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more fixes
[lectures/latex.git]
/
posic
/
thesis
/
d_tersoff.tex
diff --git
a/posic/thesis/d_tersoff.tex
b/posic/thesis/d_tersoff.tex
index
9a5e76b
..
8492963
100644
(file)
--- a/
posic/thesis/d_tersoff.tex
+++ b/
posic/thesis/d_tersoff.tex
@@
-38,9
+38,9
@@
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
done
.
+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 ${\bf r}_i$}
+ \section
[Derivative of $V_{ij}$ with respect to ${r}_i$]{\boldmath
Derivative of $V_{ij}$ with respect to ${\bf r}_i$}
\begin{eqnarray}
\nabla_{{\bf r}_i} V_{ij} & = & \nabla_{{\bf r}_i} f_C(r_{ij}) \big[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \big] + \nonumber \\
\begin{eqnarray}
\nabla_{{\bf r}_i} V_{ij} & = & \nabla_{{\bf r}_i} f_C(r_{ij}) \big[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \big] + \nonumber \\
@@
-67,7
+67,7
@@
In the following all the necessary derivatives to calculate $\nabla_{{\bf r}_i}
& = & \Big[ \frac{\cos\theta_{ijk}}{r_{ij}^2} - \frac{1}{r_{ij} r_{ik}} \Big] {\bf r}_{ij} + \Big[ \frac{\cos\theta_{ijk}}{r_{ik}^2} - \frac{1}{r_{ij} r_{ik}} \Big] {\bf r}_{ik}
\end{eqnarray}
& = & \Big[ \frac{\cos\theta_{ijk}}{r_{ij}^2} - \frac{1}{r_{ij} r_{ik}} \Big] {\bf r}_{ij} + \Big[ \frac{\cos\theta_{ijk}}{r_{ik}^2} - \frac{1}{r_{ij} r_{ik}} \Big] {\bf r}_{ik}
\end{eqnarray}
- \section
{
Derivative of $V_{ji}$ with respect to ${\bf r}_i$}
+ \section
[Derivative of $V_{ji}$ with respect to ${r}_i$]{\boldmath
Derivative of $V_{ji}$ with respect to ${\bf r}_i$}
\begin{eqnarray}
\nabla_{{\bf r}_i} V_{ji} & = & \nabla_{{\bf r}_i} f_C(r_{ji}) \big[ f_R(r_{ji}) + b_{ji} f_A(r_{ji}) \big] + \nonumber \\
\begin{eqnarray}
\nabla_{{\bf r}_i} V_{ji} & = & \nabla_{{\bf r}_i} f_C(r_{ji}) \big[ f_R(r_{ji}) + b_{ji} f_A(r_{ji}) \big] + \nonumber \\
@@
-95,7
+95,7
@@
In the following all the necessary derivatives to calculate $\nabla_{{\bf r}_i}
& = & \frac{1}{r_{ji} r_{jk}} {\bf r}_{jk} - \frac{\cos\theta_{jik}}{r_{ji}^2} {\bf r}_{ji}
\end{eqnarray}
& = & \frac{1}{r_{ji} r_{jk}} {\bf r}_{jk} - \frac{\cos\theta_{jik}}{r_{ji}^2} {\bf r}_{ji}
\end{eqnarray}
- \section
{
Derivative of $V_{jk}$ with respect to ${\bf r}_i$}
+ \section
[Derivative of $V_{jk}$ with respect to ${r}_i$]{\boldmath
Derivative of $V_{jk}$ with respect to ${\bf r}_i$}
\begin{eqnarray}
\nabla_{{\bf r}_i} V_{jk} & = & f_C(r_{jk}) f_A(r_{jk}) \nabla_{{\bf r}_i} b_{jk} \\
\begin{eqnarray}
\nabla_{{\bf r}_i} V_{jk} & = & f_C(r_{jk}) f_A(r_{jk}) \nabla_{{\bf r}_i} b_{jk} \\
@@
-128,7
+128,7
@@
This poses a more convenient method to obtain the forces
keeping in mind that all the necessary force contributions for atom $i$
are calculated and added in subsequent loops.
keeping in mind that all the necessary force contributions for atom $i$
are calculated and added in subsequent loops.
-\subsection
{
Derivative of $V_{ij}$ with respect to ${\bf r}_j$}
+\subsection
[Derivative of $V_{ij}$ with respect to ${r}_j$]{\boldmath
Derivative of $V_{ij}$ with respect to ${\bf r}_j$}
\begin{eqnarray}
\nabla_{{\bf r}_j} V_{ij} & = &
\begin{eqnarray}
\nabla_{{\bf r}_j} V_{ij} & = &
@@
-155,7
+155,7
@@
The contribution of the bond order term is given by:
\frac{\cos\theta_{ijk}}{r_{ij}^2}{\bf r}_{ij}
\end{eqnarray}
\frac{\cos\theta_{ijk}}{r_{ij}^2}{\bf r}_{ij}
\end{eqnarray}
-\subsection
{
Derivative of $V_{ij}$ with respect to ${\bf r}_k$}
+\subsection
[Derivative of $V_{ij}$ with respect to ${r}_k$]{\boldmath
Derivative of $V_{ij}$ with respect to ${\bf r}_k$}
The derivative of $V_{ij}$ with respect to ${\bf r}_k$ just consists of the
single term
The derivative of $V_{ij}$ with respect to ${\bf r}_k$ just consists of the
single term