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In this work the Perdew–Burke–Ernzerhof exchange functional coupled with the exact-exchange is applied on closed-shell atoms confined by impenetrable and penetrable walls. When the Hartree–Fock method is used as the reference, one-parameter hybrid exchange functionals qualitatively give a good description of atoms enclosed by a sphere surrounded by an infinite potential. For atoms confined by a finite potential, however, the same hybrid exchange functionals predict results appreciably different to the reference for small confinement radii. The main reason of this result is the Laplacian of the electron density involved in the exchange potential and, consequently, the effective potential diverges at the nucleus, which cannot be remedied by the inclusion of the exact exchange. Localization and delocalization exhibited by the electron density are used as arguments to explain the differences found between various exchange functionals tested in this article. We show that generalized-gradient functionals are unable to give a good description of the corresponding system when the electron density is squeezed by finite potentials over small regions and how one-parameter hybrid exchange functionals alleviate some of the encountered problems. Although the model of the confined atom is extremely simple, it can reproduce some features predicted by sophisticated methods of electronic structure designed for crystal systems, therefore, this model can be useful to test exchange functionals defined within the Kohn–Sham density functional-theory.

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Spatial confinements induce localization or delocalization on the electron density in atoms and molecules, and the hydrogen atom is not the exception to these results. In previous works, this system has been confined by an infinite and a finite potential where the wave-function exhibits an

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.