\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}

Following the idea of orthogonalized planewaves leads to the pseudopotential idea, which --- in describing only the valence electrons --- effectively removes an undesriable subspace from the investigated problem.

Let $\ket{\Psi_\text{V}}$ be the wavefunction of a valence electron with the Schr\"odinger equation 
\begin{equation}
H \ket{\Psi_\text{V}} = \left(\frac{1}{2m}p^2+V\right)\ket{\Psi_\text{V}}=
E\ket{\Psi_\text{V}} \text{ .}
\end{equation}
\ldots projection operatore $P_\text{C}$ \ldots

\subsubsection{Semilocal form of the pseudopotential}

Ionic potentials, which are spherically symmteric, suggest to treat each angular momentum $l,m$ separately leading to $l$-dependent non-local (NL) model potentials $V_l(r)$ and a total potential
\begin{equation}
V=\sum_{l,m}\ket{lm}V_l(\vec{r})\bra{lm} \text{ .}
\end{equation}
In fact, applied to a function, the potential turns out to be non-local in the angular coordinates but local in the radial variable, which suggests to call it asemilocal (SL) potential.

Problem of semilocal potantials become valid once matrix elements need to be computed.
Integral with respect to the radial component needs to be evaluated for each planewave combination, i.e.\  $N(N-1)/2$ integrals.
\begin{equation}
\bra{k+G}V\ket{k+G'} = \ldots
\end{equation}

A local potential can always be separated from the potential \ldots
\begin{equation}
V=\ldots=V_{\text{local}}(\vec{r})+\ldots
\end{equation}

\subsubsection{Norm conserving pseudopotentials}

HSC potential \ldots

\subsubsection{Fully separable form of the pseudopotential}

KB transformation \ldots

\subsection{Spin-orbit interaction}

Relativistic effects can be incorporated in the normconserving pseudopotential method up to but not including order $\alpha^2$ with $\alpha$ being the fine structure constant.
This is advantageous since \ldots
With the solutions of the all-electron Dirac equations, the new pseudopotential reads
\begin{equation}
V(\vec{r})=\sum_{l,m}\left[
\ket{l+\frac{1}{2},m+{\frac{1}{2}}}V_{l,l+\frac{1}{2}}(\vec{r})
\bra{l+\frac{1}{2},m+{\frac{1}{2}}} +
\ket{l-\frac{1}{2},m-{\frac{1}{2}}}V_{l,l-\frac{1}{2}}(\vec{r})
\bra{l-\frac{1}{2},m-{\frac{1}{2}}}
\right] \text{ .}
\end{equation}
By defining an averaged potential weighted by the different $j$ degeneracies of the $\ket{l\pm\frac{1}{2}}$ states
\begin{equation}
\bar{V}_l(r)=\frac{1}{2l+1}\left(
l V_{l,l-\frac{1}{2}}(\vec{r})+(l+1)V_{l,l+\frac{1}{2}}(\vec{r})\right)
\end{equation}
and a potential describing the difference in the potential with respect to the spin
\begin{equation}
V^{\text{SO}}_l(\vec{r})=\frac{2}{2l+1}\left(
V_{l,l+\frac{1}{2}}(\vec{r})-V_{l,l-\frac{1}{2}}(\vec{r})\right)
\end{equation}
the total potential can be expressed as
\begin{equation}
V(\vec{r})=\sum_l
\ket{l}\left[\bar{V}_l(\vec{r})+V^{\text{SO}}_l(\vec{r})LS\right]\bra{l}
\text{ ,}
\end{equation}
where the first term correpsonds to the mass velocity and Darwin relativistic corrections and the latter is associated with the spin-orbit (SO) coupling.


\subsubsection{Excursus: real space representation within an iterative treatment}

