WO2007053486A2 - System and method of computing and rendering the nature of polyatomic molecules and polyatomic molecular ions - Google Patents

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 = l -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 Schrό" dinger 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 ~ \ -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

$Ψ(x)Ψ *(x)dx ' (1)

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, (" OO 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.blacklightpower.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.blacklightpower.com/bookdownload.shtml): in prior PCT applications PCT/US02/35872; PCT/US02/06945; PCT/US02/06955; PCT/USOl/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, PA₉ 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.blacklightpower.com/techpapers.shtinl. 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.V/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 at http://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 a block diagram of an exemplary software program; and Figs. 2 and 3 illustrate pictures of an exemplary software program.

DETADLED DESCRIPTION OF INVETION

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 k' of a H₂ -type ellipsoidal MO due to the equivalent of two point charges of at the foci is given by:

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 c' is given by:

The internuclear distance is

The length of the semiminor axis of the prolate spheroidal MO b = c is given by b = Va² - c'² (15.4) And, the eccentricity, e , is c' e = — (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

The potential energy of the two nuclei is

The kinetic energy of the electrons is

And, the energy, V_1n , of the magnetic force between the electrons is

The total energy of the H₂ -type prolate spheroidal MO, E₁, (^₁Mo), is given by the sum of the energy terms:

E_T (_HlMo)= V_e +T + V_m + V (15.10)

where W₁ 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 c_B0 , 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'_B0 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 C₁ 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 N 14.53414 eV and 13.605804 eV , the Coulombic energy between the electron and proton of H ;

0.91771 , ⁷ the ratio of 14.82575 eV , '

and 13.605804 eV ;

0.87495 , the ratio of 15.55033 eV , ' and 13.605804 eV ;

' 0.85252 , the ratio of 15.95955 eV , and 13.605804 eV ;

0.85252 , ' the ratio of 15.95955 eV ,' and 13.605804 eV , and

0.86359 , ' the ratio of 15.55033 eV , ' nd 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 E_τ \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.

where IP is the rath ionization energy (positive) of the atom. The radius r , of the hybridized shell is given by:

shell is given by

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:

Then, the energy Eyatom,msp³ J of the outer electron of the atom msp³ shell is given by the sum of E_Cgulomb uztom,msp³ J and E(magnetic) : O⁵-¹⁶⁾

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. hi 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, msp' ¹J for 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 C2sp³ shell given by Eq. (14.146) is

E(C,2SP³ )= -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, hi 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 I 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

E_τ (mol.atom,msp³ λ (m is the integer of the valence shell) of the HO electrons is given by i 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:

E₇. (mol.atom, msp³ V E (atom, msp³ )- ∑ iP, (15.17) m=2 where IP_1n is the τ»th ionization energy (positive) of the atom and the sum of -IP₁ plus the hybridization energy is E[atom,msp³ \ Thus, the radiu of the hybridized shell is given

by:

where s = 1,2,3 for a single, double, and triple bond, respectively. The Coulombic energy of the outer electron of the atom msp³ shell is given by

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 given by Eq.

(15.15). Then, the energ of the outer electron of the atom msp³ shell is

given by the sum o

and E(magnetic) :

he energy change of each atom msp³ shell with the formation of the

atom-atom-bond MO is given by the difference between

nd

Using Eqs. (15.18-15.28), the final values of and and the

resulting 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 s are given in Table 15.1.

Table 15.1. The final values the resulting

^me MO due to charge donation from the HO to the MO where

refers to the bond order of the carbon-carbon bond.

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 Eynol.atom,msp³ J of the outer electron of the atom msp³ shell of each bonding atom

must be the average of

ϊox two different values of s :

In this case, E_τ(afom- atom,msp² \ 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 :

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, E_τ\atom- atom,msp³) of the C- H-bond MO is given by

0.5JE₂, [C = C,2sp³ J (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 ΗOs 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 ΗOs 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,

E_τ(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)

where c is the multiple of the BO of s,, . The radius r . of the atom msp³ shell of each

bonding atom is given by the Coulombic energy using the initial energy E_Couhmb \atom,msp³ J

and E_τ (atom -atom, msp³ \ the energy change of each atom msp³ shell with the formation of each atom-atom-bond MO:

wher The Coulombic energ

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 msp¹ shell is given by the sum of

nd E(magnetic) (Eq. (15.20)). E_τ (atom-

the energy change of each atom msp³ shell with the formation of the atom-atom-bond MO is given by the difference between

and E[atom,msp³ j given by Eq.

(15.21). Using Eq. (15.2 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 o and the resulting

of the MO comprising a linear combination of H₂-type ellipsoidal MOs and

corresponding ΗOs of single or mixed bond order where c₅ is the multiple of the bond order paramete given in Table 15.1.

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 ₃ of the C2sp³ HO of a carbon

atom of a given molecule is calculated using Eq. (14.514) by consideri

the total energy donation to each bond with which it participates in bonding. The general equation for the radius is, given by .

The Coulombic energy E_Coulomb \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)₅ 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.atotn,msp³ Λ of the outer electron of the

atom msp³ shell is given by the sum of E_Coulomb (mol.atom,msp³Λ and E(magnetic) (Eq.

(15.20)).

For example, the

O 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 C2sp³ HO of each methylene group o ontributes -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

hi 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 C2sp³ 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 and

are calculated using the total energy donation to each bond

with which an atom participates in bonding corresponding to the values of

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 o

calculated using the values of given in Tables 15.1 and 15.2.

Table 15.3.B. The final values of

calculated for heterocyclic groups using the values of given in Tables 15.1

and 15.2.

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 c' (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 n_x times the total energy of H₂ where n_x 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, E_T (H₂MO), is given by the sum of the energy terms (Eqs. (15.6-

15.11)) plus E_T (A0 / H0):

E_τ (H₂Moy V_e + T + V_m + V_p + E_τ (AO I HO) (15.36)

where 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 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 AE_{H MO} (AO I HO) due to the AO or ΗO's charge donation to the MO.

E₇ (AOI Hθ)= E(A0I H0)- &E (AO I HO) • (15.38)

To solve the bond parameters and energies, c' =

(15.2)) is substituted int

to give

The total energy is set equal to Eφasis energies) which in the most general case is given by the sum of a first integer n_x times the total energy of H₂ minus a second integer n₂ times the total energy of H₅ minus a third integer n₃ times the valence energy of E[AOj (e.g.

