Density Functional Theory and Optimal Transport with Coulomb Cost

In this talk I explain a promising and previously unnoticed link between electronic structure of molecules and optimal transportation (OT), and I give some first results. The ‘exact’ mathematical model for electronic structure, the many-electron Schroedinger equation, becomes computationally unfeasible for more than a dozen or so electrons. For larger systems, the standard model underlying a huge literature in computational physics/chemistry/materials science is density functional theory (DFT). In DFT, one only computes the single particle density instead of the full many-particle wave function. In order to obtain a closed equation, one needs a closure assumption which expresses the pair density in terms of the single-particle density rho. We show that in the semiclassical Hohenberg-Kohn limit, there holds an exact closure relation, namely the pair density is the solution to a optimal transport problem with Coulomb cost. We give an explicit characterization of the optimal transport problem for systems with 2 electrons and for systems with infinite numbers of electrons.Based on joint works with Gero Friesecke (TU Munich), ClaudiaKlueppelberg (TU Munich), Christian Mendl (TU Munich) andBrendan Pass (UAlberta).