System and method of computing and rendering the nature of dipole moments, condensed matter, and reaction kinetics

Abstract

A method and system of physically solving the charge, mass, and current density functions of organic molecules using Maxwell's equations and computing and rendering the physical nature of the chemical bond using the solutions. The solutions can be used to solve the dipole moments in molecules or induced dipole moments between species that in turn can be used to solve condensed matter parameters and reaction kinetics. 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 species can permit the solution and display of those of other species to provide utility to anticipate their reactivity and physical properties.

Description

FIELD OF THE INVENTION

This invention relates to a system and method of physically solving the charge, mass, current density functions, and dipole moments of polyatomic molecules, polyatomic molecular ions, diatomic molecules, molecular radicals, molecular ions, or any portion of these species including in condensed matter and undergoing reaction, 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 compositions of matter.

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, a classical solution to the bound electron is derived 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 physics (CP) was derived from first principles as reported previously [reference Nos. 1-13] 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 Schrödinger 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 structure of the electron is derived as a boundary-value problem wherein the electron comprises the source current of time-varying electromagnetic fields during transitions with the constraint that the bound n=1 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 and Maxwell 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. 2005/0209788A1, 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.

Applicant's previously filed WO2007/051078 discloses a method and system of physically solving the charge, mass, and current density functions of polyatomic molecules and polyatomic molecular 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. This incorporated application discloses complete flow charts and written description of a computer program and systems that can be modified using the novel equations and description below to physically solve the charge, mass, and current density functions of the specific groups of molecules and molecular ions disclosed herein and computing and rendering the nature of the specific groups of molecules and molecular ions disclosed herein.

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. Rather, atomic and molecular physics theory, derived from first principles, must successfully and consistently apply physical laws on all scales [1-13]. Stability to radiation was ignored by all past atomic models, but in this case, it is the basis of the solutions wherein the structure of the electron is first solved and the result determines the nature of the atomic and molecular electrons involved in chemical bonds. Historically, the point at which quantum mechanics 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 [1-13]. Later physics was replaced by “pure mathematics” based on the notion of the inexplicable wave-particle duality nature of electrons which lead to the Schrödinger equation wherein the consequences of radiation predicted by Maxwell's equations were ignored. Ironically, Bohr, Schrödinger, 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 [14]. 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 [1-18]. Consequently, Feynman for example, attempted to use first principles including Maxwell's equations to discover new physics to replace quantum mechanics [19].

Starting with the same essential physics as Bohr, Schrödinger, and Dirac of e₋ moving in the Coulombic field of the proton and an electromagnetic wave equation and matching electron source current rather than an energy diffusion equation originally sought by Schrödinger, advancements in the understanding of the stability of the bound electron to radiation are applied to solve for the exact nature of the electron. Rather than using the postulated Schrödinger 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 structure of the electron is derived as a boundary-value problem wherein the electron comprises the source current of time-varying electromagnetic fields during transitions with the constraint that the bound n=1 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. The physical boundary condition of nonradiation of that was imposed on the bound electron follows from a derivation by Haus [20]. 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. A simple invariant physical model arises naturally wherein the results are extremely straightforward, internally consistent, and predictive of conjugate parameters for the first time, requiring minimal math as in the case of the most famous exact equations (no uncertainty) of Newton and Maxwell on which the model is based. No new physics is needed; only the known physical laws based on direct observation are used.

The structure of the bound atomic electron was solved by first considering one-electron atoms [1-13]. Since the hydrogen atom is stable and nonradiative, the electron has constant energy. Furthermore, it is time dynamic with a corresponding current that serves as a source of electromagnetic radiation during transitions. The wave equation solutions of the radiation fields permit the source currents to be determined as a boundary-value problem. These source currents match the field solutions of the wave equation for two dimensions plus time when the nonradiation condition is applied. Then, the mechanics of the electron can be solved from the two-dimensional wave equation plus time in the form of an energy equation wherein it provides for conservation of energy and angular momentum as given in the Electron Mechanics and the Corresponding Classical Wave Equation for the Derivation of the Rotational Parameters of the Electron section of Ref. [1]. Once the nature of the electron is solved, all problems involving electrons can be solved in principle. Thus, in the case of one-electron atoms, the electron radius, binding energy, and other parameters are solved after solving for the nature of the bound electron.

For time-varying spherical electromagnetic fields, Jackson [21] gives a generalized expansion in vector spherical waves that are convenient for electromagnetic boundary-value problems possessing spherical symmetry properties and for analyzing multipole radiation from a localized source distribution. The Green function G(x′,x) which is appropriate to the equation(∇²+k²)G(x′,x)=−δ(x′−x)  (1)in the infinite domain with the spherical wave expansion for the outgoing wave Green function is

$\begin{array}{cc}G\left(x^{\prime },x\right)=\frac{e^{-ikx-x^{\prime }}}{4\pi x-x^{\prime }}=ik\sum _{l=0}^{\infty }{j}_l\left({\mathrm{kr}}_{<}\right)h_l^{\left(1\right)}\left({\mathrm{kr}}_{<}\right)\sum _{m=-l}^l{Y}_{l,m}^{*}\left({\theta }^{\prime },{\varphi }^{\prime }\right)Y_{l,m}\left(\theta ,\varphi \right)& \left(2\right)\end{array}$ Jackson [21] further gives the general multipole field solution to Maxwell's equations in a source-free region of empty space with the assumption of a time dependence e^iωnt:

$\begin{array}{cc}B=\sum _{l,m}\left[a_E\left(l,m\right)f_l\left(\mathrm{kr}\right)X_{l,m}-\frac{i}{k}{a}_M\left(l,m\right)\nabla \times g_l\left(\mathrm{kr}\right)X_{l,m}\right]E=\sum _{l,m}\left[\frac{i}{k}{a}_E\left(l,m\right)\nabla \times f_l\left(\mathrm{kr}\right)X_{l,m}+a_M\left(l,m\right)g_l\left(\mathrm{kr}\right)X_{l,m}\right]& \left(3\right)\end{array}$ where the cgs units used by Jackson are retained in this section. The radial functions f_t(kr) and g_t(kr) are of the form:g_t(kr)=A_t⁽¹⁾h_t⁽¹⁾+A_t⁽²⁾h_t⁽²⁾  (4)X_t.m is the vector spherical harmonic defined by

$\begin{array}{cc}{X}_{l,m}\left(\theta ,\varphi \right)=\frac{1}{\sqrt{l\left(l+1\right)}}{\mathrm{LY}}_{l,m}\left(\theta ,\varphi \right)\mathrm{where}& \left(5\right)\\ L=\frac{1}{i}\left(r\times \nabla \right)& \left(6\right)\end{array}$ The coefficients a_E(l,m) and a_M(l,m) of Eq. (3) specify the amounts of electric (l,m) multipole and magnetic (l,m) multipole fields, and are determined by sources and boundary conditions as are the relative proportions in Eq. (4). Jackson gives the result of the electric and magnetic coefficients from the sources as

$\begin{array}{cc}{a}_E\left(l,m\right)=\frac{4\pi k^2}{i\sqrt{l\left(l+1\right)}}\int Y_l^{m*}\left\{\begin{array}{c}\begin{array}{c}\rho \frac{\partial }{\partial r}\left[{\mathrm{rj}}_l\left(\mathrm{kr}\right)\right]+\\ \frac{\mathrm{ik}}{c}\left(r·J\right)j_l\left(\mathrm{kr}\right)-\end{array}\\ \mathrm{ik}\nabla ·\left(r\times M\right)j_l\left(\mathrm{kr}\right)\end{array}\right\}{d}^{3}x\mathrm{and}& \left(7\right)\\ a_M\left(l,m\right)=\frac{-4\pi k^2}{\sqrt{l\left(l+1\right)}}\int j_l\left(\mathrm{kr}\right)Y_l^{m*}L·\left(\frac{J}{c}+\nabla \times M\right)d^{3}x& \left(8\right)\end{array}$ respectively, where the distribution of charge ρ(x,t), current J(x,t), and intrinsic magnetization M(x,t) are harmonically varying sources: ρ(x)e^−iωt, J(x)e^−iωt, and M(x)e^−iωt.

The electron current-density function can be solved as a boundary value problem regarding the time varying corresponding source current J(x)e^−iωt that gives rise to the time-varying spherical electromagnetic fields during transitions between states with the further constraint that the electron is nonradiative in a state defined as the n=1 state. The potential energy, V(r), is an inverse-radius-squared relationship given by given by Gauss' law which for a point charge or a two-dimensional spherical shell at a distance r from the nucleus the potential is

$\begin{array}{cc}V\left(r\right)=-\frac{e^2}{4{\mathrm{\pi \varepsilon }}_{0}r}& \left(9\right)\end{array}$ Thus, consideration of conservation of energy would require that the electron radius must be fixed. Addition constraints requiring a two-dimensional source current of fixed radius are matching the delta function of Eq. (1) with no singularity, no time dependence and consequently no radiation, absence of self-interaction (See Appendix III of Ref. [1]), and exact electroneutrality of the hydrogen atom wherein the electric field is given by

$\begin{array}{cc}n·\left(E_1-E_2\right)=\frac{{\sigma }_s}{{\varepsilon }_0}& \left(10\right)\end{array}$ where n is the normal unit vector, E₁ and E₂ are the electric field vectors that are discontinuous at the opposite surfaces, σ_x is the discontinuous two-dimensional surface charge density, and E₂=0. Then, the solution for the radial electron function, which satisfies the boundary conditions is a delta function in spherical coordinates—a spherical shell [22]

$\begin{array}{cc}f\left(r\right)=\frac{1}{r^2}\delta \left(r-r_n\right)& \left(11\right)\end{array}$ where r_n is an allowed radius. This function defines the charge density on a spherical shell of a fixed radius (See FIG. 1), not yet determined, with the charge motion confined to the two-dimensional spherical surface. The integer subscript n is determined during photon absorption as given in the Excited States of the One-Electron Atom (Quantization) section of Ref. [1]. It is shown in this section that the force balance between the electric fields of the electron and proton plus any resonantly absorbed photons gives the result that r_n=nr₁ wherein n is an integer in an excited state.

Given time harmonic motion and a radial delta function, the relationship between an allowed radius and the electron wavelength is given by2πr_n=λ_n  (12)Based on conservation of the electron's angular momentum of, the magnitude of the velocity and the angular frequency for every point on the surface of the bound electron are

$\begin{array}{cc}{v}_n=\frac{h}{m_e{\lambda }_n}=\frac{h}{m_{e}2\pi r_n}=\frac{m_e{r}_n}{}& \left(13\right)\\ {\omega }_n=\frac{m_e{r}_n^2}{}& \left(14\right)\end{array}$ To further match the required multipole electromagnetic fields between transitions of states, the trial nonradiative source current functions are time and spherical harmonics, each having an exact radius and an exact energy. Then, each allowed electron charge-density (mass-density) function is the product of a radial delta function

$\left(f\left(r\right)=\frac{1}{r^2}\delta \left(r-r_n\right)\right),$ two angular functions (spherical harmonic functions Y_t^m(θ,φ)=P_l^m(cos θ)e^imφ), and a time-harmonic function e^imωnt. The spherical harmonic Y₀⁰(θ,φ)=1 is also an allowed solution that is in fact required in order for the electron charge and mass densities to be positive definite and to give rise to the phenomena of electron spin. The real parts of the spherical harmonics vary between −1 and 1. But the mass of the electron cannot be negative; and the charge cannot be positive. Thus, to insure that the function is positive definite, the form of the angular solution must be a superposition:Y₀⁰(θ,φ)+Y_l^m(θ,φ)  (15)The current is constant at every point on the surface for the s orbital corresponding to Y₀⁰(θ,φ). The quantum numbers of the spherical harmonic currents can be related to the observed electron orbital angular momentum states. The currents corresponding to s, p, d, f, etc. orbitals are

$\begin{array}{cc}l=0\rho \left(r,\theta ,\varphi ,t\right)=\frac{e}{8\pi r^2}\left[\delta \left(r-r_n\right)\right]\left[Y_0^0\left(\theta ,\varphi \right)+Y_l^m\left(\theta ,\varphi \right)\right]& \left(16\right)\\ l\ne 0\rho \left(r,\theta ,\varphi ,t\right)=\frac{e}{4\pi r^2}\left[\delta \left(r-r_n\right)\right]\left[\begin{array}{c}{Y}_0^0\left(\theta ,\varphi \right)+\\ \mathrm{Re}\left\{Y_l^m\left(\theta ,\varphi \right)e^{i{\mathrm{m\omega }}_{n}t}\right\}\end{array}\right]& \left(17\right)\end{array}$ where Y_l^m(θ,φ) are the spherical harmonic functions that spin about the z-axis with angular frequency ω_n with Y₀⁰(θ,φ) the constant function and Re{Y_l^m(θ,φ)e^inωnt}=P_l^m(cos θ)cos(mφ+mω_nt).

The Fourier transform of the electron charge-density function is a solution of the four-dimensional wave equation in frequency space (k, ω-space). Then the corresponding Fourier transform of the current-density function K(s,Θ,Φ,ω) is given by multiplying by the constant angular frequency corresponding to a potentially emitted photon.

$\begin{array}{cc}K\left(s,\Theta ,\Phi ,\omega \right)=4{\mathrm{\pi \omega }}_n\frac{\sin \left(2s_n{r}_n\right)}{2s_n{r}_n}\otimes 2\pi \stackrel{\infty }{\sum _{\upsilon =1}}\frac{{\left(-1\right)}^{\upsilon -1}{\left(\pi \sin \Theta \right)}^{2\left(\upsilon -1\right)}}{\left(\upsilon -1\right)!\left(\upsilon -1\right)!}\frac{\Gamma \left(\frac{1}{2}\right)\Gamma \left(\upsilon +\frac{1}{2}\right)}{{\left(\pi \cos \Theta \right)}^{2\upsilon +1}{2}^{\upsilon +1}}\frac{2\upsilon !}{\left(\upsilon -1\right)!}{s}^{-2\upsilon }\otimes 2\pi \sum _{\upsilon =1}^{\infty }\frac{{\left(-1\right)}^{\upsilon -1}{\left(\pi \sin \Phi \right)}^{2\left(\upsilon -1\right)}}{\left(\upsilon -1\right)!\left(\upsilon -1\right)!}\frac{\Gamma \left(\frac{1}{2}\right)\Gamma \left(\upsilon +\frac{1}{2}\right)}{{\left(\pi \cos \Phi \right)}^{2\upsilon +1}{2}^{\upsilon +1}}\frac{2\upsilon !}{\left(\upsilon -1\right)!}{s}^{-2\upsilon }\frac{1}{4\pi }\left[\delta \left(\omega -{\omega }_n\right)+\delta \left(\omega +{\omega }_n\right)\right]& \left(18\right)\end{array}$ The motion on the orbitsphere is angular; however, a radial correction exists due to special relativistic effects. When the velocity is c corresponding to a potentially emitted photons_n·v_n=s_n·c=ω_n  (19)the relativistically corrected wavelength is (Eq. (1.247) of Ref. [1])r_n=λ_n  (20)Substitution of Eq. (20) into the sinc function results in the vanishing of the entire Fourier transform of the current-density function. Thus, spacetime harmonics of

$\frac{{\omega }_n}{c}=k\mathrm{or}\frac{{\omega }_n}{c}\sqrt{\frac{\varepsilon }{{\varepsilon }_o}}=k$ for which the Fourier transform of the current-density function is nonzero do not exist. Radiation due to charge motion does not occur in any medium when this boundary condition is met. There is acceleration without radiation. (Also see Abbott and Griffiths and Goedecke [23-24]). Nonradiation is also shown directly using Maxwell's equations directly in Appendix 1 of Ref. [1]. However, in the case that such a state arises as an excited state by photon absorption, it is radiative due to a radial dipole term in its current-density function since it possesses spacetime Fourier transform components synchronous with waves traveling at the speed of light as shown in the Instability of Excited States section of Ref. [1]. The radiation emitted or absorbed during electron transitions is the multipole radiation given by Eq. (2) as given in the Excited States of the One-Electron Atom (Quantization) section and the Equation of the Photon section of Ref. [1] wherein Eqs. (4.18-4.23) give a macro-spherical wave in the far-field.

In Chapter 1 of Ref. [1], the uniform current density function Y₀⁰(θ,φ) (Eqs. (16-17)) that gives rise to the spin of the electron is generated from two current-vector fields (CVFs). Each CVF comprises a continuum of correlated orthogonal great circle current-density elements (one dimensional “current loops”). The current pattern comprising each CVF is generated over a half-sphere surface by a set of rotations of two orthogonal great circle current loops that serve as basis elements about each of the

$\left(-i_x,i_y,0i_z\right)\mathrm{and}\left(-\frac{1}{\sqrt{2}}{i}_x,\frac{1}{\sqrt{2}}{i}_y,i_z\right)-\mathrm{axis};$ the span being π radians. Then, the two CVFs are convoluted, and the result is normalized to exactly generate the continuous uniform electron current density function Y₀⁰(θ,φ) covering a spherical shell and having the three angular momentum components of

$L_{\mathrm{xy}}=+/-\frac{\hslash }{4}\left(+/-\mathrm{designates}\mathrm{both}\mathrm{the}\mathrm{positive}\mathrm{and}\mathrm{negative}\mathrm{vector}\mathrm{directions}\mathrm{along}\mathrm{an}\mathrm{axis}\mathrm{in}\mathrm{the}\mathrm{xy}\text{-}\mathrm{plane}\right)\mathrm{and}{L}_z=\frac{\hslash }{2}.$ The z-axis view of a representation of the total current pattern of the Y₀⁰(θ,φ) orbitsphere comprising the superposition of 144 current elements is shown in FIG. 2A. As the number of great circles goes to infinity the current distribution becomes continuous and is exactly uniform following normalization. A representation of the

$\left(-\frac{1}{\sqrt{2}}{i}_x,\frac{1}{\sqrt{2}}{i}_y,i_z\right)\text{-}\mathrm{axis}$ view of the total uniform current-density pattern of the Y₀⁰(φ,θ) orbitsphere with 144 vectors overlaid on the continuous bound-electron current density giving the direction of the current of each great circle element is shown in FIG. 2B. This superconducting current pattern is confined to two spatial dimensions.

Thus, a bound electron is a constant two-dimensional spherical surface of charge (zero thickness and total charge=−e), called an electron orbitsphere that can exist in a bound state at only specified distances from the nucleus determined by an energy minimum for the n=1 state and integer multiples of this radius due to the action of resonant photons as shown in the Determination of Orbitsphere Radii section and Excited States of the One-Electron Atom (Quantization) section of Ref. [1], respectively. The bound electron is not a point, but it is point-like (behaves like a point at the origin). The free electron is continuous with the bound electron as it is ionized and is also point-like as shown in the Electron in Free Space section of Ref. [1]. The total function that describes the spinning motion of each electron orbitsphere is composed of two functions. One function, the spin function (see FIG. 1 for the charge function and FIG. 2 for the current function), is spatially uniform over the orbitsphere, where each point moves on the surface with the same quantized angular and linear velocity, and gives rise to spin angular momentum. It corresponds to the nonradiative n=1, l=0 state of atomic hydrogen which is well known as an s state or orbital. The other function, the modulation function, can be spatially uniform—in which case there is no orbital angular momentum and the magnetic moment of the electron orbitsphere is one Bohr magneton—or not spatially uniform—in which case there is orbital angular momentum. The modulation function rotates with a quantized angular velocity about a specific (by convention) z-axis. The constant spin function that is modulated by a time and spherical harmonic function as given by Eq. (17) is shown in FIG. 3 for several t values. The modulation or traveling charge-density wave that corresponds to an orbital angular momentum in addition to a spin angular momentum are typically referred to as p, d, f, etc. orbitals and correspond to an l quantum number not equal to zero.

It was shown previously [1-13] that classical physics 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, 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×p, can be applied directly to the wavefunction (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 agreement between observations and predictions based on closed-form equations with fundamental constants only matches to the limit permitted by the error in the measured fundamental constants.

In contrast to the failure of the Bohr theory and the nonphysical, unpredictive, adjustable-parameter approach of quantum mechanics, multielectron atoms [1, 5] and the nature of the chemical bond [1, 6] are given by exact closed-form solutions containing fundamental constants only. Using the nonradiative electron current-density functions, the radii are determined from the force balance of the electric, magnetic, and centrifugal forces that correspond to the minimum of energy of the atomic or ionic 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 available from the internet [25]. For 400 atoms and ions the agreement between the predicted and experimental results are remarkable [5]. Here I extend these results to the nature of the chemical bond. In this regard, quantum mechanics has historically sought the lowest energy of the molecular system, but this is trivially the case of the electrons inside the nuclei. Obviously, the electrons must obey additional physical laws since matter does not exist in a state with the electrons collapsed into the nuclei. Specifically, molecular bonding is due to the physics of Newton's and Maxwell's laws together with achieving an energy minimum.

The structure of the bound molecular electron was solved by first considering the one-electron molecule H₂⁺ and then the simplest molecule H₂ [1, 6]. The nature of the chemical bond was solved in the same fashion as that of the bound atomic electron. First principles including stability to radiation requires that the electron charge of the molecular orbital is a prolate spheroid, a solution of the Laplacian as an equipotential minimum energy surface in the natural ellipsoidal coordinates compared to spheroidal in the atomic case, and the current is time harmonic and obeys Newton's laws of mechanics in the central field of the nuclei at the foci of the spheroid. There is no a priori reason why the electron position must be a solution of the three-dimensional wave equation plus time and cannot comprise source currents of electromagnetic waves that are solutions of the three-dimensional wave equation plus time. Then, the special case of nonradiation determines that the current functions are confined to two-spatial dimensions plus time and match the electromagnetic wave-equation solutions for these dimensions. In addition to the important result of stability to radiation, several more very important physical results are subsequently realized: (i) The charge is distributed on a two-dimension surface; thus, there are no infinities in the corresponding fields (Eq. (10)). Infinite fields are simply renormalized in the case of the point-particles of quantum mechanics, but it is physically gratifying that none arise in this case since infinite fields have never been measured or realized in the laboratory. (ii) The hydrogen molecular ion or molecule has finite dimensions rather than extending over all space. From measurements of the resistivity of hydrogen as a function of pressure, the finite dimensions of the hydrogen molecule are evident in the plateau of the resistivity versus pressure curve of metallic hydrogen [26]. This is in contradiction to the predictions of quantum probability functions such as an exponential radial distribution in space. Furthermore, despite the predictions of quantum mechanics that preclude the imaging of a molecule orbital, the full three-dimensional structure of the outer molecular orbital of N₂ has been recently tomographically reconstructed [27]. The charge-density surface observed is similar to that shown in FIG. 4 for H₂ which is direct evidence that MO's electrons are not point-particle probability waves that have no form until they are “collapsed to a point” by measurement. Rather they are physical, two-dimensional equipotential charge density functions as derived herein. (iii) Consistent with experiments, neutral scattering is predicted without violation of special relativity and causality wherein a point must be everywhere at once as required in the QM case. (iv) There is no electron self-interaction. The continuous charge-density function is a two-dimensional equipotential energy surface with an electric field that is strictly normal for the elliptic parameter ξ>0 according to Gauss' law and Faraday's law. The relationship between the electric field equation and the electron source charge-density function is given by Maxwell's equation in two dimensions [28,29] (Eq. (10)). This relation shows that only a two-dimensional geometry meets the criterion for a fundamental particle. This is the nonsingularity geometry that is no longer divisible. It is the dimension from which it is not possible to lower dimensionality. In this case, there is no electrostatic self-interaction since the corresponding potential is continuous across the surface according to Faraday's law in the electrostatic limit, and the field is discontinuous, normal to the charge according to Gauss' law [28-30]. (v) The instability of electron-electron repulsion of molecular hydrogen is eliminated since the central field of the hydrogen molecular ion relative to a second electron at ξ>0 which binds to form the hydrogen molecule is that of a single charge at the foci. (vi) The ellipsoidal MOs allow exact spin pairing over all time that is consistent with experimental observation. This aspect is not possible in the QM model.

Current algorithms to solve molecules are based on nonphysical models based on the concept that the electron is a zero or one-dimensional point in an all-space probability wave function Ψ(x) that permits the electron to be over all space simultaneously and give output based on trial and error or direct empirical adjustment of parameters. These models ultimately cannot be the actual description of a physical electron in that they inherently violate physical laws. They suffer from the same shortcomings that plague atomic quantum theory, infinities, instability with respect to radiation according to Maxwell's equations, violation of conservation of linear and angular momentum, lack of physical relativistic invariance, and the electron is unbounded such that the edge of molecules does not exist. There is no uniqueness, as exemplified by the average of 150 internally inconsistent programs per molecule for each of the 788 molecules posted on the NIST website [31]. Furthermore, from a physical perspective, the implication for the basis of the chemical bond according to quantum mechanics being the exchange integral and the requirement of zero-point vibration, “strictly quantum mechanical phenomena,” is that the theory cannot be a correct description of reality as described for even the simple bond of molecular hydrogen as reported previous [1, 6]. Even the premise that “electron overlap” is responsible for bonding is opposite to the physical reality that negative charges repel each other with an inverse-distance-squared force dependence that becomes infinite. A proposed solution based on physical laws and fully compliant with Maxwell's equations solves the parameters of molecules even to infinite length and complexity in closed form equations with fundamental constants only.

For the first time in history, the key building blocks of organic chemistry have been solved from two basic equations. Now, the true physical structure and parameters of an infinite number of organic molecules up to infinite length and complexity can be obtained to permit the engineering of new pharmaceuticals and materials at the molecular level. The solutions of the basic functional groups of organic chemistry were obtained by using generalized forms of a geometrical and an energy equation for the nature of the H—H bond. The geometrical parameters and total bond energies of about 800 exemplary organic molecules were calculated using the functional group composition. The results obtained essentially instantaneously match the experimental values typically to the limit of measurement [1]. The solved function groups are given in Table 1.

TABLE 1
Partial List of Organic Functional
Groups Solved by Classical Physics.
Continuous-Chain Alkanes
Branched Alkanes
Alkenes
Branched Alkenes
Alkynes
Alkyl Fluorides
Alkyl Chlorides
Alkyl Bromides
Alkyl Iodides
Alkenyl Halides
Aryl Halides
Alcohols
Ethers
Primary Amines
Secondary Amines
Tertiary Amines
Aldehydes
Ketones
Carboxylic Acids
Carboxylic Acid Esters
Amides
N-alkyl Amides
N,N-dialkyl Amides
Urea
Carboxylic Acid Halides
Carboxylic Acid Anhydrides
Nitriles
Thiols
Sulfides
Disulfides
Sulfoxides
Sulfones
Sulfites
Sulfates
Nitroalkanes
Alkyl Nitrates
Alkyl Nitrites
Conjugated Alkenes
Conjugated Polyenes
Aromatics
Napthalene
Toluene
Chlorobenzene
Phenol
Aniline
Aryl Nitro Compounds
Benzoic Acid Compounds
Anisole
Pyrrole
Furan
Thiophene
Imidizole
Pyridine
Pyrimidine
Pyrazine
Quinoline
Isoquinoline
Indole
Adenine
Fullerene (C60)
Graphite
Phosphines
Phosphine Oxides
Phosphites
Phosphates

The two basic equations that solves organic molecules, one for geometrical parameters and the other for energy parameters, were applied to bulk forms of matter containing trillions of trillions of electrons. For example, using the same alkane- and alkene-bond solutions as elements in an infinite network, the nature of the solid molecular bond for all known allotropes of carbon (graphite, diamond, C₆₀, and their combinations) were solved. By further extension of this modular approach, the solid molecular bond of silicon and the nature of semiconductor bond were solved. The nature of other fundamental forms of matter such as the nature of the ionic bond, the metallic bond, and additional major fields of chemistry such as that of silicon, organometallics, and boron were solved exactly such that the position and energy of each and every electron is precisely specified. The implication of these results is that it is possible using physical laws to solve the structure of all types of matter. Some of the solved forms of matter of infinite extent as well as additional major fields of chemistry are given in Table 2. In all cases, the agreement with experiment is remarkable [1].

TABLE 2
Partial List of Additional Molecules and Compositions
of Matter Solved by Classical Physics.
Solid Molecular Bond of the Three Allotropes of Carbon
Diamond
Graphite
Fullerene (C60)
Solid Ionic Bond of Alkali-Hydrides
Alkali-Hydride Crystal Structures
Lithium Hydride
Sodium Hydride
Potassium Hydride
Rubidium & Cesium Hydride
Potassium Hydrino Hydride
Solid Metallic Bond of Alkali Metals
Alkali Metal Crystal Structures
Lithium Metal
Sodium Metal
Potassium Metal
Rubidium & Cesium Metals
Alkyl Aluminum Hydrides
Silicon Groups and Molecules
Silanes
Alkyl Silanes and Disilanes
Solid Semiconductor Bond of Silicon
Insulator-Type Semiconductor Bond
Conductor-Type Semiconductor Bond
Boron Molecules
Boranes
Bridging Bonds of Boranes
Alkoxy Boranes
Alkyl Boranes
Alkyl Borinic Acids
Tertiary Aminoboranes
Quaternary Aminoboranes
Borane Amines
Halido Boranes Organometallic Molecular
Functional Groups and Molecules
Alkyl Aluminum Hydrides
Bridging Bonds of Organoaluminum Hydrides
Organogermanium and Digermanium
Organolead
Organoarsenic
Organoantimony
Organobismuth
Organic Ions
1° Amino
2° Amino
Carboxylate
Phosphate
Nitrate
Sulfate
Silicate
Proteins
Amino Acids
Peptide Bonds
DNA
Bases
2-deoxyribose
Ribose
Phosphate Backbone

The background theory of classical physics (CP) for the physical solutions of atoms and atomic ions is disclosed in Mills journal publications [1-13], R. Mills, The Grand Unified Theory of Classical Quantum Mechanics, January 2000 Edition, BlackLight Power, Inc., Cranbury, N.J., (“'00 Mills GUT”), provided by BlackLight Power, Inc., 493 Old Trenton Road, Cranbury, N.J., 08512; R. Mills, The Grand Unified Theory of Classical Quantum Mechanics, September 2001 Edition, BlackLight Power, Inc., Cranbury, N.J., Distributed by Amazon.com (“'01 Mills OUT”), provided by BlackLight Power, Inc., 493 Old Trenton Road, Cranbury, N.J., 08512; R. Mills, The Grand Unified Theory of Classical Quantum Mechanics, July 2004 Edition, BlackLight Power, Inc., Cranbury, N.J., (“'04 Mills GUT”), provided by BlackLight Power, Inc., 493 Old Trenton Road, Cranbury, N.J., 08512; R. Mills, The Grand Unified Theory of Classical Quantum Mechanics, January 2005 Edition, BlackLight Power, Inc., Cranbury, N.J., (“'05 Mills GUT”), provided by BlackLight Power, Inc., 493 Old Trenton Road, Cranbury, N.J., 08512; 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, (“'06 Mills GUT”), provided by BlackLight Power, Inc., 493 Old Trenton Road, Cranbury, N.J., 08512; R. Mills, The Grand Unified Theory of Classical Quantum Mechanics, October 2007 Edition, BlackLight Power, Inc., Cranbury, N.J., (“'07 Mills GUT”), provided by BlackLight Power, Inc., 493 Old Trenton Road, Cranbury, N.J., 08512; R. Mills, The Grand Unified Theory of Classical Physics, June 2008 Edition, BlackLight Power, Inc., Cranbury, N.J., (“'08 Mills GUT-CP”); in prior published PCT applications WO2005/067678; WO2005/116630; WO2007/051078; WO2007/053486; and WO2008/085,804, and U.S. Pat. No. 7,188,033; U.S. Application No. 60/878,055, filed 3 Jan. 2007; 60/880,061, filed 12 Jan. 2007; 60/898,415, filed 31 Jan. 2007; 60/904,164, filed 1 Mar. 2007; 60/907,433, filed 2 Apr. 2007; 60/907,722, filed 13 Apr. 2007; 60/913,556, filed 24 Apr. 2007; 60/986,675, filed 9 Nov. 2007; 60/988,537, filed 16 Nov. 2007; 61/018,595, filed 2 Jan. 2008; 61/027,977, filed 12 Feb. 2008; 61/029,712, filed 19 Feb. 2008; and 61/082,701, filed 22 Jul. 22, 2008, the entire disclosures of which are all incorporated herein by reference (hereinafter “Mills Prior Publications”).

SUMMARY OF THE INVENTION

The present invention, an exemplary embodiment of which is also referred to as Millsian software and systems, stems from a new fundamental insight into the nature of the atom. Applicant's theory of Classical Physics (CP) 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, CP analytical 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 CP framework. All the major functional groups that make up most organic molecules and the most common classes of molecules have been solved exactly in closed-form solutions with CP. 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 heats of formation of an infinite number of molecules, and these solutions can be used in modeling applications.

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 800 exemplary molecules, typically significantly more accuracy then those given by the current Hartree-Fock algorithm based on QM [2].

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 specific groups of molecules and molecular ions disclosed herein or any portion of these species from first principles. In an embodiment, the solution for the molecules and 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 the molecules and 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 the atoms, molecules, and molecular ions, or any portion of these species and their rendering to numerical or graphical form to apply to proteins, 2-deoxyribonucleic acid (DNA), ribonucleic acid (RNA), and proteins.

Another objective of the present invention is to apply the methods and systems of solving the nature of the atoms, molecules, and molecular ions, or any portion of these species and their rendering to numerical or graphical form the dipole moment of functional groups and by vector additivity, the dipole moment of a molecule or molecular species comprised of the functional groups.

Another objective of the present invention is to apply the methods and systems of solving the nature of the atoms, molecules, and molecular ions, any portion of these species, their dipole moments, or induced dipole moments due to interaction between species or within species and their rendering to numerical or graphical form to solve at least one of the structure, energy, and properties of condensed matter.

Another objective of the present invention is to apply the methods and systems of solving the nature of the atoms, molecules, and molecular ions, any portion of these species, their dipole moments, or induced dipole moments due to interaction between species or within species and their rendering to numerical or graphical form to solve at least one of the structure, energy, properties, and kinetics of reaction transition states and reactions involving the atoms, molecules, and molecular ions, any portion of these species.

These objectives and other objectives are obtained by a system of computing and rendering the nature of at least one specie selected from the groups of molecules and polyatomic molecules disclosed herein, 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

(i) a bond distance between two of the atoms, (ii) a bond angle between three of the atoms, (iii) a bond energy between two of the atoms, (iv) dipole moment of at least one bond, (v) orbital intercept distances and angles, (vi) charge-density functions of atomic, hybridized, and molecular orbitals, (vii) orientations distances, and energies of species in different physical states such as solid, liquid, and gas, and (viii) reaction parameters with other species.

The parameters such as bond distance, bond angle, bond energy, species orientations and reactions 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 groups of molecules and molecular ions disclosed herein can be applied to other unknown species. These solutions can be used to predict the properties of presently unknown species and engineer compositions of matter in a manner that 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 in solution 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. The approach can be applied to solid, liquid, and gases phases of a species or a species present in a mixture or solution.

Embodiments of the system for performing computing and rendering of the nature of the groups of molecules and molecular ions, or any portion of these species using the physical solutions and their phases or structures in different media 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. A complete description of how a computer can be used is disclosed in Applicant's prior incorporated WO2007/051078 application.

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. A bound electron is a constant two-dimensional spherical surface of charge (zero thickness, total charge=θ=π, and total mass=m_e), called an electron orbitsphere. The corresponding uniform current-density function having angular momentum components of

$L_{\mathrm{xy}}=\frac{\hslash }{4}\mathrm{and}{L}_z=\frac{\hslash }{2}$ give rise to the phenomenon of electron spin.

FIGS. 2A and 2B. The bound electron exists as a spherical two-dimensional supercurrent (electron orbitsphere), an extended distribution of charge and current completely surrounding the nucleus. Unlike a spinning sphere, there is a complex pattern of motion on its surface (indicated by vectors) that generates two orthogonal components of angular momentum (FIG. 1) that give rise to the phenomenon of electron spin. FIG. 2A shows a great-circle representation of the z-axis view of the total current pattern of the Y₀⁰(φ,θ) orbitsphere comprising 144 great circle current elements. FIG. 2B shows a representation of the

$\left(-\frac{1}{\sqrt{2}}{i}_x,\frac{1}{\sqrt{2}}{i}_y,i_z\right)\text{-}\mathrm{axis}$ view of the total uniform current-density pattern of the Y₀⁰(φ,θ) orbitsphere with 144 vectors overlaid on the continuous bound-electron current density giving the direction of the current of each great circle element (nucleus not to scale).

FIG. 3. The orbital function modulates the constant (spin) function, (shown for t=0; three-dimensional view).

FIGS. 4A-B. Prolate spheroidal H₂MO, an equipotential minimum energy two-dimensional surface of charge and current that is stable to radiation. (A) External surface showing the charge density that is proportional to the distance from the origin to the tangent to the surface with the maximum density of the MO closest to the nuclei, an energy minimum. (B) Prolate spheroid parameters of molecules and molecular ions where a is the semimajor axis, 2a is the total length of the molecule or molecular ion along the principal axis, b=c is the semiminor axis, 2b=2c is the total width of the molecule or molecular ion along the minor axis, c′ is the distance from the origin to a focus (nucleus), 2c′ is the internuclear distance, and the protons are at the foci.

FIG. 5. Aspirin (acetylsalicylic acid)

FIG. 6. Gray scale, translucent view of the charge density of aspirin showing the orbitals of the atoms at their radii, the ellipsoidal surface of each H or H₂-type ellipsoidal MO that transitions to the corresponding outer shell of the atom(s) participating in each bond, and the hydrogen nuclei (dark gray, not to scale).

FIG. 7. Cyclotrimethylene-trinitramine (RDX)

FIG. 8. Gray scale charge density of RDX showing the outer orbitals of the atoms at their radii and the ellipsoidal surface of each H or H₂-type ellipsoidal MO that transitions to the corresponding outer shell of the atom(s) participating in each bond.

FIG. 9. Gray scale, translucent view of the charge-densities of molecular NaH showing the inner orbitals of the Na atom at their radii, the ellipsoidal surface of the H₂-type ellipsoidal MO formed from the outer Na3s AO and the H1s AO H, and the hydrogen nucleus (dark gray, not to scale).

FIG. 10. Gray scale, translucent view of the charge density of insulin created and modeled using Millsian 2.0 run on a PC.

FIG. 11. Gray scale, translucent view of the charge density of lysozyme created and modeled using Millsian 2.0 run on a PC.

FIG. 12. Gray scale, translucent view of the charge-density of a double-stranded DNA helix created and modeled using Millsian 2.0 run on a PC.

FIG. 13. H₂O MO comprising the linear combination of two O—H-bond MOs. Each O—H-bond MO comprises the superposition of a H₂-type ellipsoidal MO and the O2p_z AO or the O2p_y AO. (A)-(C) Gray scale, translucent views of the charge density of the H₂O MO from the top, side-on with H in foreground, and side-on with O in the foreground, respectively. For each O—H bond, the ellipsoidal surface of each H₂-type ellipsoidal MO transitions to the O2p AO. The O2p shell, the O2s shell, the O1s shell, and the nuclei (not to scale) are shown. (D) Cut-away view showing the inner most O1s shell, and moving radially, the O2s shell, the O2p shell, and the H₂-type ellipsoidal MO that transitions to the O2p AO for each O—H bond.

FIG. 14. Tetrahedral unit cell structure of Type I ice using the transparent gray scale charge density of each H₂O MO comprising the linear combination of two O—H-bond MOs. Each dipole-dipole bond that is Coulombic in nature is depicted by connecting sticks.

FIG. 15. C-axis view of the ideal hexagonal lattice structure of Type I ice using the opaque gray scale charge density of each H₂O MO comprising the linear combination of two O—H-bond MOs. Each dipole-dipole bond that is Coulombic in nature is depicted by connecting sticks.

FIG. 16. An off-angle view of the ideal hexagonal lattice structure of Type I ice using the opaque gray scale charge density of each H₂O MO comprising the linear combination of two O—H-bond MOs. Each dipole-dipole bond that is Coulombic in nature is depicted by connecting sticks.

FIG. 17. Structure of steam. (A). Ensemble of gaseous water molecules undergoing elastic hard-sphere collisions. (B). H-bonded water vapor molecules using the gray scale charge density of each H₂O MO comprising the linear combination of two O—H-bond MOs.

FIG. 18. NH₃ MO comprising the linear combination of three N—H-bonds. Each N—H-bond MO comprises the superposition of a H₂-type ellipsoidal MO and the N2p_x, N2p_y, or N2p_z AO. (A) Gray scale, translucent view of the charge density of the NH₃ MO shown obliquely from the top. For each N—H bond, the ellipsoidal surface of each H₂-type ellipsoidal MO transitions to a N2p AO. The N2p shell, the N2s shell, the N1s shell, and the nuclei (dark gray, not to scale) are shown. (B) Off-center cut-away view showing the complete inner most N1s shell, and moving radially, the cross section of the N2s shell, the N2p shell, and the H₂-type ellipsoidal MO that transitions to a N2p AO for each N—H bond. (C)-(E) Gray scale, side-on, top, and bottom translucent views of the charge density of the NH₃ MO, respectively.

FIG. 19. Structure of the H₃N•••H—OH H bond. The H-bonded ammonia-water vapor molecular dimer using the gray scale charge density of each NH₃ and H₂O MO comprising the linear combination of three N—H and two O—H-bond MOs, respectively.

FIG. 20. The structure of graphite. (A). Single plane of macromolecule of indefinite size. (B). Layers of graphitic planes having an interplane spacing of 3.5 Å.

FIG. 21. The face-centered cubic crystal structures of noble gas condensates, all to the same scale. (A) The crystal structure of neon. (B) The crystal structure of argon. (C) The crystal structure of krypton. (D) The crystal structure of xenon.

FIG. 22. The charge densities of the van der Waals dipoles and face-centered cubic crystal structures of noble gas condensates, all to the same scale. (A) The charge density and crystal structure of neon. (B) The charge density and crystal structure of argon. (C) The charge density and crystal structure of krypton. (D) The charge density and crystal structure of xenon.

FIG. 23. Gray scale, translucent view of the chloride-ion-chloromethane transition state comprising the Cl^δ+—C—Cl^δ− functional group showing the orbitals of the atoms at their radii, the ellipsoidal surface of each H or H₂-type ellipsoidal MO that transitions to the corresponding outer shell of the atom(s) participating in each bond, and the hydrogen nuclei (dark gray, not to scale).

FIG. 24. Gray scale, translucent view of the negatively-charged molecular ion complex comprising the Cl⁻.C^δ+ functional group showing the orbitals of the atoms at their radii, the ellipsoidal surface of each H or H₂-type ellipsoidal MO that transitions to the corresponding outer shell of the atom(s) participating in each bond, and the hydrogen nuclei (not to scale).

FIG. 25. Overhead-view of exemplary gray scale, translucent views of the charge-densities of the inner and outer electrons of molecular-hydrogen excited states. The outer-electron orbital function modulates the time-constant (spin) function, (shown for t=0; three-dimensional view). The inner electron is essentially that of H₂⁺ (nuclei not to scale).

DESCRIPTION OF THE INVENTION

The present Invention comprises molecular modeling methods and systems for solving atomic and molecular structures based on applying the classical laws of physics, (Newton's and Maxwell's Laws) to the atomic scale. The functional groups such as amino acids and peptide bonds with charged functional groups, bases, 2-deoxyribose, ribose, phosphate backbone with charged functional groups, organic ions, and the functional groups of organic and other molecules have been solved in analytical equations. By using these functional groups as building blocks, or independent units, a potentially infinite number of molecules can be solved. As a result, the method and systems of the present invention can visualize the exact three-dimensional structure and calculate physical characteristics of many molecules, up to arbitrary length and complexity. Even complex proteins and DNA (the molecules that encode genetic information) may be solved in real-time interactively on a personal computer. By contrast, previous software based on traditional quantum methods must resort to approximations and run on powerful computers for even the simplest systems.

The Nature of the Chemical Bond of Hydrogen

The nature of the chemical bond of functional groups is solved by first solving the simplest molecule, molecular hydrogen as given in the Nature of the Chemical Bond of Hydrogen-Type Molecules section of Ref. [1]. The hydrogen molecule charge and current density functions, bond distance, and energies are solved from the Laplacian in ellipsoidal coordinates with the constraint of nonradiation [1,6].

