+Considering a system with a nondegenerate ground state there is obviously only one ground-state charge density $n_0(\vec{r})$ that correpsonds to a given potential $V(\vec{r})$.
+In 1964 Hohenberg and Kohn showed the opposite and far less obvious result \cite{hohenberg64}.
+Employing no more than the Rayleigh-Ritz minimal principle it is concluded by {\em reductio ad absurdum} that for a nondegenerate ground state the same charge density cannot be generated by different potentials.
+Thus, the charge density of the ground state $n_0(\vec{r})$ uniquely determines the potential $V(\vec{r})$ and, consequently, the full Hamiltonian and ground-state energy $E_0$.
+In mathematical terms the full many-electron ground state is a unique functional of the charge density.
+Im particular, $E$ is a functional $E[n(\vec{r})]$ of $n(\vec{r})$.
+
+The ground-state charge density $n_0(\vec{r})$ minimizes the energy functional $E[n(\vec{r})]$, its value corresponding to the ground-state energy $E_0$, which enables the formulation of a minimal principle in terms of the charge density \cite{hohenberg64,levy82}
+\begin{equation}
+E_0=\min_{n(\vec{r})}
+ \left\{
+ F[n(\vec{r})] + \int n(\vec{r}) V(\vec{r}) d\vec{r}
+ \right\}
+ \text{ ,}
+\end{equation}
+where $F[n(\vec{r})]$ is a universal functional of the charge density $n(\vec{r})$, which is composed of the kinetic energy functional $T[n(\vec{r})]$ and the interaction energy functional $U[n(\vec{r})]$.
+The challenging problem of determining the exact ground-state is now formally reduced to the determination of the $3$-dimensional function $n(\vec{r})$ via a well-defined but not explicitly known functional of the charge density.
+
+It is worth to note, that this minimal principle may be regarded as exactification of TF theory, which is rederived by the approximations
+\begin{equation}
+T=\int n(\vec{r})\frac{3}{10}k_{\text{F}}^2[n(\vec{r})]d\vec{r}
+\text{ ,}
+\end{equation}
+\begin{equation}
+U=\frac{1}{2}\int\frac{n(\vec{r})n(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}d\vec{r}'
+\text{ .}
+\end{equation}