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# Quasi-Linear Buildup of Coulomb Integrals via the Coupling Strength Parameter in the Non-Relativistic Electronic Schrödinger Equation

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 wellknown 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’s rule, virial-, Hohenberg-Kohn- and Koopmans theorems in relation to the coupling strength parameter.