$\begin{array}{cc}\left(\eta -\zeta \right)R_{\xi }\frac{\partial }{\partial \xi }\left(R_{\xi }\frac{\partial \varphi }{\partial \xi }\right)+\left(\zeta -\xi \right)R_{\eta }\frac{\partial }{\partial \eta }\left(R_{\eta }\frac{\partial \varphi }{\partial \eta }\right)+\left(\xi -\eta \right)R_{\zeta }\frac{\partial }{\partial \zeta }\left(R_{\zeta }\frac{\partial \varphi }{\partial \zeta }\right)=0& \left(21\right)\end{array}$ a. The Geometrical Parameters of the Hydrogen Molecule

As shown in FIG. 4, the nuclei are at the foci of the electrons comprising a two-dimensional, equipotential-energy, charge- and current-density surface that obeys Maxwell's equations including stability to radiation and Newton's laws of motion. The force balance equation for the hydrogen molecule is

$\begin{array}{cc}\frac{{\hslash }^2}{m_e{a}^2{b}^2}D=\frac{e^2}{8\pi {\varepsilon }_o{\mathrm{ab}}^2}D+\frac{{\hslash }^2}{2m_e{a}^2{b}^2}D& \left(22\right)\end{array}$ whereD=r(t)·i_ξ  (23)is the time dependent distance from the origin to the tangent plane at a point on the ellipsoidal MO. Eq. (22) has the parametric solutionr(t)=ia cos ωt+jb sin ωt  (24)when the semimajor axis, a, isa=a₀  (25)The internuclear distance, 2c′, which is the distance between the foci is2c′=√{square root over (2)}a₀  (26)The experimental internuclear distance is √{square root over (2)}a₀. The semiminor axis is

$\begin{array}{cc}b=\frac{1}{\sqrt{2}}{a}_o& \left(27\right)\end{array}$ The eccentricity, e, is

$\begin{array}{cc}e=\frac{1}{\sqrt{2}}& \left(28\right)\end{array}$ b. The Energies of the Hydrogen Molecule

The potential energy of the two electrons in the central field of the protons at the foci is

$\begin{array}{cc}{V}_e=\frac{-2e^2}{8\pi {\varepsilon }_o\sqrt{a^2-b^2}}\ln \frac{a+\sqrt{a^2-b^2}}{a-\sqrt{a^2-b^2}}=-67.836\mathrm{eV}& \left(29\right)\end{array}$ The potential energy of the two protons is

$\begin{array}{cc}{V}_p=\frac{e^2}{8\pi {\varepsilon }_o\sqrt{a^2-b^2}}=19.242\mathrm{eV}& \left(30\right)\end{array}$ The kinetic energy of the electrons is

$\begin{array}{cc}T=\frac{{\hslash }^2}{2m_{e}a\sqrt{a^2-b^2}}\ln \frac{a+\sqrt{a^2-b^2}}{a-\sqrt{a^2-b^2}}=33.918\mathrm{eV}& \left(31\right)\end{array}$ The energy, V_m, of the magnetic force between the electrons is

$\begin{array}{cc}{V}_m=\frac{-{\hslash }^2}{4m_{e}a\sqrt{a^2-b^2}}\ln \frac{a+\sqrt{a^2-b^2}}{a-\sqrt{a^2-b^2}}=-16.959\mathrm{eV}& \left(32\right)\end{array}$ During bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the protons. The corresponding energy Ē_osc is the difference between the Doppler and average vibrational kinetic energies:

$\begin{array}{cc}{\stackrel{\_}{E}}_{\mathrm{osc}}={\stackrel{\_}{E}}_D+{\stackrel{\_}{E}}_{\mathrm{Kvib}}=\left(V_e+T+V_m+V_p\right)\sqrt{\frac{2{\stackrel{\_}{E}}_K}{{\mathrm{Mc}}^2}}+\frac{1}{2}\hslash \sqrt{\frac{k}{\mu }}& \left(33\right)\end{array}$ The total energy is

$\begin{array}{cc}{E}_T=V_e+T+V_m+V_p+{\stackrel{\_}{E}}_{\mathrm{osc}}& \left(34\right)\\ \begin{array}{c}{E}_T=-\frac{e^2}{8\pi {\varepsilon }_o{a}_0}\left[\begin{array}{c}\left(\begin{array}{c}2\sqrt{2}-\\ \sqrt{2}+\frac{\sqrt{2}}{2}\end{array}\right)\ln \frac{\sqrt{2}+1}{\sqrt{2}-1}-\\ \sqrt{2}\end{array}\right]\left[\begin{array}{c}1+\\ \sqrt{\frac{2\hslash \sqrt{\frac{e^2}{\frac{4\pi {\varepsilon }_o{a}_0^3}{m_e}}}}{m_e{c}^2}}\end{array}\right]-\\ \frac{1}{2}\hslash \sqrt{\frac{k}{\mu }}\\ =-31.686\mathrm{eV}\end{array}& \left(35\right)\end{array}$ The energy of two hydrogen atoms isE(2H[a_H])=−27.21 eV  (36)The bond dissociation energy, E_D, is the difference between the total energy of the corresponding hydrogen atoms (Eq. (36)) and E_T (Eq. (35)).E_D=E(2H[a_H])−E_T=4.478 eV  (37)The experimental energy is E_D=4.478 eV. The calculated and experimental parameters of H₂, D₂, H₂⁺, and D₂⁺ from Ref. [6] and Chp. 11 of Ref. [1] are given in Table 3.

TABLE 3
The Maxwellian closed-form calculated and experimental
parameters of H2, D2, H2+ and D2+.
Parameter	Calculated	Experimental
H2 Bond Energy	4.478	eV	4.478	eV
D2 Bond Energy	4.556	eV	4.556	eV
H2+ Bond Energy	2.654	eV	2.651	eV
D2+ Bond Energy	2.696	eV	2.691	eV
H2 Total Energy	31.677	eV	31.675	eV
D2 Total Energy	31.760	eV	31.760	eV
H2 Ionization Energy	15.425	eV	15.426	eV
D2 Ionization Energy	15.463	eV	15.466	eV
H2+ Ionization Energy	16.253	eV	16.250	eV
D2+ Ionization Energy	16.299	eV	16.294	eV
H2+ Magnetic Moment	9.274 × 10−24 JT−1 (μB)	9.274 × 10−24 JT−1 (μB)
Absolute H2 Gas-Phase	−28.0	ppm	−28.0	ppm
NMR Shift
H2 Internuclear Distancea	0.748	Å	0.741	Å
	{square root over (2)}ao
D2 Internuclear Distancea	0.748	Å	0.741	Å
	{square root over (2)}ao
H2+ Internuclear Distance	1.058	Å	1.06	Å
	2ao
D2+ Internuclear Distancea	1.058	Å	1.0559	Å
	2ao
H2 Vibrational Energy	0.517	eV	0.516	eV
D2 Vibrational Energy	0.371	eV	0.371	eV
H2 ωeχe	120.4	cm−1	121.33	cm−1
D2 ωeχe	60.93	cm−1	61.82	cm−1
H2+ Vibrational Energy	0.270	eV	0.271	eV
D2+ Vibrational Energy	0.193	eV	0.196	eV
H2 J = 1 to J = 0 Rotational Energya	0.0148	eV	0.01509	eV
D2 J = 1 to J = 0 Rotational Energya	0.00741	eV	0.00755	eV
H2+ J = 1 to J = 0 Rotational Energy	0.00740	eV	0.00739	eV
D2+ J = 1 to J = 0 Rotational Energya	0.00370	eV	0.003723	eV
aNot corrected for the slight reduction in internuclear distance due to Ēosc.

Derivation of the General Geometrical and Energy Equations of Organic Chemistry

Organic molecules comprising an arbitrary number of atoms can be solved using 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 of Ref. [1]. Similarly, the geometrical parameters and energies of all functional groups such as those given in Table 1 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 derivation of the dimensional parameters and energies of the function groups are given in the Nature of the Chemical Bond of Hydrogen-Type Molecules, Polyatomic Molecular Ions and Molecules, More Polyatomic Molecules and Hydrocarbons, and Organic Molecular Functional Groups and Molecules sections of Ref. [1]. (Reference to equations of the form Eq. (15.number), Eq. (1.number), Eq. (13.number), and Eq. (14.number) will refer to the corresponding equations of Ref [1].) Additional derivations for other non-organic function groups given in Table 2 are derived in the following sections of Ref. [1]: Applications: Pharmaceuticals, Specialty Molecular Functional Groups and Molecules, Dipole Moments, and Interactions, Nature of the Solid Molecular Bond of the Three Allotropes of Carbon, Silicon Molecular Functional Groups and Molecules, Nature of the Solid Semiconductor Bond of Silicon, Boron Molecules, and Organometallic Molecular Functional Groups and Molecules sections.

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:

$\begin{array}{cc}{k}^{\prime }=\frac{C_1{C}_{2}2e^2}{4\pi {\varepsilon }_0}& \left(38\right)\end{array}$ 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:

$\begin{array}{cc}{c}^{\prime }=a\sqrt{\frac{h^{2}4\pi {\varepsilon }_0}{m_e{e}^{2}2C_1{C}_{2}a}}=\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}& \left(39\right)\end{array}$ The internuclear distance is

$\begin{array}{cc}2c^{\prime }=2\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}& \left(40\right)\end{array}$ The length of the semiminor axis of the prolate spheroidal MO b=c is given byb=√{square root over (a²−c′²)}  (41)And, the eccentricity, e, is

$\begin{array}{cc}e=\frac{c^{\prime }}{a}& \left(42\right)\end{array}$ From Eqs. (11.207-11.212), the potential energy of the two electrons in the central field of the nuclei at the foci is

$\begin{array}{cc}{V}_e=n_1{c}_1{c}_2\frac{-2e^2}{8\pi {\varepsilon }_o\sqrt{a^2-b^2}}\ln \frac{a+\sqrt{a^2-b^2}}{a-\sqrt{a^2-b^2}}& \left(43\right)\end{array}$ The potential energy of the two nuclei is

$\begin{array}{cc}{V}_p=n_1\frac{e^2}{8\pi {\varepsilon }_o\sqrt{a^2-b^2}}& \left(44\right)\end{array}$ The kinetic energy of the electrons is

$\begin{array}{cc}T=n_1{c}_1{c}_2\frac{{\hslash }^2}{2m_{e}a\sqrt{a^2-b^2}}\ln \frac{a+\sqrt{a^2-b^2}}{a-\sqrt{a^2-b^2}}& \left(45\right)\end{array}$ And, the energy, Y_m, of the magnetic force between the electrons is

$\begin{array}{cc}{V}_m=n_1{c}_1{c}_2\frac{-{\hslash }^2}{4m_{e}a\sqrt{a^2-b^2}}\ln \frac{a+\sqrt{a^2-b^2}}{a-\sqrt{a^2-b^2}}& \left(46\right)\end{array}$ The total energy of the H₂-type prolate spheroidal MO, E_T(H2MO), is given by the sum of the energy terms:

$\begin{array}{cc}{E}_{T^{\left(H_2\mathrm{MO}\right)}}=V_e+T+V_m+V_p& \left(47\right)\\ \begin{array}{c}{E}_{T^{\left(H_2\mathrm{MO}\right)}}=-\frac{n_1{e}^2}{8\pi {\varepsilon }_o\sqrt{a^2-b^2}}\left[\begin{array}{c}{c}_1{c}_2\left(2-\frac{a_0}{a}\right)\\ \ln \frac{a+\sqrt{a^2-b^2}}{a-\sqrt{a^2-b^2}}-1\end{array}\right]\\ =-\frac{n_1{e}^2}{8\pi {\varepsilon }_o{c}^{\prime }}\left[c_1{c}_2\left(2-\frac{a_0}{a}\right)\ln \frac{a+c^{\prime }}{a-c^{\prime }}-1\right]\end{array}& \left(48\right)\end{array}$ where n₁ is the number of equivalent bonds of the MO. 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 HOs or AOs of the bonding atoms, the factor c₂ of a H₂-type ellipsoidal MO may given by (i) one, (ii) the ratio of the Coulombic or valence energy of the AO or HO of at least one atom of the bond and 13.605804 eV, the Coulombic energy between the electron and proton of H, (iii) the ratio of the valence energy of the AO or HO of one atom and the Coulombic energy of another, (iv) the ratio of the valence energies of the AOs or HOs of two atoms, (v) the ratio of two c₂ factors corresponding to any of cases (ii)-(iv), and (vi) the product of two different c₂ factors corresponding to any of the cases (i)-(v). Specific examples of the factor c₂ of a H₂-type ellipsoidal MO given in previously [1] 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, the ratio of 14.82575 eV, −E_Coulomb(C,2sp³), and 13.605804 eV;
  • 0.87495, the ratio of 15.55033 eV, −E_Coulomb(C_ethane,2sp₃), and 13.605804 e V;
  • 0.85252, the ratio of 15.95955 eV, −E_Coulomb(C_ethylene,2sp³), and 13.605804 eV;
  • 0.85252, the ratio of 15.95955 eV, −E_Coulomb(C_benzene,2sp³), and 13.605804 eV, and
  • 0.86359, the ratio of 15.55033 eV, −E_Coulomb(C_alkane,2sp³), and 13.605804 eV.

In the generalization of the hybridization of at least two atomic-orbital shells to form a shell of hybrid orbitals, the hybridized shell comprises a linear combination of the electrons of the atomic-orbital shells. The radius of the hybridized shell is calculated from the total Coulombic energy equation by considering that the central field decreases by an integer for each successive electron of the shell and that the total energy of the shell is equal to the total Coulombic energy of the initial AO electrons. The total energy E_T(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.

$\begin{array}{cc}{E}_T\left(\mathrm{atom},{\mathrm{msp}}^3\right)=-\sum _{m=1}^n{\mathrm{IP}}_m& \left(49\right)\end{array}$ where IP_m is the mth ionization energy (positive) of the atom. The radius r_msp3, of the hybridized shell is given by:

$\begin{array}{cc}{r}_{{\mathrm{msp}}^3}=\sum _{q=Z-n}^{Z-1}\frac{-\left(Z-q\right)e^2}{8\pi {\varepsilon }_0{E}_T\left(\mathrm{atom},{\mathrm{msp}}^3\right)}& \left(50\right)\end{array}$ Then, the Coulombic energy E_Coulomb(atom,msp³) of the outer electron of the atom msp³ shell is given by

$\begin{array}{cc}{E}_{\mathrm{Coulomb}}\left(\mathrm{atom},{\mathrm{msp}}^3\right)=\frac{-e^2}{8\pi {\varepsilon }_0{r}_{{\mathrm{msp}}^3}}& \left(51\right)\end{array}$ 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:

$\begin{array}{cc}E\left(\mathrm{magnetic}\right)=\frac{2\pi {\mu }_0{e}^2{\hslash }^2}{m_e^2{r}^3}=\frac{8\pi {\mu }_o{\mu }_B^2}{r^3}& \left(52\right)\end{array}$ Then, the energy E(atom,msp³) of the outer electron of the atom msp³ shell is given by the sum of E_Coulomb(atom,msp³) and E(magnetic):

$\begin{array}{cc}E\left(\mathrm{atom},{\mathrm{msp}}^3\right)=\frac{-e^2}{8\pi {\varepsilon }_0{r}_{{\mathrm{msp}}^3}}+\frac{2\pi {\mu }_0{e}^2{\hslash }^2}{m_e^2{r}^3}& \left(53\right)\end{array}$

Consider next that the at least two atomic orbitals hybridize as a linear combination of electrons at the same energy in order to achieve a bond at an energy minimum with another atomic orbital or hybridized orbital. As a further generalization of the basis of the stability of the MO, the sharing of electrons between two or more such hybridized orbitals to form a MO permits the participating hybridized orbitals to decrease in energy through a decrease in the radius of one or more of the participating orbitals. In this case, the total energy of the hybridized orbitals is given by the sum of E(atom,msp³) 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) is the sum of the first ionization energy of the atom and the hybridization energy. An example of E(atom,msp³) for E(C,2sp³) 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. In order to further satisfy the potential, kinetic, and orbital energy relationships, each atom msp³ HO donates an excess of 25% per bond of its electron density to the atom-atom-bond MO to form an energy minimum wherein the atom-atom bond comprises one of a single, double, or triple bond. In each case, the radius of the hybridized shell is calculated from the Coulombic energy equation by considering that the central field decreases by an integer for each successive electron of the shell and the total energy of the shell is equal to the total Coulombic energy of the initial AO electrons plus the hybridization energy. The total energy E_T(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:

$\begin{array}{cc}{E}_T\left(\mathrm{mol}.\mathrm{atom},{\mathrm{msp}}^3\right)=E\left(\mathrm{atom},{\mathrm{msp}}^3\right)=\sum _{m=2}^n{\mathrm{IP}}_m& \left(54\right)\end{array}$ where IP_m is the mth ionization energy (positive) of the atom and the sum of −IP₁ plus the hybridization energy is E(atom,msp³). Thus, the radius r_msp3 of the hybridized shell due to its donation of a total charge −Qe to the corresponding MO is given by is given by:

$\begin{array}{cc}\begin{array}{c}{r}_{{\mathrm{msp}}^3}=\left(\sum _{q=Z-n}^{Z-1}\left(Z-q\right)-Q\right)\frac{-e^2}{8\pi {\varepsilon }_0{E}_T\left(\mathrm{mol}.\mathrm{atom},{\mathrm{msp}}^3\right)}\\ =\left(\sum _{q=Z-n}^{Z-1}\left(Z-q\right)-s\left(0.25\right)\right)\frac{-e^2}{8\pi {\varepsilon }_0{E}_T\left(\mathrm{mol}.\mathrm{atom},{\mathrm{msp}}^3\right)}\end{array}& \left(55\right)\end{array}$ where −e is the fundamental electron charge and s=1, 2, 3 for a single, double, and triple bond, respectively. The Coulombic energy E_Coulomb(mol.atom,msp³) of the outer electron of the atom msp³ shell is given by

$\begin{array}{cc}{E}_{\mathrm{Coulomb}}\left(\mathrm{mol}·\mathrm{atom},{\mathrm{msp}}^3\right)=\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_0{r}_{{\mathrm{msp}}^3}}& \left(56\right)\end{array}$ 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. (52). 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):

$\begin{array}{cc}E\left(\mathrm{mol}.\mathrm{atom},{\mathrm{msp}}^3\right)=\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_0{r}_{{\mathrm{msp}}^3}}=\frac{2{\mathrm{\pi \mu }}_0{e}^2{\hslash }^2}{m_e^2{r}^3}& \left(57\right)\end{array}$ E^T(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):E^T(atom-atom,msp³)=E(mol.atom,msp³)−E(atom,msp³)  (58)In the case of the C2sp³ HO, the initial parameters (Eqs. (14.142-14.146)) are

$\begin{array}{cc}\begin{array}{c}{r}_{2{\mathrm{sp}}^3}=\sum _{n=2}^5\frac{\left(Z-n\right)e^2}{8{\mathrm{\pi \varepsilon }}_0\left(e148.25751\mathrm{eV}\right)}\\ =\frac{10e^2}{8{\mathrm{\pi \varepsilon }}_0\left(e148.25751\mathrm{eV}\right)}\\ =0.91771a_0\end{array}& \left(59\right)\\ \begin{array}{c}{E}_{\mathrm{Coulomb}}=\left(C,2{\mathrm{sp}}^3\right)\\ =\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_0{r}_{2{\mathrm{sp}}^3}}\\ =\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_{0}0.91771a_0}\\ =-14.82575\mathrm{eV}\end{array}& \left(60\right)\\ \begin{array}{c}E\left(\mathrm{magnetic}\right)=\frac{2{\mathrm{\pi \mu }}_0{e}^2{\hslash }^2}{{m_e^2\left(r_3\right)}^3}\\ =\frac{8{\mathrm{\pi \mu }}_o{\mu }_B^2}{{\left(0.84317a_0\right)}^3}\\ =0.19086\mathrm{eV}\end{array}& \left(61\right)\\ \begin{array}{c}E\left(C,2{\mathrm{sp}}^3\right)=\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_0{r}_{2{\mathrm{sp}}^3}}+\frac{2{\mathrm{\pi \mu }}_0{e}^2{\hslash }^2}{{m_e^2\left(r_3\right)}^3}\\ =-14.82575\mathrm{eV}+0.19086\mathrm{eV}\\ =-14.63489\mathrm{eV}\end{array}& \left(62\right)\end{array}$ In Eq. (55),

$\begin{array}{cc}\sum _{q=Z-n}^{Z-1}\left(Z-q\right)=10& \left(63\right)\end{array}$ Eqs. (14.147) and (54) giveE_T(mol.atom,msp³)=E_T(C_ethane,2sp³)=−151.61569 eV  (64)Using Eqs. (55-65), the final values of r_C2sp3, E_Coulomb(C2sp³), and E(C2sp³), and the resulting

$E_T\left(C\stackrel{\mathrm{BO}}{—}C,C2{\mathrm{sp}}^3\right)$ of the MO due to charge donation from the HO to the MO where

$C\stackrel{\mathrm{BO}}{—}C$ refers to the bond order of the carbon-carbon bond for different values of the parameter s are given in Table 4.

TABLE 4
$\hspace{1em}\begin{array}{c}\mathrm{The}\mathrm{final}\mathrm{values}\mathrm{of}{r}_{C2{\mathrm{sp}}^3},E_{\mathrm{Coulomb}}\left(C2{\mathrm{sp}}^3\right),\mathrm{and}E\left(C2{\mathrm{sp}}^3\right)\mathrm{and}\mathrm{the}\mathrm{resulting}\\ E_T\left(C\stackrel{\mathrm{BO}}{—}C,C2{\mathrm{sp}}^3\right)\mathrm{of}\mathrm{the}\mathrm{MO}\mathrm{due}\mathrm{to}\mathrm{charge}\mathrm{donation}\mathrm{from}\mathrm{the}\mathrm{HO}\mathrm{to}\mathrm{the}\mathrm{MO}\\ \mathrm{where}C\stackrel{\mathrm{BO}}{—}C\mathrm{refers}\mathrm{to}\mathrm{the}\mathrm{bond}\mathrm{order}\mathrm{of}\mathrm{the}\mathrm{carbon}\text{-}\mathrm{carbon}\mathrm{bond}.\end{array}$
MO Bond Order (BO)	s1	s2	rC2sp3(a0) Final	ECoulomb(C2sp3) (eV) Final	E(C2sp3) (eV) Final	$\hspace{1em}\begin{array}{c}{E}_T\left(C\stackrel{\mathrm{BO}}{—}C,C2{\mathrm{sp}}^3\right)\\ \left(\mathrm{eV}\right)\end{array}$
I	1	0	0.87495	−15.55033	−15.35946	−0.72457
II	2	0	0.85252	−15.95955	−15.76868	−1.13379
III	3	0	0.83008	−16.39089	−16.20002	−1.56513
IV	4	0	0.80765	−16.84619	−16.65532	−2.02043

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 E(mol.atom,msp³) of the outer electron of the atom msp³ shell of each bonding atom must be the average of E(mol.atom,msp³) for two different values of s:

$\begin{array}{cc}E\left(\mathrm{mol}.\mathrm{atom},{\mathrm{msp}}^3\right)=\frac{\begin{array}{c}E\left(\mathrm{mol}.\mathrm{atom}\left(s_1\right),{\mathrm{msp}}^3\right)+\\ E\left(\mathrm{mol}.\mathrm{atom}\left(s_2\right),{\mathrm{msp}}^3\right)\end{array}}{2}& \left(65\right)\end{array}$ In this case, E_T(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:

$\begin{array}{cc}{E}_T\left(\mathrm{atom}—\mathrm{atom},{\mathrm{msp}}^3\right)=\frac{\begin{array}{c}{E}_T\left(\mathrm{atom}—\mathrm{atom}\left(s_1\right),{\mathrm{msp}}^3\right)+\\ E_T\left(\mathrm{atom}—\mathrm{atom}\left(s_2\right),{\mathrm{msp}}^3\right)\end{array}}{2}& \left(66\right)\end{array}$

Consider an aromatic molecule such as benzene given in the Benzene Molecule section of Ref. [1]. 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_T (atom-atom,msp³) of the C—H-bond MO is given by 0.5E^T(C═C,2sp³) (Eq. (14.247)) corresponding to one half of a double bond that matches the condition for a single-bond order for C—H that is lowered in energy due to the aromatic character of the bond.

A further general possibility is that a minimum-energy bond is achieved with satisfaction of the potential, kinetic, and orbital energy relationships by the formation of an MO comprising an allowed multiple of a linear combination of H₂-type ellipsoidal MOs and corresponding HOs or AOs that contribute a corresponding allowed multiple (e.g. 0.5, 0.75, 1) of the bond order given in Table 4. For example, the alkane MO given in the Continuous-Chain Alkanes section of Ref. [1] comprises a linear combination of factors of 0.5 of a single bond and 0.5 of a double bond.

Consider a first MO and its HOs comprising a linear combination of bond orders and a second MO that shares a HO with the first. In addition to the mutual HO, the second MO comprises another AO or HO having a single bond order or a mixed bond order. Then, in order for the two MOs to be energy matched, the bond order of the second MO and its HOs or its HO and AO is a linear combination of the terms corresponding to the bond order of the mutual HO and the bond order of the independent HO or AO. Then, in general, E_T(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:

$\begin{array}{cc}{E}_T\left(\mathrm{atom}—\mathrm{atom},{\mathrm{msp}}^3\right)=\sum _{n=1}^N{C}_{S_n}{E}_T\left(\mathrm{atom}—\mathrm{atom}\left(s_n\right),{\mathrm{msp}}^3\right)& \left(67\right)\end{array}$ where c_sn is the multiple of the BO of s_n. The radius r_msp3 of the atom msp³ shell of each bonding atom is given by the Coulombic energy using the initial energy E_Coulomb(atom,msp³) and E_T(atom-atom,msp³), the energy change of each atom msp³ shell with the formation of each atom-atom-bond MO:

$\begin{array}{cc}{r}_{{\mathrm{msp}}^3}=\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_0{a}_0\left(\begin{array}{c}{E}_{\mathrm{Coulomb}}\left(\mathrm{atom},{\mathrm{msp}}^3\right)+\\ E_T\left(\mathrm{atom}\mathrm{—atom},{\mathrm{msp}}^3\right)\end{array}\right)}& \left(68\right)\end{array}$

where E_Coulomb(C2sp³)=−14.825751 eV. The Coulombic energy E_Coulomb(mol·atom,msp³) of the outer electron of the atom msp³ shell is given by Eq. (56). 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. (52)) 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. (57)). E_T(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. (58). Using Eq. (60) for E_Coulomb(C,2sp³) in Eq. (68), the single bond order energies given by Eqs. (55-64) and shown in Table 4, and the linear combination energies (Eqs. (65-67)), the parameters of linear combinations of bond orders and linear combinations of mixed bond orders are given in Table 5.

TABLE 5
$\hspace{1em}\begin{array}{c}\mathrm{The}\mathrm{final}\mathrm{values}\mathrm{of}{r}_{C2{\mathrm{sp}}^3},E_{\mathrm{Coulomb}}\left(C2{\mathrm{sp}}^3\right),\mathrm{and}E\left(C2{\mathrm{sp}}^3\right)\mathrm{and}\mathrm{the}\mathrm{resulting}{E}_T\left(C\stackrel{\mathrm{BO}}{—}C,C2{\mathrm{sp}}^3\right)\mathrm{of}\mathrm{the}\mathrm{MO}\mathrm{comprising}\\ a\mathrm{linear}\mathrm{combination}\mathrm{of}{H}_2\text{-}\mathrm{type}\mathrm{ellipsoidal}\mathrm{MOs}\mathrm{and}\mathrm{corresponding}\mathrm{HOs}\mathrm{of}\mathrm{single}\mathrm{or}\mathrm{mixed}\mathrm{bond}\mathrm{order}\mathrm{where}{c}_{s_n}\\ \mathrm{is}\mathrm{the}\mathrm{multiple}\mathrm{of}\mathrm{the}\mathrm{bond}\mathrm{order}\mathrm{parameter}{E}_T\left(\mathrm{atom}\text{-}\mathrm{atom}\left(s_n\right),{\mathrm{msp}}^3\right)\mathrm{given}\mathrm{in}\mathrm{Table}4.\end{array}$
MO Bond Order (BO)	s1	cs1	s2	cs2	s3	cs3	rC2sp3(a0) Final	ECoulomb(C2sp3) (eV) Final	E(C2sp3) (eV) Final	$\hspace{1em}\begin{array}{c}{E}_T\left(C\stackrel{\mathrm{BO}}{—}C,C2{\mathrm{sp}}^3\right)\\ \left(\mathrm{eV}\right)\end{array}$
1/2I	1	0.5	0	0	0	0	0.89582	−15.18804	−14.99717	−0.36228
1/2II	2	0.5	0	0	0	0	0.88392	−15.39265	−15.20178	−0.56689
1/2I + 1/4II	1	0.5	2	0.25	0	0.25	0.87941	−15.47149	−15.28062	−0.64573
1/4II + 1/4(I + II)	2	0.25	1	0.25	2	0.25	0.87363	−15.57379	−15.38293	−0.74804
3/4II	2	0.75	0	0	0	0	0.86793	−15.67610	−15.48523	−0.85034
1/2I + 1/2II	1	0.5	2	0.5	0	0	0.86359	−15.75493	−15.56407	−0.92918
1/2I + 1/2III	1	0.5	3	0.5	0	0	0.85193	−15.97060	−15.77974	−1.14485
1/2I + 1/2IV	1	0.5	4	0.5	0	0	0.83995	−16.19826	−16.00739	−1.37250
1/2II + 1/2III	2	0.5	3	0.5	0	0	0.84115	−16.17521	−15.98435	−1.34946
1/2II + 1/2IV	2	0.5	4	0.5	0	0	0.82948	−16.40286	−16.21200	−1.57711
I + 1/2(I + II)	1	1	1	0.5	2	0.5	0.82562	−16.47951	−16.28865	−1.65376
1/2III + 1/2IV	3	0.5	4	0.5	0	0	0.81871	−16.61853	−16.42767	−1.79278
1/2IV + 1/2IV	4	0.5	4	0.5	0	0	0.80765	−16.84619	−16.65532	−2.02043
1/2(I + II) + II	1	0.5	2	0.5	2	1	0.80561	−16.88873	−16.69786	−2.06297

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_mol2sp3 of the C2sp³ HO of a carbon atom of a given molecule is calculated using Eq. (14.514) by considering ΣE_Tmol(MO,2sp₃), the total energy donation to all bonds with which it participates in bonding. The general equation for the radius is given by

$\begin{array}{cc}\begin{array}{c}{r}_{\mathrm{mol}2{\mathrm{sp}}^3}=\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_0\left(E_{\mathrm{Coulomb}}\left(C,2{\mathrm{sp}}^3\right)+\sum E_{T_{\mathrm{mol}}}\left(\mathrm{MO},2{\mathrm{sp}}^3\right)\right)}\\ =\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_0\left(e14.825751\mathrm{eV}+\sum E_{T_{\mathrm{mol}}}\left(\mathrm{MO},2{\mathrm{sp}}^3\right)\right)}\end{array}& \left(69\right)\end{array}$ The Coulombic energy E_Coulomb(mol.atom,msp³) of the outer electron of the atom msp³ shell is given by Eq. (56). 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. (52)) 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. (57)).

For example, the C2sp³ HO of each methyl group of an alkane contributes −0.92918 eV (Eq. (14.513)) to the corresponding single C—C bond; thus, the corresponding C2sp³ HO radius is given by Eq. (14.514). The 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. Thus, the radius (Eq. (69)), the Coulombic energy (Eq. (56)), and the energy (Eq. (57)) of each alkane methylene group are

$\begin{array}{cc}\begin{array}{c}{r}_{{\mathrm{alkaneC}}_{\mathrm{methylene}}2{\mathrm{sp}}^3}=\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_0\left(\begin{array}{c}{E}_{\mathrm{Coulomb}}\left(C,2{\mathrm{sp}}^3\right)+\sum E_{T_{\mathrm{alkane}}}\\ \left(\mathrm{methylene}C\mathrm{—C},2{\mathrm{sp}}^3\right)\end{array}\right)}\\ =\frac{e^2}{8{\mathrm{\pi \varepsilon }}_0\left(\begin{array}{c}e14.825751\mathrm{eV}+\\ e0.93918\mathrm{eV}+e0.92918\mathrm{eV}\end{array}\right)}\\ =0.81549a_0\end{array}& \left(70\right)\\ E_{\mathrm{Coulomb}}\left(C_{\mathrm{methylene}}2{\mathrm{sp}}^3\right)=\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_0\left(0.81549a_0\right)}=-16.68412\mathrm{eV}& \left(71\right)\\ \begin{array}{c}E\left(C_{\mathrm{methylene}}2{\mathrm{sp}}^3\right)=\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_0\left(0.81549a_0\right)}\\ =\frac{2{\mathrm{\pi \mu }}_0{e}^2{\hslash }^2}{{m_e^2\left(0.84317a_0\right)}^3}\\ =-16.49325\mathrm{eV}\end{array}& \left(72\right)\end{array}$

In the determination of the parameters of functional groups, heteroatoms bonding to C2sp³ HOs to form MOs are energy matched to the C2sp³ HOs. Thus, the radius and the energy parameters of a bonding heteroatom are given by the same equations as those for C2sp³ HOs. Using Eqs. (52), (56-57), (61), and (69) in a generalized fashion, the final values of the radius of the HO or AO, r_Atom.HO.AO, E_Coulomb(mol.atom,msp³), and E(C_mol2sp³) are calculated using ΣE_Tgroup(MO,2sp³), the total energy donation to each bond with which an atom participates in bonding corresponding to the values of

$E_T\left(C\stackrel{\mathrm{BO}}{—}C,C2{\mathrm{sp}}^3\right)$ of the MO due to charge donation from the AO or HO to the MO given in Tables 4 and 5.

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. (38)) is used to determine the ellipsoidal parameter c′ (Eq. (39)) 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. (48))) which is set equal to n₁ times the total energy of H₂ where n₁ 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. (40-42)). 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. (43-48)) plus E_T(AO/HO):

$\begin{array}{cc}{E}_T{(}_{H_2\mathrm{MO}})=V_e+T+V_m+V_p+E_T\left(\mathrm{AO}\text{/}\mathrm{HO}\right)& \left(73\right)\\ \begin{array}{c}{E}_T{(}_{H_2\mathrm{MO}})=-\frac{n_1{e}^2}{8{\mathrm{\pi \varepsilon }}_o\sqrt{a^2-b^2}}\left[\begin{array}{c}{c}_1{c}_2\left(2-\frac{a_0}{a}\right)\\ \ln \frac{a+\sqrt{a^2-b^2}}{a-\sqrt{a^2-b^2}}-1\end{array}\right]+\\ E_T\left(\mathrm{AO}\text{/}\mathrm{HO}\right)\\ =\frac{n_1{e}^2}{8{\mathrm{\pi \varepsilon }}_o{c}^{\prime }}\left[\begin{array}{c}{c}_1{c}_2\left(2-\frac{a_0}{a}\right)\\ \ln \frac{a+c^{\prime }}{a-c^{\prime }}-1\end{array}\right]+E_T\left(\mathrm{AO}\text{/}\mathrm{HO}\right)\end{array}& \left(74\right)\end{array}$ where n₁ 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 E_T(AO/HO) is the total energy comprising the difference of the energy E(AO/HO) of at least one atomic or hybrid orbital to which the MO is energy matched and any energy component ΔE_H2MO(AO/HO) due to the AO or HO's charge donation to the MO.E_T(AO/HO)=E(AO/HO)−ΔE_H2MO(AO/HO)  (75)

To solve the bond parameters and energies,

$c^{\prime }=a\sqrt{\frac{{\hslash }^{2}4{\mathrm{\pi \varepsilon }}_0}{m_e{e}^{2}2C_1{C}_{2}a}}=\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}$ (Eq. (39)) is substituted into E_T(H2MO) to give

$\begin{array}{cc}\begin{array}{c}{E}_T{(}_{H_2\mathrm{MO}})=-\frac{n_1{e}^2}{8{\mathrm{\pi \varepsilon }}_o\sqrt{a^2-b^2}}\left[\begin{array}{c}{c}_1{c}_2\left(2-\frac{a_0}{a}\right)\\ \ln \frac{a+\sqrt{a^2-b^2}}{a-\sqrt{a^2-b^2}}-1\end{array}\right]+\\ E_T\left(\mathrm{AO}\text{/}\mathrm{HO}\right)\\ =\frac{n_1{e}^2}{8{\mathrm{\pi \varepsilon }}_o{c}^{\prime }}\left[\begin{array}{c}{c}_1{c}_2\left(2-\frac{a_0}{a}\right)\\ \ln \frac{a+c^{\prime }}{a-c^{\prime }}-1\end{array}\right]+E_T\left(\mathrm{AO}\text{/}\mathrm{HO}\right)\\ =-\frac{n_1{e}^2}{8{\mathrm{\pi \varepsilon }}_o\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}}\left[\begin{array}{c}{c}_1{c}_2\left(2-\frac{a_0}{a}\right)\\ \ln \frac{a+\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}}{a-\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}}-1\end{array}\right]+\\ E_T\left(\mathrm{AO}\text{/}\mathrm{HO}\right)\end{array}& \left(76\right)\end{array}$ The total energy is set equal to E(basis energies) which in the most general case is given by the sum of a first integer n₁ times the total energy of H₂ minus a second integer n₂ times the total energy of H, minus a third integer n₃ times the valence energy of E(AO) (e.g. E(N)=−14.53414 eV) where the first integer can be 1, 2, 3 . . . , and each of the second and third integers can be 0, 1, 2, 3 . . . .E(basis energies)=n₁(−31.63536831 eV)−n₂(−13.605804 eV)−n₃E(AO)  (77)In the case that the MO bonds two atoms other than hydrogen, E(basis energies) is n₁ times the total energy of H₂ where n₁ 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:E(basis energies)=n₁(−31.63536831 eV)  (78)

E_T(H2MO), is set equal to E(basis energies), and the semimajor axis a is solved. Thus, the semimajor axis a is solved from the equation of the form:

$\begin{array}{cc}-\frac{n_1{e}^2}{8{\mathrm{\pi \varepsilon }}_o\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}}\left[\begin{array}{c}{c}_1{c}_2\left(2-\frac{a_0}{a}\right)\\ \ln \frac{a+\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}}{a-\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}}-1\end{array}\right]+E_T\left(\mathrm{AO}\text{/}\mathrm{HO}\right)=E\left(\mathrm{basis}\mathrm{energies}\right)& \left(79\right)\end{array}$ 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. (39-41). Then, the component energies are given by Eqs. (43-46) and (76).

The total energy of the MO of the functional group, E_T(MO), is the sum of the total energy of the components comprising the energy contribution of the MO formed between the participating atoms and E_T(atom-atom,msp³.AO), the change in the energy of the AOs or HOs upon forming the bond. From Eqs. (76-77), E_T(MO) isE_T(MO)=E(basis energies)+E_T(atom-atom,msp³.AO)  (80)

During bond formation, the electrons undergo a reentrant oscillatory orbit with vibration of the nuclei, and the corresponding energy Ē_osc is the sum of the Doppler, Ē_D, and average vibrational kinetic energies, Ē_Kvib:

$\begin{array}{cc}{\stackrel{\_}{E}}_{\mathrm{osc}}=n_1\left({\stackrel{\_}{E}}_D+{\stackrel{\_}{E}}_{\mathrm{Kvib}}\right)=n_1\left(E_{\mathrm{hv}}\sqrt{\frac{2{\stackrel{\_}{E}}_K}{m_e{c}^2}}+\frac{1}{2}\hslash \sqrt{\frac{k}{\mu }}\right)& \left(81\right)\end{array}$ where n₁ 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 Ē_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

$\begin{array}{cc}f\left(R\right)=-\frac{C_{1o}{C}_{2o}{e}^2}{4\pi {\varepsilon }_0{R}^3}\mathrm{and}& \left(82\right)\\ f^{\prime }\left(a\right)=2\frac{C_{1o}{C}_{2o}{e}^2}{4\pi {\varepsilon }_0{R}^3}& \left(83\right)\end{array}$ such that the angular frequency of the oscillation in the transition state is given by

$\begin{array}{cc}\omega =\sqrt{\frac{\left[\frac{-3}{a}f\left(a\right)-f^{\prime }\left(a\right)\right]}{m_e}}=\sqrt{\frac{k}{m_e}}=\sqrt{\frac{\frac{C_{1o}{C}_{2o}{e}^2}{4{\mathrm{\pi \varepsilon }}_0{R}^3}}{m_e}}& \left(84\right)\end{array}$ 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_1o 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_1o=C₁ and C_2o=C₂. The kinetic energy, E_K, corresponding to Ē_D is given by Planck's equation for functional groups:

$\begin{array}{cc}{\stackrel{\_}{E}}_K=\mathrm{\hslash \omega }=\hslash \sqrt{\frac{\frac{C_{1o}{C}_{2o}{e}^2}{4{\mathrm{\pi \varepsilon }}_0{R}^3}}{m_e}}& \left(85\right)\end{array}$ The Doppler energy of the electrons of the reentrant orbit is

$\begin{array}{cc}{\stackrel{\_}{E}}_D\cong E_{\mathrm{hv}}\sqrt{\frac{2{\stackrel{\_}{E}}_K}{m_e{c}^2}}=E_{\mathrm{hv}}\sqrt{\frac{2\hslash \sqrt{\frac{\frac{C_{1o}{C}_{2o}{e}^2}{4{\mathrm{\pi \varepsilon }}_0{R}^3}}{m_e}}}{m_e{c}^2}}& \left(86\right)\end{array}$ Ē_osc given by the sum of Ē_D and Ē_Kvib is

$\begin{array}{cc}\begin{array}{c}{\stackrel{\_}{E}}_{{\mathrm{osc}}^{\left(\mathrm{group}\right)}}=n_1\left({\stackrel{\_}{E}}_D+{\stackrel{\_}{E}}_{\mathrm{Kvib}}\right)\\ =n_1\left(E_{\mathrm{hv}}\sqrt{\frac{2\hslash \sqrt{\frac{\frac{C_{1o}{C}_{2o}{e}^2}{4{\mathrm{\pi \varepsilon }}_0{R}^3}}{m_e}}}{m_e{c}^2}}+E_{\mathrm{vib}}\right)\end{array}& \left(87\right)\end{array}$ E_hv of a group having n₁ bonds is given by E_T(MO)/n₁ such that

$\begin{array}{cc}{\stackrel{\_}{E}}_{\mathrm{osc}}=n_1\left({\stackrel{\_}{E}}_D+{\stackrel{\_}{E}}_{\mathrm{Kvib}}\right)=n_1\left(E_{T^{\left(\mathrm{MO}\right)}}/n_1\sqrt{\frac{2{\stackrel{\_}{E}}_K}{M{c}^2}}+\frac{1}{2}\hslash \sqrt{\frac{k}{u}}\right)& \left(88\right)\end{array}$ E_T+osc(Group) is given by the sum of E_T(MO) (Eq. (79)) and Ē_osc (Eq. (88)):

$\begin{array}{cc}\begin{array}{c}{E}_{T+{\mathrm{osc}}^{\left(\mathrm{Group}\right)}}=E_{T^{\left(\mathrm{MO}\right)}}+{\stackrel{\_}{E}}_{\mathrm{osc}}\\ =\left(\begin{array}{c}\left(\begin{array}{c}-\frac{n_1{e}^2}{8\pi {\varepsilon }_0\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}}\left[\begin{array}{c}{c}_1{c}_2\left(2-\frac{a_0}{a}\right)\\ \ln \frac{a+\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}}{a-\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}}-1\end{array}\right]+\\ E_T\left(\mathrm{AO}\text{/}\mathrm{HO}\right)+E_T\left(\mathrm{atom}\mathrm{—atom},{\mathrm{msp}}^3·\mathrm{AO}\right)\end{array}\right)\\ \left[1+\sqrt{\frac{2\hslash \sqrt{\frac{\frac{C_{1o}{C}_{2o}{e}^2}{4{\mathrm{\pi \varepsilon }}_o{R}^3}}{m_e}}}{m_e{c}^2}}\right]+n_1\frac{1}{2}\hslash \sqrt{\frac{k}{\mu }}\end{array}\right)\\ =\left(\begin{array}{c}E\left(\mathrm{basis}\mathrm{energies}\right)+\\ E_T\left(\mathrm{atom}\mathrm{—atom},{\mathrm{msp}}^3·\mathrm{AO}\right)\end{array}\right)\\ \left[1+\sqrt{\frac{2\hslash \sqrt{\frac{\frac{C_{1o}{C}_{2o}{e}^2}{4{\mathrm{\pi \varepsilon }}_0{R}^3}}{m_e}}}{m_e{c}^2}}\right]+n_1\frac{1}{2}\hslash \sqrt{\frac{k}{\mu }}\end{array}& \left(89\right)\end{array}$

The total energy of the functional group E_T(group) is the sum of the total energy of the components comprising the energy contribution of the MO formed between the participating atoms, E(basis energies), the change in the energy of the AOs or HOs upon forming the bond (E_T(atom-atom,msp³.AO)), the energy of oscillation in the transition state, and the change in magnetic energy with bond formation, E_mag. From Eq. (89), the total energy of the group E_T(Group) is

$\begin{array}{cc}{E}_{T^{\left(\mathrm{Group}\right)}}=\left(\begin{array}{c}\left(\begin{array}{c}E\left(\mathrm{basis}\mathrm{energies}\right)+\\ E_T\left(\mathrm{atom}\mathrm{—atom},{\mathrm{msp}}^3·\mathrm{AO}\right)\end{array}\right)\\ \left[1+\sqrt{\frac{2\hslash \sqrt{\frac{\frac{C_{1o}{C}_{2o}{e}^2}{4{\mathrm{\pi \varepsilon }}_o{R}^3}}{m_e}}}{m_e{c}^2}}\right]+n_1{\stackrel{\_}{E}}_{\mathrm{Kvib}}+E_{\mathrm{mag}}\end{array}\right)& \left(90\right)\end{array}$ 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

$\begin{array}{cc}{E}_{\mathrm{mag}}=c_3\frac{2{\mathrm{\pi \mu }}_0{e}^2{\hslash }^2}{m_e^2{r}^3}=c_3\frac{8{\mathrm{\pi \mu }}_o{\mu }_B^2}{r^3}& \left(91\right)\end{array}$ where r³ is the radius of the atom that reacts to form the bond and c₃ is the number of electron pairs.

