## Probability Seminar: Codina Cotar

- Date: 04/18/2012
- Time: 15:00

University of British Columbia

Density functional theory and optimal transport with Coulomb cost

Abstract

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 prove that for the case

with $N=2$ electrons this problem has a unique solution given by an

optimal map; moreover we derive an explicit formula for the optimal map

in the case when $\rho$ is radially symmetric (note: atomic ground state

densities are radially symmetric for many atoms such as He, Li, N, Ne,

Na, Mg, Cu).

In my talk I focus on how to deal with its main mathematical novelties

(cost decreases with distance; cost has a singularity on the diagonal). I

also discus the derivation of the Coulombic OT problem from the

many-electron Schroedinger equation for the case with $N\ge 3$

electrons, and give some results and explicit solutions for the

many-marginals OT problem.

Joint works with Gero Friesecke (TU Munich) and Claudia Klueppelberg (TU Munich).