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Lennard-Jones Lines: Pair Force
Franz J. Vesely
franz.vesely@univie.ac.at
Computational Physics Group, University of Vienna,
Boltzmanngasse 5, A-1090 Vienna
Back to Notes on LJL
For reference, the definition of the LJL potential again:
Let $\vec{e}_{1,2}$ be the direction vectors, $\vec{r}_{1,2}$
the position of the centers of the two line segments, and
$\vec{r}_{12}$ the vector between these centers. Let
$\lambda, \, \mu \, \epsilon [\pm h]$ ($h$ ... half length)
be parameters giving the positions of
the interacting points along $1$ and $2$. The vector between
any two such points is given by
$\vec{r}(\lambda,\mu)=
\vec{r}_{12}+\mu \cdot \vec{e}_{2}-\lambda \cdot \vec{e}_{1}$, with
modulus $ r(\lambda, \mu) =
| \vec{r}_{12}+\mu \cdot \vec{e}_{2}-\lambda \cdot \vec{e}_{1} |$.
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_{\textstyle 1}d\lambda \int \limits_{\textstyle 2}d\mu \, \,
u_{LJ}\left[ r(\lambda,\mu) \right]
$
Approximating $u_{LJ}$ by a sum of three Gaussians we have
$
u(\vec{r}_{12},\vec{e}_{1},\vec{e}_{2}) = \frac{\textstyle 1}{\textstyle L^{2}} \,
\sum \limits_{k=1}^{3} A_{k} \,
\int \limits_{-h}^{h} d \lambda
\int \limits_{-h}^{h} d \mu \,
\exp \left\{ - B_{k}
| \vec{r}_{12}+\mu \cdot \vec{e}_{2}-\lambda \cdot \vec{e}_{1} |^{2}
\right\}
$
Now denote by $\vec{p}_{1,2}$ the points of minimum distance
between the two carrier lines. We find that
$
\vec{p}_{1}= \vec{r}_{1} + \lambda_{0} \vec{e}_{1} \; ,
\;\;\;\;\;\;
\vec{p}_{2}= \vec{r}_{2} + \mu_{0} \vec{e}_{2}
$
with
$
\lambda_{0}= \frac{\textstyle c-bd }{\textstyle 1-d^{2}},
\;\;\;\;
\mu_{0}=\frac{\textstyle cd-b }{\textstyle 1-d^{2}}
$
where we have introduced
$b \equiv \vec{r}_{12} \cdot \vec{e}_{2}$,
$c \equiv \vec{r}_{12} \cdot \vec{e}_{1}$, and
$d \equiv \vec{e}_{1} \cdot \vec{e}_{2}$.
The vector connecting the proxy points is then
$
\vec{r}_{0} \equiv \vec{p}_{2}-\vec{p}_{1} =
\vec{r}_{12}+\mu_{0} \vec{e}_{2} - \lambda_{0} \vec{e}_{1}
$
Choosing new integration variables $\gamma, \delta$ originating from the proxy
points we have, with
$\gamma = \lambda - \lambda_{0}$ and
$\delta = \mu - \mu_{0}$,
$
\begin{eqnarray}
u(\vec{r}_{12},\vec{e}_{1},\vec{e}_{2}) &=&
\frac{\textstyle 1}{\textstyle L^{2}} \,
\sum \limits_{k=1}^{3} A_{k} \,
\exp \left[ - B_{k} r_{0}^{2}\right] \,
\int \limits_{\textstyle \gamma_{a}}^{\textstyle \gamma_{b}} d \gamma
\int \limits_{\textstyle \delta_{a}}^{\textstyle \delta_{b}} d \delta \,
\exp \left\{ - B_{i} \left(
\gamma^{2}+\delta^{2}-2 \gamma \delta d \right)
\right\}
\\
& \equiv &
\frac{\textstyle 1}{\textstyle L^{2}} \,
\sum \limits_{k=1}^{3} A_{k} \, v_{k}(1,2)
\end{eqnarray}
$
where
$\gamma_{a} \! = - \lambda_{0} \! - \! h$,
$\gamma_{b} \! = - \lambda_{0} \! + \! h$,
$\delta_{a} \! = - \mu_{0} \! - \! h$,
$\delta_{b} \! = - \mu_{0} \! + \! h$.
