+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{schroedinger26} 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 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.
+
+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}
+
+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}