E[N J= -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_χ (-31.63536831 eV)- n₂ (-13.605804 eV)- nβ(Aθ) (15.40) In the case that the MO bonds two atoms other than hydrogen, E(basis energies) is W₁ times the total energy of H₂ where n_χ 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_χ (-31.63536831 eV^ (15.41) set equal to Eφasis energies) , and the semimajor axis a is solved.

Thus, the semimajor axis a is solved from the equation of the form:

E(basis energies) (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). 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, 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 E₇, {atom- atom, msp³. AOJ, the change in the energy of the AOs or

ΗOs upon forming the bond. From Eqs. (15.39-15.40), E₇, (MO) is

During bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energ the sum of the Doppler, E_D , and

average vibrational kinetic energies

where n_x 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 E_D is determined by the force between the central field and the electrons in the transition state. The force and its derivative are given by

and

such that the angular frequency of the oscillation in the transition state is given by

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, c_B0 is the bond- order factor which is 1 for a single bond and when the MO comprises W₁ equivalent single bonds as in the case of functional groups. c_B0 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 C_2o 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, C,_o = C₁ and C_2o = C₂. The kinetic energy, E_x. , corresponding to E_D is given by Planck's equation for functional groups:

The Doppler energy of the electrons of the reentrant orbit is

is given by the sum o

(Eq. (15.51)):

(15.52) The total energy of the functional grou

is the sum of the total energy of the components comprising the energy contribution of the MO formed between the participating atoms, E(jbasis energies) , the change in the energy of the AOs or HOs upon forming the bond (E_τ (atom- atom,msp³.AOJ), the energy of oscillation in the transition state, and the change in magnetic energy with bond formation, From Eq. (15.52), the total energy of

the group is

-£L (15.53)

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

where r is the radius of the atom that reacts to form the bond and C₃ is the number of electron pairs.

The total bond energy of the grou 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

E_D - \E(basis energies) + E₇, [atom - atom,msp*.AOj (15.56)

In the case of organic molecules, the atoms of the functional groups are energy matched to the C2sp³ HO such that

E(AOI H0)= -14.63489 eV (15.57) For examples of E_ma from previous sections:

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 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, iven

by Eqs. (15.19) and (15.31-15.32).

(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 given by Eqs. (15.20) and (15.31-15.32). For

\E(valence)\ >13.605804 eV :

(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, iven 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 E(valence) is the ionization energy

given by Eqs. (15.20) and (15.31-15.32). For

(v) the ratio that is less than one of the magnitude of the valence-level energies, E_n(valence), of the AO or HO of the nth participating atom of two that are energy matched where Eiyalence) is the ionization energy or

given by Eqs. (15.20) and

(15.31-15.32):

(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 C₁ [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 C₁ is the hybridization factor C₁ (\ \ of the valence AOs or HOs a first pair of atoms and the hybridization factor C₁ \2 \ of the valence AO or HO a third atom or second pair to which the first two are energy matched:

(vii) the factor that is the product of the hybridization factor C₁ (\ \ of the valence AO or HO of a first atom and the hybridization factor c₂ (l \ 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 of the valence AOs or HOs a first pair of

atoms and the hybridization factor

of the valence AO or HO a third atom or second pair to which the first two are energy matched:

The hybridization factor c₂ corresponds to the force constant k (Eqs. (11.65) and (13.58)). hi 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).

Specific examples of the factors c₂ and C₂ of a H₂ -type ellipsoidal MO of Eq. (15.51) given in following sections are

d

C₂ (S3p to aryl-type C2s≠Hθ)= J&EL = ^^{0'36001 eV} = 0.65700. ^{2 V} * ^{V J} E(C,2sp ) -15.76868 eV

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 A -atom AO are determined from the polar equation of the ellipse:

The radius of the A shell is r_A, and the polar radial coordinate of the ellipse and the radius of the A shell are equal at the point of intersection such that

The polar angle θ' at the intersection point is given by

Then, the angle Θ_{A AO} the radial vector of the A AO makes with the interauclear axis is

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 — Θ_{H 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:

such that

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

The distance d_{A A0} 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:

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^y of the A- B ellipsoidal MO is given by:

The internuclear distance is

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, (#₂M?), 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 c_BO , the bond-order factor which is 1 for a single bond and when the MO comprises n_x equivalent single bonds as in the case of functional groups. c_B0 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 ox 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.AO), the energy component due to the AO or ΗO's charge donation to the terminal MO, is added to the other energy terms to give E₁, (^₂M?):

(15.82)

The radiation reaction force in the case of the vibration of A - B in the transition state corresponds to the Doppler energy, E_D , 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 E_T (H₂MO) (Eq. (15.82)) and E_gsc given Eqs. (11.213-

11.220), (11.231-11.236), and (11.239-11.240). Thus, the total energy E_T (A - B) of the A - B MO including the Doppler term is E₇

(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, C_2o 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 mjn_* μ = is the reduced mass of the nuclei given by Eq. ( 11.154). To match the boundary

Tn₁ + m₂ condition that the total energy of the A- B ellipsoidal MO is zero, E₇, (A - B j given by Eq. (15.83) is set equal to zero. Substitution of Eq. (15.81) into Eq. (15.83) gives

(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

and 'BO (15.86)

4πε_Qa

The nuclear repulsion force and its derivative are given by

(15.87)

$πε_o(a + c'j and

such that the angular frequency of the oscillation is given by

Since both terms of E_osc = E_D + E_Kvib 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

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:

In the case of the determination of the bond angle of the ACH 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 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:

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, E_Coulomb \MO.atom,msp^ι \ or the energy, E\MO.atom,mspⁱ \ the radius r A-B AorBsp , 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 participates in bonding as it forms the terminal bond. The Coulombic energy E_Coulomb [MO.atom,msp³ λ of the outer electron of the atom msp* 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

and E(magnetic) (Eq. (15.20)).

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

of the outer electron of the C2sp³ shell given by Eq. (15.19) with the radius r G — C- Czsp ₃ of each C2sp³ HO of the terminal C- C bond calculated using Eq. (15.32)

by considering 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

is

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:

In the exemplary cases of C- C, 0- 0 , and N - N where C is carbon:

hi the exemplary cases of C- N , C- O, and C- S ,

where C is carbon and α, (c to Bλ 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 E₁. (atom~ atom,msp³.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

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 is determined using Eq.

