Revisiting density-functional theory of the total current density

Markus Penz,Andre Laestadius,Erik I Tellgren

doi:10.1088/1361-648x/abf784

Markus Penz, Andre Laestadius + Show 1 more

Open Access

https://doi.org/10.1088/1361-648x/abf784

Copy DOI

Abstract

Density-functional theory (DFT) 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 DFT 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.

Highlights

The Hohenberg–Kohn theorem is commonly regarded as the theoretical foundation of density-functional theory (DFT)
The underlying crucial assumptions have been clarified by a reformulation in terms of a maximin principle
As Diener’s construction employs a nonstandard variational principle, it avoids some of the usual difficulties with the total current density as a variational parameter

Summary

INTRODUCTION

The Hohenberg–Kohn theorem is commonly regarded as the theoretical foundation of density-functional theory (DFT). A peculiar feature of CDFT is that it is the paramagnetic current density, and not the gauge-invariant total current density, that enters as a basic variable This leaves a disconnect between ground-state CDFT and the time-dependent version of the theory, which is naturally formulated using the total current density [7, 8]. A work by Diener [20] is the only candidate for a density-functional theory of the total current that does not modify the underlying Schrodinger equation Logical gaps in his formulation have been identified before [12, 13], one specific criticism was mistaken (Proposition 8 in Ref. 13, which we correct below at the end of Sec. IV). Our analysis is very general and applies even if previously identified issues [12, 13] could somehow be resolved

PRELIMINARIES

DIENER’S FORMULATION AS A MAXIMIN VARIATIONAL PRINCIPLE

DIENER’S ORIGINAL FORMULATION

CONCLUSIONS