# The Local Density Approximation in Density Functional Theory

## TL;DR

This paper provides a rigorous mathematical justification for the Local Density Approximation in Density Functional Theory, quantifying its accuracy and conditions under which it is valid.

## Contribution

It offers the first rigorous proof and quantitative estimates for the Local Density Approximation's validity in DFT.

## Key findings

- Quantitative error bounds involving gradient terms
- Justification of LDA for flat electron densities
- Conditions for the approximation's accuracy

## 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.