+\label{subsection:interact_pot}
+
+The potential energy of $N$ interacting atoms can be written in the form
+\begin{equation}
+U(\{{\bf r}\}) = \sum_i U_1({\bf r}_i) + \sum_i \sum_{j>i} U_2({\bf r}_i,{\bf r}_j) + \sum_i \sum_{j>i} \sum_{k>j>i} U_3({\bf r}_i,{\bf r}_j,{\bf r}_k) \ldots
+\end{equation}
+where $U$ is the total potential energy.
+$U_1$ is a single particle potential describing external forces.
+Examples of single particle potentials are the gravitational force or an electric field.
+$U_2$ is a two body pair potential which only depends on the distance $r_{ij}$ between the two atoms $i$ and $j$.
+If not only pair potentials are considered, three body potentials $U_3$ or multi body potentials $U_n$ can be included.
+Mainly these higher order terms are avoided since they are not easy to model and it is rather time consuming to evaluate potentials and forces originating from these many body terms.
+
+Ordinary pair potentials have a close-packed structure like face-centered cubic (FCC) or hexagonal close-packed (HCP) as a ground state.
+A pair potential is thus unable to describe properly elements with other structures than FCC or HCP.
+Silicon and carbon for instance, have a diamand/zincblende structure with four covalent bonded neighbours, which is far from a close-packed structure.
+A three body potential has to be included for these types of elements.
+
+In the following, relevant potentials for this work are discussed.