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\documentstyle[12pt]{article}
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\begin{document}
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\begin{center}
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\large{Additional documentation for the Gay-Berne ellipsoidal potential \\
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as implemented in LAMMPS}
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\end{center}
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\centerline{Mike Brown, Sandia National Labs, April 2007}
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\vspace{0.3in}
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The Gay-Berne anisotropic LJ interaction between pairs of dissimilar
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ellipsoidal particles is given by
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$$ U ( \mathbf{A}_1, \mathbf{A}_2, \mathbf{r}_{12} ) = U_r (
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\mathbf{A}_1, \mathbf{A}_2, \mathbf{r}_{12}, \gamma ) \cdot \eta_{12} (
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\mathbf{A}_1, \mathbf{A}_2, \upsilon ) \cdot \chi_{12} ( \mathbf{A}_1,
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\mathbf{A}_2, \mathbf{r}_{12}, \mu ) $$
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where $\mathbf{A}_1$ and $\mathbf{A}_2$ are the transformation
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matrices from the simulation box frame to the body frame and
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$\mathbf{r}_{12}$ is the center to center vector between the
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particles. $U_r$ controls the shifted distance dependent interaction
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based on the distance of closest approach of the two particles
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($h_{12}$) and the user-specified shift parameter gamma:
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$$ U_r = 4 \epsilon ( \varrho^{12} - \varrho^6) $$
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$$ \varrho = \frac{\sigma}{ h_{12} + \gamma \sigma} $$
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Let the shape matrices $\mathbf{S}_i=\mbox{diag}(a_i, b_i, c_i)$ be
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given by the ellipsoid radii. The $\eta$ orientation-dependent energy
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based on the user-specified exponent $\upsilon$ is given by
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$$ \eta_{12} = [ \frac{ 2 s_1 s_2 }{\det ( \mathbf{G}_{12} )}]^{
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\upsilon / 2 } , $$
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$$ s_i = [a_i b_i + c_i c_i][a_i b_i]^{ 1 / 2 }, $$
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and
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$$ \mathbf{G}_{12} = \mathbf{A}_1^T \mathbf{S}_1^2 \mathbf{A}_1 +
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\mathbf{A}_2^T \mathbf{S}_2^2 \mathbf{A}_2 = \mathbf{G}_1 +
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\mathbf{G}_2. $$
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Let the relative energy matrices $\mathbf{E}_i = \mbox{diag}
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(\epsilon_{ia}, \epsilon_{ib}, \epsilon_{ic})$ be given by
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the relative well depths (dimensionless energy scales
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inversely proportional to the well-depths of the respective
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orthogonal configurations of the interacting molecules). The
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$\chi$ orientation-dependent energy based on the user-specified
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exponent $\mu$ is given by
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$$ \chi_{12} = [2 \hat{\mathbf{r}}_{12}^T \mathbf{B}_{12}^{-1}
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\hat{\mathbf{r}}_{12}]^\mu, $$
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$$ \hat{\mathbf{r}}_{12} = { \mathbf{r}_{12} } / |\mathbf{r}_{12}|, $$
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and
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$$ \mathbf{B}_{12} = \mathbf{A}_1^T \mathbf{E}_1^2 \mathbf{A}_1 +
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\mathbf{A}_2^T \mathbf{E}_2^2 \mathbf{A}_2 = \mathbf{B}_1 +
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\mathbf{B}_2. $$
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Here, we use the distance of closest approach approximation given by the
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Perram reference, namely
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$$ h_{12} = r - \sigma_{12} ( \mathbf{A}_1, \mathbf{A}_2,
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\mathbf{r}_{12} ), $$
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$$ r = |\mathbf{r}_{12}|, $$
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and
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$$ \sigma_{12} = [ \frac{1}{2} \hat{\mathbf{r}}_{12}^T
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\mathbf{G}_{12}^{-1} \hat{\mathbf{r}}_{12}.]^{ -1/2 } $$
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Forces and Torques: Because the analytic forces and torques have not
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been published for this potential, we list them here:
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$$ \mathbf{f} = - \eta_{12} ( U_r \cdot { \frac{\partial \chi_{12}
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}{\partial r} } + \chi_{12} \cdot { \frac{\partial U_r }{\partial r} }
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) $$
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where the derivative of $U_r$ is given by (see Allen reference)
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$$ \frac{\partial U_r }{\partial r} = \frac{ \partial U_{SLJ} }{
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\partial r } \hat{\mathbf{r}}_{12} + r^{-2} \frac{ \partial U_{SLJ} }{
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\partial \varphi } [ \mathbf{\kappa} - ( \mathbf{\kappa}^T \cdot
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\hat{\mathbf{r}}_{12}) \hat{\mathbf{r}}_{12} ], $$
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$$ \frac{ \partial U_{SLJ} }{ \partial \varphi } = 24 \epsilon ( 2
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\varrho^{13} - \varrho^7 ) \sigma_{12}^3 / 2 \sigma, $$
