On the Hohenberg-Kohn and Levy-Lieb Constrained Search Proofs of Density Functional Theory

In HK, a 1-1 relationship between the density $\rho ({\bf{r}})$ and the potential $v({\bf{r}})$ is established. (The relationship between $v({\bf{r}})$ and the ground state $\Psi$ is 1-1.) The proof, valid for $v$-representable densities, shows $\rho({\bf{r}})$ to be a basic variable. The LL proof is independent of $v({\bf{r}})$, and is valid for $N$-representable densities. In,\footnote{Pan and Sahni, IJQC 110, 2833 (2010)} we have proved that in an external magnetic field ${\bf{B}}({\bf{r}})=\mathbf{\nabla} \times {\bf{A}}({\bf{r}})$, there is a 1-1 relationship between $\{\rho({\bf{r}}), {\bf{j}} ({\bf{r}})\}$, with ${\bf{j}}({\bf{r}})$ the physical current density, and the potentials $\{v({\bf{r}}), {\bf{A}}({\bf{r}})\}$. (The relationship between $\{v({\bf{r}}), {\bf{A}}({\bf{r}})\}$ and $\Psi$ is \emph{many-to-one}.) This proves that $\{\rho({\bf{r}}), {\bf{j}}({\bf{r}})\}$ are the basic variables. The LL proof independent of $\{v({\bf{r}}), {\bf{A}}({\bf{r}})\}$ follows readily. However, such a proof also follows if $\{\rho({\bf{r}}), {\bf{j}}_{p}({\bf{r}})\}$, with ${\bf{j}}_{p}({\bf{r}})$ the paramagnetic current density, are considered the basic variables. As such knowledge of the basic variables as determined via HK is a pre-requisite to any LL type proof.