Exact Density Functional for the Non-Relativistic Particle Energy in the Local External Field

TL;DR

This paper derives exact density functionals for non-relativistic particles in local external fields and discusses limitations of the Hohenberg-Kohn lemma for systems with more than two electrons, highlighting differences between fermions and bosons.

Contribution

It provides exact inhomogeneous density functionals for particle energies and reveals the non-existence of a universal density functional for systems with more than two noninteracting electrons.

Findings

Exact density functionals derived from the Schrödinger equation.

Hohenberg-Kohn lemma does not hold for systems with more than two electrons.

Density functional theory remains valid for noninteracting bosons due to Bose condensation.

Abstract

Based on the Schrodinger equation, exact expressions for the non-relativistic particle energy in the local external field and the external field potential are derived as inhomogeneous density functionals. On this basis, it is shown that, when considering more than two noninteracting electrons, the energy of such a system cannot be an inhomogeneous density functional. This means that the Hohenberg-Kohn lemma which assert that in the ground state to each inhomogeneous density corresponds only one potential of the external field cannot be a justification of the existence of the universal density functional in the general case. At the same time, statements of the density functional theory remain valid when considering any number of noninteracting ground-state bosons due to the Bose condensation effect.