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Rotation of coefficients

The discussion in the previous section has been restricted to the case where Ct is in +x3 direction of Cs. In this section we shall remove this assumption to generalise the discussion. We divide the interaction list of Cs into 6 lists: uplist, downlist, northlist, southlist, eastlist and westlist. The uplist and downlist contain target cells located in +x3 and -x3 directions of Cs, respectively. The northlist and southlist contain target cells located in +x2 and -x2 directions of Cs except those in the uplist or downlist, respectively. The eastlist and westlist contain target cells located in +x1 and -x1 directions of Cs except those in the uplist, downlist, northlist or southlist, respectively. The situation in the previous section can be described as the case where Ct is contained in the uplist of Cs. If the target cell is included in lists except the uplist of Cs 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. In general the multipole moments in the new coordinate system are obtained as follows:

  
$\displaystyle \widetilde{M}^1_{j,n,m}(O)$ = $\displaystyle {\cal A}_{ji}
\sum_{m'=-n}^{n} {\cal R}_{n,m,m'}(\mbox{\boldmath$ \nu $ },\alpha)
M^1_{i,n,m'}(O)$  
      (27)
$\displaystyle \widetilde{M}^2_{n,m}(O)$ = $\displaystyle \sum_{m'=-n}^{n} {\cal R}_{n,m,m'}(\mbox{\boldmath$ \nu $ },\alpha)
M^2_{n,m'}(O)$  

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, $\alpha$ is a rotation angle and ${\cal A}_{ij}$ is a rotation matrix. The explicit form of ${\cal R}_{n,m,m'}(\mbox{\boldmath$\space \nu $ },\alpha)$ is given by (See Biedenharn and Louck[25])
 
    $\displaystyle {\cal R}_{n,m,m'}(\mbox{\boldmath$ \nu $ },\alpha)=(-1)^{m+m'}(n+m')! (n-m')!$  
    $\displaystyle \sum_k
\left[(\alpha_0-i\alpha_3)^{n+m-k}(-i\alpha_1-\alpha_2)^{m'-m+k}\right.$  
    $\displaystyle \left. (-i\alpha_1+\alpha_2)^{k}
(\alpha_0+i\alpha_3)^{n-m'-k}\right] \left[(n+m-k)! \right.$  
    $\displaystyle \left. (m'-m+k)!k!(n-m'-k)!\right]^{-1}$ (28)

where $\alpha_0=\cos(\alpha/2)$ and $\alpha_i=-\nu_i\sin(\alpha/2)$. The summation in (33) is carried out over such k that the powers in the numerator are all non-negative.

We next describe the generalised M2L translation process in the new FMM.

1.
Rotation:

First we rotate the multipole moments via (31) and (32) so as to make the procedure presented in 4.1 applicable. The specific forms of (31) and (32) depend on the location of Ct and are described as follows:

where $\mbox{\boldmath$\space e $ }_i$ is the base vector for the Cartesian coordinates and superposed indices {U, D, N, S, E, W} correspond to the initial letters of {uplist, downlist, northlist, southlist, eastlist, westlist}, respectively.
2.
Compute the coefficients of the exponential expansion:

Compute the coefficients of the exponential expansion via (25) and (26) as follows:

  
$\displaystyle {W^{1\diamondsuit}_j(p,q)=}$
    $\displaystyle \frac{\omega_p}{M(p)d}\sum_{m=-\infty}^{\infty}(-i)^m
e^{-i m \al...
...\vert m\vert}^{\infty}
(\lambda_p/d)^n \widetilde{M}^{1\diamondsuit}_{j,n,m}(O)$  
      (36)
$\displaystyle {W^{2\diamondsuit}(p,q)=}$
    $\displaystyle \frac{\omega_p}{M(p)d}\sum_{m=-\infty}^{\infty}(-i)^m
e^{-i m \al...
...n=\vert m\vert}^{\infty}
(\lambda_p/d)^n \widetilde{M}^{2\diamondsuit}_{n,m}(O)$  

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 the centroid of Cs (O) to the centroid of Ct ( $\mbox{\boldmath$\space x $ }_0$), the coefficients of the exponential expansion is translated according to (27) and (28) as follows:

  
$\displaystyle {V^{1\diamondsuit}_j(p,q) =W^{1\diamondsuit}_j(p,q)}$
    $\displaystyle e^{ -(\lambda_p/d) (\widetilde{O\vec{x}_0})_3
+ i (\lambda_p/d) (...
...O\vec{x}_0})_1
\cos \alpha_q(p) + (\widetilde{O\vec{x}_0})_2 \sin \alpha_q(p))}$  
      (37)
$\displaystyle {V^{2\diamondsuit}(p,q) = (W^{2\diamondsuit}(p,q)
-W^{1\diamondsuit}_k(p,q)(\widetilde{O\mbox{\boldmath$ x $ }_0})_k)}$
    $\displaystyle e^{ -(\lambda_p/d) (\widetilde{O\vec{x}_0})_3
+ i (\lambda_p/d) (...
...O\vec{x}_0})_1
\cos \alpha_q(p) + (\widetilde{O\vec{x}_0})_2 \sin \alpha_q(p))}$  
      (38)
    $\displaystyle \qquad (\widetilde{O\mbox{\boldmath$ x $ }_0})_i = A^{\diamondsuit}_{ij} (\overrightarrow{O\mbox{\boldmath$ x $ }_0})_j$  

where $\diamondsuit$ is an element of $\{U, D, N, S, E, W\}$.
4.
Compute the coefficients of the local expansion:

Compute the coefficients of the local expansion from the exponential expansion according to (29) and (30) as follows:

  
$\displaystyle {\widetilde{L}^{1\diamondsuit}_{j,n,m}(\mbox{\boldmath$ x $ }_0) =}$
    $\displaystyle \sum_{p=1}^{s(\varepsilon)}\sum_{q=1}^{M(p)}V^{1\diamondsuit}_j(p,q)
(-i)^m (-\lambda_p/d)^n e^{-i m \alpha_q(p)}$  
      (39)
$\displaystyle {\widetilde{L}^{2\diamondsuit}_{n,m}(\mbox{\boldmath$ x $ }_0) =}$
    $\displaystyle \sum_{p=1}^{s(\varepsilon)} \sum_{q=1}^{M(p)}V^{2\diamondsuit}(p,q)
(-i)^m (-\lambda_p/d)^n e^{-i m \alpha_q(p)}$  

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


next up previous
Next: Algorithm for the new Up: New FMM Previous: Formulation for the new
Ken-ichi Yoshida
2001-03-26