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.
+The elemental parameters were obtained by a 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 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.
+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.
The challenging problem of determining the exact ground-state is now formally reduced to the determination of the $3$-dimensional function $n(\vec{r})$, which minimizes the energy functional.
However, the complexity associated with the many-electron problem is now relocated in the task of finding the well-defined but, in contrast to the potential energy, not explicitly known functional $F[n(\vec{r})]$.
-It is worth to note, that this minimal principle may be regarded as exactification of the TF theory, which is rederived by the approximations
+It is worth to note that this minimal principle may be regarded as exactification of the TF theory, which is rederived by the approximations
\begin{equation}
T=\int n(\vec{r})\frac{3}{10}k_{\text{F}}^2(n(\vec{r}))d\vec{r}
\text{ ,}
\subsection{Pseudopotentials}
As discussed in the last part of the previous section, an extremely large basis set of plane waves would be required to perform an all-electron calculation and a vast amount of computational time would be required to calculate the electronic wave functions.
-It is worth to stress out one more time, that this is mainly due to the orthogonalization wiggles of the wave functions of valence electrons near the nuclei.
+It is worth to stress out one more time that this is mainly due to the orthogonalization wiggles of the wave functions of valence electrons near the nuclei.
Thus, existing core states practically prevent the use of a PW basis set.
However, the core electrons, which are tightly bound to the nuclei, do not contribute significantly to chemical bonding or other physical properties of the solid.
This fact is exploited in the pseudopotential (PP) approach~\cite{cohen70} by removing the core electrons and replacing the atom and the associated strong ionic potential by a pseudoatom and a weaker PP that acts on a set of pseudo wave functions rather than the true valance wave functions.
indeed reveals a contribution to the change in total energy due to the change of the wave functions $\Phi_j$.
However, provided that the $\Phi_j$ are eigenstates of $H$, it is easy to show that the last two terms cancel each other and in the special case of $H=T+V$ the force is given by
\begin{equation}
-\vec{F}_i=-\sum_j \langle \Phi_j | \Phi_j\frac{\partial V}{\partial \vec{R}_i} \rangle
+\vec{F}_i=-\sum_j \langle \Phi_j | \frac{\partial V}{\partial \vec{R}_i} \Phi_j \rangle
\text{ .}
\end{equation}
This is called the Hellmann-Feynman theorem~\cite{feynman39}, which enables the calculation of forces, called the Hellmann-Feynman forces, acting on the nuclei for a given configuration, without the need for evaluating computationally costly energy maps.
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