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The Gay-Berne Potential
Positions: $\vec{r}_{i}$, $\vec{r}_{j}$;
directions: $\vec{u}_{i}$, $\vec{u}_{j}$;
width: $\sigma_{s}=\sigma_{0}$ (side-by-side);
length: $\sigma_{e}$ (end-to-end).
interparticle vector:
$\vec{r} \equiv \vec{r}_{ij} \equiv \vec{r}_{j}-\vec{r}_{i}$
with $r \equiv |\vec{r}|$;
$ \vec{s} \equiv \vec{r}/r$;
Then
$
U\left(\vec{u}_{i},\vec{u}_{j},\vec{r}\right)
=4 \epsilon \left(\vec{u}_{i},\vec{u}_{j},\vec{s}\right)
\left\{
\left[ \frac{\textstyle \sigma_{0}}{\textstyle r-\sigma\left(\vec{u}_{i},\vec{u}_{j},\vec{s}\right)+\sigma_{0}} \right]^{12}
- \left[ \frac{\textstyle \sigma_{0}}{\textstyle r-\sigma\left(\vec{u}_{i},\vec{u}_{j},\vec{s}\right)+\sigma_{0}} \right]^{6}
\right\}
$
where the generalized diameter and well depth parameters
are given by
$
\sigma(\vec{u}_{i},\vec{u}_{j},\vec{s})= \sigma_{0} \left/
\sqrt{1-\frac{\textstyle \chi}{\textstyle 2} \left\{\frac{\textstyle \left[\vec{s}\cdot \left(\vec{u}_{i}+\vec{u}_{j}\right) \right]^{2}}{\textstyle 1+\chi \left( \vec{u}_{i}\cdot \vec{u}_{j} \right)}
+ \frac{\textstyle \left[\vec{s}\cdot \left(\vec{u}_{i}-\vec{u}_{j}\right) \right]^{2}}{\textstyle 1-\chi \left( \vec{u}_{i}\cdot \vec{u}_{j} \right)} \right\} } \right.
$
and
$
\epsilon(\vec{u}_{i},\vec{u}_{j},\vec{s})= \epsilon_{0}
\epsilon^{\nu}(\vec{u}_{i},\vec{u}_{j})
\epsilon'^{\mu}(\vec{u}_{i},\vec{u}_{j},\vec{s})
$
with
$
\epsilon(\vec{u}_{i},\vec{u}_{j}) =
1 \left/
\sqrt{
1-\chi^{2} \; \left( \vec{u}_{i}\cdot\vec{u}_{j}\right)^{2}
} \right.
$
$
\epsilon'(\vec{u}_{i},\vec{u}_{j},\vec{s})=
1-\frac{\textstyle \chi'}{\textstyle 2}
\left\{
\frac{\textstyle \left[\vec{s}\cdot \left(\vec{u}_{i}+\vec{u}_{j}\right) \right]^{2}}{\textstyle 1+\chi' \left( \vec{u}_{i}\cdot \vec{u}_{j} \right)}
+\frac{\textstyle \left[\vec{s}\cdot \left(\vec{u}_{i}-\vec{u}_{j}\right) \right]^{2}}{\textstyle 1-\chi' \left( \vec{u}_{i}\cdot \vec{u}_{j} \right)}
\right\}
$
and
$
\chi =\frac{\textstyle (\sigma_{e}/\sigma_{s})^{2}-1}{\textstyle (\sigma_{e}/\sigma_{s})^{2}+1}
\;\; , \;\;\;\;\;\;\;\;\;\;
\chi'=\frac{\textstyle 1-(\epsilon_{e}/\epsilon_{s})^{\textstyle 1/\mu}}{\textstyle 1+(\epsilon_{e}/\epsilon_{s})^{\textstyle 1/\mu}}
$
A widely studied system is defined by the length-to-width ratio
$\sigma_{e}/\sigma_{s} = 3$. In this case the usual choice for the
parameters $\mu$, $\nu$ and $\epsilon_{e}/\epsilon_{s}$ is
$
\mu=2, \;\;\;\;\;\; \nu=1,\;\;\;\;\;\;
\epsilon_{e}/\epsilon_{s}=1/5
$
It should be noted that these parameters were identified by an optimal fit
to a site-site potential in which 4 LJ centers were placed along a line
at distances $2 \sigma_{0}/3$; in other words, the distance between
the outer LJ centers was
$2 \sigma_{0}$.
For molecules of other l/w ratios other choices of the parameters would
be appropriate. However, model simulations are often done with the same
$\mu,\nu$, and even $\epsilon_{e}/\epsilon_{s}$
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