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 (long-range) Coulomb and dipole-dipole interactions in free space. The governing potential is simply the convolution of an interaction kernel $U(\mathbf{x})$ and a density function $\rho(\mathbf{x})=|\psi(\mathbf{x})|^2$ for some complex-valued wave function $\psi(\mathbf{x})$, permitting the formal use of Fourier methods. These are hampered by the fact that the Fourier transform of the interaction kernel $\widehat{U}(\mathbf{k})$ has a singularity and/or $\widehat{\rho}(\mathbf{k})\ne0$ at the origin $\mathbf{k}={\bf 0}$ in Fourier (phase) space. Thus, accuracy is lost when using a uniform Cartesian grid in $\mathbf{k}$ 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, ...