-\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)
+\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:
+\begin{eqnarray}
+\partial_{x^i_n} b_{ij} & = &
+- \frac{1}{2n_i} \chi_{ij} \Bigg( 1 + \beta_i^{n_i} \Bigg[ \sum_{k \ne i,j} \bigg( f_C(r_{ij}) \omega_{ik} \Big( 1 + \frac{c_i^2}{d_i^2} - \frac{c_i^2}{d_i^2 + (h_i - \cos \theta_{ijk})^2} \Big) \bigg)^{n_i} \Bigg] \Bigg)^{-\frac{1}{2n_i} - 1} \times \nonumber\\
+&& \times n_i \beta_i^{n_i} \Bigg[ \sum_{k \ne i,j} f_C(r_{ik}) \omega_{ik} \Big( 1 \frac{c_i^2}{d_i^2} - \frac{c_i^2}{d_i^2 + (h_i - \cos \theta_{ijk})^2} \Big) \Bigg]^{n_i -1} \times \nonumber\\
+&& \times \sum_{k \ne i,j} \Bigg( \omega_{ik} \Big( 1 + \frac{c_i^2}{d_i^2} - \frac{c_i^2}{d_i^2 + (h_i - \cos \theta_{ijk})^2} \Big) \partial_{x^i_n} f_C(r_{ik}) + \nonumber\\
+&& + f_C(r_{ik}) \omega_{ik} (-1) \frac{c_i^2}{(d_i^2 + (h_i - \cos \theta_{ijk})^2)^2} \times \nonumber\\
+&& \times 2 \Big( h_i - \cos \theta_{ijk} \Big) \sin \theta_{ijk} \partial_{x^i_n} \theta_{ijk} \Bigg)