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

Abstract

For a many-electron system, whether the particle density $\ensuremath{\rho}(\mathbf{r})$ and the total current density $\mathbf{j}(\mathbf{r})$ are sufficent to determine the one-body potential $V(\mathbf{r})$ and vector potential $A(\mathbf{r})$ 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 functional ${\mathcal{E}}_{{V}_{0},{\mathbf{A}}_{0}}(\ensuremath{\rho},\mathbf{j})=\ensuremath{\langle}\ensuremath{\psi}(\ensuremath{\rho},\mathbf{j}),H({V}_{0},{\mathbf{A}}_{0})\ensuremath{\psi}(\ensuremath{\rho},\mathbf{j})\ensuremath{\rangle}$ can be minimal for densities that are not the ground-state densities of the fixed potentials ${V}_{0}$ and ${\mathbf{A}}_{0}$. Furthermore, for an arbitrary number of electrons and under the assumption that a Hohenberg-Kohn theorem exists formulated with $\ensuremath{\rho}$ and $\mathbf{j}$, we discuss the possibility of a variational principle in total current-density-functional theory such as that of Hohenberg-Kohn.