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small fixes
author
hackbard
<hackbard>
Mon, 11 Dec 2006 22:57:35 +0000
(22:57 +0000)
committer
hackbard
<hackbard>
Mon, 11 Dec 2006 22:57:35 +0000
(22:57 +0000)
posic/thesis/d_tersoff.tex
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diff --git
a/posic/thesis/d_tersoff.tex
b/posic/thesis/d_tersoff.tex
index
d1f0488
..
dbc798a
100644
(file)
--- a/
posic/thesis/d_tersoff.tex
+++ b/
posic/thesis/d_tersoff.tex
@@
-74,11
+74,11
@@
In the following all the necessary derivatives to calculate $\nabla_{{\bf r}_i}
& & + f_C(r_{ji}) \big[ \nabla_{{\bf r}_i} f_R(r_{ji}) + b_{ji} \nabla_{{\bf r}_i} f_A(r_{ji}) + f_A(r_{ji}) \nabla_{{\bf r}_i} b_{ji} \big]
\end{eqnarray}
\begin{eqnarray}
& & + f_C(r_{ji}) \big[ \nabla_{{\bf r}_i} f_R(r_{ji}) + b_{ji} \nabla_{{\bf r}_i} f_A(r_{ji}) + f_A(r_{ji}) \nabla_{{\bf r}_i} b_{ji} \big]
\end{eqnarray}
\begin{eqnarray}
-\nabla_{{\bf r}_i} f_R(r_{ji}) & = & - A_{ji} \lambda_{ji} \frac{{\bf r}_{ji}}{r_{ji}} \exp(-\lambda_{ji} r_{ji}) = \nabla_{{\bf r}_i} f_R(r_{ij} \\
+\nabla_{{\bf r}_i} f_R(r_{ji}) & = & - A_{ji} \lambda_{ji} \frac{{\bf r}_{ji}}{r_{ji}} \exp(-\lambda_{ji} r_{ji}) = \nabla_{{\bf r}_i} f_R(r_{ij}
)
\\
\nabla_{{\bf r}_i} f_A(r_{ji}) & = & + B_{ji} \mu_{ji} \frac{{\bf r}_{ji}}{r_{ji}} \exp(-\mu_{ji} r_{ji}) = \nabla_{{\bf r}_i} f_A(r_{ij})
\end{eqnarray}
\begin{equation}
\nabla_{{\bf r}_i} f_A(r_{ji}) & = & + B_{ji} \mu_{ji} \frac{{\bf r}_{ji}}{r_{ji}} \exp(-\mu_{ji} r_{ji}) = \nabla_{{\bf r}_i} f_A(r_{ij})
\end{eqnarray}
\begin{equation}
-\nabla_{{\bf r}_i} f_C(r_{ij}) = f_C(r_{ij}) = \left\{
+\nabla_{{\bf r}_i} f_C(r_{ij}) =
\nabla_{{\bf r}_i}
f_C(r_{ij}) = \left\{
\begin{array}{ll}
- \frac{1}{2} \sin \Big( \frac{\pi(r_{ji}-R_{ji})}{S_{ji}-R_{ji}} \Big) \frac{\pi}{S_{ji}-R_{ji}} \frac{{\bf r}_{ji}}{r_{ji}}, & R_{ji} < r_{ji} < S_{ji} \\
0, & \textrm{else.}
\begin{array}{ll}
- \frac{1}{2} \sin \Big( \frac{\pi(r_{ji}-R_{ji})}{S_{ji}-R_{ji}} \Big) \frac{\pi}{S_{ji}-R_{ji}} \frac{{\bf r}_{ji}}{r_{ji}}, & R_{ji} < r_{ji} < S_{ji} \\
0, & \textrm{else.}
@@
-99,9
+99,9
@@
In the following all the necessary derivatives to calculate $\nabla_{{\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} \\
-\nabla_{{\bf r}_i} b_{jk} & = & - \frac{\chi_{j
i
}}{2} (1+\beta^{n_j} \zeta_{jk}^{n_j})^{-\frac{1}{2n_j}-1} \beta^{n_j} \zeta_{jk}^{n_j-1} \nabla_{{\bf r}_i} \zeta_{jk} \\
+\nabla_{{\bf r}_i} b_{jk} & = & - \frac{\chi_{j
k
}}{2} (1+\beta^{n_j} \zeta_{jk}^{n_j})^{-\frac{1}{2n_j}-1} \beta^{n_j} \zeta_{jk}^{n_j-1} \nabla_{{\bf r}_i} \zeta_{jk} \\
\nabla_{{\bf r}_i} \zeta_{jk} & = & \sum_{l \neq j,k} \big( g(\theta_{jkl}) \nabla_{{\bf r}_i} f_C(r_{jl}) + f_C(r_{jl}) \nabla_{{\bf r}_i} g(\theta_{jkp}) \big) \nonumber \\
\nabla_{{\bf r}_i} \zeta_{jk} & = & \sum_{l \neq j,k} \big( g(\theta_{jkl}) \nabla_{{\bf r}_i} f_C(r_{jl}) + f_C(r_{jl}) \nabla_{{\bf r}_i} g(\theta_{jkp}) \big) \nonumber \\
- & = & f_C(r_{ji}) \nabla_{{\bf r}_i} g(\theta_{jki}) + g(\theta_
jki
) \nabla_{{\bf r}_i} f_C(r_{ji}) \\
+ & = & f_C(r_{ji}) \nabla_{{\bf r}_i} g(\theta_{jki}) + g(\theta_
{jki}
) \nabla_{{\bf r}_i} f_C(r_{ji}) \\
\nabla_{{\bf r}_i} g(\theta_{jki}) & = & - \frac{2(h_j-\cos\theta_{jki})c_j^2}{\big[d_j^2 + (h_j - \cos\theta_{jki})^2\big]^2} \nabla_{{\bf r}_i} (\cos\theta_{jki}) \\
\nabla_{{\bf r}_i} \cos \theta_{jki} & = & \nabla_{{\bf r}_i} \Big( \frac{{\bf r}_{jk} {\bf r}_{ji}}{r_{jk} r_{ji}} \Big) \nonumber \\
& = & \frac{1}{r_{jk} r_{ji}} {\bf r}_{jk} - \frac{\cos\theta_{jki}}{r_{ji}^2} {\bf r}_{ji}
\nabla_{{\bf r}_i} g(\theta_{jki}) & = & - \frac{2(h_j-\cos\theta_{jki})c_j^2}{\big[d_j^2 + (h_j - \cos\theta_{jki})^2\big]^2} \nabla_{{\bf r}_i} (\cos\theta_{jki}) \\
\nabla_{{\bf r}_i} \cos \theta_{jki} & = & \nabla_{{\bf r}_i} \Big( \frac{{\bf r}_{jk} {\bf r}_{ji}}{r_{jk} r_{ji}} \Big) \nonumber \\
& = & \frac{1}{r_{jk} r_{ji}} {\bf r}_{jk} - \frac{\cos\theta_{jki}}{r_{ji}^2} {\bf r}_{ji}