Optimal transport with Coulomb cost and the semiclassical limit of density functional theory

Abstract

We present some progress in the direction of determining the semiclassical limit of the Hoenberg-Kohn universal functional in Density Functional Theory for Coulomb systems. In particular we give a proof of the fact that for Bosonic systems with an arbitrary number of particles the limit is the multimarginal optimal transport problem with Coulomb cost and that the same holds for Fermionic systems with 2 or 3 particles. Comparisons with previous results are reported . The approach is based on some techniques from the optimal transportation theory.

Semi-classical limit of the Levy–Lieb functional in Density Functional Theory

“…An interesting question, important for applications in DFT, is to approximate P by a regular Nparticle probability density P ε with the same density ρ Pε = ρ P . A more challenging problem, considered first in [6,1] for N = 2, 3 and solved for all N 2 in this note, is to find a fermionic quantum state Γ ε with the same density ρ Γε = ρ P and a controlled kinetic energy.…”

“…Consider a radial function χ ∈ C ∞ c (R d , R) with support in the unit ball of R d , such that´R d χ 2 = 1, and denote χ ε (x) = ε −d/2 χ(x/ε). In [1], Bindini and De Pascale have introduced the following elegant regularization 2 ρ P * χ 2 ε (z k ) dP(y 1 , ..., y N ) dz 1 · · · dz N .…”

Marginals with Finite Repulsive Cost

“…When the semi-classical limit is considered, as already proved in [2,7,8,16], the stationary states reach the minimum of potential energy, i.e.,…”

Convergence of Levy–Lieb to Thomas–Fermi density functional

“…If we do not completely ignore the kinetic part, but take h → 0 and fix N , then the Levy-Lieb functional functional E N (f ) converges to the interaction functional I N (f ) in (15). Results of this kind can be found in remarkable recent works [4,2,17].…”