Current Densities in Density Functional Theory

TL;DR

This paper investigates the conditions under which a given density and current density can be simultaneously realized by a determinantal wave function in density functional theory, providing solutions and bounds in specific cases.

Contribution

It offers explicit solutions and energy bounds for realizing density and current density pairs, especially when the velocity field is curl free, and explores cases with non curl free fields.

Findings

Explicit solutions for curl free velocity fields.

Finite energy solutions exist for N≥4 with non curl free fields.

Examples of realizable and non-realizable current densities for N=2.

Abstract

It is well known that any given density rho(x)can be realized by a determinantal wave function for N particles. The question addressed here is whether any given density rho(x) and current density j(x) can be simultaneously realized by a (finite kinetic energy) determinantal wave function. In case the velocity field v(x) =j(x)/rho(x) is curl free, we provide a solution for all N, and we provide an explicit upper bound for the energy. If the velocity field is not curl free, there is a finite energy solution for all N\geq 4, but we do not provide an explicit energy bound in this case. For N=2 we provide an example of a non curl free velocity field for which there is a solution, and an example for which there is no solution. The case $N=3 with a non curl free velocity field is left open.