Local Exchange Potentials for Electronic Structure Calculations

TL;DR

This paper analyzes various local approximations to the nonlocal Hartree-Fock exchange operator, exploring their mathematical properties, variational formulations, and solutions to associated equations in electronic structure calculations.

Contribution

It provides a comprehensive mathematical analysis of local exchange potentials, including derivations, properties, and existence proofs for solutions to key equations.

Findings

Slater, KLI, CEDA, and ELP potentials are solutions to variational problems.

Rigorous derivation of the integral OEP equation.

Existence of solutions to nonlinear PDE systems for the Slater approximation.

Abstract

The Hartree-Fock exchange operator is an integral operator arising in the Hartree-Fock method and replaced by a multiplicative operator (a local potential) in Kohn-Sham density functional theory. 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) and common energy-denominator approximations (CEDA) to the OEP, and the effective local potential (ELP). In particular, we show that the Slater, KLI, CEDA potentials and the ELP can all be defined as solutions to certain variational problems. We also provide a rigorous derivation of the integral OEP equation and establish the existence of a solution to a system of coupled nonlinear partial differential equations defining the Slater approximation to the Hartree-Fock equations.