivalent to that of pyridine given in the corresponding section. The
3e aromatic C=C and C - H functional groups are also equivalent to those of pyridine and benzene given in the Aromatic and Heterocyclic Compounds section with the aromaticity
3e maintained by the electrons from nitrogen in the C=N group which is also aromatic.
The symbols of the functional groups of pyrimidine are given in Table 15.297. The corresponding designation of the structure is shown in Figure 70. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11), (15.17-15.56), and (15.146-15.147)) parameters of pyrimidine are given in Tables 15.298, 15.299, and 15.300, respectively. The total energy of pyrimidine given in Table 15.301 was calculated as the sum over the integer multiple of each ED[umlψ) of Table 15.300 corresponding to functional-group composition of the molecule. The bond angle parameters of pyrimidine determined using Eqs. (15.79-15.108) are given in Table 15.302.
Figure imgf000815_0001
Figure imgf000816_0001
Figure imgf000817_0001
Figure imgf000818_0001
PYRAZINE
Pyrazine has the formula C4H4N2 and comprises the pyrimidine molecule with para
rather than ortho aromatic nitrogen atoms. The C=N functional group is equivalent to that
3e of pyrimidine and pyridine given in the corresponding sections. The aromatic C = C and C-H functional groups are also equivalent to those of pyrimidine, pyridine, and benzene given in the Aromatic and Heterocyclic Compounds section with the aromaticity maintained by the electrons from nitrogen in the C=N group which is also aromatic.
The symbols of the functional groups of pyrazine are given in Table 15.303. The corresponding designation of the structure is shown in Figure 71. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11), (15.17-15.56), and (15.146-15.147)) parameters of pyrazine are given in Tables 15.304, 15.305, and 15.306, respectively. The total energy of pyrazine given in Table 15.307 was calculated as the sum over the integer multiple of each ED{cmuP) of Table 15.306 corresponding to functional-group composition of the molecule. The bond angle parameters of pyrazine determined using Eqs. (15.79-15.108) are given in Table 15.308.
Figure imgf000819_0001
Figure imgf000820_0001
Figure imgf000821_0001
Figure imgf000822_0001
QUINOLINE
Quinoline has the formula C9H7N and comprises the naphthalene molecule with one
CH group replaced by a nitrogen atom which gives rise to a C=N functional group. The
3e aromatic C = C and C-H functional groups are equivalent to those of naphthalene given in the corresponding section with the aromaticity maintained by the electrons from nitrogen in
3e the C=N group which is also aromatic. The C- C functional group is also equivalent to that of naphthalene. The bonding in quinoline can be further considered as a linear
3c combination of the naphthalene and pyridine groups wherein the C=N group is equivalent to that of pyridine, pyrimidine, and pyrazine as given in the corresponding sections.
The symbols of the functional groups of quinoline are given in Table 15.309. The corresponding designation of the structure is shown in Figure 72. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11), (15.17-15.56), and (15.146-15.147)) parameters of quinoline are given in Tables 15.310, 15.311, and 15.312, respectively. The total energy of quinoline given in Table 15.313 was calculated as the sum over the integer multiple of each ED{armφ) of Table 15.312 corresponding to functional-group composition of the molecule. The bond angle parameters of quinoline determined using Eqs. (15.79-15.108) are given in Table 15.314.
Figure imgf000823_0001
Figure imgf000824_0001
Figure imgf000825_0001
Figure imgf000826_0001
ISOQUINOLINE
Isoquinoline has the formula C9H7N and comprises the naphthalene molecule with
3e one CH group replaced by a nitrogen atom which gives rise to a C=N functional group. Isoquinoline is also equivalent to quinoline with the nitrogen in the meta rather than the ortho position relative to the benzene ring of the molecule. The aromatic C=C and C-H functional groups are equivalent to those of naphthalene given in the corresponding section with the aromaticity maintained by the electrons from nitrogen in the C=N group which is also aromatic. The C- C functional group is also equivalent to that of naphthalene. The bonding in isoquinoline can be further considered as a linear combination of the naphthalene
3e and pyridine groups wherein the C=N group is equivalent to that of pyridine, pyrimidine, pyrazine, and quinoline as given in the corresponding sections.
The symbols of the functional groups of isoquinoline are given in Table 15.315. The corresponding designation of the structure is shown in Figure 73. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11), (15.17-15.56), and (15.146-15.147)) parameters of isoquinoline are given in Tables 15.316, 15.317, and 15.318, respectively. The total energy of isoquinoline given in Table 15.319 was calculated as the sum over the integer multiple of each Eυ (βmup) of Table 15.318 corresponding to functional-group composition of the molecule. The bond angle parameters of isoquinoline determined using Eqs. (15.79-15.108) are given in Table 15.320.
Figure imgf000827_0001
Figure imgf000828_0001
o
Figure imgf000828_0002
Figure imgf000829_0001
Figure imgf000830_0001
INDOLE
Indole having the formula C8H7N comprises a phenyl moiety with a conjugated five- membered ring which comprises pyrrole except that one of the double bonds is part of the
3e aromatic ring. The structure is shown in Figure. 74. The aromatic C=C and C-H functional groups of the phenyl moiety are equivalent to those of benzene given in the Aromatic and Heterocyclic Compounds section. The CH, NH , and Cd = Ce groups of the pyrrole-type ring are equivalent to the corresponding groups of pyrrole, furan, and thiophene where present as given in the corresponding sections. The Cb - Cd single bond of aryl carbon to the Cd = Ce bond is also a functional group. This group is equivalent to the C - C(O) group of benzoic acids with regard to ΔEH MO(ΛO I HO) in Eq. (15.42) and
Eτ[atom — atom,msp3.AOj in Eq. (15.52) both being -1.29147 eV . This energy is a linear
combination of of the C- H group that the
Figure imgf000831_0001
Cb - Cd and C - C(O) groups replace, and that of an independent C2sp3 HO, -0.72457 eV
(Eq. (14.151)). However, as in the case of pyrrole, the indole hybridization term C1 is the aromatic c2(benzeneC2sp3 Hθ) = 0.85252 to match the aryl C2,sp3 HO, and the energy terms corresponding to oscillation in the transition state correspond to indole.
As in the case of pyrrole, the C - N - C -bond MO comprising a linear combination of two single bonds is solved in the same manner as a double bond with «, = 2 in Eqs.
(15.42) and (15.52). The hybridization factor c2 [arylC2sp3HO to N) = 0.84665 (Eq. (15.152)) matches the aromatic character of the C2,sp3 HOs to the N atom of the NH group, and C2and C2n in Eqs. (15.42) and (15.52) become that of benzene given by Eq. (15.143), C2 (benzeneC2sp3Hθ) = 0.85252. Furthermore, ΔEH UO (AO / HO) in Eq. (15.42)
and Ejiatom- atom^nsp3.AOj in Eq. (15.52) are both -2.42526 e V which is a linear
combination of — : , E.Jatom- atom,msp?'.AOj of the C- H group that the Cc - N bond replaces, and -1.85836 eV (Eq. (14.513)) which is equivalent to the corresponding component of the C- N- C -bond of pyrrole. The symbols of the functional groups of indole are given in Table 15.321. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of indole are given in Tables 15.322, 15.323, and 15.324, respectively. The total energy of indole given in Table 15.325 was calculated as the sum over the integer multiple of each E l} (Group) of Table 15.324 corresponding to functional- group composition of the molecule. The bond angle parameters of indole determined using Eqs. (15.79-15.108) are given in Table 15.326.
Figure imgf000832_0001
Figure imgf000833_0001
Figure imgf000834_0001
Figure imgf000835_0001
ADENINE
Adenine having the formula C5H5N5 comprises a pyrimidine moiety with an aniline group and a conjugated fϊve-membered ring which comprises imidazole except that one of the double bonds is part of the aromatic ring. The structure is shown in Figure. 75. The aromatic C=C , C - H , and C=N functional groups of the pyrimidine moiety are equivalent to those of pyrimidine as given in the corresponding section. The NH2 and C0 - N7 functional groups of the aniline moiety are equivalent to those of aniline as given in the corresponding section. The CH , NH , Cd ~Ne , and Ne = Ce groups of the imidazole- type ring are equivalent to the corresponding groups of imidazole as given in the corresponding section. The C - N- C functional group of the imidazole-type ring is equivalent to the corresponding group of indole having the same structure with the
C - N- C group bonding to aryl and alkenyl groups.
The symbols of the functional groups of adenine are given in Table 15.327. The geometrical (Eqs. (15.1-15.5) and (15.42)), intercept (Eqs. (15.71-15.78)), and energy (Eqs. (15.6-15.11) and (15.17-15.56)) parameters of adenine are given in Tables 15.328, 15.329, and 15.330, respectively. The total energy of adenine given in Table 15.332 was calculated as the sum over the integer multiple of each ED {GWIΨ) of Table 15.330 corresponding to functional-group composition of the molecule. The bond angle parameters of adenine determined using Eqs. (15.79-15.108) are given in Table 15.332.
Figure imgf000836_0001
Figure imgf000837_0001
Figure imgf000837_0002
Figure imgf000838_0001
Figure imgf000839_0001
While the claimed invention has been described in detail and with reference to specific embodiments thereof, it will be apparent to one of ordinary skill in the art that various changes and modifications can be made to the claimed invention without departing from the spirit and scope thereof.

Claims

I CLAIM:
1. A system of computing and rendering the nature of at least one specie selected from a group of diatomic molecules having at least one atom that is other than hydrogen, polyatomic molecules, molecular ions, polyatomic molecular ions, or molecular radicals, or any functional group therein, comprising physical, Maxwellian solutions of charge, mass, and current density functions of said specie, said system comprising: processing means for processing physical, Maxwellian equations representing charge, mass, and current density functions of said specie; and an output device in communication with the processing means for displaying said physical, Maxwellian solutions of charge, mass, and current density functions of said specie.
2. The system of claim 1 wherein the output device is a display that displays at least one of visual or graphical media.
3. The system of claim 2 wherein the display is at least one of static or dynamic.
4. The system of claim 3 wherein at least one of vibration and rotation is be displayed.
5. The system of claim 4 wherein displayed information is used to model reactivity and physical properties.
6. The system of claim 5, wherein the output device is a monitor, video projector, printer, or three-dimensional rendering device.
7. The system of claim 6 wherein displayed information is used to model other species and provides utility to anticipate their reactivity and physical properties.
8. The system of claim 7 wherein the processing means is a general purpose computer.
9. The system of claim 8 wherein the general purpose computer comprises a central processing unit (CPU), one or more specialized processors, system memory, a mass storage device such as a magnetic disk, an optical disk, or other storage device, an input means.
10. The system of claim 9, wherein the input means comprises a serial port, usb port, microphone input, camera input, keyboard or mouse.
11. The system of claim 10 wherein the processing means comprises a special purpose computer or other hardware system.
12. The system of claim 11 further comprising computer program products.
13. The system of claim 12 further comprising computer readable medium having embodied therein program code means.
14. The system of claim 13 wherein the computer readable media is any available media which can be accessed by a general purpose or special purpose computer.
15. The system of claim 14 wherein the computer readable media comprises at least one of RAM, ROM, EPROM, CD ROM, DVD or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium which can embody the desired program code means and which can be accessed by a general purpose or special purpose computer.
16. The system of claim 15 wherein the program code means comprises executable instructions and data which cause a general purpose computer or special purpose computer to perform a certain function of a group of functions.
17. The system of claim 16 wherein the program code is Mathematica programmed with an algorithm based on the physical solutions, and the computer is a PC.
18. The system of claim 17 wherein the algorithm is ParametricPlot3D[{2*Sqrt[l- z*z]*Cos[u],Sqrt[(l-z*z)]*Sin[u],z},{u,0,2* Pi},{z,-1,.9999}], and the rendering is viewed from different perspectives.
19. The system of claim 18 wherein the algorithms for viewing from different perspectives comprises Show[Out[l], ViewPoint->{x,y,z}] where x, y, and z are Cartesian coordinates.
20. The system of claim 19 wherein the physical, Maxwellian solutions of the charge, mass, and current density functions of said specie comprises a solution of the classical wave
Figure imgf000843_0001
21. The system of claim 20 wherein the boundary constraint of the wave equation solution is nonradiation according to Maxwell's equations.
22. The system of claim 21 wherein the boundary condition is met for an ellipsoidal-time harmonic function when
_ πh ___ h " meA meαh where the area of an ellipse is A = παh where 2b is the length of the semiminor axis and 1α is the length of the semimajor axis.
23. The system of claim 22 wherein the specie charge and current density functions, bond distance, and energies are solved from the Laplacian in ellipsoidal coordinates:
Figure imgf000843_0002
with the constraint of nonradiation.
24. The system of claim 23 wherein each bond of the said specie defined as a molecular orbital (MO) has the ellipsoidal charge-density function given by σ = ? *
Aπαbc & y P"
25. The system of claim 24 wherein the bonds of the said specie defined as a molecular orbital (MO) has the charge-density function comprising a linear combination of ellipsoids wherein the charge density of one said ellipsoid is given by
Figure imgf000844_0004
26. The system of claim 25 wherein the equation of motion has the parametric form r(t) = iαr cos ωt + \b sin ωt
27. The system of claim 26 wherein he force balance of the hydrogen-type molecular ion ellipsoidal MO is given by
mea2b2 &πε0 a = 2a0
28. The system of claim 27 where the force constant k of a HJ -type ellipsoidal MO due to the equivalent of two point charges of at the foci is given by: 2e2
Jc = -
4πε»
29. The system of claim 28 wherein the distance from the origin of the H2 -type-ellipsoidal- MO to each focus c ' is given by :
Figure imgf000844_0001
30. The system of claim 29 wherein the internuclear distance 2c' is given by:
Figure imgf000844_0003
31. The system of claim 30 wherein the length of the semiminor axis of the prolate spheroidal H2 -type MO b = c is given by:
Figure imgf000844_0002
32. The system of claim 31 wherein the length of the semiminor axis of the prolate spheroidal
H2 -type MO b = c is given by: c' e = — a
33. The system of claim 32 wherein the internuclear distance, 2c' , which is the distance between the foci is
2c' = 2ao; the semiminor axis is b = V3α0 , and the eccentricity, e , is
1 e = — . 2
34. The system of claim 33 wherein the potential energy of the electron in the central field of the protons at the foci is
Figure imgf000845_0001
-Ae2 , a + c' -In
8πεoc' a-C
-4e2
-In3 = -59.7575 e V
SπεoaH
The potential energy of the two protons is
γ P =- = 13.5984 eV
Sπε0ΛJa2 -b2 8πεoaH The kinetic energy of the electron is
29.8787 eF , and
Figure imgf000845_0002
The total energy, E7, , is given by the sum of the energy terms:
ET = Ve +T + Vp
Figure imgf000846_0001
—e
(41n3 -l-21n3)
SπeoaH
= -16.2803 eV wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
35. The system of claim 34 wherein during bond formation, the electron undergoes a reentrant oscillatory orbit with vibration of the protons, and the corresponding energy Eosc is the difference between the Doppler and average vibrational kinetic energies:
Figure imgf000846_0002
The total energy is
Eτ +τ+yp
Figure imgf000846_0003
= -16.2803 eF-0.118811 eF + -(0.29282 eV)
= -16.2527 eV wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
36. The system of claim 35 wherein the bond dissociation energy, ED , is the difference between the total energy of the hydrogen atom and Eτ :
ED = E(H)- E7 = 2.654 eV wherein the total energy of a hydrogen atom is
E(H)= -13.59844 eV, wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
37. The system of claim 36 wherein the H* -type ellipsoidal MO and the hydrogen atomic orbital (AO) serve as basis functions for the MOs of specie.
38. The system of claim 37 wherein the MO must (1) be a solution of Laplace's equation to give a equipotential energy surface, (2) correspond to an orbital solution of the Newtonian equation of motion in an inverse-radius-squared central field having a constant total energy, (3) be stable to radiation, and (4) conserve the electron angular momentum of h .
39. The system of claim 38 wherein the potential energy of electron in the central field of the protons at the foci is
Figure imgf000847_0001
The potential energy of the two protons is e2 e2 P Uεja2 -b2 8JiS0C1 '
The kinetic energy of the electron is
Figure imgf000847_0002
The total energy, Eτ , is given by the sum of the energy terms:
Eτ = K +τ+vp
Figure imgf000847_0003
40. The system of claim 39 wherein during bond formation, the electron undergoes a reentrant oscillatory orbit with vibration of the protons, and the corresponding energy Eosc is the difference between the Doppler and average vibrational kinetic energies:
^
Figure imgf000847_0004
The total energy is
ET = Ve +T + Vp + Eosc
Figure imgf000848_0001
41. The system of claim 40 wherein the force balance equation derived of a H2 -type ellipsoidal MO is given by
mea 2* 2 8πε ab2 2mea ,2 bL2
α = ^
42. The system of claim 41 where the force constant k of a H2 -type ellipsoidal MO due to the equivalent of two point charges of at the foci is given by:
2e2 k = -
4πεn
43. The system of claim 42 wherein the distance from the origin of the H2 -type-ellipsoidal- MO to each focus c ' is given by :
Figure imgf000848_0002
44. The system of claim 43 wherein the internuclear distance 2c ' is given by:
Figure imgf000848_0003
45. The system of claim 44 wherein the length of the semiminor axis of the prolate spheroidal H2 -type MO b = c is given by:
46. The system of claim 45 wherein the length of the semiminor axis of the prolate spheroidal H2 -type MO b = c is given by: c' e = — a
47. The system of claim 46 wherein the internuclear distance, 2c1 , which is the distance between the foci is
Figure imgf000849_0001
the semiminor axis is
b = ^ ao > aΑd the eccentricity, e , is 1
V2 "
48. The system of claim 47 wherein the potential energy of the two electrons in the central field of the protons at the foci is
Figure imgf000849_0002
The potential energy of the two protons is
Figure imgf000849_0003
The kinetic energy of the electrons is
Figure imgf000849_0004
The energy, Vn , of the magnetic force between the electrons is
and
Figure imgf000849_0005
The total energy, E7, , is given by the sum of the energy terms: ET = Ve +T + Vm +Vp Eτ ~P !31.63
Figure imgf000850_0001
wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
49. The system of claim 48 wherein during bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the protons, and the corresponding energy Eosc is the difference between the Doppler and average vibrational kinetic energies:
Figure imgf000850_0002
The total energy is
ET = Ve +T + Vm +Vp +E0SC
Figure imgf000850_0003
= -31.635 eF-0.326469 e V +-(0.56764 eV)
= -31.6776 eV wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to ± 10%, if desired.
50. The system of claim 49 wherein the bond dissociation energy, E0 , is the difference between the total energy of the hydrogen atoms and E1. : ED = E(2H[aH} -ET = 4.478 eV wherein the total energy of two hydrogen atoms is
E(2H\aH J = -27.21 eV, wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
51. The system of claim 50 wherein the H2 -type ellipsoidal MO and the hydrogen atomic orbital (AO) serve as basis functions for the MOs of the specie.
52. The system of claim 51 wherein the MO must (1) be a solution of Laplace's equation to give a equipotential energy surface, (2) correspond to an orbital solution of the Newtonian equation of motion in an inverse-radius-squared central field having a constant total energy, (3) be stable to radiation, and (4) conserve the electron angular momentum of h .
53. The system of claim 52 wherein the potential energy of the two electrons in the central field of the nuclei at the foci is
Figure imgf000851_0001
The potential energy of the two nuclei is
Figure imgf000851_0002
The kinetic energy of the electrons is
Figure imgf000851_0003
The energy, V111 , of the magnetic force between the electrons is
Figure imgf000851_0004
The total energy, E7, , is given by the sum of the energy terms:
ET = Ve +T + Vm +Vp
Figure imgf000851_0005
54. The system of claim 53 wherein during bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energy Eosc is the difference between the Doppler and average vibrational kinetic energies:
Figure imgf000851_0006
The total energy is
ET = Ve +T + Vm +Vp +Eo:
Figure imgf000852_0001
55. The system of claim 54 wherein during bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energy Eosc is the difference between the Doppler and average vibrational kinetic energies:
Eosc and
Figure imgf000852_0002
The total energy is
ET = Ve +T+ Vm + Vp +E0SC
E7,
Figure imgf000852_0003
56. The system of claim 55 wherein the energy of the MO is matched to that of the outermost atomic orbital of a bonding heteroatom in the case where a minimum energy is achieved with a direct bond to the AO.