(15.32) an 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 (C\ 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 C2sp³ HO (c₂ (2)= 0.91771 , Eq. (13.430)). E_τ (atom- atom,msp³.AθΛ of Eq. (15.90) is the maximum for the participating atoms which is -1.44915 eF , -0.92918 eF , -0.92918 eV , and -0.33582 eF 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 (T) for the fluorine atom given by Eq. (15.68) that matches the valence energy of F (E_γ(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 /, C₂ 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 (E^alence^) 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)). E_τ (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 I , 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 , 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 C2sp³ HO. In these cases, C₂ is given by Eq. (15.65) for the corresponding atom X where C 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:

With S₁ = 2c_c' __A, the internuclear distance of the C- A bond, ^₂ = 2c_c' __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 Q^₀₃ between the C - A and C - B bonds is given by

Consider the exemplary structure C_bC_a{O_a)O_b wherein C_a is bound to C_b , O_a , and O_b. In the general case that the three bonds are coplanar and two of the angles are known, say O_x and θ₂ , then the third 6>₃ can be determined geometrically:

In the general case that two of the three coplanar bonds are equivalent and one of the angles is known, say θ_χ , then the second and third can be determined geometrically:

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 C_3v 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_{orj m}__B from the origin to the nucleus of a terminal B atom is given by

the height along the z-axis from the origin to the A nucleus d_hejght is given by

the angle θ_y of each A - B bond from the z-axis is given by

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 Θ^_ABC given by Eq. (14.206) is

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 angl 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 and the distances between the A ,

B , and C atoms. The distance d_χ along the bisector o from C to the internuclear-

distance line between A and

, is given by

where 2c'_c__Λ is the internuclear distance between A and C . The atoms A , A , and B define the base of a pyramid. Then, the pyramidal angle Θ_ΔABA can be solved, from the internuclear distances between A and A , 2c'_A__Λ , and between A and B , 2c \__B , using the law of cosines (Eq. (15.98)):

Then, the distance d₂ along the bisector of Θ^_BA from B to the internuclear-distance line , is given by

The lengths d_χ , d₂, and 2c'_c__B define a triangle wherein the angle between d_χ and the internuclear distance between B and C , 2c '_C__B , is the dihedral angle Θ_ZBC/ACA that can be solved using the law of cosines (Eq. (15.98)):

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 Ηydrogen-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 (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,2sp³ V -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 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.

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 (CH₂CH₂) section. The radius ^_{eΛ /(iBβ2j} , (0.85252α₀) of the C2sp² shell of ethylene calculated from the Coulombic energy is given by Eq. (14.244). The Coulombic energy ^Ec_oubmb (^_rAyte'²^³) (-15.95955 eV ) of the outer electron of the C2^p³ shell is given by Eq.

(14.245). The energy E(C_elhy!ene,2sp³^ (-15.76868 eV ) of the outer electron of the C2sp³ shell is given by Eq. (14.246). E_τ (c = C,2sp³^ (-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\C_{e thγlem},2sp³ J and E[C,2sp³ J. 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 C2ψ³ΗOs and 75% of a H₂ -type ellipsoidal MO divided between the C2sp³ ΗOs:

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 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_benzen 2sp³ HO-shell Coulombic energ iven by Eq. (14.245). To meet

the equipotential condition of the union ₂ and C₂ of Eq. (15.42) for the

O is given by Eq. (15.62) as the ratio of 15.95955 eV , the magnitude of ^Ecl- (14.245)), and 13.605804 eV , the magnitude of the Coulombic energy between the electron and proton of H (Eq. (1.243)):

The energies of each C=C bond of benzene are also determined using the same equations as those of ethylene (Eqs. (14.251-14.25 .319-14.333) with the parameters of benzene. Ethylene serves as a basis element for t bonding of benzene wherein each of

the si onds of benzene comprises (θ.75 electrons according to Eq. (15.142).

the bonds of the eighteen electrons of the nds of benzene,

given by (6)(θ.75) times

q. (14.492)), the total energy of the C= ^3e C -bond MO of benzene including the Doppler term, minus eighteen times E /[C,2sp³) \

(Eq. (14.146)), the initial energy of each Clsp³ HO of each C that forms the onds of

bond order two. Thus, the total energy of the six

bonds of benzene with three electron per aromatic bond given by Eq. (14.493) is

The results of benzene can be generalized to the class of aromatic and heterocyclic compounds.

E_hv of an aromatic bond is given by E_Γ (HΛ (Eqs. (11.212) and (14.486)), the maximum total energy of each H₂ -type MO such that

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 /, = 0.75 with the substitution of Eq. (15.145) gives the total energy of the aromatic bond:

The total bond energy of the aromatic group

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

O I HO):

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 V_e , V , are the same as those of the hydrogen carbide radical, except that

1.13379 eV (Eq. (14.247)) is subtracted from E_T (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,msp³.Aθ)-= -1.13379 eV .

The total energy of the benzene C - H -bond MO, given by Eq. (14.467) is the sum of 0. e energy change of each ell per single bond due to

the decrease in radius with the formation of the correspondin bond MO (Eq. (14.247)), and the σ MO contribution given by Eq.

. In the corresponding

generalization of the aromatic CH group, the energy parameters are determined using Eqs. — 1 13379 eV (15.146-15.147) with Z₁ = I an — ^: . Thus, the energy

3e 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 5 = 1 in 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 = 1 in Eq.

(15.56) with E_mag 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, -E (¹²CH) (Eq. (14.477)), the total energy of the twelve electrons of the six C- H bonds of benzene, £_r (C₆H₆,C - H), given by Eq. (14.494) is

£_r (C₆H₆₅C- H)= (6)(^ (¹²CH))= 6(-3.90454 eF)= -23.42724 e F (15.148) The total bond dissociation energy of benzene, E_D [C₆H₆ \ given by Eq. (14.495) is the negative

(Eq. (14.493)) and E₇ (C₆H₆,C - H) (Eq. (14.494)):

= -((-33.83284 eV)+ (-23.42724 <ZF)) (15.149)

= 57.2601 eF

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 E_D (ρmup) 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.

Table 15.213. The s mbols of functional rou s of aromatics and hertoc clics.

