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[lectures/latex.git] / posic / thesis / basics.tex

diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
index 09143b6..d3c91aa 100644 (file)
--- a/posic/thesis/basics.tex
+++ b/posic/thesis/basics.tex
@@ -184,20 +184,20 @@ This is gradually done in the following.
 The $n$th component of the force acting on atom $i$ is
 \begin{eqnarray}
 F_n^i & = & - \frac{\partial}{\partial x_n} \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\\
+ & = & \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)
 \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}{(S_{ij} - R_{ij}) r_{ij}}
 \label{eq:d_cutoff}
 \end{equation}
 for $R_{ij} < r_{ij} < S_{ij}$ and otherwise zero.
 The derivations of the repulsive and attractive part are:
 \begin{eqnarray}
-\partial_{x_n^i} f_R(r_{ij}) & = & - \lambda_{ij} A_{ij} \exp (-\lambda_{ij} r_{ij})\\
-\partial_{x_n^i} f_A(r_{ij}) & = & \mu_{ij} B_{ij} \exp (-\mu_{ij} r_{ij}) \textrm{ .}
+\partial_{x_n^i} f_R(r_{ij}) & = & - \lambda_{ij} \frac{x_n^i - x_n^j}{r_{ij}} A_{ij} \exp (-\lambda_{ij} r_{ij})\\
+\partial_{x_n^i} f_A(r_{ij}) & = & \mu_{ij} \frac{x_n^i - x_n^j}{r_{ij}} B_{ij} \exp (-\mu_{ij} r_{ij}) \textrm{ .}
 \end{eqnarray}
 The angle $\theta_{ijk}$ can be expressed by the atom distances with the law of cosines:
 \begin{eqnarray}