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

Abstract

Abstract. 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 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 inÛ (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 ρ(x) is smooth and decays sufficiently fast as x → ∞. More precisely, the calculation requires O(N log N ) 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.Key words. Coulomb interaction, dipole-dipole interaction, interaction energy, nonuniform FFT, nonlocal, Poisson equation. AMS subject classifications. 33C10, 33F05, 44A35, 65R10, 65T50, 81Q401. Introduction. Nonlocal (long-range) interactions are encountered in modeling a variety of problems from quantum physics and chemistry to materials science and biology. A typical example is the Coulomb interaction in the nonlinear Schrödinger equation (or Schrödinger-Poisson system in three dimensions (3D)) as a "mean field limit" for N -electrons, assuming binary Coulomb interactions [10, 11, 23] and the Kohn-Sham equation of density functional theory (DFT) for electronic structure calculations in materials simulation and design [23, 38, 53, 56, 57]. Dipole-dipole interactions arise in quantum chemistry [32, 43], in dipolar Bose-Einstein condensation (BEC) [2-4, 31, 40, 46, 55, 62-64], in dipolar Fermi gases [48], and in dipole-dipole interacting Rydberg molecules [35][36][37].In physical space, the interaction kernel is both long-range and singular at the origin, requiring both accurate quadrature techniques and suitable fast algorithms. When the density function is smooth, however, it is often more convenient to use Fourier methods since the frequency content is well-controlled. Unfortunately, the Fourier transform of the interaction kernel is singular at the origin of Fourier (phase) space as well, resulting in significant numerical burdens and challenges [3, 4, 9, 13, 14, 26, 59, 66].In this paper, we present a fast and accurate algorithm for the numerical evalua-

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Spin squeezing in dipolar spinor condensates

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