57. The system of claim 56 wherein the MO is continuous with the AO containing paired electrons that do not participate in the bond, and said paired electrons provide a means for the energy matched MO to form a continuous equipotential energy surface.
58. The system of claim 57 wherein an independent MO is formed such that the AO force balance causes the remaining electrons to be at lower energy and a smaller radius.
59. The system of claim 58 wherein at least two atomic orbital hybridize as a linear combination of electrons at the same energy in order to achieve a bond at an energy minimum.
60. The system of claim 59 wherein at least two atomic orbital hybridize as a linear combination of electrons at the same energy in order to achieve a bond at an energy minimum.
61. The system of claim 60 wherein at least two atomic orbitals hybridize as a linear combination of electrons at the same energy in order to achieve a bond at an energy minimum, and the sharing of electrons between two or more such orbitals to form a MO permits the participating hybridized orbitals to decrease in energy through a decrease in the radius of one or more of the participating orbitals.
62. The system of claim 61 where the force constant k' ofa. H2 -type ellipsoidal MO due to the equivalent of two point charges of at the foci is given by: k , ^ c1C2Ie2 4πεo where C1 is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the specie and c2 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of the chemical bond.
63. The system of claim 62 where the distance from the origin of the MO to each focus c' is given by:
Figure imgf000853_0001
' the internuclear distance is
\aa,
2c' = 2Λ •*o .
2k' ' the length of the semiminor axis of the prolate spheroidal MO h = c is given by
Figure imgf000853_0002
and, the eccentricity, e , is c' e = — . a
64. The system of claim 63 wherein the potential energy of the two electrons in the central field of the nuclei at the foci is
Figure imgf000854_0001
The potential energy of the two nuclei is
V = H1 -
%πεja2 -h2 The kinetic energy of the electrons is
Figure imgf000854_0002
The energy, V1n , of the magnetic force between the electrons is
^ =
Figure imgf000854_0003
The total energy, Eτ , is given by the sum of the energy terms:
ET = Ve +T + Vm +Vp
Figure imgf000854_0004
where Ux is the number of equivalent bonds of the MO, Cx is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the specie, and c2 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of each chemical bond.
65. The system of claim 64 wherein during bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energy E010 is the difference between the Doppler and average vibrational kinetic energies:
Figure imgf000854_0005
The total energy is
ET = Ve +T + Vm +Vp +Eosc
Figure imgf000855_0001
where U1 is the number of equivalent bonds of the MO, C1 is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the specie, and c2 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of each chemical bond.
66. The system of claim 65 wherein during bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energy Eosc is the difference between the Doppler and average vibrational kinetic energies:
Figure imgf000855_0002
The total energy is
ET = Ve +T + Vm +Vp +Eosc
Figure imgf000855_0003
where W1 is the number of equivalent bonds of the MO, C1 is the fraction of the H1 -type ellipsoidal MO basis function of a chemical bond of the specie, and c2 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of each chemical bond.
67. The system of claim 66 wherein a hybridized shell comprises a linear combination of the electrons of at least two atomic-orbital shells.
68. The system of claim 67 wherein the radius of the hybridized shell is calculated from the Coulombic energy equation by considering that the central field decreases by an integer for each successive electron of the shell and the total energy of the shell is equal to the total
Coulombic energy of the initial AO electrons.
69. The system of claim 68 wherein the total energy E1. (atom, msp3 J (m is the integer of the valence shell) of the AO electrons and the hybridized shell is given by the sum of energies of successive ions of the atom over the n electrons comprising total electrons of the at least two AO shells
70. The system of claim 69 wherein the radius rms , of the hybridized shell is given by:
Figure imgf000856_0002
71. The system of claim 70 wherein the Coulombic energy ECoulomb (atom, msp3) of the outer electron of the atom msp3 shell is given by
— e
8πεnr Ψ*
72. The system of claim 71 wherein the during hybridization at least one of the spin-paired AO electrons is unpaired in the hybridized orbital (HO) and the energy change the promotion to the unpaired state is the magnetic energy E (magnetic) at the initial radius rn of the AO electron: _ . . . 2πμne2 2 E(magnetπ) = f° 2nh2 = 8πμnμl =~ 8 wfψ2 -:(O3 " (O3
73. The system of claim 72 wherein the energy E(atom,msp3 j of the outer electron of the
atom msp3 shell is given by the sum of ECoulomb (atom, msp3 ) and E(magnetic) :
Figure imgf000856_0001
74. The system of claim 73 wherein at least two atomic orbitals hybridize as a linear combination of electrons at the same energy in order to achieve a bond at an energy minimum, and the sharing of electrons between two or more such hybridized orbitals to form a MO permits the participating hybridized orbitals to decrease in energy through a decrease in the radius of one or more of the participating orbitals; the total energy of the hybridized orbitals is given by the sum of E (atom, msp3) and the next energies of successive ions of the atom over the n electrons comprising total electrons of the at least two initial AO shells; is the sum of the first energy of the atom and the hybridization energy.
75. The system of claim 74 wherein the sharing of electrons between two atom msp3 HOs to form an atom-atom-bond MO permits each participating hybridized orbital to decrease in radius and energy.
76. The system of claim 75 wherein in order to further satisfy the potential, kinetic, and orbital energy relationships, each atom msp3 HO donates an excess of 25% per bond of its electron density to the atom-atom-bond MO to form an energy minimum wherein the atom- atom bond comprises one of a single, double, or triple bond.
77. The system of claim 76 wherein the radius of the hybridized shell is calculated from the Coulombic energy equation by considering that the central field decreases by an integer for each successive electron of the shell and the total energy of the shell is equal to the total Coulombic energy of the initial AO electrons and the hybridization energy.
78. The system of claim 77 wherein the total energy Eτ ( mol. atom, msp3 j (m is the integer of the valence shell) of the HO electrons is given by the sum of energies of successive ions of the atom over the n electrons comprising total electrons of the at least two initial AO shells and the hybridization energy.
79. The system of claim 78 wherein the radius r 3 of the hybridized shell is given by:
Figure imgf000857_0001
where s = 1,2,3 for a single, double, and triple bond, respectively.
80. The system of claim 79 wherein the Coulombic energy ECoulomb (moLatom, msp3) of the outer electron of the atom msp3 shell is given by
Figure imgf000858_0003
81. The system of claim 80 wherein the during hybridization at least one of the spin-paired AO electrons is unpaired in the hybridized orbital (HO) and the energy change the promotion to the unpaired state is the magnetic energy E(magnetic) at the initial radius rn of the AO electron:
Figure imgf000858_0001
82. The system of claim 81 wherein the energy E(mol.atom,msp3 J of the outer electron of
the atom msp3 shell is given by the sum of ECoulomb (moLatom, msp3) and Eimagnetic) :
Figure imgf000858_0002
83. The system of claim 82 wherein Eτ (atom - atom^msp3 ) , the energy change of each atom msp3 shell with the formation of the atom-atom-bond MO is given by the difference between E [moLatom, msp3 ) and E (atom, msp2 ) .
E1. (atom - atom, msp3 ) = E (moLatom, msp3 )-E (atom, msp3 )
84. The system of claim 83 wherein to meet an energy matching condition for all MOs at all HOs, the energy E (moLatom, msp3) of the outer electron of the atom msp3 shell of each
bonding atom must be the average of E (moLatom, msp3 ) for two different values of s :
, ,N E(mol(sΛ, msp3) + E(mol(s2), msp3) E(moL,msp3) = -^ V U y } ^ V 2 } F ' (14.512)
85. The system of claim 84 wherein Eτ (atom - atom, msp3 ) , the energy change of each atom msp3 shell with the formation of each atom-atom-bond MO, is average of
Eτ (atom - atom, msp3 J for two different values of s :
, v Eτ ( atom- atomtsΛ,msp ) + Eτ ( atom- atomCs2),msp3) Eτ {atom -atom,msp ) = — '— —
86. The system of claim 85 wherein the radius rm 3 of the atom msp3 shell of each bonding
atom is given by the Coulombic energy using the initial energy ECoulomb (atom, msp3 ) and
Eτ (atom - atom, msp3 \ , the energy change of each atom msp3 shell with the formation of each atom-atom-bond MO :
(14.514)
Figure imgf000859_0002
87. The system of claim 86 wherein the Coulombic energy ECoulomb (mol. atom, msp' ') of the outer electron of the atom msp3 shell is given by
Ecou,omb (mol. atom, msp3 )
Figure imgf000859_0003
Sπεnr \sp
88. The system of claim 87 wherein the during hybridization at least one of the spin-paired AO electrons is unpaired in the hybridized orbital (HO) and the energy change the promotion to the unpaired state is the magnetic energy EQnagnetic) at the initial radius rnof the AO electron:
„. .
E(magnetw)
Figure imgf000859_0001
89. The system of claim 88 wherein the energy E (mol. atom, msp3) of the outer electron of the atom msp3 shell is given by the sum of ECoulomb {moLatom, msp3 J and E(magnetic) :
Figure imgf000860_0001
90. The system of claim 89 wherein E1. {atom - atom, msp3 ) , the energy change of each atom msp3 shell with the formation of the atom-atom-bond MO is given by the difference between E{mol.atom,msp3)and E{αtom,msp3\ .
E7, {atom - atom, msp3 ) = E (moLatom, msp3 \- E {atom, msp3 j
91. The system of claim 90 wherein ECouhmb {mol. atom, msp3) is one of is one of
ECoulomb {Cethylem^P ) ' ECculomb {C 'ethane > 2sP~ ' ) . ECoulcmb {C acetylene > 2sP ) » ^
Ecoulomb \palkam> 2sP ) >
Ecouiomb {atom,msp3) is one of ECmlomb {C,2sp3 ) and ECouhmb {θ,3sp3 ) ; E {mol. atom, msp3 ) is one of E {Celhylene , 2sp3 ) , E {Cetham , 2sp3 ) , E{Cacetykne,2sp3) E{Calkam,2sp3) ;
E {atom, msp3 ) is one of and E {c, 2sp3 ) and E {CI, 3sp3 ) ;
E7 {atom - atom, msp3 ) is one of E {C - C, 2sp3 ) , E {C = C,2sp3 ) , and.
E{c ≡ C,2sp3) ; atom msp3 is one of C2sp3 , Cl3sp3
E1, {atom- atom {sλ), msp3 J is ET {C-C,2sp3j and Eτ {atom- atom {s2), msp31 J is
Eτ {C = C,2sp3) , anά
rmsp^ 1S °ne 0^ rC2φ3 ' rethαne2sp'i ' ' 'ethylene! sp> ' ' " acetyleneisp* ' r alkam2sp' ' ^ rCl3sp2 '
92. The system of claim 91 wherein the energy of the MO is matched to that of the outermost atomic orbital or hybridized orbital of a bonding atom in the case where a minimum energy is achieved with a direct bond to the AO or HO.
93. The system of claim 92 wherein the potential energy of the two electrons in the central field of the nuclei at the foci is
Figure imgf000861_0001
The potential energy of the two nuclei is
Figure imgf000861_0002
The kinetic energy of the electrons is
Figure imgf000861_0003
The energy, V1n , of the magnetic force between the electrons is
-h2 , a + yla2 -b2 Vm = niClC2 rln- and
4m a-\Ja2 -b2 a-yla2 -b2 The total energy, Eτ , is given by the sum of the energy terms plus E(AO) :
ET = Ve +T + Vm +Vp
Figure imgf000861_0004
where H1 is the number of equivalent bonds of the MO, C1 is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the specie, c2 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of each chemical bond, and E (AO) is the energy of the at least one atomic orbital to which the MO is energy matched.
94. The system of claim 93 wherein during bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energy Eosc is the difference between the Doppler and average vibrational kinetic energies: EOSc = ED + EKvib , ^ά the total energy is
ET = Ve +T + Vm + Vp +Eosc +E(AO) .
95. The system of claim 94 wherein during bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energy Eosc is the difference between the Doppler and average vibrational kinetic energies:
md
Figure imgf000862_0001
the total energy is
ET = Ve +T + Vm +Vp +E0SC
Figure imgf000862_0002
where Hx is the number of equivalent bonds of the MO, C1 is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the specie, c2 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of each chemical bond, and E (AO) is the energy of the at least one atomic orbital to which the MO is energy matched.
96. The system of claim 95 wherein during bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energy Eosc is the difference between the Doppler and average vibrational kinetic energies:
Figure imgf000862_0003
the total energy is
F = V +T + V +V +E
Figure imgf000863_0001
where M1 is the number of equivalent bonds of the MO, C1 is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the specie, C2 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of 5 each chemical bond, and E(AO) is the energy of the at least one atomic orbital or hybridized atomic orbital or said orbital or hybridized orbital and the energy change with the formation of the bond by one or more of the orbital electrons to match the energies of the said orbital and the MO.
10 97. The system of claim 96 wherein E(AO) , the energy of the at least one atomic orbital or hybridized atomic orbital or said orbital or hybridized orbital and the energy change with the formation of the bond by one or more of the orbital electrons to match the energies of the said orbital and the MO, is at least one from the group of E(AO) = E(OIp shell) = -EQonization; O) = -13.6181 eV ;
15 E(AO) = E(NIp shell) = -E(ionization; N) = -14.53414 eV ;
E(AO) = E(C,2sp3) = -14.63489 eV ;
Eτ [AO) = ECoulomb (Cl,3sp3) = -14.60295 eV ;
Eτ (AO) = E(ionization\ C) + E(ionization; C+) ;
ET {AO) = E(Celhane,2sp3) = -15.35946 eV ;
20 Eτ [AO) = +E(Cethylette,2sp3 )-E(Celhylene,2ψ3) ;
Eτ [AO) = E(c,2sp3 )-2ET (C = C,2sp3) = -14.63489 eF-(-2.26758 eV) ;
Er (ΛO) = £(Cα^te,e,2,/)-£^^
E7, (AO) = E(C,2sp3)-2ET (C ≡ C,2sp3) = -14.63489 eV-(-3.13026 eV) ;
Eτ (AO) = E(Cbemem,2sp3 )- E(Cbemene,2sp3 ) ; E7 (AO) = E(C,2sp3)- ET (C = C,2sp3) = -U.63m eV -(-IΛ3379 eV) , and
E7, (AO) ~ E(Calkane,2sρ3 ) =-15.56407 eV , wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
98. The system of claim 97 wherein is substituted into E1, to give
Figure imgf000864_0001
Figure imgf000864_0002
where H1 is the number of equivalent bonds of the MO, C1 is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the specie, C2 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of each chemical bond, and E (AO) is the energy of the at least one atomic orbital or hybridized atomic orbital to which the MO is energy matched.
99. The system of claim 98 wherein E(basis energies) is given by the sum of a first integer qx times the total energy of H2 and a second integer q2 times the total energy of H , minus a third integer q3 times the total energy of E(AO) where the first integer can be 1,2,3..., and each of the second and third integers can be 0, 1, 2, 3....
100. The system of claim 99 wherein E7, is set equal to E(basis energies) , and the semimajor axis a is solved.
101. The system of claim 100 wherein the semimajor axis a is solved from the equation of the form: energies)
Figure imgf000865_0001
102. The system of claims 101 and 63 wherein the distance from the origin of the H2 -type- ellipsoidal-MO to each focus c ' , the internuclear distance 2c ' , and the length of the semiminor axis of the prolate spheroidal H2 -type MO b = c are solved from the semimajor axis a .
103. The system of claims 102 where the number of equivalent bonds of the MO i\ each comprising an H2 -type ellipsoidal MO is an integer greater than one.
104. The system of claims 103 where the fraction C1 of a H2 -type ellipsoidal MO is 1. 0.75, 0.5, and 0.75/2.
105. The system of claims 104 where the factor C2 of a H2 -type ellipsoidal MO is given by one of the list of 1 and the ratio of the ionization energy of at least one atom of the bond and 13.605804 eV , the Coulombic energy between the electron and proton of H to meet the equipotential condition of the union of the H2 -type-ellipsoidal-MO and the AO of the atom, wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
106. The system of claim 105 where the factor c2 of a H2 -type ellipsoidal MO is
0.936127 , the ratio of the ionization energy of N 14.53414 eV and 13.605804 eV , the Coulombic energy between the electron and proton of H ;
0.91771 , the ratio of 14.82575 eV , ECoulomb (C,2sp3 ) and 13.605804 eV , the
Coulombic energy between the electron and proton of H ;
0.93172 , the ratio of 14.60295 eV , ECmlamb (Cl,3sp3) given by Eq. (13.759), and 13.605804 eV , the Coulombic energy between the electron and proton of H ; 0.87495, the ratio of 15.55033 eV , ECoulomb {Celhme,2sp%) and 13.605804 eV , the
Coulombic energy between the electron and proton of H ;
0.85252 , the ratio of 15.95955 eV , ECouhmb (Celhylem,2sp3) and 13.605804 eV , the Coulombic energy between the electron and proton of H ;
0.85252 , the ratio of 15.95955 eV , ECoulomb (Cbenzene,2sp^) and 13.605804 eV , the Coulombic energy between the electron and proton of H , or
0.86359 , the ratio of 15.55033 eV , ECoulomb (Caltøe,2sp3) and 13.605804 eV , the
Coulombic energy between the electron and proton of H , wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
107. The system of claim 106 where the fraction C1 of a H2 -type ellipsoidal MO is such that the orbital energy E for each elliptical cross section of the prolate spheroidal MO is given by the sum of the kinetic T and potential V energies; E = T + V is constant; the orbit is closed such that T <| V | ; the time average of the kinetic energy, < T > , for elliptic motion in an inverse-squared field is 1/2 that of the time average of the magnitude of the potential energy, < F >| (<r >= l/2|< F >| ), and in the case that the energy of the MO is matched to at least one atomic orbital (AO), E = T + V , and for all points on the AO5 |E| = T = 1 / 2|K| .
108. The system of claim 107 where the energy of the MO is matched to at least one atomic orbital (AO) such that E = T + V , and for all points on the AO, \E\ = T = 1 / 2 \V\ .
109. A system of claim 108 of computing and rendering the nature of bound atomic and atomic ionic electrons from physical solutions of the charge, mass, and current density functions of atoms and atomic ions, which solutions are derived from Maxwell's equations using a constraint that the bound electron(s) does not radiate under acceleration, comprising: processing means for processing and solving the equations for charge, mass, and current density functions of electron(s) in a selected atom or ion, wherein the equations are derived from Maxwell's equations using a constraint that the bound electron(s) does not radiate under acceleration; and a display in communication with the processing means for displaying the current and charge density representation of the electron(s) of the selected atom or ion.
110. The system of claim 109 wherein the physical, Maxwellian solutions of the charge, mass, and current density functions of atoms and atomic ions comprises a solution of the
- 0 which is the equation of motion of the
Figure imgf000867_0001
charge.
111. The system of claim 110, wherein the time, radial, and angular solutions of the wave equation are separable.
112. The system of claim 111, wherein the boundary constraint of the wave equation solution is nonradiation according to Maxwell's equations.
113. The system of claim 112, wherein a radial function that satisfies the boundary condition is a radial delta function
114. The system of claim 113, wherein the boundary condition is met for a time harmonic function when the relationship between an allowed radius and the electron wavelength is given by
ω = h r- , and A mer h v = mj where ω is the angular velocity of each point on the electron surface, v is the velocity of each point on the electron surface, and r is the radius of the electron.
115. The system of claim 114, wherein the spin function is given by the uniform function
F0 0 (φ, θ) comprising angular momentum components of L = — and L- = — .
