Total Bond Energies of Exact Classical Solutions of Molecules Generated by Millsian 1.0 Compared to Those Computed Using Modern 3-21G and 6-31G* Basis Sets R. L. Mills, B. Holverstott, B. Good, N. Hogle, A. Makwana Millsian, Inc., 493 Old Trenton Road, Cranbury, NJ 08512, www.millsian.com ABSTRACT Mills [1-12] solved the structure of the bound electron using classical laws and subsequently developed a unification theory based on those laws called the Grand Unified Theory of Classical Physics (GUTCP) with results that match observations for the basic phenomena of physics and chemistry from the scale of the quarks to cosmos. Millsian 1.0 is a program comprising molecular modeling applications of GUTCP, 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 of all major classes of chemical bonding including those involved in most organic molecules have been solved exactly in closed-form solutions. By using these functional groups as building blocks, or independent units, a potentially infinite number of molecules can be solved. As a result, Millsian software can visualize the exact three-dimensional structure and calculate physical characteristics of almost any molecule of any length and complexity. Even complex proteins and DNA (the molecules that encode genetic information) can 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 energies of exact classical solutions of molecules generated by Millsian 1.0 and those from a modern quantum mechanics-based program, Spartan’s pre- computed database using 3-21G and 6-31G* basis sets at the Hartree-Fock level of theory, were compared to experimental values. The Millsian results were consistently within an average relative deviation of about 0.1% of the experimentally values. In contrast, the 3-21G and 6-31G* results deviated over a wide range of relative error, typically being >30-150% with a large percentage of catastrophic failures, depending on functional group type and basis set. I. INTRODUCTION In this paper, 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 [1-12]. 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 [1-12]. Later physics was replaced by “pure mathematics” based on the notion of the inexplicable wave-particle duality nature of electrons which lead to the 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 [13]. He and many founders of QM such as Sommerfeld, Bohm, and Weinstein wrongly pursued a planetary model, were unsuccessful, and resorted to the current mathematical-probability-wave model that has many problems [1-17]. Consequently, Feynman for example, attempted to use first principles including Maxwell's equations to discover new physics to replace quantum mechanics [18]. 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 2 of that was imposed on the bound electron follows from a derivation by Haus [19]. 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 [1-12]. 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 and the nonradiative n =1 state 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 [20] 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 ()∇+22kG()x', x =−δ ( x' − x ) (1) in the infinite domain with the spherical wave expansion for the outgoing wave Green function is −−ik xx' ∞ A e ()1 * GikjkrhkrYY()x',',', x ==∑∑AA()<>() A,,mm (θ φθφ ) A ( ) (2) 4π xx'− AA==−0 m Jackson [20] 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 eitωn : 3 ⎡⎤i AA BX=−∇×∑ ⎢⎥amfkrEmMm()(),,AA.. a() m gkr AA () X A,m ⎣⎦k (3) ⎡⎤i AA EXX=∇×+∑ ⎢⎥amEmMm(),, fkraAA ().. ()() mgkr AA A,m ⎣⎦k where the cgs units used by Jackson are retained in this section. The radial functions fA (kr) and gkrA () are of the form: (11) ( ) ( 2) ( 2) gkrAhAAAAA()=+ Ah (4) XA.m is the vector spherical harmonic defined by 1 XLAA,,mm()θ ,,φθφ= Y () (5) AA()+1 where 1 Lr=×∇() (6) i The coefficients amE ()A, and amM (A, ) of Eq. (3) specify the amounts of electric (A,m) multipole and magnetic ()A,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 4π∂kik2 ⎧⎫ amA, =+⋅−∇⋅× Ym* ρ ⎡⎤ rjkrrJ jkrikrM jkrdx3 (7) E () ∫ AA⎨⎬⎣⎦() ()() A ( )() A i AA()+1 ⎩⎭∂ rc and −4π k 2 ⎛⎞J amA, =⋅+∇× jkrYm*3L M dx (8) M () ∫ AA() ⎜⎟ AA()+1 ⎝⎠c respectively, where the distribution of charge ρ (x,t) , current Jx(),t , and intrinsic magnetization M ()x,t are harmonically varying sources: ρ (x)e−itω , Jx()e−itω , and M ()x e−itω . The electron current-density function can be solved as a boundary value problem regarding the time varying corresponding source current Jx( )e−itω 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 e2 Vr()=− (9) 4πε0r 4 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.