-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}} \times \nonumber\\
- & & \times \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) \label{eq:d_theta}
-\end{eqnarray}
-Using the expressions \eqref{eq:d_cutoff} and \eqref{eq:d_theta} the derivation of $b_{ij}$ with respect to $x^i_n$ can be written as:
+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: