\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 done.
+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 ${\bf r}_i$}
+ \section[Derivative of $V_{ij}$ with respect to ${r}_i$]{\boldmath Derivative of $V_{ij}$ with respect to ${\bf r}_i$}
\begin{eqnarray}
\nabla_{{\bf r}_i} V_{ij} & = & \nabla_{{\bf r}_i} f_C(r_{ij}) \big[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \big] + \nonumber \\
& = & \Big[ \frac{\cos\theta_{ijk}}{r_{ij}^2} - \frac{1}{r_{ij} r_{ik}} \Big] {\bf r}_{ij} + \Big[ \frac{\cos\theta_{ijk}}{r_{ik}^2} - \frac{1}{r_{ij} r_{ik}} \Big] {\bf r}_{ik}
\end{eqnarray}
- \section{Derivative of $V_{ji}$ with respect to ${\bf r}_i$}
+ \section[Derivative of $V_{ji}$ with respect to ${r}_i$]{\boldmath Derivative of $V_{ji}$ with respect to ${\bf r}_i$}
\begin{eqnarray}
\nabla_{{\bf r}_i} V_{ji} & = & \nabla_{{\bf r}_i} f_C(r_{ji}) \big[ f_R(r_{ji}) + b_{ji} f_A(r_{ji}) \big] + \nonumber \\
& = & \frac{1}{r_{ji} r_{jk}} {\bf r}_{jk} - \frac{\cos\theta_{jik}}{r_{ji}^2} {\bf r}_{ji}
\end{eqnarray}
- \section{Derivative of $V_{jk}$ with respect to ${\bf r}_i$}
+ \section[Derivative of $V_{jk}$ with respect to ${r}_i$]{\boldmath Derivative of $V_{jk}$ with respect to ${\bf r}_i$}
\begin{eqnarray}
\nabla_{{\bf r}_i} V_{jk} & = & f_C(r_{jk}) f_A(r_{jk}) \nabla_{{\bf r}_i} b_{jk} \\
keeping in mind that all the necessary force contributions for atom $i$
are calculated and added in subsequent loops.
-\subsection{Derivative of $V_{ij}$ with respect to ${\bf r}_j$}
+\subsection[Derivative of $V_{ij}$ with respect to ${r}_j$]{\boldmath Derivative of $V_{ij}$ with respect to ${\bf r}_j$}
\begin{eqnarray}
\nabla_{{\bf r}_j} V_{ij} & = &
\frac{\cos\theta_{ijk}}{r_{ij}^2}{\bf r}_{ij}
\end{eqnarray}
-\subsection{Derivative of $V_{ij}$ with respect to ${\bf r}_k$}
+\subsection[Derivative of $V_{ij}$ with respect to ${r}_k$]{\boldmath Derivative of $V_{ij}$ with respect to ${\bf r}_k$}
The derivative of $V_{ij}$ with respect to ${\bf r}_k$ just consists of the
single term