Revised Thomas-Fermi Approximation for Singular Potentials

TL;DR

This paper introduces a regularized Thomas-Fermi approximation for singular potentials in many-fermion systems, ensuring non-singular densities and extending finite-temperature regularization methods.

Contribution

It develops a formal, exact mapping to regularize the Thomas-Fermi approximation for singular potentials at finite temperatures.

Findings

Density becomes non-singular with the new regularized potential.

The approach generalizes previous ground-state regularizations to finite temperatures.

Provides approximate expressions for the regularized potential at different temperature regimes.

Abstract

Approximations to the many-fermion free energy density functional that include the Thomas-Fermi (TF) form for the non-interacting part lead to singular densities for singular external potentials (e.g. attractive Coulomb). This limitation of the TF approximation is addressed here by a formal map of the exact Euler equation for the density onto an equivalent TF form characterized by a modified Kohn-Sham potential. It is shown to be a "regularized" version of the Kohn-Sham potential, tempered by convolution with a finite-temperature response function. The resulting density is non-singular, with the equilibrium properties obtained from the total free energy functional evaluated at this density. This new representation is formally exact. Approximate expressions for the regularized potential are given to leading order in a non-locality parameter and the limiting behavior at high and low temperatures is described. The non-interacting part of the free energy in this approximation is the usual Thomas-Fermi functional. These results generalize and extend to finite temperatures the ground-state regularization by Parr and Ghosh (Proc. Nat. Acad. Sci. 83, 3577 (1986)) and by Pratt, Hoffman, and Harris (J. Chem. Phys. 92, 1818 (1988)) and formally systematize the finite-temperature regularization given by the latter authors.