CC (aromatic bond) C=C

CH (aromatic) CH (i) Table 15.214.. The geometrical bond parameters of aromatics and hertocyclics and ex erimental values I .

Table 15.215. The MO to HO intercept geometrical bond parameters of benzene. E_τ is E₁, {atom - atom,msp³.AOy

Table 15.217. The total bond energies of benzene calculated using the functional group composition and

REFERENCES FORDETAILED DESCRIPTION OF INVENTION

1. D. R. Lide, CRC Handbook of Chemistry and Physics, 86th Edition, CRC Press, Taylor & 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. bl 2-methvl-l-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. fiuoroethane at http://webbook.nist.gov/.

12. G. Herzberg, Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules, Kiieger 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/.

16. methyl iodide at http://webbook.nist.gov/.

17. K. P. Huber, G. Herzberg, Molecular Spectra and Molecular Structure, IV. Constants of Diatomic Molecules, Van Nostrand Reinhold Company, New York, (1979).

18. J. Crovisier, Molecular Database — Constants for molecules of astrophysical interest in the gas phase: photodissociation, microwave and infrared spectra, Ver. 4.2, Observatoire de Paris, Section de Meudon, Meudon, France, May 2002, pp. 34-37, available at http://wwwusr.obspm.fr/~crovisie/.

19. methanol at http://webbook.nist.gov/. 20. 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. 46.

21. dimethyl ether at http ://webbook.nist. gov/.

.2. T. Amano, P. F. Bernath, R. W. McKellar, "Direct observation of the V₁ and v₃ fundamental bands of NH₂ by difference frequency laser spectroscopy", J. MoI. Spectrosα, Vol. 94, (1982), pp. 100-113.

13. methylamine at http://webbook.nist.gov/.

24. D. R. Lide, CRC Handbook of Chemistry and Physics, 79th Edition, CRC Press, Boca Raton, Florida, (1998-9), pp. 9-80 to 9-85.

25. D. Lin-Vien. Ν. 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.gov/.

28. 2-butanone at http ://webbook.nist. gov/.

29. acetic acid at http://webbook.nist.gov/.

30. formic acid at http://webbook.nist. gov/.

31. D. Lin-Vien. Ν. 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. 138.

32. methyl formate at http://webbook.nist.gov/.

33. D. Lin-Vien. Ν. 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. 144.

34. D. Lin-Vien. Ν. 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. 147.

35. D. Lin-Vien. Ν. 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. 143. 36. H. J. Vledder, F. C. Mijlhoff, J. C. Leyte, C. Romers, "An electron diffraction investigation of the molecular structure of gaseous acetic anhydride", Journal of Molecular Structure, Vol. 7, Issues 3-4, (1971), pp. 421-429.

37. acetonitrile at http://webbook.nist.gov/.

38. D. R. Lide, CRC Handbook of Chemistry and Physics, 86th Edition, CRC Press, Taylor & Francis, Boca Raton, (2005-6), pp. 10-202 to 10-204.

39. D. R. Lide, CRC Handbook of Chemistry and Physics, 86th Edition, CRC Press, Taylor & Francis, Boca Raton, (2005-6), p. 9-79.

40. D. R. Lide, CRC Handbook of Chemistry and Physics, 86th Edition, CRC Press, Taylor & Francis, Boca Raton, (2005-6), pp. 9-82 to 9-86.

41. thiirane at http://webbook.nist.gov/.

42. 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. 242.

43. 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. 241.

44. R. M. Ibberson, M. T. F. Telling, S. Parsons, "Structural determination and phase transition behavior of dimethyl sulfate", Acta. Cryst, Vol. B62, (2006), pp. 280-286.

45. 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. 181.

46. 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. 480.

47. 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. 187.

48. 1 ,3 -butadiene at http://webbook.nist.gov/.

49. 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. 50. W. L 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..

>1. 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.

52. H. B. Burgi, S. C. Capelli, "Getting more out of crystal-structure analyses," Helvetica Chimica Acta, Vol. 86, (2003), pp. 1625-1640.

53. 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. 481.

54. R. Boese, D. Blaser, M. Nussbaumer, T. M. Krygowski, "Low temperature crystal and molecular structure of nitrobenzene", Structural Chemistry, Vol. 3, No. 5, (1992), pp. 363-368.

55. G. A. Sim, J. M. Robertson, T. H. Goodwin, "The crystal and molecular structure of benzoic acid", Acta Cryst, Vol. 8, (1955), pp.157- 164

56. 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. 301.

57. 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. 303.

58. 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. 207.

59. S. Martinez-Carrera, "The crystal structure of Imidazole at -150⁰C", Acta. Cryst., Vol. 20, (1966), pp. 783-789.

60. 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), pp. 296-297. 1. H. Sternglanz, C. E. Bugg, "Conformations of N⁶-monosubstituted adenine derivatives crystal structure of N⁶-methyladenine", Biochimica et Biophysica Acta, Vol. 308, (1973), pp. 1-8.

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), 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, Irix 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 1 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 #AT0MS header, and so on. #NAMES CH3 Alkane CH3

#SMILES C-

#ATOMS

1 C 0 0 0

2 H 1.9775 -0.698 0

3 H -0.9888 -0 .698 1.712

4 H -0.9888 -0 .698 -1.712

#AO

1 1 Is 0.171 2

2 1 2sp3 0.864 -1

#BONDS

1 1 2 1.649

2 1 3 1.649

3 1 4 1.649

#BONDAXES 1 1 1 90 0

#DATA

RCH bond angle: 109.44° CH bond length: 2.097 a HFORM: 12.492

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. #NAMES

Pentane

#SMILES

CCCCC

#GROUPS

1 C-

2 -C-

3 -C-

4 -C-

5 C-

#GROUP LINKS

1 1 1 2 1 180

2 22 3 1 180

3 3 2 4 1 180

4 42 5 1 180

Processing data files: As further shown in the flow diagram of Figure 1 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 2 and 3, 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.

Claims

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

equation i V²- \p(r, θ,φ,t) = 0 which is the equation of motion of the charge.

L v² a² J

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 according to claim 1, wherein the force generalized constant A:¹ of a H₂ -type ellipsoidal MO due to the equivalent of two point charges at the foci is given by:

CC 2e² r = ^^2le (i5.i)

4πε₀ 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.

