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[lectures/latex.git]
/
posic
/
thesis
/
d_tersoff.tex
diff --git
a/posic/thesis/d_tersoff.tex
b/posic/thesis/d_tersoff.tex
index
ca23535
..
b031be2
100644
(file)
--- a/
posic/thesis/d_tersoff.tex
+++ b/
posic/thesis/d_tersoff.tex
@@
-38,7
+38,7
@@
For a three body potential, if $V_{ij}$ is not equal to $V_{ji}$, the derivative
\begin{equation}
\nabla_{{\bf r}_i} E = \frac{1}{2} \big[ \sum_j ( \nabla_{{\bf r}_i} V_{ij} + \nabla_{{\bf r}_i} V_{ji} ) + \sum_k \sum_j \nabla_{{\bf r}_i} V_{jk} \big] \textrm{ .}
\end{equation}
\begin{equation}
\nabla_{{\bf r}_i} E = \frac{1}{2} \big[ \sum_j ( \nabla_{{\bf r}_i} V_{ij} + \nabla_{{\bf r}_i} V_{ji} ) + \sum_k \sum_j \nabla_{{\bf r}_i} V_{jk} \big] \textrm{ .}
\end{equation}
-In the following all the necessary derivatives to calculate $\nabla_{{\bf r}_i} E$ are written down.
+In the following
,
all the necessary derivatives to calculate $\nabla_{{\bf r}_i} E$ are written down.
\section[Derivative of $V_{ij}$ with respect to ${r}_i$]{\boldmath Derivative of $V_{ij}$ with respect to ${\bf r}_i$}
\section[Derivative of $V_{ij}$ with respect to ${r}_i$]{\boldmath Derivative of $V_{ij}$ with respect to ${\bf r}_i$}
@@
-138,7
+138,7
@@
are calculated and added in subsequent loops.
b_{ij} \nabla_{{\bf r}_j} f_A(r_{ij}) +
f_A(r_{ij}) \nabla_{{\bf r}_j} b_{ij} \big]
\end{eqnarray}
b_{ij} \nabla_{{\bf r}_j} f_A(r_{ij}) +
f_A(r_{ij}) \nabla_{{\bf r}_j} b_{ij} \big]
\end{eqnarray}
-Using the equality $\nabla_{{\bf r}_i} r_{ij}=-\nabla_{{\bf r}_j} r_{ij}$
+Using the equality $\nabla_{{\bf r}_i} r_{ij}=-\nabla_{{\bf r}_j} r_{ij}$
,
the following relations are valid:
\begin{eqnarray}
\nabla_{{\bf r}_j} f_R(r_{ij}) &=& - \nabla_{{\bf r}_i} f_R(r_{ij}) \\
the following relations are valid:
\begin{eqnarray}
\nabla_{{\bf r}_j} f_R(r_{ij}) &=& - \nabla_{{\bf r}_i} f_R(r_{ij}) \\