Section: Application Domains

Density Functionnal Theory (DFT)

The precise modeling of electron correlations continues to constitute the major obstacle in developing high-accuracy, low-cost methods for electronic structure computations in molecules and solids. The article [36] sheds a new light on the longstanding problem of how to accurately incorporate electron correlation into DFT, by deriving and analyzing the semiclassical limit of the exact Hohenberg-Kohn functional with the single-particle density $\rho$ held fixed. In this limit, in the case of two electrons, the exact functional reduces to a very interesting functional that depends on an optimal transport map $M$ associated with a given density $\rho$. The limit problem is known in the DFT literature with the optimal transport map being called a correlation function or a co-motion function , but it has not been rigorously derived, and it appears that it has not previously been interpreted as an optimal transport problem. The article [36] thereby links for the first time DFT, which is a large and very active research area in physics and chemistry, to optimal transportation theory, which has recently become a very active area in mathematics. Numerics are still widely open [26] .