23. The system according to claim 1, wherein the distance from the origin of the MO to each focus c ' is given by:

the internuclear distance is

the length of the semiminor axis of the prolate spheroidal MO b = c is given by

and, the eccentricity, e , is c' e = — (15.5). a

24. The system according to claim 23, wherein a potential energy of the electrons in the central field of the nuclei at the foci is

where U₁ 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 c_BO „ the bond-order factor, which is 1 for a single bond, 4 for an independent double bond and 9 for an independent triplet bond.

25. The system according to claim 1, wherein the potential energy of the two nuclei is

26. The system according to claim 1, wherein the kinetic energy of the electrons is

where f\ 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 c_B0 , the bond-order factor, which is 1 for a single bond, 4 for an independent double bond and 9 for an independent triplet bond.

27. The system according to claim 1, wherein the energy, V_n, , of the magnetic force between the electrons is

11 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 c_m , the bond-order factor, which is 1 for a single bond, 4 for an independent double bond and 9 for an independent triplet bond.

28. The system according to claim 1, wherein total energy of the H₂ -type prolate spheroidal MO₅ E₇. [H₇MO) , is given by the sum of the energy terms:

where τ\ 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 c_B0 , the bond-order factor, which is 1 for a single bond, 4 for an independent double bond and 9 for an independent triplet bond.

29. The system according to claim 1, wherein the total energy E_τ [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:

where IP_1n is the m th ionization energy (positive) of the atom.

30. The system according to claim 29, wherein the radius r , of the hybridized shell is given by:

31. The system according to claim 29, wherein the Coulombic energy E_Coulomb (atom, msp³ J of the outer electron of the atom msp³ shell is given by

32. The system of claim 31, 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:

then, the energy E (atom, msp³ J of the outer electron of the atom msp³ shell is given by the sum of E_Couhmb [atom,msp³ \ and E{magnetic) :

33. The system according to claim 29, wherein the total energy E₁, [mol.atom,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:

where IP_m is the m th ionization energy (positive) of the atom and the sum of -IP₁ plus the hybridization energy i

34. The system of claim 33, wherein the radius r , of the hybridized shell is given by:

where s = 1,2,3 for a single, double, and triple bond, respectively.

35. The system of claim 34, wherein the Coulombic energy

f the outer electron of the atom msp³ shell is given by

36. The system of claim 35, 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 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 E_Coulomb ( mol. atom, msp³ j and E(magnetic) :

(15.20)

E₇, [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

and

37. The system of claim 31, wherein

s one of:

wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.

38. The system of claim 37, wherein

and using Eqs. (15.18-15.28), the final values of r_C2sp, , E_Coulomb (C2sp²), and E(C2sp³) , and

BO the resulting EJ 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:

wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.

39. The system of claim 33, 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

of the outer electron of the atom msp³ shell of each bonding atom is the average of EunoLatom, msp³ λ for two different values of s :

40. The system of claim 39, wherein in the case, E_τ (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 :

41. The system of claim 39, 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:

where c_s is the multiple of the BO of S_n . The radiu of the atom msp³¹ shell of each

bonding atom is given by the Coulombic energy using the initial energ

and E₁. (atom - atom, msp³ ) , the energy change of each atom msp³ shell with the formation of each atom-atom-bond MO:

where

42. The system of claim 41, wherein the Coulombic energy E_Coulomb (mol.atom,msp³) of the outer electron of the atom msp³ shell is given by Eq. (15.19).

43. The system of claim 42, 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. Then, the energy E[mol.atom,msp³} of the outer electron of the atom msp³ shell is given by the sum of E_Coulcmb [moLatom, msp³ ) and E(magnetic) (Eq. (15.20)).

44. The system of claim 42, 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[atom,msp³} given by Eq. (15.21).

45. The system of claim 44, wherein using the equation (15.23) for 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.

46. The system of claim 29, wherein the radius of the C2sp³ HO of a carbon atom of a

given specie is calculated using Eq. (14.514) by consideri e total

energy donation to each bond with which it participates in bonding.

47. The system of claim 1, wherein equation for the radius is given by

48. The system of 29, wherein the Coulombic energy E_Cmlomb {mol.atom,msp³^ of the outer electron of the atom msp³ shell is given by Eq. (15.19).

49. The system of claim 29, 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 E_Coulomb (mol. atom, msp³) and E(magnetic) (Eq. (15.20)).

50. The system of claim 49, 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 C_nH_2n+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.

51. The system of claim 50, wherein the radius (Eq. (15.32)), the Coulombic energy (Eq. (15.19)), and the energy (Eq. (15.20)) of each alkane methylene group are

wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.

52. The system of claim 51, wherein in the determination of the parameters of functional groups, heteroatoms bonding to C2sp³ ΗOs 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.

53. The system of claim 52, 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_{Λtom HO AO} , ) are calculated using

the total energy donation to each bond with which an atom participates in bonding

corresponding to the values of f 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 ulated using the values of

iven 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.

54. The system of claim 53, 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.

55. The system of claim 54, wherein the force constant Ar¹ (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 Y\ times the total energy of H₂ where «, 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, wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.

56. The system of claim 55, 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.

57. The system of claim 1, wherein the total energy, E₇ [H₁Mo) , is given by the sum of the energy terms (Eqs. (15.6-15.11)) plus E_τ (AO I HO) :

E_τ [H₁MO) = V_e + T + V_m + V_p +E_T (AOIHO) (15.36)

(15.37) where τ\ 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 (AOI HO) is the total energy comprising the difference of the energy

E (AOI HO) of at least one atomic or hybrid orbital to which the MO is energy matched and any energy component tsE_H^M0 (AO I HO) due to the AO or ΗO's charge donation to the MO.

58. The system of claim 57, wherein

E₇ (AOI HO) = E(AOI HO)- AE_N^₀ (AO I HO) (15.38)

59. The system of claim 58, wherein as specific examples, E₇ (AOIHO) is one from the group of

E₇ (AOIHO) = E(OIp shell) = -EQonization; O) = -13.6181 eV ;

E₇ (AOIHO) = E(N2p shell) = -E (ionization; N) = -14.53414 eV ;

herein the calculated and measured

values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.

60. The system of claim 1, wherein to solve the bond parameters and energies,

(Eq- (15.2)) is substituted into E_τ (H₂MO) to give

(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 Ti_x times the total energy of H₂ minus a second integer n₂ times the total energy of H, minus a third integer H₃ 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 O₅1, 2, 3....

