While eqs ( )-( ) are consistent with eqs ( )-( ) for nonoverlapping particles r ≥ 2a, they contain a regularization term for overlapping particles r < 2a. This regularization guarantees that M is positive-definite for all possible configurations in the ω → 0 limit and ensures that M never becomes ill-conditioned. To solve eq (13), we truncate the moment expansion to some maximum multipole order and then invert the truncated M to find the unknown particle moments. If the multipole expansion is truncated at the dipole moment, the linear system is