New parallel optimal-parameter fast multipole method (OPFMM)

The three key translation equations of the fast multipole method (FMM) are deduced from the general polypolar expansions given earlier. Simplifications are introduced for the rotation-based FMM that lead to a very compact FMM formalism. The optimum-parameter searching procedure, a stable and efficient way of obtaining the optimum set of FMM parameters, is established with complete control over the tolerable error (ε). This new procedure optimizes the linear scaling with respect to the number of particles for a given ε. In addition, a new parallel FMM algorithm, which requires virtually no internode communication, is suggested that is suitable for the parallel construction of Fock matrices in electronic structure calculations.