0.1 Definitions

Positions: $\vec{r}_{i}$, $\vec{r}_{j}$ with $\vec{r} \equiv \vec{r}_{ij} \equiv \vec{r}_{j}-\vec{r}_{i}$ and $r \equiv \vert\vec{r}\vert$;
Interparticle unit vector: $\vec{s} \equiv \vec{r}/r$
Directions (unit vectors): $\vec{u}_{i}$, $\vec{u}_{j}$
Width: $\sigma_{s}=\sigma_{0}$ (side-by-side), Length: $\sigma_{e}$ (end-to-end).

Then

$$U\left(\vec{u}_{i},\vec{u}_{j},\vec{r}\right) =4 \epsilon \l... ...i},\vec{u}_{j},\vec{s}\right)+\sigma_{0}} \right]^{6} \right\}$$ (1)

where the generalized diameter and well depth parameters are given by

$$\sigma(\vec{u}_{i},\vec{u}_{j},\vec{s})= \sigma_{0} \left/ \... ...eft( \vec{u}_{i}\cdot \vec{u}_{j} \right)} \right\} } \right.$$ (2)

and

$$\epsilon(\vec{u}_{i},\vec{u}_{j},\vec{s})= \epsilon_{0} \eps... ...,\vec{u}_{j}) \epsilon'^{\mu}(\vec{u}_{i},\vec{u}_{j},\vec{s})$$ (3)

with

$$\epsilon(\vec{u}_{i},\vec{u}_{j}) = 1 \left/ \sqrt{ 1-\chi^{2} \; \left( \vec{u}_{i}\cdot\vec{u}_{j}\right)^{2} } \right.$$ (4)

$$\epsilon'(\vec{u}_{i},\vec{u}_{j},\vec{s})= 1-\frac{\chi'}{2... ...{1-\chi' \left( \vec{u}_{i}\cdot \vec{u}_{j} \right)} \right\}$$ (5)

and

$$\chi=\frac{(\sigma_{e}/\sigma_{s})^{2}-1}{(\sigma_{e}/\sigma... .../\epsilon_{s})^{1/\mu}}{1+(\epsilon_{e}/\epsilon_{s})^{1/\mu}}$$ (6)

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

$$\mu=2, \;\;\;\;\;\; \nu=1,\;\;\;\;\;\; \epsilon_{e}/\epsilon_{s}=1/5$$ (7)

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 $\epsilon_{e}/\epsilon_{s}$.

For $\sigma_{e}/\sigma_{s} \longrightarrow 1$ the model should become identical to the simple LJ model; it does so only if the parameter $\epsilon_{e}/\epsilon_{s}$ is also modified so as to approach $1$. A simple linear interpolation formula would be

$$\frac{\epsilon_{e}}{\epsilon_{s}}= \frac{1}{5} + (3-\frac{\sigma_{e}}{\sigma_{s}}) \frac{2}{5}$$ (8)

The problem with this formula is that for lengths $\sigma_{s}/\sigma_{e}$ larger than $3.5$ it proceeds to unphysical negative values of $\epsilon_{e}/\epsilon_{s}$. In fact, this ratio should, for longer and longer molecules, asymptotically tend to zero, since the negative side-by-side potential energy increases while the end-to-end energy remains finite. An interpolation formula that accounts for this is

$$\frac{\epsilon_{e}}{\epsilon_{s}}= 0.4 \frac{\sigma_{s}}... ...{e}} + 0.6 \left( \frac{\sigma_{s}}{\sigma_{e}}\right)^{2}$$ (9)