Density functional theory and optimal transportation with Coulomb cost

TL;DR

This paper connects density functional theory with optimal transport theory, revealing new forms of exchange-correlation functionals and providing mathematical proofs of optimal transport maps for electron densities.

Contribution

It introduces a novel perspective on exchange-correlation functionals using optimal transport, proving existence and uniqueness of optimal maps for any electron number and density.

Findings

Exact exchange-correlation functional reduces to an optimal transport-based functional in the two-electron semiclassical limit.

Proved existence and uniqueness of optimal transport maps for any number of electrons and densities.

Explicitly determined the optimal map for radially symmetric densities.

Abstract

We present here novel insight into exchange-correlation functionals in density functional theory, based on the viewpoint of optimal transport. We show that in the case of two electrons and in the semiclassical limit, the exact exchange-correlation functional reduces to a very interesting functional of novel form, which depends on an optimal transport map $T$ associated with a given density $\rho$. Since the above limit is strongly correlated, the limit functional yields insight into electron correlations. We prove the existence and uniqueness of such an optimal map for any number of electrons and each $\rho$, and determine the map explicitly in the case when $\rho$ is radially symmetric.