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Rotation of coefficients

The discussion in the previous section has been restricted to the case where the target cell is in the uplist of Cs. We now consider the general case. If the target cell is included in lists other than the uplist we rotate the coordinate system so that the target cell is in the positive $\tilde x_3$ direction viewed from the source cell, where $\tilde x_i$ denotes the new axis. Accordingly, the multipole moments in the new coordinate system are obtained as follows:

 
$\displaystyle \widetilde{M}_{n,m}(O) = \sum_{m'=-n}^{n} {\cal R}_{n,m,m'}(\mbox{\boldmath$ \nu $ },\alpha)
M_{n,m'}(O)$     (4.11)

where ${\cal R}_{n,m,m'}(\mbox{\boldmath$\space \nu $ },\alpha)$ is the coefficient of rotation, $\mbox{\boldmath$\space \nu $ }$ is a unit vector parallel to the rotation axis and $\alpha$ is a rotation angle. The explicit form of the coefficient ${\cal R}_{n,m,m'}(\mbox{\boldmath$\space \nu $ },\alpha)$ is given by (See Biedenharn and Louck[6])
 
$\displaystyle {\cal R}_{n,m,m'}(\mbox{\boldmath$ \nu $ },\alpha)$ = (-1)m+m'(n+m')! (n-m')!  
    $\displaystyle \sum_k
\frac{(\alpha_0-i\alpha_3)^{n+m-k}(-i\alpha_1-\alpha_2)^{m...
...pha_1+\alpha_2)^{k}(\alpha_0+i\alpha_3)^{n-m'-k}}{(n+m-k)!(m'-m+k)!k!(n-m'-k)!}$ (4.12)

where $\alpha_0=\cos(\alpha/2)$ and $\alpha_i=-\nu_i\sin(\alpha/2)$. The summation is over such k that the numbers in the parentheses in the denominator are all non-negative. We next describe the M2L translation process for the general case.
1.
Rotation:
We first rotate the multipole moments via (4.11). The specific forms of (4.11) vary as follows, as the location of Ct changes: where $\mbox{\boldmath$\space e $ }_i$ is the base vector for the Cartesian coordinates.
2.
Compute the coefficients of the exponential expansion:
Compute the coefficients of the exponential expansion via (4.6) as follows:
 
$\displaystyle X^{\diamondsuit}(k,j;O)=\frac{\omega_k}{M(k)d}\sum_{m=-\infty}^{\...
...{n=\vert m\vert}^{\infty} (\lambda_k/d)^n \widetilde{M}^{\diamondsuit}_{n,m}(O)$     (4.19)

where $\diamondsuit$ is an element of $\{U,D,N,S,E,W\}$ .
3.
Translation of the coefficients of the exponential expansion:
As the centre of the exponential expansion is shifted from O (the centroid of Cs) to $\mbox{\boldmath$\space x $ }_0$ (the centroid of Ct), the coefficients of the exponential expansion are transformed according to (4.7) as follows:
 
$\displaystyle X^{\diamondsuit}(k,j;\mbox{\boldmath$ x $ }_0) = X^{\diamondsuit}...
...0})_1 \cos \alpha_j + (\widetilde{O\mbox{\boldmath$ x $ }_0})_2 \sin \alpha_j)}$     (4.20)

where $\diamondsuit$ is an element of $\{U,D,N,S,E,W\}$ and the components $(\widetilde{O\mbox{\boldmath$\space x $ }_0})_i$ are obtained by applying the corresponding rotation of coordinates to $\overrightarrow{O\mbox{\boldmath$\space x $ }_0}$.
4.
Compute the coefficients of the local expansion:
Compute the coefficients of the local expansion from the exponential expansion according to (4.10) as follows:
 
$\displaystyle \widetilde{L}^{\diamondsuit}_{n,m}(\mbox{\boldmath$ x $ }_0) = \s...
...dsuit} (k,j;\mbox{\boldmath$ x $ }_0)
(-i)^m (-\lambda_k/d)^n e^{-i m \alpha_j}$     (4.21)

where $\diamondsuit$ is an element of $\{U,D,N,S,E,W\}$. Then rotate $\tilde L_{n,m}^{\diamondsuit}$ as follows:


next up previous contents
Next: Algorithm for the new Up: Crack problems for three-dimensional Previous: New FM-BIEM
Ken-ichi Yoshida
2001-07-28