Improve the efficiency of the Cartesian tensor based fast multipole method for Coulomb interaction using the traces

To compute the non-oscillating mutual interaction for a system with N points, the fast multipole method (FMM) has an efficiency that scales linearly with the number of points. Specifically, for Coulomb interaction, FMM can be constructed using either the spherical harmonic functions or the totally symmetric Cartesian tensors. In this paper, we will present that the efficiency of the Cartesian tensor based FMM for the Coulomb interaction can be significantly improved by implementing the traces of the Cartesian tensors in calculation to reduce the independent elements of the n-th rank totally symmetric Cartesian tensor from (n+1)(n+2)/2 to 2n+1. The computation complexity for the operations in FMM are analyzed and expressed as polynomials of the highest rank of the Cartesian tensors. For most operations, the complexity is reduced by one order. Numerical examples regarding the convergence and the efficiency of the new algorithm are demonstrated. A reduction of computation time up to 50% has been observed for moderate number of points and rank of tensors.