Hohenberg-Kohn theorem including electron spin

Abstract

The Hohenberg-Kohn theorem is generalized to the case of a finite system of $N$ electrons in external electrostatic $\mathcal{E}(\mathbf{r})=\ensuremath{-}\ensuremath{\nabla}v(\mathbf{r})$ and magnetostatic $\mathbf{B}(\mathbf{r})=\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}\mathbf{A}(\mathbf{r})$ fields in which the interaction of the latter with both the orbital and spin angular momentum is considered. For a nondegenerate ground state a bijective relationship is proved between the gauge invariant density $\ensuremath{\rho}(\mathbf{r})$ and physical current density $\mathbf{j}(\mathbf{r})$ and the potentials ${v(\mathbf{r}),\mathbf{A}(\mathbf{r})}$. The possible many-to-one relationship between the potentials ${v(\mathbf{r}),\mathbf{A}(\mathbf{r})}$ and the wave function is explicitly accounted for in the proof. With the knowledge that the basic variables are ${\ensuremath{\rho}(\mathbf{r}),\mathbf{j}(\mathbf{r})}$, and explicitly employing the bijectivity between ${\ensuremath{\rho}(\mathbf{r}),\mathbf{j}(\mathbf{r})}$ and ${v(\mathbf{r}),\mathbf{A}(\mathbf{r})}$, the further extension to $N$-representable densities and degenerate states is achieved via a Percus-Levy-Lieb constrained-search proof. A ${\ensuremath{\rho}(\mathbf{r}),\mathbf{j}(\mathbf{r})}$-functional theory is developed. Finally, a Slater determinant of equidensity orbitals which reproduces a given ${\ensuremath{\rho}(\mathbf{r}),\mathbf{j}(\mathbf{r})}$ is constructed.