Hohenberg–Kohn theorems in the presence of magnetic field

Abstract

In this article, we examine Hohenberg–Kohn theorems for Current Density Functional Theory, that is, generalizations of the classical Hohenberg–Kohn theorem that includes both electric and magnetic fields. In the Vignale and Rasolt formulation (Vignale and Rasolt, Phys. Rev. Lett. 1987, 59, 2360), which uses the paramagnetic current density, we address the issue of degenerate ground states and prove that the ensemble‐representable particle and paramagnetic current density determine the degenerate ground states. For the formulation that uses the total current density, we note that the proof suggested by Diener (Diener, J. Phys.: Condens. Matter. 1991, 3, 9417) is unfortunately not correct. Furthermore, we give a proof that the magnetic field and the ensemble‐representable particle density determine the scalar and vector potentials up to a gauge transformation. This generalizes the result of Grayce and Harris (Grayce and Harris, Phys. Rev. A 1994, 50, 3089) to the case of degenerate ground states. We moreover prove the existence of a positive wavefunction that is the ground state of infinitely many different Hamiltonians. © 2014 Wiley Periodicals, Inc.

Nonexistence of a Hohenberg-Kohn variational principle in total current-density-functional theory

“…For N = 1, a Hohenberg-Kohn theorem exists formulated with the total current density [5,6], i.e., ρ(r) and j(r) determine V (r) and A(r) up to a gauge transformation. In particular, A(r) = a(ρ, j; r) − ∇χ(r) for some function χ(r).…”

“…Since the proofs in [3] and [4] have been shown to contain errors (cf. [5,6]), the existence of such a theorem remains an open question except for the one-electron case. For a system with only one electron, however, it can be shown that the particle density and the total current density determine the potentials up to a gauge transformation [5,6].…”

“…[5,6]), the existence of such a theorem remains an open question except for the one-electron case. For a system with only one electron, however, it can be shown that the particle density and the total current density determine the potentials up to a gauge transformation [5,6]. Moreover, it is well-known that the same statement made with the paramagnetic current density in place of the total current density is not true.…”

“…• For N = 1 and fixed potentials V 0 and A 0 , the energy functional E V 0 ,A 0 (ρ, j) is welldefined since ρ(r) and j(r) determine V 0 (r) and A 0 (r) up to a gauge transformation [5,6]. However, E V 0 ,A 0 (ρ, j) can be minimized by other densities then the ground-state densities of H(V 0 , A 0 ).…”

Density-functional theory for internal magnetic fields

“…The most conservative extensions to date are the magnetic-field density-functional theory (BDFT) due to Grayce and Harris [5] as well as the paramagnetic current density functional theory (CDFT) due to Vignale and Rasolt [6]. Less clear-cut are the attempts to formulate current density functional theories based on the physical current density due to Diener [7] as well as Pan and Sahni [8], which both suffer from gaps or mistakes in the proposed proofs of central results [9][10][11][12][13][14]. Another possible route to incorporate external magnetic fields is via a relativistic density-functional theory based on the Dirac equation, which is likely to involve the physical current density as a basic variable.…”