Research Article Open Access

Multilevel Evaluation of Coulomb Lattice Sums of Charge Systems

I. Suwan1, A. Brandt2 and V. Ilyin3
  • 1 Arab American University, Palestinian Authority
  • 2 , United States
  • 3 Taras Shevchenko Kyiv National University, Ukraine

Abstract

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.

Journal of Mathematics and Statistics
Volume 8 No. 3, 2012, 361-372

DOI: https://doi.org/10.3844/jmssp.2012.361.372

Submitted On: 7 February 2012 Published On: 11 September 2012

How to Cite: Suwan, I., Brandt, A. & Ilyin, V. (2012). Multilevel Evaluation of Coulomb Lattice Sums of Charge Systems. Journal of Mathematics and Statistics, 8(3), 361-372. https://doi.org/10.3844/jmssp.2012.361.372

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Keywords

  • Long range
  • fast summation
  • multilevel
  • local part
  • madelung constant
  • smooth part