1 \part{Theory of the solid state}
3 \chapter{Atomic structure}
5 \chapter{Electronic structure}
7 \section{Noninteracting electrons}
9 \subsection{Bloch's theorem}
11 \section{Nearly free and tightly bound electrons}
13 \subsection{Tight binding model}
15 \section{Interacting electrons}
17 \subsection{Density functional theory}
19 \subsubsection{Hohenberg-Kohn theorem}
21 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 $V(\vec{r})$.
22 In 1964, Hohenberg and Kohn showed the opposite and far less obvious result \cite{hohenberg64}.
23 For a nondegenerate ground state, the ground-state charge density uniquely determines the external potential in which the electrons reside.
24 The proof presented by Hohenberg and Kohn proceeds by {\em reductio ad absurdum}.
26 Suppose two potentials $V_1$ and $V_2$ exist, which yield the same electron density $n(\vec{r})$.
27 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 $E_2$.
28 Then, due to the variational principle (see \ref{sec:var_meth}), one can write
30 E_1=\langle \Psi_1 | H_1 | \Psi_1 \rangle < \langle \Psi_2 | H_1 | \Psi_2 \rangle
32 Expressing $H_1$ by $H_2+H_1-H_2$
34 \langle \Psi_2 | H_1 | \Psi_2 \rangle =
35 \langle \Psi_2 | H_2 | \Psi_2 \rangle +
36 \langle \Psi_2 | H_1 -H_2 | \Psi_2 \rangle
38 and the fact that the two Hamiltonians, which describe the same number of electrons, differ only in the potential
40 H_1-H_2=V_1(\vec{r})-V_2(\vec{r})
44 E_1<E2+\int n(\vec{r}) \left( V_1(\vec{r})-V_2(\vec{r}) \right) d\vec{r}
46 By switching the indices ...