## 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

The extended non-relativistic electronic
Hamiltonian, H_{Ñ}+ H_{ne}+ aH_{ee}, 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_{ee})
in total electronic energy – apart from virial theorem and the highly detailed
and well-known algorithm for V_{ee}, 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_{ij}^{-1}
was overwritten as r_{ij}^{-1} ® ar_{ij}^{-1}, and used “a” as input. The most important finding
beside that the repulsion energy V_{ee}(a) is a quasi-linear function
of “a”, is that the extended 1^{st} Hohenberg-Kohn theorem (Y_{0}(a=1) Û H_{ne} Û Y_{0}(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.