116. The system of claim 115, wherein the atomic and atomic ionic charge and current density functions of bound electrons are described by a charge-density (mass-density) function which is the product of a radial delta function, two angular functions (spherical harmonic functions), and a time harmonic function: p(r,θ,φ,t) Y(θ,φ)k(t)
Figure imgf000868_0001
wherein the spherical harmonic functions correspond to a traveling charge density wave confined to the spherical shell which gives rise to the phenomenon of orbital angular momentum.
117. The system of claim 116, wherein based on the radial solution, the angular charge and current-density functions of the electron, A(θ,φ,t) , must be a solution of the wave equation in two dimensions (plus time),
Figure imgf000868_0002
where p(r,θ,φ,t) Y(θ,φ)k(t)
Figure imgf000868_0003
Figure imgf000868_0004
where v is the linear velocity of the electron.
118. The system of claim 117, wherein the charge-density functions including the time- function factor are
Jl = O
Figure imgf000869_0001
Jl ≠ 0
Figure imgf000869_0004
where Y"' (θ, ^) are the spherical harmonic functions that spin about the z-axis with angular frequency ωn with Y° (θ,φ) the constant function where to keep the form of the spherical
Figure imgf000869_0005
harmonic as a traveling wave about the z-axis, ωn = mωn .
119. The system of claim 118, wherein the spin and angular moment of inertia, I, angular momentum, L, and energy, E, for quantum number i are given by Jl = O
Figure imgf000869_0006
Figure imgf000869_0002
i ≠O
Figure imgf000869_0003
L£ = rrih
^-1Z tola! ~ ^z spin "*" As orbital
Figure imgf000870_0006
Figure imgf000870_0001
\ ~ 'rotational, orbital ] ~ '
5 120. The system of claim 119, wherein the force balance equation for one-electron atoms and ions is
Figure imgf000870_0002
1 Z where aH is the radius of the hydrogen atom. 0
121. The system of claim 120, wherein from Maxwell's equations, the potential energy V , kinetic energy T , electric energy or binding energy Eele are
V =
Figure imgf000870_0003
-Z2 X 4.3675 X 10"18 J = -Z2 X 27.2 eV Aπεorλ 4πεoaH
5
Figure imgf000870_0004
2^2
E . = - Z g = -Z2 X 2.1786 X 10~ls J = -Z2 Z 13.598 eF ,
wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
0
122. The system of claim 121, wherein the force balance equation solution of two electron atoms is a central force balance equation with the nonradiation condition is given by
Figure imgf000870_0005
which gives the radius of both electrons as
Figure imgf000871_0001
123. The system of claim 122, wherein the ionization energy for helium, which has no electric field beyond r\ is given by
Ionization Energy(He) = —E(electric) + Eζtnagnetic) where,
Figure imgf000871_0003
Figure imgf000871_0002
For 3 < Z
Ionization Energy = -Electric Energy Magnetic Energy .
124. The system of claim 123, wherein the electrons of multielectron atoms all exist as orbitspheres of discrete radii which are given by rn of the radial Dirac delta function, δ(r -rn) .
125. The system of claim 124, wherein electron orbitspheres may be spin paired or unpaired depending on the force balance which applies to each electron wherein the electron configuration is a minimum of energy.
126. The system of claim 125, wherein the minimum energy configurations are given by solutions to Laplace's equation.
127. The system of claim 126, wherein the electrons of an atom with the same principal and i quantum numbers align parallel until each of the m jj levels are occupied, and then pairing occurs until each of the m JJ levels contain paired electrons.
128. The system of claim 127, wherein the electron configuration for one through twenty- electron atoms that achieves an energy minimum is: Is < 2s < 2p < 3s < 3p < 4s.
129. The system of claim 128, wherein the corresponding force balance of the central Coulombic, paramagnetic, and diamagnetic forces was derived for each n-electron atom that was solved for the radius of each electron.
130. The system of claim 129, wherein the central Coulombic force is that of a point charge at the origin since the electron charge-density functions are spherically symmetrical with a time dependence that is nonradiative.
131. The system of claim 130, wherein the ionization energies are obtained using the calculated radii in the determination of the Coulombic and any magnetic energies.
132. The system of claim 131, wherein the general equation for the radii of s electrons is given by
Figure imgf000872_0001
rm in units of aQ where positive root must be taken in order that rn > 0 ;
Z is the nuclear charge, n is the number of electrons, rm is the radius of the proceeding filled shell(s) given by
Figure imgf000873_0001
or the preceding s shell(s);
r3
Figure imgf000873_0002
r3 in units of a0 or the 2p shell, and
Figure imgf000873_0003
r12 m units of aQ or the 3p shell; the parameter A corresponds to the diamagnetic force, Fdjamαgne/jc :
Figure imgf000874_0001
the parameter B corresponds to the paramagnetic force, Fmag. 2 :
Figure imgf000874_0002
the parameter C corresponds to the diamagnetic force, FΛ ne/te 3 :
Figure imgf000874_0003
the parameter D corresponds to the paramagnetic force, Fmag :
Figure imgf000874_0004
the parameter E corresponds to the diamagnetic force, ¥dιamagnellc 2 , due to a relativistic effect with an electric field for r > /*„ :
Figure imgf000874_0005
diamagnetic 2 and
Figure imgf000874_0006
diamagnetic 2
Figure imgf000874_0007
wherein the parameters of atoms filling the Is, 2s, 3 s, and 4s orbitals are
Atom Electron Ground Orbital Diamag. Paramag. Diamag. Param Diamag.
Type Configuration State Arrangement Force Force Force ag. Force
Term of Factor Factor Factor Force Factor s Electrons A B C Factor β
(s state) D
Neutral \$ λ 2 o
»1/2
I e 0 0 0
Atom
H
Neutral \s 2 1S0
2 e 0
Atom
He
Neutral 2sl 2S '1,/2
3 e 0 0 0
Atom
Li
Neutral 2s2 1S0
4 e 0 0
Atom
Be
Neutral \s 22s22p6?>sl Ml li e 1 0 8 0 0
Atom
Na
Neutral Is22s22p63s2 %
12 e 1 3 12 1 0
Atom
Mg
Neutral Ly^s22p63s23p6Asl 2S112
19 e 2 0 12 0 0
Figure imgf000875_0001
Atom K
Neutral Is22s22p63s23p64s2 1S0
2Oe 1 3 24 1 0
Atom
Ca
Ie Ion i/ 2sm 0 0 0 0 0
2 e Ion \s 2 X
0
3 e Ion 2s1 X2 1 0 0 0 1
4elon 2s2 1S0 1 0 0 1 1
11 e Ion ^22s22p63s1 2S1n 1 4 8 0 J2
1 "I" '
2
12eloni/2522j!76352 1^0 1 6 0 0 ,/2
+ 2 19 e Ion I5 22/ 2p63s23p64sl 2Sm 3 0 24 0 2-V2
20QΪon\s^2s22p63s23p64s2 1S0 2 0 24 0 2— >/2
Figure imgf000876_0001
133. The system of claim 132, with the radii, rn , wherein the ionization energy for atoms having an outer s-shell are given by the negative of the electric energy, E(electric) , given by:
Figure imgf000877_0003
except that minor corrections due to the magnetic energy must be included in cases wherein the s electron does not couple to p electrons as given by
Figure imgf000877_0004
1 Ionization Energy = -Electric Energy Magnetic Energy
38 eV
^ ^
Figure imgf000877_0005
= 8.9216 eV + 0.03226 eV + 0.33040 eV = 9.28430 eV
E (Ionization) = -Electric Energy Magnetic Energy — E1, , wherein the calculated and
measured values and constants recited in the equations herein can be adjusted, for example, up to ± 10%, if desired.
134. The system of claim 133, wherein the radii and energies of the 2p electrons are solved using the forces given by
Figure imgf000877_0001
Z mern r3
Figure imgf000877_0002
Figure imgf000877_0006
Figure imgf000878_0001
and the radii r, are given by
Figure imgf000878_0002
/* ! in units of a0
135. The system of claim 134, wherein the electric energy given by
Eilonization) = —Electric Energy = —
8πεorn gives the corresponding ionization energies.
136. The system of claim 135, wherein for each n-electron atom having a central charge of Z times that of the proton and an electron configuration Is22s22p"~4 , there are two indistinguishable spin-paired electrons in an orbitsphere with radii rx and r2 both given by:
Figure imgf000878_0003
two indistinguishable spin-paired electrons in an orbitsphere with, radii r3 and r4 both given by:
Figure imgf000879_0001
rx in units of a0 and n - 4 electrons in an orbitsphere with radius rn given by
Figure imgf000879_0002
r3 w MJΪΪΪS o/ α0 the positive root must be taken in order that rn > 0 ; the parameter A corresponds to the diamagnetic force, Fώαmagne(,c : Fdlamas"e"c
Figure imgf000880_0001
and the parameter B corresponds to the paramagnetic force, F, mag 2 *
Figure imgf000880_0002
wherein the parameters of five through ten-electron atoms are
Atom Type Electron Ground Orbital Diama Param
Configuration State Arrangement gnetic agneti Term of Force c
2p Electrons Factor Force (2p state) A Factor B
Neutral 5 e Atom 1/2/2/ 2Py2
B 0
Neutral 6 e Atom Is2Is2Ip2 3P0 ^
C 3
Neutral 7 e Atom Is22s22p3 X2
N
Neutral 8 e Atom 1/2/2/ 3P2
O 1 2
Neutral 9 e Atom h22s22p5 2P3°/2 1
F 3 3
Neutral 1O e Atom 1/2/2/ 1S0
Ne 0 3
5 e lon 1/2/2/ 2P1 10/2 _5 3
6 e lon 1/2/2/ 5 3
7 e lon 1/2/2/ 4 ς.O ύ3/2 5_
Figure imgf000881_0001
3 8 e lon Is ,222os^z2op_4 5_
3
9 e lon Is22s22p5 2 pO r3/2 5^ 3
10 e Ion 1/2/2/ 1S0 £
Figure imgf000882_0003
3 12
137. The system of claim 136, wherein the ionization energy for the boron atom is given by
Figure imgf000882_0004
= 8.147170901 eV+ 0.15548501 eV = 8.30265592 eV wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
138. The system of claim 137, wherein the ionization energies for the n-electron atoms having the radii, rn ,are given by the negative of the electric energy, E(electric) , given by
E(Ionizatiori) =
Figure imgf000882_0001
139. The system of claim 138, wherein the radii of the 3p electrons are given using the forces given by
(Z~ή)e2 R L el,e =
Aπεorn
Figure imgf000882_0002
mag 2
Figure imgf000883_0001
Figure imgf000883_0002
Figure imgf000883_0005
Figure imgf000883_0003
and the radii r12 are, given by
Figure imgf000883_0004
r10 w Mw/ti1 o/ a0
140. The system of claim 139, wherein the ionization energies are given by electric energy given by:
E{Ionization) = -Electric Energy = — .
%πεorn
141. The system of claim 140, wherein for each n-electron atom having a central charge of Z times that of the proton and an electron configuration Is22s22p63s23p"~12 , there are two indistinguishable spin-paired electrons in an orbitsphere with radii rλ and r2 both given by:
Figure imgf000884_0001
two indistinguishable spin-paired electrons in an orbitsphere with radii r3 and r4 both given by:
Figure imgf000884_0002
rx in units of ao three sets of paired indistinguishable electrons in an orbitsphere with radius rw given by:
Figure imgf000884_0003
K in units of a0 two indistinguishable spin-paired electrons in an orbitsphere with radius rn given by:
Figure imgf000885_0001
r10 /» z/»z7s o/ α0 and n -12 electrons in a 3p orbitsphere with radius rn given by
Figure imgf000885_0002
rn in units of a0 where the positive root must be taken in order that rn > 0 ; the parameter A corresponds to the diamagnetic force, ¥diamagnetic :
¥diama8"elic
Figure imgf000885_0003
paramagnetic force, Fmαg 2 :
Figure imgf000885_0004
Figure imgf000885_0005
Figure imgf000886_0001
wherein the parameters of thirteen through eighteen-electron atoms are
Atom Electron Ground Orbital Diamagn Paramag
Type Configuration State Arrangement eettiicc nneettiicc
Term of F Foorrccee FFoorrccee
3p Electrons F Faaccttoorr FFaaccttoorr
(3p state) A A BB
Neutral Is2 2s22p6 3s2 2 p0
1/2 1
13 e 3 0
Atom
Al
Neutral \s2 2s22p6 3s2 3/ 3P 0 7
14 e 3
Atom
Si
Neutral Is2 2s22p6 3s2 4 rτQ
3/ "3/2 5
15 e 3
Atom
P
Neutral Is2 2s22p6 3s2 3/ 3P2 4
16 e 3
Atom
S
Neutral Is2 2s22p6 3s2 2 pO
3/ 3/2 2
17 e 3
Atom
Cl
Neutral Is2 2s22p6 3s2 3/ ύ0 1
18 e 3
Atom
Ar
13 e Ion Is2 2s22p6 3s2 2 p0
3P1 5
12
Figure imgf000887_0001
U e lon h*2s22p63s23p2 3P0 I 3 16
15 e Ion 1/2^2/3/3P3 X2 0 24
16 e Ion 1^2^2/3/3/ 3P2 I 3 24
17 e Ion i/2/2/3523/ 2P3°2 2 3 32
18 e lon i522522p63523/ 1S0
Figure imgf000888_0001
0 40
142. The system of claim 141, wherein the ionization energies for the n-electron 3p atoms are given by electric energy given by:
E(Ionizatioή) = -Electric Energy =
Figure imgf000888_0002
143. The system of claim 142, wherein the ionization energy for the aluminum atom is given by
Figure imgf000888_0003
wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to ± 10%, if desired.
144. A system of claim 1 of computing the nature of bound atomic and atomic ionic electrons from physical solutions of the charge, mass, and current density functions of atoms and atomic ions, which solutions are derived from Maxwell's equations using a constraint that the bound electron(s) does not radiate under acceleration, comprising: processing means for processing and solving the equations for charge, mass, and current density functions of electron(s) in selected atoms or ions, wherein the equations are derived from Maxwell's equations using a constraint that the bound electrons) does not radiate under acceleration; and output means for outputting the solutions of the charge, mass, and current density functions of the atoms and atomic ions.
145. A system of claim 1 comprising the steps of; a.) inputting electron functions that are derived from Maxwell's equations using a constraint that the bound electron(s) does not radiate under acceleration; b.) inputting a trial electron configuration; c.) inputting the corresponding centrifugal, Coulombic, diamagnetic and paramagnetic forces, d.) forming the force balance equation comprising the centrifugal force equal to the sum of the Coulombic, diamagnetic and paramagnetic forces; e.) solving the force balance equation for the electron radii; f.) calculating the energy of the electrons using the radii and the corresponding electric and magnetic energies; g.) repeating Steps a-f for all possible electron configurations, and h.) outputting the lowest energy configuration and the corresponding electron radii for that configuration.
146. The system of claim 145, wherein the output is rendered using the electron functions.
147. The system of claim 146, wherein the electron functions are given by at least one of the group comprising:
Jl = 0
p(r,θ,φ,t) = -^[δ(r -rn)][YQ o (θ,φ) + Yι"> (θ,φ)] Jl ≠ O
Figure imgf000890_0002
where Y"' (θ,φ) are the spherical harmonic functions that spin about the z-axis with angular frequency ωn with F0 0 (θ, φ) the constant function. Rejζ" (θ,φ)e"°"'} = P£ m (cosθ)cos(mφ + ώnή where to keep the form of the spherical harmonic as a traveling wave about the z-axis, ωn - mωn .
148. The system of claim 147, wherein the forces are given by at least one of the group comprising:
Figure imgf000890_0001
Figure imgf000891_0001
Figure imgf000891_0007
Figure imgf000891_0002
Figure imgf000891_0008
Figure imgf000891_0003
Figure imgf000891_0004
Figure imgf000891_0005
0
Figure imgf000891_0009
149. The system of claim 148, wherein the radii are given by at least one of the group comprising:
Figure imgf000891_0006
5
Figure imgf000892_0001
Figure imgf000892_0002
Figure imgf000893_0001
150. The system of claim 149, wherein the electric energy of each electron of radius rn is given by at least one of the group comprising:
Figure imgf000894_0001
wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
151. The system of claim 150, wherein the radii of s electrons are given by
Figure imgf000895_0001
rm in units of a0 where positive root must be taken in order that rn > 0 ;
Z is the nuclear charge, n is the number of electrons, rm is the radius of the proceeding filled shell(s) given by
Figure imgf000895_0002
for the preceding s shell(s);
Figure imgf000896_0001
for the 2p shell, and
Figure imgf000896_0002
for the 3p shell; the parameter A corresponds to the diamagnetic force, FΛαmαgrø,,c :
Figure imgf000896_0004
the parameter B corresponds to the paramagnetic force, F 2 :
Figure imgf000896_0003
the parameter C corresponds to the diamagnetic force, F diamagnetic 3 "
Figure imgf000896_0005
the parameter D corresponds to the paramagnetic force, F mag :
Figure imgf000897_0001
the parameter E corresponds to the diamagnetic force, ¥ώamagnetlc 2 , due to a relativistic effect with an electric field for r > rn :
Figure imgf000897_0002
wherein the parameters of atoms filling the Is, 2s, 3s, and 4s orbitals are
Atom Electron Ground Orbital Dia Para Dia Para Diamag
Type Configuration State Arrangemen mag. mag. mag. mag. .
Term t Fore Fore Fore Fore Force of e e e e Factor s Electrons Fact Fact Fact Fact E (s state) or or or or A B C D
Neutral \sl ύl/2
I e 0 0 0 0
Atom
H
Neutral I*2 %
2 e 0 0 0 1
Atom
He
Neutral 2sl 2 o "1/2
3 e 0 0 0 0
Atom
Li
Neutral 2s2
4 e 0 0 1
Atom
Be
Neutral Is22s22p63s1 "1/2 l i e 0 8 0 0
Atom
Na
Neutral Is22s22p63s2 "0
12 e 12 1 0
Figure imgf000898_0001
Atom
Mg Neutral Is2Hs22p63s23p64sl 2S1n
19 e 2 0 12 0 0
Atom K
Neutral Is22s22 p63s23 p64s2 1S0
2Oe 1 3 24 1 0
Atom Ca Ie Ion is1 2S1n
0 0 0 0 0
2 e Ion 1/ 1Sn
0 0
3 e Ion 2sx 3\n 1 0 0 0 1
4elon 2s2 1S0 1 0 0 1 1
11 e Ion is22,y22p63s1 2S112 1 4 8 0 i+72
+ 2 12 e Ion 22s22p63s2 1S0 1 6 0 0 Jϊ
2
19 e Ion 1/2^2/3/3/4S1 2S112 3 0 24 0 2-V2
20eIoni522.s22/763/3/452 1^0
Figure imgf000899_0001
2 0 24 0 2-V2
152. The system of claim 151, with the radii, rn , wherein the ionization energy for atoms having an outer s-shell are given by the negative of the electric energy, E (electric) , given by:
Figure imgf000900_0001
except that minor corrections due to the magnetic energy must be included in cases wherein the s electron does not couple to p electrons as given by
Ionization Energy (He) = -E(electric) + E(magnetic)\ 1 — —cos— | +a
Ionization Energy = -Electric Energy Magnetic Energy
Au
Eiionization; Li)
Figure imgf000900_0002
= 5.3178 eV + 0.0860 eV = 5.4038 eV E(Ionization) = E(Electric) + E7.
E "(vionization; . ; and
Figure imgf000900_0003
= 8.9216 eV + 0.03226 eV + 0.33040 eV = 9.28430 eV
E(Ionizatioή) = -Electric Energy Magnetic Energy - Eτ
wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
153. The system of claim 152, wherein the radii and energies of the 2p electrons are solved using the forces given by
Figure imgf000900_0004
Figure imgf000901_0001
diamagnetic 2
Figure imgf000901_0002
and the radii r3 are given by
Figure imgf000901_0003
T1 in units of a0
154. The system of claim 153, wherein the electric energy given by
Eilonizatioή) = —Electric Energy = —
Bπεorn gives the corresponding ionization energies.
