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The discussion in the previous section has been restricted to the case
where the target cell is in the uplist of C_{s}. 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
direction viewed from the source cell,
where
denotes the new axis.
Accordingly, the multipole moments in the new coordinate system are
obtained as follows:



(4.11) 
where
is the coefficient of
rotation,
is a unit vector parallel to the rotation axis
and
is a rotation angle. The explicit form of the coefficient
is given by (See Biedenharn and
Louck[6])

= 
(1)^{m+m'}(n+m')! (nm')! 




(4.12) 
where
and
.
The summation is over such k that the numbers in the parentheses in
the denominator are all nonnegative.
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
C_{t} changes:
where
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:



(4.19) 
where
is an element of
.
 3.
 Translation of the coefficients of the exponential expansion:
As the centre of the exponential expansion is shifted from O (the
centroid of C_{s}) to
(the centroid of C_{t}), the
coefficients of the exponential expansion are transformed according
to (4.7) as follows:



(4.20) 
where
is an element of
and the
components
are obtained by applying the corresponding
rotation of coordinates to
.
 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:



(4.21) 
where
is an element of
.
Then rotate
as follows:
Next: Algorithm for the new
Up: Crack problems for threedimensional
Previous: New FMBIEM
Kenichi Yoshida
20010728