  The non-relativistic electronic Hamiltonian, Hkin + Hne + aHee, is linear in
coupling strength parameter (a), but its eigenvalues (electronic energies) have
only quasi-linear dependence on it. Detailed analysis is given on the
participation of electron-electron repulsion energy (Vee) in total electronic
energy (Etotal electr,k) in addition to the well-known virial theorem and
standard algorithm for vee(a=1)=Vee calculated during the standard- and post
HF-SCF routines. Using a particular modification in the SCF part of the
Gaussian package, we have analyzed the ground state solutions via the parameter
a. Technically, with a single line in the SCF algorithm, operator was changed
as 1/rij-> a/rij with input a. The most important findings are, 1, vee(a) is
quasi-linear function of a, 2, the extension of 1st Hohenberg-Kohn theorem
(PSI0(a=1)<=>Hne<=>Y0(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 Y0 is the accurate form) to the physical case a=1.
Moreover, we have generalized the emblematic Hund rule, virial-,
Hohenberg-Kohn- and Koopmans theorems in relation to the coupling strength
parameter.
