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

Abstract

For a many-electron system, whether the particle density ρ(r) and the total current density j(r) are sufficient to determine the one-body potential V (r) and vector potential A(r), is still an open question. For the one-electron case, a Hohenberg-Kohn theorem exists formulated with the total current density. Here we show that the generalized Hohenberg-Kohn energy functionalcan be minimal for densities that are not the ground-state densities of the fixed potentials V 0 and A 0 . Furthermore, for an arbitrary number of electrons and under the assumption that a Hohenberg-Kohn theorem exists formulated with ρ and j, we show that a variational principle for Total Current Density Functional Theory as that of HohenbergKohn for Density Functional Theory does not exist. The reason is that the assumed map from densities to the vector potential, written (ρ, j) → A(ρ, j; r), enters explicitly in E V 0 ,A 0 (ρ, j).

Uniform magnetic fields in density-functional theory

“…The existence of a HK functional of the physical current density thus cannot be excluded. However, such a functional would require a different approach than in standard DFT since it would not have a straightforward variation principle [34,50,51].…”

“…The proof of the above theorem follows every step in Sec.III.A of Ref. 51, which still applies and where we let the minimizing L l = ψ k |L|ψ k . Note that ψ k and the infinitely many current densities {j l } are denoted ψ 0 and {j ε } ε>0 , respectively, in Ref.…”

Kohn–Sham Theory with Paramagnetic Currents: Compatibility and Functional Differentiability

“…Recent work in current-density-functional theory (CDFT) has been devoted to the extension of the HK theorem, the HK variational principle, and the KS iteration scheme to include current densities, [12][13][14] as well as to highlight the complexity of such a generalization. 3,15,16 Other approaches are feasible as well, e.g., the magnetic-field density-functional theory (BDFT) of Grace and Harris, 17 where a semi-universal functional is employed instead. There exists also a convexified formulation, in which BDFT and paramagnetic CDFT are related to each other by partial Legendre-Fenchel transformations.…”

“…3,15 However, even if such a result could be shown, a HK variational principle does not exist for the total current density. 16 Circumventing these problems may require the Maxwell-Schrödinger variational principle in place of the standard one. 23 For the CDFT that makes use of the paramagnetic current density, it is well-known that there are counterexamples that rule out any analogue of the HK theorem.…”

Force balance approach for advanced approximations in density functional theories

“…14 and references therein), there is an ambiguity concerning which current-density quantity one should use 15,16 . Indeed, if the physical current density is used in the ground-state case, the usual variational approach is not even applicable and a definition of the xc potential in the usual way is not available any more 17 . To overcome this restriction one commonly employs the paramagnetic current density instead to set up a current-density-functional theory (CDFT) for ground states, which makes the theory gauge dependent 18 .…”