+The cosine of the angle $\theta_{ijk}$ can be expressed by the atom distances with the law of cosines
+\begin{equation}
+\cos \theta_{ijk} = \Big( (r_{ij}^2 + r_{ik}^2 - r_{jk}^2)/(2 r_{ij} r_{ik}) \Big)
+\end{equation}
+or by the definition of the scalar product
+\begin{equation}
+\cos \theta_{ijk} = \frac{\vec{r}_{ij} \vec{r}_{ik}}{r_{ij} r_{ik}} \textrm{ .}
+\end{equation}
+The derivation of the angle $\theta_{ijk}$ with respect to $x^i_n$ is given by
+\begin{equation}
+\partial_{x^i_n} \cos \theta_{ijk} = \Big( r_{ik} r_{ij} - \vec{r}_{ij} \vec{r}_{ik} \frac{r_{ik}}{r_{ij}} \Big) (x_n^i - x_n^j) + \Big( r_{ik} r_{ij} - \vec{r}_{ij} \vec{r}_{ik} \frac{r_{ij}}{r_{ik}} \Big) (x_n^i - x_n^k)
+\label{eq:d_costheta}
+\end{equation}
+
+Using the expressions \eqref{eq:d_cutoff} and \eqref{eq:d_costheta} the derivation of $b_{ij}$ with respect to $x^i_n$ can be written as: