-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 derivation of the Lennard-Jones potential in respect to $x_i$ (the $i$th component of the distance vector $\vec{r}$)
-\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 $F^j$ on particle $j$ is obtained by
-\begin{equation}
-F_i^j = - \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}