155. The system of claim 154, wherein for each n-electron atom having a central charge of Z times that of the proton and an electron configuration Is22s22p"~4 , there are two indistinguishable spin-paired electrons in an orbitsphere with radii T1 and r2 both given by:
Figure imgf000902_0003
two indistinguishable spin-paired electrons in an orbitsphere with radii r3 and r4 both given by:
Figure imgf000902_0001
T1 /'« w«to o/ ao and « — 4 electrons in an orbitsphere with radius rn given by
Figure imgf000902_0002
K in units of a0 the positive root must be taken in order that rn > 0 ; the parameter A corresponds to the diamagnetic force, Vdlamagnetjc :
F diamagnetic
Figure imgf000903_0001
and the parameter B corresponds to the paramagnetic force, ¥mag 2 :
Figure imgf000903_0002
wherein the parameters of five through ten-electron atoms are
Atom Type Electron Ground Orbital Diama Param
Configuration State Arrangement gnetic agneti Term of Force c
2p Electrons Factor Force (2p state) A Factor B
Neutral 5 e Atom \s 2s 2nO
IS2 ΔS22 ΔpJ)l 2Py2
B 0
Neutral 6 e Atom Is2Is2Ip2 3P0 1
C 3
Neutral 7 e Atom \s 22s22p3 X2
N
Neutral 8 e Atom Is22s22p4 O
Neutral 9 e Atom ls ,2zo2s.,22<2-.p5 2pO -^3/2 2 F 3
Neutral 1Oe Atom \s 22s22p6 1S0 iVe 0
5 elon Is22s22pl 2nO
1/2
6 elon Is22s22p2 3P0 5
3
7 elon lif 2τWn2lop_3* 44SO-0
3/2 5
3
Figure imgf000904_0001
8 e lon ls22s22pA 3P2 I 3 6
9 e lon Is22s22p5 2P3°t2 S 3 9
1O e Ion Is22s22p6 1S0 5
Figure imgf000905_0003
3 12
156. The system of claim 155, wherein the ionization energy for the boron atom is given by
Figure imgf000905_0004
= 8.147170901 eV + 0.15548501 eV = 8.30265592 eV wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
157. The system of claim 156, wherein the ionization energies for the n-electron atoms having the radii, rn ,are given by the negative of the electric energy, E(electric) , given by
Figure imgf000905_0005
158. The system of claim 157, wherein the radii of the 3p electrons are given using the forces given by
(Z-n)e2
F diamagnet liie
Figure imgf000905_0001
Figure imgf000905_0002
Figure imgf000906_0001
Λ mern 'Yl
z, mern rn and the radii r12 are given by
Figure imgf000906_0002
r10 i« Mfjjt5 of a0
159. The system of claim 158, wherein the ionization energies are given by electric energy given by:
E{lonizatioή) = —Electric Energy
Figure imgf000906_0003
160. The system of claim 159, wherein for each n-electron atom having a central charge of Z times that of the proton and an electron configuration Is22s22p63s23p"~12 , there are two indistinguishable spin-paired electrons in an orbitsphere with radii rγ and r2 both given by:
Figure imgf000907_0003
two indistinguishable spin-paired electrons in an orbitsphere with radii r3 and r4 both given by:
r4
Figure imgf000907_0001
T1 m units of ao three sets of paired indistinguishable electrons in an orbitsphere with radius r10 given by.
Figure imgf000907_0002
K in units of Ct0 two indistinguishable spin-paired electrons in an orbitsphere with radius rn given by:
Figure imgf000908_0001
r10 /« units of α0 and « -12 electrons in a 3p orbitsphere with radius rn given by
Figure imgf000908_0002
r12 m MΠΪΪ5 of a0 where the positive root must be taken in order that rn > 0 ; the parameter A corresponds to the diamagnetic force, Fώaraagne((c :
B corresponds to
Figure imgf000908_0003
the paramagnetic force, Fmαg 2 :
Figure imgf000908_0004
Figure imgf000908_0005
Figure imgf000909_0001
wherein the parameters of thirteen to eighteen-electron atoms are
Atom Electron Ground Orbital Diamagn Paramag
Type Configuration State Arrangement etic netic
Term of Force Force
3p Electrons Factor Factor
(3p state) A B
Neutral Is22s22p63s2 2 pO
Ip1 1/2 11
13 e 3 0
Atom
Al
Neutral Is22s22p63s2 3P 0 7
14 e 3 0
Atom
Si
Neutral Is22s22 pβ 3s2 3/ "3/2 5
15 e 3 2
Atom
P
Neutral Is22s22 p63s2 3P 2 4
16 e 3 1
Atom
Neutral Is22s22p63s2 3/ ■r3/2 2
17 e 3 2
Atom
Cl
Neutral I$22s22p63s2 V i c 1
18 e 3 4
Atom
Ar
13 e Ion Is22s22p63s2 V 2 pO
■M/z 5
3
Figure imgf000910_0001
14 e Ion 1^2^2/3/3/ _1 3 ι6
l5 Q lon is^2s22p63s23p3
0 24
16 e Ion Is22s22 p63s23 p4 I^
3 24
17 e lon i522s22/3s23.p5 2 3 32
lS cIon ιs*2s22p63s23p6
Figure imgf000911_0001
0 40
161. The system of claim 160 wherein the bond comprises a H2 -type MO with two paired electron wherein the formation of the H2 -type MO by the contribution of an electron from each participating atom results in a diamagnetic force between the remaining atomic electrons and the H2 -type MO which causes the H2 -type MO to move to greater principal axes than would result with the Coulombic force alone.
162. The system of claim 161 wherein the integer increase of the central field and the resulting increased Coulombic as well as magnetic central forces on the remaining atomic electrons of each atom decrease the radius of the corresponding shell such that the energy minimum is achieved that is lower than that of the reactant atoms.
163. The system of claim 162 wherein the general equation for the central Coulomb force on the outer-most shell (nth where n = Z - 1 ) electron due to the nucleus and the inner electrons is given by:
Figure imgf000911_0002
for r >/•„_!.
164. The system of claim 163 wherein the general equation for H?diamagnellc due to the p - orbital contribution is given by
Figure imgf000912_0001
165. The system of claim 164 wherein the general equation for Fmag 2 is given by
Figure imgf000912_0002
166. The system of claim 165 wherein the general equation for Fdtωnagπelic 2 due to the binding of the p-orbital electron having an electric field of +1 outside of its radius is given by
Figure imgf000912_0003
167. The system of claim 166 wherein the general equation for ¥mag 3 , due to the contribution of a 2p electron from each binding atom in the formation of the σ MO that gives rise to a paramagnetic force on the remaining two 2p electrons that pair, is given by
Figure imgf000912_0004
168. The system of claim 167 wherein the parameters A , B , and C are zero or a positive integer such that the resulting energy of the molecule is minimized and the electron angular momentum is conserved in the formation of the specie.
169. The system of claim 168 wherein the radius of the 2p shell is calculated by equating the outward centrifugal force to the sum of the electric and diamagnetic and paramagnetic forces:
Figure imgf000913_0001
170. The system of claim 169 wherein the radius of the 2p shell is calculated by equating the outward centrifugal force to the sum of the electric and diamagnetic and paramagnetic
forces where the velocity given by vn = and s = — gives:
«Λ 2
Figure imgf000913_0002
171. The system of claim 170 wherein the general equation for the radius of the shell of the remaining electrons is given by
Figure imgf000913_0003
r3 in units of a0
172. The system of claim 171 wherein the radii r3 are given by
Figure imgf000914_0001
T1 in units of a0
173. The system of claim 172, wherein for each n-electron atom having a central charge of Z times that of the proton and an electron configuration Is2Is2Ip"'* , there are two indistinguishable spin-paired electrons in an orbitsphere with radii T1 and r2 both given by:
Figure imgf000914_0002
and two indistinguishable spin-paired electrons in an orbitsphere with radii r3 and r4 both given by:
Figure imgf000915_0001
rx in units of ao
174. The system of claim 173 wherein the sum Eτ [molecule, AOs) of the Coulonibic energy change of the AO electrons of both atoms of a bond is given by using the initial radius rn of each atom and the final radius rn_x of the binding shell of each atom and by considering that the central Coulombic field decreases by an integer for each successive electron of the shell:
Eτ (molecule, AOs)
Figure imgf000915_0002
where the subscript designates atom 1 and atom 2 of the bond.
175. The system of claim 174 wherein the forces on the electrons of the MO are the Coulombic force:
F, Coulomb Dx
%πεoab ιξ the spin pairing force:
Figure imgf000916_0001
the diamagnetic force: neh2
F 1 d,iamagnelicMO\ Λ 2 T 2 -Du ξ
where ne is the total number of electrons that interact with the binding σ -MO electron, the force on the pairing electron of the σ MO: ,
F 1 d,,iamagneticMO2
Figure imgf000916_0002
where |z| is the magnitude of the angular momentum of each atom at a focus that is the source of the diamagnetism at the σ -MO5 and the centrifugal force:
QentφgaiMo ~ ' wherein the force balance of the centrifugal force equated
Figure imgf000916_0003
to the Coulombic and magnetic forces is solved for the length of the semimajor axis.
176. The system of claim 175 wherein the force balance equation for the σ -MO with
Figure imgf000916_0004
a = \ 2 + - \an
177. The system of claim 176 wherein the force balance equation for the σ -MO with
Figure imgf000916_0005
178. The system of claim 177 wherein the force balance equation for the σ -MO ne = 2 , and \∑\ = Ti :
Figure imgf000917_0004
Figure imgf000917_0001
179. The system of claim 178 wherein the Coulombic force is
Figure imgf000917_0005
the spin pairing force is
Figure imgf000917_0006
the force on the electrons of the MO due to two paired electrons in at least one shell with » = 2 is
Figure imgf000917_0002
the force VdiamagmticM02 ^s given by the sum of the contributions over the components of 15 angular momentum L of the atoms at the foci acting on the electrons of the σ -MO:
Figure imgf000917_0003
the centrifugal force is
Figure imgf000917_0007
20 180. The system of claim 179 wherein the force balance equation for the σ -MO of the
carbon nitride radical comprising carbon with charge Z1 = 6 and IZ11 = h and |Z2| and
Figure imgf000917_0008
nitrogen with Z1 = I and |Z3j = % is
Figure imgf000918_0001
181. The system of claim 180 wherein the Coulombic force is
Figure imgf000918_0005
the spin pairing force is
Figure imgf000918_0006
the force on the electrons of the MO due to two paired electrons in at least one shell with «„ = 2 is
Figure imgf000918_0002
the force VάamaglιetιcM02 is given by the sum of the contributions over the components of angular momentum L\ of the atoms at the foci acting on the electrons of the σ -MO:
, and
Figure imgf000918_0003
the centrifugal force is
F, centnfugalMO .
Figure imgf000918_0004
182. The system of claim 181 wherein the force balance equation for the σ -MO of the
carbon nitride radical comprising carbon with charge Z1 = 6 and Ji1J = h and Ji2J = and
Figure imgf000918_0007
nitrogen with Z2 = 7 and Jl31 = h is
Figure imgf000919_0001
183. The system of claim 182 wherein the Coulombic force is
Figure imgf000919_0006
the spin pairing force is
Figure imgf000919_0002
the force on the electrons of the MO due to two paired electrons in at least one shell with
«„ = 2 is
Figure imgf000919_0003
the force 'FdιamagnelιcMO2 is given by the sum of the contributions over the components of angular momentum \∑ of the atoms at the foci acting on the electrons of the σ -MO, and the sum of the contributions from atom 1 with Z = Z, and atom 2 with Z = Z, with |Z.| = h is
Figure imgf000919_0004
the centrifugal force is
Figure imgf000919_0007
184. The system of claim 183 wherein force balance equation for the σ -MO of the carbon monoxide molecule is
Figure imgf000919_0005
Figure imgf000920_0001
185. The system of claim 184 wherein the Coulombic force is
Figure imgf000920_0005
the spin pairing force is
Figure imgf000920_0006
the force on the electrons of the MO due to two paired electrons in at least one shell with n = 2 is
Figure imgf000920_0007
the force F diamagneticMO 2 of the nitric oxide radical comprising nitrogen with charge Z1 = 7 and
Iz1I = h and |X2| and oxygen with Z2 = 8 and |I3| = % given by the corresponding sum
Figure imgf000920_0008
of the contributions is
diamagnelicMOl
Figure imgf000920_0002
the centrifugal force is
Figure imgf000920_0003
186. The system of claim 185 wherein the general force balance equation for the σ -MO of the nitric oxide radical is the same as that of CN :
Figure imgf000920_0004
Figure imgf000921_0001
where Z1 = 7 and Z2 = 8 for NO and Z1 = 6 and Z2 = 7 for CiV .
187. The system of claims 186 and 63 wherein the distance from the origin of the H2 -type- ellipsoidal-MO to each focus c ' , the internuclear distance 2e ' , and the length of the semiminor axis of the prolate spheroidal H2 -type MO b = c are solved from the semimajor axis a .
188. The system of claim 187 wherein the potential energy of the two electrons in the central field of the nuclei at the foci is
Figure imgf000921_0002
The potential energy of the two nuclei is
Figure imgf000921_0003
The kinetic energy of the electrons is
Figure imgf000921_0004
The energy, V1n , of the magnetic force between the electrons is
Figure imgf000921_0005
the total energy, E1. , is given by the sum of the energy terms plus E(AO) :
ET = Ve +T + Vm +Vp
Er
Figure imgf000921_0006
where H1 is the number of equivalent bonds of the MO, C1 is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the specie, C2 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of each chemical bond, and E(AO) is the energy of the at least one atomic orbital to which the MO is energy matched.
189. The system of claim 188 wherein during bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energy Eosc is the difference between the Doppler and average vibrational kinetic energies:
the total energy is
F = V +T + V +V + E
190. The system of claim 189 wherein during bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energy Eosc is the difference between the Doppler and average vibrational kinetic energies:
Figure imgf000922_0001
the total energy is
ET = Ve +T + Vm +Vp +Eosc
Figure imgf000922_0002
where R is b or a , H1 is the number of equivalent bonds of the MO, C1 is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the specie, c2 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of each chemical bond, and E (AO) is the energy of the at least one atomic orbital to which the MO is energy matched.
191. The system of claim 190 wherein the bond comprises a H2 -type MO with four paired electron wherein the formation of the H2 -type MO by the contribution of two electrons from each participating atom results in a diamagnetic force between the remaining atomic electrons and the H2 -type MO which causes the H2 -type MO to move to greater principal axes than would result with the Coulombic force alone.
192. The system of claim 191 wherein the integer increase of the central field and the resulting increased Coulombic as well as magnetic central forces on the remaining atomic electrons of each atom decrease the radius of the corresponding shell such that the energy minimum is achieved that is lower than that of the reactant atoms.
193. The system of claim 192 wherein the general equation for the central Coulomb force on the outer-most shell (nth where n = Z - 2 ) electron due to the nucleus and the inner electrons is given by:
3e2 .
** ele = 4πε0r* for r > r n-\ -
194. The system of claim 193 wherein the general equation for FdiamagneUc due to the p - orbital contribution is given by
Figure imgf000923_0001
195. The system of claim 194 wherein the general equation for Fmag 2 is given by
Z m/>3
196. The system of claim 195 wherein the general equation for Vdiamagnellc 2 due to me binding of the p-orbital electron having an electric field of +2 outside of its radius is given by
Figure imgf000924_0001
197. The system of claim 196 wherein the general equation for Fmag 3 , due to the contribution of a 2p electron from each binding atom in the formation of the σ MO that gives rise to a paramagnetic force on the remaining two 2p electrons that pair, is given by
Figure imgf000924_0004
198. The system of claim 197 wherein the parameters A , B , and C are zero or a positive integer such that the resulting energy of the molecule is minimized and the electron angular momentum is conserved in the formation of the specie.
199. The system of claim 198 wherein the radius of the 2p shell is calculated by equating the outward centrifugal force to the sum of the electric and diamagnetic and paramagnetic forces:
Figure imgf000924_0002
200. The system of claim 199 wherein the radius of the 2p shell is calculated by equating the outward centrifugal force to the sum of the electric and diamagnetic and paramagnetic
forces where the velocity given by vn = and s = — gives:
WX 2
Figure imgf000924_0003
201. The system of claim 200 wherein the general equation for the radius of the shell of the remaining electrons is given by
Figure imgf000925_0001
r3 in units of aQ
202. The system of claim 201 wherein the radii r3 are given by
Figure imgf000925_0002
V1 in units of ao
203. The system of claim 202, wherein for each n-electron atom having a central charge of Z times that of the proton and an electron configuration \s22s22p"~A , there are two indistinguishable spin-paired electrons in an orbitsphere with radii rx and r2 both given by:
Figure imgf000926_0001
and two indistinguishable spin-paired electrons in an orbitsphere with radii r3 and r4 both given by.
Figure imgf000926_0002
T1 z« wnto o/ α0
204. The system of claim 203 wherein the force balance of a double-bond MO corresponds to that of a second pair of two electrons binding to a molecular ion having +2e at each focus and a first bound pair such that the forces are the same as those of a molecule ion having +e at each focus.
205. The system of claim 204 wherein the forces on the electrons of the MO are the Coulombic force:
„2
F, Coulomb DL
&πεQab 2 ^Λξ ;
the spin pairing force:
Figure imgf000927_0001
the diamagnetic force:
Figure imgf000927_0003
where ne is the total number of electrons that interact with the binding σ -MO electron; the force on the pairing electron of the σ MO:
Figure imgf000927_0002
where \L is the magnitude of the angular momentum of each atom at a focus that is the source of the diamagnetism at the σ -MO, and the centrifugal force:
Figure imgf000927_0004
the force balance of the centrifugal force equated to the Coulombic and magnetic forces is solved for the length of the semimajor axis.
206. The system of claim 205 wherein the forces on the electrons of the NO2 MO are the Coulombic force:
Figure imgf000927_0005
the spin pairing force:
Figure imgf000927_0006
the diamagnetic force FdmmaglιellcMOι for each σ -MO of the NO2 molecule due to the two paired electrons in the O2p shell with ne - 2 :
Figure imgf000927_0007
which is also the corresponding force of NO ;
^ώamagnetπMoi of the nitrogen dioxide molecule comprising nitrogen with charge Z1 = 7 and JZ1 J = h and |Z2| = and the two oxygen atoms, each with Z2 = 8 and JZ31 = h given
Figure imgf000928_0005
by the corresponding sum of the contributions:
F ώamagιieticMO2
Figure imgf000928_0001
which is also the corresponding force of NO and CN except the term due to oxygen is twice that of NO due to the two oxygen atoms of NO2 , and the centrifugal force
F L centnfiigalMO ~ and
Figure imgf000928_0002
the force balance of the centrifugal force equated to the Coulombic and magnetic forces is solved for the length of the semimajor axis.
207. The system of claim 206 wherein the force balance equation for the σ -MOs of NO2 with Z1 = 7 and Z2 = 8 is
Figure imgf000928_0004
Figure imgf000928_0003
208. The system of claims 207 and 63 wherein the distance from the origin of the H2 -type- ellipsoidal-MΟ to each focus c' , the internuclear distance 2c' , and the length of the semiminor axis of the prolate spheroidal H2 -type MO b = c are solved from the semimajor axis a .
209. The system of claim 208 wherein the sum E7. {molecule, AOs) of the Coulombic energy change of the AO electrons of the participating atoms or ions of the bonds of the specie is the sum of the contributions over all such atoms or ions given by using the initial radius rn of the atom or ion and the final radius rn_2 of the binding shell of the atom or ion and by considering that the central Coulombic field decreases by an integer for each successive electron of the shell:
ET(molecule,AOs)
Figure imgf000929_0001
where the subscript designates the exemplary atom 1 of the bond.
210. The system of claim 209 wherein the sum Eτ {molecule, AOs) of the Coulombic energy change of the AO electron of a participating atom of a bond wherein all of the electrons of the shell of the AO are contributed to the bond is given by the sum of the corresponding ionization energies of the AO electrons.
