+%\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.
+
+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 potential cutoff function to limit the range of the potential.
+It is designed to have a smooth transition of the potential at distances $R_{ij}$ and $S_{ij}$.