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 Fig. \ref{img:tersoff_angle}.
+This is illustrated in Fig.~\ref{img:tersoff_angle}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=8cm]{tersoff_angle.eps}
\caption[Insertion positions for interstitial defect atoms in the diamond lattice.]{Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}), \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) and bond-centered ({\color{cyan}$\bullet$}) interstitial defect atom in the diamond lattice. The black dots correspond to the lattice atoms and the blue lines indicate the covalent bonds of the perfect diamond structure.}
\label{fig:basics:ins_pos}
\end{figure}
-The respective estimated interstitial insertion positions for various interstitial structures in a diamond lattice are displayed in Fig. \ref{fig:basics:ins_pos}.
+The respective estimated interstitial insertion positions for various interstitial structures in a diamond lattice are displayed in Fig.~\ref{fig:basics:ins_pos}.
The labels of the interstitial types indicate their positions in the interstitial lattice.
In a dumbbell (DB) configuration two atoms share a single lattice site along a certain direction that is also comprehended in the label of the defect.
For the DB configurations the nearest atom of the bulk lattice is slightly displaced by $(0,0,-1/8)$ and $(-1/8,-1/8,0)$ of the unit cell length respectively.
\end{figure}
One possibility to compute the migration path from one stable configuration into another one is provided by the constrained relaxation technique (CRT) \cite{kaukonen98}.
The atoms involving great structural changes in the diffusion process are moved stepwise from the starting to the final position and relaxation after each step is only allowed in the plane perpendicular to the direction of the vector connecting its starting and final position.
-This is illustrated in Fig. \ref{fig:basics:crto}.
+This is illustrated in Fig.~\ref{fig:basics:crto}.
The number of steps required for smooth transitions depends on the shape of the potential energy surface.
No constraints are applied to the remaining atoms to allow for the relaxation of the surrounding lattice.
To prevent the remaining lattice to shift according to the displacement of the defect, however, some atoms far away from the defect region should be fixed in all three coordinate directions.
For some structures even the expected final configurations are not obtained.
Thus, the method mentioned above is adjusted adding further constraints in order to obtain smooth transitions with respect to energy and structure.
In the modified method all atoms are stepwise displaced towards their final positions.
-In addition to this, relaxation of each individual atom is only allowed in the plane perpendicular to the last individual displacement vector, as displayed in Fig. \ref{fig:basics:crtm}.
+In addition to this, relaxation of each individual atom is only allowed in the plane perpendicular to the last individual displacement vector, as displayed in Fig.~\ref{fig:basics:crtm}.
In the modified version respective energies could be higher than the real ones due to the additional constraints that prevent a more adequate relaxation until the final configuration is reached.
-Structures of maximum configurational energy do not necessarily constitute saddle point configurations, i .e. the method does not guarantee to find the true minimum energy path.
+Structures of maximum configurational energy do not necessarily constitute saddle point configurations, i.e.\ the method does not guarantee to find the true minimum energy path.
Whether a saddle point configuration and, thus, the minimum energy path is obtained by the CRT method, needs to be verified by calculating the respective vibrational modes.
Modifications used to add the CRT feature to the \textsc{vasp} code and a short instruction on how to use it can be found in appendix \ref{app:patch_vasp}.
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