Vol. 13, No. 2, 2018

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ISSN: 2157-5452 (e-only)
ISSN: 1559-3940 (print)
A numerical study of the extended Kohn–Sham ground states of atoms

Eric Cancès and Nahia Mourad

Vol. 13 (2018), No. 2, 139–188
Abstract

In this article, we consider the extended Kohn–Sham model for atoms subjected to cylindrically symmetric external potentials. The variational approximation of the model and the construction of appropriate discretization spaces are detailed together with the algorithm to solve the discretized Kohn–Sham equations used in our code. Using this code, we compute the occupied and unoccupied energy levels of all the atoms of the first four rows of the periodic table for the reduced Hartree–Fock (rHF) and the extended Kohn–Sham Xα models. These results allow us to test numerically the assumptions on the negative spectra of atomic rHF and Kohn–Sham Hamiltonians used in our previous theoretical works on density functional perturbation theory and pseudopotentials. Interestingly, we observe accidental degeneracies between s and d shells or between p and d shells at the Fermi level of some atoms. We also consider the case of an atom subjected to a uniform electric field. For various magnitudes of the electric field, we compute the response of the density of the carbon atom confined in a large ball with Dirichlet boundary conditions, and we check that, in the limit of small electric fields, the results agree with the ones obtained with first-order density functional perturbation theory.

Keywords
density functional theory, electronic structure of atoms, extended Kohn–Sham model, Stark effect
Mathematical Subject Classification 2010
Primary: 35P30, 35Q40, 65Z05, 81V45
Milestones
Received: 7 February 2017
Revised: 25 February 2018
Accepted: 25 March 2018
Published: 5 June 2018
Authors
Eric Cancès
Centre d’Enseignement et de Recherche en Mathématiques et Calcul Scientifique
Ecole des Ponts ParisTech and Institut National de Recherche en Informatique et en Automatique Paris
Université Paris-Est
Marne-la-Vallée
France
Nahia Mourad
Centre d’Enseignement et de Recherche en Mathématiques et Calcul Scientifique
Ecole des Ponts ParisTech
Université Paris-Est
Marne-la-Vallée
France