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

Abstract

In a recent work, Bindini and De Pascale have introduced a regularization of N -particle symmetric probabilities which preserves their one-particle marginals. In this short note, we extend their construction to mixed quantum fermionic states. This enables us to prove the convergence of the Levy-Lieb functional in Density Functional Theory, to the corresponding multi-marginal optimal transport in the semiclassical limit. Our result holds for mixed states of any particle number N , with or without spin.

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.,…”

Functional derivative of the zero-point-energy functional from the strong-interaction limit of density-functional theory

“…In Fig. 2, we provide the profile of δF ZPE /δρ(x) for the test densities (39). The plots show that the shape of the curve can vary drastically depending on the density chosen.…”

“…In particular, in all the densities we chose (excluding ρ 3 ) the functional derivative Figure 2. Functional derivative as from (35) for the first three densities (39). Hartree atomic units.…”

Convergence of Levy–Lieb to Thomas–Fermi density functional

“…which can be proved rigorously. In fact, when w is the Coulomb potential in R 3 , the lower bound in (16) is a direct consequence of the Lieb-Oxford inequality [22] I N (f ) (17) and the upper bound in (16) can be achieved easily by choosing a Slater determinant (a wave function of the form Ψ N = u 1 ∧ u 2 ∧ ... ∧ u N ) with the density N f . The proof of (16) for more general w can be extracted from the proof of our main result below.…”