+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.
+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.
+
+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.
+
+\subsection{Born-Oppenheimer approximation}