Vol. 2, No. 1, 2020

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The local density approximation in density functional theory

Mathieu Lewin, Elliott H. Lieb and Robert Seiringer

Vol. 2 (2020), No. 1, 35–73
DOI: 10.2140/paa.2020.2.35
Abstract

We give the first mathematically rigorous justification of the local density approximation in density functional theory. We provide a quantitative estimate on the difference between the grand-canonical Levy–Lieb energy of a given density (the lowest possible energy of all quantum states having this density) and the integral over the uniform electron gas energy of this density. The error involves gradient terms and justifies the use of the local density approximation in the situation where the density is very flat on sufficiently large regions in space.

Keywords
Schrödinger operators, statistical mechanics, density functional theory, uniform electron gas
Mathematical Subject Classification 2010
Primary: 35Q40, 81V55, 82B10
Milestones
Received: 3 April 2019
Revised: 15 July 2019
Accepted: 20 August 2019
Published: 9 November 2019
Authors
Mathieu Lewin
CNRS & CEREMADE
Université Paris-Dauphine
PSL University
Paris
France
Elliott H. Lieb
Departments of Mathematics and Physics
Princeton University
Princeton, NJ
United States
Robert Seiringer
Institute of Science and Technology Austria
Klosterneuburg
Austria