One--particle Effective Potential for Helium Atom

A single-particle pseudo--potential that splits the effect of the electron--electron repulsive potential of Helium (He) atom into two non--interacting identical particle potentials is numerically computed. This is done by minimizing the expectation value of the difference between the approximate and exact Hamiltonians over the Hilbert space of He atom. The one--particle potential is expanded in a spatial basis--set which leads to an overdetermined system of linear equation that was solved using a least square approximation. The method involves a self--consistent iterative scheme where a converged solution valid for any state of the atom can be calculated. The total ground state energy for these two non--interacting particles under the calculated potential is found to be -2.861 68, which is the Hartree--Fock limit for the He atom.