-\chapter{Festk"orperphysik}
+\part{Theory of the solid state}
+
+\chapter{Atomic structure}
+
+\chapter{Electronic structure}
+
+\section{Noninteracting electrons}
+
+\subsection{Bloch's theorem}
+
+\section{Nearly free and tightly bound electrons}
+
+\subsection{Tight binding model}
+
+\section{Interacting electrons}
+
+\subsection{Density functional theory}
+
+\subsubsection{Hohenberg-Kohn theorem}
+
+The Hamiltonian of a many-electron problem has the form
+\begin{equation}
+H=T+V+U\text{ ,}
+\end{equation}
+where
+\begin{eqnarray}
+T & = & \langle\Psi|\sum_{i=1}^N\frac{-\hbar^2}{2m}\nabla_i^2|\Psi\rangle\\
+ & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} d\vec{r}' \,
+ \langle \Psi | \vec{r} \rangle \langle \vec{r} |
+ \nabla_i^2
+ | \vec{r}' \rangle \langle \vec{r}' | \Psi \rangle\\
+ & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} d\vec{r}' \,
+ \langle \Psi | \vec{r} \rangle \nabla_{\vec{r}_i}
+ \langle \vec{r} | \vec{r}' \rangle
+ \nabla_{\vec{r}'_i} \langle \vec{r}' | \Psi \rangle\\
+ & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} d\vec{r}' \,
+ \nabla_{\vec{r}_i} \langle \Psi | \vec{r} \rangle
+ \delta_{\vec{r}\vec{r}'}
+ \nabla_{\vec{r}'_i} \langle \vec{r}' | \Psi \rangle\\
+ & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} \,
+ \nabla_{\vec{r}_i} \Psi^*(\vec{r}) \nabla_{\vec{r}_i} \Psi(\vec{r})
+ \text{ ,} \\
+V & = & \int V(\vec{r})\Psi^*(\vec{r})\Psi(\vec{r})d\vec{r} \text{ ,} \\
+U & = & \frac{1}{2}\int\frac{1}{\left|\vec{r}-\vec{r}'\right|}
+ \Psi^*(\vec{r})\Psi^*(\vec{r}')\Psi(\vec{r}')\Psi(\vec{r})
+ d\vec{r}d\vec{r}'
+\end{eqnarray}
+represent the kinetic energy, the energy due to the external potential and the energy due to the mutual Coulomb repulsion.
+
+\begin{remark}
+As can be seen from the above, two many-electron systems can only differ in the external potential and the number of electrons.
+The number of electrons is determined by the electron density.
+\begin{equation}
+N=\int n(\vec{r})d\vec{r}
+\end{equation}
+Now, if the external potential is additionally determined by the electron density, the density completely determines the many-body problem.
+\end{remark}
+
+Considering a system with a nondegenerate ground state, there is obviously only one ground-state charge density $n_0(\vec{r})$ that corresponds to a given potential $V(\vec{r})$.
+\begin{equation}
+n_0(\vec{r})=\int \Psi_0^*(\vec{r},\vec{r}_2,\vec{r}_3,\ldots,\vec{r}_N)
+ \Psi_0(\vec{r},\vec{r}_2,\vec{r}_3,\ldots,\vec{r}_N)
+ d\vec{r}_2d\vec{r}_3\ldots d\vec{r}_N
+\end{equation}
+In 1964, Hohenberg and Kohn showed the opposite and far less obvious result \cite{hohenberg64}.
+
+{\begin{theorem}
+For a nondegenerate ground state, the ground-state charge density uniquely determines the external potential in which the electrons reside.
+\end{theorem}
+
+\begin{proof}
+The proof presented by Hohenberg and Kohn proceeds by {\em reductio ad absurdum}.
+Suppose two potentials $V_1$ and $V_2$ exist, which yield the same electron density $n(\vec{r})$.
+The corresponding Hamiltonians are denoted $H_1$ and $H_2$ with the respective ground-state wavefunctions $\Psi_1$ and $\Psi_2$ and eigenvalues $E_1$ and $E_2$.
+Then, due to the variational principle (see \ref{sec:var_meth}), one can write
+\begin{equation}
+E_1=\langle \Psi_1 | H_1 | \Psi_1 \rangle <
+\langle \Psi_2 | H_1 | \Psi_2 \rangle \text{ .}
+\label{subsub:hk01}
+\end{equation}
+Expressing $H_1$ by $H_2+H_1-H_2$, the last part of \eqref{subsub:hk01} can be rewritten:
+\begin{equation}
+\langle \Psi_2 | H_1 | \Psi_2 \rangle =
+\langle \Psi_2 | H_2 | \Psi_2 \rangle +
+\langle \Psi_2 | H_1 -H_2 | \Psi_2 \rangle
+\end{equation}
+The two Hamiltonians, which describe the same number of electrons, differ only in the potential
+\begin{equation}
+H_1-H_2=V_1(\vec{r})-V_2(\vec{r})
+\end{equation}
+and, thus
+\begin{equation}
+E_1<E2+\int n(\vec{r}) \left( V_1(\vec{r})-V_2(\vec{r}) \right) d\vec{r}
+\text{ .}
+\label{subsub:hk02}
+\end{equation}
+By switching the indices of \eqref{subsub:hk02} and adding the resulting equation to \eqref{subsub:hk02}, the contradiction
+\begin{equation}
+E_1 + E_2 < E_2 + E_1 +
+\underbrace{
+\int n(\vec{r}) \left( V_1(\vec{r})-V_2(\vec{r}) \right) d\vec{r} +
+\int n(\vec{r}) \left( V_2(\vec{r})-V_1(\vec{r}) \right) d\vec{r}
+}_{=0}
+\end{equation}
+is revealed, which proofs the Hohenberg Kohn theorem. \qed
+\end{proof}
+
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