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$$ \frac{ \partial U_{SLJ} }{ \partial r } = 24 \epsilon ( 2
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\varrho^{13} - \varrho^7 ) / \sigma, $$
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and
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$$ \mathbf{\kappa} = \mathbf{G}_{12}^{-1} \cdot \mathbf{r}_{12}. $$
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The derivate of the $\chi$ term is given by
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$$ \frac{\partial \chi_{12} }{\partial r} = - r^{-2} \cdot 4.0 \cdot [
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\mathbf{\iota} - ( \mathbf{\iota}^T \cdot \hat{\mathbf{r}}_{12} )
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\hat{\mathbf{r}}_{12} ] \cdot \mu \cdot \chi_{12}^{ ( \mu -1 ) / \mu
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}, $$
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and
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$$ \mathbf{\iota} = \mathbf{B}_{12}^{-1} \cdot \mathbf{r}_{12}. $$
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The torque is given by:
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$$ \mathbf{\tau}_i = U_r \eta_{12} \frac{ \partial \chi_{12} }{
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\partial \mathbf{q}_i } + \chi_{12} ( U_r \frac{ \partial \eta_{12} }{
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\partial \mathbf{q}_i } + \eta_{12} \frac{ \partial U_r }{ \partial
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\mathbf{q}_i } ), $$
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$$ \frac{ \partial U_r }{ \partial \mathbf{q}_i } = \mathbf{A}_i \cdot
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(- \mathbf{\kappa}^T \cdot \mathbf{G}_i \times \mathbf{f}_k ), $$
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$$ \mathbf{f}_k = - r^{-2} \frac{ \delta U_{SLJ} }{ \delta \varphi }
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\mathbf{\kappa}, $$
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and
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$$ \frac{ \partial \chi_{12} }{ \partial \mathbf{q}_i } = 4.0 \cdot
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r^{-2} \cdot \mathbf{A}_i (- \mathbf{\iota}^T \cdot \mathbf{B}_i
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\times \mathbf{\iota} ). $$
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For the derivative of the $\eta$ term, we were unable to find a matrix
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expression due to the determinant. Let $a_{mi}$ be the mth row of the
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rotation matrix $A_i$. Then,
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$$ \frac{ \partial \eta_{12} }{ \partial \mathbf{q}_i } = \mathbf{A}_i
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\cdot \sum_m \mathbf{a}_{mi} \times \frac{ \partial \eta_{12} }{
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\partial \mathbf{a}_{mi} } = \mathbf{A}_i \cdot \sum_m \mathbf{a}_{mi}
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\times \mathbf{d}_{mi}, $$
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where $d_mi$ represents the mth row of a derivative matrix $D_i$,
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$$ \mathbf{D}_i = - \frac{1}{2} \cdot ( \frac{2s1s2}{\det (
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\mathbf{G}_{12} ) } )^{ \upsilon / 2 } \cdot {\frac{\upsilon}{\det (
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\mathbf{G}_{12} ) }} \cdot \mathbf{E}, $$
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where the matrix $E$ gives the derivate with respect to the rotation
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matrix,
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$$ \mathbf{E} = [ e_{my} ] = \frac{ \partial \eta_{12} }{ \partial
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\mathbf{A}_i }, $$
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and
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$$ e_{my} = \det ( \mathbf{G}_{12} ) \cdot \mbox{trace} [
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\mathbf{G}_{12}^{-1} \cdot ( \hat{\mathbf{p}}_y \otimes \mathbf{a}_m +
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\mathbf{a}_m \otimes \hat{\mathbf{p}}_y ) \cdot s_{mm}^2 ]. $$
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Here, $p_v$ is the unit vector for the axes in the lab frame $(p1=[1, 0,
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0], p2=[0, 1, 0], and p3=[0, 0, 1])$ and $s_{mm}$ gives the mth radius of
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the ellipsoid $i$.
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\end{document}
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\documentstyle[12pt]{article}
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\begin{document}
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\begin{center}
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\large{Additional documentation for the RE-squared ellipsoidal potential \\
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as implemented in LAMMPS}
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\end{center}
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\centerline{Mike Brown, Sandia National Labs, October 2007}
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\vspace{0.3in}
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Let the shape matrices $\mathbf{S}_i=\mbox{diag}(a_i, b_i, c_i)$ be
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given by the ellipsoid radii. Let the relative energy matrices
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$\mathbf{E}_i = \mbox{diag} (\epsilon_{ia}, \epsilon_{ib},
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\epsilon_{ic})$ be given by the relative well depths
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(dimensionless energy scales inversely proportional to the well-depths
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of the respective orthogonal configurations of the interacting molecules).
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Let $\mathbf{A}_1$ and $\mathbf{A}_2$ be the transformation matrices
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from the simulation box frame to the body frame and $\mathbf{r}$
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be the center to center vector between the particles. Let $A_{12}$ be
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the Hamaker constant for the interaction given in LJ units by
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$A_{12}=4\pi^2\epsilon_{\mathrm{LJ}}(\rho\sigma^3)^2$.
