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The non-relativistic electronic Hamiltonian, Hkin + Hne + aHee, is linear incoupling strength parameter (a), but its eigenvalues (electronic energies) haveonly quasi-linear dependence on it. Detailed analysis is given on theparticipation of electron-electron repulsion energy (Vee) in total electronicenergy (Etotal electr,k) in addition to the well-known virial theorem andstandard algorithm for vee(a=1)=Vee calculated during the standard- and postHF-SCF routines. Using a particular modification in the SCF part of theGaussian package, we have analyzed the ground state solutions via the parametera. Technically, with a single line in the SCF algorithm, operator was changedas 1/rij-> a/rij with input a. The most important findings are, 1, vee(a) isquasi-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 latterallows an algebraic transfer from the simpler solution of case a=0 (where thesingle 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 strengthparameter.

Quasi-linear buildup of Coulomb integrals via the coupling strength parameter in the non-relativistic electronic Schrodinger equation