The constants $\epsilon$ and $\sigma$ are usually determined by fitting to experimental data.
$\epsilon$ accounts to the depth of the potential well, where $\sigma$ is regarded as the radius of the particle, also known as the van der Waals radius.
-Writing down the derivation of the Lennard-Jones potential in respect to $x_i$ (the $i$th component of the distance vector $\vec{r}$)
+Writing down the derivation of the Lennard-Jones potential in respect to $x_i$ (the $i$th component of the distance vector ${\bf r}$)
\begin{equation}
\frac{\partial}{\partial x_i} U^{LJ}(r) = 4 \epsilon x_i \Big( -12 \frac{\sigma^{12}}{r^{14}} + 6 \frac{\sigma^6}{r^8} \Big)
\label{eq:lj-d}
\subsubsection{The Tersoff potential}
-Ther Tersoff potential \cite{tersoff1} \ldots
-
+Tersoff proposed an empirical interatomic potential for covalent systems.
+The Tersoff potential explicitly incorporates the dependence of bond order on local envirenments, permitting an improved description of covalent materials.
+Tersoff applied the potential to silicon \cite{tersoff_silicon1,tersoff_silicon2,tersoff_silicon3}, carbon \cite{tersoff_carbon} and also to multicomponent systems like $SiC$ \cite{tersoff_multi}.
+The basic idea is that, in real systems, the bond order depends upon the local environment.
+An atom with many neighbours forms weaker bonds than an atom with few neighbours.
+
+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 potential cutoff function to limit the range of the potential.
+It is designed to have a smooth transition of the potential at distances $R_{ij}$ and $S_{ij}$.
\begin{equation}
-V_{ij} = f_C(r_{ij}) [ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) ]
+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.
\end{equation}
-The total energy is then given by
+The function $b_{ij}$ represents a measure of the bond order, monotonically 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] \\
+b_{ij} & = & \chi_{ij} \Big( 1 + \beta_i^{n_i} \Big[ \sum_{k \ne i,j} f_C (r_{ik}) \omega_{ik} \big[ 1 + c_i^2/d_i^2 - c_i^2/[d_i^2 + (h_i - \cos \theta_{ijk})^2] \big] \Big] \Big)^{-1/2n_i}
+\end{eqnarray}
+where $\theta_{ijk}$ is the bond angle between bonds $ij$ and $ik$.
+This is illustrated in Figure \ref{img:tersoff_angle}.
+
+\printimg{!h}{width=8cm}{tersoff_angle.eps}{Angle between bonds of atoms $i,j$ and $i,k$.}{img:tersoff_angle}
+
+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}
-E = \frac{1}{2} \sum_{i \ne j} V_{ij} \, \textrm{.}
+\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}
-f_R(r_{ij}) = A_{ij} \exp (- \lambda_{ij} r_{ij} ) \\
+\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
\begin{equation}
-f_A(r_{ij}) = -B_{ij} \exp (- \mu_{ij} r_{ij} ) \\
+\cos \theta_{ijk} = \frac{\vec{r}_{ij} \vec{r}_{ik}}{r_{ij} r_{ik}} \textrm{ .}
\end{equation}
-
-The function $f_C$ is the potential cutoff function designed to have a smooth transition between $R_{ij}$ and $S_{ij}$.
+The derivation of the angle $\theta_{ijk}$ with respect to $x^i_n$ is given by
\begin{equation}
-f_C(r_{ij}) = \left\{
- \begin{array}{ll}
- 1 & r_{ij} < R_{ij} \\
- \frac{1}{2} + \frac{1}{2} \cos [ \pi (r_{ij} - R_{ij})/(S_{ij} - R_{ij}) ] & R_{ij} < r_{ij} < S_{ij} \\
- 0 & r_{ij} > S_{ij}
- \end{array} \right.
+\partial_{x^i_n} \cos \theta_{ijk} = \Big( r_{ik} r_{ij} - \vec{r}_{ij} \vec{r}_{ik} \frac{r_{ik}}{r_{ij}} \Big) (x_n^i - x_n^j) + \Big( r_{ik} r_{ij} - \vec{r}_{ij} \vec{r}_{ik} \frac{r_{ij}}{r_{ik}} \Big) (x_n^i - x_n^k)
+\label{eq:d_costheta}
\end{equation}
+Using the expressions \eqref{eq:d_cutoff} and \eqref{eq:d_costheta} the derivation of $b_{ij}$ with respect to $x^i_n$ can be written as:
+\begin{eqnarray}
+\partial_{x^i_n} b_{ij} & = &
+- \frac{1}{2n_i} \chi_{ij} \Bigg( 1 + \beta_i^{n_i} \Bigg[ \sum_{k \ne i,j} \bigg( f_C(r_{ik}) \omega_{ik} \Big( 1 + \frac{c_i^2}{d_i^2} - \frac{c_i^2}{d_i^2 + (h_i - \cos \theta_{ijk})^2} \Big) \bigg)^{n_i} \Bigg] \Bigg)^{-\frac{1}{2n_i} - 1} \times \nonumber\\
+&& \times n_i \beta_i^{n_i} \sum_{k \ne i,j} \Bigg( \Bigg[ f_C(r_{ik}) \omega_{ik} \Big( 1 + \frac{c_i^2}{d_i^2} - \frac{c_i^2}{d_i^2 + (h_i - \cos \theta_{ijk})^2} \Big) \Bigg]^{n_i -1} \times \nonumber\\
+&& \times \Bigg[ \omega_{ik} \Big( 1 + \frac{c_i^2}{d_i^2} - \frac{c_i^2}{d_i^2 + (h_i - \cos \theta_{ijk})^2} \Big) \partial_{x^i_n} f_C(r_{ik}) - \nonumber\\
+&& - f_C(r_{ik}) \omega_{ik} \frac{2 c_i^2 (h_i - \cos \theta_{ijk})}{(d_i^2 + (h_i - \cos \theta_{ijk})^2)^2} \partial_{x^i_n} \cos \theta_{ijk} \Bigg]
+\end{eqnarray}
+
\subsubsection{The Brenner potential}
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