EP1941415A2 - Systeme et procede de calcul et de rendu de la nature de molecules polyatomiques et d'ions moleculaires polyatomiques - Google Patents

Systeme et procede de calcul et de rendu de la nature de molecules polyatomiques et d'ions moleculaires polyatomiques

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Publication number
EP1941415A2
EP1941415A2 EP06827305A EP06827305A EP1941415A2 EP 1941415 A2 EP1941415 A2 EP 1941415A2 EP 06827305 A EP06827305 A EP 06827305A EP 06827305 A EP06827305 A EP 06827305A EP 1941415 A2 EP1941415 A2 EP 1941415A2
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Prior art keywords
energy
given
atom
bond
electrons
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EP1941415A4 (fr
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Randell L. Mills
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Brilliant Light Power Inc
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BlackLight Power Inc
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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

Definitions

  • 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

L'invention porte sur un procédé et sur un système visant à résoudre du point de vue physique les fonctions de charge, masse et densité de courant de molécules polyatomiques, d'ions moléculaires polyatomiques, de molécules diatomiques, de radicaux moléculaires, d'ions moléculaires ou d'une partie quelconque de ces espèces à l'aide des équations de Maxwell, et calculer et réaliser le rendu de la nature physique de la liaison chimique en utilisant les solutions. Les résultats peuvent être affichés sur un support visuel ou graphique. L'affichage peut être statique ou dynamique en fonction du déplacement des électrons et du déplacement par vibration, rotation et translation de l'espèce. Les informations affichées sont utiles pour anticiper la réactivité et les propriétés physiques. Connaître la nature de la liaison chimique d'au moins une espèce peut permettre la solution et l'affichage d'autres espèces et peut être utile pour anticiper leur réactivité et leurs propriétés physiques.
EP06827305A 2005-10-28 2006-10-30 Systeme et procede de calcul et de rendu de la nature de molecules polyatomiques et d'ions moleculaires polyatomiques Withdrawn EP1941415A4 (fr)

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