Density functional theory and optimal transportation with Coulomb cost

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.