+As discussed above, $b_{ij}$ represents a measure of the bond order, monotonously decreasing with the coordination of atoms $i$ and $j$.
+It is of the form:
+\begin{eqnarray}
+b_{ij} & = & \chi_{ij} (1 + \beta_i^{n_i} \zeta^{n_i}_{ij})^{-1/2n_i} \\
+\zeta_{ij} & = & \sum_{k \ne i,j} f_C (r_{ik}) \omega_{ik} g(\theta_{ijk}) \\
+g(\theta_{ijk}) & = & 1 + c_i^2/d_i^2 - c_i^2/[d_i^2 + (h_i - \cos \theta_{ijk})^2]
+\end{eqnarray}
+where $\theta_{ijk}$ is the bond angle between bonds $ij$ and $ik$.
+This is illustrated in Figure \ref{img:tersoff_angle}.
+\begin{figure}[t]
+\begin{center}
+\includegraphics[width=8cm]{tersoff_angle.eps}
+\end{center}
+\caption{Angle between bonds of atoms $i,j$ and $i,k$.}
+\label{img:tersoff_angle}
+\end{figure}
+The angular dependence does not give a fixed minimum angle between bonds since the expression is embedded inside the bond order term.
+The relation to the above discussed bond order potential becomes obvious if $\chi=1, \beta=1, n=1, \omega=1$ and $c=0$.
+Parameters with a single subscript correspond to the parameters of the elemental system \cite{tersoff_si3,tersoff_c} while the mixed parameters are obtained by interpolation from the elemental parameters by the arithmetic or geometric mean.
+The elemental parameters were obtained by fit with respect to the cohesive energies of real and hypothetical bulk structures and the bulk modulus and bond length of the diamond structure.
+New parameters for the mixed system are $\chi$, which is used to finetune the strength of heteropolar bonds, and $\omeag$, which is set to one for the C-Si interaction but is available as a feature to permit the application of the potential to more drastically different types of atoms in the future.
+
+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
+\begin{equation}
+\nabla_{{\bf r}_i} E = \frac{1}{2} \big[ \sum_j ( \nabla_{{\bf r}_i} V_{ij} + \nabla_{{\bf r}_i} V_{ji} ) + \sum_k \sum_j \nabla_{{\bf r}_i} V_{jk} \big] \textrm{ .}
+\end{equation}
+The force is then given by
+\begin{equation}
+F^i = - \nabla_{{\bf r}_i} E \textrm{ .}
+\end{equation}
+Details of the Tersoff potential derivative are presented in appendix \ref{app:d_tersoff}.
+
+\subsubsection{Improved analytical bond order potential}
+
+Although the Tersoff potential is one of the most widely used potentials there are some shortcomings.
+Describing the Si interaction Tersoff was unable to find a single parameter set to describe well both, bulk and surface properties.
+Due to this and since the first approach labeled T1 \cite{tersoff_si1} turned out to be unstable \cite{dodson87}, two further parametrizations exist, T2 \cite{tersoff_si2} and T3 \cite{tersoff_si3}.
+While T2 describes well surface properties, T3 yields improved elastic constants and should be used for describing bulk properties.
+However, T3, which is used in the Si/C potential, suffers from an underestimation of the dimer binding energy.
+Similar behavior is found for the C interaction.
+
+For this reason, Erhart and Albe provide a reparametrization of the Tersoff potential based on three independently fitted potentials for the Si-Si, C-C and Si-C interaction \cite{albe_sic_pot}.
+The functional form is nearly the same like the one proposed by Tersoff.
+Differences in the energy functional and the force evaluation routine are pointed out in appendix \ref{app:d_tersoff}.
+For Si ...
+
+\subsection{Statistical ensembles}
+\label{subsection:statistical_ensembles}
+
+By default ... NVE
+
+However, we need to control T -> NVT
+.. and p -> NpT ...
+
+
+\section{Denstiy functional theory}
+\label{section:dft}
+
+\subsection{Hohenberg-Kohn theorem}
+
+\subsection{Born-Oppenheimer (adiabatic) approximation}
+
+\subsection{Effective potential}
+
+\subsection{Kohn-Sham system}
+
+\subsection{Approximations for exchange and correlation}
+
+\subsection{Pseudopotentials}
+
+\section{Modeling of defects}
+\label{section:basics:defects}
+
+\section{Migration paths and diffusion barriers}
+\label{section:basics:migration}