-\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 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}
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