Search

A single-particle pseudo-potential that splits the effect of the electron–electron repulsive potential of Helium (He) atom into two noninteracting 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 noninteracting particles under the calculated potential is found to be − 2.861 68, which is the Hartree–Fock limit for the He atom.

A numerical solution of the Kohn–Sham (KS) differential equation within the local density approximation is presented. The present method involves solving for the Hartree potential from its differential form which is the Poisson equation. Radial differential equations are derived for closed-shell atoms and are solved as initial value problems. A self-consistent procedure for solving the resulting radial KS equations based on our new algorithm for solving differential equations is also discussed. Numerical tests are done on the Helium atom and comparison with results obtained from spectral exponential and Gaussian type basis functions are shown.