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pair_tersoff_zbl.tex
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rLAMMPS lammps
pair_tersoff_zbl.tex
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\documentclass[12pt]{article}
\begin{document}
\begin{eqnarray*}
E & = & \frac{1}{2} \sum_i \sum_{j \neq i} V_{ij} \\
V_{ij} & = & (1 - f_F(r_{ij})) V^{ZBL}_{ij} + f_F(r_{ij}) V^{Tersoff}_{ij} \\
f_F(r_{ij}) & = & \frac{1}{1 + e^{-A_F(r_{ij} - r_C)}}\\
\\
\\
V^{ZBL}_{ij} & = & \frac{1}{4\pi\epsilon_0} \frac{Z_1 Z_2 \,e^2}{r_{ij}} \phi(r_{ij}/a) \\
a & = & \frac{0.8854\,a_0}{Z_{1}^{0.23} + Z_{2}^{0.23}}\\
\phi(x) & = & 0.1818e^{-3.2x} + 0.5099e^{-0.9423x} + 0.2802e^{-0.4029x} + 0.02817e^{-0.2016x}\\
\\
\\
V^{Tersoff}_{ij} & = & f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right] \\
f_C(r) & = & \left\{ \begin{array} {r@{\quad:\quad}l}
1 & r < R - D \\
\frac{1}{2} - \frac{1}{2} \sin \left( \frac{\pi}{2} \frac{r-R}{D} \right) &
R-D < r < R + D \\
0 & r > R + D
\end{array} \right. \\
f_R(r) & = & A \exp (-\lambda_1 r) \\
f_A(r) & = & -B \exp (-\lambda_2 r) \\
b_{ij} & = & \left( 1 + \beta^n {\zeta_{ij}}^n \right)^{-\frac{1}{2n}} \\
\zeta_{ij} & = & \sum_{k \neq i,j} f_C(r_{ik}) g(\theta_{ijk})
\exp \left[ {\lambda_3}^m (r_{ij} - r_{ik})^m \right] \\
g(\theta) & = & \gamma_{ijk} \left( 1 + \frac{c^2}{d^2} -
\frac{c^2}{\left[ d^2 +
(\cos \theta - \cos \theta_0)^2\right]} \right)
\end{eqnarray*}
\end{document}
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