To derive an expression for the force acting from 2 on 1 we have to
find the gradients
of the terms $v_{k}(1,2)$ with respect to $\vec{r}_{1}$:
$
\vec{F}_{12} \equiv \sum \limits_{k} \frac{\textstyle A_{k}}{\textstyle L^{2}} \,
\left\{ - \vec{\nabla}_{1} \, v_{k}(1,2) \right\}
$
with
$
v_{k}(1,2)=
\exp \left[ - B_{k} r_{0}^{2}\right] \,
\int \limits_{\textstyle \gamma_{a}}^{\textstyle \gamma_{b}} d \gamma
\int \limits_{\textstyle \delta_{a}}^{\textstyle \delta_{b}} d \delta \,
\exp \left\{ - B_{k} \left(
\gamma^{2}+\delta^{2}-2 \gamma \delta d \right)
\right\}
$
The following positional derivatives will be needed:
$
\vec{\nabla}_{1} \, r_{0}^{2} = - 2 \, \vec{r}_{0}
$
$
\vec{\nabla}_{1} \gamma_{a,b}
= \vec{\nabla}_{1} ( - \lambda_{0})
= \vec{\nabla}_{1} \left[ - \frac{\textstyle c-db}{\textstyle 1-d^{2}} \right]
= \frac{\textstyle \vec{e}_{1} - d \vec{e}_{2}}{\textstyle 1-d^{2}}
$
$
\vec{\nabla}_{1} \delta_{a,b}
= \vec{\nabla}_{1} ( - \mu_{0})
= \vec{\nabla}_{1} \left[ - \frac{\textstyle dc-b}{\textstyle 1-d^{2}} \right]
= \frac{\textstyle d \vec{e}_{1} - \vec{e}_{2}}{\textstyle 1-d^{2}}
$
Now write
$
\begin{eqnarray}
\vec{\nabla}_{1} v_{k}(1,2) &=&
\vec{\nabla}_{1}
\left[
\exp \{ - B_{k} r_{0}^{2}\}
\right] \,
\int \limits_{\textstyle \gamma_{a}}^{\textstyle \gamma_{b}} d \gamma
\int \limits_{\textstyle \delta_{a}}^{\textstyle \delta_{b}} d \delta \,
\exp \left\{ - B_{k} \left(
\gamma^{2}+\delta^{2}-2 \gamma \delta d \right)
\right\}
\\
&+&
\exp \{ - B_{k} r_{0}^{2}\}
\int \limits_{\textstyle \gamma_{a}}^{\textstyle \gamma_{b}} d \gamma
\vec{\nabla}_{1} \,
\left[
\int \limits_{\textstyle \delta_{a}}^{\textstyle \delta_{b}} d \delta \,
\exp \left\{ - B_{k} \left(
\gamma^{2}+\delta^{2}-2 \gamma \delta d \right)
\right\}
\right]
\\
&+&
\exp \{ - B_{k} r_{0}^{2}\}
\int \limits_{\textstyle \delta_{a}}^{\textstyle \delta_{b}} d \delta \,
\vec{\nabla}_{1} \,
\left[
\int \limits_{\textstyle \gamma_{a}}^{\textstyle \gamma_{b}} d \gamma
\exp \left\{ - B_{k} \left(
\gamma^{2}+\delta^{2}-2 \gamma \delta d \right)
\right\}
\right]
\\
&\equiv&
\vec{w}_{k}^{I} \;\;\; + \;\;\; \vec{w}_{k}^{II} \;\;\; + \;\;\; \vec{w}_{k}^{III}
\end{eqnarray}
$
Term $I$:
$
\vec{w}_{k}^{I} =
2 B_{k} \vec{r}_{0} \,
\exp \{ - B_{k} r_{0}^{2}\} \,
\int \limits_{\textstyle \gamma_{a}}^{\textstyle \gamma_{b}} d \gamma
