Franz J. Vesely > Lennard-Jones Lines (Force)
 
 





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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

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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) $


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vesely feb-07