# Optimal transport with Coulomb cost and the semiclassical limit of Density Functional Theory

## TL;DR

This paper investigates the semiclassical limit of the Hohenberg-Kohn functional in Density Functional Theory for Coulomb systems, establishing a connection with multimarginal optimal transport problems for bosonic and small fermionic systems.

## Contribution

It proves that the semiclassical limit corresponds to a multimarginal optimal transport problem with Coulomb cost for bosonic systems and for fermionic systems with 2 or 3 particles.

## Key findings

- Limit is the multimarginal optimal transport with Coulomb cost for bosons.
- Limit is the multimarginal optimal transport with Coulomb cost for fermions with 2 or 3 particles.
- Techniques from optimal transportation theory are used in the proof.

## Abstract

We present some progress in the direction of determining the semiclassical limit of the Hoenberg-Kohn universal functional in Density Functional Theory for Coulomb systems. In particular we give a proof of the fact that for Bosonic systems with an arbitrary number of particles the limit is the multimarginal optimal transport problem with Coulomb cost and that the same holds for Fermionic systems with 2 or 3 particles. Comparisons with previous results are reported . The approach is based on some techniques from the optimal transportation theory.

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Source: https://tomesphere.com/paper/1702.04957