$\begin{array}{cc}{E}_{T^{\left(\mathrm{Group}\right)}}=\left(\begin{array}{c}\left(\begin{array}{c}E\left(\mathrm{basis}\mathrm{energies}\right)+\\ E_T\left(\mathrm{atom}\mathrm{—atom},{\mathrm{msp}}^3·\mathrm{AO}\right)\end{array}\right)\\ \left[1+\sqrt{\frac{2\hslash \sqrt{\frac{\frac{C_{1o}{C}_{2o}{e}^2}{4{\mathrm{\pi \varepsilon }}_o{R}^3}}{m_e}}}{m_e{c}^2}}\right]+\\ n_1{\stackrel{\_}{E}}_{\mathrm{Kvib}}+c_3\frac{8{\mathrm{\pi \mu }}_o{\mu }_B^2}{r_n^3}\end{array}\right)& \left(92\right)\end{array}$ The total bond energy of the group E_D(Group) is the negative difference of the total energy of the group (Eq. (92)) and the total energy of the starting species given by the sum of c₄E_initial(c₄ AO/HO) and c₅E_initial(c₅ AO/HO):

$\begin{array}{cc}{E}_{D^{\left(\mathrm{Group}\right)}}=-\left(\begin{array}{c}\left(\begin{array}{c}E\left(\mathrm{basis}\mathrm{energies}\right)+\\ E_T\left(\mathrm{atom—atom},{\mathrm{msp}}^3·\mathrm{AO}\right)\end{array}\right)\\ \left[1+\sqrt{\frac{2\hslash \sqrt{\frac{\frac{C_{1o}{C}_{2o}{e}^2}{4{\mathrm{\pi \varepsilon }}_o{R}^3}}{m_e}}}{m_e{c}^2}}\right]+n_1{\stackrel{\_}{E}}_{\mathrm{Kvib}}+\\ c_3\frac{8{\mathrm{\pi \mu }}_o{\mu }_B^2}{r_n^3}-\left(\begin{array}{c}{c}_4{E}_{\mathrm{initial}}\left(AO\text{/}\mathrm{HO}\right)+\\ c_5{E}_{\mathrm{initial}}\left(c_5\mathrm{AO}\text{/}\mathrm{HO}\right)\end{array}\right)\end{array}\right)& \left(93\right)\end{array}$ In the case of organic molecules, the atoms of the functional groups are energy matched to the C2sp³ HO such thatE(AO/HO)=−14.63489 eV  (94)For example, of E_mag of the C2sp₃ HO is:

$\begin{array}{cc}\begin{array}{c}{E}_{\mathrm{mag}}\left(C2{\mathrm{sp}}^3\right)=c_3\frac{8{\mathrm{\pi \mu }}_o{\mu }_B^2}{r^3}\\ =c_3\frac{8{\mathrm{\pi \mu }}_o{\mu }_B^2}{{\left(0.91771a_0\right)}^3}\\ =c_{3}0.14803\mathrm{eV}\end{array}& \left(95\right)\end{array}$

Each molecule, independently of its complexity and size, is comprised of functional groups wherein each present occurs an integer number of times in the molecule. The total bond energy of the molecule is then given by the integer-weighted sum of the energies of the functions groups corresponding to the composition of the molecule. Thus, integer formulas can be constructed easily for molecules for a given class such as straight-chain hydrocarbons considered as an example infra. The results demonstrate how simply and instantaneously molecules are solved using the classical exact solutions. In contrast, quantum mechanics requires that wavefunction are nonlinear, and any sum must be squared. The results of Millsian disprove quantum mechanics in this regard, and the linearity and superposition properties of Millsian represent a breakthrough with orders of magnitude reduction in complexity in solving molecules as well as being accurate physical representations rather than pure mathematical curve-fits devoid of a connection to reality.

Total Energy of Continuous-Chain Alkanes

E_D(C_nH_2n+2), the total bond dissociation energy of C_nH_2n+2, is given as the sum of the energy components due to the two methyl groups, n−2 methylene groups, and n−1 C—C bonds where each energy component is given by Eqs. (14.590), (14.625), and (14.641), respectively. Thus, the total bond dissociation energy of C_nH_2n+2 is

$\begin{array}{cc}\begin{array}{c}{E}_D\left(C_n{H}_{2n+2}\right)={E_D\left(C\mathrm{—C}\right)}_{n-1}+2E_{D_{\mathrm{alkane}}}\left(\mathrm{CH}_3\right)+\\ \left(n-2\right)E_{D_{\mathrm{alkane}}}\left(\mathrm{CH}_2\right)\\ =\left(n-1\right)\left(4.32754\mathrm{eV}\right)+2\\ \left(12.49186\mathrm{eV}\right)+\left(n-2\right)\left(7.83016\mathrm{eV}\right)\end{array}& \left(96\right)\end{array}$

The experimental total bond dissociation energy of C₂H_2n+2, E_Desp(C_nH_2n+2), is given by the negative difference between the enthalpy of its formation (ΔH_f(C_nH_2n+2(gas))) and the sum of the enthalpy of the formation of the reactant gaseous carbons (ΔH_f(C(gas))) and hydrogen (ΔH_f(H(gas))) atoms:

$\begin{array}{cc}\begin{array}{c}{E}_{D_{\exp }}\left(C_n{H}_{2n+2}\right)=-\left\{\begin{array}{c}\Delta H_f\left(C_n{H}_{2n+2}\left(\mathrm{gas}\right)\right)-\\ \left[\begin{array}{c}n\Delta H_f\left(C\left(\mathrm{gas}\right)\right)+\\ \left(2n+2\right)\Delta H_f\left(H\left(\mathrm{gas}\right)\right)\end{array}\right]\end{array}\right\}\\ =-\left\{\begin{array}{c}\Delta H_f\left(C_n{H}_{2n+2}\left(\mathrm{gas}\right)\right)-\\ \left[\begin{array}{c}n7.42774\mathrm{eV}+\\ \left(2n+2\right)2.259353\mathrm{eV}\end{array}\right]\end{array}\right\}\end{array}& \left(97\right)\end{array}$ where the heats of formation atomic carbon and hydrogen gas are given by [32-33]ΔH_f(C(gas))=716.68 kJ/mole (7.42774 eV/molecule)  (98)ΔH_f(H(gas))=217.998 k/mole (2.259353 eV/molecule)  (99)The comparison of the results predicted by Eq. (96) and the experimental values given by using Eqs. (97-99) with the data from Refs. [32-33] is given in Table 6.

TABLE 6
Summary results of n-alkanes.
		Calculated	Experimental
		Total Bond	Total Bond	Relative
Formula	Name	Energy (eV)	Energy (eV)	Error
C3H8	Propane	41.46896	41.434	−0.00085
C4H10	Butane	53.62666	53.61	−0.00036
C5H12	Pentane	65.78436	65.77	−0.00017
C6H14	Hexane	77.94206	77.93	−0.00019
C7H16	Heptane	90.09976	90.09	−0.00013
C8H18	Octane	102.25746	102.25	−0.00006
C9H20	Nonane	114.41516	114.40	−0.00012
C10H22	Decane	126.57286	126.57	−0.00003
C11H24	Undecane	138.73056	138.736	0.00004
C12H26	Dodecane	150.88826	150.88	−0.00008
C18H38	Octadecane	223.83446	223.85	0.00008

The following list of references, which are also incorporated herein by reference in their entirety, are referred to in the above sections using [brackets]:

REFERENCES

• 1. R. Mills, The Grand Unified Theory of Classical Physics; June 2008 Edition, posted at http://www.blacklightpower.com/theory/bookdownload.shtml.
• 2. R. L. Mills, B. Holverstott, B. Good, N. Hogle, A. Makwana, J. Paulus, “Total Bond Energies of Exact Classical Solutions of Molecules Generated by Millsian 1.0 Compared to Those Computed Using Modern 3-21G and 6-310G* Basis Sets”, submitted.
• 3. R. L. Mills, “Classical Quantum Mechanics”, Physics Essays, Vol. 16, No. 4, December, (2003), pp. 433-498.
• 4. R. Mills, “Physical Solutions of the Nature of the Atom, Photon, and Their Interactions to Form Excited and Predicted Hydrino States”, in press.
• 5. R. L. Mills, “Exact Classical Quantum Mechanical Solutions for One- Through Twenty-Electron Atoms”, Physics Essays, Vol. 18, (2005), pp. 321-361.
• 6. R. L. Mills, “The Nature of the Chemical Bond Revisited and an Alternative Maxwellian Approach”, Physics Essays, Vol. 17, (2004), pp. 342-389.
• 7. R. L. Mills, “Maxwell's Equations and QED: Which is Fact and Which is Fiction”, Vol. 19, (2006), pp. 225-262.
• 8. R. L. Mills, “Exact Classical Quantum Mechanical Solution for Atomic Helium Which Predicts Conjugate Parameters from a Unique Solution for the First Time”, in press.
• 9. R. L. Mills, “The Fallacy of Feynman's Argument on the Stability of the Hydrogen Atom According to Quantum Mechanics,” Annales de la Fondation Louis de Broglie, Vol. 30, No. 2, (2005), pp. 129-151.
• 10. R. Mills, “The Grand Unified Theory of Classical Quantum Mechanics”, Int. J. Hydrogen Energy, Vol. 27, No. 5, (2002), pp. 565-590.
• 11. 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.
• 12. R. Mills, “The Hydrogen Atom Revisited”, Int. J. of Hydrogen Energy, Vol. 25, Issue 12, December, (2000), pp. 1171-1183.
• 13. R. Mills, “The Grand Unified Theory of Classical Quantum Mechanics”, Global Foundation, Inc. Orbis Scientiac entitled The Role of Attractive and Repulsive Gravitational Forces in Cosmic Acceleration of Particles The Origin of the Cosmic Gamma Ray Bursts, (29th Conference on High Energy Physics and Cosmology Since 1964) Dr. Behram N. Kursunoglu, Chairman, Dec. 14-17, 2000, Lago Mar Resort, Fort Lauderdale, Fla., Kluwer Academic/Plenum Publishers, New York, pp. 243-258.
• 14. P. Pearle, Foundations of Physics, “Absence of radiationless motions of relativistically rigid classical electron”, Vol. 7, Nos. 11/12, (1977), pp. 931-945.
• 15. V. F. Weisskopf, Reviews of Modern Physics, Vol. 21, No. 2, (1949), pp. 305-315.
• 16. A. Einstein, B. Podolsky, N. Rosen, Phys. Rev., Vol. 47, (1935), p. 777.
• 17. 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.
• 18. F. Laloë, Do we really understand quantum mechanics? Strange correlations, paradoxes, and theorems, Am. J. Phys. 69 (6), June 2001, 655-701.
• 19. F. Dyson, “Feynman's proof of Maxwell equations”, Am. J. Phys., Vol. 58, (1990), pp. 209-211.
• 20. H. A. Haus, “On the radiation from point charges”, American Journal of Physics, Vol. 54, (1986), 1126-1129.
• 21. J. D. Jackson, Classical Electrodynamics, Second Edition, John Wiley & Sons, New York, (1975), pp. 739-779.
• 22. J. D. Jackson, Classical Electrodynamics, Second Edition, John Wiley & Sons, New York, (1975), p. 111.
• 23. T. A. Abbott and D. J. Griffiths, Am. J. Phys., Vol. 153, No. 12, (1985), pp. 1203-1211.
• 24. G. Goedecke, Phys. Rev 135B, (1964), p. 281.
• 25. http://www.blacklightpower.com/theory/theory.shtml.
• 26. W. J. Nellis, “Making Metallic Hydrogen,” Scientific American, May, (2000), pp. 84-90.
• 27. J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin, J. C. Kieffer, P. B. Corkum, D. M. Villeneuve, “Tomographic imaging of molecular orbitals”, Nature, Vol. 432, (2004), pp. 867-871.
• 28. J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, 1941), p. 195.
• 29. J. D. Jackson, Classical Electrodynamics, 2^nd Edition (John Wiley & Sons, New York, (1975), pp. 17-22.
• 30. H. A. Haus, J. R. Melcher, “Electromagnetic Fields and Energy,” Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, (1985), Sec. 5.3.
• 31. NIST Computational Chemistry Comparison and Benchmark Data Base, NIST Standard Reference Database Number 101, Release 14, September, (2006), Editor R. D. Johnson III, http://srdata.nist.gov/cccbdb.
• 32. D. R. Lide, CRC Handbook of Chemistry and Physics, 79th Edition, CRC Press, Boca Raton, Fla., (1998-9), pp. 9-63.
• 33. D. R. Lide, CRC Handbook of Chemistry and Physics, 79th Edition, CRC Press, Boca Raton, Fla., (1998-9), pp. 5-1 to 5-60.The equation numbers and sections-referenced herein infra. are those disclosed in R. Mills, The Grand Unified Theory of Classical Physics; June 2008 Edition, posted at http://www.blacklightpower.com/theory/bookdownload.shtml which is herein incorporated by reference in its entirety.General Considerations of the Bonding in Pharmaceutical and Specialty Molecules

Pharmaceutical and specialty molecules comprising an arbitrary number of atoms can be solved using similar principles and procedures as those used to solve general organic molecules of arbitrary length and complexity. Pharmaceuticals and specialty molecules can be considered to be comprised of functional groups such those of 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, ureas, acid halides, acid anhydrides, nitriles, thiols, sulfides, disulfides, sulfoxides, sulfones, sulfite, sulfates, nitro alkanes, nitrites, nitrates, conjugated polyenes, aromatics, heterocyclic aromatics, substituted aromatics, and others given in the Organic Molecular Functional Groups and Molecules section. The solutions of the functional groups can be conveniently obtained by using generalized forms of the geometrical and energy equations. The functional-group solutions can be made into a linear superposition and sum, respectively, to give the solution of any pharmaceutical or specialty molecule comprising these groups. The total bond energies of exemplary pharmaceutical or specialty molecules such as aspirin, RDX, and NaH are calculated using the functional group composition and the corresponding energies derived in the previous sections as well as those of any new component functional groups derived herein.

Aspirin (Acetylsalicylic Acid)

Aspirin comprises salicylic acid (ortho-hydroxybenzoic acid) with the H of the phenolic OH group replaced by an acetyl group. Thus, aspirin comprises the benzoic acid C—C(O)—OH moiety that comprises C═O and OH functional groups that are the same as those of carboxylic acids given in the corresponding section. The single bond of aryl carbon to the carbonyl carbon atom, C—C(O), is also a functional group given in the Benzoic Acid Compounds section. The aromatic

$C\stackrel{3e}{=}C$ and C—H functional groups are equivalent to those of benzene given in Aromatic and Heterocyclic Compounds section. The phenolic ester C—O functional group is equivalent to that given in the Phenol section. The acetyl O—C(O)—CH₃ moiety comprises (i) C═O and C—C functional groups that are the same as those of carboxylic acids and esters given in the corresponding sections, (ii) a CH₃ group that is equivalent to that of alkanes given in the corresponding sections, (iii) and a C—O bridging the carbonyl carbon and the phenolic ester which is equivalent to that of esters given in the corresponding section.

The symbols of the functional groups of aspirin are given in Table 7. The corresponding designations of aspirin are shown in FIG. 5. The geometrical (Eqs. (15.1-15.5) and (15.51)), intercept (Eqs. (15.80-15.87)), and energy (Eqs. (15.6-15.11) and (15.17-15.65)) parameters of aspirin are given in Tables 8, 9, and 10, respectively. The total energy of aspirin given in Table 11 was calculated as the sum over the integer multiple of each E_D(Group) of Table 10 corresponding to functional-group composition of the molecule. The bond angle parameters of aspirin determined using Eqs. (15.88-15.117) are given in Table 12. The color scale, translucent view of the charge density of aspirin comprising the concentric shells of atoms with the outer shell bridged by one or more H₂-type ellipsoidal MOs or joined with one or more hydrogen MOs is shown in FIG. 6.

TABLE 7
The symbols of functional groups of aspirin.
Functional Group           Group Symbol
CC (aromatic bond)
CH (aromatic)              CH
Aryl C—C(O)                C—C(O) (i)
Alkyl C—C(O)               C—C(O) (ii)
C═O (aryl carboxylic acid) C═O
Aryl (O)C—O                C—O (i)
Alkyl (O)C—O               C—O (ii)
Aryl C—O                   C—O (iii)
OH group                   OH
CH3 group                  CH3

TABLE 8
The geometrical bond parameters of aspirin
and experimental values of similar molecules [1].
Parameter	Group	CH Group	C—C(O) (i) Group	C—C(O) (ii) Group	C═O Group
a (a0)	1.47348	1.60061	1.95111	2.04740	1.29907
c′ (a0)	1.31468	1.03299	1.39682	1.43087	1.13977
Bond	1.39140	1.09327	1.47833	1.51437	1.20628
Length
2c′ (Å)
Exp. Bond	1.399	1.101	1.48[2]	1.520	1.214
Length	(benzene)	(benzene)	(benzoic	(acetic	(acetic
(Å)			acid)	acid)	acid)
b, c (a0)	0.66540	1.22265	1.36225	1.46439	0.62331
e	0.89223	0.64537	0.71591	0.69887	0.87737
	C—O (i)	C—O (ii)	C—O (iii)	OH	C—H (CH3)
Parameter	Group	Group	Group	Group	Group
a (a0)	1.73490	1.73490	1.68220	1.26430	1.64920
c′ (a0)	1.31716	1.31716	1.29700	0.91808	1.04856
Bond	1.39402	1.39402	1.37268	0.971651	1.10974
Length
2c′ (Å)
Exp. Bond	1.393	1.393	1.364	0.972	1.08
Length	(methyl	(avg.	(phenol)	formic	(methyl
(Å)	formate)	methyl		acid	formate)
		formate)			1.107
					(C—H
					propane)
					1.117
					(C—H
					butane)
b, c (a0)	1.12915	1.12915	1.07126	0.86925	1.27295
e	0.75921	0.75921	0.77101	0.72615	0.63580

TABLE 9
The MO to HO intercept geometrical bond parameters of aspirin. ET is ET (atom-atom, msp3 · AO).
		ET	ET	ET	ET	Final Total
		(eV)	(eV)	(eV)	(eV)	Energy	rinitial	rfinal
Bond	Atom	Bond 1	Bond 2	Bond 3	Bond 4	C2sp3 (eV)	(a0)	(a0)
C—H (CcH)	Cc	−0.85035	−0.85035	−0.56690	0	−153.88327	0.91771	0.79597
	Cc	−0.85035	−0.85035	−0.56690	0	−153.88327	0.91771	0.79597
CbCa(O)O—H	O	−0.92918	0	0	0		1.00000	0.86359
CbCa(O)—OH	O	−0.92918	0	0	0		1.00000	0.86359
CbCa(O)—OH	Ca	−0.92918	−1.34946	−0.64574	0	−154.54007	0.91771	0.76652
CbCa(OH)═O	O	−1.34946	0	0	0		1.00000	0.84115
OCe(CfH3)═O
CbCa(OH)═O	Ca	−1.34946	−0.64574	−0.92918	0	−154.54007	0.91771	0.76652
Cb—Ca(O)OH	Ca	−0.64574	−1.34946	−0.92918	0	−154.54007	0.91771	0.76652
Cb—Ca(O)OH	Cb	−0.64574	−0.85035	−0.85035	0	−153.96212	0.91771	0.79232
	Cb	−0.64574	−0.85035	−0.85035	0	−153.96212	0.91771	0.79232
	Cd	−0.74804	−0.85035	−0.85035	0	−154.06442	0.91771	0.78762
	O	−0.74804	−0.92918	0	0		1.00000	0.82445
	O	−0.92918	−0.74804	0	0		1.00000	0.82445
O—Ce(O)CfH3	Ce	−0.92918	−1.34946	−0.92918	0	−154.82352	0.91771	0.75447
OCe(CfH3)═O	Ce	−1.34946	−0.92918	−0.92918	0	−154.82352	0.91771	0.75447
O(O)Ce—CfH3	Ce	−0.92918	−1.34946	−0.92918	0	−154.82352	0.91771	0.75447
OCe(O)—CfH3	Cf	−0.92918	0	0	0	−152.54487	0.91771	0.86359
	ECoulomb(C2sp3)	E(C2sp3)
	(eV)	(eV)	θ′	θ1	θ2	d1	d2
Bond	Final	Final	(°)	(°)	(°)	(a0)	(a0)
C—H (CcH)	−17.09334	−16.90248	74.42	105.58	38.84	1.24678	0.21379
	−17.09334	−16.90248	134.24	45.76	58.98	0.75935	0.55533
CbCa(O)O—H	−15.75493		115.09	64.91	64.12	0.55182	0.36625
CbCa(O)—OH	−15.75493		101.32	78.68	48.58	1.14765	0.16950
CbCa(O)—OH	−17.75013	−17.55927	93.11	86.89	42.68	1.27551	0.04165
CbCa(OH)═O	−16.17521		137.27	42.73	66.31	0.52193	0.61784
OCe(CfH3)═O
CbCa(OH)═O	−17.75013	−17.55927	134.03	45.97	62.14	0.60699	0.53278
Cb—Ca(O)OH	−17.75013	−17.55927	70.34	109.66	32.00	1.65466	0.25784
Cb—Ca(O)OH	−17.17218	−16.98131	73.74	106.26	33.94	1.61863	0.22181
	−17.17218	−16.98132	134.09	45.91	58.79	0.76344	0.55124
	−17.27448	−17.08362	100.00	80.00	46.39	1.16026	0.13674
	−16.50297		102.93	77.02	48.60	1.11250	0.18449
	−16.50297		98.22	81.78	46.27	1.19921	0.11795
O—Ce(O)CfH3	−18.03358	−17.84271	91.96	88.04	41.90	1.29138	0.02578
OCe(CfH3)═O	−18.03358	−17.84271	133.47	46.53	61.46	0.62072	0.51905
O(O)Ce—CfH3	−18.03358	−17.84272	56.25	123.75	25.37	1.85002	0.41915
OCe(O)—CfH3	−15.75493	−15.56407	72.27	107.73	34.17	1.69388	0.26301

TABLE 10
The energy parameters (eV) of functional groups of aspirin.
Parameters	Group	CH Group	C—C(O) (i) Group	C—C(O) (ii) Group	C═O Group
fl	0.75	1
n1	2	1	1	1	2
n2	0	0	0	0	0
n3	0	0	0	0	0
C1	0.5	0.75	0.5	0.5	0.5
C2	0.85252	1	1	1	1
c1	1	1	1	1	1
c2	0.85252	0.91771	0.91771	0.91771	0.85395
c3	0	1	0	0	2
c4	3	1	2	2	4
c5	0	1	0	0	0
C1o	0.5	0.75	0.5	1	0.5
C2o	0.85252	1	1	1	1
Ve (eV)	−101.12679	−37.10024	−32.15216	−30.19634	−111.25473
Vp (eV)	20.69825	13.17125	9.74055	9.50874	23.87467
T (eV)	34.31559	11.58941	8.23945	7.37432	42.82081
Vm (eV)	−17.15779	−5.79470	−4.11973	−3.68716	−21.41040
E(AO/HO) (eV)	0	−14.63489	−14.63489	−14.63489	0
ΔEH2MO(AO/HO) (eV)	0	−1.13379	−1.29147	0	−2.69893
ET (AO/HO) (eV)	0	−13.50110	−13.34342	−14.63489	2.69893
ET (H2MO) (eV)	−63.27075	−31.63539	−31.63530	−31.63534	−63.27074
ET (atom-atom, msp3 · AO) (eV)	−2.26759	−0.56690	−1.29147	−1.85836	−2.69893
ET (MO) (eV)	−65.53833	−32.20226	−32.92684	−33.49373	−65.96966
ω (1015 rad/s)	49.7272	26.4826	10.7262	23.3291	59.4034
EK (eV)	32.73133	17.43132	7.06019	15.35563	39.10034
Ē D (eV)	−0.35806	−0.26130	−0.17309	−0.25966	−0.40804
ĒKvib (eV)	0.19649	0.35532	0.10502	0.10502	0.21077
	[3]	Eq. (13.458)	[4]	[4]	[5]
Ēosc (eV)	−0.25982	−0.08364	−0.12058	−0.20715	−0.30266
Emag (eV)	0.14803	0.14803	0.14803	0.14803	0.11441
ET (Group) (eV)	−49.54347	−32.28590	−33.04742	−33.70088	−66.57498
Einitial (C4 AO/HO) (eV)	−14.63489	−14.63489	−14.63489	−14.63489	−14.63489
Einitial (C5 AO/HO) (eV)	0	−13.59844	0	0	0
ED (Group) (eV)	5.63881	3.90454	3.77764	4.43110	7.80660
	C—O (i)	C—O (ii)	C—O (iii)	OH	CH3
Parameters	Group	Group	Group	Group	Group
fl
n1	1	1	1	1	3
n2	0	0	0	0	2
n3	0	0	0	0	0
C1	0.5	0.5	0.5	0.75	0.75
C2	1	1	1	1	1
c1	1	1	1	0.75	1
c2	0.85395	0.85395	0.79329	1	0.91771
c3	0	0	0	1	0
c4	2	2	2	1	1
c5	0	0	0	1	3
C1o	0.5	0.5	0.5	0.75	0.75
C2o	1	1	1	1	1
Ve (eV)	−35.08488	−35.08488	−34.04658	−40.92709	−107.32728
Vρ (eV)	10.32968	10.32968	10.49024	14.81988	38.92728
T (eV)	10.11150	10.11150	10.11966	16.18567	32.53914
Vm (eV)	−5.05575	−5.05575	−5.05983	−8.09284	−16.26957
E(AO/HO) (eV)	−14.63489	−14.63489	−14.63489	−13.6181	−15.56407
ΔEH2MO (AO/HO) (eV)	−2.69893	−2.69893	−1.49608	0	0
ET (AO/HO) (eV)	−11.93596	−11.93596	−13.13881	−13.6181	−15.56407
ET (H2MO) (eV)	−31.63541	−31.63541	−31.63532	−31.63247	−67.69451
ET (atom-atom, msp3 · AO) (eV)	−1.85836	−1.85836	−1.49608	0	0
ET (MO) (eV)	−33.49373	−33.49373	−33.13145	−31.63537	−67.69450
ω (1015 rad/s)	24.3637	12.7926	13.3984	44.1776	24.9286
EK (eV)	16.03660	8.42030	8.81907	29.07844	16.40846
ĒD (eV)	−0.26535	−0.19228	−0.19465	−0.33749	−0.25352
ĒKvib (eV)	0.14010	0.14965	0.12808	0.46311	0.35532
	[6]	[7]	[8]	[9-10]	(Eq. (13.458))
Ēosc (eV)	−0.19530	−0.11745	−0.13061	−0.10594	−0.22757
Ēmag (eV)	0.14803	0.14803	0.14803	0.11441	0.14803
ET (Group) (eV)	−33.68903	−33.61118	−33.26206	−31.74130	−67.92207
Einitial (c4 AO/HO) (eV)	−14.63489	−14.63489	−14.63489	−13.6181	14.63489
Einitial (c5 AO/HO) (eV)	0	0	0	−13.59844	−13.59844
ED (Group) (eV)	4.41925	4.34141	3.99228	4.41035	12.49186

TABLE 11
The total bond energies of salicylic acid and aspirin calculated
using the functional group composition and the energies of Table 10.
Formula	Name	Group	CH Group	C—C(O) (i) Group	C—C(O) (ii) Group	C═O Group	C—O (i) Group	C—O (ii) Group
C7H6O3	Salicylic acid	6	4	1	0	1	1	0
C9H8O4	Aspirin	6	4	1	1	2	1	1
					Calculated	Experimental
		C—O (iii)	OH	C—H (CH3)	Total Bond	Total Bond	Relative
Formula	Name	Group	Group	Group	Energy (eV)	Energy (eV)	Error
C7H6O3	Salicylic acid	1	2	0	78.26746 [11]	78.426	0.00202
C9H8O4	Aspirin	1	1	1	102.92809

TABLE 12
The bond angle parameters of aspirin and experimental values [1].
			2c′		Atom 1		Atom 2
	2c′	2c′	Terminal		Hybridization		Hybridization
Atoms of	Bond 1	Bond 2	Atoms	ECoulombic	Designation	ECoulombic	Designation	c2	c2
Angle	(a0)	(a0)	(a0)	Atom 1	(Table 13)	Atom 2	(Table 13)	Atom 1	Atom 2
∠CCC	2.62936	2.62936	4.5585	−17.17218	34	−17.17218	34	0.79232	0.79232
(aromatic)
∠CCH
∠CCO
(aromatic)
∠CaObH	2.63431	1.83616	3.6405	−14.82575	1	−14.82575	1	1	0.91771
∠CbCa(O)	2.82796	2.27954	4.4721	−17.17218	34	−13.61806	O	0.79232	0.85395
									(Eq. (15.114))
∠CbCaO	2.82796	2.63431	4.6690	−16.40067	19	−13.61806	O	0.82959	0.85395
									(Eq. (15.114))
∠(O)CaO	2.27954	2.63431	4.3818	−16.17521	12	−15.75493	7	0.84115	0.86359
				(O)		O
∠CfCe(O)	2.86175	2.27954	4.5826	−16.68411	24	−13.61806	O	0.81549	0.85395
									(Eq. (15.133))
∠CfCeO	2.86175	2.63431	4.4944	−15.75493	7	−13.61806	O	0.86359	0.85395
									(Eq. (15.133))
∠OCeO	2.27954	2.63431	4.3818	−16.17521	12	−15.75493	7	0.84115	0.86359
				(O)		O
∠CdOCe	2.59399	2.63431	4.3589	−17.27448	38	−18.03358	53	0.78762	0.75447
				Cd		Ce
Methyl	2.09711	2.09711	3.4252	−15.75493	7	H	H	0.86359	1
∠HCfH
Atoms of					ET	θv	θ1	θ2	Cal. θ	Exp. θ
Angle	C1	C2	c1	c′2	(eV)	(°)	(°)	(°)	(°)	(°)
∠CCC	1	1	1	0.79232	−1.85836				120.19	120 [12-14]
(aromatic)										(benzene)
∠CCH							120.19		119.91	120 [12-14]
∠CCO										(benzene)
(aromatic)
∠CaObH	0.75	1	0.75	0.91771	0				107.71
∠CbCa(O)	1	1	1	0.82313	−1.65376				121.86	122 [2]
										(benzoic acid)
∠CbCaO	1	1	1	0.84177	−1.65376				117.43	118 [2]
										(benzoic acid)
∠(O)CaO	1	1	1	0.85237	−1.44915				126.03	122 [2]
										(benzoic acid)
∠CfCe(O)	1	1	1	0.83472	−1.65376				125.70	126.6 [1]
										(acetic acid)
∠CfCeO	1	1	1	0.85877	−1.44915				109.65	110.6 [1]
										(acetic acid)
∠OCeO	1	1	1	0.85237	−1.44915				126.03
∠CdOCe	1	1	1	0.77105	−1.85836				112.96	114 [1]
										(methyl formate)
Methyl	1	1	0.75	1.15796	0				109.50
∠HCfH
ET is ET(atom−atom, msp3 · AO).

TABLE 13
$\hspace{1em}\begin{array}{c}\mathrm{The}\mathrm{final}\mathrm{values}\mathrm{of}{r}_{\mathrm{Atom},\mathrm{HO},\mathrm{AO}},E_{\mathrm{Coulomb}}\left(\mathrm{mol}.\mathrm{atom},{\mathrm{msp}}^3\right),\mathrm{and}E\left(C_{\mathrm{mol}}C2{\mathrm{sp}}^3\right)\\ \mathrm{calculated}\mathrm{using}\mathrm{the}\mathrm{values}\mathrm{of}{E}_T\left(C\stackrel{\mathrm{BO}}{—}C,C2{\mathrm{sp}}^3\right)\mathrm{given}\mathrm{in}\mathrm{Tables}4\mathrm{and}5.\end{array}$
Atom Hy- brid- iza- tion Des- igna- tion	$E_T\left(C\stackrel{\mathrm{BO}}{—}C,C2{\mathrm{sp}}^3\right)$	$E_T\left(C\stackrel{\mathrm{BO}}{—}C,C2{\mathrm{sp}}^3\right)$	$E_T\left(C\stackrel{\mathrm{BO}}{—}C,C2{\mathrm{sp}}^3\right)$	$E_T\left(C\stackrel{\mathrm{BO}}{—}C,C2{\mathrm{sp}}^3\right)$	$E_T\left(C\stackrel{\mathrm{BO}}{—}C,C2{\mathrm{sp}}^3\right)$	rAtom,HO,AO Final	ECoulomb (mol.atom, msp3) (eV) Final	E(Cmol2sp3) (eV) Final
1	0	0	0	0	0	0.91771	−14.82575	−14.63489
2	−0.36229	0	0	0	0	0.89582	−15.18804	−14.99717
3	−0.46459	0	0	0	0	0.88983	−15.29034	−15.09948
4	−0.56689	0	0	0	0	0.88392	−15.39265	−15.20178
5	−0.72457	0	0	0	0	0.87495	−15.55033	−15.35946
6	−0.85034	0	0	0	0	0.86793	−15.6761	−15.48523
7	−0.92918	0	0	0	0	0.86359	−15.75493	−15.56407
8	−0.54343	−0.54343	0	0	0	0.85503	−15.91261	−15.72175
9	−0.18144	−0.92918	0	0	0	0.85377	−15.93607	−15.74521
10	−1.13379	0	0	0	0	0.85252	−15.95955	−15.76868
11	−1.14485	0	0	0	0	0.85193	−15.9706	−15.77974
12	−0.46459	−0.82688	0	0	0	0.84418	−16.11722	−15.92636
13	−1.34946	0	0	0	0	0.84115	−16.17521	−15.98435
14	−1.3725	0	0	0	0	0.83995	−16.19826	−16.00739
15	−0.46459	−0.92918	0	0	0	0.83885	−16.21952	−16.02866
16	−0.72457	−0.72457	0	0	0	0.836	−16.2749	−16.08404
17	−0.5669	−0.92918	0	0	0	0.8336	−16.32183	−16.13097
18	−0.82688	−0.72457	0	0	0	0.83078	−16.37721	−16.18634
19	−1.56513	0	0	0	0	0.83008	−16.39089	−16.20002
20	−0.64574	−0.92918	0	0	0	0.82959	−16.40067	−16.20981
21	−1.57711	0	0	0	0	0.82948	−16.40286	−16.212
22	−0.72457	−0.92918	0	0	0	0.82562	−16.47951	−16.28865
23	−0.85035	−0.85035	0	0	0	0.82327	−16.52645	−16.33559
24	−1.79278	0	0	0	0	0.81871	−16.61853	−16.42767
25	−1.13379	−0.72457	0	0	0	0.81549	−16.68411	−16.49325
26	−0.92918	−0.92918	0	0	0	0.81549	−16.68412	−16.49325
27	−0.56690	−0.54343	−0.85034	0	0	0.81052	−16.78642	−16.59556
28	−2.02043	0	0	0	0	0.80765	−16.84619	−16.65532
29	−1.13379	−0.92918	0	0	0	0.80561	−16.88872	−16.69786
30	−0.56690	−0.56690	−0.92918	0	0	0.80561	−16.88873	−16.69786
31	−0.85035	−0.85035	−0.46459	0	0	0.80076	−16.99104	−16.80018
32	−0.85035	−0.42517	−0.92918	0	0	0.79891	−17.03045	−16.83959
33	−0.5669	−0.72457	−0.92918	0	0	0.78916	−17.04641	−16.85554
34	−1.13379	−1.13379	0	0	0	0.79597	−17.09334	−16.90248
35	−1.34946	−0.92918	0	0	0	0.79546	−17.1044	−16.91353
36	−0.46459	−0.92918	−0.92918	0	0	0.79340	−17.14871	−16.95784
37	−0.64574	−0.85034	−0.85034	0	0	0.79232	−17.17217	−16.98131
38	−0.85035	−0.5669	−0.92918	0	0	0.79232	−17.17218	−16.98132
39	−0.72457	−0.72457	−0.92918	0	0	0.79085	−17.20408	−17.01322
40	−0.75586	−0.75586	−0.92918	0	0	0.78798	17.26666	17.07580
41	−0.74804	−0.85034	−0.85034	0	0	0.78762	17.27448	17.08362
42	−0.82688	−0.72457	−0.92918	0	0	0.78617	−17.30638	−17.11552
43	−0.72457	−0.92918	−0.92918	0	0	0.78155	−17.40868	−17.21782
44	−0.92918	−0.72457	−0.92918	0	0	0.78155	−17.40869	−17.21783
45	−0.54343	−0.54343	−0.5669	−0.92918	0	0.78155	−17.40869	−17.21783
46	−0.92918	−0.85034	−0.85034	0	0	0.77945	−17.45561	−17.26475
47	−0.42517	−0.42517	−0.85035	−0.92918	0	0.77945	−17.45563	−17.26476
48	−0.82688	−0.92918	−0.92918	0	0	0.77699	−17.51099	−17.32013
49	−0.92918	−0.92918	−0.92918	0	0	0.77247	−17.6133	−17.42244
50	−0.85035	−0.54343	−0.5669	−0.92918	0	0.76801	−17.71561	−17.52475
51	−1.34946	−0.64574	−0.92918	0	0	0.76652	−17.75013	−17.55927
52	−0.85034	−0.54343	−0.60631	−0.92918	0	0.76631	−17.75502	−17.56415
53	−1.1338	−0.92918	−0.92918	0	0	0.7636	−17.81791	−17.62705
54	−0.46459	−0.85035	−0.85035	−0.92918	0	0.75924	−17.92022	−17.72936
55	−0.82688	−1.34946	−0.92918	0	0	0.75877	−17.93128	−17.74041
56	−0.92918	−1.34946	−0.92918	0	0	0.75447	−18.03358	−17.84272
57	−1.13379	−1.13379	−1.13379	0	0	0.74646	−18.22712	−18.03626
58	−1.79278	−0.92918	−0.92918	0	0	0.73637	−18.47690	−18.28604

Cyclotrimethylene-Trinitramine (C₃H₆N₆O₆)

The compound cyclotrimethylene-trinitramine, commonly referred to as Cyclonite or by the code designation RDX, is a well-known explosive. RDX comprises three methylene (CH₂) groups joined by six alkyl C—N secondary amine functional groups given in the corresponding section. Each of the three N's of the six-membered ring shown in FIG. 7 is bonded to a NO₂ functional group given in the Nitroalkanes section by a N—N functional group. The latter requires hybridization of the nitrogen atoms in order to match the energies of the bridged groups.

Similar to the case of carbon, silicon, and aluminum, the bonding in the nitrogen of the N—N functional group involves four sp₃ hybridized orbitals formed from the outer 2p and 2s shells. In RDX, bonds form between two N2sp³ HOs (N—N functional group), between a N2sp³ HO and a C2sp³ HO (C—N functional group), and between a N2sp³ HO and a O2p AO (each N—O bond of the NO₂ functional group). The geometrical and energy equations of the N—N functional group are given in the Derivation of the General Geometrical and Energy Equations of Organic Chemistry section wherein the energy is matched to E(C,2sp₃)=−14.63489 e V (Eq. (15.25)).

The 2sp³ hybridized orbital arrangement after Eq. (13.422) is

$\begin{array}{cc}\frac{\uparrow \downarrow }{0,0}\stackrel{2{\mathrm{sp}}^3\mathrm{state}}{\frac{\uparrow }{1,-1}\frac{\uparrow }{1,0}}\frac{\uparrow }{1,1}& \left(16.1\right)\end{array}$ where the quantum numbers (l,m_t) are below each electron. The total energy of the state is given by the sum over the five electrons. The sum E_T(N,2sp₃) of experimental energies [15] of N, N⁺, N₂₊, N₃₊, and N⁴⁺ is

$\begin{array}{cc}\begin{array}{c}{E}_T\left(N,2{\mathrm{sp}}^3\right)=-\left(\begin{array}{c}97.8902\mathrm{eV}+77.4735\mathrm{eV}+\\ 47.44924\mathrm{eV}+29.6013\mathrm{eV}+\\ 14.53414\mathrm{eV}\end{array}\right)\\ =-266.94838\mathrm{eV}\end{array}& \left(16.2\right)\end{array}$ By considering that the central field decreases by an integer for each successive electron of the shell, the radius r_2sp3 of the N2sp³ shell may be calculated from the Coulombic energy using Eq. (15.13):

$\begin{array}{cc}\begin{array}{c}{r}_{2{\mathrm{sp}}^3}=\sum _{n=2}^6\frac{\left(Z-n\right)e^2}{8{\mathrm{\pi \varepsilon }}_0\left(e266.94838\mathrm{eV}\right)}\\ =\frac{15e^2}{8{\mathrm{\pi \varepsilon }}_0\left(e266.94838\mathrm{eV}\right)}\\ =0.76452a_0\end{array}& \left(16.3\right)\end{array}$ where Z=7 for nitrogen. Using Eq. (15.14), the Coulombic energy E_Coulomb(N,2sp³) of the outer electron of the N2sp³ shell is

$\begin{array}{cc}\begin{array}{c}{E}_{\mathrm{Coulomb}}\left(N,2{\mathrm{sp}}^3\right)=\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_0{r}_{2{\mathrm{sp}}^3}}\\ =\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_{0}0.76452a_0}\\ =-17.79656\mathrm{eV}\end{array}& \left(16.4\right)\end{array}$

In RDX, the C2sp³ HO has a hybridization factor of 0.91771 (Eq. (13.430)) with a corresponding energy of E(C,2sp³)=−14.63489 eV (Eq. (15.25)), and the N HO has an energy of E(N,2sp³)=−17.79656 eV (Eq. (16.4)). To meet the equipotential, minimum-energy condition of the union of the N2sp³ and C2sp³ HOs, C₂=1 in Eqs. (15.2-15.5), (15.51), and (15.61) for the N—N-bond MO, and c₃ given by Eqs. (15.77) and (15.79) is

$\begin{array}{cc}\begin{array}{c}{c}_2(\begin{array}{c}C2{\mathrm{sp}}^3\mathrm{HO}\mathrm{to}{N}_{b}2{\mathrm{sp}}^3\mathrm{HO}\\ \mathrm{to}{N}_{a}2{\mathrm{sp}}^3\mathrm{HO}\end{array})=\frac{E\left(C,2{\mathrm{sp}}^3\right)}{E\left(N,2{\mathrm{sp}}^3\right)}{c}_2\left(C2{\mathrm{sp}}^3\mathrm{HO}\right)\\ =\frac{-14.63489\mathrm{eV}}{-17.79656\mathrm{eV}}\left(0.91771\right)\\ =0.75468\end{array}& \left(16.5\right)\end{array}$ The energy of the N—N-bond MO is the sum of the component energies of the H₂-type ellipsoidal MO given in Eq. (15.51). Since the energy of the MO is matched to that of the C2sp² HO, E(AO/HO) in Eqs. (15.51) and (15.61) is E(C,2sp³)=−14.63489 eV given by Eq. (15.25) and E_T(atom-atom,msp³.AO) is 0 eV.

The symbols of the functional groups of RDX are given in Table 14. The geometrical (Eqs. (15.1-15.5) and (15.51)), intercept (Eqs. (15.80-15.87)), and energy (Eqs. (15.6-15.11) and (15.17-15.65)) parameters of RDX are given in Tables 15, 16, and 17, respectively. The total energy of RDX given in Table 18 was calculated as the sum over the integer multiple of each E_D(Group) of Table 17 corresponding to functional-group composition of the molecule. The bond angle parameters of RDX determined using Eqs. (15.88-15.117) are given in Table 19. The color scale charge density of RDX comprising atoms with the outer shell bridged by one or more H₂-type ellipsoidal MOs or joined with one or more hydrogen MOs is shown in FIG. 8.

TABLE 14
The symbols of functional groups of RDX.
Functional Group Group Symbol
NO2 group        NO2
N—N              N—N
C—N (alkyl)      C—N
CH2 group        C—H (CH2)

TABLE 15
The geometrical bond parameters of RDX and experimental values [1].
	NO2	N—N	C—N	C—H (CH2)
Parameter	Group	Group	Group	Group
a (a0)	1.33221	1.68711	1.94862	1.67122
c′ (a0)	1.15421	1.29889	1.39593	1.05553
Bond Length
2c′ (Å)	1.22157	1.37468	1.47739	1.11713
Exp. Bond Length	1.224 (nitromethane)	1.390 [16] (RDX)	1.468 [16] (RDX)	1.107 (C—H propane)
(Å)	1.22 avg. [16] (RDX)			1.117 (C—H butane)
				1.092 [161] (RDX)
b, c (a0)	0.66526	1.07668	1.35960	1.29569
e	0.86639	0.76989	0.71637	0.63159

TABLE 16
The MO to HO intercept geometrical bond parameters of RDX.
		ET	ET	ET	ET	Final Total
		(eV)	(eV)	(eV)	(eV)	Energy	rinitial	rfinal
Bond	Atom	Bond 1	Bond 2	Bond 3	Bond 4	C2sp3 (eV)	(a0)	(a0)
NbNa(O)═O	Oa	−0.92918	0	0	0		1.00000	0.86359
NbNa(O)═O	Na	−0.92918	−0.92918	0	0		0.93084	0.81549
CH2Nb—NaO2	Na	−0.92918	−0.92918	0	0		0.93084	0.81549
CH2Nb—NaO2	Nb	−0.56690	−0.56690	0	0		0.93084	0.85252
C—H (CH2)	Ca	−0.56690	−0.56690	0	0	−152.74948	0.91771	0.85252
—H2Ca—NbNa	Nb	−0.56690	−0.56690	0	0		0.93084	0.85252
—H2Ca—NbNa	Ca	−0.56690	−0.56690	0	0	−152.74948	0.91771	0.85252
	ECoulomb	E(C2sp3)
	(eV)	(eV)	θ′	θ1	θ2	d1	d2
Bond	Final	Final	(°)	(°)	(°)	(a0)	(a0)
NbNa(O)═O	−15.75493		135.25	44.75	66.05	0.54089	0.61333
NbNa(O)═O	−16.68411		133.16	46.84	63.41	0.59640	0.55781
CH2Nb—NaO2	−16.68411		101.80	78.20	47.85	1.13213	0.16676
CH2Nb—NaO2	−15.95954		104.60	75.40	50.02	1.08404	0.21485
C—H (CH2)	−15.95954	−15.76868	73.60	106.40	39.14	1.29624	0.24071
—H2Ca—NbNa	−15.95954		80.95	99.05	38.26	1.53008	0.13415
—H2Ca—NbNa	−15.95954	−15.76868	80.95	99.05	38.26	1.53008	0.13415
ET is ET(atom-atom, msp3 · AO).

