---
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.

    *© 2019 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. Final version to appear in *Pure and Applied Analysis*.*
address:
- 'CNRS & CEREMADE, Université Paris-Dauphine, PSL University, 75016 Paris, France'
- 'Departments of Mathematics and Physics, Jadwin Hall, Princeton University, Washington Rd., Princeton, NJ 08544, USA'
- 'IST Austria (Institute of Science and Technology Austria), Am Campus 1, 3400 Klosterneuburg, Austria'
author:
- Mathieu Lewin
- 'Elliott H. Lieb'
- Robert Seiringer
title: The Local Density Approximation in Density Functional Theory
---

Introduction
============

Density Functional Theory (DFT) [@DreGro-90; @ParYan-94; @EngDre-11; @BurWag-13; @PriGroBur-15] is the most efficient approximation of the many-body Schrödinger equation for electrons. 