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\documentstyle[12pt]{article}
\begin{document}
\begin{center}
\large{Additional documentation for the RE-squared ellipsoidal potential \\
as implemented in LAMMPS}
\end{center}
\centerline{Mike Brown, Sandia National Labs, October 2007}
\vspace{0.3in}
Let the shape matrices $\mathbf{S}_i=\mbox{diag}(a_i, b_i, c_i)$ be
given by the ellipsoid radii. Let the relative energy matrices
$\mathbf{E}_i = \mbox{diag} (\epsilon_{ia}, \epsilon_{ib},
\epsilon_{ic})$ be given by the relative well depths
(dimensionless energy scales inversely proportional to the well-depths
of the respective orthogonal configurations of the interacting molecules).
Let $\mathbf{A}_1$ and $\mathbf{A}_2$ be the transformation matrices
from the simulation box frame to the body frame and $\mathbf{r}$
be the center to center vector between the particles. Let $A_{12}$ be
the Hamaker constant for the interaction given in LJ units by
$A_{12}=4\pi^2\epsilon_{\mathrm{LJ}}(\rho\sigma^3)^2$.
\vspace{0.3in}
The RE-squared anisotropic interaction between pairs of
ellipsoidal particles is given by
$$ U=U_A+U_R, $$
$$ U_\alpha=\frac{A_{12}}{m_\alpha}(\frac\sigma{h})^{n_\alpha}
(1+o_\alpha\eta\chi\frac\sigma{h}) \times \prod_i{
\frac{a_ib_ic_i}{(a_i+h/p_\alpha)(b_i+h/p_\alpha)(c_i+h/p_\alpha)}}, $$
$$ m_A=-36, n_A=0, o_A=3, p_A=2, $$
$$ m_R=2025, n_R=6, o_R=45/56, p_R=60^{1/3}, $$
$$ \chi = 2 \hat{\mathbf{r}}^T \mathbf{B}^{-1}
\hat{\mathbf{r}}, $$
$$ \hat{\mathbf{r}} = { \mathbf{r} } / |\mathbf{r}|, $$
$$ \mathbf{B} = \mathbf{A}_1^T \mathbf{E}_1 \mathbf{A}_1 +
\mathbf{A}_2^T \mathbf{E}_2 \mathbf{A}_2 $$
$$ \eta = \frac{ \det[\mathbf{S}_1]/\sigma_1^2+
det[\mathbf{S}_2]/\sigma_2^2}{[\det[\mathbf{H}]/
(\sigma_1+\sigma_2)]^{1/2}}, $$
$$ \sigma_i = (\hat{\mathbf{r}}^T\mathbf{A}_i^T\mathbf{S}_i^{-2}
\mathbf{A}_i\hat{\mathbf{r}})^{-1/2}, $$
$$ \mathbf{H} = \frac{1}{\sigma_1}\mathbf{A}_1^T \mathbf{S}_1^2 \mathbf{A}_1 +
\frac{1}{\sigma_2}\mathbf{A}_2^T \mathbf{S}_2^2 \mathbf{A}_2 $$
Here, we use the distance of closest approach approximation given by the
Perram reference, namely
$$ h = |r| - \sigma_{12}, $$
$$ \sigma_{12} = [ \frac{1}{2} \hat{\mathbf{r}}^T
\mathbf{G}^{-1} \hat{\mathbf{r}}]^{ -1/2 }, $$
and
$$ \mathbf{G} = \mathbf{A}_1^T \mathbf{S}_1^2 \mathbf{A}_1 +
\mathbf{A}_2^T \mathbf{S}_2^2 \mathbf{A}_2 $$
\vspace{0.3in}
The RE-squared anisotropic interaction between a
ellipsoidal particle and a Lennard-Jones sphere is defined
as the $\lim_{a_2->0}U$ under the constraints that
$a_2=b_2=c_2$ and $\frac{4}{3}\pi a_2^3\rho=1$:
$$ U_{\mathrm{elj}}=U_{A_{\mathrm{elj}}}+U_{R_{\mathrm{elj}}}, $$
$$ U_{\alpha_{\mathrm{elj}}}=(\frac{3\sigma^3c_\alpha^3}
{4\pi h_{\mathrm{elj}}^3})\frac{A_{12_{\mathrm{elj}}}}
{m_\alpha}(\frac\sigma{h_{\mathrm{elj}}})^{n_\alpha}
(1+o_\alpha\chi_{\mathrm{elj}}\frac\sigma{h_{\mathrm{elj}}}) \times
\frac{a_1b_1c_1}{(a_1+h_{\mathrm{elj}}/p_\alpha)
(b_1+h_{\mathrm{elj}}/p_\alpha)(c_1+h_{\mathrm{elj}}/p_\alpha)}, $$
$$ A_{12_{\mathrm{elj}}}=4\pi^2\epsilon_{\mathrm{LJ}}(\rho\sigma^3), $$
with $h_{\mathrm{elj}}$ and $\chi_{\mathrm{elj}}$ calculated as above
by replacing $B$ with $B_{\mathrm{elj}}$ and $G$ with $G_{\mathrm{elj}}$:
$$ \mathbf{B}_{\mathrm{elj}} = \mathbf{A}_1^T \mathbf{E}_1 \mathbf{A}_1 + I, $$
$$ \mathbf{G}_{\mathrm{elj}} = \mathbf{A}_1^T \mathbf{S}_1^2 \mathbf{A}_1.$$
\vspace{0.3in}
The interaction between two LJ spheres is calculated as:
$$
U_{\mathrm{lj}} = 4 \epsilon \left[ \left(\frac{\sigma}{|\mathbf{r}|}\right)^{12} -
\left(\frac{\sigma}{|\mathbf{r}|}\right)^6 \right]
$$
\vspace{0.3in}
The analytic derivatives are used for all force and torque calculation.
\end{document}
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