+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}
+
+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.