211. The system of claim 210 wherein energy of the double bond MO is match to the participating AOs and the potential energy of the four electrons in the central field of the nuclei at the foci is
Figure imgf000929_0002
The potential energy of the two nuclei is
Figure imgf000929_0005
The kinetic energy of the electrons is
Figure imgf000929_0003
The energy, Vm , of the magnetic force between the electrons is
Figure imgf000929_0004
The total energy, E1. , is given by the sum of the energy terms plus E (AO) : F = V + T + V + V
Figure imgf000930_0001
where H1 is the number of equivalent bonds of the MO, C1 is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the specie, c2 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of each chemical bond, and E (AO) is the energy of the at least one atomic orbital to which the MO is energy matched.
212. The system of claim 211 wherein during bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energy Eosc is the difference between the Doppler and average vibrational kinetic energies:
E0Sc = ED +EKvib , md the total energy is
ET = Ve +T + Vm +Vp + E(AO) + E0SC .
213. The system of claim 212 wherein during bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energy Eosc is the difference between the Doppler and average vibrational kinetic energies:
Figure imgf000930_0002
where H1 is the number of equivalent bonds of the MO, C1 is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the specie, C2 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of each chemical bond, and E (AO) is the energy of the at least one atomic orbital to which the
MO is energy matched.
214. The system of claim 213 wherein the energy components of WQ V6 , Vp, T , V111 , and
E1, , except that the terms based on charge are multiplied by four and the kinetic energy term is multiplied by two due to each σ -MO double bond.
215. The system of claim 214 wherein the potential energy of the four electrons of double bond in the central field of the nuclei at the foci is
Figure imgf000931_0001
The potential energy of the two nuclei is
Figure imgf000931_0002
The kinetic energy of the electrons is
Figure imgf000931_0003
The energy, V1n , of the magnetic force between the electrons is
Figure imgf000931_0004
The total energy, E7 , is given by the sum of the energy terms:
ET = Ve +T + Vm +Vp
Figure imgf000931_0005
where r\ is the number of equivalent bonds of the MO, C1 is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the specie, and c2 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of each chemical bond.
216. The system of claim 215 wherein during bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energy Eosc is the difference between the Doppler and average vibrational kinetic energies:
E0Sc = ED +EKvib , and the total energy is
ET = Ve + T + Vm +Vp + E0SC .
217. The system of claim 216 wherein during bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energy Eosc is the difference between the Doppler and average vibrational kinetic energies:
Figure imgf000932_0001
R1 is the number of equivalent bonds of the MO, C1 is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the specie, and C2 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of each chemical bond.
218. The system of claim 217 wherein the total energy of the specie is the sum of the total energy of the components comprising the energy contribution of the MO formed between the participating atoms, the change in the energy of the AOs or HOs upon forming the bond, the change in magnetic energy with bond formation, and the energy of oscillation in the transition state.
219. The system of claim 218 wherein the total energy of the specie is the sum over all of the component groups wherein the total energy of each said group is the sum of the total energy of the components comprising the energy contribution of the MO formed between the participating atoms, the change in the energy of the AOs or HOs upon forming the bond, the change in magnetic energy with bond formation, and the energy of oscillation in the transition state.
220. The system of claim 219 wherein the change in magnetic energy is given by
Figure imgf000933_0001
where rM 3 is the radius of the atom that reacts to form the bond.
221. The system of claim 220 wherein the bond energy of the molecular or molecular ion is difference in the energy of the total energy of the starting species and the total energy of the specie.
222. The system of claim 221 wherein the bond energy of a specific bond of the molecular or molecular ion is difference in the energy of the sum of the energies of the atoms and any change in energy of any groups formed with the starting atoms and the total energy of the bond of the specie.
223. The system of claim 222 wherein the total energy of a molecule, E1. [molecule] , is given by the sum of: the sum of the energies of the electrons donated to each bond, the sum of the energies of electrons of at least one other atom donated to the bonds, the sum over the participating atoms of each AO contribution due to the decrease in radius with the formation of each bond, and the σ MO energy contribution per bond.
224. The system of claim 223 wherein the total energy of a molecule, E7. [molecule] , is given by the sum of: the sum of the energies of the electrons donated to each bond such that all of the electrons of the shell are donated, the sum of the energies of electrons of at least one other atom donated to the bonds, the sum over the participating atoms of each AO contribution due to the decrease in radius with the formation of each bond, and the σ MO energy contribution per bond.
225. The system of claim 223 wherein the total energy of CO2 , E1. (CO2 ) , is given by the sum of Eiionization; C) and E(ionization; C+) , the sum of the energies of the first and second electrons of carbon donated to each double bond, the sum of E (ionization; O) and two times Eiionization; O+) , the energies of the first and second electrons of oxygen donated to the double bonds, two times Eτ (0, 2p) , the O2p AO contribution due to the decrease in radius with the formation of each bond and two times Eτ (C = O,σ) , the σ MO contribution:
Figure imgf000934_0001
wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
226. The system of claim 225 wherein the total energy of a molecule, Eτ+osc (molecule) is given by the sum of E1. (molecule) and Eosc .
227. The system of claim 226 wherein during bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energy Eosc is the difference between the Doppler and average vibrational kinetic energies: Ksc ,
Figure imgf000935_0001
where the angular frequency of the reentrant oscillation in the transition state is
Figure imgf000935_0002
the kinetic energy, Eκ , is given by Planck's:
Figure imgf000935_0003
the Doppler energy of the electrons of the reentrant orbit is
Figure imgf000935_0004
Eosc is given by the sum of and EKvib , the vibrational energy:
EOSc [molecule]
Figure imgf000935_0005
where R is b or α , H1 is the number of equivalent bonds of the MO, cBO is the bond-order factor which is 1 for a single bond, 4 for a double bond, and 9 for a triplet bond, C1 is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the specie, and c2 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of each chemical bond.
228. The system of claim 227 wherein Ehv is given by Eτ [molecule) 12 in the case of a double bond such that Eosc
Figure imgf000936_0001
229. The system of claim 228 wherein Ehv of a molecule having r\ bonds is given by Ex [molecule) I Yix such that
Ksc
Figure imgf000936_0002
230. The system of claim 229 wherein E1n, of a molecule having r\ bonds is given by Ex [H2) such that
Figure imgf000936_0005
where E1, [H2) = -31.63537 eV is the total energy of the hydrogen molecule, wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
231. The system of claim 230 wherein during bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energy Eosc is the difference between the Doppler and average vibrational kinetic energies:
Figure imgf000936_0006
where the angular frequency of the reentrant oscillation in the transition state is determined by the force between the central field and the electrons in the transition state; said force and its derivative are given by
Figure imgf000936_0003
and
Figure imgf000936_0004
such that the angular frequency of the oscillation is given by
Figure imgf000937_0001
where R is b or a , cB0 is the bond-order factor which is 1 for a single bond, 4 for a double bond, and 9 for a triplet bond, Cx is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the specie, and C2 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of each chemical bond.
232. The system of claim 231 wherein the nucleus of the B atom and the nucleus of the A atom comprise the foci of each H2 -type ellipsoidal MO of the A- B bond and the parameters of the point of intersection of each H2 -type ellipsoidal MO and the A -atom AO are determined from the polar equation of the ellipse: 1 + e r = rn l+ecos#'
233. The system of claim 232 wherein the radius of the A shell is rA , and the polar radial coordinate of the ellipse and the radius of the A shell are equal at the point of intersection such that
Figure imgf000937_0003
the polar angle θ ' at the intersection point is given by
Figure imgf000937_0002
234. The system of claim 233 wherein the angle ΘAAO the radial vector of the A AO makes with the internuclear axis is
6U =180°-<9«
235. The system of claim 234 wherein the distance from the point of intersection of the orbitals to the internuclear axis must be the same for both component orbitals such that the angle ωt = θHiM0 between the internuclear axis and the point of intersection of each H2 -type ellipsoidal MO with the A radial vector obeys the following relationship: rA sinθA A0 = bsmθff2M0 such that θ - cin-i r« sin^0 .
the distance dH uo along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by d H2Mo = ^ cos ^2MO , and the distance dAAO along the internuclear axis from the origin of the A atom to the point of intersection of the orbitals is given by
®A AO = C ~ ®H2MO •
236. The system of claim 235 where the bond angle is determined from the zero energy condition of the total energy of the potential bond between any pair of terminal atoms.
237. The system of claim 236 where the force constant k ' of a H2 -type ellipsoidal MO due to the equivalent of two point charges of at the foci is given by:
4πε0 where C1 is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the specie and c2 is the factor that results in an equipotential energy match of the participating at least two atomic orbitals of the chemical bond.
238. The system of claim 237 where the distance from the origin of the MO to each focus c ' is given by:
Figure imgf000938_0001
the internuclear distance is
Figure imgf000939_0001
the length of the semiminor axis of the prolate spheroidal MO b = c is given by
Figure imgf000939_0002
and, the eccentricity, e , is c' e = ~. a
239. The system of claim 238 wherein the potential energy of the two electrons in the central field of the nuclei at the foci is
Figure imgf000939_0003
The potential energy of the two nuclei is
2 γ ~ •
P Usoja2 ~b2 '
The kinetic energy of the electrons is
Figure imgf000939_0004
The energy, Vn, , of the magnetic force between the electrons is
Figure imgf000939_0005
The total energy, E7 , is given by the sum of the energy terms:
ET = Ve + T + V1n
Figure imgf000939_0006
where cx is the traction of the H2 -type ellipsoidal MO basis function of a chemical bond of the specie, and C2 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of each chemical bond.
240. The system of claim 239 wherein during bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energy Eosc is the difference between the Doppler and average vibrational kinetic energies:
Figure imgf000940_0001
The total energy is
ET = Ve +T + Vm +Vp +Eosc
Figure imgf000940_0002
where C1 is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the specie, and C1 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of each chemical bond.
1 k
241. The system of claim 240 wherein the vibrational energy —h i — is given by
2 \ μ
Figure imgf000940_0003
where μ is the reduced mass of the nuclei.
242. The system of claim 241 wherein the energy components of are Ve , Vp, T , V1n , and Eτ , except that the terms based on charge are multiplied by four and the kinetic energy term is multiplied by two due to a σ -MO double bond.
243. The system of claim 242 wherein c ' = is substituted into the
Figure imgf000940_0004
equation for E7, which is set equal to zero, and the semimajor axis is solved.
244. The system of claims 243 and 63 wherein the distance from the origin of the H2 -type- ellipsoidal-MO to each focus c\ the internuclear distance 2c' , and the length of the semiminor axis of the prolate spheroidal H2 -type MO b = c are solved from the semimajor axis a .
245. The system of claim 244 wherein with 2cA'_B defined as the internuclear distance of each A-B bond and C = 2c 'B_B defined as the internuclear distance of the two terminal B atoms, the bond angle between the A-B bonds is given by the law of cosines is
Figure imgf000941_0001
246. The system of claim 245 wherein the specie comprises more than two bonds with one A-B bound along an axis defined as the vertical or z-axis and any two A-B bonds form an isosceles triangle; the angle of the bonds from the defined axis is determined from the geometrical relationships: the distance dongm_B from the origin to the nucleus of a terminal B atom is given by
onm~B 2sin60° ' the height along the z-axis from the origin to A nucleus dheιgk is given by
Figure imgf000941_0002
the angle θv of each A-B bond from the z-axis is given by
6> = tan -1 I ®orιgm-B d h. eight
247. The system of claim 246 wherein the vibrational energies are determined by the forces between the central field and the electrons and those between the nuclei; said electron- central-field force and its derivative are given by
Figure imgf000941_0003
and
Figure imgf000942_0001
' said nuclear repulsion force and its derivative are given by
/(2C) = -
8πεo (2c') and
Figure imgf000942_0002
such that the angular frequency of the oscillation is given by
Figure imgf000942_0003
where μ = — - — — is the reduced mass of the nuclei, R is b or a, cB0 is the bond-order mx + m2 factor which is 1 for a single bond, 4 for a double bond, and 9 for a triplet bond, C1 is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the specie, and c2 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of each chemical bond.
248. The system of claim 247 wherein the transition-state vibrational energy, Evώ (θ) , given by Planck's equation is:
Evώ (0) = hω
249. The system of claim 248 wherein the energy vυ of state υ is given by vυ = υω0 -υ(υ -I)(O0X0 , u = 0,1,2,3... where hcωl
4Dn ω0 is the frequency of the υ = 1 -> υ - 0 transition, and -D0 is the bond dissociation energy.
250. The system of claim 249 wherein ω0 is given by ωQ = Evlb (0) -2ω0x0 , and
Figure imgf000943_0001
such that ^ + o _^£v/5 (o) = O , and he he
Figure imgf000943_0002
251. The system of claim 250 wherein Be , the rotational parameter, for A- B is given by:
e 2Iehe where / = μr2 , r = 2c ' , and μ is the reduced mass.
252. The system of claim 251 wherein the vibrational energy levels of the A-A and A-B bonds of the specie are solved as sets of coupled atomic harmonic oscillators wherein each atom of a chain of bonds is further coupled to at least one additional harmonic oscillators by using the Lagrangian, the differential equation of motion, and the eigenvalue solutions wherein the spring constants are derived from the central forces.
253. The system of claim 252 wherein the vibrational energy levels of the C - C bonds of CnH2n+2 are solved as n-\ sets of coupled carbon harmonic oscillators wherein each carbon is further coupled to two or three equivalent H harmonic oscillators by using the Lagrangian, the differential equation of motion, and the eigenvalue solutions wherein the spring constants are derived from the central forces.
254. The system of claim 253 wherein ED (CnH2n+2 ) , the total bond dissociation energy of CnH2n+2 , is given as the sum of the energy components due to the two methyl groups, n - 2 methylene groups, and n-\ C-C bonds.
255. The system of claim 254 wherein the total bond dissociation energy of C11H2n+2 is given by
Figure imgf000944_0001
= («-l)(4.32754 eF) + 2(12.49186 eF) + (n-2)(7.83016 eF) wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
256. The system of claim 255 wherein the exact solution for the dimensional parameters, charge density functions, and energies of molecules are determined using the results for the determination of CnH2n+2 and the functional groups as basis sets that are linearly combined.
257. The system of claim 256 wherein one or more of the hydrogen atoms of the solution for CnH2n+2 are substituted with one or more of the previously solved functional groups or derivative functional groups to give the parameters of a desired molecule.
258. The system of claim 257 wherein the parameters of a given molecule are given by energy matching each group to CnH2n+2.
259. The system of claim 258 wherein substitution of one or more H 's of CnH2n+2 with functional groups from the list of CH3 , other CnH2n+2 groups, Tf2C = CH2 , HC ≡ CH , F , Cl , O, OH, NH , NH2 , CN , NO, NO2 , CO , CO2 , and C6H6 give the solutions of branched alkanes, alkenes, and alkynes, alkyl halides, ethers, alcohols, amides, amines, nitriles, alkyl nitrosos, alkyl nitrates, aldehydes, ketones, carboxylic acids, esters, and substituted aromatics.
260. A composition of matter comprising a plurality of atoms, the improvement comprising a novel property or use discovered by calculation of at least one of a bond distance between two of the atoms, a bond angle between three of the atoms, and a bond energy between two of the atoms, orbital intercept distances and angles, charge-density functions of atomic, hybridized, and molecular orbitals, the bond distance, bond angle, and bond energy being calculated from physical solutions of the charge, mass, and current density functions of atoms and atomic ions, which solutions are derived from Maxwell's equations using a constraint that a bound electron(s) does not radiate under acceleration.
261. A composition of matter according to claim 260, wherein the novel property is a new pharmaceutical use.
262. A composition of matter according to claim 261 that is novel, wherein the novel property is stability at room temperature of a new arrangement of atoms or ions.
263. A novel composition of matter discovered by calculating at least one of a bond distance between two of the atoms, a bond angle between three of the atoms, and a bond energy between two of the atoms, orbital intercept distances and angles, charge-density functions of atomic, hybridized, and molecular orbitals, the bond distance, bond angle, and bond energy being calculated from physical solutions of the charge, mass, and current density functions of atoms and atomic ions, which solutions are derived from Maxwell's equations using a constraint that a bound electron(s) does not radiate under acceleration.
264. A system of determining at least one of a bond distance between two of the atoms, a bond angle between three of the atoms, and a bond energy between two of the atoms, orbital intercept distances and angles, charge-density functions of atomic, hybridized, and molecular orbitals, wherein the bond distance, bond angle, and bond energy being calculated from physical solutions of the charge, mass, and current density functions of atoms and atomic ions, which solutions are derived from Maxwell's equations using a constraint that the bound electron(s) does not radiate under acceleration.
265. The system of claim 264, further comprising discovering a new composition of matter.
266. The system of claim 1 , wherein the nature of said specie includes the nature of a chemical bond thereof.
267. The system of claim 1 , wherein at least one bond angle between three atoms is computed and displayed.
268. The system of claim 1 , wherein at least one bond distance between two atoms is computed and displayed.
269. The system of claim 1, wherein at least one geometric component is computed and displayed.
270. The system of claim 1, wherein at least one potential energy between an electron and a nucleus is computed.
271. The system of claim 1 , wherein at least one potential energy between two nuclei is computed.
272. The system of claim 1 , wherein at least one kinetic energy of an electron is computed.
273. The system of claim 1, wherein at least one magnetic energy between electrons is computed.
274. The system of claim 1 , wherein at least one total energy of a bond is computed.
275. The system of claim 1, wherein at least one change in atomic energy between atoms due to bonding is computed.
276. The system of claim 1, further comprising at least one database of functional groups and at least one database of molecules, molecular ions, and/or molecular radicals, characterized in that the functional groups, molecules, molecular ions, and molecular radicals having the nature of their chemical bonds computed by the Maxwillian solutions of charge, mass, and current density, or both databases being combined into one database or split into multiple databases, and the processing means being in communication with the databases or combined database.
277. The system of claim 1, further comprising at least one database of functional groups having the nature of their chemical bonds computed by the Maxwillian solutions of charge, mass, and current density.
278. The system of claim 276, further comprising means for selecting a desired first functional group, characterized in that the selected functional group is displayed with open bonds being displayed, means for selecting an open bond, means for selecting a second functional group to be bound to the first functional group, characterized in that the second functional group is displayed with open bonds being displayed, means for selecting an open bond on the second functional group, and means for combining the selected bonds, whereby at least a portion of a desired molecule is displayed.
279. The system of claim 276, further comprising selecting and combining functional groups until a desired molecule is displayed.
280. The system of claim 277, further comprising means for selecting a desired first functional group, characterized in that the selected functional group is displayed with open bonds being displayed, means for selecting an open bond, means for selecting a second functional group to be bound to the first functional group, characterized in that the second functional group is displayed with open bonds being displayed, means for selecting an open bond on the second functional group, and means for combining the selected bonds, whereby at least a portion of a desired molecule is displayed.
281. The system of claim 277, further comprising selecting and combining functional groups until a desired molecule is displayed.
282. The system of claim 1, further comprising a means for inputting the chemical structure of a desired molecule and parsing the inputted molecule into functional groups.
283. The system of claim 282, further comprising a database of functional groups having the nature of their chemical bonds computed by the Maxwillian solutions of charge, mass, and current density, characterized in that the processing means combines the functional groups and the desired molecule is displayed.
284. The system of claim 282, wherein the chemical structure is imputed using a Simplified Molecular Input Line Entry System.
285. The system of claim 283, wherein the chemical structure is imputed using a Simplified Molecular Input Line Entry System.
286. The system of claim 282, wherein the Simplified Molecular Input Line Entry System is SMILES.
287. The system of claim 283, wherein the Simplified Molecular Input Line Entry System is SMILES.
288. The system according to claim 1, wherein the force generalized constant ky of a H2 - type ellipsoidal MO due to the equivalent of two point charges at the foci is given by.
Figure imgf000948_0001
where C1 is the fraction of the H2 -type ellipsoidal molecular orbital basis function of a chemical bond of the specie and C2 is the factor that results in an equipotential energy match of the participating at least two molecular or atomic orbitals of the chemical bond.