In the following, the spin-orbit part is evaluated in real space.
Since spin is treated in another subspace, it can be treated separately.
The matrix elements of the orbital angular momentum part of the potential in KB form for orbital angular momentum $l$ read
\begin{equation}
\bra{\vec{r'}}(r\times p)\ket{\chi_{lm}}E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\vec{r''}}
\text{ .}
\end{equation}
With
\begin{eqnarray}
\bra{\vec{r'}}p\ket{\chi_{lm}} & = & -i\hbar\nabla_{\vec{r'}} \braket{\vec{r'}}{\chi_{lm}}
=-i\hbar\nabla_{\vec{r'}}\,\chi_{lm}(\vec{r'}) \\
r\ket{\vec{r'}} & = & r'\ket{\vec{r'}}
\end{eqnarray}
we get
\begin{equation}
-i\hbar(\vec{r'}\times \nabla_{\vec{r'}})\braket{\vec{r'}}{\chi_{lm}}
E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\vec{r''}}
\text{ .}
\label{eq:solid:so_me}
\end{equation}
To further evaluate this expression, the KB projectors
\begin{equation}
\chi_{lm}=\frac{\ket{\delta V_l^{\text{SO}}\Phi_{lm}}}
{\braket{\delta V_l^{\text{SO}}\Phi_{lm}}
        {\Phi_{lm}\delta V_l^{\text{SO}}}^{1/2}}
\end{equation}
must be known in real space (with respect to $\vec{r'}$).
\begin{equation}
\braket{\vec{r'}}{\chi_{lm}}=
\frac{\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}}{
\braket{\delta V_l^{\text{SO}}\Phi_{lm}}{\Phi_{lm}\delta V_l^{\text{SO}}}
^{1/2}}
\end{equation}
and
\begin{equation}
\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}=
\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}Y_{lm}(\Omega_{r'})
\text{ .}
\label{eq:solid:so_r1}
\end{equation}
In this expression, only the spherical harmonics are complex functions.
Thus, the complex conjugate with respect to $\vec{r''}$ is given by
\begin{equation}
\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}=
\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}Y^*_{lm}(\Omega_{r''})
\text{ .}
\label{eq:solid:so_r2}
\end{equation}
Using the orthonormality property 
\begin{equation}
\int Y^*_{l'm'}(\Omega_r)Y_{lm}(\Omega_r) d\Omega_r = \delta_{ll'}\delta_{mm'}
\label{eq:solid:y_ortho}
\end{equation}
of the spherical harmonics, the squared norm of the $\chi_{lm}$ reduces to
\begin{eqnarray}
\braket{\delta V_l^{\text{SO}}\Phi_{lm}}{\Phi_{lm}\delta V_l^{\text{SO}}} &=&
\int \braket{\delta V_l^{\text{SO}}\Phi_{lm}}{\vec{r'}}
\braket{\vec{r'}}{\Phi_{lm}\delta V_l^{\text{SO}}} d\vec{r'}\\
&=&\int 
{\delta V_l^{\text{SO}}}^2(r')\frac{u_l^2(r')}{r'^2}Y^*_{lm}(\Omega_{r'})
Y_{lm}(\Omega_{r'})
r'^2 dr' d\Omega_{r'} \\
&=&\int_{r'}
{\delta V_l^{\text{SO}}}^2(r') u_l^2(r') dr'
\int_{\Omega_{r'}}Y^*_{lm}(\Omega_{r'})Y_{lm}(\Omega_{r'}) d\Omega_{r'}\\
&=&\int_{r'}{\delta V_l^{\text{SO}}}^2(r') u_l^2(r') dr' \text{ .}
\end{eqnarray}
To obtain a final expression for the matrix elements \eqref{eq:solid:so_me}, the product of \eqref{eq:solid:so_r1} and \eqref{eq:solid:so_r2} must be further evaluated.
\begin{eqnarray}
\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}
\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}&=&
\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}Y_{lm}(\Omega_{r'})
\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}Y^*_{lm}(\Omega_{r''})\nonumber\\
&=&
\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}
Y^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'})
\end{eqnarray}
and if all states with magnetic quantum numbers $m=-l,-l+1,\ldots,l-1,l$ that contribute to the potential for angular momentum $l$ are considered
\begin{equation}
\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}
\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}=
\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}
\sum_mY^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'}) \text{ ,}
\end{equation}
which can be rewritten as
\begin{equation}
\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}
\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}=
\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}
\frac{2l+1}{4\pi}P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)
\end{equation}
using the vector addition theorem
\begin{equation}
P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)=
\frac{4\pi}{2l+1}\sum_mY^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'})
\end{equation}
In total, the matrix elements of the potential for angular momentum $l$ can be calculated as
\begin{eqnarray}
\bra{\vec{r'}}V^{\text{KB,SO}}\ket{\vec{r''}}&=&
\bra{\vec{r'}}(r\times p)\ket{\chi_{lm}}E^{\text{SO,KB}}_l
\braket{\chi_{lm}}{\vec{r''}}\\
&=&
-i\hbar(\vec{r'}\times \nabla_{\vec{r'}})
\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)\times\nonumber\\
&&\times\frac{E^{\text{SO,KB}}_l\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}}
             {\int_{r}{\delta V_l^{\text{SO}}}^2(r) u_l^2(r) dr} \,
\frac{2l+1}{4\pi}\\
&=&
-i\hbar
\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
P'_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)\times\nonumber\\
&&\left(\frac{\vec{r'}\times\vec{r''}}{r'r''}\right)
\frac{E^{\text{SO,KB}}_l\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}}
        {\int_{r}{\delta V_l^{\text{SO}}}^2(r) u_l^2(r) dr} \,
\frac{2l+1}{4\pi}
\end{eqnarray}
If the potential at position $\vec{r}$ is considered a sum of atomic potentials $v_{\alpha}(\vec{r}-\vec{\tau}_{\alpha n})$ (atom $n$ of species $\alpha$)
\begin{equation}
V(\vec{r})=\sum_{\alpha}\sum_n v_{\alpha}(\vec{r}-\vec{\tau}_{\alpha n})
\end{equation}
and the SO projectors are likewise centered on atoms, the SO potential contribution reads
\begin{equation}
\end{equation}
The $E_l^{\text{SO,KB}}$ are given by
\begin{equation}
E_l^{\text{SO,KB}}=
\frac{\braket{\delta V_lu_l}{u_l\delta V_l}}
     {\bra{u_l}\delta V_l\ket{u_l}}=
\frac{\int_{r}\delta V^2_l(r)u^2_l(r)}r^2dr
     {\int_{r'}\int_{r''}\braket{u_l}{r'}\bra{r'}\delta V_l
\ket{r''}\braket{r''}{u_l}}=
\end{equation}
Finally, to evaluate $V^{\text{SO}}_l\ket{\Psi}$, the integral \ldots
\begin{equation}
\end{equation}

