Chapter 3.5 Extensions of the Ewald method for Coulomb interactions in crystals

Reciprocal space

Second Online Edition (2010)

Part 3. Dual bases in crystallographic computing

  1. T. A. Darden

Published Online: 1 JUN 2010

DOI: 10.1107/97809553602060000772

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Darden, T. A. 2010. Extensions of the Ewald method for Coulomb interactions in crystals. International Tables for Crystallography. B:3:3.5:458–481.

Author Information

  1. Laboratory of Structural Biology, National Institute of Environmental Health Sciences, 111 T. W. Alexander Drive, Research Triangle Park, NC 27709, USA

Publication History

  1. Published Online: 1 JUN 2010


The electrostatic energy per unit cell of a large but finite crystal of point charges that is immersed in a dielectric medium can be efficiently calculated using the Ewald-sum technique (assuming a neutral unit cell). The form of this sum is derived here in a manner that generalizes to other interactions, such as long-range dispersion interactions, that depend on higher inverse powers of distance. The Ewald method is then generalized to finite crystals of model charge densities using Hermite Gaussian basis functions. Finally, efficient methods to accelerate the Ewald sum for these charge densities to near-linear scaling (i.e. scaling as N log N with system size N) are derived and tested.


  • Ewald method;
  • Coulomb interactions;
  • lattice sums;
  • polarization response;
  • smooth particle-mesh Ewald method;
  • Fourier Poisson method;
  • B-splines