\int \limits_{\textstyle \delta_{a}}^{\textstyle \delta_{b}} d \delta \,
\exp \left\{ - B_{k} \left(
\gamma^{2}+\delta^{2}-2 \gamma \delta d \right)
\right\}
=
2 B_{k} \vec{r}_{0} \, v_{k}(1,2)
$
Term $II$:
$
\begin{eqnarray}
\vec{w}_{k}^{II}& = &
\exp \{ - B_{k} r_{0}^{2} \} \,
\frac{\textstyle d \vec{e}_{1} - \vec{e}_{2}}{\textstyle 1-d^{2}} \,
\left[
\exp \left\{ -B_{k} \delta_{b}^{2} (1-d^{2}) \right\} \,
\int \limits_{\textstyle \gamma_{a}}^{\textstyle \gamma_{b}} d \gamma \,
\exp \left\{
- B_{k} \left(\gamma - \delta_{b} d \right)^{2}
\right\}
-
\exp \left\{ -B_{k} \delta_{a}^{2} (1-d^{2}) \right\} \,
\int \limits_{\textstyle \gamma_{a}}^{\textstyle \gamma_{b}} d \gamma \,
\exp \left\{
- B_{k} \left(\gamma - \delta_{a} d \right)^{2}
\right\}
\right]
\\
& = &
\sqrt{ \frac{\textstyle \pi}{\textstyle 4 B_{k}} } \,
\exp \{ - B_{k} r_{0}^{2} \} \,
\frac{\textstyle d \vec{e}_{1} - \vec{e}_{2}}{\textstyle 1-d^{2}} \,
\left[ \;\;
\exp \left\{ -B_{k} \delta_{b}^{2} (1-d^{2}) \right\} \,
\left(
erf \sqrt{B_{k}} (\gamma_{b}-\delta_{b}d)
-
erf \sqrt{B_{k}} (\gamma_{a}-\delta_{b}d)
\right)
\right.
\\
&&
\left.
\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;
\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;
\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;
-
\exp \left\{ -B_{k} \delta_{a}^{2} (1-d^{2}) \right\} \,
\left(
erf \sqrt{B_{k}} (\gamma_{b}-\delta_{a}d)
-
erf \sqrt{B_{k}} (\gamma_{a}-\delta_{a}d)
\right)
\right]
\end{eqnarray}
$
Term $III$:
$
\begin{eqnarray}
\vec{w}_{k}^{III}& = &
\exp \{ - B_{k} r_{0}^{2} \} \,
\frac{\textstyle \vec{e}_{1} - d \vec{e}_{2}}{\textstyle 1-d^{2}} \,
\left[
\exp \left\{ -B_{k} \gamma_{b}^{2} (1-d^{2}) \right\} \,
\int \limits_{\textstyle \delta_{a}}^{\textstyle \delta_{b}} d \delta \,
\exp \left\{
- B_{k} \left(\delta - \gamma_{b} d \right)^{2}
\right\}
-
\exp \left\{ -B_{k} \gamma_{a}^{2} (1-d^{2}) \right\} \,
\int \limits_{\textstyle \delta_{a}}^{\textstyle \delta_{b}} d \delta \,
\exp \left\{
- B_{k} \left(\delta - \gamma_{a} d \right)^{2}
\right\}
\right]
\\
& = &
\sqrt{ \frac{\textstyle \pi}{\textstyle 4 B_{k}} } \,
\exp \{ - B_{k} r_{0}^{2} \} \,
\frac{\textstyle \vec{e}_{1} - d \vec{e}_{2}}{\textstyle 1-d^{2}} \,
\left[ \;\;
\exp \left\{ -B_{k} \gamma_{b}^{2} (1-d^{2}) \right\} \,
\left(
erf \sqrt{B_{k}} (\delta_{b}-\gamma_{b}d)
-
erf \sqrt{B_{k}} (\delta_{a}-\gamma_{b}d)
\right)
\right.
