### 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) |
---|---|

Pages (from-to) | B777-B794 |

Journal | SIAM Journal on Scientific Computing |

Volume | 36 |

Issue number | 5 |

DOIs | |

State | Published - 2014 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics

### Cite this

*SIAM Journal on Scientific Computing*,

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

**Fast and accurate evaluation of nonlocal coulomb and dipole-dipole interactions via the nonuniform FFT.** / Jiang, Shidong; Greengard, Leslie; Bao, Weizhu.

Research output: Contribution to journal › Article

*SIAM Journal on Scientific Computing*, vol. 36, no. 5, pp. B777-B794. https://doi.org/10.1137/130945582

}

TY - JOUR

T1 - Fast and accurate evaluation of nonlocal coulomb and dipole-dipole interactions via the nonuniform FFT

AU - Jiang, Shidong

AU - Greengard, Leslie

AU - Bao, Weizhu

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

KW - Coulomb interaction

KW - Dipole-dipole interaction

KW - Interaction energy

KW - Nonlocal

KW - Nonuniform FFT

KW - Poisson equation

UR - http://www.scopus.com/inward/record.url?scp=84911435399&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84911435399&partnerID=8YFLogxK

U2 - 10.1137/130945582

DO - 10.1137/130945582

M3 - Article

VL - 36

SP - B777-B794

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

SN - 1064-8275

IS - 5

ER -