From 222b641f21554bfb1e8b02efd71e8c3fc3a03eb4 Mon Sep 17 00:00:00 2001
From: hackbard <hackbard>
Date: Fri, 17 Nov 2006 07:11:38 +0000
Subject: [PATCH] ...

---
 posic/thesis/basics.tex | 9 +++++++--
 1 file changed, 7 insertions(+), 2 deletions(-)

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.
-- 
2.20.1