-%\subsubsection{The Lennard-Jones potential}
-%
-%The L-J potential is a realistic two body pair potential and is of the form
-%\begin{equation}
-%U^{LJ}(r) = 4 \epsilon \Big[ \Big( \frac{\sigma}{r} \Big)^{12} - \Big( \frac{\sigma}{r} \Big)^6 \Big] \, \textrm{,}
-%\label{eq:lj-p}
-%\end{equation}
-%where $r$ denotes the distance between the two atoms.
-%
-%The attractive tail for large separations $(\sim r^{-6})$ is essentially due to correlations between electron clouds surrounding the atoms. The attractive part is also known as {\em van der Waals} or {\em London} interaction.
-%It can be derived classically by considering how two charged spheres induce dipol-dipol interactions into each other, or by considering the interaction between two oscillators in a quantum mechanical way.
-%
-%The repulsive term $(\sim r^{-12})$ captures the non-bonded overlap of the electron clouds.
-%It does not have a true physical motivation, other than the exponent being larger than $6$ to get a steep rising repulsive potential wall at short distances.
-%Chosing $12$ as the exponent of the repulsive term it is just the square of the attractive term which makes the potential evaluable in a very efficient way.
-%
-%The constants $\epsilon$ and $\sigma$ are usually determined by fitting to experimental data.
-%$\epsilon$ accounts to the depth of the potential well, where $\sigma$ is regarded as the radius of the particle, also known as the van der Waals radius.
-%
-%Writing down the derivative of the Lennard-Jones potential in respect to $x_i$ (the $i$th component of the distance vector ${\bf r}$)
-%\begin{equation}
-%\frac{\partial}{\partial x_i} U^{LJ}(r) = 4 \epsilon x_i \Big( -12 \frac{\sigma^{12}}{r^{14}} + 6 \frac{\sigma^6}{r^8} \Big)
-%\label{eq:lj-d}
-%\end{equation}
-%one can easily identify $\sigma$ by the equilibrium distance of the atoms $r_e=\sqrt[6]{2} \sigma$.
-%Applying the equilibrium distance into \eqref{eq:lj-p} $\epsilon$ turns out to be the negative well depth.
-%The $i$th component of the force is given by
-%\begin{equation}
-%F_i = - \frac{\partial}{\partial x_i} U^{LJ}(r) \, \textrm{.}
-%\label{eq:lj-f}
-%\end{equation}
-%
-%\subsubsection{The Stillinger Weber potential}
-%
-%The Stillinger Weber potential \cite{stillinger_weber} \ldots
-%
-%\begin{equation}
-%U = \sum_{i,j} U_2({\bf r}_i,{\bf r}_j) + \sum_{i,j,k} U_3({\bf r}_i,{\bf r}_j,{\bf r}_k)
-%\end{equation}
-%
-%\begin{equation}
-%U_2(r_{ij}) = \left\{
-% \begin{array}{ll}
-% \epsilon A \Big( B (r_{ij} / \sigma)^{-p} - 1\Big) \exp \Big[ (r_{ij} / \sigma - 1)^{-1} \Big] & r_{ij} < a \sigma \\
-% 0 & r_{ij} \ge a \sigma
-% \end{array} \right.
-%\end{equation}
-%
-%\begin{equation}
-%U_3({\bf r}_i,{\bf r}_j,{\bf r}_k) =
-%\epsilon \Big[ h(r_{ij},r_{ik},\theta_{jik}) + h(r_{ji},r_{jk},\theta_{ijk}) + h(r_{ki},r_{kj},\theta_{ikj}) \Big]
-%\end{equation}
-%
-%\begin{equation}
-%h(r_{ij},r_{ik},\theta_{jik}) =
-%\lambda \exp \Big[ \gamma (r_{ij}/\sigma -a)^{-1} + \gamma (r_{ik}/\sigma - a)^{-1} \Big] \Big( \cos \theta_{jik} + \frac{1}{3} \Big)^2
-%\end{equation}
-
-\subsubsection{The Tersoff potential}
-
-Tersoff proposed an empirical interatomic potential for covalent systems.
-The Tersoff potential explicitly incorporates the dependence of bond order on local envirenments, permitting an improved description of covalent materials.
-Tersoff applied the potential to silicon \cite{tersoff_si1,tersoff_si2,tersoff_si3}, carbon \cite{tersoff_c} and also to multicomponent systems like $SiC$ \cite{tersoff_m}.
-The basic idea is that, in real systems, the bond order depends upon the local environment.
-An atom with many neighbours forms weaker bonds than an atom with few neighbours.
-
+\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.
+Based on pseudopotential theory the bond order term $b_{ijk}$ limitting the attractive pair interaction is of the form $b_{ijk}\propto Z^{-\delta}$ where $Z$ is the coordination number and $\delta$ a constant \cite{abell85}, which is $\frac{1}{2}$ in the seond-moment approximation within the tight binding scheme \cite{horsfield96}.
+
+Tersoff incorporated the concept of bond order in a three-body potential formalism.