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\vspace{0.3in}
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The RE-squared anisotropic interaction between pairs of
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ellipsoidal particles is given by
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$$ U=U_A+U_R, $$
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$$ U_\alpha=\frac{A_{12}}{m_\alpha}(\frac\sigma{h})^{n_\alpha}
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(1+o_\alpha\eta\chi\frac\sigma{h}) \times \prod_i{
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\frac{a_ib_ic_i}{(a_i+h/p_\alpha)(b_i+h/p_\alpha)(c_i+h/p_\alpha)}}, $$
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$$ m_A=-36, n_A=0, o_A=3, p_A=2, $$
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$$ m_R=2025, n_R=6, o_R=45/56, p_R=60^{1/3}, $$
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$$ \chi = 2 \hat{\mathbf{r}}^T \mathbf{B}^{-1}
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\hat{\mathbf{r}}, $$
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$$ \hat{\mathbf{r}} = { \mathbf{r} } / |\mathbf{r}|, $$
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$$ \mathbf{B} = \mathbf{A}_1^T \mathbf{E}_1 \mathbf{A}_1 +
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\mathbf{A}_2^T \mathbf{E}_2 \mathbf{A}_2 $$
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$$ \eta = \frac{ \det[\mathbf{S}_1]/\sigma_1^2+
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det[\mathbf{S}_2]/\sigma_2^2}{[\det[\mathbf{H}]/
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(\sigma_1+\sigma_2)]^{1/2}}, $$
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$$ \sigma_i = (\hat{\mathbf{r}}^T\mathbf{A}_i^T\mathbf{S}_i^{-2}
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\mathbf{A}_i\hat{\mathbf{r}})^{-1/2}, $$
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$$ \mathbf{H} = \frac{1}{\sigma_1}\mathbf{A}_1^T \mathbf{S}_1^2 \mathbf{A}_1 +
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\frac{1}{\sigma_2}\mathbf{A}_2^T \mathbf{S}_2^2 \mathbf{A}_2 $$
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Here, we use the distance of closest approach approximation given by the
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Perram reference, namely
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$$ h = |r| - \sigma_{12}, $$
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$$ \sigma_{12} = [ \frac{1}{2} \hat{\mathbf{r}}^T
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\mathbf{G}^{-1} \hat{\mathbf{r}}]^{ -1/2 }, $$
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and
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$$ \mathbf{G} = \mathbf{A}_1^T \mathbf{S}_1^2 \mathbf{A}_1 +
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\mathbf{A}_2^T \mathbf{S}_2^2 \mathbf{A}_2 $$
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\vspace{0.3in}
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The RE-squared anisotropic interaction between a
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ellipsoidal particle and a Lennard-Jones sphere is defined
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as the $\lim_{a_2->0}U$ under the constraints that
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$a_2=b_2=c_2$ and $\frac{4}{3}\pi a_2^3\rho=1$:
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$$ U_{\mathrm{elj}}=U_{A_{\mathrm{elj}}}+U_{R_{\mathrm{elj}}}, $$
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$$ U_{\alpha_{\mathrm{elj}}}=(\frac{3\sigma^3c_\alpha^3}
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{4\pi h_{\mathrm{elj}}^3})\frac{A_{12_{\mathrm{elj}}}}
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{m_\alpha}(\frac\sigma{h_{\mathrm{elj}}})^{n_\alpha}
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(1+o_\alpha\chi_{\mathrm{elj}}\frac\sigma{h_{\mathrm{elj}}}) \times
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\frac{a_1b_1c_1}{(a_1+h_{\mathrm{elj}}/p_\alpha)
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(b_1+h_{\mathrm{elj}}/p_\alpha)(c_1+h_{\mathrm{elj}}/p_\alpha)}, $$
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$$ A_{12_{\mathrm{elj}}}=4\pi^2\epsilon_{\mathrm{LJ}}(\rho\sigma^3), $$
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with $h_{\mathrm{elj}}$ and $\chi_{\mathrm{elj}}$ calculated as above
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by replacing $B$ with $B_{\mathrm{elj}}$ and $G$ with $G_{\mathrm{elj}}$:
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$$ \mathbf{B}_{\mathrm{elj}} = \mathbf{A}_1^T \mathbf{E}_1 \mathbf{A}_1 + I, $$
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$$ \mathbf{G}_{\mathrm{elj}} = \mathbf{A}_1^T \mathbf{S}_1^2 \mathbf{A}_1.$$
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\vspace{0.3in}
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The interaction between two LJ spheres is calculated as:
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$$
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U_{\mathrm{lj}} = 4 \epsilon \left[ \left(\frac{\sigma}{|\mathbf{r}|}\right)^{12} -
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\left(\frac{\sigma}{|\mathbf{r}|}\right)^6 \right]
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$$
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\vspace{0.3in}
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The analytic derivatives are used for all force and torque calculation.
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\end{document}
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