Revisiting density-functional theory of the total current density

TL;DR

This paper critically examines a previous formulation of density-functional theory involving total current density, revealing fundamental flaws and clarifying the theoretical framework.

Contribution

It reinterprets Diener's variational approach, proves its limitations, and clarifies the correct theoretical structure for total current density in density-functional theory.

Findings

Diener's formulation cannot reproduce the correct ground-state energy.

The proposed Hohenberg-Kohn map contains an irreparable mistake.

The paper clarifies the theoretical basis for total current density in DFT.

Abstract

Density-functional theory requires an extra variable besides the electron density in order to properly incorporate magnetic-field effects. In a time-dependent setting, the gauge-invariant, total current density takes that role. A peculiar feature of the static ground-state setting is, however, that the gauge-dependent paramagnetic current density appears as the additional variable instead. An alternative, exact reformulation in terms of the total current density has long been sought but to date a work by Diener is the only available candidate. In that work, an unorthodox variational principle was used to establish a ground-state density-functional theory of the total current density as well as an accompanying Hohenberg-Kohn-like result. We here reinterpret and clarify Diener's formulation based on a maximin variational principle. Using simple facts about convexity implied by the resulting variational expressions, we prove that Diener's formulation is unfortunately not capable of reproducing the correct ground-state energy and, furthermore, that the suggested construction of a Hohenberg-Kohn map contains an irreparable mistake.