TABLE 17
The energy parameters (eV) of functional groups of RDX.
	NO2	N—N	C—N	CH2
Parameters	Group	Group	Group	Group
n1	2	1	1	2
n2	0	0	0	1
n3	0	0	0	0
C1	0.5	0.5	0.5	0.75
C2	1	1	1	1
c1	1	1	1	1
c2	0.85987	0.75468	0.91140	0.91771
c3	0	0	0	1
c4	4	2	2	1
c5	0	0	0	2
C1o	0.5	0.5	1	0.75
C2o	1	1	1	1
Ve (eV)	−106.90919	−32.25503	−31.98456	−70.41425
Vp (eV)	23.57588	10.47496	9.74677	25.78002
T (eV)	40.12475	9.55926	8.20698	21.06675
Vm (eV)	−20.06238	−4.77963	−4.10349	−10.53337
E(AO/HO) (eV)	0	−14.63489	−14.63489	−15.56407
ΔEH2MO(AO/HO) (eV)	0	0	−1.13379	0
ET(AO/HO) (eV)	0	−14.63489	−13.50110	−15.56407
ET(H2MO) (eV)	−63.27093	−31.63533	−31.63540	−49.66493
ET(atom-atom, msp3 · AO) (eV)	−3.71673	0	−1.13379	0
ET(MO) (eV)	−66.98746	−31.63537	−32.76916	−49.66493
ω (1015 rad/s)	19.0113	26.1663	26.0778	24.2751
EK (eV)	12.51354	17.22313	17.16484	15.97831
ĒD (eV)	−0.23440	−0.25974	−0.26859	−0.25017
ĒKvib (ev)	0.19342 [17]	0.12770 [18]	0.11159 [19]	0.35532
				(Eq. (13.458))
Ēosc (ev)	−0.13769	−0.19588	−0.21280	−0.14502
Emag (eV)	0.11441	0.14803	0.14803	0.14803
ET(Group) (eV)	−67.26284	−31.83125	−32.98196	−49.80996
Einitial(c4AO/HO) (eV)	−14.63489	−14.63489	−14.63489	−14.63489
Einitial(c5AO/HO) (eV)	0	0	0	−13.59844
ED(Group) (eV)	8.72329	2.56147	3.71218	7.83016
Exp. ED(Group) (eV)		Est. 2.86, 2.08 [20]	3.69 [20]

TABLE 18
The total bond energy of gaseous-state RDX calculated using the functional
group composition and the energies of Table 17.
						Calculated	Experimental
		NO2	N—N	C—N	CH2	Total bond	Total Bond
Formula	Name	Group	Group	Group	Group	Energy (eV)	Energy (eV)	Relative Error
C3H6N6O6	RDX	3	3	6	3	79.61783

TABLE 19
The bond angle parameters of RDX and experimental values [1].
			2c′		Atom 1		Atom 2
	2c′	2c′	Terminal		Hybridization		Hybridization
Atoms of	Bond 1	Bond 2	Atoms	ECoulombic	Designation	ECoulombic	Designation	c2	c2
Angle	(α0)	(α0)	(α0)	Atom 1	(Table 13)	Atom 2	(Table 13)	Atom 1	Atom 2
∠OaNOb	2.30843	2.30843	4.1231	−16.68411	24	−16.68411	24	0.81549	0.81549
				Oa		Ob
∠NbNaOa	2.59778	2.27630	4.0988	−17.79656		−13.61806		0.75468	0.85987
				Nb (Eq. (16.4))		Oa		(Eq. (16.5))	(Eq. (15.159))
∠CNbNa	2.79186	2.59778	4.5826	−16.32183	16	−14.53414		0.83360	0.91140
									(Eq. (15.135))
∠CNC	2.79186	2.79186	4.6260	−17.04640	31	−17.04640	31	0.79816	0.79816
Methylene	2.11106	2.11106	3.4252	−15.75493	7	H	H	0.86359	1
∠HCaH
∠HCN	2.09711	2.79186	4.0661	−14.82575	1	−14.53414	N	0.91771	0.93383
									(Eq. (15.136))
Atoms of					ET	θv	θ1	θ2	Cal. θ	Exp. θ
Angle	C1	C2	c1	c2′	(eV)	(°)	(°)	(°)	(°)	(°)
∠OaNOb	1	1	1	0.81549	−1.44915				126.52	125.3
										(nitromethane)
∠NbNaOa	1	1	1	0.80727	−1.44915				114.32	116.8
										[16] (RDX)
∠CNbNa	1	1	1	0.87250	−1.44915				116.43	116.6
										[16] (RDX)
∠CNC	1	1	1	0.79816	−1.85836				111.89	111.8
										(dimethylamine)
Methylene	1	1	0.75	1.15796	0				108.44	107
∠HCaH										(propane)
∠HCN	0.75	1	0.75	1.01756	0				111.76	112
										(dimethylamine)

Sodium Hydride Molecule (NaH)

Alkali hydride molecules each comprising an alkali metal atom and a hydrogen atom can be solved using similar principles and procedures as those used to solve organic molecules. The solutions of these molecules can be conveniently obtained by using generalized forms of the force balance equation given in the Force Balance of the σ MO of the Carbon Nitride Radical section and the geometrical and energy equations given in the Derivation of the General Geometrical and Energy Equations of Organic Chemistry section.

The bonding in the sodium atom involves the outer 3s atomic orbital (AO), and the Na—H bond forms between the Na3s AO and the H1s AO. The energy of the reactive outer electron of the sodium atom is significantly less than the Coulombic energy between the electron and proton of H given by Eq. (1.243). Consequently, the outer electron comprising the Na3s AO and the H1s AO form a σ-MO, and the inner AOs of Na remain unaltered. The MO semimajor axis of molecular sodium hydride is determined from the force balance equation of the centrifugal, Coulombic, and magnetic forces as given in the Polyatomic Molecular Ions and Molecules section and the More Polyatomic Molecules and Hydrocarbons section. Then, the geometric and energy parameters of the MO are calculated using Eqs. (15.1-15.117) wherein the distance from the origin of the H₂-type-ellipsoidal-MO to each focus c′, the internuclear distance 2c′, and the length of the semiminor axis of the prolate spheroidal H₂-type MO b=c are solved from the semimajor axis a.

The force balance of the centrifugal force equated to the Coulombic and magnetic forces is solved for the length of the semimajor axis. The Coulombic force on the pairing electron of the MO is

$\begin{array}{cc}{F}_{\mathrm{Coulomb}}=\frac{e^2}{8{\mathrm{\pi \varepsilon }}_0{\mathrm{ab}}^2}{\mathrm{Di}}_{\xi }& \left(16.6\right)\end{array}$ The spin pairing force is

$\begin{array}{cc}{F}_{\mathrm{spin}-\mathrm{pairing}}=\frac{{\hslash }^2}{2m_e{a}^2{b}^2}{\mathrm{Di}}_{\xi }& \left(16.7\right)\end{array}$ The diamagnetic force is:

$\begin{array}{cc}{F}_{\mathrm{diamagneticMO}1}=-\frac{n_e{\hslash }^2}{4m_e{a}^2{b}^2}{\mathrm{Di}}_{\xi }& \left(16.8\right)\end{array}$ where n_e is the total number of electrons that interact with the binding σ-MO electron. The diamagnetic force F_{diamagneticMO2} on the pairing electron of the σ MO is given by the sum of the contributions over the components of angular momentum:

$\begin{array}{cc}{F}_{\mathrm{diamagnetic}\mathrm{MO}2}=-\sum _{i,j}\frac{L_i\hslash }{Z_{j}2m_e{a}^2{b}^2}{\mathrm{Di}}_{\xi }& \left(16.9\right)\end{array}$ where |L| is the magnitude of the angular momentum of each atom at a focus that is the source of the diamagnetism at the σ-MO. The centrifugal force is

$\begin{array}{cc}{F}_{\mathrm{centrifugalMO}}=-\frac{{\hslash }^2}{m_e{a}^2{b}^2}{\mathrm{Di}}_{\xi }& \left(16.10\right)\end{array}$

The force balance equation for the σ-MO of the Na—H-bond MO with n_e=2 an

$\begin{array}{cc}\begin{array}{c}L=\left(2+\sqrt{\frac{3}{4}}\right)\hslash \mathrm{is}\frac{{\hslash }^2}{m_e{a}^2{b}^2}D\\ =\frac{e^2}{8{\mathrm{\pi \varepsilon }}_0{\mathrm{ab}}^2}D+\frac{{\hslash }^2}{2m_e{a}^2{b}^2}D-\left(\frac{2}{2}+\frac{2}{Z}+\frac{\sqrt{\frac{3}{4}}}{Z}\right)\\ \frac{{\hslash }^2}{2m_e{a}^2{b}^2}D\end{array}& \left(16.11\right)\\ a={\left(2+\frac{2}{Z}+\frac{\sqrt{\frac{3}{4}}}{Z}\right)}_{a_0}& \left(16.12\right)\end{array}$ With Z=11, the semimajor axis of the Na—H-bond MO isa=2.26055a₀  (16.13)

Using the semimajor axis, the geometric and energy parameters of the MO are calculated using Eqs. (15.1-15.117) in the same manner as the organic functional groups given in the Organic Molecular Functional Groups and Molecules section. For the Na—H-bond MO of the NaH, c₁=1, c₂=1 and C₂=1 in both the geometry relationships (Eqs. (15.2-15.5)) and the energy equation (Eq. (15.61)). In NaH the molecule, the Na3s AO has an energy of E(Na3s)=−5.139076 eV [15] and the H AO has an energy of E(H)=−13.59844 eV [15]. To meet the equipotential condition of the union of the Na3s AO and the H1s AO, c₂ and C₂ of Eqs. (15.2-15.5) and Eq. (15.61) for the Na—H-bond MO given by Eq. (15.77) is

$\begin{array}{cc}\begin{array}{c}{C}_2\left(\mathrm{Na}\mathrm{3s}\mathrm{AO}\mathrm{to}H1s\mathrm{AO}\right)=c_2\left(\mathrm{Na}3s\mathrm{AO}\mathrm{to}H1s\mathrm{AO}\right)\\ =\frac{-5.139076\mathrm{eV}}{-13.59844\mathrm{eV}}\\ =0.37792\end{array}& \left(16.14\right)\end{array}$

The energy of the MO is matched to that of the Na2p AO with which in intersects such that E(AO/HO) is E(Na2p)=−47.2864 eV [15]; thus, E_initial(c₄ AO/HO) (eV) is given by the sum of E(Na2p)=−47.2864 eV and E(Na3s)=−5.139076 eV.

The symbol of the functional group of molecular NaH is given in Table 20. The geometrical (Eqs. (15.1-15.5) and (16.11-16.14)), intercept (Eqs. (15.80-15.87)), and energy (Eqs. (15.61-15.65) and (16.13-16.14)) parameters of molecular NaH are given in Tables 21, 22, and 23, respectively. The color scale, translucent view of the charge-densities of molecular NaH comprising the concentric shells of the inner AOs of the Na atom and an outer MO formed from the outer Na3s AO and the His AO are shown in FIG. 9.

TABLE 20
The symbol of the functional group of molecular NaH.
Functional Group Group Symbol
NaH group        Na—H

TABLE 21
The geometrical bond parameters of molecular
NaH and experimental values [20].
                 Na—H
Parameter        Group
a (a0)           2.26055
c′ (a0)          1.72939
Bond Length      1.83031
2c′ (Å)
Exp. Bond Length 1.88654
(Å)              (NaH)
b, c (a0)        1.45577
e                0.76503

TABLE 22
The MO to Na2p AO intercept geometrical bond parameters of NaH.
		ET	ET	ET	ET
		(eV)	(eV)	(eV)	(eV)
Bond	Atom	Bond 1	Bond 2	Bond 3	Bond 4
Na—H (NaH)	Na	0	0	0	0
	Final Total			ECoulomb(Na2p)	E(Na2p)
	Energy Na2p	rinitial	rfinal	(eV)	(eV)
Bond	(eV)	(a0)	(a0)	Final	Final
Na—H (NaH)		2.65432	0.56094		−47.2864
	θ′	θ1	θ2	d1	d2
Bond	(°)	(°)	(°)	(a0)	(a0)
Na—H (NaH)	28.66	151.34	10.65	2.22161	0.49221

TABLE 23
The energy parameters (eV) of the Na—H functional
group of molecular NaH.
                              Na—H
Parameters                    Group
n1                            1
n2                            0
n3                            0
C1                            0.37792
C2                            1
c1                            1
c2                            1
c3                            0
c4                            1
c5                            1
C1o                           0.37792
C2o                           1
Ve (eV)                       −31.72884
Vp (eV)                       7.86738
T (eV)                        7.01795
Vm (eV)                       −3.50898
E(AO/HO) (eV)                 −47.2864
ΔEH2MO(AO/HO) (eV)            0
ET(AO/HO) (eV)                −47.2864
ET(H2MO) (eV)                 −67.63888
ET(atom-atom, msp3 · AO) (eV) 0
ET(MO) (eV)                   −67.63888
ω (1015 rad/s)                14.4691 [20]
EK (eV)                       9.52384
ĒD (eV)                       −0.41296
ĒKvib (ev)                    0.14534
Ēosc (ev)                     −0.34029
Emag (eV)                     0.11441
ET(Group) (eV)                −67.97917
Einitial(c4AO/HO) (eV)        −52.425476
Einitial(c5AO/HO) (eV)        −13.59844
ED(Group) (eV)                1.95525
Exp. ED(Group) (eV)           1.92451 (Na—H [21])

Bond and Dipole Moments

The bond moment of a functional group may be calculated by considering the charge donation between atoms of the functional group. Since the potential of an MO is that of a point charge at infinity (Eq. (11.36)), an asymmetry in the distribution of charge between nonequivalent HOs or AOs of the MO occurs to maintain an energy match of the MO with the bridged orbitals. The charge must redistribute between the spherical orbitals to achieve a corresponding current-density that maintains constant current at the equivalent-energy condition according to the energy-matching factor such as c₂ or C₂ of Eqs. (15.51) and (15.61). Since the orbital energy and radius are reciprocally related, the contribution scales as the square of the ratio (over unity) of the energy of the resultant net positively-charged orbital and the initial matched energy of the resultant net negatively-charged orbital of the bond multiplied by the energy-matching factor (e.g. c₂ or C₂). The partial charges on the HOs or AOs corresponding to the charge contribution are equivalent to point charges centered on the nuclei. Due to symmetry, the bond moment μ of each functional group is along the internuclear axis and is calculated from the partial changes at the separation distance, the internuclear distance.

Using the reciprocal relationship between the orbital energies and radii, the dependence of the orbital area on the radius squared, and the relationship of the partial charge q to the areas with energy matching for each electron of the MO, the bond moment μ along the internuclear axis of A-B wherein A is the net positively-charged atom is given by

$\begin{array}{cc}\mu =\mathrm{qd}=n_1\mathrm{ce}\left(1-{\left(\frac{E_A\left(\mathrm{valence}\right)}{E_B\left(\mathrm{valence}\right)}\right)}^2\right)2c^{\prime }& \left(16.15\right)\end{array}$ wherein n₁ is the number of equivalent bonds of the MO, c is energy-matching factor su as c₁, c₂, C₁, or C₂ of Eqs. (15.51) and (15.61) where c₁ and C₂ may correspond to both electrons of a MO localized on one AO or HO such as when the magnitude of the valence or Coulombic energy of the AO or HO is less than that of E_Coulomb(H)=−13.605804 eV or when the orbital may contain paired or shared electrons in a linear combination with the partner orbital, and d is the charge-separation distance, the internuclear distance 2c′. E_B(valence) is the initial matched energy of the resultant net negatively-charged orbital of the bond that is further lowered by bonding (Eqs. (15.32) and (15.16)) to atom A having an energy E_A(valence). Typically, E_B(valence) of a carbon-heteroatom bond is −14.63489 eV, the initial C2sp³ HO (Eq. (15.25)) energy to which the heteroatom is energy matched. Functional group bond moments determined using Eq. (16.15) are given in Table 24.

$\begin{array}{cc}E\left(\mathrm{atom},{\mathrm{msp}}^3\right)=\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_0{r}_{{\mathrm{msp}}^3}}+\frac{2{\mathrm{\pi \mu }}_0{e}^2{\hslash }^2}{m_e^2{r}^3}& \left(15.16\right)\\ \begin{array}{c}E\left(C,2{\mathrm{sp}}^3\right)=\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_0{r}_{2{\mathrm{sp}}^3}}+\frac{2{\mathrm{\pi \mu }}_0{e}^2{\hslash }^2}{{m_e^2\left(r_3\right)}^3}\\ =-14.82575\mathrm{eV}+0.19086\mathrm{eV}\\ =-14.63489\mathrm{eV}\end{array}& \left(15.25\right)\\ \begin{array}{c}{r}_{\mathrm{mol}2{\mathrm{sp}}^3}=\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_0\left(E_{\mathrm{Coulomb}}\left(C,2{\mathrm{sp}}^3\right)+\sum E_{T_{\mathrm{mol}}}\left(\mathrm{MO},2{\mathrm{sp}}^3\right)\right)}\\ =\frac{e^2}{8{\mathrm{\pi \varepsilon }}_0\left(e14.825751\mathrm{eV}+\sum E_{T_{\mathrm{mol}}}\left(\mathrm{MO},2{\mathrm{sp}}^3\right)\right)}\end{array}& \left(15.32\right)\\ -\frac{n_1{e}^2}{8{\mathrm{\pi \varepsilon }}_0\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}}\left[\begin{array}{c}{c}_1{c}_2\left(2-\frac{a_0}{a}\right)\ln \\ \frac{a+\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}}{a-\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}}-1\end{array}\right]+E_T\left(\mathrm{AO}\text{/}\mathrm{HO}\right)=E\left(\mathrm{basis}\mathrm{energies}\right)& \left(15.51\right)\\ \begin{array}{c}{E}_{T+\mathrm{osc}}\left(\mathrm{Group}\right)=E_T\left(\mathrm{MO}\right)+{\stackrel{\_}{E}}_{\mathrm{osc}}\\ =\left(\begin{array}{c}\left(\begin{array}{c}-\frac{n_1{e}^2}{8{\mathrm{\pi \varepsilon }}_0\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}}\left[\begin{array}{c}{c}_1{c}_2\left(2-\frac{a_0}{a}\right)\ln \\ \frac{a+\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}}{a-\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}}-1\end{array}\right]+\\ E_T\left(\mathrm{AO}\text{/}\mathrm{HO}\right)+E_T\left(\mathrm{atom}-\mathrm{atom},{\mathrm{msp}}^3·\mathrm{AO}\right)\end{array}\right)\\ \left[1+\sqrt{\frac{2\hslash \sqrt{\frac{C_{1o}{C}_{2o}{e}^2}{\frac{4{\mathrm{\pi \varepsilon }}_o{R}^3}{m_e}}}}{m_e{c}^2}}\right]+n_1\frac{1}{2}\hslash \sqrt{\frac{k}{\mu }}\end{array}\right)\\ =\left(E\left(\mathrm{basis}\mathrm{energies}\right)+E_T\left(\mathrm{atom}-\mathrm{atom},{\mathrm{msp}}^3·\mathrm{AO}\right)\right)\\ \left[1+\sqrt{\frac{2\hslash \sqrt{\frac{C_{1o}{C}_{2o}{e}^2}{\frac{4{\mathrm{\pi \varepsilon }}_o{R}^3}{m_e}}}}{m_e{c}^2}}\right]+n_1\frac{1}{2}\hslash \sqrt{\frac{k}{\mu }}\end{array}& \left(15.61\right)\end{array}$

TABLE 24
The bond moments of functional groups compared to experimental value [22-87]
wherein the parameters correspond to those given previously except as indicated.
Functional Groupa	n1	(c1)c2	(C1)C2	EB(valence)	EA(valence)	$\frac{q}{e}$	Bond Length 2c′ (Å)	Bond Moment μ (D)	Exp. Bond Moment μ (D)
H—C (alkyl)	1	0.91771	1	14.63489	15.35946	0.070	1.11713	0.37	0.4
H—C (aromatic)	1	0.91771	1	15.95955	15.95955	0	1.09327	0	0
H—Nb (amine)	1	0.78896	1	13.59844	15.81768	0.279	1.00343	1.34	1.31
H—Nc (ammonia)	1	0.74230	1	13.59844	15.81768	0.262	1.03677	1.30	1.31
H—Od (alcohol)	1	0.91771	1	13.59844	15.81768	0.324	0.97165	1.51	1.51
H—Oe (water)	1	0.71419	1	13.59844	15.81768	0.323	0.97157	1.51	1.51
C—N	1	0.91140	1	14.53414	14.82575	0.037	1.46910	0.26	0.22
C—O	1	0.85395	1	14.63489	15.56407	0.112	1.41303	0.76	0.74
C—Ff	1	1.09254b	1	14.63489	15.98435	0.211	1.38858	1.41	1.41
C—Cl	1	1	(2)0.81317	14.63489	15.35946	0.165	1.79005	1.42	1.46
C—Br	1	1	(2)0.74081	14.63489	15.35946	0.150	1.93381	1.40	1.38
C—Ig	1	1	(2)0.65537	14.63489	15.28545	0.119	2.13662	1.22	1.19
C═O	2	0.85395	1	14.63489	16.20002	0.385	1.20628	2.23	2.3
C≡N	3	0.91140	1	14.63489	16.20002	0.616	1.16221	3.44	3.5
H—Sh	1	0.69878	1	14.63489	15.81768	0.118	1.34244	0.76	0.69
C—S	1	1	0.91771	14.63489	15.35946	0.093	1.81460	0.81	0.9
S—O	1	1	0.77641	14.63489	15.76868	0.125	1.56744	0.94	1.0
S═Oi	2	0.82897	1	10.36001	11.57099	0.410	1.49118	2.94	2.93
N—O	1	1.06727	1	14.53414	14.82575	0.043	1.40582	0.29	0.30
N═O (nitro)	2	0.91140	1	14.63489	15.95955	0.345	1.22157	2.02	2.01
C—P	1	1	0.73885	14.63489	15.35946	0.075	1.86534	0.67	0.69
P—O	1	0.79401	1	14.63489	15.35946	0.081	1.61423	0.62	0.60
P═Oj	2	1.25942	1	14.63489	15.76868	0.405	1.46521	2.85	2.825
Si—H	1	1	0.75800	10.25487	11.37682	0.131	1.48797	0.94	0.99
Si—C	1	1	0.70071	14.63489	15.35946	0.071	1.87675	0.64	0.60
Si—Ok	1	1	1.32796	10.25487	10.87705	0.166	1.72480	1.38	1.38
B—Hl	1	1.14361	1	11.80624	12.93364	0.172	1.20235	0.99	1.0
B—C	1	0.80672	1	14.63489	15.35946	0.082	1.57443	0.62	0.69
B—O (alkoxy)	1	1	0.79562	11.80624	12.93364	0.159	1.37009	1.05	0.93
B—N	1	1	0.81231	11.89724	14.53414	0.400	1.36257	2.62	2.68
B—Fm	1	0.85447	1	14.88734	17.42282	0.316	1.29621	1.97	1.903
B—Cl	1	1	0.91044	11.80624	12.93364	0.182	1.76065	1.54	1.58
aThe more positive atom is on the left.
bc2 from Eqs. (15.77), (15.79), and Eq. (13.430) and EA(valence) is given by 1/2 two H2-type ellipsoidal MOs (Eq. (11.212)).
cc2 from Eqs. (15.77), (15.79), and the product of 0.936127 (Eq. (13.248)) and 0.92235 given by 13.59844 eV/(13.59844 eV + 0.25 · ED) where ED is the N—H bond energy ED(14NH3) = 4.57913 eV given by Eq. (13.404) and the energy of H is 13.59844 eV; EA(valence) is given by 1/2 two H2-type ellipsoidal MOs (Eq. (11.212)).
dEA(valence) is given by 1/2 two H2-type ellipsoidal MOs (Eq. (11.212)).
ec2 from Eqs. (15.77) given by 13.59844 eV/(13.59844 eV + 0.25 · ED) where ED is the O—H bond energy ED(H16OH) = 5.1059 eV given by Eq. (13.222)) and the energy of H is 13.59844 eV; EA(valence) is given by 1/2 two H2-type ellipsoidal MOs (Eq. (11.212)).
fEq. (15.129) with the inverse energy ratio of E(F) = −17.42282 eV and E(C, 2sp3) = −14.63489 eV corresponding to higher binding energy of the former.
gEA(valence) is given by 15.35946 eV − 1/2Emag (Eqs. (14.150) and (15.67)).
hc1 from Eqs. (15.79), (15.145), and (13.430); EA(valence) is given by 1/2 two H2-type ellipsoidal MOs (Eq. (11.212)).
ic2 from the reciprocal of Eq. (15.147), EA(valence) is given by Eq. (15.139), and EB(valence) is E(S) = −10.36001 eV.
jc2 from the reciprocal of Eq. (15.182).
kc2 from the reciprocal of Eq. (20.49).
lc2 from the reciprocal of Eq. (22.29).
mc2 from Eq. (15.77) using E(F) = −17.42282 eV and E(BB—Fborane, 2sp3) = −14.88734 eV (Eq. (22.61)).

The dipole moment of a given molecule is then given by the vector sum of the bond moments in the molecule. Thus, the dipole moment is given by taking into account the magnitude and direction of the bond moment of each functional group wherein the function-group bond moment stays constant from molecule to molecule and is in the vector direction of the internuclear axis. The dipole moments of water and ammonia to compare to the experimental values are given from the corresponding moments in Table 24. The calculated dipole moment of H₂O is

$\begin{array}{cc}{\mu }_{H_{2}O}=2\left(1.51\right)\cos \left(\frac{106°}{2}\right)=1.8128D& \left(16.16\right)\end{array}$ where the angle between the O—H bond is 106° given by Eq. (13.242). The experimental dipole moment of H₂O is [23]μ_H2O=1.8546D  (16.17)The calculated dipole moment of NH₃ isμ_NH3=3(1.30)cos(68°)=1.467D  (16.18)where the angle between each N—H bond and the z-axis is 68° given by Eq. (13.417). The experimental dipole moment of NH₃ is [23]μ_NH3=1.4718D  (16.19)

The charge distributions of the functional groups given in Table 24 facilitate the rendering of the charge distribution of molecules of unlimited complexity comprised of these functional groups. What was previously impossible to achieve using supercomputers can be readily accomplished on a personal computer (PC). The rendering of the true charge densities of the exemplary proteins insulin and lysozyme are shown in color scale, translucent view in FIGS. 10 and 11, respectively. The color scale, translucent view of the charge-density of an exemplary double-stranded RNA helix is shown in FIG. 12.

Nature of the Dipole Bond: Dipole-Dipole, Hydrogen, and Van Der Waals Bonding

The boundless number and length of permutations of the functional groups can form a correspondingly infinite number of molecules. The intermolecular forces instill upon molecules their inherent properties such as state—being solid, liquid, or gas, the temperatures at which phase transitions occur, and the energy content change required to change the state. However, the types of bonding are relatively few even though the breadth of molecular compositions is infinite. Since all molecules comprise nuclei that behave on the scale of molecules as electrostatic point charges, and electrically charged electrons exist as charge and current densities that obey Maxwell's equations, the binding is determined by electrical and electrodynamics forces. These typically dominate over any magnetic forces since the latter is a relativistic effect of the former and is thus negligible as the norm. Thus, essentially all molecular bonding is Coulombic in nature. The extreme case involves ions, and ionic bonding between charged functional groups of molecules obeys the same physical principles as inorganic ions as given in the Nature of the Solid Ionic Bond of Alkali Hydrides and Halides section. Similarly, the charge-density distributions of negatively-charged electrons relative to the positively-charged nuclei of neutral molecules gives rise to Coulombic-based bonding that can be grouped into two main categories, bonding that comprises permanent dipole-dipole interactions further including an extreme case, hydrogen bonding, and bonding regarding reversible mutually induced dipole fields in near-neighbor molecules called van der Waals bonding.

The H bond is exemplary of the extreme of dipole-dipole interactions as the source of bond energy and rises from the extremely high dipole moments of H bound to F, O, or N as shown in the Bond and Dipole Moments section. The bond energies of these types of bonds are large due to the very high Coulombic energy associated with the dipole-dipole interaction between H-bonded molecules compared to those having much lower dipole moments. Still H-bond energies are typically small by the standards of covalent bonds. The differences are also reflected in the relative bond lengths. In water for example, the O—H bond distance and energy are 2c′=0.970±0.005 Å (Eq. (13.186)) and E_D(H¹⁶OH)=5.1059 eV (Eq. 13.222), respectively; whereas, those of the hydrogen bond of water are 2c_O•••H′=1.78 Å (Eq. (16.27) and E_{vapor,0° C.}=0.233 eV/H-bond (Eq. (16.57)), respectively. On the other end of the spectrum, van der Waals bonds are also Coulombic in nature and are between dipoles. However, the dipoles are mutually induced rather than permanent, and the mutual induction is typically small. Thus, the bond distances are on the order of angstroms and the energies in the 10's of meV's range. The bonding between molecules gives rise to condensed matter, and the classical theory of condensed matter based on these forms of bonding is treated next.

Condensed Matter Physics

Condensed matter comprises liquids and solids of atoms and molecules. It is shown infra that the geometrical parameters, energies, and properties of the latter can be solved using the same equations as those used to solved the geometrical parameters and component energies of the individual molecules as given in the Organic Molecular Functional Groups and Molecules section.

The structure and properties of liquids can be solved by first solving the unit cell of the corresponding condensed solid. The unit cell may be solved by first determining the packing that minimizes the lattice energy. In nature, there are a small, finite number of packing arrangements. The particular arrangement relates to the most efficient one giving the most objects packed into a given space with the size and shape limitations. The water molecule, for example, is small compared to the unit cell of ice; so, it will naturally assume a tetrahedral structure and hexagonal packing given the geometry of its electric dipoles with a partial positive on the H's and partial negative on the O. In general, a reiterative algorithm is used that optimizes the packing of the molecules and tests that packing against the unit cell parameters and lattice energy until an optimum is found. The lattice parameters can be verified by X-ray crystallography and neutron diffraction. The lattice energy can be measured using calorimetry; so, the model can be directly tested.

Bonding in neutral condensed solids and liquids arises from interactions between molecules wherein the molecules of the lattice have multipoles that give rise to corresponding Coulombic or magnetic interactions. Typically, the multipoles are electric or magnetic dipoles. Consider the former case. Since the separated partial charges that give rise to bond moments are equivalent to point charges centered on the bond nuclei as given in the Bond and Dipole Moments section, the maximum interaction energy between interacting species can be calculated using Coulomb's law with the corresponding partial monopole charges and separation distance. The energy from the interaction of the partial charges increases as the separation decreases, but concomitantly, the energy of a bond that may form between the interacting species increases as well. The equilibrium separation distance corresponds to the occurrence of the balance between the Coulombic potential energy of the interacting atoms and the energy of the bond whose formation involves the interacting atoms. Thus, the balance is at the energy threshold for the formation of a nascent bond that would replace the interacting partial charges while also destabilizing the standard bonds of the interacting molecules. Then, an optimal lattice structure corresponds to an energy minimum with an associated energy. The minimum energy structure corresponds to the highest density of interacting dipoles in their minimum energy state. A convenient method to calculate the lattice energy is to determine the electric or magnetic field in the material having an electric or magnetic polarization density, and in turn, the energy can be calculated from the energy of each dipole in the corresponding field using the electrostatic or magnetostatic form of Gauss' or Amperes' equation, respectively.

Once the a, b, and c parameters of the unit cell are solved from the energy (force) balance between the electric monopoles and the nascent bond energy, the unit cell is determined. Then, the unit cell can be proliferated to arbitrary scale to render the solid. Typically, only one lattice parameter needs to be determined since the additional distances can be determined from geometrical relations based on the unit cell structure. The lattice energy may be calculated from the potential between dipoles using the cell parameters. The dielectric constant and other properties may also be calculated using Maxwell's equations and other first principles.

The structures of liquids can be modeled as linear combinations of unit cells comprising perturbations of the solid unit cell. In one approach, increasing disorder is added to the solid structure in the transition from solid to liquid to gas. Complete disorder or statistical gas behavior applies in the ideal gas limit. Thus, liquid states may be modeled by adding more cells with increasing loss of order of the solid unit cell as the temperature of the liquid is increased. The disorder is due to population of translational, rotational, and vibrational levels to match the internal energy at a given temperature. Consider thermodynamics. In principle, it is possible to classically calculate the fields over all space, the exact field interactions, and the position, trajectory, momentum, and energy of every particle of a material at each instance. Then, the material properties can be determined from these parameters. However, in practice, it is impossible computationally. For the same reason, simple underlying physical principles are applied to derive statistical properties for large ensembles of particles as given in the Statistical Mechanics section. The same statistical thermodynamic methods may be applied to modeling liquids and gases using the exact solutions of the individual molecules. Using the molecular geometrical parameters, charge distributions, and corresponding interactions as input, unit cells can be computed based on the solid unit cell. Working with increasing numbers of unit cells of increasing randomness and populating the unit cells based on appropriate statistical models such as Boltzmann statistics for increasing enthalpy input and temperature, accurate models of liquids are provided. The corresponding liquid properties can be solved from each liquid structure.

A preferred approach to solving the energy and geometric parameters of ice, considered next, is to solve the separation distance of the electric monopoles comprising a partial positive on each H and a partial negative charge on each O as the balance between the Coulombic attraction energy between the partial charges and the repulsion energy due to the formation of a nascent H—O bond between the hydrogen-bonded atoms. The nascent bond substitutes for the hydrogen bond while also removing electron density and stability from the standard water molecule bonds. Thus, it offsets the Coulombic energy and establishes the equilibrium minimum approach distance of the interacting atoms of the water molecules. Then, using Gauss's law, the energy per water molecule is calculated as the dipole energy in the electric field of the lattice of electric dipoles.

Geometrical Parameters and Energies of the Hydrogen Bond of H₂O in the Ice Phase

The extraordinary properties of water are determined by hydrogen (H) bonds, designated by the dotted bond O—H•••O, each between a participating H of one water molecule and an O of another. The structure of each phase of water is then determined by the number of H bonds on average per water molecule. As shown in the Bond and Dipole Moments section, the O—H bond has a bond moment μ of 1.51 D corresponding to a partial charge on each H of +0.323e and a component of partial charge on each O per bond moment of −0.323e. The thermodynamic basis of the H bond is the minimization of the Coulombic energy between the H and O of the hydrogen bond, limited by the formation of a nascent bond between these atoms that destabilizes the initial O—H bond. The sum of the torques and forces are zero at force balance to achieve a hexagonal crystal structure that is an energy minimum. The maximum electrostatic energy of the partial charges is calculated for the components along the H-bond axis. This energy is balanced by the total energy of the nascent bond that can form between the H•••O atoms of the H bond. The bond length of the H bond, the internuclear distance between the H and O of the H•••O bond, is calculated by a similar method as that used to determine the bond angle given in the Bond Angle of H₂O section.

The H₂O MO comprises a linear combination of two O—H-bond MOs. Each O—H-bond MO comprises the superposition of a H₂-type ellipsoidal MO and the O2p_z AO or the O2p_y AO with a relative H partial orbital contribution to the MO of 0.75; otherwise, the O2p orbitals are the same as those of the oxygen atom. The solution of the geometrical parameters and component energies are given in the Water Molecule (H₂O) section and the color scale charge density of the H₂O MO is show in FIG. 13.

Rather than consider the possible bond between the two H atoms of the O—H bonds in the determination of the bond angle, consider that the hydrogen bond may achieve a partial bond order or partial three-centered O—H—O bond as given in the Bridging Bonds of Organoaluminum Hydrides (Al—H—Al and Al—C—Al) and Bridging Bonds of Boranes (B—H—B and B—B—B) sections, and the H can become mobile between water molecules corresponding to H exchange. Such exchange of O•••H—O to O—H•••O bonding would decrease the initial O—H-bond strength since electron density would be shifted from the O—H bonds to the O•••H bond. Concomitantly, the Coulombic energy of the H bond would be eliminated. Thus, the equilibrium distance r_e or internuclear bond distance of O•••H designated as 2c′_O•••H=r_e is determined by the condition that the total energy of the nascent H₂-type ellipsoidal MO formed from the atoms of the O•••H bond is equal to the maximum Coulombic energy between the partial charges of the H and O atoms of the H bond.

The O—H bond moments superimpose at the central O. The minimum energy corresponds to the maximum separation of the δ⁻ of each bond moment on the O atom that occurs in space and time with π phase. The corresponding distance is the hypotenuse of the right triangle having the distance 2c′_O•••H between the H and O nuclei of the H•••O bond as one side and the radius of the oxygen atom, r_O2p=₀(Eq. (10.162)), as the other. Then, the maximum Coulomb energy E_Coulomb(H-bond) between the atoms of the O•••H bond due to the two separated δ⁻'s on the oxygen atom with the δ⁺ centered on the nucleus of hydrogen is

$\begin{array}{cc}{E}_{\mathrm{Coulomb}}\left(H\text{-}\mathrm{bond}\right)=\frac{-2{\delta }^2{e}^2}{4{\mathrm{\pi \varepsilon }}_0\sqrt{{\left(2c_{O-H}^{\prime }\right)}^2+{\left(r_{O2p}\right)}^2}}& \left(16.20\right)\end{array}$ Since each H bond is between two H₂O molecules and there are four H bonds per H₂O molecule, the Coulomb energy per H₂O E_Coulomb(H₂O) is equivalent to two times E_Coulomb(H-bond) (Eq. (16.20)):

$\begin{array}{cc}{E}_{\mathrm{Coulomb}}\left(H_{2}O\right)=\frac{-4{\delta }^2{e}^2}{4{\mathrm{\pi \varepsilon }}_0\sqrt{{\left(2c_{\mathrm{O—H}}^{\prime }\right)}^2+{\left(r_{O2p}\right)}^2}}& \left(16.21\right)\end{array}$

Eq. (16.21) is the energy to be equated to that of the nascent covalent bonds involving the atoms of the H bonds of the water molecule. Using Eq. (15.3), the internuclear distance of this bond, 2c_O•••H′=r_e, in terms of the corresponding semimajor axis a_O•••H is

$\begin{array}{cc}2c_{\mathrm{O—H}}^{\prime }=2\sqrt{\frac{a_{\mathrm{O—H}}{a}_0}{2C_1{C}_2}}& \left(16.22\right)\end{array}$ The length of the semiminor axis of the prolate spheroidal MO b=c is given byb_O•••H=√{square root over ((a_O•••H)²−(c_O•••H′)²)}  (16.23)And, the eccentricity, e, is

$\begin{array}{cc}{e}_{\mathrm{O—H}}=\frac{c_{\mathrm{O—H}}^{\prime }}{a_{\mathrm{O—H}}}& \left(16.24\right)\end{array}$

The semimajor axis a_O•••H of the O•••H bond is determined using the general equation for determination of the bond angle between terminal atoms given by Eqs. (15.93) and (15.99) with Eqs. (15.46-15.47) except that the MO energy is matched to the Coulombic energy of the H bond (Eq. (16.21) with substitution of Eq. (15.3)) rather than being set equal to zero for zero interaction energy in the case of the bond-angle determination:

$\begin{array}{cc}\frac{-4{\delta }^2{e}^2}{4{\mathrm{\pi \varepsilon }}_0\sqrt{\begin{array}{c}{\left(2\sqrt{\frac{a_{\mathrm{O—H}}{a}_0}{2C_1{C}_2}}\right)}^2+\\ {\left(r_{O2p}\right)}^2\end{array}}}=\hspace{1em}[\begin{array}{c}\left(\begin{array}{c}-\frac{n_1{e}^2}{8{\mathrm{\pi \varepsilon }}_0\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}}\left[\begin{array}{c}{c}_1{c}_2\left(2-\frac{a_0}{a}\right)\ln \\ \frac{a+\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}}{a-\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}}-1\end{array}\right]+\\ E_T\left(\mathrm{AO}/\mathrm{HO}\right)+E_T\left(\mathrm{atom}\mathrm{—atom},{\mathrm{msp}}^3·\mathrm{AO}\right)\end{array}\right)\\ \left[1+\sqrt{\frac{2\hslash \sqrt{\frac{\frac{C_{1o}{C}_{2o}{e}^2}{4{\mathrm{\pi \varepsilon }}_o{R}^3}}{m_e}}}{m_e{c}^2}}\right]+\\ n_1\frac{1}{2}\hslash \sqrt{\frac{\frac{c_1{c}_2{e}^2}{8{\mathrm{\pi \varepsilon }}_o{a}^3}-\frac{e^2}{8{{\mathrm{\pi \varepsilon }}_o\left(a+\sqrt{\frac{{\mathrm{aa}}_0}{2C_1{C}_2}}\right)}^3}}{\mu }}\end{array}]& \left(16.25\right)\end{array}$ where n₁ is the number of equivalent bonds of the MO, c₁ is the fraction of the H₂-type ellipsoidal MO basis function, c₂ is the factor that results in an equipotential energy match of the participating at least two atomic orbitals of each chemical bond, C_1o 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, E_T(AO/HO) is the total energy comprising the difference of the energy E(AO/HO) of at least one atomic or hybrid orbital to which the MO is energy matched and any energy component ΔE_H2MO (AO/HO) due to the AO or HO's charge donation to the MO, E_T(atom-atom,msp³.AO) is the change in the energy of the AOs or HOs upon forming the bond, and μ is the reduced mass.

For the determination of the H-bond distance, the energy parameters are the same as those of water given in the Water Molecule (H₂O) section except that any parameters due to matching AO's, E_T (AO/HO) and E_T(atom-atom,msp³.AO), is zero since only the energies of the MO electrons to form the O•••H MO are considered. The partial charge δ=q/e from Table 24 is 0.323, and the reduced mass is

$\mu =\frac{16}{17}.$ The parameters are summarized in Table 16.18 and Eq. (16.26).

TABLE 25
The energy parameters (eV) of the O···H functional
group of the hydrogen bond of Type I ice.
                              O···H
Parameters                    Group
δ                             0.323
n1                            2
C1                            0.75
C2                            1
c1                            0.75
c2                            1
c1o                           1.5
C2o                           1
Ve (eV)                       −20.30177
Vp (eV)                       16.15958
T (eV)                        2.38652
Vm (eV)                       −1.19326
E(AO/HO) (eV)                 0
ΔEH2MO(AO/HO) (eV)            0
ET(AO/HO) (eV)                0
ET(H2MO) (eV)                 −2.94892
ET(atom−atom, msp3 · AO) (eV) 0
ET(MO) (eV)                   −2.94892
ω (1015 rad/s)                6.55917
EK (eV)                       4.31736
ĒD (eV)                       −0.012122
ĒKvib (eV)                    0.03263
Ēosc (eV)                     0.004191
ET(Group) (eV)                −2.94054

Substitution of the parameters of Table 25, the internuclear distance 2c_O—H′ given by Eq. (13.185), and R given by Eq. (16.23) and (16.22) into Eq. (16.25) gives

$\begin{array}{cc}\frac{-4{\left(0.323\right)}^2{e}^2}{4{\mathrm{\pi \varepsilon }}_0\sqrt{\begin{array}{c}{\left(2\sqrt{\frac{a_{\mathrm{O—H}}{a}_0}{2\left(0.75\right)}}\right)}^2+\\ {\left(5.2917706\times {10}^{-11}m\right)}^2\end{array}}}=\hspace{1em}\left\{\begin{array}{c}\left(\frac{-e^2}{4{\mathrm{\pi \varepsilon }}_0\sqrt{\frac{a_{\mathrm{O—H}}{a}_0}{2\left(0.75\right)}}}\left(\begin{array}{c}\left(\frac{3}{2}-\frac{3}{8}\frac{a_0}{a_{\mathrm{O—H}}}\right)\\ \ln \frac{a_{\mathrm{O—H}}+\sqrt{\frac{a_{\mathrm{O—H}}{a}_0}{2\left(0.75\right)}}}{a_{\mathrm{O—H}}-\sqrt{\frac{a_{\mathrm{O—H}}{a}_0}{2\left(0.75\right)}}}-1\end{array}\right)\right)\\ \left(1+2\sqrt{\frac{2\hslash \sqrt{\frac{3}{2}\frac{\frac{e^2}{4{{\mathrm{\pi \varepsilon }}_0\left(\sqrt{\begin{array}{c}{\left(a_{\mathrm{O—H}}\right)}^2-\\ {\left(2\sqrt{\frac{a_{\mathrm{O—H}}{a}_0}{2\left(0.75\right)}}\right)}^2\end{array}}\right)}^3}}{m_e}}}{m_e{c}^2}}\right)+\\ 2\left(\frac{1}{2}\right)\hslash \sqrt{\frac{\frac{0.75e^2}{8{{\mathrm{\pi \varepsilon }}_o\left(a_{\mathrm{O—H}}\right)}^3}-\frac{e^2}{8{{\mathrm{\pi \varepsilon }}_o\left(a_{\mathrm{O—H}}+\sqrt{\frac{a_{\mathrm{O—H}}{a}_0}{2\left(0.75\right)}}\right)}^3}}{\frac{16}{17}}}\end{array}\right\}& \left(16.26\right)\end{array}$ From the energy relationship given by Eq. (16.26) and the relationships between the axes given by Eqs. (16.22-16.24), the dimensions of the O•••H MO can be solved.

