### 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 ψ(x) and a density function ψ(x) = |ψ(x)|2 for some complexvalued wave function ψψ(x), permitting the formal use of Fourier methods. These are hampered by the fact that the Fourier transform of the interaction kernel ρ U(k) has a singularity and/or ρ(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 U (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| → 8. 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.

Original language | English (US) |
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Pages (from-to) | B777-B794 |

Journal | SIAM Journal on Scientific Computing |

Volume | 36 |

Issue number | 5 |

DOIs | |

State | Published - 2014 |

### Keywords

- Coulomb interaction
- Dipole-dipole interaction
- Interaction energy
- Nonlocal
- Nonuniform FFT
- Poisson equation

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

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## Cite this

*SIAM Journal on Scientific Computing*,

*36*(5), B777-B794. https://doi.org/10.1137/130945582