+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.
+Usually 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 and zincblende structure with four covalently bonded neighbors, which is far from a close-packed structure.
+A three body potential has to be included for these types of elements.
+
+\subsubsection{The Tersoff bond order potential}
+
+Tersoff proposed an empirical interatomic potential for covalent systems \cite{tersoff_si1,tersoff_si2}.
+The Tersoff potential explicitly incorporates the dependence of bond order on local environments, permitting an improved description of covalent materials.
+Due to the covalent character Tersoff restricted the interaction to nearest neighbor atoms accompanied by an increases in computational efficiency for the evaluation of forces and energy based on the short-range potential.
+Tersoff applied the potential to silicon \cite{tersoff_si1,tersoff_si2,tersoff_si3}, carbon \cite{tersoff_c} and also to multicomponent systems like silicon carbide \cite{tersoff_m}.
+The basic idea is that, in real systems, the bond order, i.e. the strength of the bond, depends upon the local environment \cite{abell85}.
+Atoms with many neighbors form weaker bonds than atoms with only a few neighbors.
+Although the bond strength intricately depends on geometry the focus on coordination, i.e. the number of neighbors forming bonds, is well motivated qualitatively from basic chemistry since for every additional formed bond the amount of electron pairs per bond and, thus, the strength of the bonds is decreased.
+If the energy per bond decreases rapidly enough with increasing coordination the most stable structure will be the dimer.
+In the other extreme, if the dependence is weak, the material system will end up in a close-packed structure in order to maximize the number of bonds and likewise minimize the cohesive energy.
+This suggests the bond order to be a monotonously decreasing function with respect to coordination and the equilibrium coordination being determined by the balance of bond strength and number of bonds.
+
+Tersoff incorporated the concept of bond order based on pseudopotential theory \cite{abell85} in a three-body potential formalism.
+The interatomic potential is taken to have the form
+\begin{eqnarray}
+E & = & \sum_i E_i = \frac{1}{2} \sum_{i \ne j} V_{ij} \textrm{ ,} \\
+V_{ij} & = & f_C(r_{ij}) [ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) ] \textrm{ .}
+\end{eqnarray}
+$E$ is the total energy of the system, constituted either by the sum over the site energies $E_i$ or by the bond energies $V_{ij}$.
+The indices $i$ and $j$ correspond to the atoms of the system with $r_{ij}$ being the distance from atom $i$ to atom $j$.
+The functions $f_R$ and $f_A$ represent a repulsive and an attractive pair potential.
+The repulsive part is due to the orthogonalization energy of overlapped atomic wave functions.
+The attractive part is associated with the bonding.
+\begin{eqnarray}
+f_R(r_{ij}) & = & A_{ij} \exp (- \lambda_{ij} r_{ij} ) \\
+f_A(r_{ij}) & = & -B_{ij} \exp (- \mu_{ij} r_{ij} )
+\end{eqnarray}
+The function $f_C$ is the a cutoff function to limit the range of interaction to nearest neighbors.
+It is designed to have a smooth transition of the potential at distances $R_{ij}$ and $S_{ij}$.