61. The system of claim 60, wherein

Eφasis energies) = U₁ (-31.63536831 eV)-n₂ (-13.605804 eV)-n_iE(AO) (15.40) in the case that the MO bonds two atoms other than hydrogen, E{basis energies) is H₁ times the total energy of H₂ where r\ is the number of equivalent bonds of the MO and the energy of H₂ , -31.6353683 X eV , Eq. (11.212) is the minimum energy possible for a prolate spheroidal MO:

Eφasis energies) = n, (-31.63536831 eV) (15.41)

E₁, [H₁UO) , is set equal to Eφasis energies) , and the semimajor axis a is solved.

62. The system of claims 23 or 60, wherein the semimajor axis a is solved from the equation of the form:

(15.42) The distance from the origin of the H₂ -type-ellipsoidal-MO to each focus c ' , 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).

63. The system of claim 62, wherein the component energies are given by Eqs. (15.6-15.9) and (15.39).

64. The system of claim 57, wherein the total energy of the MO of the functional group, E₇. (MO) , is the sum of the total energy of the components comprising the energy contribution of the MO formed between the participating atoms and E₇. (atom - atom,mspⁱ.AOj , the change in the energy of the AOs or HOs upon forming the bond.

65. The system of claim 64, wherein from Eqs. (15.39-15.40), E₁. (MO) is

E₇, (MO) - Eφasis energies) + E_T [atom - atom, msp³.Aθ) (15.43)

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 E_osc is the sum of the Doppler, E_n , and average vibrational kinetic energies, E_mb :

where ι\ 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.

67. The system of claim 66, wherein the angular frequency of the reentrant oscillation in the transition state corresponding to E_D is determined by the force between the central field and the electrons in the transition state.

68. The system of claim 67,wherein the force and its derivative are given by

and

'¹M-²"- ⁽¹⁵-^46> such that the angular frequency of the oscillation in the transition state is given by

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, c_B0 is 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, c_B0 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 C₂₀ = C₂ , the kinetic energy, E_κ , corresponding to E₀ is given by Planck's equation for functional groups:

69. The system of claim 68, wherein the Doppler energy of the electrons of the reentrant orbit is

given by the sum o

E_1n, of a group having «, bonds is given by E₇ [Mo)In_x such that

£_r+osc {0^) is given by the sum of E_x (MO) (Eq. (15.42)) and E_osc (Eq. (15.51)):

(15.52).

70. The system of claim 69, 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, Eφasis energies) , the change in the energy of the AOs or HOs upon forming the bond ( E₁. (atom - atom, msp³.Aθ) ), the energy of oscillation in the transition state, and the change in magnetic energy with bond formation,

71. The system of claim 70, wherein from Eq. (15.52), the total energy of the group E₇. (βmup) is

72. The system of claim 71, wherein 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

where r³ is the radius of the atom that reacts to form the bond and c₃ is the number of electron pairs.

(15.55).

73. The system of claim 72, wherein the total bond energy of the group E_D {GW_W) 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 c_iE_jllWal (c₄ AOI HO) and c₅E_jmUal (c₅ AOIHO) :

(15.56).

74. The system of claim 70, wherein in the case of organic molecules, the atoms of the functional groups are energy matched to the C2sp³ HO such that

E(AOIHO) = -14.63489 eV (15.57) For examples of E_mag from previous sections:

75. The system of claim 74, 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:

(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, E_Cmlonb (MO.atom,msp^{3 >}J

given by Eqs. (15.19) and (15.31-15.32). For

>l3.605804 eV :

(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) given by Eqs. (15.20) and (15.31-15.32). For

(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

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

given by Eqs. (15.20) and (15.31-15.32). For

(v) the ratio that is less than one of the magnitude of the valence-level energies, E_n(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\ 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 (w) 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:

(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).

76. The system of claim 75, wherein specific examples of the factors C₂ and C₂ of a H₂ -type ellipsoidal MO of Eq. (15.51) given in following sections are

wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to ± 10%, if desired.

77. 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:

78. The system of claim 77, wherein the radius of the A shell is r_Λ , and the polar radial coordinate of the ellipse and the radius of the A shell are equal at the point of intersection such that

79. The system of claim 78, wherein the polar angle θ ' at the intersection point is given by

80. The system of claim 79, wherein the angle Θ_{A AO} the radial vector of the A AO makes with the internuclear axis is .

81. he 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 - Θ_{H 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:

such that

82. The system of claim 81, wherein the distanc ong the internuclear axis from the

origin of H₂ -type ellipsoidal MO to the point of intersection of the orbitals is given by

83. The system of claim 82, wherein the distance d_{A Λ0} along the internuclear axis from the origin of the A atom to the point of intersection of the orbitals is given by

84. 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.

85. The system of claim 84, 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:

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).

86. The system of claim 85, wherein the distance from the origin of the MO to each focus c ' of the A-B ellipsoidal MO is given by:

87. The system of claim 85, wherein the internuclear distance is

88. The system of claim 85, wherein the length of the semiminor axis of the prolate spheroidal A-B MO b ~ c is given by Eq. (15.4).

89. The system of claim 85, wherein the component energies and the total energy, E₁. (H₂M6) , 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 c_B0 , the bond-order factor which is 1 for a single bond and when the MO comprises W₁ equivalent single bonds as in the case of functional groups. c_B0 is 4 for an independent double bond as in the case of the CO₂ and NO₂ molecules.

90. The system of claim 85, wherein 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.

91. The system of claim 85, 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.

92. The system of claim 84, wherein when A-B comprises atoms other than H,

E₇. [atom - atom, msp¹.AO} , the energy component due to the AO or ΗO's charge donation to the terminal MO, is added to the other energy terms to give E_τ [H₁Mo) :

93. The system of claim 84, the radiation reaction force in the case of the vibration of A -B in the transition state corresponds to the Doppler energy, E_D , given by Eq. (11.181) that is dependent on the motion of the electrons and the nuclei.

94. The system of claim 93, wherein the total energy that includes the radiation reaction of the A-B MO is given by the sum of E_T (H₂MO) (Eq. (15.82)) and E_osc given Eqs. (11.213- 11.220), (11.231-11.236), and (11.239-11.240).

