Free-basis-set method to describe the helium atom confined by a spherical box with finite and infinite potentials

Abstract

The finite difference method based on a non-regular mesh is used to solve Hartree–Fock and Kohn–Sham equations for the helium atom confined by finite and infinite potentials. The reliability of this approach is shown when this is contrasted with the Roothaan’s approach, which depends on a basis set and therefore its exponents must be optimized for each confinement imposed over the helium atom. The comparison between our numerical approach and the Roothaan’s approach was done by using total and orbitals energies from the Hartree–Fock method where there are several sources of comparison. By the side of the Kohn–Sham method there are a few published results and consequently the results reported here can be used as benchmark for future comparisons. The electron density, through the Shannon’s entropy, was also used for the comparison between our approach and other reports. This entropy shows that the helium atom confined by an infinite potential can be described almost with any approach, Hartree–Fock or Kohn–Sham give almost same results. This conclusion cannot be applied for finite potential since Hartree–Fock and Kohn–Sham methods present large differences between themselves. This study represents the first step to develop a numerical code free of basis sets to obtain the electronic structure of many-electron atoms.