The most convenient way to solve Eq. (16.26) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures isa_O•••H=4.25343a₀=2.25082×10¹⁰ m  (16.27)The component energy parameters at this condition are given in Table 25. Substitution of Eq. (16.27) into Eq. (16.22) givesc_O•••H′=1.68393a₀=8.91097×10⁻¹¹ m  (16.28)and internuclear distance of the H bond:2c_O•••H′=3.36786a₀=1.78219×10⁻¹⁰ m=1.78219 Å  (16.29)The internuclear distance of the O—H given by Eq. (13.185) is2c′=1.83601a₀=9.71574×10⁻¹¹ m  (16.30)The internuclear distance 2c_O•••H′ of the O—H bond added to 2c_O•••H′ gives the internuclear distance 2c_O•••HO′ between the oxygen atoms of the group O—H•••O:2c_O•••HO′=2c_O•••H′+2c_O—H′  (16.31)Substitution of 2c_O•••H′ (Eq. (16.29)) and 2c_O—H′ (Eq. (13.185)) into Eq. (16.31) gives the nearest-neighbor separation, the internuclear distance 2c_O•••HO′ between the oxygen atoms of the O—H•••O bond in Type I ice:

$\begin{array}{cc}\begin{array}{c}2c_{\mathrm{O—H}}^{\prime }=2c_{\mathrm{O—H}}^{\prime }+2c_{\mathrm{O—H}}^{\prime }\\ =1.78219\times {10}^{-10}m+9.71574\times {10}^{-11}m\\ =2.75377\times {10}^{-10}m\\ =2.75377\AA \end{array}& \left(16.32\right)\end{array}$

The experimental oxygen nearest-neighbor separation distance 2c_O•••HO′ is [88]2c_O•••HO′=2.75 Å  (16.33)The experimental internuclear distance of the O—H bond of H₂O is [89]2c′=9.70±0.005×10⁻¹¹ m  (16.34)Using Eqs. (16.33) and (16.34), the experimental H bond distance 2c_O•••HO′ in Type I ice is [88-89]2c_O•••H′=1.78 Å  (16.35)

The other H-bond MO parameters can also be determined by the relationships among the parameters. Substitution of Eqs. (16.27) and (16.28) into Eq. (16.23) givesb_O•••H=c_O•••H=3.90590a₀=2.06691×10⁻¹⁰ m  (16.36)Substitution of Eqs. (16.27) and (16.28) into Eq. (16.24) givese_O•••H=0.39590 (16.37)

Since water is a hexagonal crystal system in common with the carbon allotrope diamond, the internuclear distance of the two terminal O atoms of a set of three H₂O's corresponding to the hexagonal lattice parameter a₁ is calculated using the same approach as that given by Eqs. (17.1-17.3) using the law of cosines:s₁²+s₂²−2s₁s₂ cosine θ=s₃²  (16.38)where s₃=a₁ is the hypotenuse of the isosceles triangle having equivalent sides of length equal to 2c_O•••HO′. With the bond angle between three water molecules formed by the two corresponding H bonds given by θ_{∠H2O,H2O,H2O}=109.5 ([90] and s₁=s₂=2c_O•••HO′ given by Eq. (16.32), the distance between the oxygen atoms of the terminal water molecules along the hypotenuse, s₃=2c_H2O—H2O′=a₁, is

$\begin{array}{cc}\begin{array}{c}{a}_l=2c_{H_2{\mathrm{O—H}}_{2}O}^{\prime }\\ =\sqrt{2{\left(2c_{\mathrm{O—HO}}^{\prime }\right)}^2\left(1-\mathrm{cosine}\left(109.5°\right)\right)}\\ =\sqrt{2{\left(2.75377\AA \right)}^2\left(1-\mathrm{cosine}\left(109.5°\right)\right)}\\ =4.49768\AA \end{array}& \left(16.39\right)\end{array}$

Due to the tetrahedral structure shown in FIG. 14, four water molecules form a pyramidal structure with a central H₂O(1) at the apex designated as on the z-axis, and the three other water molecules, H₂O(n) n=2, 3, 4, form the base in the xy-plane. As further shown in FIG. 14, a fifth H₂O(5) is positioned a distance 2c_O•••HO′ along the z-axis. Twice the height along the z-axis from the base of the pyramid to the fifth H₂O comprises the Type I ice unit cell parameter c which is determined next using Eqs. (13.412-13.417).

Since any two O—H•••O bonds having the internuclear distance 2c_O•••HO′ between the oxygen atoms of in Type I ice form an isosceles triangle having the hypotenuse a₁ between the terminal oxygen's, the distance d_origin-O from the origin of the pyramidal base to the nucleus of a terminal oxygen atom is given by

$\begin{array}{cc}{d}_{\mathrm{origin}-O}=\frac{a_l}{2\sin 60°}& \left(16.40\right)\end{array}$ Substitution of Eq. (16.39) into Eq. (16.40) givesd_origin-O=2.59674a₀  (16.41)The height d_height along the z-axis of the pyramid from the origin to the O nucleus of H₂O(1) is given byd_height=√{square root over ((2c_O•••HO′)₂−(d_{origin•••O})²)}  (16.42)Substitution of Eqs. (16.32) and (16.41) into Eq. (16.42) givesd_height=0.91662a₀  (16.43)The angle θ_v of each O—H•••O bond from the z-axis is given by

$\begin{array}{cc}{\theta }_v={\tan }^{-1}\left(\frac{d_{\mathrm{origin}-O}}{d_{\mathrm{height}}}\right)& \left(16.44\right)\end{array}$ Substitution of Eqs. (16.41) and (16.43) into Eq. (16.44) givesθ_v=70.560  (16.45)Using Eqs. (16.32) and (16.43), the hexagonal lattice parameter c₁ for Type I ice given by twice the height along the z-axis from the base of the pyramid to the fifth water, H₂O(5), is

$\begin{array}{cc}\begin{array}{c}{c}_l=2\left(c_{\mathrm{O—H}}^{\prime }+d_{\mathrm{height}}\right)\\ =2\left(2.75377\AA +0.91662\AA \right)\\ =7.34077\AA \end{array}& \left(16.46\right)\end{array}$ The experimental lattice parameters a₁ and c₁ for Type I ice are [90, 91]a₁=4.49 Åa₁=4.5212 Å  (16.47)and [91, 92]c₁=7.31 Åc₁=7.3666 ÅThe tetrahedral unit cell and the ideal hexagonal lattice structure of Type I ice are shown in FIGS. 14-16, using the color scale charge density of each water molecule.

A convenient method to calculate the lattice energy is to determine the electric field in ice having an electric polarization density corresponding to the aligned molecular water dipoles moments, and in turn, the energy can be calculated from the energy of each dipole in the corresponding field using the electrostatic form of Gauss' equation. The electric field inside of a material having a uniform polarization density P₀ given by Eq. (6.3.3.15) of Haus and Melcher [93] is

$\begin{array}{cc}E\left(H_{2}O\right)=\frac{P_0}{3{\varepsilon }_0}\left(-\cos \theta i_r+\sin \theta i_{\theta }\right)& \left(16.49\right)\end{array}$ The polarization density P₀ given by Eq. (6.3.3.3) of Haus and Melcher [93] isP₀=Nμ_H2O  (16.50)where μ_H2O is the dipole moment of water and N is the number density of water dipoles given by the density ρ_ice divided by the molecular weight MW and multiplied by the Avogadro constant N_A:

$\begin{array}{cc}N=\frac{{\rho }_{\mathrm{ice}}}{\mathrm{MW}}{N}_A& \left(16.51\right)\end{array}$ Substitution of Eqs. (16.50) and (16.51) into Eq. (16.49) gives

$\begin{array}{cc}E\left(H_{2}O\right)=\frac{{\mu }_{H_{2}O}\frac{{\rho }_{\mathrm{ice}}}{\mathrm{MW}}{N}_A}{3{\varepsilon }_0}\left(-\cos \theta i_r+\sin \theta i_{\theta }\right)& \left(16.52\right)\end{array}$ The energy of forming the condensed phase is that of the alignment of the water dipoles each comprised of two O—H component dipoles where the angular dependence along the z-axis in ice is unity, and this condition applies even in the case of the local order in water. The corresponding energy U(H₂O) per water dipole due to the polarization electric field of the lattice of hexagonal dipoles is given by

$\begin{array}{cc}U\left(H_{2}O\right)=2{\mu }_{H_{2}O}·E\left(H_{2}O\right)=\frac{-2{\left({\mu }_{H_{2}O}\right)}^2\frac{{\rho }_{\mathrm{ice}}}{\mathrm{MW}}{N}_A}{3{\varepsilon }_0}& \left(16.53\right)\end{array}$ Substitution of the density of ice

$\rho =\frac{0.92g}{1\times {10}^{-6}{m}^3}$ [92], the MW=18 g/mole, N_A=6.0221415×10²³ molecules/mole, and the water dipole moment given by Eq. (16.16) with the predicted and experimental hexagonal bond angle of ice, θ_∠H2O=109.5° [90]:μ_H2O=2(1.51)cos(109.5/2°)=5.79898×10⁻³⁰ C·m  (16.54)into Eq. (16.53) gives

$\begin{array}{cc}\begin{array}{c}U\left(H_{2}O\right)=\frac{\begin{array}{c}-2{\left(\begin{array}{c}5.79898\times \\ {10}^{-30}C·m\end{array}\right)}^2\frac{0.92g}{\frac{1\times {10}^{-6}{m}^3}{18g\text{/}\mathrm{mole}}}6.0221415\times \\ {10}^{23}\mathrm{molecules}\text{/}\mathrm{mole}\end{array}}{3{\varepsilon }_0}\\ =-0.48643\mathrm{eV}\left(-46.934\mathrm{kJ}\text{/}\mathrm{mole}\right)\end{array}& \left(16.55\right)\end{array}$ U(H₂O) is also the negative of E_{vapor,0° C.}, the energy of water initially at 0° C. or the energy of vaporization of water at 0° C.:E_{vapor,0° C.}=−U(H₂O)=0.48643 eV (46.934 kJ/mole)  (16.56)The experimental energy of vaporization of water at 0° C. (Type I ice) is [94]E_{vapor,0° C.}=45.054 kJ/mole  (16.57)The calculated results based on first principles and given by analytical equations are summarized in Table 26.

TABLE 26
The calculated and experimental geometrical and energy
parameters of the H bond of water of Type I ice.
			Ref. for
Parameter	Calculated	Experimental	Exp.
H Bond Length 2c′O···H	1.78219	Å	1.78	Å	88, 89
Nearest Neighbor
Separation Distance
2c′O···H	2.75377	Å	2.75	Å	88
			4.49	Å	90
H2O Lattice Parameter a1	4.49768	Å	4.5212	Å	91
H2O Lattice Parameter c1	7.34077	Å	7.31	Å	92
			7.3666	Å	91
Energy of Vaporization	46.934	kJ/mole	45.054	kJ/mole	94
of Water at 0° C.

As the temperature increases, the corresponding molecular kinetic energy can excite a vibrational mode along the H bond axis. Concomitantly, the O—H bond elongates and decreases in energy. As a consequence, the hydrogen bond achieves a partial bond order or partial three-centered O—H—O bond, and the H can undergo exchange between water molecules. The time-average effect of exchange is to decrease the statistical equilibrium separation distance of water molecules. In competition with the separation-distance decreasing effect of exchange is the increasing effect due to collisional impact and recoil as a function of increasing temperature. The former effect dominates from the temperature of ice to 4° C. at which point water assumes a maximum density. Thereafter, the momentum imparted with water-water collisions overwhelms the decrease due to exchange, and the molecular separation statistically increases with temperature until a totally gaseous state is achieved at atmospheric pressure at 100° C. Unit cells with increasing entropy can be derived from the ice unit cell by populating translational, rotational, and vibrational levels of molecules within the cells to match the internal energy at a given temperature. Using statistical mechanical models such as Boltzmann statistics to populate an increasing number of basis units cells of increasing disorder and based on the ice unit cell, the behavior of water as a function of temperature can be modeled over the range of states from ice to liquid to steam. The structure of each phase of water is then determined by the number of H bonds on average per water molecule. Based on the 10% energy change in the heat of vaporization in going from ice at 0° C. to water at 100° C. [94], the average number of H bonds per water molecule in boiling water is 3.6. The H bond distance is calculated next using the enthalpy to form steam from boiling water.

Geometrical Parameters and Energies of the Hydrogen Bond of H₂O in the Vapor Phase

Two or more water molecules can interact along the O•••H or H bond axis. In the gas phase, the maximum energy of interaction between water molecules of steam is equivalent to the negative of the heat of vaporization of water at the boiling point, 100° C.; otherwise, water vapor would form the corresponding condensed state. For the determination of the H-bond distance, the energy parameters, partial charge, and reduced mass are the same as those of the water molecules of ice given in Eq. (16.26) except that the negative of the experimental E_{vapor,100° C.}=0.42137 eV (40.657 kJ/mole) [94] is equated to the nascent covalent bond energy. The parameters are summarized in Table 27 and Eq. (16.58).

TABLE 27
The energy parameters (eV) of the O···H functional
group of the hydrogen bond of water vapor.
                              O···H
Parameters                    Group
δ                             0.323
n1                            2
C1                            0.75
C2                            1
c1                            0.75
c2                            1
c1o                           1.5
C2o                           1
Ve (eV)                       −15.20020
Vp (eV)                       14.08285
T (eV)                        1.35707
Vm (eV                        −0.67853
E(AO/HO) (eV)                 0
ΔEH2MO(AO/HO) (eV)            0
ET(AO/HO) (eV)                0
ET(H2MO) (eV)                 −0.43882
ET(atom-atom, msp3 · AO) (eV) 0
ET(MO) (eV)                   −0.43882
ω (1015 rad/s)                4.20131
EK (eV)                       2.76538
ĒD (eV)                       0.001444
ĒKvib (eV)                    0.02033
Ēosc (eV)                     0.008724
ET(Group) (eV)                −0.42137

Substitution of the parameters of Table 16.20 and −E_{vapor,0° C.} (Eq. (16.57)) into Eq. (16.26) gives

$\begin{array}{cc}e\left(0.42137\mathrm{eV}\right)=\{\begin{array}{c}\left(\frac{-e^2}{4{\mathrm{\pi \varepsilon }}_0\sqrt{\frac{a_{O-H}{a}_0}{2\left(0.75\right)}}}\left(\left(\frac{3}{2}-\frac{3}{8}\frac{a_0}{a_{O-H}}\right)\ln \frac{a_{O-H}+\sqrt{\frac{a_{O-H}{a}_0}{2\left(0.75\right)}}}{a_{O-H}-\sqrt{\frac{a_{O-H}{a}_0}{2\left(0.75\right)}}}-1\right)\right)\\ \left(1+2\sqrt{\frac{2\hslash \sqrt{\frac{3}{2}\frac{e^2}{\frac{4{{\mathrm{\pi \varepsilon }}_0\left(\sqrt{{\left(a_{O-H}\right)}^2-{\left(2\sqrt{\frac{a_{O-H}{a}_0}{2\left(0.75\right)}}\right)}^2}\right)}^3}{m_e}}}}{m_e{c}^2}}\right)+\\ 2\left(\frac{1}{2}\right)\hslash \sqrt{\frac{\frac{0.75e^2}{8{{\mathrm{\pi \varepsilon }}_o\left(a_{O-H}\right)}^3}-\frac{e^2}{8{{\mathrm{\pi \varepsilon }}_o\left(a_{O-H}+\sqrt{\frac{a_{O-H}{a}_0}{2\left(0.75\right)}}\right)}^3}}{\frac{16}{17}}}\end{array}\}& \left(16.58\right)\end{array}$ From the energy relationship given by Eq. (16.58) and the relationships between the axes given by Eqs. (16.22-16.24), the dimensions of the O•••H MO can be solved.

The most convenient way to solve Eq. (16.58) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures isa_O•••H=5.60039a₀=2.96360×10⁻¹⁰ m  (16.59)The component energy parameters at this condition are given in Table 27. Substitution of Eq. (16.59) into Eq. (16.22) givesc_O•••H′=1.93225a₀=1.02250×10⁻¹⁰ m  (16.60)and internuclear distance of the H bond:2c_O•••H′=3.86450a₀=2.04501×10⁻¹⁰ m  (16.61)The experimental H bond distance 2c_O•••H′ in the gas phase is [95]2c_O•••H′=2.02×10⁻¹⁰ m  (16.62)and [96]2c_O•••H′=2.05×10⁻¹⁰ m  (16.63)The other H-bond MO parameters can also be determined by the relationships among the parameters. Substitution of Eqs. (16.59) and (16.60) into Eq. (16.23) givesb_O•••H=c_O•••H=5.25650a₀=2.78162×10⁻¹⁰ m  (16.64)Substitution of Eqs. (16.59) and (16.60) into Eq. (16.24) givese_O•••H=0.34502  (16.65)Substitution of 2c_O•••H′ (Eq. (16.61)) and 2c_O—H′ (Eq. (13.185)) into Eq. (16.31) gives the nearest neighbor separation, the internuclear distance 2c_O•••HO′ between the oxygen atoms of the O—H•••O bond in Type I ice:

$\begin{array}{cc}\begin{array}{c}2c_{O-\mathrm{HO}}^{\prime }=2c_{O-H}^{\prime }+2c_{O-H}^{\prime }\\ =2.04501\times {10}^{-10}m+9.71574\times {10}^{-11}m\\ =3.01658\times {10}^{-10}m\\ =3.01658\AA \end{array}& \left(16.66\right)\end{array}$ Using Eqs. (16.31), (16.34), and (16.63), the experimental nearest neighbor separation 2c_O—HO′ is [89, 96]

$\begin{array}{cc}\begin{array}{c}2c_{O-\mathrm{HO}}^{\prime }=2c_{O-H}^{\prime }+2c_{O-H}^{\prime }\\ =2.05\times {10}^{-10}m+9.70\times {10}^{-11}m\\ =3.02\times {10}^{-10}m\\ =3.02\AA \end{array}& \left(16.67\right)\end{array}$ H-bonded water vapor molecules in steam are shown in FIGS. 17A-B using the color scale charge density of each water molecule.The calculated results based on first principles and given by analytical equations are summarized in Table 28.

TABLE 28
The calculated and experimental geometrical
and energy parameters of the H bond of steam.
				Ref. for
	Parameter	Calculated	Experimental	Exp.
	H Bond Length 2c′O···H	2.04501 Å	2.02 Å	95, 96
			2.05 Å
	Nearest Neighbor
	Separation Distance
	2c′O···H	3.01658 Å	3.02 Å	89, 96

Geometrical Parameters and Energies of the Hydrogen Bond of H₂O and NH₃

Similar to the water molecule, the ammonia molecule has a strong dipole moment along each of its N—H-bonds. The NH₃ MO comprises the linear combination of three N—H-bond MOs. Each N—H-bond MO comprises the superposition of a H-type ellipsoidal MO and the N2p_x, N2p_y, or N2p_z AO with a relative H partial orbital contribution to the MO of 0.75. The solution of the geometrical parameters and component energies are given in the Ammonia (NH₃) section, and the color scale charge density of the NH₃ MO is show in FIG. 18.

Due to the interacting dipoles, hydrogen bonds also form between the nitrogen of ammonia and the hydrogen of water molecules. Water hydrogen bonds to ammonia molecules by interaction along the N•••HO or H bond axis. As shown in the Bond and Dipole Moments section, each N—H bond of ammonia has a bond moment μ of 1.30 D corresponding to a N component of partial charge of −0.262e, and the O—H bond has a bond moment μ of 1.51 D corresponding to a H partial charge of +0.323e. The thermodynamic basis of the H bond is the minimization of the Coulombic energy between the hydrogen bonded H of H₂O and N of ammonia, limited by the formation of a nascent N—H bond between these atoms that destabilizes the initial O—H bond of the water molecule partner. As in the case of ice, the maximum electrostatic energy of the partial charges is calculated for the components along the H-bond axis. This energy is balanced by the total energy of the nascent bond that can form between the N•••H atoms of the H bond. The bond length of the H bond, the internuclear distance between the N and H of the N•••H bond, is calculated using Eq. (16.25) by a similar method as that used to calculate the O•••H bond distance of ice. According to the method given in the Geometrical Parameters and Energies of the hydrogen Bond of H₂O section, the equilibrium distance r_e or internuclear bond distance of N•••H designated as 2c_N•••H′=r_e is determined by the condition that the total energy of the nascent H₂-type ellipsoidal MO formed from the atoms of the N•••H bond is equal to the maximum Coulombic energy between the partial charges of the N and H atoms of the H bond.

The maximum Columbic energy corresponds to the minimum separation distance of N and H atoms corresponding to the alignment along the N•••H bond axis. The corresponding distance from the δ⁺ of the H₂O Hand the NH₃ N is the distance 2c_N•••H′ between the N and H nuclei of the N•••H bond. Then, the maximum Coulomb energy E_Coulomb(H-bond) between the atoms of the N•••H bond due to the δ⁻ on the nitrogen atom with the δ⁺ centered on the nucleus of hydrogen is

$\begin{array}{cc}{E}_{\mathrm{Coulomb}}\left(H\text{-}\mathrm{bond}\right)=\frac{-{\delta }_N^{-}{\delta }_H^{+}{e}^2}{4{\mathrm{\pi \varepsilon }}_{0}2c_{N-H}^{\prime }}& \left(16.68\right)\end{array}$ Eq. (16.68) is the energy to be equated to that of the nascent bonds involving the atoms of the H bond.

For the determination of the H-bond distance, the energy parameters of the nascent N—H bond are the same as those of ammonia given in the Ammonia (NH₃) section except that any parameter due to matching AO's, E_T(AO/HO) and E_T(atom-atom,msp³.AO), is zero since only the energies of the MO electrons to form the N•••H MO are considered. The energy of Eq. (16.68) is multiplied by three to match the total energy of the three N—H bond MOs of ammonia. The partial charges δ=q/e from Table 24 are −0.262 and +0.323, and the reduced mass is

$\mu =\frac{14}{15}.$ The parameters are summarized in Table 29 and Eq. (16.69).

TABLE 29
The energy parameters (eV) of the N···H functional group
of the hydrogen bond of the ammonia-water molecular dimer.
                              N···H
Parameters                    Group
δN−                           0.262
δH+                           0.323
n1                            3
C1                            0.75
C2                            0.93613
c1                            0.75
c2                            1
C1o                           1.5
C2o                           1
Ve (eV)                       −23.60741
Vp (eV)                       20.75035
T (eV)                        2.17246
Vm (eV                        −1.08623
E(AO/HO) (eV)                 0
ΔEH2MO(AO/HO) (eV)            0
ET(AO/HO) (eV)                0
ET(H2MO) (eV)                 −1.77083
ET(atom-atom, msp3 · AO) (eV) 0
ET(MO) (eV)                   −1.77083
ω (1015 rad/s)                4.44215
EK (eV)                       2.92390
ĒD (eV)                       −0.00599
ĒKvib (eV)                    0.021843
Ēosc (eV)                     0.00493
ET(Group) (eV)                1.75603
ET(Group) (eV) N—H            0.58534

Substitution of the parameters of Table 29 into Eq. (16.25) with R=a_N•••H gives

$\begin{array}{cc}\frac{-3\left(0.262\right)\left(0.323\right)e^2}{4{\mathrm{\pi \varepsilon }}_0\left(2\sqrt{\frac{a_{N-H}{a}_0}{2\left(0.75\right)\left(0.93613\right)}}\right)}=\{\begin{array}{c}\left(\frac{-3e^2}{8{\mathrm{\pi \varepsilon }}_0\sqrt{\frac{a_{N-H}{a}_0}{2\left(0.75\right)\left(0.93613\right)}}}\left(\begin{array}{c}\left(\frac{3}{2}-\frac{3}{8}\frac{a_0}{a_{N-H}}\right)\\ \ln \frac{a_{N-H}+\sqrt{\frac{a_{N-H}{a}_0}{2\left(0.75\right)\left(0.93613\right)}}}{a_{N-H}-\sqrt{\frac{a_{N-H}{a}_0}{2\left(0.75\right)\left(0.93613\right)}}}-1\end{array}\right)\right)\\ \left(1+3\sqrt{\frac{2\hslash \sqrt{\frac{3}{2}\frac{e^2}{\frac{4{{\mathrm{\pi \varepsilon }}_0\left(\sqrt{{\left(a_{N-H}\right)}^2-{\left(2\sqrt{\frac{a_{N-H}{a}_0}{2\left(0.75\right)}}\right)}^2}\right)}^3}{m_e}}}}{m_e{c}^2}}\right)+\\ 3\left(\frac{1}{2}\right)\hslash \sqrt{\frac{\frac{0.75e^2}{8{{\mathrm{\pi \varepsilon }}_o\left(a_{N-H}\right)}^3}-\frac{e^2}{8{{\mathrm{\pi \varepsilon }}_o\left(a_{N-H}+\sqrt{\frac{a_{N-H}{a}_0}{2\left(0.75\right)\left(0.93613\right)}}\right)}^3}}{\frac{14}{15}}}\end{array}\}& \left(16.69\right)\end{array}$ From the energy relationship given by Eq. (16.69) and the relationships between the axes given by Eqs. (16.22-16.24), the dimensions of the N•••H MO can be solved.

The most convenient way to solve Eq. (16.69) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures isa_N•••H=5.43333a₀=2.87519×10⁻¹⁰ m  (16.70)The component energy parameters at this condition are given in Table 29. Substitution of Eq. (16.70) into Eq. (16.22) givesc_N•••H′=1.96707a₀=1.04093×10⁻¹⁰ m  (16.71)and internuclear distance of the H bond:2c_N•••H′=3.93414a₀=2.08186×10⁻¹⁰ m=2.08186 Å  (16.72)The experimental H bond distance 2c_N•••H′ in the gas phase is [96, 97]2c_N•••HO′=2.02×10⁻¹⁰ m  (16.73)The other H-bond MO parameters can also be determined by the relationships among the parameters. Substitution of Eqs. (16.70) and (16.71) into Eq. (16.23) givesb_N•••H=c_N•••H=5.06475a₀=2.68015×10⁻¹⁰ m  (16.74)Substitution of Eqs. (16.70) and (16.71) into Eq. (16.24) givese_N•••H=0.36204  (16.75)The addition of 2c_N•••H′ (Eq. (16.72)) and 2c_O—H′ (Eq. (13.185)) gives the nearest neighbor separation, the internuclear distance 2c_N•••HO′ between the nitrogen and oxygen atoms of the N•••H—O bond of the ammonia-water molecular dimer.

$\begin{array}{cc}\begin{array}{c}2c_{N-\mathrm{HO}}^{\prime }=2c_{N-H}^{\prime }+2c_{O-H}^{\prime }\\ =2.08186\times {10}^{-10}m+9.71574\times {10}^{-11}m\\ =3.05343\times {10}^{-10}m\\ =3.05343\AA \end{array}& \left(16.76\right)\end{array}$ The addition of the experimental 2c_N•••H′ (Eq. (16.73)) and 2c_O—H′ (Eq. (13.185)) gives the experimental nearest neighbor separation 2c_N•••HO′ [96, 89]:

$\begin{array}{cc}\begin{array}{c}2c_{N-\mathrm{HO}}^{\prime }=2c_{N-H}^{\prime }+2c_{O-H}^{\prime }\\ =2.02\times {10}^{-10}m+9.70\times {10}^{-11}m\\ =2.99\times {10}^{-10}m\\ =2.99\AA \end{array}& \left(16.77\right)\end{array}$ H-bonded ammonia-water molecular dimer is shown in FIG. 19 using the color scale charge density of each molecule.The energy of forming the dimer in the gas phase is that of the alignment of the ammonia dipole moment in the electric field of the H—O water dipole. Using μ_NH3=1.467D=4.89196×10⁻³⁰ C·m (Eq. (16.18)), μ_H—O,H2O=1.51 D=5.02385×10⁻³⁰ C·m (Table 24), and the N•••H distance, 2c_N•••H′=2.08186×10⁻¹⁰ m (Eq. (16.72)), the N•••H bond dissociation energy E_D(N•••H) of the ammonia-water molecular dimer is

$\begin{array}{cc}\begin{array}{c}{E}_D\left(N\dots H\right)={\mu }_{H_{3}N}·\frac{2{\mu }_{H-O,H_{2}O}}{4{{\mathrm{\pi \varepsilon }}_0\left(2c_{N-H}^{\prime }\right)}^3}\\ =\frac{\begin{array}{c}\left(4.89196\times {10}^{-30}C·m\right)\\ \left(5.02385\times {10}^{-30}C·m\right)\end{array}}{4{{\mathrm{\pi \varepsilon }}_0\left(2.08186\times {10}^{-10}m\right)}^2}\\ =29.48\mathrm{kJ}\end{array}& \left(16.78\right)\end{array}$ The experimental N•••H bond dissociation energy between amino N and hydroxyl H is approximately [98]E_D(N•••H)=29 kJ  (16.79)The calculated results based on first principles and given by analytical equations are summarized in Table 30.

TABLE 30
The calculated and experimental geometrical and energy parameters
of the H-bonded ammonia-water vapor molecular dimer.
			Ref. for
Parameter	Calculated	Experimental	Exp.
H Bond Length 2c′N···H	2.08186	Å	2.02	Å	96, 97
Nearest Neighbor	3.05343	Å	2.99	Å	96, 89
Separation Distance
2c′N···HO
N···H Bond Dissociation	29.48	kJ/mole	29	kJ/mole	98
Energy

Geometrical Parameters Due to the Interplane Van Der Waals Cohesive Energy of Graphite

Eq. (16.25) can be applied to other solids such as graphite. Graphite is an allotrope of carbon that comprises planar sheets of covalently bound carbon atoms arranged in hexagonal aromatic rings of a macromolecule of indefinite size. The structure of graphite is shown in FIGS. 20A and B. The structure shown in FIG. 20 has been confirmed directly by TEM imaging, and the Pi cloud predicted by quantum mechanics has been dispatched [99].

As given in the Graphite section, the structure of the indefinite network of aromatic hexagons of a sheet of graphite is solved using a linear combination of aromatic

$C\stackrel{3e}{=}C$ aromatic bonds comprising (0.75)(4)=3 electrons according to Eq. (15.161). In graphite, the minimum energy structure with equivalent carbon atoms wherein each carbon forms bonds with three other such carbons requires a redistribution of charge within an aromatic system of bonds. Considering that each carbon contributes four bonding electrons, the sum of electrons of graphite at a vertex-atom comprises four from the vertex atom plus two from each of the two atoms bonded to the vertex atom where the latter also contribute two each to the juxtaposed bond. These eight electrons are distributed equivalently over the three bonds of the group such that the electron number assignable to each bond is 8/3. Thus, the

$C\stackrel{8/3c}{=}C$ functional group of graphite comprises the aromatic bond with the exception that the electron-number per bond is 8/3. The sheets, in turn, are bound together by weaker intermolecular van der Waals forces. The geometrical and energy parameters of graphite are calculated using Eq. (16.25) with the van der Waals energy equated to the nascent bond energy.

The van der Waals energy is due to mutually induced nonpermanent dipoles in near-neighbor bonds. Albeit, the

$C\stackrel{8/3e}{\stackrel{}{——}}C$ functional group is symmetrical such that it lacks a permanent dipole moment, a reversible dipole can be induced upon van der Waals bonding. The parameters of the

$C\stackrel{8/3e}{\stackrel{}{——}}C$ functional group are the same as those of the aromatic

$C\stackrel{3e}{=}C$ functional group, the basis functional group of aromatics, except that the bond order is

$8/3\left(e.g.2c_{C\stackrel{8/3e}{=}C}^{\prime }=2c_{C\stackrel{3e}{=}C}^{\prime }\right).$ Using Eq. (16.15) wherein C₂ of Eq. (15.51) for the aromatic

$C\stackrel{3e}{=}C-\mathrm{bond}\mathrm{MO}$ is C₂(aromaticC2sp³HO)=c₂(aromaticC2sp³HO)=0.85252 (Eq. (15.162)) and E_Coulomb(C_benzene,2sp³) is 15.95955 eV (Eq. (14.245)), E(C,2sp³)=−14.63489 eV (Eq. (14.143)) and 2c′=1.39140×10⁻¹⁰ m (Table 15.214), the van der Waals dipole of graphite is given in Table 31.

TABLE 31
$\hspace{1em}\begin{array}{c}\mathrm{The}\mathrm{parameters}\mathrm{and}\mathrm{van}\mathrm{der}\mathrm{Waals}\mathrm{dipole}\mathrm{bond}\mathrm{moment}\\ \mathrm{of}\mathrm{the}C\stackrel{8/3e}{=}C\mathrm{functional}\mathrm{group}\mathrm{of}\mathrm{graphite}.\end{array}$
Functional Group	n1	(c1)c2	(C1)C2	EB(valence)	EA(valence)	$\frac{q}{e}$	Bond Length 2c′ (Å)	Bond Moment μ (D)
$C\stackrel{8/3e}{=}C$	$\frac{8}{3}$	0.85252	1	14.82575	15.95955	0.36101	1.3914	2.41270

The interaction between a dipole in one plane with the nearest neighbor in another plane is zero in the case that the aromatic rings of one layer are aligned such that they would superimpose as the interlayer separation goes to zero. But, the energy of interaction is nonzero when one plane is translated relative to a neighboring plane. A minimum equal-energy is achieved throughout the graphite structure when each layer is displaced by

$2c_{\begin{array}{c}3e\\ C=C\end{array}}^{\prime },$ the bond length of

$C\stackrel{3e}{——}C$ along an intra-planar C₂ axis relative to the next as shown in FIG. 20B. Then, a pair of dipoles exists for each dipole of a given plane with one dipole above and one below in neighboring planes such that all planes can be equivalently bound by van der Waals forces. In this case, the distance r_{μ1. . . μ2} between dipole μ₁ in one plane and its nearest neighbor μ₂ above or below on a neighboring and

$\begin{array}{cc}2c_{C\stackrel{3e}{=}C}^{\prime }-\mathrm{displaced}\mathrm{plane}\mathrm{is}{r}_{{\mu }_1\dots {\mu }_2}=\sqrt{{\left(2c_{C\stackrel{3e}{=}C}^{\prime }\right)}^2+{\left(2c_{\mathrm{C\dots C}}^{\prime }\right)}^2}& \left(16.80\right)\end{array}$ where 2c_C•••C′ is the interplane distance. The alignment angle θ_{μ1. . . μ2} between the dipoles is

$\begin{array}{cc}\begin{array}{c}{\theta }_{{\mu }_1\dots {\mu }_2}={\sin }^{-1}\frac{2c_{C\dots C}^{\prime }}{r_{{\mu }_1\dots {\mu }_2}}\\ ={\sin }^{-1}\frac{2c_{C\dots C}^{\prime }}{\sqrt{{\left(2c_{C\stackrel{3e}{=}C}^{\prime }\right)}^2+{\left(2c_{C\dots C}^{\prime }\right)}^2}}\end{array}& \left(16.81\right)\end{array}$ The van der Waals energy is the potential energy between interacting neighboring pairs of

$C\stackrel{8/3e}{=}C$ induced dipoles. Using Eqs. (16.80-16.81),

${\mu }_{C\stackrel{8/3e}{=}C}=2.41270D=8.04790\times {10}^{-30}C·m\left(\mathrm{Table}31\right),\mathrm{and}\mathrm{the}C\stackrel{8/3e}{=}C\mathrm{distance},2c_{C\stackrel{8/3e}{=}C}^{\prime }=1.39140\times {10}^{-10}m\left(\mathrm{Table}15.214\right),$ the van der Waals energy of graphite between two planes at a vertex atom is

$\begin{array}{cc}\begin{array}{c}{E}_{\mathrm{van}\mathrm{der}\mathrm{Waals}}\left(\mathrm{graphite}\right)=\left(3\right)\frac{2{\left({\mu }_{C\stackrel{813e}{=}C}\right)}^2}{4\pi {{\varepsilon }_0\left(r_{{\mu }_1\dots {\mu }_2}\right)}^3}\cos {\theta }_{{\mu }_1\dots {\mu }_2}\\ =\left(\frac{6{\left(8.04790\times {10}^{-30}C·m\right)}^2}{\begin{array}{c}4\pi {{\varepsilon }_0\left(\begin{array}{c}{\left(1.39140\times {10}^{-10}m\right)}^2+\\ {\left(2\sqrt{\frac{a_{C\dots C}{a}_0}{2C_1{C}_2}}\right)}^2\end{array}\right)}^{1.5}\\ \cos {\sin }^{-1}\frac{2\sqrt{\frac{a_{C\dots C}{a}_0}{2C_1{C}_2}}}{\sqrt{\begin{array}{c}{\left(1.39140\times {10}^{-10}m\right)}^2+\\ {\left(2\sqrt{\frac{a_{C\dots C}{a}_0}{2C_1{C}_2}}\right)}^2\end{array}}}\end{array}}\right)\end{array}& \left(16.82\right)\end{array}$ where there are three bonds at each vertex atom.

The graphite inter-plane distance of 3.5 Å [100] is calculated using Eq. (16.25) with the van der Waals energy (Eq. (16.82)) between dipoles of two neighboring planes equated to the nascent bond energy. The energy matching parameter c₂ is the same that of the graphite sheet corresponding to the aromatic carbons as given in the Graphite section, and the reduced mass is μ=6. The parameters are summarized in Table 32 and Eq. (16.83).

TABLE 32
The energy parameters (eV) of the graphite interplanar
functional group (Caromatic···Caromatic).
                              Caromatic···Caromatic
Parameters                    Group
n1                            1
C1                            0.5
C2                            1
c1                            1
c2                            0.85252
C1o                           0.5
C2o                           1
Ve (eV)                       −4.35014
Vp (eV)                       4.10093
T (eV)                        0.19760
Vm (eV)                       −0.09880
E(AO/HO) (eV)                 0
ΔEH2MO(AO/HO) (eV)            0
ET(AO/HO) (eV)                0
ET(H2MO) (eV)                 −0.15042
ET(atom-atom, msp3 · AO) (eV) 0
ET(MO) (eV)                   −0.15042
ω (1015 rad/s)                0.800466
EK (eV)                       0.52688
ĒD (eV)                       −0.00022
ĒKvib (eV)                    0.00317
Ēosc (eV)                     0.00137
ET(Group) (eV)                −0.14905

Substitution of the parameters of Table 32 and the interlayer cohesive energy of graphite (Eq. (16.82)) into Eq. (16.25) with R=a_C•••C gives

$\begin{array}{cc}\frac{-6{\left(8.04790\times {10}^{-30}C·m\right)}^2}{4\pi {{\varepsilon }_0\left(\begin{array}{c}{\left(1.39140\times {10}^{-10}m\right)}^2+\\ {\left(2\sqrt{\frac{a_{C\dots C}{a}_0}{2\left(0.5\right)}}\right)}^2\end{array}\right)}^{1.5}\hspace{1em}}\hspace{1em}\hspace{1em}\cos {\sin }^{-1}\frac{2\sqrt{\frac{a_{C\dots C}{a}_0}{2\left(0.5\right)}}}{\sqrt{\begin{array}{c}{\left(1.39140\times {10}^{-10}m\right)}^2+\\ {\left(2\sqrt{\frac{a_{C\dots C}{a}_0}{2\left(0.5\right)}}\right)}^2\end{array}}}=\left\{\begin{array}{c}\left(\frac{-e^2}{8\pi {\varepsilon }_0\sqrt{\frac{a_{C\dots C}{a}_0}{2\left(0.5\right)}}}\left(\begin{array}{c}\left(0.85252\right)\left(2-\frac{1}{2}\frac{a_0}{a_{C\dots C}}\right)\\ \ln \frac{a+\sqrt{\frac{a_{C\dots C}{a}_0}{2\left(0.5\right)}}}{a-\sqrt{\frac{a_{C\dots C}{a}_0}{2\left(0.5\right)}}}-1\end{array}\right)\right)\\ \left(1+2\sqrt{\frac{2\hslash \sqrt{\frac{\left(0.5\right)\frac{e^2}{4\pi {{\varepsilon }_o\left(a_{C\dots C}\right)}^3}}{m_e}}}{m_e{c}^2}}\right)+\\ \left(\frac{1}{2}\right)\hslash \sqrt{\frac{\frac{\left(0.85252\right)e^2}{8\pi {{\varepsilon }_o\left(a_{C\dots C}\right)}^3}-\frac{e^2}{8\pi {{\varepsilon }_o\left(a_{C\dots C}+\sqrt{\frac{a_{C\dots C}{a}_0}{2\left(0.5\right)}}\right)}^3}}{6}}\end{array}\right\}& \left(16.83\right)\end{array}$ From the energy relationship given by Eq. (16.83) and the relationships between the axes given by Eqs. (16.22-16.24), the dimensions of the C•••C MO can be solved.

The most convenient way to solve Eq. (16.83) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures isa_C•••C=11.00740a₀=5.82486×10⁻¹⁰ m  (16.84)The component energy parameters at this condition are given in Table 32. Substitution of Eq. (16.84) into Eq. (16.22) givesc_C•••C′=3.31774a₀=1.75567×10⁻¹⁰ m  (16.85)and internuclear distance of the graphite interplane bond at vacuum ambient pressure:2c_C•••C′=6.63548a₀=3.51134×10⁻¹⁰ m=3.51134 Å  (16.86)The experimental graphite interplane distance 2c_C•••C′ is [100]2c_C•••C′=3.5×10⁻¹⁰ m=3.5 Å  (16.87)The other interplane bond MO parameters can also be determined by the relationships among the parameters. Substitution of Eqs. (16.84) and (16.85) into Eq. (16.23) givesb_C•••C=c_C•••C=10.49550a₀=5.55398×10⁻¹⁰ m  (16.88)Substitution of Eqs. (16.84) and (16.85) into Eq. (16.25) givese_C•••C=0.30141  (16.89)Using Eqs. (16.80) and (16.86), the distance r_{μ1. . . μ2} between dipole μ₁ on one plane and its nearest neighbor μ₂ above or below on a juxtaposed and

$\begin{array}{cc}2c_{C\stackrel{3e}{=}C}^{\prime }-\mathrm{displaced}\mathrm{plane}\mathrm{is}{r}_{{\mu }_1\dots {\mu }_2}=3.77697\times {10}^{-10}m& \left(16.90\right)\end{array}$ Using Eqs. (16.81) and (16.86), the alignment angle θ_{μ1. . . μ2} between the dipoles isθ_{μ1. . . μ2}=68.38365°  (16.91)Using Eqs. (16.82) and (16.90-91), the van der Waals energy per carbon atom isE_{van der Waals}(graphite/C)=0.04968 eV  (16.92)The experimental van der Waals energy per carbon atom is [101]E_{van der Waals}(graphite/C)=0.052 eV  (16.93)The calculated results based on first principles and given by analytical equations are summarized in Table 33.

TABLE 33
The calculated and experimental geometrical parameters and
interplane van der Waals cohesive energy of graphite.
Parameter	Calculated	Experimental	Ref. for Exp.
Graphite Interplane	3.51134	Å	3.5	Å	100
Distance 2c′C...C
van der Waals Energy	0.04968	eV	0.052	eV	101
per Carbon Atom

Graphite has a high cohesive energy due to its significant van der Waals dipole bond moment of 2.41270D. Other species such as atoms and molecules having mirror symmetry and consequently no permanent dipole moment also form reversible van der Waals dipole bond moments. Different phases can be achieved according to the extent of the van der Waals dipole bonding as the internal energy as a function of temperature and pressure changes analogously to the H-bonded system water that can exist as ice, water, and steam. Thus, the factors in the van der Waals bonding can give rise to numerous material behaviors. In the case of atoms such as noble gas atoms and certain diatomic molecules such as hydrogen, the moments, their interaction energies, and the corresponding nascent bond energies are much smaller. Thus, except at cryogenic temperatures, these elements exist as gases, and even at temperatures approaching absolute zero, solidification of helium has not been achieved in the absence of high pressure. This is due to the nature of the induced dipoles and van der Waals phenomena in helium. Since this system is a good example of van der Waals forces in atoms, it will be treated next.

Geometrical Parameters and Energies Due to the Interatomic Van Der Waals Cohesive Energy of Liquid Helium

Noble gases such as helium are typically gaseous and comprised of non-interacting atoms having no electric or magnetic multipoles. But, at very low temperatures it is possible to form diffuse diatomic molecules, or alternatively, these gases may be condensed with the formation of mutually induced van der Waals dipole interactions. As a measure of the nascent bond between two noble gas atoms used to calculate the limiting separation for condensation, consider that the experimental bond energies of diatomic molecules of helium and argon, for example, are only 49.7 meV and 49 meV, respectively [21]. This is a factor of about 100 smaller than the bond energy of a carbon-carbon bond that is the form of nascent bond in graphite. Thus, the corresponding energy of the interspecies interaction is smaller and the van der Waals spacing is larger, except wherein the nascent bond energy as a function of separation distance mitigates this relationship to some extent. The nature of the helium bonding is solved using the same approach as that of other functional groups given in the Organic Molecular Functional Groups and Molecules section.

