Generalization of the Hohenberg-Kohn theorem to the case of the presence of a magnetic field

We generalize the HK theorem for the nondegenerate ground state of electrons in an external electrostatic field ${\bf{E}}({\bf{r}}) = -$ {\boldmath $\nabla$} $v ({\bf{r}})$ to the presence of an additional external magnetostatic field ${\bf{B}} ({\bf{r}}) =$ {\boldmath $\nabla$} $\times {\bf{A}} ({\bf{r}})$. We prove that the nondegenerate ground state wave function $\Psi$ is a functional of the ground state density $\rho ({\bf{r}})$, the physical current density ${\bf{j}} ({\bf{r}})$, and a gauge function $\alpha ({\bf{R}})$, with ${\bf{R}} = \{{\bf{r}} \}$. In other words, the basic variables, viz. those that uniquely determine the external potentials $ \{v ({\bf{r}}), {\bf{A}} ({\bf{r}}) \}$, are $\{ \rho ({\bf{r}}), {\bf{j}} ({\bf{r}}) \}$. As the choice of $\alpha ({\bf{R}})$ is arbitrary, it is possible to construct a $\{ \rho ({\bf{r}}), {\bf{j}} ({\bf{r}}) \}$ functional theory, as well as the corresponding Kohn-Sham and quantal density functional theories.