\\
&&
\left.
\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;
\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;
\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;
-
\exp \left\{ -B_{k} \gamma_{a}^{2} (1-d^{2}) \right\} \,
\left(
erf \sqrt{B_{k}} (\delta_{b}-\gamma_{a}d)
-
erf \sqrt{B_{k}} (\delta_{a}-\gamma_{a}d)
\right)
\right]
\end{eqnarray}
$
For a more compact notation we introduce the function
$
t_{k}(\alpha,\beta) \equiv
\exp \left\{ -B_{k} \alpha^{2} (1-d^{2}) \right\} \,
\left(
erf \sqrt{B_{k}} (\beta-\alpha d)
\right)
$
such that
$
\begin{eqnarray}
\vec{w}_{k}^{II} &=&
\frac{\textstyle d \vec{e}_{1} - \vec{e}_{2}}{\textstyle 1-d^{2}} \,
\sqrt{ \frac{\textstyle \pi}{\textstyle 4 B_{k}} } \,
\exp \{ - B_{k} r_{0}^{2} \} \,
\left[
\left(
t_{k}(\delta_{b},\gamma_{b})
-
t_{k}(\delta_{b},\gamma_{a})
\right)
-
\left(
t_{k}(\delta_{a},\gamma_{b})
+
t_{k}(\delta_{a},\gamma_{a})
\right)
\right]
\\
\vec{w}_{k}^{III} &=&
\frac{\textstyle \vec{e}_{1} - d \vec{e}_{2}}{\textstyle 1-d^{2}} \,
\sqrt{ \frac{\textstyle \pi}{\textstyle 4 B_{k}} } \,
\exp \{ - B_{k} r_{0}^{2} \} \,
\left[
\left(
t_{k}(\gamma_{b},\delta_{b})
-
t_{k}(\gamma_{b},\delta_{a})
\right)
-
\left(
t_{k}(\gamma_{a},\delta_{b})
+
t_{k}(\gamma_{a},\delta_{a})
\right)
\right]
\end{eqnarray}
$
Finally, we have for the pair force
$
\begin{eqnarray}
\vec{F}_{12}
& = &
- \sum \limits_{k} \frac{\textstyle A_{k}}{\textstyle L^{2}} \,
\left\{
\vec{r}_{0} \, \left( 2 B_{k} \, v_{k} \right)
+ \frac{\textstyle d \vec{e}_{1} - \vec{e}_{2}}{\textstyle 1-d^{2}} \,
D_{k} \,
\left[
t_{k}(\delta_{b},\gamma_{b})
-
t_{k}(\delta_{b},\gamma_{a})
-
t_{k}(\delta_{a},\gamma_{b})
+
t_{k}(\delta_{a},\gamma_{a})
\right]
\right.
\\
&&
\left.
\;\;\;\;\;\;\;\; \;\;\;\;\;
\;\;\;\;\;\;\;\; \;\;\;\;\;
\;\;\;\;\;\;
+ \frac{\textstyle \vec{e}_{1} - d \vec{e}_{2}}{\textstyle 1-d^{2}} \,
D_{k} \,
\left[
t_{k}(\gamma_{b},\delta_{b})
-
t_{k}(\gamma_{b},\delta_{a})
-
t_{k}(\gamma_{a},\delta_{b})
+
t_{k}(\gamma_{a},\delta_{a})
\right]
\right\}
\end{eqnarray}
$
where
$
D_{k} \equiv \sqrt{ \frac{\textstyle \pi}{\textstyle 4 B_{k}} } \,
\exp \{ - B_{k} r_{0}^{2} \} \,
$
and
$
t_{k}(\alpha,\beta) \equiv
\exp \left\{ -B_{k} \alpha^{2} (1-d^{2}) \right\} \,
\left(
erf \sqrt{B_{k}} (\beta-\alpha d)
\right)
$
Back to Notes on LJL
vesely feb-07