Franz J. Vesely > Lectures > LJ Sticks

 < >

LENNARD-JONES LINES?

 Definition: Consider two lines of finite length $L$ containing a homogeneous density of LJ centers. Let $\vec{e}_{1,2}$ be the direction vectors, $\vec{r}_{12}$ the vector between the centers of the two line segments, and $\lambda, \, \mu$ parameters giving the positions of the interacting points along $1$ and $2$. The squared distance between any two such points is given by $r^{2}= \left| r_{12}+\mu \vec{e}_{2}-\lambda \vec{e}_{1} \right|^{2}$. The total interaction energy between the two lines is then $u(\vec{r}_{12},\vec{e}_{1},\vec{e}_{2}) = \frac{\textstyle 1}{\textstyle L^{2}} \int \limits_{1} d\lambda \int \limits_{2} d\mu \, \, u_{LJ} ( r )$ with $u_{LJ}(r)=4 \,(r^{-12}-r^{-6})$.

Let's try:
• At fixed $\lambda$ the ($r^{-6}$) integral over $\mu$ is
$I^{-}(\vec{r}_{12},\vec{e}_{1},\vec{e}_{2},\lambda) = \int \limits_{-h}^{h}d\mu \, \left| r_{12}+\mu \vec{e}_{2}-\lambda \vec{e}_{1} \right|^{-6}$
where $h$ is the half length of the stick.

• Find the "proxy points" of shortest relative distance $r_{0}$ on the two carrier lines.

• Using new parameters $\gamma$, $\delta$ originating at the proxy points we have
$I^{-}(\gamma) = \int \limits_{\delta_{a}}^{\delta_{b}}d\delta \, \left[ \delta^{2}+p \delta + q^{2} \right]^{-3}$
where $\delta_{b,a}=-\mu_{0} \pm h$, $p=-2 \gamma \rho$, and $q^{2}=\gamma^{2}+r_{0}^{2}$; the correlation $\rho$ is just the scalar product of the direction vectors, $\rho = e_{1} \cdot e_{2}$.

• There is in fact an analytic solution to this:
$\begin{eqnarray} I^{-}(\gamma) &=& \left[ 3 \, \frac{p+2\delta}{(4q^{2}-p^{2})^{2}[q^{2}+\delta(p+\delta)]} +\frac{1}{2} \, \frac{p+2\delta}{(4q^{2}-p^{2})[q^{2}+\delta(p+\delta)]^{2}} \right. \\ && \left. +\frac{12}{(4q^{2}-p^{2})^{5/2}} \arctan \frac{p+2\delta}{(4q^{2}-p^{2})^{1/2}} \right]_{\delta=\delta_{a}}^{\delta=\delta_{b}} \end{eqnarray}$

• However, the second integration in

$I^{\pm} \equiv \int \limits_{\gamma_{a}}^{\gamma_{b}} d\gamma \, I^{\pm}(\gamma)$

with $\gamma_{b,a}=-\lambda_{0} \pm h$ can in general not be performed in closed form.

 This is what happens when we try ... But then again... (Thanks to J. Auersperg)
 < >

vesely nov-2006