\section{Denstiy functional theory}
\label{section:dft}
-In quantum-mechanical modeling the problem of describing a many-body problem is manifested in the high-dimensional Schr\"odinger equation for the wave function $\Psi({\vec{R}},{\vec{r}})$ that depends on the coordinates of the nuclei and electrons.
+Dirac declared that chemistry has come to an end, its content being entirely contained in the powerul equation published by Schr\"odinger in 1926 \cite{schroeder26} marking the beginning of wave mechanics.
+Following the path of Schr\"odinger the problem in quantum-mechanical modeling of describing the many-body problem, i.e. a system of a large amount of interacting particles, is manifested in the high-dimensional Schr\"odinger equation for the wave function $\Psi({\vec{R}},{\vec{r}})$ that depends on the coordinates of all nuclei and electrons.
The Schr\"odinger equation contains the kinetic energy of the ions and electrons as well as the electron-ion, ion-ion and electron-electron interaction.
-This cannot be solved exactly and there are several layers of approximations to reduce the number of parameters.
-In density functional theory (DFT) the problem is recasted to the charge density $n(\vec{r})$ instead of using the description by a wave function.
-Formally DFT can be regarded as an exactification of both, the Thomas Fermi and Hartree theory.
+This cannot be solved exactly and finding approximate solutions requires several layers of simplification in order to reduce the number of free parameters.
+Approximations that consider a truncated Hilbert space of single-particle orbitals yield promising results, however, with increasing complexity and demand for high accuracy the amount of Slater determinats to be evaluated massively increases.
-Since {\textsc vasp} \cite{kresse96} is used in this work, theory and implementation of sophisticated algorithms of DFT codes is not subject of this work.
-Thus, the content of the following sections is restricted to the very basic idea of DFT.
+In contrast, instead of using the description by the many-body wave function, the key point in density functional theory (DFT) is to recast the problem to a description utilizing the charge density $n(\vec{r})$, which constitutes a quantity in real space depending only on the three spatial coordinates.
+In the following sections the basic idea of DFT will be outlined.
+As will be shown, DFT can formally be regarded as an exactification of the Thomas Fermi theory \cite{thomas27,fermi27} and the self-consistent Hartree equations \cite{hartree28}.
+A nice review is given in the Nobel lecture of Kohn \cite{kohn99}, one of the inventors of DFT.
\subsection{Born-Oppenheimer approximation}
-The first approximation ...
+Born and Oppenheimer proposed a simplification enabling the effective decoupling of the electronic and ionic degrees of freedom \cite{born27}.
+Within the Born-Oppenheimer (BO) approximation the light electrons are assumed to move much faster and, thus, follow adiabatically to the motion of the heavy nuclei, if the latter are only slightly deflected from their equilibrium positions.
+Thus, on the timescale of electronic motion the ions appear at fixed positions.
+On the other way round, on the timescale of nuclear motion the electrons appear blurred in space adding an extra term to the ion-ion potential.
+The simplified Schr\"odinger equation no longer contains the kinetic energy of the ions.
+The momentary positions of the ions enter as fixed parameters and, therefore, the ion-ion interaction may be regarded as a constant added to the electronic energies.
+The Schr\"odinger equation describing the remaining electronic problem reads
+\begin{equation}
+\left[-\frac{\hbar^2}{2m}\sum_j\nabla^2_j-
+\sum_{j,l} \frac{Z_le^2}{|\vec{r}_j-\vec{R}_l|}+
+\frac{1}{2}\sum_{j\neq j'}\frac{e^2}{|\vec{r}_j-\vec{r}_{j'}|}
+\right] \Psi = E \Psi
+\text{ ,}
+\end{equation}
+where $Z_l$ are the atomic numbers of the nuclei and $\Psi$ is the many-electron wave function, which depends on the positions and spins of the electrons.
+Accordingly, there is only a parametrical dependence on the ionic coordinates $\vec{R}_l$.
+However, the remaining number of free parameters is still too high and need to be further decreased.
+
+\subsection{Hohenberg-Kohn theorem and variational principle}
-\subsection{Hohenberg-Kohn theorem}
+Investigating the energetics of Cu$_x$Zn$_{1-x}$ alloys, which for different compositions exhibit different transfers of charge between the Cu and Zn unit cells due to their chemical difference and, thus, varying electrostatic interactions contributing to the total energy, the work of Hohenberg and Kohn had a natural focus on the distribution of charge.
+Although it was clear that the Thomas Fermi (TF) theory only provides a rough approximation to the exact solution of the many-electron Schr\"odinger equation the theory was of high interest since it provides an implicit relation of the potential and the electron density distribution.
+This raised the question how to establish a connection between TF expressed in terms of $n(\vec{r})$ and the exact Schr\"odinger equation expressed in terms of the many-electron wave function $\Psi({\vec{r}})$ and whether a complete description in terms of the charge density is possible in principle.
+The answer to this question, whether the charge density completely characterizes a system, became the starting point of modern DFT.
+
+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.
-\subsection{Effective potential}
+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}
\subsection{Kohn-Sham system}
+Now find $F[n]$ ...
+
\subsection{Approximations for exchange and correlation}
\subsection{Pseudopotentials}
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