doc/src/Eqs/pair_tersoff_mod.tex

   1 \documentclass[12pt]{article}
   2
   3 \begin{document}
   4
   5 \begin{eqnarray*}
   6   E & = & \frac{1}{2} \sum_i \sum_{j \neq i} V_{ij} \\
   7   V_{ij} & = & f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right] \\
   8   f_C(r) & = & \left\{ \begin{array} {r@{\quad:\quad}l}
   9     1 & r < R - D \\
  10     \frac{1}{2} - \frac{9}{16} \sin \left( \frac{\pi}{2} \frac{r-R}{D} \right) - \frac{1}{16} \sin \left( \frac{3\pi}{2} \frac{r-R}{D} \right) &
  11       R-D < r < R + D \\
  12     0 & r > R + D
  13     \end{array} \right. \\
  14   f_R(r) & = & A \exp (-\lambda_1 r) \\
  15   f_A(r) & = & -B \exp (-\lambda_2 r) \\
  16   b_{ij} & = & \left( 1 + {\zeta_{ij}}^\eta \right)^{-\frac{1}{2n}} \\
  17   \zeta_{ij} & = & \sum_{k \neq i,j} f_C(r_{ik}) g(\theta_{ijk})
  18                    \exp \left[ \alpha (r_{ij} - r_{ik})^\beta \right] \\
  19   g(\theta) & = & c_1 + g_o(\theta) g_a(\theta) \\
  20   g_o(\theta) & = & \frac{c_2 (h - \cos \theta)^2}{c_3 + (h - \cos \theta)^2} \\
  21   g_a(\theta) & = & 1 + c_4 \exp \left[ -c_5 (h - \cos \theta)^2 \right] \\
  22 \end{eqnarray*}
  23
  24 \end{document}