# First Order Expansion in the Semiclassical Limit of the Levy–Lieb Functional

## TL;DR

This paper proves a first-order expansion of the Levy–Lieb functional in the semiclassical limit, a key concept in Density Functional Theory.

## Contribution

The novel contribution is proving the first-order expansion conjecture using optimal transport and Dirichlet penalization methods.

## Key findings

- A general asymptotic lower bound is established using zero point oscillation functional.
- An asymptotic upper bound is proven for two electrons in one dimension.
- The problem is interpreted through singular perturbation of optimal transport.

## Abstract

We prove the conjectured first order expansion of the Levy–Lieb functional in the semiclassical limit, arising from Density Functional Theory (DFT). In particular, we prove a general asymptotic first order lower bound in terms of the zero point oscillation functional and the corresponding asymptotic upper bound in the case of two electrons in one dimension. This is accomplished by interpreting the problem as the singular perturbation of an Optimal Transport problem via a Dirichlet penalization.