Helium is a two-electron neutral atom with both electrons paired as mirror-image current densities in a shell of radius 0.566987a₀ (Eq. (7.35)). Thus, in isolation or at sufficient separation, there is no energy between helium atoms. The absence of any force such as so-called long-range London forces having a r⁻ⁿ; n>2 dependency is confirmed by elastic electron scattering from helium atoms as shown in the Electron Scattering Equation for the Helium Atom Based on the Orbitsphere Model section. However, reversible mutual van der Waals dipoles may be induced by collisions when the atoms are in close proximity such that helium gas can condense into a liquid. The physics is similar to the case of graphite except that the dipoles are atomic rather than molecular, and in both cases the limiting separation is based on the formation of a nascent bond to replace the dipole-dipole interaction. Thus, Eq. (16.25) can also be applied to atoms such as helium.

The van der Waals bonding in the helium atom involves hybridizing the one 1s AO into 1s¹ HO orbitals containing two electrons. The total energy of the state is given by the sum over the two electrons. The sum E_T(He,1s¹) of experimental energies [15] of He and He⁺ is

$\begin{array}{cc}\begin{array}{c}{E}_T\left(\mathrm{He},1s^1\right)=54.41776\mathrm{eV}+24.587387\mathrm{eV}\\ =79.005147\mathrm{eV}\end{array}& \left(16.94\right)\end{array}$ By considering that the central field decreases by an integer for each successive electron of the shell, the radius r_1s1 of the He1s¹ shell may be calculated from the Coulombic energy using Eq. (15.13):

$\begin{array}{cc}\begin{array}{c}{r}_{1s^1}=\sum _{n=0}^1\frac{\left(Z-n\right)e^2}{8\pi {\varepsilon }_0\left(e79.005147\mathrm{eV}\right)}\\ =\frac{3e^2}{8\pi {\varepsilon }_0\left(e79.005147\mathrm{eV}\right)}\\ =0.51664a_0\end{array}& \left(16.95\right)\end{array}$ where Z=2 for helium. Using Eq. (15.14), the Coulombic energy E_Coulomb(He,1s¹) of the outer electron of the van der Waals bound He1s¹ shell is

$\begin{array}{cc}\begin{array}{c}{E}_{\mathrm{Coulomb}}\left(\mathrm{He},1s^1\right)=\frac{-e^2}{8\pi {\varepsilon }_0{r}_{1s^1}}\\ =\frac{-e^2}{8\pi {\varepsilon }_{0}0.51664a_0}\\ =-26.335049\mathrm{eV}\end{array}& \left(16.96\right)\end{array}$ To meet the equipotential condition of the union of the two He1s¹ HOs in a nascent bond, c₂ of Eqs. (15.2-15.5) and Eq. (15.61) for the nascent He—He-bond MO is given by Eq. (15.75) as the ratio of the valance energy of the He AO, E(He)=−24.587387 eV and the magnitude of E_Coulomb(He,1s¹) (Eq. (16.96)):

$\begin{array}{cc}{c}_2\left(\mathrm{He}\mathrm{—He},\mathrm{He}1s^1\mathrm{HO}\right)=\frac{24.587387\mathrm{eV}}{26.33505\mathrm{eV}}=0.93364& \left(16.97\right)\end{array}$ The opposite charges distributions act as symmetrical point charges at the point of maximum separation, each being centered at ½ the He-atom radius from the origin. Using the parameters of Eq. (16.97) and 2c′=0.51664a₀=2.73395×10⁻¹¹ m (Eq. (16.95)), the van der Waals dipole of helium is given in Table 34.

TABLE 34
The parameters and van der Waals dipole bond
moment of the He functional group of liquid helium.
Functional Group	n1	(c1)c2	(C1)C2	EB(valence)	EA(valence)	$\frac{q}{e}$	Bond Length 2c′ (Å)	Bond Moment μ (D)
He	1	0.93364	1	24.587387	26.33505	0.13744	0.273395	0.18049

As in the case with graphite, the van der Waals energy is the potential energy between interacting neighboring induced dipoles. Using μ_He=0.18049 D=6.02040×10⁻³¹ C·m (Table 34), the van der Waals energy is

$\begin{array}{cc}\begin{array}{c}{E}_{\mathrm{van}\mathrm{der}\mathrm{Waals}}\left(\mathrm{He}\right)=2\frac{2{\left({\mu }_{\mathrm{He}}\right)}^2}{4\pi {{\varepsilon }_0\left(r_{\mathrm{He}\dots \mathrm{He}}\right)}^3}\\ =\left(\frac{2{\left(6.02040\times {10}^{-31}C·m\right)}^2}{4\pi {{\varepsilon }_0\left(2\sqrt{\frac{a_{\mathrm{He}\dots \mathrm{He}}{a}_0}{2C_1{C}_2}}\right)}^3}\right)\end{array}& \left(16.98\right)\end{array}$ where there are two bonds at each vertex atom.

The helium interatomic distance is calculated using Eq. (16.25) with the van der Waals energy (Eq. (16.98)) between neighboring dipoles equated to the nascent bond energy. The energy matching parameter c₂ is the same that of the helium dipole, and the reduced mass is μ=2. The parameters are summarized in Table 35 and Eq. (16.99).

TABLE 35
The energy parameters (eV) of the helium functional group (He . . . He).
                              He . . . He
Parameters                    Group
n1                            1
C1                            0.5
C2                            0.93364−1
c1                            1
c2                            0.93364
C1o                           0.5
C2o                           0.93364−1
Ve (eV)                       −3.96489
Vp (eV)                       3.88560
T (eV)                        0.15095
Vm (eV)                       −0.07548
E(AO/HO) (eV)                 0
ΔEH2MO(AO/HO) (eV)            0
ET(AO/HO) (eV)                0
ET(H2MO) (eV)                 −0.00382
ET(atom-atom, msp3 · AO) (eV) 0
ET(MO) (eV)                   −0.00382
ω (1015 rad/s)                0.635696
EK (eV)                       0.41843
ĒD (eV)                       0.00000
ĒKvib (eV)                    0.00443
Ēosc (eV)                     0.00221
ET(Group) (eV)                −0.00160

Substitution of the parameters of Table 35 and the interatomic cohesive energy of helium (Eq. (16.89)) into Eq. (16.25) with R=a_{He•••He} gives

$\begin{array}{cc}\frac{-4{\left(6.02040\times {10}^{-31}C·m\right)}^2}{4\pi {{\varepsilon }_0\left(2\sqrt{\frac{a_{\mathrm{He}\dots \mathrm{He}}{a}_0}{2\left(0.5\right){\left(0.93364\right)}^{-1}}}\right)}^3}=\left\{\begin{array}{c}\left(\begin{array}{c}\frac{-e^2}{8\pi {\varepsilon }_0\sqrt{\frac{a_{\mathrm{He}\dots \mathrm{He}}{a}_0}{2\left(0.5\right){\left(0.93364\right)}^{-1}}}}\\ \left(\begin{array}{c}\left(0.93364\right)\left(2-\frac{1}{2}\frac{a_0}{a_{\mathrm{He}\dots \mathrm{He}}}\right)\\ \ln \frac{a+\sqrt{\frac{a_{\mathrm{He}\dots \mathrm{He}}{a}_0}{2\left(0.5\right){\left(0.93364\right)}^{-1}}}}{a-\sqrt{\frac{a_{\mathrm{He}\dots \mathrm{He}}{a}_0}{2\left(0.5\right){\left(0.93364\right)}^{-1}}}}-1\end{array}\right)\end{array}\right)\\ \left(1+2\sqrt{\frac{2\hslash \sqrt{\frac{\left(0.5\right){\left(0.93364\right)}^{-1}\frac{e^2}{4\pi {{\varepsilon }_0\left(a_{\mathrm{He}\dots \mathrm{He}}\right)}^3}}{m_e}}}{m_e{c}^2}}\right)+\\ \left(\frac{1}{2}\right)\hslash \sqrt{\frac{\begin{array}{c}\frac{\left(0.93364\right)e^2}{8\pi {{\varepsilon }_o\left(a_{\mathrm{He}\dots \mathrm{He}}\right)}^3}-\\ \frac{e^2}{8\pi {{\varepsilon }_o\left(a_{\mathrm{He}\dots \mathrm{He}}+\sqrt{\frac{a_{\mathrm{He}\dots \mathrm{He}}{a}_0}{2\left(0.5\right){\left(0.93364\right)}^{-1}}}\right)}^3}\end{array}}{2}}\end{array}\right\}& \left(16.99\right)\end{array}$ From the energy relationship given by Eq. (16.99) and the relationships between the axes given by Eqs. (16.22-16.24), the dimensions of the He•••He MO can be solved.

The most convenient way to solve Eq. (16.99) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures isa_{He•••He}=13.13271a₀=6.94953×10⁻¹⁰ m  (16.100)The component energy parameters at this condition are given in Table 35. Substitution of Eq. (16.100) into Eq. (16.22) givesc_{He•••He}′=3.50160a₀=1.85297×10⁻¹⁰ m  (16.101)and internuclear distance between neighboring helium atoms:2c_{He•••He}′=7.00320a₀=3.70593×10⁻¹⁰ m=3.70593 Å  (16.102)The experimental helium interatomic distance 2c_C•••C′ at 4.24K and <2.25 K are [102]2c_{He•••He}′(4.24K)=3.72×10⁻¹⁰ m=3.72 Å2c_{He•••He}′(<2.25K)=3.70×10⁻¹⁰ m=3.70 Å  (16.103)The other interatomic bond MO parameters can also be determined by the relationships among the parameters. Substitution of Eqs. (16.100) and (16.101) into Eq. (16.23) givesb_{He•••He}=c_{He•••He}=12.65729a_o=6.69795×10⁻¹⁰ m  (16.104)Substitution of Eqs. (16.100) and (16.101) into Eq. (16.25) givese_{He•••He}=0.26663  (16.105)Using Eqs. (16.89) and (16.102) and the relationship that there are two van der Waals bonds per helium atom and two atoms per bond, the van der Waals energy per helium atom isE_{van der Waals}(liquid He/He)=0.000799 eV  (16.106)The experimental van der Waals energy calculated from the heat of vaporization per helium atom is [103]E_{van der Waals}(liquid He)=E_vapor,4.221K=0.0829 kJ/mole=0.000859 eV/He  (16.107)At 1.7 K, the viscosity of liquid helium is close to zero, and a characteristic roton scattering dominates over phonon scattering at this temperature and below [104]. The van der Waals bond energy is also equivalent to the roton energy [105, 106]E_roton(liquid He)=8.7 K=0.00075 eV  (16.108)and the roton is localized within a region of radius ≈3.7-4.0 Å [104, 106-108] that matches the He•••He van der Waals bond distance (Eq. (16.102)). The origin of the roton energy and its cross section as belonging to the van der Waals bond resolves its nature. Independent of this result, the modem view of the roton is that it is not considered associated with the excitation of vorticity as it was historically; rather it is considered to be due to short-wavelength phonon excitations [105]. Its role in scattering free electrons in superfluid helium is discussed in the Free Electrons in Superfluid Helium are Real in the Absence of Measurement Requiring a Connection of ψ to Physical Reality section. The calculated results based on first principles and given by analytical equations are summarized in Table 36.

TABLE 36
The calculated and experimental geometrical parameters and
interatomic van der Waals cohesive energy of liquid helium.
			Ref. for
Parameter	Calculated	Experimental	Exp.
Liquid Helium	3.70593	Å	3.72	Å (T = 4.24 K)	102
Interatomic
Distance 2c′C...C			3.70	(T < 2.25 K)
Roton Length	3.70593	Å	3.7-4.0	Å	104,
Scale					106-108
van der Waals	0.000799	eV	0.000859	eV	103
Energy per Helium
Atom (4.221 K)
Roton Energy	0.000799	eV	0.00075	eV	105, 106

Helium, exhibits unique behavior due to its possible phases based on the interplay of the factors that determine the van der Waals bonding at a given temperature and pressure to achieve an energy minimum. In extreme cases of sufficient ultra-low temperatures with the atoms driven in phase with an external excitation field such that the formation of a van der Waals-dipole-bound macromolecular state or other forms of bonding, such as metallic bonding in the case of alkali metals or van der Waals bonding in meta-stable helium atoms, are suppressed, a pure statistical thermodynamic state called a Bose-Einstein condensate [109] (BEC)¹ can form having a predominant population of the atoms in a single, lowest-energy translational state in the trap. Since helium has only two electrons in an outer s-shell having a small diameter, the dipole moment too very weak to form transverse dipoles associated with packing. Specifically, with the angular dependence of packed dipoles interactions, the van der Waals energy E_{van der Waals}(He) (Eqs. (16.98) and (16.99)) between neighboring dipoles becomes less than the vibrational energy in the transition state (Ē_Kvib term of Eq. (16.99) from Eq. (15.53)). Consequently, helium can only mutually induce and form linear dipole-dipole bonds having end-to-end interactions as an energy minimum. Interposed atoms can form a non-bonded phase having correlated translational motion and obeying Bose-Einstein statistics. This phase forms a Bose-Einstein condensate (BEC) as an energy minimum wherein the translations are synchronous. Since a phase comprised of linearly ordered unit cells held together by dipole interactions, specifically van der Waals dipole interactions, can exist with a BEC phase, super-fluidity can arise wherein the lines of bound dipoles move without friction relative to the BEC phase having correlated-translational motion. The linear bonding is also the origin of quantized vortex rings that enter as quantized vortex lines to form rings. ¹ The BEC is incorrectly interpreted as a single large atom having a corresponding probability wave function of quantum mechanics. Since excitation occurs in units of ℏ in order of to conserve angular momentum as shown previously for electronic (Chapter 2), vibrational (Chapter 11), rotational (Chapter 12), and translational excitation (Chapter 3) and Bose-Einstein statistics arise from an underlying deterministic physics (Chapter 24), this state comprised of an ensemble of individual atoms is predicted classically using known equations [110]. As in the case of the coherent state of photons in a laser cavity (Chapter 4), the coherency of the BEC actually disproves the inherent Heisenberg Uncertainty Principle (HUP) of quantum mechanics since the atomic positions and energies are precisely determined simultaneously. Furthermore, it is possible to form a BEC comprising molecules in addition to atoms [111] wherein the molecules lack zero-order vibration in contradiction to the HUP. The classical physics underlying Bose-Einstein statistics was covered in the Statistical Mechanics section.

The van der Waals bonds undergo breakage and fbrmation and exist on a time-average basis depending on the internal energy and pressure as in the case of liquid water. The van der Waals bonding exhibits a maximum extent as the temperature is lowered below the boiling point, and the BEC phase comprises the balance of the atoms as the temperature is further lowered to absolute zero. Helium cannot form a solid without application of high pressure to decrease the interatomic separation and permit energetically favorable transverse dipole interactions as well as linear ones. In contrast, other noble gases such as Ne, Ar, Kr, and Xe each possess additional shells including an outer p-shell having a relatively larger radius that gives rise to a significant bond moment supportive of dipole packing interactions; thus, these gases can form solids without the application of high pressure.

Geometrical Parameters and Energies Due to the Interatomic Van Der Waals Cohesive Energy of Solid Neon

Neon is a ten-electron neutral atom having the electron configuration 1s²2s₂2p⁶ with the electrons of each shell paired as mirror-image current densities in a shell wherein the radius of the outer shell is r₁₀=0.63659a₀ (Eq. (10.202)). Thus, in isolation or at sufficient separation, there is no energy between neon atoms. However, reversible mutual van der Waals dipoles may be induced by collisions when the atoms are in close proximity such that neon gas can condense into a liquid and further solidify at sufficiently low temperatures due to the strong dipole moment that accommodates close packing. As in the case of helium, the dipoles are atomic rather than molecular, and the limiting separation is based on the formation of a nascent bond to replace the dipole-dipole interaction. Thus, Eq. (16.25) can also be applied to neon atoms.

The van der Waals bonding in the neon atom involves hybridizing the three 2p AOs into 2p³ HO orbitals containing six electrons. The total energy of the state is given by the sum over the six electrons. The sum E_T(Ne,2p³) of experimental energies [15] of Ne, Ne⁺, Ne²⁺, Ne³⁺, Ne⁴⁺, and Ne⁵⁺ is

$\begin{array}{cc}\begin{array}{c}{E}_T\left(\mathrm{Ne},2p^3\right)=\left(\begin{array}{c}157.93\mathrm{eV}+126.21\mathrm{eV}+97.12\mathrm{eV}+\\ 63.45\mathrm{eV}+40.96296\mathrm{eV}+\\ 21.56454\mathrm{eV}\end{array}\right)\\ =507.2375\mathrm{eV}\end{array}& \left(16.109\right)\end{array}$ By considering that the central field decreases by an integer for each successive electron of the shell, the radius r_2p3 of the Ne2p³ shell may be calculated from the Coulombic energy using Eq. (15.13):

$\begin{array}{cc}\begin{array}{c}{r}_{2p^3}=\sum _{n=4}^9\frac{\left(Z-n\right)e^2}{8\pi {\varepsilon }_0\left(e507.2375\mathrm{eV}\right)}\\ =\frac{21e^2}{8\pi {\varepsilon }_0\left(e507.2375\mathrm{eV}\right)}\\ =0.56329a_0\end{array}& \left(16.110\right)\end{array}$ where Z=10 for neon. Using Eq. (15.14), the Coulombic energy E_Coulomb(Ne,2p³) of the outer electron of the van der Waals bound Ne2p³ shell is

$\begin{array}{cc}\begin{array}{c}{E}_{\mathrm{Coulomb}}\left(\mathrm{Ne},2p^3\right)=\frac{-e^2}{8\pi {\varepsilon }_0{r}_{2p^3}}\\ =\frac{-e^2}{8\pi {\varepsilon }_{0}0.56329a_0}\\ =-24.154167\mathrm{eV}\end{array}& \left(16.111\right)\end{array}$ To meet the equipotential condition of the union of the two Ne2p³ HOs in a nascent bond, c₂ of Eqs. (15.2-15.5) and Eq. (15.61) for the nascent Ne—Ne-bond MO is given by Eq. (15.75) as the ratio of the valance energy of the Ne AO, E(Ne)=−21.56454 eV and the magnitude of E_Coulomb(Ne,2p³) (Eq. (16.111)):

$\begin{array}{cc}{c}_2\left(\mathrm{Ne}\mathrm{—Ne},\mathrm{Ne}2p^3\mathrm{HO}\right)=\frac{21.56454\mathrm{eV}}{24.154167\mathrm{eV}}=0.89279& \left(16.112\right)\end{array}$ The opposite charges distributions act as symmetrical point charges at the point of maximum separation, each being centered at ½ the Ne-atom radius from the origin. Using the parameters of Eq. (16.112) and 2c′=0.56329a₀=2.98080×10⁻¹¹ m (Eq. (16.110)), the van der Waals dipole of neon is given in Table 37.

TABLE 37
The parameters and van der Waals dipole bond
moment of the Ne functional group of solid neon.
Functional Group	n1	(c1)c2	(C1)C2	EB(valence)	EA(valence)	$\frac{q}{e}$	Bond Length 2c′ (Å)	Bond Moment μ (D)
Ne	1	0.89279	1	21.56454	24.15417	0.22730	0.298080	0.32544

The minimum-energy packing of neon dipoles is face-centered cubic also called cubic close packing. In this case, each neon atom has 12 nearest neighbors and the angle between the aligned dipoles is

$\frac{\pi }{4}$ radians. As in the case with graphite, the van der Waals energy is the potential energy between interacting neighboring induced dipoles. Using μ_Ne=0.32544 D=1.08554×10⁻¹⁰C·m (Table 37), the van der Waals energy is

$\begin{array}{cc}\begin{array}{c}{E}_{\mathrm{van}\mathrm{der}\mathrm{Waals}}\left(\mathrm{Ne}\right)=12\frac{2{\left({\mu }_{\mathrm{Ne}}\right)}^2}{4{{\mathrm{\pi \varepsilon }}_0\left(r_{\mathrm{Ne}\dots \mathrm{Ne}}\right)}^3}\cos \left(\frac{\pi }{4}\right)\\ =\left(\frac{24{\left(1.08554\times {10}^{-30}C·m\right)}^2}{4{{\mathrm{\pi \varepsilon }}_0\left(2\sqrt{\frac{a_{\mathrm{Ne}\dots \mathrm{Ne}}{a}_0}{2C_1{C}_2}}\right)}^3}\right)\cos \left(\frac{\pi }{4}\right)\end{array}& \left(16.113\right)\end{array}$

The neon interatomic distance is calculated using Eq. (16.25) with the van der Waals energy (Eq. (16.113)) between neighboring dipoles equated to the nascent bond energy. The energy matching parameter c₂ is the same that of the neon dipole, and the reduced mass is μ=10. The parameters are summarized in Table 38 and Eq. (16.114).

TABLE 38
The energy parameters (eV) of the neon functional group (Ne . . . Ne).
                              Ne . . . Ne
Parameters                    Group
n1                            1
C1                            0.5
C2                            0.89279−1
c1                            1
c2                            0.89279
C1o                           0.5
C2o                           0.89279−1
Ve (eV)                       −4.40464
Vp (eV)                       4.27694
T (eV)                        0.19429
Vm (eV)                       −0.09714
E(AO/HO) (eV)                 0
ΔEH2MO(AO/HO) (eV)            0
ET(AO/HO) (eV)                0
ET(H2MO) (eV)                 −0.03055
ET(atom-atom, msp3 · AO) (eV) 0
ET(MO) (eV)                   −0.03055
ω (1015 rad/s)                0.810674
EK (eV)                       0.53360
ĒD (eV)                       −0.00004
ĒKvib (eV)                    0.00240
Ēosc (eV)                     0.00116
ET(Group) (eV)                −0.02939

Substitution of the parameters of Table 38 and the interatomic cohesive energy of neon (Eq. (16.113)) into Eq. (16.25) with R=a_{Ne•••Ne} gives

$\begin{array}{cc}\frac{-24{\left(1.08554\times {10}^{-30}C·m\right)}^2}{4{{\mathrm{\pi \varepsilon }}_0\left(2\sqrt{\frac{a_{\mathrm{Ne}\dots \mathrm{Ne}}{a}_0}{2\left(0.5\right){\left(0.89279\right)}^{-1}}}\right)}^3}\cos \left(\frac{\pi }{4}\right)=\left\{\begin{array}{c}\left(\begin{array}{c}\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_0\sqrt{\frac{a_{\mathrm{Ne}\dots \mathrm{Ne}}{a}_0}{2\left(0.5\right){\left(0.89279\right)}^{-1}}}}\\ \left(\begin{array}{c}\left(0.89279\right)\left(2-\frac{1}{2}\frac{a_0}{a_{\mathrm{Ne}\dots \mathrm{Ne}}}\right)\\ \ln \frac{a+\sqrt{\frac{a_{\mathrm{Ne}\dots \mathrm{Ne}}{a}_0}{2\left(0.5\right){\left(0.89279\right)}^{-1}}}}{a-\sqrt{\frac{a_{\mathrm{Ne}\dots \mathrm{Ne}}{a}_0}{2\left(0.5\right){\left(0.89279\right)}^{-1}}}}-1\end{array}\right)\end{array}\right)\\ \left(1+2\sqrt{\frac{2\hslash \sqrt{\frac{\left(0.5\right){\left(0.89279\right)}^{-1}\frac{e^2}{4{{\mathrm{\pi \varepsilon }}_0\left(a_{\mathrm{Ne}\dots \mathrm{Ne}}\right)}^3}}{m_e}}}{m_e{c}^2}}\right)+\\ \left(\frac{1}{2}\right)\hslash \sqrt{\frac{\begin{array}{c}\frac{\left(0.89279\right)e^2}{8{{\mathrm{\pi \varepsilon }}_o\left(a_{\mathrm{Ne}\dots \mathrm{Ne}}\right)}^3}-\\ \frac{e^2}{8{{\mathrm{\pi \varepsilon }}_o\left(a_{\mathrm{Ne}\dots \mathrm{Ne}}+\sqrt{\frac{a_{\mathrm{Ne}\dots \mathrm{Ne}}{a}_0}{2\left(0.5\right){\left(0.89279\right)}^{-1}}}\right)}^3}\end{array}}{10}}\end{array}\right\}& \left(16.114\right)\end{array}$ From the energy relationship given by Eq. (16.114) and the relationships between the axes given by Eqs. (16.22-16.24), the dimensions of the Ne•••Ne MO can be solved.

The most convenient way to solve Eq. (16.114) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures isa_{Ne•••Ne}=11.33530a₀=5.99838×10⁻¹⁰ m  (16.115)The component energy parameters at this condition are given in Table 38. Substitution of Eq. (16.115) into Eq. (16.22) givesc_{Ne•••Ne}′=3.18120a₀1.68342×10⁻¹⁰ m  (16.116)and internuclear distance between neighboring neon atoms:2c_{Ne•••Ne}′=6.36239a₀=3.36683×10⁻¹⁰ m=3.36683 Å  (16.117)The experimental neon interatomic distance 2c_C•••C′ at the melting point of 24.48 K is [112, 113]2c_{Ne•••Ne}′(24.48K)=3.21×10⁻¹⁰ m=3.21 Å  (16.118)The other interatomic bond MO parameters can also be determined by the relationships among the parameters. Substitution of Eqs. (16.115) and (16.116) into Eq. (16.23) givesb_{Ne•••Ne}c_{Ne•••Ne}=10.87975a₀=5.75732×10⁻¹⁰ m  (16.119)Substitution of Eqs. (16.115) and (16.116) into Eq. (16.25) givese_{Ne•••Ne}=0.28065  (16.120)

A convenient method to calculate the lattice energy is to determine the electric field in solid neon having an electric polarization density corresponding to the aligned dipoles moments, and in turn, the energy can be calculated from the energy of each dipole in the corresponding field using the electrostatic form of Gauss' equation. Substitution of the density of solid neon at the melting point

$\begin{array}{cc}\rho =\frac{1.433g}{1\times {10}^{-6}{m}^3},& \left[113\right]\end{array}$ the MW=20.179 g/mole, N_A=6.0221415×10²³ molecules/mole, and the neon dipole moment given in Table 37 into Eq. (16.53) gives:

$\begin{array}{cc}\begin{array}{c}U\left(\mathrm{Ne}\right)=\frac{-2{\left({\mu }_{\mathrm{Ne}}\right)}^2\frac{{\rho }_{\mathrm{solid}\mathrm{Ne}}}{\mathrm{MW}}{N}_A}{3{\varepsilon }_0}\\ \frac{\begin{array}{c}-2{\left(1.08554\times {10}^{-30}C·m\right)}^2\frac{\frac{1.433g}{1\times {10}^{-6}{m}^3}}{20.179g\text{/}\mathrm{mole}}\\ 6.0221415\times {10}^{23}\mathrm{molecules}\text{/}\mathrm{mole}\end{array}}{3{\varepsilon }_0}\\ =-0.02368\mathrm{eV}\left(-2.285\mathrm{kJ}\text{/}\mathrm{mole}\right)\end{array}& \left(16.121\right)\end{array}$ U(Ne) is also the negative of E_{van der Waals}, the van der Waals energy per neon atom:E_{van der Waals}(solid Ne/Ne)=0.02368 eV=2.285 kJ/mole  (16.122)The experimental van der Waals energy calculated from the heat of vaporization and fusion per neon atom at the boiling point and triple point, respectively, is [103]E_{van der Waals}(solid Ne)=E_vapor+E_fusion=0.02125 eV/Ne=2.0502 kJ/mole  (16.123)The calculated results based on first principles and given by analytical equations are summarized in Table 39. Using neon the atomic radius (Eq. (16.110)) and the nearest-neighbor distance (Eq. (16.117)), the lattice structure of neon is shown in FIG. 21A. The charge density of the van der Waals dipoles of the crystalline lattice is shown in FIG. 22A.

TABLE 39
The calculated and experimental geometrical parameters and
interatomic van der Waals cohesive energy of solid neon.
			Ref. for
Parameter	Calculated	Experimental	Exp.
Solid Neon Interatomic	3.36683	Å	3.21	Å	113
Distance 2c′C...C			(T = 24.48	K)
van der Waals Energy	0.02368	eV	0.02125	eV	103
per Neon Atom

Geometrical Parameters and Energies Due to the Interatomic Van Der Waals Cohesive Energy of Solid Argon

Argon is an eighteen-electron neutral atom having the electron configuration 1s₂2s²2p⁶3s²3p⁶ with the electrons of each shell paired as mirror-image current densities in a shell wherein the radius of the outer shell is r₁₈=0.86680a₀ (Eq. (10.386)). Thus, in isolation or at sufficient separation, there is no energy between argon atoms. However, reversible mutual van der Waals dipoles may be induced by collisions when the atoms are in close proximity such that argon gas can condense into a liquid and further solidify at sufficiently low temperatures due to the strong dipole moment that accommodates close packing. As in the case of helium, the dipoles are atomic rather than molecular, and the limiting separation is based on the formation of a nascent bond to replace the dipole-dipole interaction. Thus, Eq. (16.25) can also be applied to argon atoms.

The van der Waals bonding in the argon atom involves hybridizing the three 3p AOs into 3p³ HO orbitals containing six electrons. The total energy of the state is given by the sum over the six electrons. The sum E_T(Ar,3p³) of experimental energies [15] of Ar, Ar⁺, Ar²⁺, Ar³⁺, Ar⁴⁺, and Ar⁵⁺ is

$\begin{array}{cc}\begin{array}{c}{E}_T\left(\mathrm{Ar},3p^3\right)=\left(\begin{array}{c}91.009\mathrm{eV}+75.02\mathrm{eV}+59.81\mathrm{eV}+\\ \begin{array}{c}40.74\mathrm{eV}+27.62966\mathrm{eV}+\\ 15.75961\mathrm{eV}\end{array}\end{array}\right)\\ =309.96827\mathrm{eV}\end{array}& \left(16.124\right)\end{array}$ By considering that the central field decreases by an integer for each successive electron of the shell, the radius r_3p3 of the Ar3p³ shell may be calculated from the Coulombic energy using Eq. (15.13):

$\begin{array}{cc}\begin{array}{c}{r}_{3p^3}=\sum _{n=12}^{17}\frac{\left(Z-n\right)e^2}{8{\mathrm{\pi \varepsilon }}_0\left(e309.96827\mathrm{eV}\right)}\\ =\frac{21e^2}{8{\mathrm{\pi \varepsilon }}_0\left(e309.96827\mathrm{eV}\right)}\\ =0.92178a_0\end{array}& \left(16.125\right)\end{array}$ where Z=18 for argon. Using Eq. (15.14), the Coulombic energy E_Coulomb(Ar,3p³) of the outer electron of the van der Waals bound Ar3p³ shell is

$\begin{array}{cc}\begin{array}{c}{E}_{\mathrm{Coulomb}}\left(\mathrm{Ar},3p^3\right)=\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_0{r}_{3p^3}}\\ =\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_{0}0.92178a_0}\\ =-14.760394\mathrm{eV}\end{array}& \left(16.126\right)\end{array}$ To meet the equipotential condition of the union of the two Ar3p³ HOs in a nascent bond, c₂ of Eqs. (15.2-15.5) and Eq. (15.61) for the nascent Ar— Ar-bond MO is given by Eq. (15.75) as the ratio of the valance energy of the Ar AO, E(Ar)=−15.75961 eV and the magnitude of E_Coulomb(Ar,3p³) (Eq. (16.126)):

$\begin{array}{cc}{c}_2\left(\mathrm{Ar}-\mathrm{Ar},\mathrm{Ar}3p^3\mathrm{HO}\right)=\frac{14.760394\mathrm{eV}}{15.75961\mathrm{eV}}=0.93660& \left(16.127\right)\end{array}$ Since the outer Ar3p³ HO shell is at a lower energy and greater radius than the non-polarized 3p shell, the inner shells are polarized as well. The dipole of the outer shell can polarize the inner shells to the limit that the sum of the primary and secondary dipoles is twice the primary scaled by the energy matching factors of the van der Waals bond given in Eq. (16.15). Thus, the limiting dipole due to polarization of the inner shells is given by

$\begin{array}{cc}\begin{array}{c}{\mu }_{\mathrm{Ar}}<2_{c_1}^{-1}q{C}_{2}2c^{\prime }=2{\left(0.93660\right)}^{-1}\left(0.13110\right){e\left(0.93660\right)}^{-1}\\ \left(4.87784\times {10}^{-11}m\right)\\ =2.49410\times {10}^{-30}C·m\\ =0.74771D\end{array}& \left(16.128\right)\end{array}$ The condition of Eq. (16.128) is matched by the participation of the outer four shells as given in Table 40. At each shell, opposite charges distributions act as symmetrical point charges at the point of maximum separation, each being centered at ½ the shell radius from the origin. Using the parameters of Eq. (16.127) and 2c′=0.92178a₀=4.87784×10⁻¹¹ m (Eq. (16.125)) as well as the radii of the inner shells of argon (Table 10.17), the van der Waals dipole of argon is given in Table 40 as the sum of the moments of each participating shell.

TABLE 40
The parameters and van der Waals dipole bond moment of the Ar functional group of solid argon.
Functional Group	n1	(c1)c2	(C1)C2	EB(valence)	EA(valence)	$\frac{q}{e}$	Bond Length 2c′ (Å)	Bond Moment μ (D)
Ar	1	0.93660	1	14.76039	15.75961	0.13110	Ar3p3 HO 0.48778	0.74366
							Ar3s AO 0.41422
							Ar2p AO 0.15282
							Ar2s AO 0.12615

The minimum-energy packing of argon dipoles is face-centered cubic also called cubic close packing. In this case, each argon atom has 12 nearest neighbors and the angle between the aligned dipoles is

$\frac{\pi }{4}$ radians. As in the case with graphite, the van der Waals energy is the potential energy between interacting neighboring induced dipoles. Using μ_Ar=0.74366 D=2.48058×10⁻³⁰ C·m (Table 40), the van der Waals energy is

$\begin{array}{cc}\begin{array}{c}{E}_{\mathrm{van}\mathrm{der}\mathrm{Waals}}\left(\mathrm{Ar}\right)=12\frac{2{\left({\mu }_{\mathrm{Ar}}\right)}^2}{4{{\mathrm{\pi \varepsilon }}_0\left(r_{\mathrm{Ar}\dots \mathrm{Ar}}\right)}^3}\cos \left(\frac{\pi }{4}\right)\\ =\left(\frac{24{\left(2.48058\times {10}^{-30}C·m\right)}^2}{4{{\mathrm{\pi \varepsilon }}_0\left(2\sqrt{\frac{a_{\mathrm{Ar}\dots \mathrm{Ar}}{a}_0}{2C_1{C}_2}}\right)}^3}\right)\cos \left(\frac{\pi }{4}\right)\end{array}& \left(16.129\right)\end{array}$

The argon interatomic distance is calculated using Eq. (16.25) with the van der Waals energy (Eq. (16.129)) between neighboring dipoles equated to the nascent bond energy. The energy matching parameter c₂ is the same that of the argon dipole, and the reduced mass is μ=20. The parameters are summarized in Table 41 and Eq. (16.130).

TABLE 41
The energy parameters (eV) of the argon functional
group (Ar . . . Ar).
                              Ar . . . Ar
Parameters                    Group
n1                            1
C1                            0.5
C2                            0.93660−1
c1                            1
c2                            0.93660
C1o                           0.5
C2o                           0.93660−1
Ve (eV)                       −4.18356
Vp (eV)                       3.97600
T (eV)                        0.16731
Vm (eV)                       −0.08365
E(AO/HO) (eV)                 0
ΔEH2MO(AO/HO) (eV)            0
ET(AO/HO) (eV)                0
ET(H2MO) (eV)                 −0.12391
ET(atom-atom, msp3 · AO) (eV) 0
ET(MO) (eV)                   −0.12391
ω (1015 rad/s)                0.683262
EK (eV)                       0.44974
ĒD (eV)                       −0.00016
ĒKvib (eV)                    0.00153
Ēosc (eV)                     0.00060
ET(Group) (eV)                −0.12331

Substitution of the parameters of Table 1 and the interatomic cohesive energy of argon (Eq. (16.129)) into Eq. (16.25) with R=a_{Ar•••Ar} gives

$\begin{array}{cc}\frac{-24{\left(2.48058\times {10}^{-30}C·m\right)}^2}{4{{\mathrm{\pi \varepsilon }}_0\left(2\sqrt{\frac{a_{\mathrm{Ar}\dots \mathrm{Ar}}{a}_0}{2\left(0.5\right){\left(0.93660\right)}^{-1}}}\right)}^3}\cos \left(\frac{\pi }{4}\right)=\left\{\begin{array}{c}\left(\begin{array}{c}\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_0\sqrt{\frac{a_{\mathrm{Ar}\dots \mathrm{Ar}}{a}_0}{2\left(0.5\right){\left(0.93660\right)}^{-1}}}}\\ \left(\begin{array}{c}\left(0.93660\right)\left(2-\frac{1}{2}\frac{a_0}{a_{\mathrm{Ar}\dots \mathrm{Ar}}}\right)\ln \\ \frac{a+\sqrt{\frac{a_{\mathrm{Ar}\dots \mathrm{Ar}}{a}_0}{2\left(0.5\right){\left(0.93660\right)}^{-1}}}}{a-\sqrt{\frac{a_{\mathrm{Ar}\dots \mathrm{Ar}}{a}_0}{2\left(0.5\right){\left(0.93660\right)}^{-1}}}}-1\end{array}\right)\end{array}\right)\\ \begin{array}{c}\left(1+2\sqrt{\frac{2\hslash \sqrt{\frac{\left(0.5\right){\left(0.93660\right)}^{-1}\frac{e^2}{4{{\mathrm{\pi \varepsilon }}_0\left(a_{\mathrm{Ar}\dots \mathrm{Ar}}\right)}^3}}{m_e}}}{m_e{c}^2}}\right)+\\ \left(\frac{1}{2}\right)\hslash \sqrt{\frac{\begin{array}{c}\frac{\left(0.93660\right)e^2}{8{{\mathrm{\pi \varepsilon }}_o\left(a_{\mathrm{Ar}\dots \mathrm{Ar}}\right)}^3}-\\ \frac{e^2}{8{{\mathrm{\pi \varepsilon }}_o\left(a_{\mathrm{Ar}\dots \mathrm{Ar}}+\sqrt{\frac{a_{\mathrm{Ar}\dots \mathrm{Ar}}{a}_0}{2\left(0.5\right){\left(0.93660\right)}^{-1}}}\right)}^3}\end{array}}{20}}\end{array}\end{array}\right\}& \left(16.130\right)\end{array}$ From the energy relationship given by Eq. (16.130) and the relationships between the axes given by Eqs. (16.22-16.24), the dimensions of the Ar•••Ar MO can be solved.

The most convenient way to solve Eq. (16.130) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures isa_{Ar•••Ar}=12.50271a₀=6.61615×10⁻¹⁰ m  (16.131)The component energy parameters at this condition are given in Table 41. Substitution of Eq. (16.131) into Eq. (16.22) givesc_{Ar•••Ar}′=3.42199a₀=1.81084×10⁻¹⁰ m  (16.132)and internuclear distance between neighboring argon atoms:2c_{Ar•••Ar}′(0 K)=6.84397a₀3.62167×10⁻¹⁰ m=3.62167 Å  (16.133)The experimental argon interatomic distance 2c_C•••C′ is [114]2c_{Ar•••Ar}′(4.2 K)=3.71×10⁻¹⁰ m=3.71 Å  (16.134)The other interatomic bond MO parameters can also be determined by the relationships among the parameters. Substitution of Eqs. (16.131) and (16.132) into Eq. (16.23) givesb_{Ar•••Ar}=c_{Ar•••Ar}=12.02530a₀=6.36351×10⁻¹⁰ m  (16.135)Substitution of Eqs. (16.131) and (16.132) into Eq. (16.25) givese_{Ar•••Ar}=0.27370  (16.136)

A convenient method to calculate the lattice energy is to determine the electric field in solid argon having an electric polarization density corresponding to the aligned dipoles moments, and in turn, the energy can be calculated from the energy of each dipole in the corresponding field using the electrostatic form of Gauss' equation. Substitution of the density of solid argon at 402 K

$\begin{array}{cc}\rho =\frac{1.83g}{1\times {10}^{-6}{m}^3},& \left[114\right]\end{array}$ the MW=39.948 g/mole, N_A=6.0221415×10²³ molecules/mole, and the argon dipole moment given in Table 40 into Eq. (16.53) gives:

$\begin{array}{cc}\begin{array}{c}U\left(\mathrm{Ar}\right)=\frac{-2{\left({\mu }_{\mathrm{Ar}}\right)}^2\frac{{\rho }_{\mathrm{solid}\mathrm{Ar}}}{\mathrm{MW}}{N}_A}{3{\varepsilon }_0}\\ \frac{\begin{array}{c}-2{\left(2.48058\times {10}^{-30}C·m\right)}^2\frac{\frac{1.83g}{1\times {10}^{-6}{m}^3}}{39.948g\text{/}\mathrm{mole}}\\ 6.0221415\times {10}^{23}\mathrm{molecules}\text{/}\mathrm{mole}\end{array}}{3{\varepsilon }_0}\\ =-0.07977\mathrm{eV}\left(-7.697\mathrm{kJ}\text{/}\mathrm{mole}\right)\end{array}& \left(16.137\right)\end{array}$ U(Ar) is also the negative of E_{van der Waals}, the van der Waals energy per argon atom:E_{van der Waals}(solid Ar,4.2 K/Ar)=0.07977 eV=7.697 kJ/mole  (16.138)The experimental van der Waals energy is the cohesive energy [115]:E_{van der Waals}(solid Ar,0 K)=0.08022 eV/Ar=7.74 kJ/mole  (16.139)The calculated results based on first principles and given by analytical equations are summarized in Table 42. Using argon the atomic radius (Eq. (16.125)) and the nearest-neighbor distance (Eq. (16.133)), the lattice structure of argon is shown in FIG. 21B. The charge density of the van der Waals dipoles of the crystalline lattice is shown in FIG. 22B.

TABLE 42
The calculated and experimental geometrical parameters and
interatomic van der Waals cohesive energy of solid argon.
			Ref. for
Parameter	Calculated	Experimental	Exp.
Solid Argon Interatomic	3.62167	Å	3.71	Å	114
Distance 2c′C...C	(T = 0	K)	(T = 4.2	K)
van der Waals Energy	0.07977	eV	0.08022	eV	115
per Argon Atom	(T = 4.2	K)	(T = 0	K)

Geometrical Parameters and Energies Due to the Interatomic Van Der Waals Cohesive Energy of Solid Krypton

Krypton is a thirty-six-electron neutral atom having the electron configuration 1s₂2s²2p₆3s₂3p⁶3d¹⁰4s²4p⁶ with the electrons of each shell paired as mirror-image current densities in a shell wherein the radius of the outer shell is r₃₆=0.97187a₀ (Eq. (10.102)). Thus, in isolation or at sufficient separation, there is no energy between krypton atoms. However, reversible mutual van der Waals dipoles may be induced by collisions when the atoms are in close proximity such that krypton gas can condense into a liquid and further solidify at sufficiently low temperatures due to the strong dipole moment that accommodates close packing. As in the case of helium, the dipoles are atomic rather than molecular, and the limiting separation is based on the formation of a nascent bond to replace the dipole-dipole interaction. Thus, Eq. (16.25) can also be applied to krypton atoms.

The van der Waals bonding in the krypton atom involves hybridizing the three 4p AOs into 4p³ HO orbitals containing six electrons. The total energy of the state is given by the sum over the six electrons. The sum E_T(Kr,4p³) of experimental energies [15, 116-119] of Kr,

$\begin{array}{cc}\begin{array}{c}{E}_T\left(\mathrm{Kr},4p^3\right)=\left(\begin{array}{c}78.5\mathrm{eV}+64.7\mathrm{eV}+52.5\mathrm{eV}+\\ 36.950\mathrm{eV}+24.35984\mathrm{eV}+\\ 13.99961\mathrm{eV}\end{array}\right)\\ =271.00945\mathrm{eV}\end{array}& \left(16.140\right)\end{array}$ By considering that the central field decreases by an integer for each successive electron of the shell, the radius r_4p3 of the Kr4p³ shell may be calculated from the Coulombic energy using Eq. (15.13):

$\begin{array}{cc}\begin{array}{c}{r}_{4p^3}=\sum _{n=30}^{35}\frac{\left(Z-n\right)e^2}{8\pi {\varepsilon }_0\left(e271.00945\mathrm{eV}\right)}\\ =\frac{21e^2}{8\pi {\varepsilon }_0\left(e271.00945\mathrm{eV}\right)}\\ =1.05429a_0\end{array}& \left(16.141\right)\end{array}$ where Z=36 for krypton. Using Eq. (15.14), the Coulombic energy E_Coulomb(Kr,4p³) of the outer electron of the van der Waals bound Kr4p³ shell is

$\begin{array}{cc}\begin{array}{c}{E}_{\mathrm{Coulomb}}\left(\mathrm{Kr},4p^3\right)=\frac{-e^2}{8\pi {\varepsilon }_0{r}_{r{p}^3}}\\ =\frac{-e^2}{8\pi {\varepsilon }_{0}1.05429a_0}\\ =-12.905212\mathrm{eV}\end{array}& \left(16.142\right)\end{array}$ To meet the equipotential condition of the union of the two Kr4p³ HOs in a nascent bond, c₂ of Eqs. (15.2-15.5) and Eq. (15.61) for the nascent Kr—Kr-bond MO is given by Eq. (15.75) as the ratio of the valance energy of the Kr AO, E(Kr)=−13.99961 eV and the magnitude of E_Coulomb(Kr,4p₃) (Eq. (16.142)):

$\begin{array}{cc}{c}_2\left(\mathrm{Kr}\mathrm{—Kr},\mathrm{Kr}4p^3\mathrm{HO}\right)=\frac{12.905212\mathrm{eV}}{13.99961\mathrm{eV}}=0.92183& \left(16.143\right)\end{array}$ Since the outer Kr4p³ HO shell is at a lower energy and greater radius than the non-polarized 4p shell, the inner shells are polarized as well. The dipole of the outer shell can polarize the inner shells to the limit that the sum of the primary and secondary dipoles is twice the primary scaled by the energy matching factors of the van der Waals bond given in Eq. (16.15). Thus, the limiting dipole due to polarization of the inner shells is given by

$\begin{array}{cc}\begin{array}{c}{\mu }_{\mathrm{Kr}}<2c_1^{-1}q{C}_{2}2c^{\prime }=2\left(0.16298\right){e\left(0.92183\right)}^{-1}\\ \left(5.57905\times {10}^{-11}m\right)\\ =3.42870\times {10}^{-30}C·m=1.02790D\end{array}& \left(16.144\right)\end{array}$ The condition of Eq. (16.144) is matched by the participation of the outer three shells as given in Table 43. At each shell, opposite charges distributions act as symmetrical point charges at the point of maximum separation, each being centered at ½ the shell radius from the origin. Using the parameters of Eq. (16.143) and 2c′=1.05429a₀=5.57905×10⁻¹¹ m (Eq. (16.141)) as well as the radii of the inner shells of krypton (Eq. (10.102)), the van der Waals dipole of krypton is given in Table 16.36 as the sum of the moments of each participating shell.

