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M2M and L2L translations

To describe the M2M and L2L translation procedures, we shall use the notation $ M$ for either $ M^T_r$ or $ M^L$, and $ L$ for either $ L^T_r$ or $ L^L$, respectively. Also, $ k$ will stand for the corresponding wave number $ k_T$ or $ k_L$.

When the centre of multipole expansion is shifted from $ y_0$ to $ y_1$, we perform the M2M translation in two steps:

Also, when the centre of local expansion is shifted from $ x_0$ to $ x_1$, we perform the L2L translation in two steps: