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Eric Cancès; Gabriel Stoltz; Gustavo E. Scuseria; Viktor N. Staroverov; Ernest R. Davidson
Local Exchange Potentials for Electronic Structure Calculations
MathS In Action, 2 no. 1 (2009), p. 1-42, doi: 10.5802/msia.2
Article PDF | Analyses MR 2520849
Class. Math.: 35P30, 81Q05, 35J60
Mots clés: Hartree-Fock model, Density Functional theory, nonlinear eigenvalue problem

Résumé - Abstract

The Hartree-Fock exchange operator is an integral operator arising in the Hartree-Fock model as well as in some instances of the density functional theory. In a number of applications, it is convenient to approximate this integral operator by a multiplication operator, i.e. by a local potential. This article presents a detailed analysis of the mathematical properties of various local approximations to the nonlocal Hartree-Fock exchange operator including the Slater potential, the optimized effective potential (OEP), the Krieger-Li-Iafrate (KLI) approximation and the common-energy denominator approximation (CEDA) to the OEP, and the effective local potential (ELP). In particular, we show that the Slater, KLI, CEDA and ELP potentials all can be defined as solutions of certain variational problems, and we provide a rigorous derivation of the OEP integral equation. We also establish an existence result for a coupled system of nonlinear partial differential equations introduced by Slater to approximate the Hartree-Fock equations.

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