TABLE 43
The parameters and van der Waals dipole bond moment of the Kr functional group (FG) of solid krypton.
FG	n1	(c1)c2	(C1)C2	EB(valence)	EA(valence)	$\frac{q}{e}$	Ion/IP/Z [116-119]	Bond Length 2c′ (Å) (Eqs. (16.141) and (10.102))	Bond Moment μ (D)
Kr	1	0.92183	1	12.90521	13.99961	0.16298	Kr6+	Kr4p3 HO	1.01129
							111.07	0.55790
								Kr4s AO
							Kr8+	0.45405
							231.59	Kr3d AO
								0.27991

The minimum-energy packing of krypton dipoles is face-centered cubic also called cubic close packing. In this case, each krypton atom has 12 nearest neighbors and the angle between the aligned dipoles is

$\frac{\pi }{4}$ radians. As in the case with graphite, the van der Waals energy is the potential energy between interacting neighboring induced dipoles. Using μ_Kr=1.01129 D=3.37329×10⁻³⁰ C·m (Table 43), the van der Waals energy is

$\begin{array}{cc}\begin{array}{c}{E}_{\mathrm{van}\mathrm{der}\mathrm{Waals}}\left(\mathrm{Kr}\right)=12\frac{2{\left({\mu }_{\mathrm{Kr}}\right)}^2}{4\pi {{\varepsilon }_0\left(r_{\mathrm{Kr}\dots \mathrm{Kr}}\right)}^3}\cos \left(\frac{\pi }{4}\right)\\ =\left(\frac{24{\left(3.37329\times {10}^{-30}C·m\right)}^2}{4\pi {{\varepsilon }_0\left(2\sqrt{\frac{a_{\mathrm{Kr}\dots \mathrm{Kr}}{a}_0}{2C_1{C}_2}}\right)}^3}\right)\cos \left(\frac{\pi }{4}\right)\end{array}& \left(16.145\right)\end{array}$

The krypton interatomic distance is calculated using Eq. (16.25) with the van der Waals energy (Eq. (16.145)) between neighboring dipoles equated to the nascent bond energy. The energy matching parameter c₂ is the same that of the krypton dipole, and the reduced mass is μ=42. The parameters are summarized in Table 44 and Eq. (16.146).

TABLE 44
The energy parameters (eV) of the krypton functional
group (Kr . . . Kr).
                              Kr . . . Kr
Parameters                    Group
n1                            1
C1                            0.5
C2                            0.92183
c1                            1
c2                            0.92183
C1o                           0.5
C2o                           0.92183
Ve (eV)                       −3.75058
Vp (eV)                       3.52342
T (eV)                        0.13643
Vm (eV)                       −0.06821
E(AO/HO) (eV)                 0
ΔEH2MO(AO/HO) (eV)            0
ET(AO/HO) (eV)                0
ET(H2MO) (eV)                 −0.15895
ET(atom-atom, msp3 · AO) (eV) 0
ET(MO) (eV)                   −0.15895
ω (1015 rad/s)                0.550731
EK (eV)                       0.36250
ĒD (eV)                       −0.00019
ĒKvib (eV)                    0.00091
Ēosc (eV)                     0.00026
ET(Group) (eV)                −0.15869

Substitution of the parameters of Table 44 and the interatomic cohesive energy of krypton (Eq. (16.145)) into Eq. (16.25) with R=a_{Kr•••Kr} gives

$\begin{array}{cc}\left(16.146\right)\frac{-24{\left(3.37329\times {10}^{-31}C·m\right)}^2}{4\pi {{\varepsilon }_0\left(2\sqrt{\frac{a_{\mathrm{Kr}\dots \mathrm{Kr}}{a}_0}{2\left(0.5\right)\left(0.92183\right)}}\right)}^3}\cos \left(\frac{\pi }{4}\right)=\left\{\begin{array}{c}\left(\frac{-e^2}{8\pi {\varepsilon }_0\sqrt{\frac{a_{\mathrm{Kr}\dots \mathrm{Kr}}{a}_0}{2\left(0.5\right)\left(0.92183\right)}}}\left(\begin{array}{c}\left(0.92183\right)\left(2-\frac{1}{2}\frac{a_0}{a_{\mathrm{Kr}\dots \mathrm{Kr}}}\right)\\ \ln \frac{a+\sqrt{\frac{a_{\mathrm{Kr}\dots \mathrm{Kr}}{a}_0}{2\left(0.5\right)\left(0.92183\right)}}}{a-\sqrt{\frac{a_{\mathrm{Kr}\dots \mathrm{Kr}}{a}_0}{2\left(0.5\right)\left(0.92183\right)}}}-1\end{array}\right)\right)\\ \left(1+2\sqrt{\frac{2\hslash \sqrt{\frac{\left(0.5\right)\left(0.92183\right)\frac{e^2}{4\pi {{\varepsilon }_o\left(a_{\mathrm{Kr}\dots \mathrm{Kr}}\right)}^3}}{m_e}}}{m_e{c}^2}}\right)+\\ \left(\frac{1}{2}\right)\hslash \sqrt{\frac{\begin{array}{c}\frac{\left(0.92183\right)e^2}{8\pi {{\varepsilon }_o\left(a_{\mathrm{Kr}\dots \mathrm{Kr}}\right)}^3}-\\ \frac{e^2}{8\pi {{\varepsilon }_o\left(a_{\mathrm{Kr}\dots \mathrm{Kr}}+\sqrt{\frac{a_{\mathrm{Kr}\dots \mathrm{Kr}}{a}_0}{2\left(0.5\right)\left(0.92183\right)}}\right)}^3}\end{array}}{42}}\end{array}\right\}& \end{array}$ From the energy relationship given by Eq. (16.146) and the relationships between the axes given by Eqs. (16.22-16.24), the dimensions of the Kr•••Kr MO can be solved.

The most convenient way to solve Eq. (16.146) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures isa_{Kr•••Kr}=13.74580a₀=7.27396×10⁻¹⁰ m  (16.147)The component energy parameters at this condition are given in Table 44. Substitution of Eq. (16.147) into Eq. (16.22) givesc_{Kr•••Kr}′=3.86154a₀=2.04344×10⁻¹⁰ m  (16.148)and internuclear distance between neighboring krypton atoms:2c_{Kr•••Kr}′=(0 K)=7.72308a₀=4.08688×10⁻¹⁰ m=4.08688 Å  (16.149)The experimental krypton interatomic distance 2c_C•••C′ is [113]2c_{Kr•••Kr}′(0 K)=3.992×10⁻¹⁰ m=3.992 Å  (16.150)The other interatomic bond MO parameters can also be determined by the relationships among the parameters. Substitution of Eqs. (16.147) and (16.148) into Eq. (16.23) givesb_{Kr•••Kr}c_{Kr•••Kr}=13.19225a₀=6.98104×10⁻¹⁰ m  (16.151)Substitution of Eqs. (16.147) and (16.148) into Eq. (16.25) givese_{Kr•••Kr}=0.28092  (16.152)

A convenient method to calculate the lattice energy is to determine the electric field in solid krypton having an electric polarization density corresponding to the aligned dipoles moments, and in turn, the energy can be calculated from the energy of each dipole in the corresponding field using the electrostatic form of Gauss' equation. Substitution of the density of solid krypton at 4.2 K

$\begin{array}{cc}\rho =\frac{3.094g}{1\times {10}^{-6}{m}^3},& \left[113\right]\end{array}$ the MW=83.80 g/mole, N_A=6.0221415×10²³ molecules/mole, and the krypton dipole moment given in Table 43 into Eq. (16.53) gives:

$\begin{array}{cc}\begin{array}{c}U\left(\mathrm{Kr}\right)=\frac{-2{\left({\mu }_{\mathrm{Kr}}\right)}^2\frac{{\rho }_{\mathrm{solid}\mathrm{Kr}}}{\mathrm{MW}}{N}_A}{3{\varepsilon }_0}\\ \frac{\begin{array}{c}-2{\left(3.37329\times {10}^{-30}C·m\right)}^2\frac{\frac{3.094g}{1\times {10}^{-6}{m}^3}}{83.80g\text{/}\mathrm{mole}}\\ 6.0221415\times {10}^{23}\mathrm{molecules}\text{/}\mathrm{mole}\end{array}}{3{\varepsilon }_0}\\ =-0.11890\mathrm{eV}\left(-11.472\mathrm{kJ}\text{/}\mathrm{mole}\right)\end{array}& \left(16.153\right)\end{array}$ U(Ar) is also the negative of E_{van der Waals}, the van der Waals energy per krypton atom:E_{van der Waals}(solid Kr,0 K/Kr)=0.11890 eV=11.472 kJ/mole  (16.154)The experimental van der Waals energy is the cohesive energy [120]:E_{van der Waals}(solid Kr,0 K/Kr)=0.11561 eV=11.15454 kJ/mole  (16.155)The calculated results based on first principles and given by analytical equations (0 K) are summarized in Table 45. Using krypton the atomic radius (Eq. (16.141)) and the nearest-neighbor distance (Eq. (16.149)), the lattice structure of krypton is shown in FIG. 21C. The charge density of the van der Waals dipoles of the crystalline lattice is shown in FIG. 22C.

TABLE 45
The calculated and experimental geometrical parameters and interatomic
van der Waals cohesive energy (0 K) of solid krypton.
			Ref. for
Parameter	Calculated	Experimental	Exp.
Solid Krypton Interatomic	4.08688	Å	3.992	Å	113
Distance 2c′C...C
van der Waals Energy	0.11890	eV	0.11561	eV	120
per Krypton Atom

Geometrical Parameters and Energies Due to the Interatomic Van Der Waals Cohesive Energy of Solid Xenon

Xenon is a fifty-four-electron neutral atom having the electron configuration 1s²2s²2p⁶3s²3p⁶3d¹⁰4s²4p⁶4d¹⁰5s²5p⁶ with the electrons of each shell paired as mirror-image current densities in a shell wherein the radius of the outer shell is r₅₄=1.12168a₀ (Eq. (10.102)). Thus, in isolation or at sufficient separation, there is no energy between xenon atoms. However, reversible mutual van der Waals dipoles may be induced by collisions when the atoms are in close proximity such that xenon gas can condense into a liquid and further solidify at sufficiently low temperatures due to the strong dipole moment that accommodates close packing. As in the case of helium, the dipoles are atomic rather than molecular, and the limiting separation is based on the formation of a nascent bond to replace the dipole-dipole interaction. Thus, Eq. (16.25) can also be applied to xenon atoms.

The van der Waals bonding in the xenon atom involves hybridizing the three 5p AOs into 5p³ HO orbitals containing six electrons. The total energy of the state is given by the sum over the six electrons. The sum E_T(Xe,5p₃) of experimental energies [15, 121-122] of Xe, Xe⁺, Xe²⁺, Xe³⁺, Xe⁴⁺, and Xe⁵⁺ is

$\begin{array}{cc}\begin{array}{c}{E}_T\left(\mathrm{Xe},5p^3\right)=\left(\begin{array}{c}66.703\mathrm{eV}+54.14\mathrm{eV}+40.9\mathrm{eV}+\\ 31.050\mathrm{eV}+20.975\mathrm{eV}+\\ 12.129842\mathrm{eV}\end{array}\right)\\ =225.89784\mathrm{eV}\end{array}& \left(16.156\right)\end{array}$ By considering that the central field decreases by an integer for each successive electron of the shell, the radius r_5p3 of the Xe5p³ shell may be calculated from the Coulombic energy using Eq. (15.13):

$\begin{array}{cc}\begin{array}{c}{r}_{{\mathrm{Sp}}^3}=\sum _{n=48}^{53}\frac{\left(Z-n\right)e^2}{8\pi {\varepsilon }_0\left(e225.897842\mathrm{eV}\right)}\\ =\frac{21e^2}{8\pi {\varepsilon }_0\left(e225.897842\mathrm{eV}\right)}\\ =1.26483a_0\end{array}& \left(16.157\right)\end{array}$ where Z=54 for xenon. Using Eq. (15.14), the Coulombic energy E_Coulomb(Xe,5p³) of the outer electron of the van der Waals bound Xe5p³ shell is

$\begin{array}{cc}\begin{array}{c}{E}_{\mathrm{Coulomb}}\left(\mathrm{Xe},5p^3\right)=\frac{-e^2}{8\pi {\varepsilon }_0{r}_{5p^3}}\\ =\frac{-e^2}{8\pi {\varepsilon }_{0}1.26483a_0}\\ =-10.757040\mathrm{eV}\end{array}& \left(16.158\right)\end{array}$ To meet the equipotential condition of the union of the two Xe5p³ HOs in a nascent bond, c₂ of Eqs. (15.2-15.5) and Eq. (15.61) for the nascent Xe—Xe-bond MO is given by Eq. (15.75) as the ratio of the valance energy of the Xe AO, E(Xe)=−12.129842 eV and the magnitude of E_Coulomb(Xe,5p³) (Eq. (16.158)):

$\begin{array}{cc}{c}_2\left(\mathrm{Xe}\mathrm{—Xe},\mathrm{Xe}5p^3\mathrm{HO}\right)=\frac{10.75704\mathrm{eV}}{12.129842\mathrm{eV}}=0.88682& \left(16.159\right)\end{array}$ Since the outer Xe5p³ HO shell is at a lower energy and greater radius than the non-polarized 5p shell, the inner shells are polarized as well. The dipole of the outer shell can polarize the inner shells to the limit that the sum of the primary and secondary dipoles is twice the primary scaled by the energy matching factors of the van der Waals bond given in Eq. (16.15). Thus, the limiting dipole due to polarization of the inner shells is given by

$\begin{array}{cc}\begin{array}{c}{\mu }_{\mathrm{Xe}}<2c_1^{-1}q{C}_{2}2c^{\prime }=2\left(0.24079\right)e\left(0.88682\right)\\ \left(6.69318\times {10}^{-11}m\right)\\ =5.16444\times {10}^{-30}C·m=1.54826D\end{array}& \left(16.160\right)\end{array}$ The condition of Eq. (16.160) is matched by the participation of the outer two shells as given in Table 46. At each shell, opposite charges distributions act as symmetrical point charges at the point of maximum separation, each being centered at ½ the shell radius from the origin. Using the parameters of Eq. (16.159) and 2c′=1.26483a_o=6.69318×10⁻¹¹ m (Eq. (16.157)) as well as the radius of the inner 5s shell of xenon (Eq. (10.102)), the van der Waals dipole of xenon is given in Table 46 as the sum of the moments of each participating shell.

TABLE 46
The parameters and van der Waals dipole bond moment of the Kr functional group (FG) of solid xenon.
FG	n1	(c1)c2	(C1)C2	EB(valence)	EA(valence)	$\frac{q}{e}$	Ion/IP/Z [121-122]	Bond Length 2c′ (Å) (Eqs. (16.157) and (10.102))	Bond moment μ (D)
Xe	1	0.88682	1	10.75704	12.12984	0.24079	Xe6+	Xe5p3 HO	1.41050
							91.67	0.66932
								Xe5s AO
								0.55021

The minimum-energy packing of xenon dipoles is face-centered cubic also called cubic close packing. In this case, each xenon atom has 12 nearest neighbors and the angle between the aligned dipoles is

$\frac{\pi }{4}$ radians. As in the case with graphite, the van der Waals energy is the potential energy between interacting neighboring induced dipoles. Using μ_Xe=1.41050 D=4.70492×10⁻³⁰ C·m (Table 46), the van der Waals energy is

$\begin{array}{cc}\begin{array}{c}{E}_{\mathrm{van}\mathrm{der}\mathrm{Waals}}\left(\mathrm{Xe}\right)=12\frac{2{\left({\mu }_{\mathrm{Xe}}\right)}^2}{4\pi {{\varepsilon }_0\left(r_{\mathrm{Xe}\dots \mathrm{Xe}}\right)}^3}\cos \left(\frac{\pi }{4}\right)\\ =\left(\frac{24{\left(4.70492\times {10}^{-30}C·m\right)}^2}{4\pi {{\varepsilon }_0\left(2\sqrt{\frac{a_{\mathrm{Xe}\dots \mathrm{Xe}}{a}_0}{2C_1{C}_2}}\right)}^3}\right)\cos \left(\frac{\pi }{4}\right)\end{array}& \left(16.161\right)\end{array}$

The xenon interatomic distance is calculated using Eq. (16.25) with the van der Waals energy (Eq. (16.161)) between neighboring dipoles equated to the nascent bond energy. The energy matching parameter c₂ is the same that of the xenon dipole, and the reduced mass is μ=65. The parameters are summarized in Table 47 and Eq. (16.162).

TABLE 47
The energy parameters (eV) of the xenon functional
group (Xe . . . Xe).
                              Xe . . . Xe
Parameters                    Group
n1                            1
C1                            0.5
C2                            0.88682
c1                            1
c2                            1
C1o                           0.5
C2o                           0.88682
Ve (eV)                       −3.49612
Vp (eV)                       3.20821
T (eV)                        0.10960
Vm (eV)                       −0.05480
E(AO/HO) (eV)                 0
ΔEH2MO(AO/HO) (eV)            0
ET(AO/HO) (eV)                0
ET(H2MO) (eV)                 −0.23311
ET(atom-atom, msp3 · AO) (eV) 0
ET(MO) (eV)                   −0.23311
ω (1015 rad/s)                0.432164
EK (eV)                       0.28446
ĒD (eV)                       −0.00025
ĒKvib (eV)                    0.00062
Ēosc (eV)                     0.00006
ET(Group) (eV)                −0.23305

Substitution of the parameters of Table 47 and the interatomic cohesive energy of xenon (Eq. (16.161)) into Eq. (16.25) with R=a_{Xe•••Xe} gives

$\begin{array}{cc}\frac{-24{\left(4.70492\times {10}^{-31}C·m\right)}^2}{4\pi {{\varepsilon }_0\left(2\sqrt{\frac{a_{\mathrm{Xe}\dots \mathrm{Xe}}{a}_0}{2\left(0.5\right)\left(0.88682\right)}}\right)}^3}\cos \left(\frac{\pi }{4}\right)=\{\begin{array}{c}\left(\frac{-e^2}{8\pi {\varepsilon }_0\sqrt{\frac{a_{\mathrm{Xe}\dots \mathrm{Xe}}{a}_0}{2\left(0.5\right)\left(0.88682\right)}}}\left(\begin{array}{c}\left(2-\frac{1}{2}\frac{a_0}{a_{\mathrm{Xe}\dots \mathrm{Xe}}}\right)\\ \ln \frac{a+\sqrt{\frac{a_{\mathrm{Xe}\dots \mathrm{Xe}}{a}_0}{2\left(0.5\right)\left(0.88682\right)}}}{a-\sqrt{\frac{a_{\mathrm{Xe}\dots \mathrm{Xe}}{a}_0}{2\left(0.5\right)\left(0.88682\right)}}}-1\end{array}\right)\right)\\ \left(1+2\sqrt{\frac{2\hslash \sqrt{\frac{\left(0.5\right)\left(0.88682\right)\frac{e^2}{4\pi {{\varepsilon }_o\left(a_{\mathrm{Xe}\dots \mathrm{Xe}}\right)}^3}}{m_e}}}{m_e{c}^2}}\right)+\hspace{1em}\left(\frac{1}{2}\right)\\ \hslash \sqrt{\frac{\frac{e^2}{8\pi {{\varepsilon }_o\left(a_{\mathrm{Xe}\dots \mathrm{Xe}}\right)}^3}-\frac{e^2}{8\pi {{\varepsilon }_o\left(a_{\mathrm{Xe}\dots \mathrm{Xe}}+\sqrt{\frac{a_{\mathrm{Xe}\dots \mathrm{Xe}}{a}_0}{2\left(0.5\right)\left(0.88682\right)}}\right)}^3}}{65}}\end{array}\}& \left(16.162\right)\end{array}$ From the energy relationship given by Eq. (16.162) and the relationships between the axes given by Eqs. (16.22-16.24), the dimensions of the Xe•••Xe MO can be solved.

The most convenient way to solve Eq. (16.162) is by the reiterative technique using a computer. The result to within the round-off error with five-significant figures isa_{Xe•••Xe}=15.94999a₀=8.44037×10⁻¹⁰ m  (16.163)The component energy parameters at this condition are given in Table 47. Substitution of Eq. (16.163) into Eq. (16.22) givesc_{Xe•••Xe}′=4.24093a₀=2.24420×10⁻¹⁰ m  (16.164)and internuclear distance between neighboring xenon atoms:2c_{Xe•••Xe}′(0 K)=8.48187a₀=4.48841×10⁻¹⁰ m=4.48841 Å  (16.165)The experimental xenon interatomic distance 2c_C•••C′ at the melting point of 161.35 K is [112, 113]2c_{Xe•••Xe}′(161.35 K)=4.492×10⁻¹⁰ m=4.492 Å  (16.166)The other interatomic bond MO parameters can also be determined by the relationships among the parameters. Substitution of Eqs. (16.163) and (16.164) into Eq. (16.23) givesb_{Xe•••Xe}=c_{Xe•••Xe}=15.37585a₀=8.13655×10⁻¹⁰ m  (16.167)Substitution of Eqs. (16.163) and (16.164) into Eq. (16.25) givese_{Xe•••Xe}=0.26589  (16.168)

A convenient method to calculate the lattice energy is to determine the electric field in solid xenon having an electric polarization density corresponding to the aligned dipoles moments, and in turn, the energy can be calculated from the energy of each dipole in the corresponding field using the electrostatic form of Gauss' equation. Substitution of the density of solid xenon at 0 K

$\begin{array}{cc}\rho =\frac{3.780g}{1\times {10}^{-6}{m}^3},& \left[113\right]\end{array}$ the MW=131.29 g mole, N_A=6.0221415×10²³ molecules/mole, and the xenon dipole moment given in Table 46 into Eq. (16.53) gives:

$\begin{array}{cc}\begin{array}{c}U\left(\mathrm{Xe}\right)=\frac{-2{\left({\mu }_{\mathrm{Xe}}\right)}^2\frac{{\rho }_{\mathrm{solid}\mathrm{Xe}}}{\mathrm{MW}}{N}_A}{3{\varepsilon }_0}\\ \frac{\begin{array}{c}-2{\left(\begin{array}{c}4.70492\times \\ {10}^{-30}C·m\end{array}\right)}^2\frac{\frac{3.780g}{1\times {10}^{-6}{m}^3}}{131.29g\text{/}\mathrm{mole}}6.0221415\times \\ {10}^{23}\mathrm{molecules}\text{/}\mathrm{mole}\end{array}}{3{\varepsilon }_0}\\ =-0.18037\mathrm{eV}\left(-17.403\mathrm{kJ}\text{/}\mathrm{mole}\right)\end{array}& \left(16.169\right)\end{array}$ U(Xe) is also the negative of E_{van der Waals}, the van der Waals energy per xenon atom:E_{van der Waals}(solid Xe,0 K/Xe)=0.18037 eV=17.403 kJ/mole  (16.170)The experimental van der Waals energy is the cohesive energy [123]:E_{van der Waals}(solid Xe,0 K)=0.16608 eV/Xe=16.02472 kJ/mole  (16.171)The calculated results based on first principles and given by analytical equations are summarized in Table 48. Using xenon the atomic radius (Eq. (16.157)) and the nearest-neighbor distance (Eq. (16.165)), the lattice structure of xenon is shown in FIG. 21D. The charge density of the van der Waals dipoles of the crystalline lattice is shown in FIG. 228D.

TABLE 48
The calculated and experimental geometrical parameters and
interatomic van der Waals cohesive energy of solid xenon.
			Ref. for
Parameter	Calculated	Experimental	Exp.
Solid Xenon	4.4884	Å	4.492	Å	113
Interatomic Distance	(T = 0	K)	(T = 161.35	K)
2c′C...C
van der Waals	0.18037	eV	0.16608	eV	123
Energy per Xenon Atom
(0 K)

Reaction Kinetics and Thermodynamics

Reaction kinetics may be modeled using the classical solutions of reacting species and their interactions during collisions wherein the bond order of the initial and final bonds undergo a decreasing and increasing bond order, respectively, with conservation of charge and energy. Collisions can be modeled starting with the simple hard sphere model with conservation of energy and momentum. The energy distribution may be modeled using the appropriate statistical thermodynamics model such as Maxwell-Boltzmann statistics. Low-energy collisions are elastic, but for sufficiently high energy, a reaction may occur. Hot reacting species such as molecules at the extreme of the kinetic energy distribution can achieve the transition state, the intermediate species at the cross over point in time and energy between the reactants and products. The rate function to form the transition state may depend on the collisional orientation as well as the collisional energy. Bond distortion conserves the energy and momentum of the collision from the trajectories of the reactants. For sufficient distortion due to a sufficiently energetic collision at an appropriate relative orientation, a reaction occurs wherein the products exiting the collision event are different from the reactants entering the collision. The initial reactant energy and momentum as well as those arising from any bonding energy changes are conserved in the translational, rotational, and vibrational energies of the products. The bond energy changes are given by the differences in the energies of the product and reactants molecules wherein the geometrical parameters, energies, and properties of the latter can be solved using the same equations as those used to solved the geometrical parameters and component energies of the individual molecules as given in the Organic Molecular Functional Groups and Molecules section. The bond energy changes at equilibrium determine the extent of a reaction according to the Gibbs free energy of reaction. Whereas, the corresponding dynamic reaction-trajectory parameters of translational, rotational, and vibrational energies as well as the time dependent electronic energy components such as the electron potential and kinetic energies of intermediates correspond to the reaction kinetics. Each aspect will be treated next in turn.

Consider the gas-phase reaction of two species A and B comprising the reactants that form one or more products C_n where n is an integer:A+B⇄C₁+ . . . +C_N  (16.172)Arising from collisional probabilities, the concentrations (denoted [A],[B], . . . ) as a function of time can be fitted to a second-order rate law

$\begin{array}{cc}-\frac{d\left[A\right]}{dt}=k\left[A\right]\left[B\right]-k^{\prime }\prod _{i=1}^n\left[C_i\right]& \left(16.173\right)\end{array}$ where k and k′ are the forward and reverse rate constants. The equilibrium constant K corresponding to the balance between the forward and reverse reactions is given by the quotient of the forward and reverse rate constants:

$\begin{array}{cc}K=\frac{k}{k^{\prime }}& \left(16.174\right)\end{array}$ The relationship between the temperature-dependent equilibrium constant and the standard Gibbs free energy of reaction ΔG_T⁰(T) at temperature T is

$\begin{array}{cc}K=Q_K\left(T\right)e^{\frac{-\Delta G_T^0\left(T\right)}{\mathrm{RT}}}& \left(16.175\right)\end{array}$ where R is the ideal gas constant,

$\begin{array}{cc}{Q}_K\left(T\right)=\frac{\prod _{i=1}^n\left[C_i\right]}{\left[A\right]\left[B\right]}& \left(16.176\right)\end{array}$ is the reaction quotient at the standard state, andΔG_T⁰(T)=ΔH_T⁰(T)−TΔS_T⁰  (16.177)where ΔH_T⁰(T) and ΔS_T⁰ are the standard-state enthalpy and entropy of reaction, respectively. Rearranging Eq. (16.175) gives the free energy change upon reaction:

$\begin{array}{cc}\Delta G=\mathrm{RT}\ln \frac{Q_K}{K}& \left(16.178\right)\end{array}$ If the instantaneous free energy change is zero, then the reaction is at equilibrium. An exergonic or work-producing reaction corresponds to the cases with ΔG_T⁰(T) or ΔG negative, and endergonic or work consuming reactions corresponds to positive values. The enthalpy of reaction or heat of reaction at constant pressure is negative for an exothermic (heat releasing) reaction, and is positive for an endothermic (heat absorbing) reaction. The enthalpy of reaction may be calculated by Hess's law as the difference of the sum of the heats of formation of the products minus the sum of the heats of formation of the reactants wherein the individual heats of the molecules are solved using the equations given in the Organic Molecular Functional Groups and Molecules section.Transition State Theory

Transition state theory (TST) has been widely validated experimentally. It entails the application classical trajectory calculations that allow the study of the dynamics at the microscopic level such as differential cross sections, total cross sections, and product energy distributions, as well as at the macroscopic level for the determination of thermal rate constants by solving the classical equations of motion with the formation of the transition state. The reaction trajectory parameters give rise to terms of a classical thermodynamic kinetics equation discovered in 1889 by Arrhenius and named after him. The data of the variation of the rate constant k with temperature of many reactions fit the Arrhenius equation given by

$\begin{array}{cc}k=A{e}^{\frac{-E_a}{\mathrm{RT}}}& \left(16.179\right)\end{array}$ where E_a is the activation energy and A is a preexponential or frequency factor that may have a relatively small temperature dependence compared to the exponential term of Eq. (16.179). For reactions that obey the Arrhenius equation, when ln k is plotted versus 1/T in a so-called Arrhenius plot, the slope is the constant −E_a/R, and the intercept is A. Eq. (16.179) confirms that typically two colliding molecules require a certain minimum kinetic energy of relative motion to sufficiently distort initial reactant bonds and concomitantly allow nascent bonds to form. The cross over species from reactants to products called the transition state will proceed through the minimum energy complex involving the reactants. Thus, the activation energy can be interpreted as the minimum energy that the reactants must have in order to form the transition state and transform to product molecules. E_a can be calculated from the total energy of the transition state relative to that of the reactants and is achieved when the thermal energy of the reactants overcomes the energy deficit between the energy of the reactants and that of the transition state. The preexponential factor corresponds to the collision frequency and energy of collisions upon which the formation of the transition state is dependent.

For bimolecular reactions, transition state theory yields [124]

$\begin{array}{cc}\begin{array}{cc}k\left(T\right)=\frac{1}{\left(k_{B}T\right)h}\gamma \left(T\right)K°\exp & \left(-\Delta G_T^{\mathrm{‡°}}/\mathrm{RT}\right)\end{array}& \left(16.180\right)\end{array}$ where ΔG_T^‡° is the quasi-thermodynamic free energy of activation, γ(T) is a transmission coefficient, K° is the reciprocal of the concentration, h is Planck's constant, and k_B is the Boltzmann constant. The factor

$\frac{1}{\left(k_{B}T\right)h}$ is obtained by dynamical classical equations of motion involving species trajectories having a statistical mechanical distribution. Specifically, the reactant molecular distribution is typically a Maxwell-Boltzmann distribution. The classical derivation of the preexponential term of the Arrhenius equation can be found in textbooks and review articles such as section 2.4 of Ref. [124]. Typically the A term can be accurately determined from the Maxwell-Boltzmann-distribution-constrained classical equations of motion by sampling or by using Monte Carlo methods on many sets (usually more than ten thousand) of initial conditions for the coordinates and momenta involving the trajectories. The translational levels are a continuous distribution, and the rotational and vibrational levels are quantized according to the classical equations given, for example, in the Vibration of the Hydrogen Molecular Ion section and the Diatomic Molecular Rotation section.S_N² Reaction of Cl⁻ with CH₃Cl

Consider the S_N2 (bimolecular nucleophilic substitution) gas-phase reaction of Cl⁻ with chloromethane through a transition state:Cl⁻+CH₃Cl→ClCH₃+Cl⁻  (16.181)The corresponding Arrhenius equation for the reaction given by Eq. (16.179) is

$\begin{array}{cc}k\left(T\right)=\frac{k_{B}T}{h}\frac{Q^{‡}}{{\Phi }^R}{e}^{\frac{-\Delta E^{‡}}{k_{B}T}}& \left(16.182\right)\end{array}$ where k_B is the Boltzmann constant, h is Planck's constant, ΔE^‡ is the activation energy of the transition state ‡, T is the temperature, Φ^R is the reaction partition per unit volume, and Q^‡ is the coordinate independent transition-state partition function. The preexponential factor

$\frac{k_{B}T}{h}\frac{Q^{‡}}{{\Phi }^R}$ has previously been calculated classically and shown to be in agreement with the experimental rate constant [125]. Then, only the transition state need be calculated and its geometry and energy compared to observations to confirm that classical physics is predictive of reaction kinetics. The activation energy can be calculated by determining the energy at the point that the nascent bond with the chloride ion is the same as that of the leaving chlorine wherein the negative charge is equally distributed on the chlorines. The rearrangement of bonds and the corresponding electron MOs of the reactants and products can be modeled as a continuous transition of the bond orders of the participating bonds from unity to zero and vice versa, respectively, wherein the transition state is a minimum-energy molecule having bonds between all of the reactants, Cl⁻ and CH₃Cl.Transition State

The reaction proceeds by back-side attack of Cl⁻ on CH₃Cl. Based on symmetry, the reaction pathway passes through a D_3h configuration having Cl^δ−—C—Cl^δδ on the C₃ axis. The hydrogen atoms are in the σ_h plane with the bond distances the same as those of the CH₃ functional group given in the Alkyl Chlorides section, since this group is not involved in the substitution reaction. The transition-state group Cl^δ−—C—Cl^δδ is treated as a three-centered-bond functional group that comprises a linear combination of Cl⁻ and the C—Cl group of chloromethane (C—Cl (i) given in Table 15.33). It is solved using the Eq. (15.51) with the total energy matched to the sum of the H₂-type ellipsoidal MO total energy, −31.63536831 e V given by Eq. (11.212) as in the case of chloromethane, and the energy of the two outer electrons of Cl⁻, E(Cl⁻)=−IP₁−IP₂−12.96764 eV−3.612724 eV=−16.58036 eV [15, 126]. These electrons are contributed to form the back-side-attack bond. Then, the corresponding parameter E_T(AO/HO) (eV) is −14.63489 eV−16.58036 eV=−31.21525 eV due to the match of the MO energy to both E(C,2sp₃)=−14.63489 eV (Eq. (15.25)) and E(Cl⁻), and E_{initial(c5AO/HO)} (eV) is −16.58036 eV corresponding to the initial energy of the Cl⁻ electrons. Also, due to the two C— Cl bonds of the Cl^δ−—C—Cl^δ− functional group n₁=2. Otherwise all of the parameters of Eq. (15.51) remain the same as those of chloromethane given in Table 15.36. The geometrical (Eqs. (15.1-15.5) and (15.51)), intercept (Eqs. (15.80-15.87)), and energy (Eqs. (15.6-15.11) and (15.17-15.65)) parameters are given in Tables 49, 50, and 51, respectively. The color scale, translucent view of the charge density of the chloride-ion-chloromethane transition state comprising the Cl^δ−—C—Cl^δ− functional group is shown in FIG. 23. The transition state bonding comprises two paired electrons in each Cl^δ−—C MO with two from Cl⁻, one from Cl and one from CH₃. As a symmetrical three-centered bond, the central bonding species are two Cl bound to a central CH₃⁺ per Cl^δ−—C MO with a continuous current onto the C—H MO at the intersection of each Cl^δ−—C MO with the CH₃⁺ group. Due to the four electrons and the valence of the chlorines, the latter possess a partial negative charge of −0.5e distributed on each Cl^δd−—C MO such that the far field is equivalent to that of the corresponding point charge at each Cl nucleus.

TABLE 49
The geometrical bond parameters of the
Clδ−—C—Clδ− and CH3 functional groups of
the chloride-ion-chloromethane transition state.
	Clδ−—C—Clδ−	C—H (CH3)
Parameter	Group	Group
a (a0)	3.70862	1.64920
c′ (a0)	2.13558	1.04856
Bond Length 2c′ (Å)	2.26020	1.10974
Literature Bond Length (Å)	2.3-2.4 [125, 127]	1.06-1.07 [125]
b, c (a0)	3.03202	1.27295
e	0.57584	0.63580

TABLE 50
The MO to HO and AO intercept geometrical bond parameters of the
Clδ−—C—Clδ− and CH3 functional groups of the chloride-ion-chloromethane
transition state.
		ET	ET	ET	ET	Final Total
		(eV)	(eV)	(eV)	(eV)	Energy C2sp3	rinitial
Bond	Atom	Bond 1	Bond 2	Bond 3	Bond 4	(eV)	(a0)
Clδ−—C—Clδ−	C	−0.36229	−0.36229	0	0	−152.34026	0.91771
Claδ−—C—Clbδ−	Claδ−	−0.36229	0	0	0		2.68720
Claδ−—C—Clbδ−	Clbδ−	−0.36229	0	0	0		1.05158
C—H (CH3)	C	−0.36229	−0.36229	0	0	−152.34026	0.91771
		ECoulomb(C2sp3)	E(C2sp3)
	rfinal	(eV)	(eV)	θ′	θ1	θ2	d1	d2
Bond	(a0)	Final	Final	(°)	(°)	(°)	(a0)	(a0)
Clδ−—C—Clδ−	0.87495	−15.55033	−15.35946
Claδ−—C—Clbδ−	0.89582	−15.18804
Claδ−—C—Clbδ−	0.89582	−15.18804
C—H (CH3)	0.87495	−15.55033	−15.35946	78.85	101.15	42.40	1.21777	0.16921
ET is ET(atom-atom, msp3 · AO).

TABLE 51
The energy parameters (eV) of the Clδ−—C—Clδ− and CH3 functional
groups of the chloride-ion-chloromethane transition state.
	Clδ−—C—Clδ−	CH3
Parameters	Group	Group
n1	2	3
n2	0	2
n3	1	0
C1	0.5	0.75
C2	0.81317	1
c1	1	1
c2	1	0.91771
c3	1	0
c4	2	1
c5	1	3
C1o	0.5	0.75
C2o	0.81317	1
Ve (eV)	−33.44629	−107.32728
Vp (eV)	12.74200	38.92728
T (eV)	4.50926	32.53914
Vm (eV)	−2.25463	−16.26957
E(AO/HO) (eV)	−31.21525	−15.56407
ΔEH2MO(AO/HO) (eV)	−1.44915	0
ET(AO/HO) (eV)	−29.76611	−15.56407
E(n3AO/HO) (eV)	−16.58036	0
ET(H2MO) (eV)	−48.21577	−67.69451
ET(atom-atom, msp3 · AO) (eV)	−1.44915	0
ET(MO) (eV)	−49.66491	−67.69450
ω (1015 rad/s)	3.69097	24.9286
EK (eV)	2.42946	16.40846
ĒD (eV)	−0.07657	−0.25352
ĒKvib (eV)	0.08059 [5]	0.35532
		(Eq. (13.458))
Ēosc (eV)	−0.03628	−0.22757
Emag (eV)	0.14803	0.14803
ET(Group) (eV)	−49.73747	−67.92207
Einitial(c4AO/HO) (eV)	−14.63489	−14.63489
Einitial(c5AO/HO) (eV)	−16.58036	−13.59844
ED(Group) (eV)	3.73930	12.49186

The bond energy of the C—Cl group of chloromethane from Table 15.36 is E_D(Group) (eV)=3.77116 eV compared to the bond energy of the Cl^δ−—C—Cl^δ− functional group of the chloride-ion-chloromethane transition state of E_D(Group) (eV)=3.73930 eV (Table 16.44). Since the energies of the CH₃ functional groups are unchanged, the chloride-ion-chloromethane transition state is ΔE=+0.03186 eV (+0.73473 kcal/mole) higher in energy than chloromethane. Experimentally, the transition state is about 1±1 kcal/mole higher [128]. Using this energy as the corresponding activation energy ΔE^‡ of Eq. (16.182) with the classically determined preexponential factor

$\frac{k_{B}T}{h}\frac{Q^{‡}}{{\Phi }^R}$ predicts the experimental reaction rate very well [125].Negatively-Charged Molecular Ion Complex C

In addition to the nature and energy of the transition state designated by ‡, experimental gas-phase rate constants indicate that the reaction of Cl⁻ with CH₃Cl passes through a bound state comprising the attachment of Cl₋ to the positive dipole of CH₃Cl [125, 127-128] (the dipole moment of the C—Cl functional group is given in the Bond and Dipole Moments section). This negatively-charged molecular ion complex designated exists as a more stable state in between the reactants and the transition state, and by equivalence of the chlorines, it also exists between the transition state and the products. Experimentally is 12.2±2 kcal/mole more stable than the isolated reactants and products, Cl⁻ and CH₃Cl. Thus, an energy well corresponding to occurs on either side of the energy barrier of the transition state ‡ that is about 1±1 kcal/mole above the reactants and products [125, 128]. Thus, the combination of the depth of this well and the barrier height yields an intrinsic barrier to nucleophilic substitution given by the reaction of Eq. (16.181) of 13.2 4±2.2 kcal/mole [125, 128].

The negatively-charged molecular ion complex comprises the functional groups of CH₃Cl (C—Cl (i) and CH₃ given in Table 15.33 of the Alkyl Chlorides section) and a Cl₋.C^δ+ functional group wherein Cl⁻ is bound to the CH₃Cl moiety by an ion-dipole bond. As given in the case of the dipole-dipole bonding of ice, liquid water, and water vapor as well as the van der Waals bonding in graphite and noble gasses given in the Condensed Matter Physics section, the bond energy and bond distance of the Cl⁻.C^δ+ functional group are determined by the limiting energy and distance of the formation of a corresponding nascent Cl⁻—CH₃Cl covalent bond that destabilizes the C—Cl bond of the CH₃Cl moiety by involving charge density of its electrons in the formation the nascent bond. Subsequently, the higher energy Cl^δ−—C—Cl^δ− functional group of the transition state is formed.

The energy and geometric parameters of the Cl⁻.C^δ+ functional group are solved using Eq. (15.51) with the total energy matched to the H₂-type ellipsoidal MO total energy, −31.63536831 eV. The parameter E_T(AO/HO) (eV) is −14.63489 eV-3.612724 eV=−18.24761 eV due to the match of the MO energy to both E(C,2sp³)=−14.63489 eV (Eq. (15.25)) and the outer electron of E(Cl⁻) (−IP₁=3.612724 eV) [126] that forms the nascent bond by the involving the electrons of the C—Cl group of the CH₃Cl moiety. Then, E_{initial(c5AO/HO)} (eV) is −3.612724 eV corresponding to the initial energy of the outer Cl⁻ electron. Also, E_T(atom-atom,msp³.AO) in Eq. (15.61) is −1.85836 eV due to the charge donation from the C HO to the MO based on the energy match between the C2sp³ HOs corresponding to the energy contribution of methylene, −0.92918 eV (Eq. (14.513). E_mag=0 since the Cl⁻ electrons are paired upon dissociation, and the vibrational energy of the transition state is appropriate for Cl⁻.C^δ+. Otherwise, all of the parameters of Eq. (15.51) remain the same as those of chloromethane given in Table 15.36. The geometrical (Eqs. (15.1-15.5) and (15.51)), intercept (Eqs. (15.80-15.87)), and energy (Eqs. (15.6-15.11) and (15.17-15.65)) parameters are given in Tables 52, 53, and 54, respectively. The color scale, translucent view of the charge density of the negatively-charged molecular ion complex comprising the Cl⁻.C^δ+ functional group is shown in FIG. 24. The bonding in the complex comprises two paired electrons in the Cl⁻.C^δ+ MO with ½ of the charge density from Cl⁻ and the other half from CH₃. The central bonding species are a Cl bound to a central CH₃⁺ with a continuous current onto the C—H MO at the intersection of the Cl⁻.C^δ+ MO with the CH₃⁺ group. Due to the two electrons and the valence of the chlorine, the latter possess a negative charge of −e distributed on the Cl⁻.C^δ+ MO such that the far field is equivalent to that of the corresponding point charge at the Cl nucleus. The bonding in the CH₃Cl moiety is equivalent to that of chloromethane except that the C—H bonds are in a plane to accommodate the Cl⁻.C^δ+ MO.

