ABSTRACT
Multipole operations, M2M, M2L, and L2L are most computationally efficient when the origin translation happens along the z-axis. However, this condition only occurs in a few box-pairs in a real 3D system. Most box-pairs have their translation axis arbitrarily oriented relative to the z-axis, and that makes the multipole translation along the z-axis inapplicable. Flexibility in the choice of coordinate system provides the mechanism to achieve the required parallel alignment of the z-axis with the translation vector. Implementation of this idea requires developing a coordinate system rotation procedure to align the z-axis with the axis of the multipole translation, which connects the box centers. This chapter addresses the problem of developing a computationally efficient procedure for rotation of coordinate system of multipole expansion. Recurrence relations named m- and k-set provide numerically stable computation of the Wigner matrix. Scaling of the matrix elements simplifies the recurrence relations, and additionally leads to more efficient rotation equations. The significant reduction in the number of floating-point operations makes the bar-scaled m-set and the tilde-scaled k-set the preferred mechanisms for computation of the Wigner matrix. The chapter provides an example of computer implementation of computation of the Wigner matrix.
