Comparative Study Between the Hartree-Fock and Kohn-Sham Models for the Lowest Singlet and Triplet States of the Confined Helium Atom

Abstract

In this work, the exchange-only Kohn-Sham (KS) model is compared with the Hartree-Fock (HF) model for the lowest singlet and triplet states of the confined helium atom. The confinement on this atom is obtained by imposing Dirichlet boundary conditions. The HF equations are solved according to the Roothaan approach coupled with modified Slater Type Orbitals, where the boundary conditions are imposed. The solution of the KS equations is obtained with a numerical code adapted to work with this sort of boundary condition. For the KS exchange functional we use the local density approximation corrected by the Perdew and Zunger self-interaction approximation. It is shown that while the Perdew and Zunger proposal of incorporating the self-interaction correction is quite appropriate for the lowest singlet state of the helium atom this approach shows large discrepancies with regard to the HF method for the lowest triplet state, particularly in regions of reduced confinement radii. Thus, when electrons of the same spin are confined within small regions, the self-interaction correction scheme of Perdew and Zunger becomes inappropriate. The HF results reported in this work and those obtained with a Hylleraas expansion indicate that the correlation energy for the lowest singlet and triplet system states is almost constant with regard to the confinement radius. For the open-shell system the correlation energy is quite small and consequently the HF model can give a good description of this system.