+Tersoff incorporated the concept of bond order in a three-body potential formalism.
+The interatomic potential is taken to have the form
+\begin{eqnarray}
+E & = & \sum_i E_i = \frac{1}{2} \sum_{i \ne j} V_{ij} \textrm{ ,} \\
+V_{ij} & = & f_C(r_{ij}) [ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) ] \textrm{ .}
+\end{eqnarray}
+$E$ is the total energy of the system, constituted either by the sum over the site energies $E_i$ or by the bond energies $V_{ij}$.
+The indices $i$ and $j$ correspond to the atoms of the system with $r_{ij}$ being the distance from atom $i$ to atom $j$.
+The functions $f_R$ and $f_A$ represent a repulsive and an attractive pair potential.
+The repulsive part is due to the orthogonalization energy of overlapped atomic wave functions.
+The attractive part is associated with the bonding.
+\begin{eqnarray}
+f_R(r_{ij}) & = & A_{ij} \exp (- \lambda_{ij} r_{ij} ) \\
+f_A(r_{ij}) & = & -B_{ij} \exp (- \mu_{ij} r_{ij} )
+\end{eqnarray}
+The function $f_C$ is the a cutoff function to limit the range of interaction to nearest neighbors.
+It is designed to have a smooth transition of the potential at distances $R_{ij}$ and $S_{ij}$.
+\begin{equation}
+f_C(r_{ij}) = \left\{
+ \begin{array}{ll}
+ 1, & r_{ij} < R_{ij} \\
+ \frac{1}{2} + \frac{1}{2} \cos \Big[ \pi (r_{ij} - R_{ij})/(S_{ij} - R_{ij}) \Big], & R_{ij} < r_{ij} < S_{ij} \\
+ 0, & r_{ij} > S_{ij}
+ \end{array} \right.
+\label{eq:basics:fc}
+\end{equation}
+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 $\omega$, 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-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-C interaction.
+
+For this reason, Erhart and Albe provide a reparametrization of the Tersoff potential based on three independently fitted parameter sets for the Si-Si, C-C and Si-C interaction \cite{albe_sic_pot}.
+The functional form is similar to the one proposed by Tersoff.
+Differences in the energy functional and the force evaluation routine are pointed out in appendix \ref{app:d_tersoff}.
+Concerning Si the elastic properties of the diamond phase as well as the structure and energetics of the dimer are reproduced very well.
+The new parameter set for the C-C interaction yields improved dimer properties while at the same time delivers a description of the bulk phase similar to the Tersoff potential.
+The potential succeeds in the description of the low as well as high coordinated structures.
+The description of elastic properties of SiC is improved with respect to the potentials available in literature.
+Defect properties are only fairly reproduced but the description is comparable to previously published potentials.
+It is claimed that the potential enables modeling of widely different configurations and transitions among these and has recently been used to simulate the inert gas condensation of Si-C nanoparticles \cite{erhart04}.
+Therefore the Erhart/Albe (EA) potential is considered the superior analytical bond order potential to study the SiC precipitation and associated processes in Si.
+
+\subsection{Verlet integration}
+\label{subsection:integrate_algo}
+
+A numerical method to integrate Newton's equation of motion was presented by Verlet in 1967 \cite{verlet67}.
+The idea of the so-called Verlet and a variant, the velocity Verlet algorithm, which additionaly generates directly the velocities, is explained in the following.
+Starting point is the Taylor series for the particle positions at time $t+\delta t$ and $t-\delta t$
+\begin{equation}
+\vec{r}_i(t+\delta t)=
+\vec{r}_i(t)+\delta t\vec{v}_i(t)+\frac{\delta t^2}{2m_i}\vec{f}_i(t)+
+\frac{\delta t^3}{6}\vec{b}_i(t) + \mathcal{O}(\delta t^4)
+\label{basics:verlet:taylor1}
+\end{equation}
+\begin{equation}
+\vec{r}_i(t-\delta t)=
+\vec{r}_i(t)-\delta t\vec{v}_i(t)+\frac{\delta t^2}{2m_i}\vec{f}_i(t)-
+\frac{\delta t^3}{6}\vec{b}_i(t) + \mathcal{O}(\delta t^4)
+\label{basics:verlet:taylor2}
+\end{equation}
+where $\vec{v}_i=\frac{d}{dt}\vec{r}_i$ are the velocities, $\vec{f}_i=m\frac{d}{dt^2}\vec{r}_i$ are the forces and $\vec{b}_i=\frac{d}{dt^3}\vec{r}_i$ are the third derivatives of the particle positions with respect to time.
+The Verlet algorithm is obtained by summarizing and substracting equations \eqref{basics:verlet:taylor1} and \eqref{basics:verlet:taylor2}
+\begin{equation}
+\vec{r}_i(t+\delta t)=
+2\vec{r}_i(t)-\vec{r}_i(t-\delta t)+\frac{\delta t^2}{m_i}\vec{f}_i(t)+
+\mathcal{O}(\delta t^4)
+\end{equation}
+\begin{equation}
+\vec{v}_i(t)=\frac{1}{2\delta t}[\vec{r}_i(t+\delta t)-\vec{r}_i(t-\delta t)]+
+\mathcal{O}(\delta t^3)
+\end{equation}
+the truncation error of which is of order $\delta t^4$ for the positions and $\delta t^3$ for the velocities.
+The velocities, although not used to update the particle positions, are not synchronously determined with the positions but drag behind one step of discretization.
+The Verlet algorithm can be rewritten into an equivalent form, which updates the velocities and positions in the same step.
+The so-called velocity Verlet algorithm is obtained by combining \eqref{basics:verlet:taylor1} with equation \eqref{basics:verlet:taylor2} displaced in time by $+\delta t$