-In the following, relevant potentials for this work are discussed.
-
-%\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.