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LENNARD-JONES LINES
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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 $u_{LJ}(r)=4 \,(r^{-12}-r^{-6})$.
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Cannot be integrated in closed form, but:
LJ Lines Paper (JCP 2006)
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vesely apr-2007
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