-In order to calculate the forces the derivation of the potential with respect to $x^i_n$ (the $n$th component of the position vector of atom $i$ $\equiv$ ${\bf r}_i$) has to be known.
-This is gradually done in the following.
-The $n$th component of the force acting on atom $i$ is
-\begin{eqnarray}
-F_n^i & = & - \frac{\partial}{\partial x_n^i} \sum_{j \neq i} V_{ij} \nonumber\\
- & = & \sum_{j \neq i} \Big( \partial_{x_n^i} f_C(r_{ij}) \big[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \big] + \nonumber\\
-& & + f_C(r_{ij}) \big[ \partial_{x_n^i} f_R(r_{ij}) + b_{ij} \partial_{x_n^i} f_A(r_{ij}) + f_A(r_{ij}) \partial_{x_n^i} b_{ij} \big] \Big) \textrm{ .}
-\end{eqnarray}
-For the implementation it is helpful to seperate the two and three body terms.
-\begin{eqnarray}
-F_n^i & = & \sum_{j \neq i} \Big( f_R(r_{ij}) \partial_{x_n^i} f_C(r_{ij}) + f_C(r_{ij}) \partial_{x_n^i} f_R(r_{ij}) \Big) + \nonumber\\
-& + & \sum_{j \neq i} \Big( \partial_{x_n^i} f_C(r_{ij}) b_{ij} f_A(r_{ij}) + f_C(r_{ij}) \big[ b_{ij} \partial_{x_n^i} f_A(r_{ij}) + f_A(r_{ij}) \partial_{x_n^i} b_{ij} \big] \Big)
-\end{eqnarray}
-The cutoff function $f_C$ derivated with repect to $x^i_n$ is
-\begin{equation}
-\partial_{x^i_n} f_C(r_{ij}) =
- - \frac{1}{2} \sin \Big( \pi (r_{ij} - R_{ij}) / (S_{ij} - R_{ij}) \Big) \frac{\pi (x^i_n - x^j_n)}{(S_{ij} - R_{ij}) r_{ij}}
-\label{eq:d_cutoff}
-\end{equation}
-for $R_{ij} < r_{ij} < S_{ij}$ and otherwise zero.
-The derivations of the repulsive and attractive part are:
-\begin{eqnarray}
-\partial_{x_n^i} f_R(r_{ij}) & = & - \lambda_{ij} \frac{x_n^i - x_n^j}{r_{ij}} A_{ij} \exp (-\lambda_{ij} r_{ij})\\
-\partial_{x_n^i} f_A(r_{ij}) & = & \mu_{ij} \frac{x_n^i - x_n^j}{r_{ij}} B_{ij} \exp (-\mu_{ij} r_{ij}) \textrm{ .}
-\end{eqnarray}
-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
+Here comes an explanation, energy per bond monotonically decreasing with the amount of bonds and so on and so on \ldots
+
+The force acting on atom $i$ is given by the derivative of the potential energy.
+For a three body potential ($V_{ij} \neq V{ji}$) the derivation is of the form