From: hackbard Date: Fri, 17 Nov 2006 07:11:38 +0000 (+0000) Subject: ... X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=222b641f21554bfb1e8b02efd71e8c3fc3a03eb4;p=lectures%2Flatex.git ... --- diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex index 1de8827..2f1bf6b 100644 --- a/posic/thesis/basics.tex +++ b/posic/thesis/basics.tex @@ -185,12 +185,17 @@ 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) +& & + 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} \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}{(S_{ij} - R_{ij}) 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.