289. The system according to claim 1 , wherein the distance from the origin of the MO to each focus c ' is given by:
Figure imgf000949_0001
the intermiclear distance is
Figure imgf000949_0002
the length of the semiminor axis of the prolate spheroidal MO b = c is given by
(15.4)
and, the eccentricity, e , is
C e = — (15.5). a
290. The system according to claim 289, wherein a potential energy of the electrons in the central field of the nuclei at the foci is
Figure imgf000949_0003
where H1 is the number of equivalent bonds of the MO for functional groups and in the case of independent MOs not in contact with the bonding atoms, the terms based on charge are multiplied by cB0 , the bond-order factor, which is 1 for a single bond, 4 for an independent double bond and 9 for an independent triplet bond.
291. The system according to claim 1, wherein the potential energy of the two nuclei is
K =^Z 8πε0Λl Ua ,22 -=b= z.22ϊ (15.7).
292. The system according to claim 1, wherein the kinetic energy of the electrons is
Figure imgf000950_0001
where Yix is the number of equivalent bonds of the MO for functional groups and in the case of independent MOs not in contact with the bonding atoms, the terms based on charge are multiplied by cB0 , the bond-order factor, which is 1 for a single bond, 4 for an independent double bond and 9 for an independent triplet bond.
293. The system according to claim 1 , wherein the energy, Vm , of the magnetic force between the electrons is
Figure imgf000950_0002
where W1 is the number of equivalent bonds of the MO for functional groups and in the case of independent MOs not in contact with the bonding atoms, the terms based on charge are multiplied by cB0 , the bond-order factor, which is 1 for a single bond, 4 for an independent double bond and 9 for an independent triplet bond.
294. The system according to claim 1, wherein total energy of the H2 -type prolate spheroidal MO, E7, (HZMO) , is given by the sum of the energy terms:
Eτ (Høo) = Vβ +T + Vm +VD (15.10)
Figure imgf000950_0003
where r\ is the number of equivalent bonds of the MO for functional groups and in the case of independent MOs not in contact with the bonding atoms, the terms based on charge are multiplied by C50 , the bond-order factor, which is 1 for a single bond, 4 for an independent double bond and 9 for an independent triplet bond.
295. The system according to claim 1 , wherein the total energy E7, (atom, msp3 \ ( m is the integer of the valence shell) of the AO electrons and the hybridized shell is given by the sum of energies of successive ions of the atom over the n electrons comprising total electrons of the at least one AO shell according to the formula:
(15.12 where IP1n is the m th ionization energy (positive) of the atom.
296. The system according to claim 295, wherein the radius r s , of the hybridized shell is given by:
(15.13).
Figure imgf000951_0004
297. The system according to claim 295, wherein the Coulombic energy ECoulomb (atom, msp' Λ of the outer electron of the atom msp3 shell is given by
—e
Ecouiomb {atom,msp3 ) = - (15.14).
Sπεnr msp
298. The system of claim 296, wherein in the case that during hybridization at least one of the spin-paired AO electrons is unpaired in the hybridized orbital (HO), the energy change for the promotion to the unpaired state is the magnetic energy E(magnetic) at the initial radius r of the AO electron:
(15.15)
Figure imgf000951_0001
then, the energy E(atom,msp3) ofthe outer electron of the atom msp3 shell is given by the
sum of ECoulomb {atom,msp3 ) and EQnagnetic) :
(15.16).
Figure imgf000951_0002
299. The system according to claim 295, wherein the total energy E1, (moLatom, msp3 ) (m is the integer of the valence shell) of the HO electrons is given by the sum of energies of successive ions of the atom over the n electrons comprising total electrons of the at least one initial AO shell and the hybridization energy:
Eτ (mol.atom,msp3} ~ E{atom,msp3 }-∑IPm (15.17) m=2
where IP1n is the m th ionization energy (positive) of the atom and the sum of -/P1 plus the hybridization energy is E (atom, msp3 ) .
300. The system of claim 299, wherein the radius r , of the hybridized shell is given by:
(15.18)
Figure imgf000952_0001
where s = 1,2,3 for a single, double, and triple bond, respectively.
301. The system of claim 299, wherein the Coulombic energy ECoulomb (moLatom, msp3 j of the outer electron of the atom msp3 shell is given by
-e
ECouion,b (™ol.atom, msp3 ) = - (15.19). msp
302. The system of claim 298, wherein in the case that during hybridization at least one of the spin-paired AO electrons is unpaired in the hybridized orbital (HO), the energy change for the promotion to the unpaired state is the magnetic energy E(magnetic) at the initial radius r ofthe AO electron given by Eq. (15.15). Then, the energy E{mol.atom,msp3} of
the outer electron of the atom msp3 shell is given by the sum of ECoulomb {mol.atom,msp3j and E(magnetic) :
E(molatom,msp3) (15.20)
Figure imgf000952_0002
E7, (atom - atom, msp3 J , the energy change of each atom msp3 shell with the formation of
the atom-atom-bond MO is given by the difference between E\mol.atom, msp1 ) and
E\atom,msp3\ :
Eτ( atom -atom, msp3 ) = E[mol. atom, msp' 1J- E( atom, msp3) (15.21).
303. The system of claim 295, wherein ECoulomb (mol. atom, msp31J is one of:
^Coulomb [C ethylene >2sp J, E Coulomb\C elι,ane,2sp j, ECoulomb\Cacetylem,2sp J, and
^Coulomb \palkam^SP )>
Eccuiomb {atom, msp3 ) is one of ECmlomb (C, 2sp3 ) and ECoulomb (a, 3sp3 ) ; E{mol.atom,msp3) is one of E(Cethylem,2sp3), E(Cethane,2sp3),
E{C→lem,2sp3)E(Calkme,2sp3);
E(atom,msp3} is one of and E{C,2sp3} andE(c/,3-fp3); Eτ {atom - atom, msp3 ) is one of E (c - C, 2sp3 ) , E(C = C, 2sp3 ) , and.
E(C≡C,2sp3); atom msp3 is one of C2sp3 , Cl3sp3
Eτ( atom -atom ^s1), msp3) is ET(C-C,2sp3) and Eτ[atom - atom(s2),msp3) is
ET(C = C, 2sp3), and
V 1S °ne rC2sp3 ' rethanelSp^ ' Veih≠enelspi ' Tacetyhnelspz ' ^Olhimlsp* ' ^ ^IZsp* "
In the case of the C2sp3 HO, the initial parameters (Εqs. (14.142-14.146)) are
(15.22)
(15.23)
(15-24)
Figure imgf000953_0001
Figure imgf000954_0001
= -14.82575 eF + 0.19086 eF (15.25)
= -14.63489 eV wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
304. The system of claim 300, wherein
Z-I
∑ (Z-g) = 10 (15.26) q=Z~n
Equations (14.147) and (15.17) give
E1, (molatom, msp3 ) = E1, [Cethme , 2sp3 ) = -151.61569 eV (15.27)
and using Eqs. (15.18-15.28), the final values of rQU^ , ECoulomb (C2sp3), and E[C2sp3) , and
BO the resulting ET \ C - C, C2sp of the MO due to charge donation from the HO to the MO
BO where C - C refers to the bond order of the carbon-carbon bond for different values of the parameter s are given in Table 15.1.
Table 15.1:
Figure imgf000954_0002
wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
305. The system of claim 295, wherein a minimum-energy bond with the constraint that it must meet the energy matching condition for all MOs at all HOs or AOs, the energy E(mol.atom,msp3j of the outer electron of the atom msp3 shell of each bonding atom is
the average of EunoLatom, msp3 ) for two different values of s :
Figure imgf000955_0001
306. The system of claim 305, wherein in the case, E7. (atom - atom, msp3 ) , the energy change of each atom msp3 shell with the formation of each atom-atom-bond MO, is average for two different values of s :
(15.29).
307. The system of claim 305, wherein a first MO and its HOs comprising a linear combination of bond orders and a second MO that shares a HO with the first. In addition to the mutual HO, the second MO comprises another AO or HO having a single bond order or a mixed bond order, in order for the two MOs to be energy matched, the bond order of the second MO and its HOs or its HO and AO is a linear combination of the terms corresponding to the bond order of the mutual HO and the bond order of the independent HO or AO, and in general, E1, (atom - atom, msp3 ) , the energy change of each atom msp3 shell with the formation of each atom-atom-bond MO, is a weighted linear sum for different values of s that matches the energy of the bonded MOs, HOs, and AOs:
(15.30)
Figure imgf000955_0003
where c^ is the multiple of the BO of sn. The radius rms 3 of the atom msp3 shell of each
bonding atom is given by the Coulombic energy using the initial energy ECmlomb (atom, msp3 )
and E7, ( atom - atom, msp3 ) , the energy change of each atom msp3 shell with the formation of each atom-atom-bond MO:
Figure imgf000956_0001
where ECouloab (C2sp3) = -14.825751 eV .
308. The system of claim 307, wherein the Coulombic energy ECoulomb (rnol. atom, msp3 J of the outer electron of the atom msp3 shell is given by Eq. (15.19).
309. The system of claim 308, wherein in the case that during hybridization, at least one of the spin-paired AO electrons is unpaired in the hybridized orbital (HO), the energy change for the promotion to the unpaired state is the magnetic energy Eimagnetic) (Eq. (15.15)) at the initial radius r of the AO electron. Then, the energy E (mol. atom, msp3) of the
outer electron of the atom msp3 shell is given by the sum of ECoulomb (mol. atom, msp3 ) and E(magnetic) (Eq. (15.20)).
310. The system of claim 308, wherein, Eτ (atom - atom, msp3 ) , the energy change of each atom msp3 shell with the formation of the atom-atom-bond MO is given by the difference between E (mol. atom, msp3 j and E( atom, msp3 ) given by Eq. (15.21).
311. The system of claim 310, wherein using the equation (15.23) for ECmlomb (C, 2sp3 J in equation (15.31), the single bond order energies given by Eqs. (15.18-15.27) and shown in Table 15.1, and the linear combination energies (Eqs. (15.28-15.30)), the parameters of linear combinations of bond orders and linear combinations of mixed bond orders are given in Table 15.2, wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to ± 10%, if desired.
312. The system of claim 295, wherein the radius r , of the C2sp3 HO of a carbon atom of a given specie is calculated using Eq. (14.514) by considering
Figure imgf000957_0002
, the total energy donation to each bond with which it participates in bonding.
313. The system of claim 1 , wherein equation for the radius is given by
(15.32).
Figure imgf000957_0001
314. The system of 295, wherein the Coulombic energy ECoulomb (tnol. atom, msp3} of the outer electron of the atom msp3 shell is given by Eq. (15.19).
315. The system of claim 295, in the case that during hybridization, at least one of the spin-paired AO electrons is unpaired in the hybridized orbital (HO), the energy change for the promotion to the unpaired state is the magnetic energy E(magnetic) (Eq. (15.15)) at the initial radius r of the AO electron. Then, the energy E(mol.atom,msp3\ of the outer
electron of the atom msp3 shell is given by the sum of ECou!omb (moLatom, msp3 \ and E(magnetic) (Eq. (15.20)).
316. The system of claim 315, wherein for the C2sp3 HO of each methyl group of an alkane contributes -0.92918 eV (Eq. (14.513)) to the corresponding single C-C bond; the corresponding C2sp3 HO radius is given by Eq. (14.514). The C2sp3 HO of each methylene group of CnH2n+2 contributes -0.92918 eV to each of the two corresponding
C-C bond MOs, wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
317. The system of claim 316, wherein the radius (Eq. (15.32)), the Coulombic energy (Eq. (15.19)), and the energy (Eq. (15.20)) of each alkane methylene group are
Figure imgf000958_0002
= 0.81549α0 (15.33)
Figure imgf000958_0001
wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
318. The system of claim 316, wherein in the determination of the parameters of functional groups, heteroatoms bonding to C2sp3 HOs to form MOs are energy matched to the C2sp3 HOs, the radius and the energy parameters of a bonding heteroatom are given by the same equations as those for C2sp3 HOs.
319. The system of claim 318, wherein using Eqs. (15.15), (15.19-15.20), (15.24), and (15.32) in a generalized fashion, the final values of the radius of the HO or AO, r Atom.Ho.Ao > Ecouiomb (molatom, msp3 ) , and £ (Cmol 2sp3 ) are calculated using
Σ E 1 r group ( \MO, 2sp3 ) J , the total energy donation to each bond with which an atom f BO \ participates in bonding corresponding to the values of E7, C - C,C2sp3 of the MO due
to charge donation from the AO or HO to the MO given in Tables 15.1 and 15.2 and the final values of rMom HO AQ , ECmlomb (molatom, msp2 ) , and E (Cmol 2sp3 ) calculated using
( BO \ the values of E7, C - C,C2sp3 given in Tables 15.1 and 15.2 are shown in Tables
15.3A and 15.3B in the specification, wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to ± 10%, if desired.
320. The system of claim 319, wherein the energy of the MO is matched to each of the participating outermost atomic or hybridized orbitals of the bonding atoms wherein the energy match includes the energy contribution due to the AO or HO's donation of charge to the MO.
321. The system of claim 320, wherein the force constant k' (Eq. (15.1)) is used to determine the ellipsoidal parameter c' (Eq. (15.2)) of the each H2 -type-ellipsoidal-MO in terms of the central force of the foci, c' is substituted into the energy equation (from Eq. (15.11))) which is set equal to H1 times the total energy of H2 where r\ is the number of equivalent bonds of the MO and the energy of H2, -31.63536831 eF , Eq. (11.212) is the minimum energy possible for a prolate spheroidal MO5 wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
322. The system of claim 321, wherein the energy equation and the relationship between the axes, the dimensions of the MO are solved, the energy equation has the semimajor axis a as it only parameter, the solution of the semimajor axis a allows for the solution of the other axes of each prolate spheroid and eccentricity of each MO (Eqs. (15.3-15.5)), and the parameter solutions then allow for the component and total energies of the MO to be determined.
323. The system of claim 1, wherein the total energy, E7. (HZMO) , is given by the sum of the energy terms (Eqs. (15.6-15.11)) plus E1. (AOIHO) : ET (H2uo) = Ve +T + Vm +Vp +ET (AO/HO) (15.36)
E1, [H1MO) IHO)
Figure imgf000959_0001
(15.37) where n, is the number of equivalent bonds of the MO, C1 is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the group, c2 is the factor that results in an equipotential energy match of the participating at least two atomic orbitals of each chemical bond, and E7. (AO I HO) is the total energy comprising the difference of the energy
E (AO I HO) of at least one atomic or hybrid orbital to which the MO is energy matched and any energy component ΔEHiM0 (AOI HO) due to the AO or HO' s charge donation to the MO.
324. The system of claim 323, wherein
Eτ (AO I HO) = E (AO I HO) -AEHiM0 (AO I HO) (15.38)
325. The system of claim 324, wherein as specific examples, E7, (AO I HO) is one from the group of Eτ (AOIHO) = E(OIp shell) = -E(ionization; O) = -13.6181 eV ;
E1. (AOIHO) = E(NIp shell) = -Eiionization; N) = -14.53414 eV ;
Eτ (AOIHO) = E(c,2sp3) = -14.63489 eV ; Eτ (AOIHO) = ECoulomb (Cl,3sp3 ) = -14.60295 eV ;
ET (AO I HO) = E(ionization; C) + EQonization; C+) ;
Eτ (AOIHO) = E(Celhane,2sp3) = -15.35946 eV ;
Eτ (AOIHO) = +E(Cethyletie,2sp3 )-E(Cethylene,2sp3 ) ;
Eτ (AOIHO) = E(c,2sp3 )- 2ET (C = C,2sp3 ) = -14.63489 eV -(-2.26758 eV) ; ET (AOIHO) = E(Cacetylene,2Sp 3)- E(Cacet ^
Eτ (AOIHθ) = E(c,2sp3)-2Eτ (C ≡ C,2sp3 ) = -U.63489 eV-(-3Λ3026 eV) ;
ET (AOIHO) = E(cben2ene,2sP 3)-E(Cbm,2sp3) ;
Eτ (AOIHO) = E(c,2sp3)- ET (C = C,2sp3) = -14.63489 eV-(-\.13379 eV) , and
E7. (AOIHO) = E(Calkane,2sp3} =-15.56407 eV , wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
326. The system of claim 1 , wherein to solve the bond parameters and energies,
(Eq. (15.2)) is substituted into E7. [H1Mo) to give
Figure imgf000961_0001
Figure imgf000961_0002
(15.39) wherein the total energy is set equal to Eφasis energies) which in the most general case is given by the sum of a first integer r\ times the total energy of H2 minus a second integer n2 times the total energy of H, minus a third integer n3 times the valence energy of E(AO) (e.g. E(N) = -14.53414 eV) where the first integer can be 1,2,3..., and each of the second and third integers can be 0, 1, 2, 3....
327. The system of claim 326, wherein
Eφasis energies) = ^ (-31.63536831 eV)-n2 (-13.605804 eV)- nβ(Aθ)
(15.40) in the case that the MO bonds two atoms other than hydrogen, Eφasis energies) is H1 times the total energy of H2 where H1 is the number of equivalent bonds of the MO and the energy of H2, -31.63536831 eV , Eq. (11.212) is the minimum energy possible for aprolate spheroidal MO:
Eφasis energies) = H1 (-31.63536831 eV) (15.41)
E1, [H2Mo) , is set equal to Eφasis energies) , and the semimajor axis a is solved.
328. The system of claims 289 or 326, wherein the semimajor axis a is solved from the equation of the form: = E(basis energies)
Figure imgf000962_0001
(15.42) The distance from the origin of the H2 -type-ellipsoidal-MO to each focus c ' , the internuclear distance 2c' , and the length of the semiminor axis of the prolate spheroidal H2 -type MO b = c are solved from the semimajor axis a using Eqs. (15.2-15.4).
329. The system of claim 328, wherein the component energies are given by Eqs. (15.6- 15.9) and (15.39).
330. The system of claim 323, wherein the total energy of the MO of the functional group, E1, (MO) , is the sum of the total energy of the components comprising the energy contribution of the MO formed between the participating atoms and
E1. (atom - atom, msp3.Aθ) , the change in the energy of the AOs or HOs upon forming the bond.
331. The system of claim 330, wherein from Eqs. (15.39-15.40), E1, (MO) is
E1, (MO) = Eφasis energies) + E1, (atom - atom, msp3.AOj (15.43)
332. The system of claim 331 , wherein during bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energy Eosc is the sum of the Doppler, ED , and average vibrational kinetic energies, EKvib :
Figure imgf000962_0002
where H1 is the number of equivalent bonds of the MO , k is the spring constant of the equivalent harmonic oscillator, and μ is the reduced mass.
333. The system of claim 332, wherein the angular frequency of the reentrant oscillation in the transition state corresponding to ED is determined by the force between the central field and the electrons in the transition state.
334. The system of claim 333, wherein the force and its derivative are given by
Figure imgf000963_0001
and
Figure imgf000963_0002
such that the angular frequency of the oscillation in the transition state is given by
ω (15.47)
Figure imgf000963_0003
where R is the semimajor axis a or the semiminor axis b depending on the eccentricity of the bond that is most representative of the oscillation in the transition state, cBO is the bond- order factor which is 1 for a single bond and when the MO comprises Hx equivalent single bonds as in the case of functional groups, cB0 is 4 for an independent double bond as in the case of the CO2 and NO2 molecules and 9 for an independent triplet bond, C10 is the fraction of the H2 -type ellipsoidal MO basis function of the oscillatory transition state of a chemical bond of the group, and C2o is the factor that results in an equipotential energy match of the participating at least two atomic orbitals of the transition state of the chemical bond. Typically, Clo = C1 and C2n = C2 , the kinetic energy, Eκ , corresponding to ED is given by Planck's equation for functional groups:
Figure imgf000963_0004
335. The system of claim 334, wherein the Doppler energy of the electrons of the reentrant orbit is
Figure imgf000963_0005
Eosc given by the sum of ED and EKvjb is
EOSo
Figure imgf000964_0001
E1n, of a group having H1 bonds is given by E7, (MO) Inx such that
Figure imgf000964_0002
Eτ+Osc (croup) is given by the sum of Eτ (MO) (Eq. (15.42)) and Eosc (Eq. (15.51)):
Figure imgf000964_0003
(15.52).
