Hohenberg-Kohn Theorems in Electrostatic and Uniform Magnetostatic Fields

TL;DR

This paper extends the Hohenberg-Kohn theorems to include electrostatic and uniform magnetostatic fields, establishing bijective relationships between potentials and densities for electrons with and without spin.

Contribution

It generalizes the HK theorems to systems with magnetic fields and fixed angular momentum, providing new variational principles and explicit proofs for these cases.

Findings

Proved HK theorems for electrostatic fields with fixed electron number.

Extended HK theorems to include uniform magnetostatic fields with fixed angular momentum.

Developed variational principles for these extended theorems.

Abstract

The Hohenberg-Kohn (HK) theorems of bijectivity between the external scalar potential and the gauge invariant nondegenerate ground state density, and the consequent Euler variational principle for the density, are proved for arbitrary electrostatic field and the constraint of fixed electron number. The HK theorems are generalized for spinless electrons to the added presence of an external uniform magnetostatic field by introducing the new constraint of fixed canonical orbital angular momentum. Thereby a bijective relationship between the external scalar and vector potentials, and the gauge invariant nondegenerate ground state density and physical current density, is proved. A corresponding Euler variational principle in terms of these densities is also developed. These theorems are further generalized to electrons with spin by imposing the added constraint of fixed canonical orbital and spin angular momentum. The proofs differ from the original HK proof, and explicitly account for the many-to-one relationship between the potentials and the nondegenerate ground state wave function.