Basic Variables in Density Functional Theory in the Presence of a Magnetic Field

We have shown$^{\dag}$ via a unitary or equivalently a gauge transformation that for a system of $N$ electrons in an external field ${\vec{\cal{F}}}^{ext} = - {\vec{\nabla}} v({\vec{r}})$, the wave function $\Psi$ is in general a functional of the ground state density $\rho ({\vec{r}})$ and a gauge function $\alpha ({\vec{R}})$; ${\vec{R}} = {\vec{r}}_{1}, \ldots , {\vec{r}}_{N} $, i.e. $\Psi = \Psi [\rho, \alpha]$. The functions $\alpha ({\vec{R}})$ are arbitrary, the choice $\alpha ({\vec{R}}) = 0$ being equally valid. It is the presence of $\alpha ({\vec{R}})$ that ensures the wave function functional is gauge variant. Similarly, in the presence of a magnetic field ${\vec{B}} ({\vec{r}}) = {\vec{\nabla}} \times {\vec{A}} ({\vec{r}})$, we show that in general the wave function is a functional of the density $\rho ({\vec{r}})$, the physical current density ${\vec{j}}_{\vec{A}} ({\vec{r}})$, and a gauge function $\alpha ({\vec{R}}): \Psi = \Psi[\rho, {\vec{j}}_{\vec{A}}, \alpha]$. Again, the $\alpha ({\vec{R}})$ are arbitrary, the choice $\alpha ({\vec{R}}) = 0$ being valid. Hence, it is possible to construct a theory in which the basic variables are $\rho ({\vec{r}})$ and ${\vec{j}}_{\vec {A}} ({\vec{r}})$. The generalized Hohenberg-Kohn theorems, as well as the equations for the noninteracting fermion Kohn-Sham system that reproduces the $\rho ({\vec{r}})$ and ${\vec{j}}_{\vec{A}} ({\vec{r}})$ of the interacting system of electrons, are derived. \\ $^{\dag}$X.-Y. Pan and V. Sahni, Int. J. Quantum Chem. \textbf {108}, 2756 (2008).