Journal of Mathematics and Statistics

Multilevel Evaluation of Coulomb Lattice Sums of Charge Systems

I. Suwan, A. Brandt and V. Ilyin

DOI : 10.3844/jmssp.2012.361.372

Journal of Mathematics and Statistics

Volume 8, Issue 3

Pages 361-372


Problem statement: Due to the long range nature of interactions of the N-body systems, direct computation of the Coulomb potential energy involves O(N2) operations. To decrease such complexity, a simple Multilevel Summation method has been developed. Approach: In the frame of the Multilevel Summation method, the two-body interaction is decomposed into two parts: a local part and a smooth part. The local part vanishes beyond some cut-off distance; hence, its contribution to the potential energy is calculated in O(N) operations. In contrast to some common fast summation methods, the smooth part is calculated in real space on a sequence of grids with increasing meshsize in O(N) operations. Results: The method is tested on the calculation of the Madelung constants of ionic crystals in one, two and three dimensional cases. For a cut-off distance equals three times the meshsize of the ionic crystal, an error less than 0.01% is obtained. Conclusion: In computing the coulomb lattice sums of charge systems consisting of N bodies, the Multilevel Summation method decreases the complexity to O(N) operations.


© 2012 I. Suwan, A. Brandt and V. Ilyin. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.