\part{Theory of the solid state}

\chapter{Atomic structure}

\chapter{Reciprocal lattice}

Example of primitive cell ...

\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}[Hohenberg / Kohn]
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}

\section{Electron-ion interaction}

\subsection{Pseudopotential theory}

The basic idea of pseudopotential theory is to only describe the valence electrons, which are responsible for the chemical bonding as well as the electronic properties for the most part.

\subsubsection{Orthogonalized planewave method}

Due to the orthogonality of the core and valence wavefunctions, the latter exhibit strong oscillations within the core region of the atom.
This requires a large amount of planewaves $\ket{k}$ to adequatley model the valence electrons.

In a very general approach, the orthogonalized planewave (OPW) method introduces a new basis set
\begin{equation}
\ket{k}_{\text{OPW}} = \ket{k} - \sum_t \ket{t}\bra{t}k\rangle \text{ ,}
\end{equation} 
with $\ket{t}$ being the eigenstates of the core electrons.
The new basis is orthogonal to the core states $\ket{t}$.
\begin{equation}
\braket{t}{k}_{\text{OPW}} =
\braket{t}{k} - \sum_{t'} \braket{t}{t'}\braket{t'}{k} =
\braket{t}{k} - \braket{t}{k}=0
\end{equation}
The number of planewaves required for reasonably converged electronic structure calculations is tremendously reduced by utilizing the OPW basis set.

\subsubsection{Pseudopotential method}

\subsubsection{Norm conserving pseudopotentials}

\begin{equation}
V=\ket{lm}V_l(r)\bra{lm}
\end{equation}

\subsubsection{Fully separable form of the pseudopotential}

\subsection{Spin orbit interaction}


\subsubsection{Perturbative treatment}

\subsubsection{Non-perturbative method}

