Quasi-linear dependence of Coulomb forces on coupling strength parameter in the non-relativistic electronic Schrödinger equation and its consequences in Hund’s rule, Mølle -Plesset perturbation- , virial - , Hohenberg-Kohn - and Koopmans theorem
<p> The extended non-relativistic electronic Hamiltonian, H<sub>Ñ</sub>+ H<sub>ne</sub>+ aH<sub>ee</sub>, is linear in coupling strength parameter (a), but its eigenvalues (interpreted as electronic energies) have only quasi-linear dependence on “a”. No detailed analysis has yet been published on the ratio or participation of electron-electron repulsion energy (V<sub>ee</sub>) in total electronic energy – apart from virial theorem and the highly detailed and well-known algorithm for V<sub>ee</sub>, which is calculated during the standard HF-SCF and post-HF-SCF routines. Using a particular modification of the SCF part in the Gaussian package we have analyzed the ground state solutions via the parameter “a”. Technically, this modification was essentially a modification of a single line in an SCF algorithm, wherein the operator r<sub>ij</sub><sup>-1</sup> was overwritten as r<sub>ij</sub><sup>-1</sup> ® ar<sub>ij</sub><sup>-1</sup>, and used “a” as input. The most important finding beside that the repulsion energy V<sub>ee</sub>(a) is a quasi-linear function of “a”, is that the extended 1<sup>st</sup> Hohenberg-Kohn theorem (Y<sub>0</sub>(a=1) Û H<sub>ne</sub> Û Y<sub>0</sub>(a=0)) and its consequences in relation to “a”. The latter allows an algebraic transfer from the simpler solution of case a=0 (where the single Slater determinant is the accurate form) to the realistic wanted case a=1. Moreover, we have generalized the emblematic theorems in the title in relation to the coupling strength parameter. </p>
Quasi-Linear Buildup of Coulomb Integrals via the Coupling Strength Parameter in the Non-Relativistic Electronic Schrödinger Equation