95. The system of claim 94, wherein the total energy E_Γ (A -B) ofthe A-B MO including the Doppler term is

(15.83) where C_λo is the fraction ofthe H₂ -type ellipsoidal MO basis function ofthe oscillatory transition state ofthe 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 ofthe transition state ofthe chemical bond, and

_ Tn₁Tn₂

M = is the reduced mass ofthe nuclei given by Eq. (11.154).

Tn₁ +m₂

96. The system of claim 95, wherein to match the boundary condition that the total energy of the A-B ellipsoidal MO is zero, E_T (A-B) given by Eq. (15.83) is set equal to zero and substitution of Eq. (15.81) into Eq. (15.83) gives

97. The system of claim 96, 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)).

98. The system of claim 96, wherein the electron-central-field force and its derivative are given by

and

99. The system of claim 97, wherein the nuclear repulsion force and its derivative are given by

and

such that the angular frequency of the oscillation is given by

100. The system of claim 99, wherein since both terms of E_∞c = E_D + E_KvΛ 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

101. The system of claim 100, 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.

102. The system of claim 101, wherein Eq. (15.90) is solved by the reiterative technique using a computer.

103. The system of claim 100, wherein a factor c₂ of a given atom in the determination of for calculating the zero of the total A- B bond energy is given by Eqs. (15.62-15.65).

104. The system of claim 103, wherein in the case of a H -H terminal bond of an alkyl or alkenyl group, c₂' is the ratio of C₁ ofEq. (15.62) for the H-H bond which is one and c₂ of the carbon of the corresponding C- H bond:

wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.

105. The system of claim 100, wherein in the case of the determination of the bond angle of the ACH 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:

106. The system of claim 100, 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 , 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 a proton 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.

107. The system of claim 100, wherein in the determination of the hybridization factor c₂ of Eq. (15.90) from Eqs. (15.62-15.70), the Coulombic energy, E_Coulomb (MO.atom,msp³ ) , or the energy, E [MO.atom, msp³ ) , the radius r_A__{B AorBs >} ⁰^ ^^e A ox 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 E₇^ (MO, 2sp³ ) , the total energy donation to each bond with which it participates in bonding as it forms the terminal bond.

108. The system of claim 107, wherein the Coulombic energy E_Coulomb (MO.atom,msp³} of the outer electron of the atom msp³ shell is given by Eq. (15.19).

109. The system of claim 108, 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 E_Coulcmb \MO.atom,msp³) and E(magnetic) (Eq. (15.20)).

110. The system of claim 99, 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 ener

the outer electron of the C2sp³ shell given by Eq. (15.19) with the radius r_{r ^}, ₃ of each C2sp³ HO of the terminal C-C bond calculated using Eq. (15.32) by considerin 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 E_τ (atom -atom, msp³. Aθ) in Eq. (15.90) is

E_T (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.

111. The system of claim 100, 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:

In the exemplary cases of C - C , O -O, and N-N where C is carbon:

Lithe exemplary cases of C-N , C-O, and C-S,

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.

112. The system of claim 97, wherein the corresponding E₇ (atom - atom, msp³.Aθ) 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

va ues an constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.

113. The system of claim 100, wherein in the case that the terminal bond is X - X where X is a halogen atom, C₁ is one, and

is the average (Eq. (15.93)) of the hybridization factors of the participating halogen atoms given by Eqs. (15.62-15.63) where

114. The system of claim 113, wherein the factor C₁ of Eq. (15.90) is one for all halogen atoms.

115. The system of claim 113, 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.

116. The system of claim 113, wherein for each of the halogens, Cl , Br , and I , 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 C2sp* 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.

117. The system of claim 116, wherein E₇, (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.

118. The system of claim 100, 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 C₁ and C₁ of Eq. (15.90) are one for all halogen atoms.

119. The system of claim 118, 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 Clsp' 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.

120. The system of claim 119, 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.

121. The system of claim 119, 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.

122. The system of claim 121, wherein the C₂ of the terminal-atom bond matches that used to determine the energies of the corresponding C-X -bond MO.

123. The system of claim 122, 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 C2sp^% 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.

124. The system of claim 123, wherein E_τ

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.

125. The system of claim 119, 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.

126. The system of claim 119, 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.

127. The system of claim 126, 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.

128. The system of claim 127, 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 C2sp³ HO.

129. The system of claim 128, 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 .

130. The system of claim 100, 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 - 2,?₁s₂cosine6> = S₃ ² (15.96).

131. The system of claim 130, wherein with S₁ = 2c_c' __A , the internuclear distance of the C-A bond, S₂ = 2c_c'__B , the internuclear distance of each C -B bond, and S₃ = 2c \__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

no

132. The system of claim 131, wherein the structure C_bC_a{O_a)O_b wherein C_a is bound to C_b , O_a , and O_b , the three bonds are coplanar and two of the angles are known, say O₁ and O₂ , then the third θ₃ can be determined geometrically:

133. The system of claim 131, wherein in the general case that two of the three coplanar bonds are equivalent and one of the angles is known, say θ_λ , then the second and third can be determined geometrically:

134. 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.

135. The system of claim 134, wherein the C_3v 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 d_orlgln__B from the origin to the nucleus of a terminal B atom is given by

the height along the z-axis from the origin to the A nucleus d_hejghl is given by

the angle θ_v of each A-B bond from the z-axis is given by

136. The system of claim 135, 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 0^₀ given by Eq. (14.206) is

137. The system of claim 134, 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 etween 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 and the distances between the A , B , and C atoms.

138. The system of claim 137, wherein the distance d_λ along the bisector o from

C to the internuclear-distance line between A and is given by

where 2c '_C__Λ is the internuclear distance between A and C .

139. The system of claim 138, wherein the atoms A , A , and B define the base of a pyramid.

140. The system of claim 139, wherein the pyramidal angle Θ_ZABA can be solved from the internuclear distances between A an

nd between A and B sing

the law of cosines (Eq. (15.98)):

141. The system of claim 140, wherein the distance d₂ along the bisector of Θ_ZABA from B to the internuclear-distance line given by

142. The system of claim 141, wherein the lengths d_x , d₂ , and 2c'_c__B define a triangle wherein the angle between d_x and the internuclear distance between B and C , 2c'_c__B , is the dihedral angle Θ_ΔBCIACA that can be solved using the law of cosines (Eq. (15.98)):

143. 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

Ηydrogen-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-S- S^{iven b}Y 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.

