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X2L translation formula

Substituting the following formula into the right-hand side of (J.3):

\begin{eqnarray*}\lefteqn{e^{-(\lambda_k/d) ((\overrightarrow{\mbox{\boldmath$ x...
...boldmath$ x $ }_0\mbox{\boldmath$ x $ }})_2 \sin \alpha_j(k))^n,

and using the following identity (See (4.9)):

\begin{eqnarray*}\frac{((\overrightarrow{O\mbox{\boldmath$ x $ }})_3 - i (\overr...
...alpha_j(k)} \ R_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }}),

we rewrite the right-hand side of (J.3) as

\begin{eqnarray*}\lefteqn{ \sum_{k=1}^{s}\sum_{j=1}^{M(k)}
... $ }_0\mbox{\boldmath$ x $ }})
L_{n,m}(\mbox{\boldmath$ x $ }_0)

where $L_{n,m}(\mbox{\boldmath$\space x $ }_0)$ is the coefficient of the local expansion expressed in terms of $X(k,j;\mbox{\boldmath$\space x $ }_0)$ as follows:
$\displaystyle L_{n,m}(\mbox{\boldmath$ x $ }_0)$ = $\displaystyle \sum_{k=1}^{s}\sum_{j=1}^{M(k)}
(-\lambda_k/d)^n (-i)^m e^{- i m \alpha_j(k)}X(k,j;\mbox{\boldmath$ x $ }_0)$  
  = $\displaystyle \sum_{k=1}^{s}(-\lambda_k/d)^n \sum_{j=1}^{M(k)}
(-i)^m e^{- i m \alpha_j(k)}X(k,j;\mbox{\boldmath$ x $ }_0)$ (J.4)

The cost for X2L translation can be roughly estimated as follows:

\begin{eqnarray*}\mbox{Cost} &=& \sum_{m=-p}^p \sum_{k=1}^s \sum_{j=1}^{M(k)} C_...
...(C_0pS\mbox{exp}) + O(C_1p^2s) \approx O(p^3) + O(p^3) = O(p^3),

where C0 and C1 are the computational cost related to $(-i)^m e^{- i m \alpha_j(k)}X(k,j;\mbox{\boldmath$\space x $ }_0)$ and $(-\lambda_k/d)^n$ in (J.4), respectively and C2 is the negligible cost.

Ken-ichi Yoshida