In this talk, we will present recent mathematical results about the Lieb functional in Density Functional Theory, which could potentially lead the way to interesting numerical methods exploiting the possibilities of exascale architectures. More precisely, the Lieb functional, for a given electronic density, can be viewed as a generalized form of optimal transport problem for which the electronic density plays the role of a marginal. A numerical discretization of this problem can be obtained by imposing afinite number of moment constraints instead of this full marginal constraints. Using the so-called Tchakhaloff's theorem, it can be shown that a minimizer of the approximate Lieb problem can be obtained as a finite-rank projector, the rank of which is at most equal to the number of moment constraints of the problem. In other words, such minimizer has a very sparse structure which can be exploited for numerics.
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