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lammps/doc/src/Eqs/pair_dipole_sf.tex
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\documentclass[12pt]{article}
\begin{document}
\begin{eqnarray*}
E_{LJ} & = & 4\epsilon \left\{ \left[ \left( \frac{\sigma}{r} \right)^{\!12} -
\left( \frac{\sigma}{r} \right)^{\!6} \right] +
\left[ 6\left( \frac{\sigma}{r_c} \right)^{\!12} -
3\left(\frac{\sigma}{r_c}\right)^{\!6}\right]\left(\frac{r}{r_c}\right)^{\!2}
- 7\left( \frac{\sigma}{r_c} \right)^{\!12} +
4\left( \frac{\sigma}{r_c} \right)^{\!6}\right\} \\
E_{qq} & = & \frac{q_i q_j}{r}\left(1-\frac{r}{r_c}\right)^{\!2} \\
E_{pq} & = & E_{ji} = -\frac{q}{r^3} \left[ 1 -
3\left(\frac{r}{r_c}\right)^{\!2} +
2\left(\frac{r}{r_c}\right)^{\!3}\right] (\vec{p}\bullet\vec{r}) \\
E_{qp} & = & E_{ij} = \frac{q}{r^3} \left[ 1 -
3\left(\frac{r}{r_c}\right)^{\!2} +
2\left(\frac{r}{r_c}\right)^{\!3}\right] (\vec{p}\bullet\vec{r}) \\
E_{pp} & = & \left[1-4\left(\frac{r}{r_c}\right)^{\!3} +
3\left(\frac{r}{r_c}\right)^{\!4}\right]\left[\frac{1}{r^3}
(\vec{p_i} \bullet \vec{p_j}) - \frac{3}{r^5}
(\vec{p_i} \bullet \vec{r}) (\vec{p_j} \bullet \vec{r})\right] \\
\end{eqnarray*}
\begin{eqnarray*}
F_{LJ} & = & \left\{\left[48\epsilon \left(\frac{\sigma}{r}\right)^{\!12} -
24\epsilon \left(\frac{\sigma}{r}\right)^{\!6} \right]\frac{1}{r^2} -
\left[48\epsilon \left(\frac{\sigma}{r_c}\right)^{\!12} - 24\epsilon
\left(\frac{\sigma}{r_c}\right)^{\!6} \right]\frac{1}{r_c^2}\right\}\vec{r}\\
F_{qq} & = & \frac{q_i q_j}{r}\left(\frac{1}{r^2} -
\frac{1}{r_c^2}\right)\vec{r} \\
F_{pq} &=& F_{ij } = -\frac{3q}{r^5} \left[ 1 -
\left(\frac{r}{r_c}\right)^{\!2}\right](\vec{p}\bullet\vec{r})\vec{r} +
\frac{q}{r^3}\left[1-3\left(\frac{r}{r_c}\right)^{\!2} +
2\left(\frac{r}{r_c}\right)^{\!3}\right] \vec{p} \\
F_{qp} &=& F_{ij} = \frac{3q}{r^5} \left[ 1 -
\left(\frac{r}{r_c}\right)^{\!2}\right] (\vec{p}\bullet\vec{r})\vec{r} -
\frac{q}{r^3}\left[1-3\left(\frac{r}{r_c}\right)^{\!2} +
2\left(\frac{r}{r_c}\right)^{\!3}\right] \vec{p} \\
F_{pp} & = &\frac{3}{r^5}\Bigg\{\left[1-\left(\frac{r}{r_c}\right)^{\!4}\right]
\left[(\vec{p_i}\bullet\vec{p_j}) - \frac{3}{r^2} (\vec{p_i}\bullet\vec{r})
(\vec{p_j} \bullet \vec{r})\right] \vec{r} + \\
& & \left[1 -
4\left(\frac{r}{r_c}\right)^{\!3}+3\left(\frac{r}{r_c}\right)^{\!4}\right]
\left[ (\vec{p_j} \bullet \vec{r}) \vec{p_i} + (\vec{p_i} \bullet \vec{r})
\vec{p_j} -\frac{2}{r^2} (\vec{p_i} \bullet \vec{r})
(\vec{p_j} \bullet \vec{r})\vec{r}\right] \Bigg\} \\
\end{eqnarray*}
\end{document}
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