forked from lijiext/lammps
34 lines
1.3 KiB
TeX
34 lines
1.3 KiB
TeX
\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}^3 (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}
|