Local Effective Hartree-Fock Potentials Obtained by the Depurated Inversion Method

In this work we show the results of a numerical experiment performed on the Hartree-Fock (HF) wave functions in order to understand the relationship between the positions of the orbital nodes and the inflection points (zeros of their second derivative). This analysis is equivalent to investigating the existence of a physical one-electron local potential representing the interactions between the electrons. We found that with successive improvements in the quality of the numerical methods, the nodes and the inflection points systematically become closer. When the nodes coincide exactly with the inflection points, the existence of an effective local potential would be proven. However, this requirement cannot be fulfilled unless an explicit constraint (missing in the standard method) is incorporated into the HF procedure. The depurated inversion method (DIM) was devised to obtain detailed n/-orbital potentials for atoms and molecules. The method is based on the inversion of Kohn-Sham-type equations, followed by a further careful optimization which eliminates singularities and also ensures the fulfillment of the appropriate boundary conditions. The orbitals resulting from these potentials have their internal inflection points located exactly at the nodes. In this way, the DIM can be employed to obtain effective potentials that accurately reproduce the HF orbitals.