336. The system of claim 335, wherein the total energy of the functional group E7, [group) is the sum of the total energy of the components comprising the energy contribution of the MO formed between the participating atoms, Eibasis energies) , the change in the energy of the AOs or HOs upon forming the bond ( E7. (atom - atom, msp3.AO) ), the energy of oscillation in the transition state, and the change in magnetic energy with bond formation,
E mag •
337. The system of claim 336, wherein from Eq. (15.52), the total energy of the group
Eτ {Group) is
Figure imgf000965_0001
(15.53).
338. The system of claim 337, wherein the change in magnetic energy Emag which arises due to the formation of unpaired electrons in the corresponding fragments relative to the bonded group is given by
Figure imgf000965_0002
where r3 is the radius of the atom that reacts to form the bond and c3 is the number of electron pairs.
Figure imgf000965_0003
(15.55).
339. The system of claim 338, wherein the total bond energy of the group ED {Group) is the negative difference of the total energy of the group (Eq. (15.55)) and the total energy of the starting species given by the sum of ciEmM (c4 AO I HO) and c5Einiml (cs AO I HO) :
Figure imgf000966_0001
(15.56).
340. The system of claim 336, wherein in the case of organic molecules, the atoms of the functional groups are energy matched to the C2sp3 HO such that
E(AOZHO) = -14.63489 eV (15.57)
For examples of Emag from previous sections:
Figure imgf000966_0002
341. The system of claim 340, wherein in the general case of the solution of an organic functional group, the geometric bond parameters are solved from the semimajor axis and the relationships between the parameters by first using Eq. (15.42) to arrive at a , the remaining parameters are determined using Eqs. (15.1-15.5), the energies are given by
Eqs. (15.52-15.59), and to meet the equipotential condition for the union of the H2 -type- ellipsoidal-MO and the HO or AO of the atom of a functional group, the factor c2 of a .H2-type ellipsoidal MO in principal Eqs. (15.42) and (15.52) may given by
(i) one:
C2 = I (15.61)
(ii) the ratio that is less than one of 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243), and the magnitude of the Coulombic energy of the participating AO or HO of the atom, ECmlomb (MO.atom,msp3}
given by Eqs. (15.19) and (15.31-15.32). For ECoulomb [MO.atom,msρz) >13.605804 eF :
Figure imgf000967_0001
(iii) the ratio that is less than one of 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243), and the magnitude of the valence energy, Eiyalence) , of the participating AO or HO of the atom where Eiyalence) is the ionization energy or E( MO.atom,msp* \ given by Eqs. (15.20) and (15.31-15.32). For \E(yalence)\> 13.605804 eV :
Figure imgf000967_0002
(iv) the ratio that is less than one of the magnitude of the Coulombic energy of the participating AO or HO of a first atom, ECoulamb (MO.atom,msp3 \ given by Eqs. (15.19) and (15.31-15.32), and the magnitude of the valence energy, E(valence) , of the participating AO or HO of a second atom to which the first is energy matched where Eiyalence) is the ionization energy or E[MO. atom, msp3 ) given by Eqs. (15.20) and (15.31-15.32). For
Ecouiomb (MO. atom, msp3 )| > E{valence) :
Figure imgf000968_0001
(v) the ratio that is less than one of the magnitude of the valence-level energies, En(valence) , of the AO or HO of the nth participating atom of two that are energy matched where Eiyalence) is the ionization energy or
Figure imgf000968_0002
given by Eqs. (15.20) and (15.31-15.32):
E2(yalence)
(vi) the factor that is the ratio of the hybridization factor c2 (l) of the valence AO or HO of a first atom and the hybridization factor C2 (2) of the valence AO or HO of a second atom to which the first is energy matched where c2 (n) is given by Eqs. (15.62-15.68); alternatively c2 is the hybridization factor c2 (l) of the valence AOs or HOs a first pair of atoms and the hybridization factor c2 (2) of the valence AO or HO a third atom or second pair to which the first two are energy matched:
<>=W) αi69>
(vii) the factor that is the product of the hybridization factor c2 (l) of the valence AO or HO of a first atom and the hybridization factor C2 (2) of the valence AO or HO of a second atom to which the first is energy matched where c2 (n) is given by Eqs. (15.62-15.69); alternatively c2 is the hybridization factor c2 (l) of the valence AOs or HOs a first pair of atoms and the hybridization factor c2 (2) of the valence AO or HO a third atom or second pair to which the first two are energy matched: C2 = c2 (l)c2 (2) (15.70)
The hybridization factor C2 corresponds to the force constant k (Eqs. (11.65) and (13.58)). In the case that the valence or Coulombic energy of the AO or HO is less than 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243), then C2 corresponding to k' (Eq. (15.1)) is given by Eqs. (15.62-15.70).
342. The system of claim 341, wherein specific examples of the factors c2 and C2 of a H2"tyPe ellipsoidal MO of Eq. (15.51) given in following sections are
;
;
Figure imgf000969_0001
Figure imgf000970_0001
Figure imgf000971_0001
wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
343. The system of claim 1 , wherein the parameters of the point of intersection of each H2 -type ellipsoidal MO and the A -atom AO are determined from the polar equation of the ellipse:
(15.71).
Figure imgf000971_0003
V
344. The system of claim 343, wherein the radius of the A shell is rΛ , and the polar radial coordinate of the ellipse and the radius of the A shell are equal at the point of intersection such that
(15.72).
Figure imgf000971_0004
345. The system of claim 344, wherein the polar angle θ ' at the intersection point is given by
Figure imgf000971_0002
346. The system of claim 345, wherein the angle ΘA AO the radial vector of the A AO makes with the internuclear axis is 0^o = 180°-0' (15.74).
347. The system of claim 345, wherein the distance from the point of intersection of the orbitals to the internuclear axis is the same for both component orbitals such that the angle ωt — ΘH M0 between the internuclear axis and the point of intersection of each H2 - type ellipsoidal MO with the A radial vector obeys the following relationship:
Figure imgf000972_0001
such that
Figure imgf000972_0002
348. The system of claim 347, wherein the distance dHiM0 along the internuclear axis from the origin of H2 -type ellipsoidal MO to the point of intersection of the orbitals is given by d Hluo = a cos QH1MO (15.77).
349. The system of claim 347, wherein the distance dA A0 along the internuclear axis from the origin of the A atom to the point of intersection of the orbitals is given by dΛ AO = c'-dH2MO (15.78).
350. The system of claim 1, wherein in ACB MO comprising a linear combination of C-A -bond and C- B -bond MOs where C is the general central atom and a bond is possible between the A and B atoms of the C-A and C-B bonds, the ZACB bond angle is determined by the condition that the total energy of the H2 -type ellipsoidal MO between the terminal A and B atoms is zero.
351. The system of claim 350, wherein the force constant k ' of a H2 -type ellipsoidal MO due to the equivalent of two point charges of at the foci is given by:
C C 2e2 Jc' = 1- 2 (15.79)
Aπεo where C1 is the fraction of the H2 -type ellipsoidal MO basis function of a chemical bond of the specie which is 0.75 (Eq. (13.59)) for a terminal A- H {A is H or other atom) and 1 otherwise and C2 is the factor that results in an equipotential energy match of the participating at least two atomic orbitals of the chemical bond and is equal to the corresponding factor of Eqs. (15.42) and (15.52).
352. The system of claim 351, wherein the distance from the origin of the MO to each focus c1 of the A- B ellipsoidal MO is given by:
Figure imgf000973_0001
353. The system of claim 351, wherein the internuclear distance is
Figure imgf000973_0002
354. The system of claim 351, wherein the length of the semiminor axis of the prolate spheroidal A-B MO b = c is given by Eq. (15.4).
355. The system of claim 351, wherein the component energies and the total energy,
Eτ [H1UO) , of the A- B bond are given by the energy equations (Eqs. (11.207-11.212),
(11.213-11.217), and (11.239)) of H2 except that the terms based on charge are multiplied by cB0 , the bond-order factor which is 1 for a single bond and when the MO comprises r\ equivalent single bonds as in the case of functional groups. cB0 is 4 for an independent double bond as in the case of the CO2 and JVO2 molecules.
356. The system of claim 355, wherein the kinetic energy term is multiplied by c \0 which is 1 for a single bond, 2 for a double bond, and 9/2 for a triple bond, the electron energy terms are multiplied by C1 , the fraction of the H2 -type ellipsoidal MO basis function of a terminal chemical bond which is 0.75 (Eq. (13.233)) for a terminal A-H {A is H or other atom) and 1 otherwise.
357. The system of claim 355, wherein the electron energy terms are further multiplied by c2' , the hybridization or energy-matching factor that results in an equipotential energy match of the participating at least two atomic orbitals of each terminal bond.
358. The system of claim 350, wherein when A-B comprises atoms other than H , E7, [atom - atom, msp3.Aθ} , the energy component due to the AO or HO's charge donation to the terminal MO5 is added to the other energy terms to give E7. {H2MO} :
Figure imgf000974_0001
(15.82).
359. The system of claim 350, the radiation reaction force in the case of the vibration of A-B in the transition state corresponds to the Doppler energy, ED , given by Eq. (11.181) that is dependent on the motion of the electrons and the nuclei.
360. The system of claim 359, wherein the total energy that includes the radiation reaction of the A-B MO is given by the sum of E7. (H1Mo) (Eq. (15.82)) and E030 given Eqs. (11.213-11.220), (11.231-11.236), and (11.239-11.240).
361. The system of claim 360, wherein the total energy E7, (A - B) of the A-B MO including the Doppler term is
Figure imgf000974_0002
(15.83) where C10 is the fraction of the H2 -type ellipsoidal MO basis function of the oscillatory transition state of the A-B bond which is 0.75 (Eq. (13.233)) in the case of H bonding to a central atom and 1 otherwise, C20 is the factor that results in an equipotential energy match of the participating at least two atomic orbitals of the transition state of the chemical bond, and
μ = — L- — is the reduced mass of the nuclei given by Eq. (11.154).
W1 + m2
362. The system of claim 361, wherein to match the boundary condition that the total energy of the A -B ellipsoidal MO is zero, E1, (A-B) given by Eq. (15.83) is set equal to zero and substitution of Eq. (15.81) into Eq. (15.83) gives
Figure imgf000975_0001
(15.84).
363. The system of claim 362, wherein the vibrational energy-term of Eq. (15.84) is determined by the forces between the central field and the electrons and those between the nuclei (Eqs. (11.141-11.145)).
364. The system of claim 362, wherein the electron-central-field force and its derivative are given by
f{a) = -cB0 cγc2e
(15.85)
4πε0a and
/ „2 f<(a) = 2cB0 ^j (15.86).
4πε0a
365. The system of claim 364, wherein the nuclear repulsion force and its derivative are given by
/(* + *')% 8πεo f(a + c ,'Ϋ) (15.87) and
Figure imgf000976_0001
such that the angular frequency of the oscillation is given by
ω = (15.89).
Figure imgf000976_0002
366. The system of claim 365, wherein since both terms of Eosc = ED + EKyib are small due to the large values of a and c' , an approximation of Eq. (15.84) which is evaluated to determine the bond angles of functional groups is given by
Figure imgf000976_0003
(15.90).
367. The system of claim 366, wherein from the energy relationship given by Eq. (15.90) and the relationship between the axes given by Eqs. (15.2-15.5), the dimensions of the A - B MO can be solved.
368. The system of claim 367, wherein Eq. (15.90) is solved by the reiterative technique using a computer.
369. The system of claim 366, wherein a factor c2 of a given atom in the determination of c2 for calculating the zero of the total A-B bond energy is given by Eqs. (15.62-15.65).
370. The system of claim 369, wherein in the case of a H - H terminal bond of an alkyl or alkenyl group, c2 is the ratio of C2 of Eq. (15.62) for the H-H bond which is one and C2 of the carbon of the corresponding C-H bond:
Figure imgf000977_0001
wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to ± 10%, if desired.
371. The system of claim 366, wherein in the case of the determination of the bond angle of the ^4CH MO comprising a linear combination of C-A -bond and C-H -bond MOs where A and C are general, C is the central atom, and c2 for an atom is given by Eqs.
(15.62-15.70), C2 of the A-H terminal bond is the ratio of c2 of the A atom for the A-H terminal bond and C2 of the C atom of the corresponding C -H bond: c2(A(A-H)msp3) 2 C2(C(C -H)(mspl) '
372. The system of claim 366, wherein in the case of the determination of the bond angle of the COH MO of an alcohol comprising a linear combination of C -O -bond and O -H -bond MOs where C 3 O, and H are carbon, oxygen, and hydrogen, respectively, C2 of the C- H terminal bond is 0.91771 since the oxygen and hydrogen atoms are at the Coulomb potential of aproton and an electron (Eqs. (1.236) and (10.162), respectively) that is energy matched to the C2sp3 HO, wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
373. The system of claim 366, wherein in the determination of the hybridization factor c2' of Eq. (15.90) from Eqs. (15.62-15.70), the Coulombic energy, ECoulomb (MO.atom,msp3) ,
or the energy, E [MO.atom, msp3 ) , the radius f"Λ_B AorBs 3 of the A or B AO or HO of the
heteroatom of the A-B terminal bond MO such as the C2sp3 HO of a terminal C-C bond is calculated using Eq. (15.32) by considering ]T ETm/ [MO, 2sp3 ) , the total energy donation to each bond with which it participates in bonding as it forms the terminal bond.
374. The system of claim 373, wherein the Coulombic energy ECotthmb [MO.atom, msp3 ) of the outer electron of the atom msp3 shell is given by Eq. (15.19).
375. The system of claim 374, wherein in the case that during hybridization, at least one of the spin-paired AO electrons is unpaired in the hybridized orbital (HO), the energy change for the promotion to the unpaired state is the magnetic energy E (magnetic) (Eq. (15.15)) at the initial radius r of the AO electron, and the energy E \ MO.atom, msp3 ) of the outer
electron of the atom msp3 shell is given by the sum of ECoulomb (MO.atom,msp3) and Eimagnetic) (Eq. (15.20)).
376. The system of claim 365, wherein in the specific case of the terminal bonding of two carbon atoms, the c2 factor of each carbon given by Eq. (15.62) is determined using the
Coulombic energy ECoulomb (c~C C2sp3) of the outer electron of the C2sp3 shell given by Eq. (15.19) with the radius r „ „„ . of each C2sp3 HO of the terminal C-C bond
calculated using Eq. (15.32) by considering Y1 E7. (MO, 2sp3 ) , the total energy donation to each bond with which it participates in bonding as it forms the terminal bond including the contribution of the methylene energy, 0.92918 eV (Eq. (14.513)), corresponding to the terminal C-C bond. The corresponding ET (atom -atomimsp3.AO\ hi Eq. (15.90) is
ET (C-C C2sp3 ) = -1.85836 eV , wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
377. The system of claim 366, wherein in the case that the terminal atoms are carbon or other heteroatoms, the terminal bond comprises a linear combination of the HOs or AOs; thus, c2' is the average of the hybridization factors of the participating atoms corresponding to the normalized linear sum: c2' ~-(c2' (atom ϊ) + c2' (atom 2)) (15.93)
2 In the exemplary cases of C-C , O -O, and TV - N where C is carbon:
Figure imgf000979_0001
In the exemplary cases of C-N , C-O, and C-S ,
Figure imgf000979_0002
where C is carbon and c2 (C to B) is the hybridization factor of Eqs. (15.52) and (15.84) that matches the energy of the atom B to that of the atom C in the group.
378. The system of claim 363, wherein the corresponding E7. (atom - atom, msp3.AO) term in Eq. (15.90) depends on the hybridization and bond order of the terminal atoms in the molecule, but typical values matching those used in the determination of the bond energies (Eq. (15.56)) are
Eτ (C -O C2sp\O2p) = -1.44915 eV ; E7, (C-O C2sp\O2p) = -1.65376 eV ;
ET (C-N C2sp\N2p) = -1.449l5 eV,' ET [C-S C2sp\S2p) = -0.72457 eV ; Er (0-0 O2p.O2p) = -1.44915 eV ; E7, (O- O O2p.O2p) = -1.65376 eV ;
E7, (N-N N2p.N2p) = -1.44915 eV ; E7, (N-O N2p.O2p) = -1.44915 eV ;
Ex (F-F F2p.F2p) = -1.44915 eV ; E7 (O-Cl CBp.O3p) = -0.92918 eV ;
E7, (Br -Br Br4p.Br4p) = -0.92918 eV ; E7 (I-I ISpJSp) = -0.36229 eV ;
ET (C-F C2sp\F2p) = -l.%5836 eV ; ET (C -Cl C2sp3.Cl3p) = -0.92918 eV ;
E7. (C - Br C2sp\Br4p) = -0.72457 eV ; ET (C-I C2sp3.I5p) = -0.36228 eV , and
E7 7, ( vO -Cl O2p f.Cl3p r) i = -0.92918 eV ^ w ,herei .n + t,he ca ,lcu ,lat .ed A and , measured , va ,lues and , constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
379. The system of claim 366, wherein in the case that the terminal bond is X - X where X is a halogen atom, C1 is one, and c2 is the average (Eq. (15.93)) of the hybridization factors of the participating halogen atoms given by Eqs. (15.62-15.63) where ^coulomb (MO.atom, msp3 J is determined using Eq. (15.32) and
ECou!omb (MO.atom, msp3 ) = 13.605804 eV for X = I .
380. The system of claim 379, wherein the factor C1 of Eq. (15.90) is one for all halogen atoms.
381. The system of claim 379, wherein the factor C2 of fluorine is one since it is the only halogen wherein the ionization energy is greater than that 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243), wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
382. The system of claim 379, wherein for each of the halogens, Cl , Br , and / , C2 is the hybridization factor of Eq. (15.52) given by Eq. (15.70) with C2 (l) being that of the halogen given by Eq. (15.68) that matches the valence energy of X (E1 (valence)) to that of the C2sp3 HO ( E2 (valence) = -14.63489 e V , Eq. (15.25)) and to the hybridization of C2,!?p3 HO (c2 (2) = 0.91771 , Eq. (13.430)), wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
383. The system of claim 382, wherein Eτ (atom - atom, msp3.Aθ) of Eq. (15.90) is the maximum for the participating atoms which is -1.44915 eV , -0.92918 eV , -0.92918 eV , and -0.33582 eV for F , Cl , Br , and / , respectively, wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
384. The system of claim 366, wherein in the case that the terminal bond is C - X where
C is a carbon atom and X is a halogen atom, the factors cx and C1 of Eq. (15.90) are one for all halogen atoms.
385. The system of claim 384, wherein for X = F , c2' is the average (Eq. (15.95)) of the hybridization factors of the participating carbon and F atoms where c2 for carbon is given by Eq. (15.62) and C2 for fluorine matched to carbon is given by Eq. (15.70) with c2 (l) for the fluorine atom given by Eq. (15.68) that matches the valence energy of F (E1 (valence) = -17.42282 eV) to that of the C2sp3 HO (E2 (valence) = -14.63489 eV , Eq. (15.25)) and to the hybridization of Clsp3 HO (c2 (2) = 0.91771 , Eq. (13.430)), wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
386. The system of claim 385, wherein the factor C2 of fluorine is one since it is the only halogen wherein the ionization energy is greater than that 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243), wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
387. The system of claim 385, wherein for each of the other halogens, Cl , Br , and / , c2' is the hybridization factor of the participating carbon atom since the halogen atom is energy matched to the carbon atom.
388. The system of claim 387, wherein the C2 of the terminal-atom bond matches that used to determine the energies of the corresponding C-X -bond MO.