TABLE 52
The geometrical bond parameters of the Cl−•Cδ+,
C—Cl, and CH3 functional groups of the
negatively-charged molecular ion complex  .
	Cl−•Cδ+	C—H (CH3)	C—Cl (i)
Parameter	Group	Group	Group
a (a0)	2.66434	1.64920	2.32621
c′ (a0)	1.81011	1.04856	1.69136
Bond Length	1.91574	1.10974	1.79005
2c′ (Å)
Literature Bond	>1.80 curve	1.06-1.07 [1]	1.785 [1]
Length (Å)	fit [127]		(methyl chloride)
b, c (a0)	1.95505	1.27295	1.59705
e	0.67938	0.63580	0.72709

TABLE 53
The MO to HO and AO intercept geometrical bond parameters of the
Cl− · Clδ+, C—Cl, and CH3 functional groups of the negatively-
charged molecular ion complex  .
		ET	ET	ET	ET	Final Total
		(eV)	(eV)	(eV)	(eV)	Energy C2sp3
Bond	Atom	Bond 1	Bond 2	Bond 3	Bond 4	(eV)
Cl− · Cδ+	C	−0.82688	−0.72457	0	0
Cl− · Cδ+	Cl−	−0.82688	0	0	0
C—Cl	C	−0.82688	−0.72457	0	0	−153.16714
C—Cl	Cl	−0.72457	0	0	0
C—H (CH3)	C	−0.82688	−0.72457	0	0	−153.16714
			ECoulomb(C2sp3)	E(C2sp3)
	rinitial	rfinal	(eV)	(eV)
Bond	(a0)	(a0)	Final	Final
Cl− · Cδ+	0.91771	0.83078	−16.37720	−16.18634
Cl− · Cδ+	2.68720	0.86923	−15.65263
C—Cl	0.91771	0.83078	−16.37720	−16.18634
C—Cl	1.05158	0.87495	−15.55033
C—H (CH3)	0.91771	0.83078	−16.37720	−16.18634
	θ′	θ1	θ2	d1	d2
Bond	(°)	(°)	(°)	(a0)	(a0)
Cl− · Cδ+
Cl− · Cδ+	16.80	163.20	7.38	2.64225	0.83214
C—Cl	63.91	116.09	27.85	2.05675	0.36539
C—Cl	69.62	110.38	30.90	1.99599	0.30463
C—H (CH3)	73.30	106.70	38.69	1.28725	0.23869
ET is ET(atom-atom, msp3 · AO).

TABLE 54
The energy parameters (eV) of the Cl−•C
δ+, C—Cl, and CH3 functional groups of the
negatively-charged molecular ion complex  .
	Cl−•Cδ+	CH3	C—Cl (i)
Parameters	Group	Group	Group
n1	1	3	1
n2	0	2	0
n3	0	0	0
C1	0.5	0.75	0.5
C2	0.81317	1	0.81317
c1	1	1	1
c2	1	0.91771	1
c3	0	0	1
c4	2	1	2
c5	1	3	0
C1o	0.5	0.75	0.5
C2o	0.81317	1	0.81317
Ve (eV)	−24.89394	−107.32728	−29.68411
Vp (eV)	7.51656	38.92728	8.04432
T (eV)	4.67169	32.53914	6.38036
Vm (eV)	−2.33584	−16.26957	−3.19018
E(AO/HO) (eV)	−18.24761	−15.56407	−14.63489
ΔEH2MO(AO/HO) (eV)	−1.65376	0	−1.44915
ET(AO/HO) (eV)	−16.59386	−15.56407	−13.18574
ET(H2MO) (eV)	−31.63537	−67.69451	−31.63536
ET(atom-atom, msp3 ·	−1.65376	0	−1.44915
AO) (eV)
ET(MO) (eV)	−33.28913	−67.69450	−33.08452
ω (1015 rad/s)	6.06143	24.9286	7.42995
EK (eV)	3.98974	16.40846	4.89052
ĒD (eV)	−0.13155	−0.25352	−0.14475
ĒKvib (eV)	0.02790 [129]	0.35532	0.08059 [5]
		(Eq. (13.458))
Ēosc (eV)	−0.11760	−0.22757	−0.10445
Emag (eV)	0	0.14803	0.14803
ET(Group) (eV)	−33.40672	−67.92207	−33.18897
Einitial(c4AO/HO) (eV)	−14.63489	−14.63489	−14.63489
Einitial(c5AO/HO) (eV)	−3.612724	−13.59844	0
ED(Group) (eV)	0.52422	12.49186	3.77116

The bond energies of the CH₃Cl moiety are unchanged to the limit of the formation of the Cl⁻.C^δ+ functional group of the negatively-charged molecular ion complex . Thus, the energy of stabilization of forming the ion-dipole complex is equivalent to the bond energy of the Cl⁻.C^δ+ functional group. Experimentally is 12.2±2 kcal/mole more stable than the isolated reactants and products [125, 127-128], Cl⁻ and CH₃Cl. The bond energy of the Cl⁻.C^δ+ functional group of the negatively-charged molecular ion complex of E_D(Group)=12.08900 kcal/mole (0.52422 eV) given in Table 16.47 matches the experimental stabilization energy very well. A simulation of the reaction of Eq. (16.181) is available on the internet [130].

Excited States of the Hydrogen Molecule

Force Balance of the Excited States of the Hydrogen Molecule

In the mathematical limit, as the eccentricity goes to zero the hydrogen molecule becomes the helium atom. The excited states of the hydrogen molecule are determined by the same physics as those of the helium atom. It was shown in the Excited States of Helium section that the inner atomic orbital is essentially that of He⁺ for all excited states with the exact result upon ionization. The infinite H₂ excited state corresponds to a free electron with the inner MO and protons comprising H₂⁺. Implicit in the calculation of the energy of the outer electron of each H₂ excited state is that the inner electron has the geometrical parameters, component energies, and the total energy of H₂⁺ as shown to very good approximation for the inner atomic electron of helium exited states. For H₂, the excited-state photon's two-dimensional ellipsoidal electric field at the outer electron superimposes that of the field of the nuclei at the foci of the inner MO and inner MO charge such that the resultant electric field has a magnitude e/n in the direction of i_ξ at the outer electron where n=2, 3, 4, . . . for excited states. Then, the force balance of the outer excited-state electron is given by balance between the centrifugal force, the central Coulombic force corresponding to the effective central field due to the superposition of photon field at the outer electron and the net field of the protons at the foci of the inner MO, and the magnetic forces for the particular spin and orbital state. The geometrical parameters for H₂ are determined from the semimajor axis given by the force balance and the relationships among the ellipsoidal parameters. The energies corresponding to the excited electron are given by the prolate spheroidal energy equations given in the Derivation of the General Geometrical and Energy Equations of Organic Chemistry section except for a ½ correction corresponding to a single electron, and the electric terms are scaled according to the effective central field of 1/n.

Singlet Excited States

l=0

The force balance between the electric, magnetic, and centrifugal forces of the outer electron given by Eqs. (9.10) and (11.285) is

$\begin{array}{cc}\frac{{\hslash }^2}{m_e{a}^2{b}^2}D=\frac{1}{n}\frac{e^2}{8{\mathrm{\pi \varepsilon }}_o{\mathrm{ab}}^2}D+\frac{1}{n}\frac{2m}{3}\frac{1}{2}\frac{{\hslash }^2}{2m_e{a}^2{b}^2}D& \left(12.37\right)\end{array}$ where the geometrical factor due to the rotation about the semimajor axis is given by Eq. (11.391) and m is a positive or negative integer due to the symmetry of the angular momentum components as given in the Force Balance of Hydrogen-Type Molecules section. The parametric solution given by Eq. (11.83) occurs when semimajor axis, a, is

$\begin{array}{cc}a=a_0\left(2n-\frac{m}{3}\right)& \left(12.38\right)\end{array}$ The internuclear distance, 2c′, which is the distance between the foci is given by Eq. (11.79) where p=1/n.

$\begin{array}{cc}2c^{\prime }=2\sqrt{\frac{{\mathrm{aa}}_0}{2p}}=2a_0\sqrt{\frac{n\left(2n-\frac{m}{3}\right)}{2}}& \left(12.39\right)\end{array}$ The semiminor axis is given by Eq. (11.80).

$\begin{array}{cc}b=\sqrt{a^2-c^{\prime 2}}=a_0\left(2n-\frac{m}{3}\right)\sqrt{1-\frac{n}{2\left(2n-\frac{m}{3}\right)}}& \left(12.40\right)\end{array}$ The eccentricity, e is given by Eq. (11.67).

$\begin{array}{cc}e=\frac{c^{\prime }}{a}=\sqrt{\frac{n}{2\left(2n-\frac{m}{3}\right)}}& \left(12.41\right)\end{array}$ l≠0

The exited singlet states of the hydrogen molecule for l≠0 are solved using the same approach as those of the excited states of the helium atom given in the corresponding section, wherein the force balance due to the a_Mag(l,m) terms corresponding to prolate spheroid geometry rather than spherical are also associated Legendre functions or spherical harmonics with regard to the semimajor axis as given by Li, Kang, and Leong [131].

The magnetic forces comprise the component of Eq. (12.37) corresponding to the nondynamic current and the a_Mag(l,m) component due to the time dynamic modulation current and its interaction with electron spin. The force balance between the electric, magnetic, and centrifugal forces of the outer electron given by Eqs. (12.37) and (9.52) is

$\begin{array}{cc}\frac{{\hslash }^2}{m_e{a}^2{b}^2}D=\frac{1}{n}\frac{e^2}{8{\mathrm{\pi \varepsilon }}_o{\mathrm{ab}}^2}D+\frac{1}{n}\frac{m}{3}\frac{{\hslash }^2}{2m_e{a}^2{b}^2}D-\frac{1}{n}\frac{\frac{3}{2}}{\left(2l+1\right)!!}{\left(\frac{l+1}{l}\right)}^{1/2}\frac{1}{l+2}\frac{1}{2}\frac{{\hslash }^2}{m_e{a}^2{b}^2}\left(1-\sqrt{\frac{l}{l+1}}\right)D& \left(12.42\right)\end{array}$ where the √{square root over (3/4)} and r⁻³ terms are replaced by one and Da⁻²b⁻² as given in the Force Balance of Hydrogen-Types Molecules section. The parametric solution given by Eq. (11.83) occurs when semimajor axis, a, is

$\begin{array}{cc}a=a_0\left(2n-\frac{m}{3}+\frac{\frac{3}{2}}{\left(2l+1\right)!!}{\left(\frac{l+1}{l}\right)}^{1/2}\frac{1}{l+2}\left(1-\sqrt{\frac{l}{l+1}}\right)\right)& \left(12.43\right)\end{array}$ The internuclear distance, 2c′, which is the distance between the foci is given by Eq. (11.79) where p=1/n.

$\begin{array}{cc}2c^{\prime }=2\sqrt{\frac{{\mathrm{aa}}_0}{2p}}=2a_0\sqrt{\frac{n\left(\begin{array}{c}n-\frac{m}{3}+\frac{\frac{3}{2}}{\left(2l+1\right)!!}{\left(\frac{l+1}{l}\right)}^{1/2}\\ 2\frac{1}{l+2}\left(1-\sqrt{\frac{l}{l+1}}\right)\end{array}\right)}{2}}& \left(12.44\right)\end{array}$ The semiminor axis is given by Eq. (11.80).

$\begin{array}{cc}\begin{array}{c}b=\sqrt{a^2-c^{\prime 2}}\\ =a_0\left(\begin{array}{c}\left(\begin{array}{c}2n-\frac{m}{3}+\frac{\frac{3}{2}}{\left(2l+1\right)!!}{\left(\frac{l+1}{l}\right)}^{1/2}\\ \frac{1}{l+2}\left(1-\sqrt{\frac{l}{l+1}}\right)\end{array}\right)\\ \sqrt{1-\frac{n}{2\left(\begin{array}{c}2n-\frac{m}{3}+\frac{\frac{3}{2}}{\left(2l+1\right)!!}{\left(\frac{l+1}{l}\right)}^{1/2}\\ \frac{1}{l+2}\left(1-\sqrt{\frac{l}{l+1}}\right)\end{array}\right)}}\end{array}\right)\end{array}& \left(12.45\right)\end{array}$ The eccentricity, e, is given by Eq. (11.67).

$\begin{array}{cc}e=\frac{c^{\prime }}{a}=\sqrt{\frac{n}{2\left(\begin{array}{c}2n-\frac{m}{3}+\frac{\frac{3}{2}}{\left(2l+1\right)!!}{\left(\frac{l+1}{l}\right)}^{1/2}\\ \frac{1}{l+2}\left(1-\sqrt{\frac{l}{l+1}}\right)\end{array}\right)}}& \left(12.46\right)\end{array}$ Triplet Excited Statesl=0

The force-balance equation and semimajor-axis solution of triplet excited states for l=0 are equivalent to those of the corresponding singlet excited states given by Eqs. (12.37-12.38). However, due to the triplet spin state, the magnetic force in Eq. (12.37) is increased by a factor of two as in the case of the corresponding helium exited states given in Eq. (9.31). Thus, m is replaced by 2 m. Then, the force balance between the electric, magnetic, and centrifugal forces of the outer electron is

$\begin{array}{cc}\frac{{\hslash }^2}{m_e{a}^2{b}^2}D=\frac{1}{n}\frac{e^2}{8{\mathrm{\pi \varepsilon }}_o{\mathrm{ab}}^2}D+\frac{1}{n}\frac{4m}{3}\frac{1}{2}\frac{{\hslash }^2}{2m_e{a}^2{b}^2}D& \left(12.47\right)\end{array}$ The parametric solution given by Eq. (11.83) occurs when semimajor axis, a, is

$\begin{array}{cc}a=a_0\left(2n-\frac{2m}{3}\right)& \left(12.48\right)\end{array}$ The internuclear distance, 2c′, which is the distance between the foci is given by Eq. (11.79) where p=1/n.

$\begin{array}{cc}2c^{\prime }=2\sqrt{\frac{{\mathrm{aa}}_0}{2p}}=2a_0\sqrt{\frac{n\left(2n-\frac{2m}{3}\right)}{2}}& \left(12.49\right)\end{array}$ The semiminor axis is given by Eq. (11.80).

$\begin{array}{cc}b=\sqrt{a^2-c^{\mathrm{\prime 2}}}=a_0\left(2n\frac{2m}{3}\right)\sqrt{1-\frac{n}{2\left(2n-\frac{2m}{3}\right)}}& \left(12.50\right)\end{array}$ The eccentricity, e, is given by Eq. (11.67).

$\begin{array}{cc}e=\frac{c^{\prime }}{a}=\sqrt{\frac{n}{2\left(2n-\frac{2m}{3}\right)}}& \left(12.51\right)\end{array}$ l≠0

The magnetic forces of triplet excited molecular states having l≠0 comprise the nondynamic-current component of Eq. (12.42) with the parameter m of the magnetic force of Eq. (12.37) increased by a factor of two and the a_Mag(l,m) component due to the time dynamic modulation current and its interaction with electron spin. The latter is solved using the same approach as that of the triplet excited states of the helium atom given in the corresponding section. The force balance between the electric, magnetic, and centrifugal forces of the outer electron given by Eqs. (12.47) and (9.63) is

$\begin{array}{cc}\frac{{\hslash }^2}{m_e{a}^2{b}^2}D=\frac{1}{n}\frac{e^2}{8{\mathrm{\pi \varepsilon }}_o{\mathrm{ab}}^2}D+\frac{1}{n}\frac{2m}{3}\frac{{\hslash }^2}{2m_e{a}^2{b}^2}D+\frac{1}{n}\frac{\frac{3}{2}}{\left(2l+1\right)!!}{\left(\frac{l+1}{l}\right)}^{1/2}\frac{1}{l+2}\frac{1}{2}\frac{{\hslash }^2}{m_e{a}^2{b}^2}\left(2-\sqrt{\frac{l}{l+1}}\right)D& \left(12.52\right)\end{array}$ where the √{square root over (3/4)} and r⁻³ terms are replaced by one and Da⁻²b⁻² as given in the Force Balance of Hydrogen-Types Molecules section. The parametric solution given by Eq. (11.83) occurs when semimajor axis, a, is

$\begin{array}{cc}a=a_0\left(\begin{array}{c}2n-\frac{2m}{3}-\frac{\frac{3}{2}}{\left(2l+1\right)!!}{\left(\frac{l+1}{l}\right)}^{1/2}\\ \frac{1}{l+2}\left(2-\sqrt{\frac{l}{l+1}}\right)\end{array}\right)& \left(12.53\right)\end{array}$ The internuclear distance, 2c′, which is the distance between the foci is given by Eq. (11.79) with the 2 factor and p=1/n.

$\begin{array}{cc}2c^{\prime }=2\sqrt{\frac{{\mathrm{aa}}_0}{2p}}=2a_0\sqrt{\frac{n\left(2n-\frac{2m}{3}-\frac{\frac{3}{2}}{\left(2l+1\right)!!}{\left(\frac{l+1}{l}\right)}^{1/2}\frac{1}{l+2}\left(2-\sqrt{\frac{l}{l+1}}\right)\right)}{2}}& \left(12.54\right)\end{array}$ The semiminor axis is given by Eq. (11.80).

$\begin{array}{cc}b=\sqrt{a^2-c^{\mathrm{\prime 2}}}=a_0\left(\begin{array}{c}\left(\begin{array}{c}2n-\frac{2m}{3}-\frac{\frac{3}{2}}{\left(2l+1\right)!!}{\left(\frac{l+1}{l}\right)}^{1/2}\\ \frac{1}{l+2}\left(2-\sqrt{\frac{l}{l+1}}\right)\end{array}\right)\\ \sqrt{1-\frac{n}{2\left(\begin{array}{c}2n-\frac{2m}{3}-\frac{\frac{3}{2}}{\left(2l+1\right)!!}{\left(\frac{l+1}{l}\right)}^{1/2}\\ \frac{1}{l+2}\left(2-\sqrt{\frac{l}{l+1}}\right)\end{array}\right)}}\end{array}\right)& \left(12.55\right)\end{array}$ The eccentricity, e, is given by Eq. (11.67).

$\begin{array}{cc}e=\frac{c^{\prime }}{a}=\sqrt{\frac{n}{2\left(\begin{array}{c}2n-\frac{2m}{3}-\frac{\frac{3}{2}}{\left(2l+1\right)!!}{\left(\frac{l+1}{l}\right)}^{1/2}\\ \frac{1}{l+2}\left(2-\sqrt{\frac{l}{l+1}}\right)\end{array}\right)}}& \left(12.56\right)\end{array}$ Energies of the Excited States of the Hydrogen Molecule

The component energies of the outer electron of the hydrogen molecule of the excited state corresponding to quantum number n are given by Eqs. (11.290-11.293) and (11.233-11.236) where the energies are each multiplied by a factor of ½ since the outer MO comprises only one electron, and those corresponding to charge are multiplied by effective-charge factor of 1/n:

$\begin{array}{cc}{V}_e=\frac{1}{n}\frac{1}{2}\frac{-2e^2}{8{\mathrm{\pi \varepsilon }}_o\sqrt{a^2-b^2}}\ln \frac{a+\sqrt{a^2-b^2}}{a-\sqrt{a^2-b^2}}& \left(12.57\right)\\ V_p=0& \left(12.58\right)\\ T=\frac{1}{2}\frac{{\hslash }^2}{2m_{e}a\sqrt{a^2-b^2}}\ln \frac{a+\sqrt{a^2-b^2}}{a-\sqrt{a^2-b^2}}& \left(12.59\right)\\ V_m=\frac{1}{n}\frac{1}{2}\frac{-{\hslash }^2}{4m_{e}a\sqrt{a^2-b^2}}\ln \frac{a+\sqrt{a^2-b^2}}{a-\sqrt{a^2-b^2}}& \left(12.60\right)\\ \begin{array}{c}{\stackrel{\_}{E}}_{\mathrm{osc}}\left(H_2\right)={\stackrel{\_}{E}}_D+{\stackrel{\_}{E}}_{\mathrm{Kvib}}\\ =-\left(V_e+T+V_m+V\right)\\ \sqrt{\frac{2\hslash \sqrt{\frac{\frac{1}{n^4}\frac{1}{2}\frac{e^2}{4{\mathrm{\pi \varepsilon }}_o{a}_0^3}}{m_e}}}{m_e{c}^2}}+{\stackrel{\_}{E}}_{\mathrm{Kvib}}\end{array}& \left(12.61\right)\end{array}$ where with regard to Eq. (12.61), the angular frequency of reentrant oscillation ω and corresponding energies E_K, Ē_D, and Ē_osc are given by Eqs. (11.233-11.236) with p=1n and the factor of ½ was applied since the outer MO comprises only one electron. The potential energy, V_p, due to proton-proton repulsion (Eq. 12.58)) is zero. The repulsive term applies only to the total energy of H₂⁺ which is implicit in the calculation of the energy of the outer electron of the H₂ excited state as in the case with the energy of the helium exited states given in the Excited States of Helium section. The total energy, E_T, for the hydrogen molecular excited state given by Eqs. (11.239-11.240) is

$\begin{array}{cc}{E}_T=V_e+T+V_m+V_p+{\stackrel{\_}{E}}_{\mathrm{osc}}& \left(12.62\right)\\ E_T=-\{\begin{array}{c}\left(\frac{-e^2}{8{\mathrm{\pi \varepsilon }}_o}-\frac{n{\hslash }^2}{4m_{e}a}+\frac{{\hslash }^2}{8m_{e}a}\right)\frac{1}{n\sqrt{a^2-b^2}}\ln \\ \frac{a+\sqrt{a^2-b^2}}{a-\sqrt{a^2-b^2}}\left[1+\sqrt{\frac{2\hslash \sqrt{\frac{\frac{e^2}{n^{4}8{\mathrm{\pi \varepsilon }}_o{a}_0^3}}{m_e}}}{m_{e}c}}\right]-{\stackrel{\_}{E}}_{\mathrm{Kvib}}\end{array}\}& \left(12.63\right)\end{array}$ The negative of Eq. (12.63) is the ionization energy of the excited state of H₂. The energy T_c from the n=1 state (also referred to as the state X) to the energy of the n^th excited state is given by the sum of E_T given by Eq. (12.63) and IP₁ of H₂ given by Eq. (11.298):T₃(H₂)=E_T+15.4248 eV  (12.64)

The geometrical (Eqs. (12.37-12.54) and energy (Eqs. (12.55-12.61)) parameters of singlet and triplet excited states of molecular hydrogen are given in Tables 55 and 56, respectively, where Ê_Kvib was given to very good approximation by ω_e of H₂⁺ (the n=∞ state) since there is a close match with ω_e of each excited state [132]. The color scale, translucent views of the charge densities of exemplary ellipsoidal spherical harmonics that modulate the time-independent spin function are shown in FIG. 25. The modulation functions propagate about the major axis as spatially and temporally harmonic charge-density waves.

TABLE 55
The geometrical and energy parameters of the singlet excited states
of molecular hydrogen compared to the experimental energies [9].
n	m	l	a (a0)	a (m)	b, c (m)	c′ (m)	2c′ (m)	e	Ve (eV)	Vp (eV)
2	4	1	2.73570	1.44767E−10	1.15312E−10	8.75257E-11	1.75051E−10	0.60460	−5.76118	0
2	1	1	3.73570	1.97685E−10	1.69169E−10	1.02279E−10	2.04558E−10	0.51739	−4.03193	0
2	0	0	4.00000	2.11671E−10	1.83312E−10	1.05835E−10	2.11671E−10	0.50000	−3.73688	0
2	−2	0	4.66667	2.46949E−10	2.18897E−10	1.14315E−10	2.28631E−10	0.46291	−3.15548	0
3	4	0	4.66667	2.46949E−10	2.03426E−10	1.40007E−10	2.80014E−10	0.56695	−2.20446	0
3	4	1	4.73570	2.50603E−10	2.07146E−10	1.41039E−10	2.82078E−10	0.56280	−2.16761	0
3	3	0	5.00000	2.64589E−10	2.21371E−10	1.44921E−10	2.89842E−10	0.54772	−2.03734	0
3	3	2	5.00562	2.64886E−10	2.21673E−10	1.45003E−10	2.90005E−10	0.54742	−2.03474	0
3	3	1	5.06904	2.68242E−10	2.25081E−10	1.45918E−10	2.91836E−10	0.54398	−2.00588	0
3	2	0	5.33333	2.82228E−10	2.39270E−10	1.49674E−10	2.99348E−10	0.53033	−1.89402	0
3	2	1	5.40237	2.85881E−10	2.42973E−10	1.50639E−10	3.01279E−10	0.52693	−1.86685	0
3	1	0	5.66667	2.99867E−10	2.57134E−10	1.54280E−10	3.08561E−10	0.51450	−1.76971	0
3	1	1	5.73570	3.03520E−10	2.60830E−10	1.55217E−10	3.10434E−10	0.51139	−1.74599	0
3	−3	1	7.06904	3.74077E−10	3.32025E−10	1.72316E−10	3.44633E−10	0.46064	−1.38755	0
4	4	1	6.73570	3.56438E−10	2.98872E−10	1.94226E−10	3.88452E−10	0.54491	−1.13268	0
4	1	2	7.67229	4.06000E−10	3.49094E−10	2.07290E−10	4.14580E−10	0.51057	−0.97861	0
4	1	1	7.73570	4.09356E−10	3.52488E−10	2.08145E−10	4.16290E−10	0.50847	−0.96969	0
4	−1	0	8.33333	4.40981E−10	3.84438E−10	2.16036E−10	4.32071E−10	0.48990	−0.89305	0
4	−1	1	8.40237	4.44634E−10	3.88126E−10	2.16929E−10	4.33857E−10	0.48788	−0.88497	0
n	m	l	T (eV)	Vm (eV)	ET(H2MO) (eV)	ĒKvib (eV)	ω (1015 rad/s)	EK (eV)	ĒD (eV)
2	4	1	2.10592	−0.52648	−4.18174	0.28479	7.30819	4.81038	−1.81447E−02
2	1	1	1.07930	−0.26982	−3.22245	0.28479	7.30819	4.81038	−1.39823E−02
2	0	0	0.93422	−0.23355	−3.03621	0.28479	7.30819	4.81038	−1.31742E−02
2	−2	0	0.67618	−0.16904	−2.64835	0.28479	7.30819	4.81038	−1.14913E−02
3	4	0	0.70858	−0.11810	−1.61398	0.28479	3.24809	2.13795	−4.66874E−03
3	4	1	0.68657	−0.11443	−1.59546	0.28479	3.24809	2.13795	−4.61517E−03
3	3	0	0.61120	−0.10187	−1.52801	0.28479	3.24809	2.13795	−4.42004E−03
3	3	2	0.60974	−0.10162	−1.52663	0.28479	3.24809	2.13795	−4.41606E−03
3	3	1	0.59357	−0.09893	−1.51124	0.28479	3.24809	2.13795	−4.37155E−03
3	2	0	0.53269	−0.08878	−1.45011	0.28479	3.24809	2.13795	−4.19471E−03
3	2	1	0.51834	−0.08639	−1.43490	0.28479	3.24809	2.13795	−4.15070E−03
3	1	0	0.46845	−0.07808	−1.37933	0.28479	3.24809	2.13795	−3.98998E−03
3	1	1	0.45661	−0.07610	−1.36548	0.28479	3.24809	2.13795	−3.94992E−03
3	−3	1	0.29443	−0.04907	−1.14219	0.28479	3.24809	2.13795	−3.30401E−03
4	4	1	0.33632	−0.04204	−0.83840	0.28479	1.82705	1.20259	−1.81892E−03
4	1	2	0.25510	−0.03189	−0.75539	0.28479	1.82705	1.20259	−1.63883E−03
4	1	1	0.25070	−0.03134	−0.75032	0.28479	1.82705	1.20259	−1.62783E−03
4	−1	0	0.21433	−0.02679	−0.70551	0.28479	1.82705	1.20259	−1.53061E−03
4	−1	1	0.21065	−0.02633	−0.70066	0.28479	1.82705	1.20259	−1.52008E−03
									Relative
n	m	l	Ēosc(eV)	IP1(H2) (eV)	Cal. Te (eV)	Exp. Te (eV)	State	Exp. Te (cm−1)	Error
2	4	1	0.12425	15.424814	11.3673	11.36819	B	91689.9	0.00008
2	1	1	0.12841	15.424814	12.3308	12.40385	C	100043.0	0.00589
2	0	0	0.12922	15.424814	12.5178	12.40631	E	100062.8	−0.00899
2	−2	0	0.13091	15.424814	12.9074	12.82999	F	103480	−0.00603
3	4	0	0.13773	15.424814	13.9486	13.96780	K	112657	0.00138
3	4	1	0.13778	15.424814	13.9671	13.98466	G	112793	0.00125
3	3	0	0.13798	15.424814	14.0348	14.01839	I	113065	−0.00117
3	3	2	0.13798	15.424814	14.0362	14.02818	Q	113144	−0.00057
3	3	1	0.13803	15.424814	14.0516	14.06042	J	113404	0.00063
3	2	0	0.13820	15.424814	14.1129	14.12043	D	113888	0.00053
3	2	1	0.13825	15.424814	14.1282	14.12055	H	113889	−0.00054
3	1	0	0.13841	15.424814	14.1839	14.19631	L	114500	0.00087
3	1	1	0.13845	15.424814	14.1978	14.21540	M	114654	0.00124
3	−3	1	0.13909	15.424814	14.4217	14.41551	N	116268	−0.00043
4	4	1	0.14058	15.424814	14.7270	14.71581	R	118690	−0.00076
4	1	2	0.14076	15.424814	14.8102	14.81549	T	119494	0.00036
4	1	1	0.14077	15.424814	14.8153	14.81772	P	119512	0.00017
4	−1	0	0.14087	15.424814	14.8602	14.85591	S	119820	−0.00029
4	−1	1	0.14088	15.424814	14.8650	14.85975	O	119851	−0.00036
								Avg. Rel. Error	−0.00035

TABLE 56
The geometrical and energy parameters of the triplet excited states
of molecular hydrogen compared to the experimental energies [9].
n	m	l	a (a0)	a (m)	b, c (m)	c′ (m)	2c′ (m)	e	Ve (eV)	Vp (eV)
2	1	1	3.02860	1.60266E−10	1.31165E−10	9.20919E−11	1.84184E−10	0.57462	−5.11612	0
2	1	0	3.33333	1.76392E−10	1.47580E−10	9.66141E−11	1.93228E−10	0.54772	−4.58402	0
3	4	1	3.02860	1.60266E−10	1.13859E−10	1.12789E−10	2.25578E−10	0.70376	−3.72248	0
3	2	2	4.63043	2.45032E−10	2.01471E−10	1.39462E−10	2.78925E−10	0.56916	−2.22432	0
3	2	0	4.66667	2.46949E−10	2.03426E−10	1.40007E−10	2.80014E−10	0.56695	−2.20446	0
3	1	1	5.02860	2.66102E−10	2.22908E−10	1.45335E−10	2.90670E−10	0.54616	−2.02419	0
3	1	2	5.29710	2.80310E−10	2.37326E−10	1.49165E−10	2.98329E−10	0.53214	−1.90861	0
3	1	0	5.33333	2.82228E−10	2.39270E−10	1.49674E−10	2.99348E−10	0.53033	−1.89402	0
4	4	1	5.02860	2.66102E−10	2.06512E−10	1.67818E−10	3.35637E−10	0.63065	−1.59277	0
4	3	1	5.69526	3.01380E−10	2.42762E−10	1.78596E−10	3.57193E−10	0.59260	−1.37400	0
4	3	2	5.96376	3.15589E−10	2.57285E−10	1.82758E−10	3.65516E−10	0.57910	−1.30225	0
4	2	1	6.36193	3.36659E−10	2.78763E−10	1.88760E−10	3.77520E−10	0.56069	−1.20882	0
4	2	0	6.66667	3.52785E−10	2.95161E−10	1.93228E−10	3.86456E−10	0.54772	−1.14600	0
4	1	1	7.02860	3.71937E−10	3.14600E−10	1.98404E−10	3.96808E−10	0.53343	−1.07948	0
4	1	0	7.33333	3.88063E−10	3.30941E−10	2.02659E−10	4.05319E−10	0.52223	−1.02923	0
5	3	1	7.69526	4.07216E−10	3.34593E−10	2.32104E−10	4.64208E−10	0.56998	−0.80341	0
5	3	2	7.96376	4.21424E−10	3.49065E−10	2.36119E−10	4.72237E−10	0.56029	−0.77238	0
5	2	1	8.36193	4.42494E−10	3.70488E−10	2.41949E−10	4.83898E−10	0.54679	−0.73059	0
6	−4	0	14.66667	7.76126E−10	6.92214E−10	3.51016E−10	7.02033E−10	0.45227	−0.33334	0
n	m	l	T (eV)	Vm (eV)	ET(H2MO) (eV)	ĒKvib (eV)	ω (1015 rad/s)	EK (eV)	ĒD (eV)
2	1	1	1.68927	−0.42232	−3.84916	0.28479	7.30819	4.81038	−1.67016E−02
2	1	0	1.37520	−0.34380	−3.55261	0.28479	7.30819	4.81038	−1.54149E−02
3	4	1	1.84367	−0.30728	−2.18609	0.28479	3.24809	2.13795	−6.32367E−03
3	2	2	0.72056	−0.12009	−1.62386	0.28479	3.24809	2.13795	−4.69732E−03
3	2	0	0.70858	−0.11810	−1.61398	0.28479	3.24809	2.13795	−4.66874E−03
3	1	1	0.60380	−0.10063	−1.52102	0.28479	3.24809	2.13795	−4.39984E−03
3	1	2	0.54047	−0.09008	−1.45822	0.28479	3.24809	2.13795	−4.21816E−03
3	1	0	0.53269	−0.08878	−1.45011	0.28479	3.24809	2.13795	−4.19471E−03
4	4	1	0.63349	−0.07919	−1.03847	0.28479	1.82705	1.20259	−2.25298E−03
4	3	1	0.48251	−0.06031	−0.95181	0.28479	1.82705	1.20259	−2.06497E−03
4	3	2	0.43672	−0.05459	−0.92012	0.28479	1.82705	1.20259	−1.99621E−03
4	2	1	0.38002	−0.04750	−0.87630	0.28479	1.82705	1.20259	−1.90115E−03
4	2	0	0.34380	−0.04298	−0.84518	0.28479	1.82705	1.20259	−1.83363E−03
4	1	1	0.30717	−0.03840	−0.81070	0.28479	1.82705	1.20259	−1.75884E−03
4	1	0	0.28070	−0.03509	−0.78362	0.28479	1.82705	1.20259	−1.70007E−03
5	3	1	0.26101	−0.02610	−0.56850	0.28479	1.16931	0.76966	−9.86698E−04
5	3	2	0.24247	−0.02425	−0.55416	0.28479	1.16931	0.76966	−9.61808E−04
5	2	1	0.21843	−0.02184	−0.53401	0.28479	1.16931	0.76966	−9.26832E−04
6	−4	0	0.06818	−0.00568	−0.27084	0.28479	0.812021	0.53449	−3.91731E−04
									Relative
n	m	l	Ēosc (eV)	IP1(H2) (eV)	Cal. Te (eV)	Exp. Te (eV)	State	Exp. Te (cm−1)	Error
2	1	1	0.12570	15.424814	11.7013	11.87084	c	95744	0.01428
2	1	0	0.12698	15.424814	11.9992	11.89489	a	95938	−0.00877
3	4	1	0.13607	15.424814	13.3748	13.36275	e	107777	−0.00090
3	2	2	0.13770	15.424814	13.9387	13.97338	d	112702	0.00249
3	2	0	0.13773	15.424814	13.9486	13.98181	h	112770	0.00238
3	1	1	0.13800	15.424814	14.0418	13.98268	g	112777	−0.00423
3	1	2	0.13818	15.424814	14.1048	14.01132	i	113008	−0.00667
3	1	0	0.13820	15.424814	14.1129	14.03488	j	113198	−0.00556
4	4	1	0.14014	15.424814	14.5265	14.47007	f	116708	−0.00390
4	3	1	0.14033	15.424814	14.6133	14.66658	V	118293	0.00363
4	3	2	0.14040	15.424814	14.6451	14.67625	k	118371	0.00212
4	2	1	0.14050	15.424814	14.6890	14.68915	p	118475	0.00001
4	2	0	0.14056	15.424814	14.7202	14.69250	s	118502	−0.00189
4	1	1	0.14064	15.424814	14.7547	14.70155	r	118575	−0.00362
4	1	0	0.14070	15.424814	14.7819	14.79379	m	119319	0.00080
5	3	1	0.14141	15.424814	14.9977	14.99651	n	120954	−0.00008
5	3	2	0.14143	15.424814	15.0121	15.01449	q	121099	0.00016
5	2	1	0.14147	15.424814	15.0323	15.03879	t	121295	0.00043
6	−4	0	0.14201	15.424814	15.2960	15.31031	u	123485	0.00094
								Avg. Rel. Error	−0.00044

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Claims

1. A system comprising at least one processor,at least one non-transitory computer readable medium, containing instructions that when executed by the at least one processor cause the at least one processor to perform operations for computing and rendering a nature of a chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of molecules, compounds, and materials and solving the dipole moment of at least one bond, the operations comprising:inputting data into the system regarding the atomic composition, positions, and excitation state of the molecule, compound, or material;calculating a solution to Maxwellian equations, wherein the solution is a two dimensional spheroidal surface, wherein charge, mass, and current density functions are determined by evaluating the two dimensional spheroidal surface of molecules, compounds, and materials, and solving the dipole moment of at least one bond, andoutputting the nature of the chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions and the corresponding energy components of molecules, compounds, and materials to an output device in communication with the at least one processor;wherein at least one processor calculates bond moment μ of a functional group by considering the charge donation between atoms of the functional group wherein the potential of an MO is that of a point charge at infinity such that an asymmetry in the distribution of charge between nonequivalent HOs or AOs of the MO occurs to maintain an energy match of the MO with the bridged orbitals and the charge redistribution between the spherical orbitals achieves a corresponding current-density that maintains constant current at the equivalent-energy condition according to an energy-matching factor,wherein the dipole moment of a given molecule is given by the vector sum of the bond moments in the molecule wherein the dipole moment is given by taking into account the magnitude and direction of the bond moment of each functional group wherein the function-group bond moment stays constant from molecule to molecule and is in the vector direction of the internuclear axis, andwherein at least one processor creates a molecular model based on the solutions derived from the Maxwellian equations and the dipole moment, and displays a three dimensional representation of the molecular model on the output device.

2. The system of claim 1 comprising an input that comprises at least one of a serial port, usb port, microphone input, camera input, keyboard, an mouse;wherein the at least one processor is a component of a general purpose computer that comprises at least one of a central processing unit (CPU), one or more specialized processors, system memory, a mass storage device such as a magnetic disk, an optical disk, and other storage device;computer program products or computer readable medium having embodied therein program code means wherein the computer readable media is any available media which can be accessed by a general purpose or special purpose computer 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 the general purpose or special purpose computer wherein the program code means comprises executable instructions and data which cause the general purpose computer or special purpose computer to perform a certain function of a group of functions, andthe output device that is a display comprising a monitor, video projector, printer, or three-dimensional rendering device that displays at least one of visual or graphical media wherein at least one of the group of static or dynamic images, vibration and rotation, and reactivity and physical properties are displayed.

3. The system of claim 1 wherein the at least one processor executes instructions to superimpose functional groups comprising at least one of the group of those of 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, ureas, acid halides, acid anhydrides, nitriles, thiols, sulfides, disulfides, sulfoxides, sulfones, sulfites, sulfates, nitro alkanes, nitrites, nitrates, conjugated polyenes, aromatics, heterocyclic aromatics, substituted aromatics.

4. The system of claim 1 wherein the energy matching factor is c1, c2, C1, or C2 of Eqs. (15.51) and (15.61):

5. The system of claim 4 wherein the orbital energy and radius are reciprocally related such that the contribution scales as the square of the ratio (over unity) of the energy of the resultant net positively-charged orbital and the initial matched energy of the resultant net negatively-charged orbital of the bond multiplied by the energy-matching factor.

6. The system of claim 5 wherein the partial charges on the HOs or AOs corresponding to the charge contribution are equivalent to point charges centered on the nuclei; due to symmetry, the bond moment p of each functional group is along the internuclear axis whereby it is calculated from the partial changes at the separation distance, the internuclear distance.

7. The system of claim 6, wherein the bond moment of each functional group is along the internuclear axis of A-B, wherein A is the net positively-charged atom, and is calculated using the reciprocal relationship between the orbital energies and radii, the dependence of the orbital area on the radius squared, and the relationship of the partial charge q to the areas with energy matching for each electron of the MO, such that the bond moment is given by

8. The system of claim 1 wherein the functional group bond moments determined using Eq. (16.15) are substantially given by

9. The system of claim 1 wherein interatomic and molecular binding is determined by electrical and electrodynamics forces wherein Coulombic-based bonding can be grouped into two main categories, bonding that comprises permanent dipole-dipole interactions further including an extreme case, hydrogen bonding, and bonding regarding reversible mutually induced dipole fields in near-neighbor collision-partner molecules called van der Waals bonding.

10. The system of claim 9 wherein structure and properties of liquids and solids are solved by first solving the unit cell of the condensed solid based on an energy minimum of the molecular interactions and their dependence on the packing.

11. The system of claim 10 wherein bonding in neutral condensed solids and liquids arises from Coulombic interactions between partial charges corresponding to dipoles of the molecules and atoms.

12. The system of claim 11 wherein the energy from the interaction of the partial charges increases as the separation decreases, but concomitantly, the energy of a bond that may form between the interacting species increases as well such that the equilibrium separation distance corresponds to the occurrence of the balance between the Coulombic potential energy of the interacting atoms and the energy of the nascent bond whose formation involves the interacting atoms.

13. The system of claim 12 wherein balance is at the energy threshold for the formation of a nascent bond that would replace the interacting partial charges while also destabilizing the standard bonds of the interacting molecules or cancel the Coulombic potential energy of interacting atoms wherein the general equation for the balance of the Coulombic energy and the nascent bond energy is given by

14. The system of claim 13 wherein the a, b, and c parameters of the unit cell are solved, then the unit cell can be proliferated to arbitrary scale to render the solid.

15. The system of claim 14 wherein the liquid is given as linear combinations of units cells based on the solid cell whose structures and populations are based on statistical thermodynamical principles.

16. The system of claim 15 wherein the electric field in the material having an electric polarization density is determined, and in turn, the lattice energy is calculated from the energy of each dipole in the corresponding field using the electrostatic form of Gauss' equation.

17. The system of claim 16 wherein the polarization density corresponding to the aligned dipoles moments determines the electric field E:

18. The system of claim 1 wherein reaction kinetics are modeled using thermal rate constants by solving the classical equations of motion with the formation of the transition state and any intermediate reaction complexes between the reactants and products on the trajectory between them.

19. The system of claim 18 wherein the transition state is the minimum energy complex involving the reactants; the activation energy Ea can be interpreted as the minimum energy that the reactants must have in order to form the transition state and transform to product molecules, and Ea is calculated from the total energy of the transition state relative to that of the reactants.

20. The system of claim 19 wherein the parameters of the transition state and any intermediate reaction complexes is solved using the equations of the corresponding functional group with the boundary conditions for the transition state and any intermediate reaction complexes.

21. The system of claim 20 wherein the equations of the functional groups are at least one of

22. A non-transitory computer readable medium containing instructions that when executed by at least one processor cause the at least one processor to perform operations for computing and rendering a nature of a chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of molecules, compounds, and materials and solving the dipole moment of at least one bond, the operations comprising: inputting data into a system containing at least one processor regarding the atomic composition, positions, and excitation state of the molecule, compound, or material;calculating a solution to Maxwellian equations, wherein the solution is a two dimensional spheroidal surface, wherein charge, mass, and current density functions are determined by evaluating the two dimensional spheroidal surface of molecules, compounds, and materials, and solving the dipole moment of at least one bond, andoutputting the nature of the chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions and the corresponding energy components of molecules, compounds, and materials to an output device in communication with the at least one processor;wherein at least one processor calculates bond moment μ of a functional group by considering the charge donation between atoms of the functional group wherein the potential of an MO is that of a point charge at infinity such that an asymmetry in the distribution of charge between nonequivalent HOs or AOs of the MO occurs to maintain an energy match of the MO with the bridged orbitals and the charge redistribution between the spherical orbitals achieves a corresponding current-density that maintains constant current at the equivalent-energy condition according to an energy-matching factor,wherein the dipole moment of a given molecule is given by the vector sum of the bond moments in the molecule wherein the dipole moment is given by taking into account the magnitude and direction of the bond moment of each functional group wherein the function-group bond moment stays constant from molecule to molecule and is in the vector direction of the internuclear axis, andwherein at least one processor creates a molecular model based on the solutions derived from the Maxwellian equations and the dipole moment, and displays a three dimensional representation of the molecular model on the output device.

23. A method for computing and rendering a nature of a chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions of molecules, compounds, and materials and solving the dipole moment of at least one bond, the method comprising: inputting data into a system containing at least one processor regarding the atomic composition, positions, and excitation state of the molecule, compound, or material;calculating a solution to Maxwellian equations, wherein the solution is a two dimensional spheroidal surface, wherein charge, mass, and current density functions are determined by evaluating the two dimensional spheroidal surface of molecules, compounds, and materials, and solving the dipole moment of at least one bond, andoutputting the nature of the chemical bond comprising physical, Maxwellian solutions of charge, mass, and current density functions and the corresponding energy components of molecules, compounds, and materials to an output device in communication with the at least one processor;wherein at least one processor calculates bond moment μ of a functional group by considering the charge donation between atoms of the functional group wherein the potential of an MO is that of a point charge at infinity such that an asymmetry in the distribution of charge between nonequivalent HOs or AOs of the MO occurs to maintain an energy match of the MO with the bridged orbitals and the charge redistribution between the spherical orbitals achieves a corresponding current-density that maintains constant current at the equivalent-energy condition according to an energy-matching factor,wherein the dipole moment of a given molecule is given by the vector sum of the bond moments in the molecule wherein the dipole moment is given by taking into account the magnitude and direction of the bond moment of each functional group wherein the function-group bond moment stays constant from molecule to molecule and is in the vector direction of the internuclear axis, andwherein at least one processor creates a molecular model based on the solutions derived from the Maxwellian equations and the dipole moment, and displays a three dimensional representation of the molecular model on the output device.