Fast and Accurate Evaluation of Nonlocal Coulomb and Dipole-Dipole Interactions via the Nonuniform FFT

We present a fast and accurate algorithm for the evaluation of nonlocal (longrange) Coulomb and dipole-dipole interactions in free space. The governing potential is simply the convolution of an interaction kernel U(x) and a density function ρ(x) = |ψ(x)|2 for some complexvalued wave function ψ(x), permitting the formal use of Fourier methods. These are hampered by the fact that the Fourier transform of the interaction kernel ̂ U(k) has a singularity and/or ρ̂(k) = 0 at the origin k = 0 in Fourier (phase) space. Thus, accuracy is lost when using a uniform Cartesian grid in k which would otherwise permit the use of the FFT for evaluating the convolution. Here, we make use of a high-order discretization of the Fourier integral, accelerated by the nonuniform fast Fourier transform (NUFFT). By adopting spherical and polar phase-space discretizations in three and two dimensions, respectively, the singularity in Û(k) at the origin is canceled so that only a modest number of degrees of freedom are required to evaluate the Fourier integral, assuming that the density function (ρx) is smooth and decays sufficiently quickly as |x| → ∞. More precisely, the calculation requires O(N logN) operations, where N is the total number of discretization points in the computational domain. Numerical examples are presented to demonstrate the performance of the algorithm.