144. They system of claim 143, wherein the bond energy is not equal to the component energy of each bond as it exists in the specie, although, they are close.

145. The system of claim 143, wherein the total energy of each group is its contribution to the total energy of the specie as a whole.

146. The system of claim 143, 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.

147. The system of claim 143, 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.

148. The system of claim 143, 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)).

149. The system of claim 143, wherein the intercept angles are determined from Eqs. (15.71-15.78) using the final radius of the HO of each atom.

150. The system of claim 143, 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.

151. The system of claim 150, 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.

152. The system of claim 151, 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.

153. The system of claim 150, wherein the donation of electron density to the AOs and HOs reduces the energy.

154. The system of claim 143, 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.

155. 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.

156. The system of claim 1 , wherein the new stable compositions of matter are predicted as well as the structures of combinatorial chemistry reactions.

157. 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.

158. 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.

159. 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.

160. The system of claim 159, the fields and interactions can be determined using a finite- element-analysis approach of Maxwell's equations.

161. 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 H -type ellipsoidal MO divided between the C2sp³ ΗOs:

162. The system of claim 162, 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.

163. The system of claim 163, 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.

164. The system of claim 161, wherein hybridization with 25% electron donation to each C = C -bond gives rise to the C_ben__em2sp³ HO-shell Coulombic energy

Eq I. ( \14.245) /.

165. The system of claim 161, wherein to meet the equipotential condition of the union of

3e the six C2sp³ 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 E_Cmlomb (C_bemem,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 (benzeneC2sp³Hθ)= c₂ (benzeneC2sp³ Hθ)= ¹³-^{605804 gF} ₌ 0.85252 (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

166. The system of claim 161, 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

167. The system of claim 161, 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).

168. The system of claim 161, wherein the total energy of the bonds of the eighteen

electrons of the C=C bonds of benzene, E₇.

is given by (6)(θ.75) times

3e

E_τ+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[£!,2sp³) (Eq. (14.146)), the initial

energy of each C2sp³ HO of each C that forms the C=C bonds of bond order two.

3e

169. The system of claim 161, wherein the total energy of the, six C=C bonds of benzene with three electron per aromatic bond given by Eq. (14.493) is

E_τ [c₆H₆,C

= (ό)(θ.75)(-66.O5796 eV)- is(-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.

170. The system of claim 161, wherein the results of benzene can be generalized to the class of aromatic and heterocyclic compounds. E_hv of an aromatic bond is given by E_τ (H₂) (EqS. (11.212) and (14.486)), the maximum total energy of each H₂-type M0 such that

wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.

171. The system of claim 170, 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).

172. The system of claim 171, wherein the multiplication of the total energy given by Eq. (15.55) by f_χ = 0.75 with the substitution of Eq. (15.145) gives the total energy of the aromatic bond:

wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.

173. The system of claim 161, wherein the total bond energy of the aromatic group

E_D (ρwup) 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 c_AE_jnjljal (c_A AO I Hθ\ and

HO /):

(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.

174. The system of claim 161, 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 V_e , V , T , V_m , and E_τ are the same

as those of the hydrogen carbide radical, except that E₁, (c = C,2sp³j= -1.13379 eV (Eq. (14.247)) is subtracted from E₇, (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.

175. The system of claim 174, 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.AO J= -1.13379 eV wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.

176. The system of claim 174, wherein the total energy of the benzene C- H -bond MO, E₇, [C - H] , given by Eq. (14.467) is the sum of 0.5E (C =

the energy benzene V 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 E₇, (CH\ the σ

MO contribution given by Eq. (14.441). hi the corresponding generalization of the aromatic CH group, the energy parameters are determined using Eqs. (15.146-15.147)

with Z₁ = 1 and E_τ (atom - atom,msp³.AOj= — ^'■ , wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.

177. The system of claim 176, 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 = lin Eqs. (15.18-15.21), and further gives a minimum energy for the molecule.

3e

178. The system of claim 177, wherein the 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 E_mag given by Eq. (15.58).

179. The system of claim 178, wherein each of the C- H bonds of benzene comprises two electrons according to Eq. (14.439).

180. The system of claim 179, wherein from the energy of each C- H bond,

(Eq. (14.477)), the total energy of the twelve electrons of the six C - H bonds of benzene, E_τ (C₆H₆,C - H), given by Eq. (14.494) is

E₁ (C₆H₆,C- H)= (β)(-E_Dt (¹²CH))= 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.

181. The system of claim 174, wherein the total bond dissociation energy of benzene,

E_D (C₆H₆ ), given by Eq. (14.495) is the negative sum of E₇, 1 C₆H₆,C=C I (Eq.

(14.493)) and E_τ (C₆H_&,C - H) (Eq. (14.494)):

= -((-33.83284 eV)+ (-23.42724 evty (15.149)

= 57.2601 eV wherein the calculated and measured values and constants recited in the equations herein can be adjusted, for example, up to + 10%, if desired.

182. The system of claim 181, 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.

183. 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.

184. The system of claim 183, further comprising a data transfer system for inputting numerical data into or out of a computational components and storage components of the main system.

185. The system of claim 183, further comprising a spreadsheet containing solutions of the bond parameters with output in a standard spreadsheet format.

186. The system of claim 185, further comprising a data-handling program to transfer data from the spreadsheets into the main program.

187. The system of claim 183, wherein output may be at least one of graphical, simulation, text, and numerical data.

188. The system of claim 187, 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

189. The system of claim 183, wherein the charge, current, energy, and geometrical parameters are output to be inputs to other programs that can be used in further applications.

190. The system of claim 189, 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.

191. The system of claim 190, 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.

192. The system of claim 190, 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.

193. The system of claim 192, 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.

194. The system of claim 183, 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.

195. The system of claim 183, 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.

196. A method of using any of the systems of claims 1 - 195.

197. A use of any system of claims 1 - 195.

198. 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.

199. A composition of matter according to claim 198, wherein the novel property is a new pharmaceutical use.

200. A composition of matter according to claim 198 that is novel, wherein the novel property is stability at room temperature of a new arrangement of atoms or ions.

201. 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.

202. 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 electrons) does not radiate under acceleration.

203. The system of claim 202, further comprising discovering a new composition of matter.

204. A system of claim 1-195, wherein the calculated and measured values and constants recited therein can be adjusted up to + 10% as desired.