Testing one-parameter hybrid exchange functionals in confined atomic systems

Abstract

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.

Shannon Entropy in Confined He-Like Ions within a Density Functional Formalism

“…Recently, in a penetrable confinement calculation within HF, some results on S r have been presented [72]. The same has also been calculated from a DFT-based study with hybrid exchange functional [52]. Results for He in free-limit from both these studies, presented in footnote show quite decent agreement with ours.…”

“…DFT calculation[64], using the BLYP functional 2. For the free atom, using N-normalized HF density; S r : 4.01[72], 4.06[91], 4.0100[92], 4.0107[52]; S p : 6.45[91]; S t : 10.52[91].Our exchange-only (X-only) results are 4.00, 6.45, and 10.52 respectively 3. Correlated S r values in the free He atom: (i) unit-normalized density: 2.7051028[93], 2.705[94]; (ii) N-normalized density: 4.40106; (variational Monte Carlo), 4.0256 (diffusion Monte Carlo)[92].…”

“…local density approximation (LDA) [47] and Becke-88 exchange potential [50]), the desired Kohn-Sham (KS) equation was solved satisfying the Dirichlet boundary condition for many-electron systems via the numerical shooting method [51]. The usefulness of a one-parameter hybrid exchange functional (including a fraction of exact exchange and the Perdew-Burke-Ernzerhof functional) for treatment of confined atoms has been presented recently [52]. In another attempt, ground and 1s2s 3 S, 1 S states of a confined He atom were reported [53] taking into account the LDA-approximated exchange-correlation (XC) (with Perdew-Wang parametrization for correlation [54]) with and without self-interaction correction.…”

“…In confined two-electron isoelectronic series (H − , He, Li + , Be 2+ ), S has been reported for only ground states [64] by means of BLYP calculations. Some reports [52,[68][69][70][71][72][73] are available on S in penetrable confinement in atoms. For example, it was observed [68] that, in CHA, up to a certain value of r c , S r decreases with r c .…”

Shannon Entropy for the Hydrogen Atom Confined by Four Different Potentials

“…This model has been used to mimic the environment where an atom is immersed [1,[3][4][5][6] or to test the quality of theoretical models that provide the electronic structure of atomic systems [7,8]. For example, the performance of exchange-correlation functionals has been tested when an atom is submitted to extreme conditions [9,10] or sophisticate methods have been tested over atoms with a few electrons to deliver accurate total energies or electron densities [11,12]. From these applications, unexpected results have been found for some atoms confined by different potentials.…”