Hohenberg-Kohn Theorem Including Electron Spin in the Presence of a Magnetostatic Field

We consider a system of $N$ electrons in the presence of an external electrostatic ${\mathbf{\cal{E}}}({\bf{r}}) = - {\mathbf{\nabla}} v({\bf{r}})$ and magnetostatic ${\bf{B}} ({\bf{r}}) = {\mathbf{\nabla}} \times {\bf{A}} ({\bf{r}})$ fields, and include the interaction of the latter with both the orbital and spin angular momentum. The relationship between the potentials $\{ v ({\bf{r}}), {\bf{A}} ({\bf{r}}) \}$ and the nondegenerate ground state $\Psi$ is many-to-one. Explicitly accounting for this, we prove as in the case\footnote{XYP and VS, IJQC 2013, DOI: 10.1002/qua.24532} when only the orbital interaction is considered, that for $\Psi$ real, there is the one-to-one relationship: $\{ v ({\bf{r}}), {\bf{A}} ({\bf{r}}) \} \leftrightarrow \{ \rho ({\bf{r}}), {\bf{j}} ({\bf{r}}) \}$, where $\rho ({\bf{r}})$ and ${\bf{j}} ({\bf{r}})$ are the corresponding density $\rho ({\bf{r}})$ and physical current density ${\bf{j}} ({\bf{r}})$. Thus, $\{ \rho ({\bf{r}}), {\bf{j}} ({\bf{r}}) \}$ are the basic variables of the system. At present, except for the one electron system, no proof of bijectivity exists for the case of $\Psi$ complex.