Where 0 ≤ α < ∞ is a parameter that determines the partitioning of the work between the short-and longranged terms, and the second line expresses the longranged operator in terms of reciprocal space quantities: a natural basis for such a long-ranged, but singularity free quantity. As in FMM, the first term is readily treated using direct summation, while the long-ranged term requires evaluation of structure factors S(m) and their complex conjugate S(-m). The standard Fourier transform treatment requires O(N 2 ) effort due to nested loops that run both over particles whose density is to be considered, and over wavevectors m for the Fourier transform; judicious choice of α as a function of system size yields scaling of O(N 3/2 ) for both terms in eqn. (4). This Ewald formulation 9 was further improved by one of the current authors by introducing particle-mesh Ewald (PME), which proceeds via a rectilinear grid intermediate that allows the O(N ) discretization of the densities in one step, followed by a Fourier transform that can now be performed in O(N log(N )) effort using the Fast Fourier Transform (FFT) due to the regular nature of the grid. Incidentally, the FFT was also named among top ten algorithms. Despite inferior asymptotic scaling to FMM, PME has found widespread popularity since its creation, primarily due to the excellent performance of FFTs on non-parallel hardware, which results in a low cost prefactor. However, parallelization of FFTs requires a large amount of communication, which limits scalability in modern supercomputers.