\section{Form of the Tersoff potential and its derivative}
-The Tersoff potential \cite{tersoff_m} is of the form
+The Tersoff potential~\cite{tersoff_m} is of 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{ .}
0, & r_{ij} > S_{ij}
\end{array} \right.
\end{equation}
-with $\theta_{ijk}$ being the bond angle between bonds $ij$ and $ik$ as shown in Figure \ref{img:tersoff_angle}.\\
+with $\theta_{ijk}$ being the bond angle between bonds $ij$ and $ik$ as shown in Figure~\ref{img:tersoff_angle}.\\
\\
For a three body potential, if $V_{ij}$ is not equal to $V_{ji}$, the derivative is of the form
\begin{equation}
\nabla_{{\bf r}_i} E = \frac{1}{2} \big[ \sum_j ( \nabla_{{\bf r}_i} V_{ij} + \nabla_{{\bf r}_i} V_{ji} ) + \sum_k \sum_j \nabla_{{\bf r}_i} V_{jk} \big] \textrm{ .}
\end{equation}
-In the following all the necessary derivatives to calculate $\nabla_{{\bf r}_i} E$ are written down.
+In the following, all the necessary derivatives to calculate $\nabla_{{\bf r}_i} E$ are written down.
\section[Derivative of $V_{ij}$ with respect to ${r}_i$]{\boldmath Derivative of $V_{ij}$ with respect to ${\bf r}_i$}
b_{ij} \nabla_{{\bf r}_j} f_A(r_{ij}) +
f_A(r_{ij}) \nabla_{{\bf r}_j} b_{ij} \big]
\end{eqnarray}
-Using the equality $\nabla_{{\bf r}_i} r_{ij}=-\nabla_{{\bf r}_j} r_{ij}$
+Using the equality $\nabla_{{\bf r}_i} r_{ij}=-\nabla_{{\bf r}_j} r_{ij}$,
the following relations are valid:
\begin{eqnarray}
\nabla_{{\bf r}_j} f_R(r_{ij}) &=& - \nabla_{{\bf r}_i} f_R(r_{ij}) \\
\subsection{Code realization}
-The implementation of the force evaluation shown in the following is applied to the potential designed by Erhart and Albe \cite{albe_sic_pot}.
+The implementation of the force evaluation shown in the following is applied to the potential designed by Erhart and Albe~\cite{albe_sic_pot}.
There are slight differences compared to the original potential by Tersoff:
\begin{itemize}
\item Difference in sign of the attractive part.