Estimating correlation energy and basis set error for Hartree-Fock-SCF calculation by scaling during the SCF subroutine with inserting only a few program lines, using analytical integration and no extra CPU time and no extra disc space

For electronic ground state energy, the HF-SCF procedure minimizes the energy functional 〈S|H∇|S〉 + 〈S|HRr|S〉 + 〈S|Hrr|S〉〉 〈Ψ0|H|Ψ0〉 ≡ Eelectr,0 for a normalized single Slater determinant approximate wavefunction (denoted as S). The H∇, HRr and Hrr are the well known kinetic-, nuclear-electron attraction- and electron-electron repulsion energy operators of the electronic Hamiltonian, H, respectively. The 〈S|H|S〉 can never reach the value Eelectr,0 (variation principle), causing about 1% non-negligible energy error, called correlation energy (Ecorr). We re-correct this error with scaling during the SCF subroutine by minimizing the new functional (1 + kc)〈S|H∇|S〉 + 〈S|HRr|S〉 + (1 + kee)〈S|Hrr|S〉 to estimate Eelectr,0 better. The very flexible kc and kee, were fitted to accurate G3 electronic ground state molecular energies. They negligibly depend on atomic numbers, number of electrons and system size, as well as transferable. Numerical results and tests include HF-SCF and B3LYP with STO-3G and 6-31G** bases.