389. The system of claim 388, wherein C2 is the hybridization factor of Eq. (15.52) given by Eq. (15.70) with c2 (l) for the halogen atom given by Eq. (15.68) that matches the valence energy of X (E1 (valence) ) to that of the C2sp3 HO (E2 (valence) = -14.63489 eV , Eq. (15.25)) and to the hybridization of C2sp3 HO (c2 (2) = 0.91771 , Eq. (13.430)), wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
390. The system of claim 385, wherein E1, (atom~atom,msp\AO) of Eq. (15.90) is the maximum for the participating atoms which is -1.85836 eV , -0.92918 eV , -0.72457 eV , and -0.33582 eV for F , Cl , Br , and / , respectively, wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
391. The system ofclaim 385, wherein in the case that the terminal bond is H-X corresponding to the angle of the atoms HCX where C is a carbon atom and X is a halogen atom, the factors C1 and C1 of Eq. (15.90) are 0.75 for all halogen atoms.
392. The system ofclaim 385, wherein for X = F , c2 is given by Eq. (15.69) with c2 of the participating carbon and F atoms given by Eq. (15.62) and Eq. (15.65), respectively.
393. The system ofclaim 392, wherein the factor C2 of fluorine is one since it is the only halogen wherein the ionization energy is greater than that 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H given by Eq. (1.243), wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
394. The system ofclaim 393, wherein for each of the other halogens, Cl , Br , and I , c2 is also given by Eq. (15.69) with c2 of the participating carbon given by Eq. (15.62) and
C2 of the participating X atom given by C2 = 0.91771 (Eq. (13.430)) since the X atom is energy matched to the C2spl HO.
395. The system ofclaim 394, wherein C2 is given by Eq. (15.65) for the corresponding atom X where C2 matches the energy of the atom X to that of H .
396. The system of claim 366, wherein the distance between the two atoms A and B of the general molecular group ACB when the total energy of the corresponding A -B MO is zero, the corresponding bond angle can be determined from the law of cosines: s 2 + S2 2 - 2S1S2COSmCO = s 2 (15.96).
397. The system of claim 396, wherein with S1 = 2cc' _A , the internuclear distance of the
C-A bond, s2 = 2cc'_B , the internuclear distance of each C-B bond, and J3 = 2c 'A_B , the internuclear distance of the two terminal atoms, the bond angle Θ^CB between the C- A and C- B bonds is given by (2c> c_A)2 +{2c'c_Bf -2(2c< c_A)(2c< c_B)cosmeθ =
Figure imgf000983_0001
(15.97)
Figure imgf000983_0002
398. The system of claim 397, wherein the structure CbCa(Oa)Ob wherein Ca is bound to Cb , Oa , and Ob , the three bonds are coplanar and two of the angles are known, say θx and θ2 , then the third θ3 can be determined geometrically: 03 = 360-0! -02 (15.99)
399. The system of claim 397, wherein in the general case that two of the three coplanar bonds are equivalent and one of the angles is known, say θx , then the second and third can be determined geometrically: , (360-4) (15 100)
400. The system of claim 1 , wherein in the general case where the group comprises three A-B bonds having B as the central atom at the apex of a pyramidal structure formed by the three bonds with the A atoms at the base in the xy-plane.
401. The system of claim 400, wherein the C3v axis centered on B is defined as the vertical or z-axis, and any two A-B bonds form an isosceles triangle, and the angle of the bonds and the distances from and along the z-axis are determined from the geometrical relationships given by Eqs. (13.412-13.416): the distance dorlgin_B from the origin to the nucleus of a terminal B atom is given by
Figure imgf000984_0003
(15-101) the height along the z-axis from the origin to the A nucleus dhejght is given by
Figure imgf000984_0001
the angle θv of each A-B bond from the z-axis is given by
(15.103).
Figure imgf000984_0002
402. The system of claim 401, wherein in the case where the central atom B is further bound to a fourth atom C and the B-C bond is along the z-axis. Then, the bond 0^BC given by Eq. (14.206) is
^ac = 180-0, (15.104).
403. The system of claim 400, wherein in the plane defined by a general ACA MO comprising a linear combination of two C -A -bond MOs where C is the central atom, the dihedral angle ΘZBCIACA between the ACA -plane and a line defined by a third bond with C , specifically that corresponding to a C - B -bond MO, is calculated from the bond angle θ^^ and the distances between the A , B , and C atoms.
404. The system of claim 403, wherein the distance dλ along the bisector of 0^04 from C to the internuclear-distance line between A and A , 2c \_A , is given by
dx = 2c\_A cos ^4- (15.105)
where 2c 'C_A is the internuclear distance between A and C .
405. The system of claim 404, wherein the atoms A , A , and B define the base of a pyramid.
406. The system of claim 405, wherein the pyramidal angle θzABA can be solved from the internuclear distances between A and A , 2cx A_A , and between A and B , 2c \_B , using the law of cosines (Eq. (15.98)):
(15.106).
Figure imgf000985_0002
407. The system of claim 406, wherein the distance d2 along the bisector of Θ/1ABA from B to the internuclear-distance line 2c\_A , is given by
d2 = 2c\_B cos^M- (15.107).
408. The system of claim 407, wherein the lengths dx , d2 > and 2c 'c_β define a triangle wherein the angle between dx and the internuclear distance between B and C , 2c'c_s , is the dihedral angle ΘZBCIACΛ that can be solved using the law of cosines (Eq. (15.98)):
Figure imgf000985_0001
409. The system of claim 1, wherein the specie are solved using the solutions of organic chemical functional groups as basis elements wherein the structures and energies where linearly added to achieve the molecular solutions, each functional group can be treated as a building block to form any desired molecular solution from the corresponding linear combination, each functional group element was solved using the atomic orbital and hybrid orbital spherical orbitsphere solutions bridged by molecular orbitals comprised of the H2 -type prolate spheroidal solution given in the Nature of the Chemical Bond of
Hydrogen-Type Molecules section, the energy of each MO was matched at the HO or AO by matching the hybridization and total energy of the MO to the AOs and HOs, the energy E mag (e-§- §iven by Eq- (15.58)) for a C2sp3 HO and Eq.(15.59) for an O2p AO) was subtracted for each set of unpaired electrons created by bond breakage.
410. They system of claim 409, wherein the bond energy is not equal to the component energy of each bond as it exists in the specie, although, they are close.
411. The system of claim 409, wherein the total energy of each group is its contribution to the total energy of the specie as a whole.
412. The system of claim 409, wherein the determination of the bond energies for the creation of the separate parts must take into account the energy of the formation of any radicals and any redistribution of charge density within the pieces and the corresponding energy change with bond cleavage.
413. The system of claim 409, wherein the vibrational energy in the transition state is dependent on the other groups that are bound to a given functional group, which will effect the functional-group energy, however because the variations in the energy based on the balance of the molecular composition are typically of the order of a few hundreds of electron volts at most, they are neglected.
414. The system of claim 409, wherein the energy of each functional-group MO bonding to a given carbon HO is independently matched to the HO by subtracting the contribution to the change in the energy of the HO from the total MO energy given by the sum of the MO contributions and E(C,2sp3) = -14.63489 eV (Eq. (13.428)).
415. The system of claim 409, wherein the intercept angles are determined from Eqs. (15.71-15.78) using the final radius of the HO of each atom.
416. The system of claim 409, wherein a final carbon-atom radius is determined using Eqs. (15.32) wherein the sum of the energy contributions of each atom to all the MOs in which it participates in bonding is determined.
417. The system of claim 416, wherein the final radius is used in Eqs. (15.19) and (15.20) to calculate the final valence energy of the HO of each atom at the corresponding final radius.
418. The system of claim 417, wherein the radius of any bonding heteroatom that contributes to a MO is calculated in the same manner, and the energy of its outermost shell is matched to that of the MO by the hybridization factor between the carbon-HO energy and the energy of the heteroatomic shell.
419. The system of claim 416, wherein the donation of electron density to the AOs and HOs reduces the energy.
420. The system of claim 419, wherein the donation of the electron density to the MO's at each AO or HO is that which causes the resulting energy to be divided equally between the participating AOs or HOs to achieve energy matching.
421. The system of claim 1 , wherein the molecular solutions are used to design synthetic pathways and predict product yields based on equilibrium constants calculated from the heats of formation.
422. The system of claim 1, wherein the new stable compositions of matter are predicted as well as the structures of combinatorial chemistry reactions.
423. The system of claim 1, wherein pharmaceutical applications include the ability to graphically or computationally render the structures of drugs that permit the identification of the biologically active parts of the specie to be identified from the common spatial charge-density functions of a series of active species.
424. The system of claim 1, wherein novel drugs are designed according to geometrical parameters and bonding interactions with the data of the structure of the active site of the drug.
425. The system of claim 1 , wherein to calculate conformations, folding, and physical properties, the exact solutions of the charge distributions in any given specie are used to calculate the fields, and from the fields, the interactions between groups of the same specie or between groups on different species are calculated wherein the interactions are distance and relative orientation dependent.
426. The system of claim 425, the fields and interactions can be determined using a finite- element-analysis approach of Maxwell's equations.
427. The system of claim 1, wherein in the case where three sets of C = C -bond MOs form bonds between the two carbon atoms of each molecule to form a six-member ring such that the six resulting bonds comprise eighteen paired electrons, and each bond comprises a linear combination of two MOs wherein each comprises two C2sp3 HOs and 75% of a H1 -type ellipsoidal MO divided between the C2sp3 HOs:
Figure imgf000988_0001
428. The system of claim 427, wherein the linear combination of each H2 -type ellipsoidal
MO with each C2sp3 HO further comprises an excess 25% charge-density contribution per bond from each C2sp3 HO to the C = C -bond MO to achieve an energy minimum.
429. The system of claim 427, wherein the dimensional parameters of each bond C = C- bond are determined using Eqs. (15.42) and (15.1-15.5) in a form that are the same equations as those used to determine the same parameters of the C = C -bond MO of ethylene (Eqs. (14.242-14.268)) while matching the boundary conditions of the structure of benzene.
430. The system of claim 427, wherein hybridization with 25% electron donation to each
C = C -bond gives rise to the Cbemm 2$p3 HO-shell Coulombic energy
*cw* (Cwne> V) given by Eq. (14.245).
431. The system of claim 427, wherein to meet the equipotential condition of the union of
the six C2sρ3 HOs, c2 and C2 of Eq. (15.42) for the aromatic C=C-bond MO is given by Eq. (15.62) as the ratio of 15.95955 eV , the magnitude of ECoulomb (Cbemene,2sp3) (Eq. (14.245)), and 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H (Eq. (1.243)):
C2 (benzeneC2spiHθ)= C2 (benzeneC2spi Hθ)= 13-605804 eV_ = Q.85252 (15.143)
wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
3e
432. The system of claim 427, wherein the energies of each C=C bond of benzene are determined using the same equations as those of ethylene (Eqs. (14.251-14.253) and (14.319-14.333) with the parameters of benzene.
3e
433. The system of claim 427, wherein ethylene serves as a basis element for the C=C
3e bonding of benzene wherein each of the six C=C bonds of benzene comprises (0.75)(4)= 3 electrons according to Eq. (15.142).
434. The system of claim 427, wherein the total energy of the bonds of the eighteen
electrons of the C=C bonds of benzene, E7, \C6H6,C=C J, is given by (6)(θ.75) times
Eτ+osc (c = c^ (Εcl- (14.492)), the total energy of the C=C -bond MO of benzene including the Doppler term, minus eighteen times E\C,2sp3) (Eq. (14.146)), the initial
3e energy of each C2sp3 HO of each C that forms the C=C bonds of bond order two.
3e
435. The system of claim 427, wherein the total energy of the six C=C bonds of benzene with three electron per aromatic bond given by Eq. (14.493) is
E1, (\ c6H6,clc = (6χo.75)Er+osc (c = c)- (6χ3)E(c,2ψ3)
= (6)(θ.75)(-66.O5796 eV)- lδ(-14.63489 eV) (15.144) = -297.26081 eV - (-263.42798 eV) = -33.83284 eV wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
436. The system of claim 427, wherein the results of benzene can be generalized to the class of aromatic and heterocyclic compounds. E}n of an aromatic bond is given by
E7. (HΛ (Eqs. (11.212) and (14.486)), the maximum total energy of each H2-type MO such that
E (15.145),
Figure imgf000990_0001
wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to ± 10%, if desired.
437. The system of claim 435, wherein the factor of 0.75 corresponding to the three electrons per aromatic bond of bond order two given in the Benzene Molecule (C6H6) section modifies Eqs. (15.52-15.56).
438. The system of claim 437, wherein the multiplication of the total energy given by Eq. (15.55) by fχ = 0.75 with the substitution of Eq. (15.145) gives the total energy of the aromatic bond:
Figure imgf000990_0002
wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to ± 10%, if desired.
439. The system of claim 427, wherein the total bond energy of the aromatic group ED φrøap) is the negative difference of the total energy of the group (Eq. (15.146)) and the total energy of the starting species given by the sum of c4Etnlllal (c4 AO / Hθ) and
Figure imgf000991_0001
I HO) /:
Figure imgf000991_0002
(15.147), wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
440. The system of claim 427, wherein benzene is considered as comprising chemical bonds between six CH radicals wherein each radical comprises a chemical bond between carbon and hydrogen atoms, energy components of Ve , V , T , Vm , and E1. are the same
as those of the hydrogen carbide radical, except that Eτ [C = C,2sp3 J= -1.13379 eV (Eq. (14.247)) is subtracted from E1, (CH) of Eq. (13.495) to match the energy of each
C - H -bond MO to the decrease in the energy of the corresponding C2sp3 HO, wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
441. The system of claim 440, wherein in the corresponding generalization of the aromatic CH group, the geometrical parameters are determined using Eq. (15.42) and Eqs. (15.1-
15.5) with E1, [atom - atom, msp3.^40j= -1.13379 eF wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
442. The system of claim 440, wherein the total energy of the benzene C - H -bond MO, E7, [C - H] , benzene given by Eq. (14.467) is the sum of 0.5E7 l, ( \c = C52sp3 \ J the energy
change of each C2,sp3 shell per single bond due to the decrease in radius with the 3β formation of the corresponding C= C -bond MO (Eq. (14.247)), and E (CH\ the σ
MO contribution given by Eq. (14.441). In the corresponding generalization of the aromatic CH group, the energy parameters are determined using Eqs. (15.146-15.147)
- -11..1133337799 eeVV atom - atom,msp3.AO )= — '■ , wherein the calculated and
2 measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
443. The system of claim 442, wherein the energy contribution to the single aromatic CH
3e bond is one half that of the C=C double bond contribution, which matches the energies
3e of the CH and C=C aromatic groups, conserves the electron number with the equivalent charge density as that of s = 1 in Eqs. (15.18-15.21), and further gives a minimum energy for the molecule.
3e
444. The system of claim 443, wherein the breakage of the aromatic C=C bonds to give CH groups creates unpaired electrons in these fragments that corresponds to C3 = I in Eq.
(15.56) with Emag given by Eq. (15.58).
445. The system of claim 444, wherein each of the C - H bonds of benzene comprises two electrons according to Eq. (14.439).
446. The system of claim 445, wherein from the energy of each C - H bond,
-E ^n (12CH) benzene V J (Eq. (14.477)), the total energy of the twelve electrons of the six C - H
bonds of benzene, Eτ (C6H6,C - H), given by Eq. (14.494) is
Eτ (C6H6,C- H)= (6jζ-EDi (2CH)^)= 6(-3.90454 eV)= -23.42724 eV (15.148), wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
447. The system of claim 440, wherein the total bond dissociation energy of benzene,
ED (C6H6 ), given by Eq. (14.495) is the negative sum of Eτ I C6H6,C=C \ (Eq.
(14.493)) and E7 (C6H6,C - H) (Eq. (14.494)):
Figure imgf000993_0001
= -((-33.83284 eV)+ (-23.42724 eP)) (15.149)
= 57.2601 eF wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.
448. The system of claim 447, wherein using the parameters given in Tables 15.214 and 15.216 in the general equations (Eqs. (15.42), (15.1-15.5), and (15.146-15.147)) reproduces the results for benzene given in the Benzene Molecule (C6H6 ) section as shown in Tables 15.214 and 15.216.
449. A system of computing and rendering the nature of at least one specie selected from a group of diatomic molecules having at least one atom that is other than hydrogen, polyatomic molecules, molecular ions, polyatomic molecular ions, or molecular radicals, or any functional group therein, comprising physical, Maxwellian solutions of charge, mass, and current density functions of said specie, said system comprising: processing means for processing physical, Maxwellian equations representing charge, mass, and current density functions of said specie to produce at least one Maxwellian solution; and an output means for outputting the Maxwellian solution.
450. The system of claim 449, further comprising a data transfer system for inputting numerical data into or out of a computational components and storage components of the , main system.
451. The system of claim 449, further comprising a spreadsheet containing solutions of the bond parameters with output in a standard spreadsheet format.
452. The system of claim 451, further comprising a data-handling program to transfer data from the spreadsheets into the main program.
453. The system of claim 449, wherein output may be at least one of graphical, simulation, text, and numerical data.
454. The system of claim 453, wherein the output may be the calculation of at least one of: (1) a bond distance between two of the atoms; (2) a bond angle between three of the atoms; (3) a bond energy between two of the atoms; (4) orbital intercept distances and angles; and (5) charge-density functions of atomic, hybridized, and molecular orbitals, wherein the bond distance, bond angle, and bond energy are calculated from physical solutions of the charge, mass, and current density functions of atoms and atomic ions, which solutions are derived from Maxwell's equations using a constraint that a bound electron(s) does not radiate under acceleration
455. The system of claim 449, wherein the charge, current, energy, and geometrical parameters are output to be inputs to other programs that can be used in further applications.
456. The system of claim 455, wherein the data of heats of formation can be input to another program to be used to predict stability (existence of compounds) equilibrium constants and to predict synthetic pathways.
457. The system of claim 456, wherein novel composition of matters can be discovered by calculating at least one of a bond distance between two of the atoms, a bond angle between three of the atoms, and a bond energy between two of the atoms, orbital intercept distances and angles, charge-density functions of atomic, hybridized, and molecular orbitals, the bond distance, bond angle, and bond energy being calculated from physical solutions of the charge, mass, and current density functions of atoms and atomic ions, which solutions are derived from Maxwell's equations using a constraint that a bound electron(s) does not radiate under acceleration.
458. The system of claim 456, wherein the charge and current density functions can be used to predict the electric and magnetic fields of the species to determine other properties due to the interaction of the fields between species.
459. The system of claim 458, wherein finite-element analysis is used to predict or calculate the interaction and resulting properties, such as the freezing point, boiling point, density, viscosity, and refractive index.
460. The system of claim 449, wherein the output data can be used to give thermodynamic, spectroscopic, and other properties, aid in drug design and other applications with or without direct visualization.
461. The system of claim 449, wherein the data can be input into other programs of the system, which calculate thermodynamic and other properties, or performs a simulation, such as a chemical reaction or molecular dynamics.
462. A method of using any of the systems or compositions of matter of claims 1-461.
463. A use of any system or composition of matter of claims 1 -461.
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WO2018093538A1 (en) * 2016-11-16 2018-05-24 Merrithew Paul System and method of calculating the structure and properties of chemicals
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US7689367B2 (en) 2004-05-17 2010-03-30 Blacklight Power, Inc. Method and system of computing and rendering the nature of the excited electronic states of atoms and atomic ions
EP2100266A2 (en) * 2007-01-03 2009-09-16 Blacklight Power, Inc. System and method of computing and rendering the nature of molecules, molecular ions, compounds and materials
EP2100266A4 (en) * 2007-01-03 2011-05-25 Blacklight Power Inc System and method of computing and rendering the nature of molecules, molecular ions, compounds and materials
WO2018093538A1 (en) * 2016-11-16 2018-05-24 Merrithew Paul System and method of calculating the structure and properties of chemicals
WO2020087826A1 (en) * 2018-10-30 2020-05-07 浙江大学 Simulation analysis method for particle system based on combined hyperellipsoid model
CN109543268A (en) * 2018-11-14 2019-03-29 大连理工大学 The recognition methods of TBM propulsive force major influence factors based on kriging model
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WO2023142596A1 (en) * 2022-01-29 2023-08-03 刘畅源 Method for calculating indistinguishable resonance nuclear reaction cross section and cross section path integral

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