-The angle $\theta_{ijk}$ can be expressed by the atom distances with the law of cosines:
-\begin{eqnarray}
-\theta_{ijk} & = & \arccos \Big( (r_{ij}^2 + r_{ik}^2 - r_{jk}^2)/(2 r_{ij} r_{ik}) \Big) \\
-\partial_{x^i_n} \theta_{ijk} & = &
-\frac{-1}{\sqrt{1 - ((r_{ik}^2+r_{ij}^2-r_{jk}^2)/2r_{ik}r_{ij})^2}}
-\Big( \frac{4 r_{ik}r_{ij} (2 x^i_n - x^k_n - x^j_n) + 2(x^j_n - x^i_n)\frac{r_{ik}}{r_{ij}} + 2(x^k_n - x^i_n)\frac{r_{ij}}{r_{ik}} }{4 r^2_{ik} r^2_{ij}}\Big)
-\end{eqnarray}
-
+The force is then given by
+\begin{equation}
+F^i = - \nabla_{{\bf r}_i} E \textrm{ .}
+\end{equation}
+The details of the Tersoff potential derivative can be seen in appendix \ref{app:d_tersoff}.