Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 3.4, p. 385   | 1 | 2 |

Section 3.4.1. Introduction

D. E. Williamsa

aDepartment of Chemistry, University of Louisville, Louisville, Kentucky 40292, USA

3.4.1. Introduction

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The electrostatic energy of an ionic crystal is often represented by taking a pairwise sum between charge sites interacting via Coulomb's law (the [n = 1] sum). The individual terms may be positive or negative, depending on whether the pair of sites have charges of the same or different signs. The Coulombic energy is very long-range, and it is well known that convergence of the Coulombic lattice-energy sum is extremely slow. For simple structure types Madelung constants have been calculated which represent the Coulombic energy in terms of the cubic lattice constant or a nearest-neighbour distance. Glasser & Zucker (1980[link]) give tables of Madelung constants and review the subject giving references dating back to 1884. If the ionic crystal structure is not of a simple type usually no Madelung constant will be available and the Coulombic energy must be obtained for the specific crystal structure being considered. In carrying out this calculation, accelerated-convergence treatment of the Coulombic lattice sum is indispensable to achieve accuracy with a reasonable amount of computational effort. A model of a molecular crystal may include partial net atomic charges or other charge sites such as lone-pair electrons. The [(n = 1)] sum also applies between these site charges.

The dispersion energy of ionic or molecular crystals may be represented by an [(n = 6)] sum over atomic sites, with possible inclusion of [(n = 8, 10, \ldots)] terms for higher accuracy. The dispersion-energy sum has somewhat better convergence properties than the Coulombic sum. Nevertheless, accelerated-convergence treatment of the dispersion sum is strongly recommended since its use can yield at least an order of magnitude improvement in accuracy for a given calculation effort. The repulsion energy between non-bonded atoms in a crystal may be represented by an exponential function of short range, or possibly by an [(n = 12)] function of short range. The convergence of the repulsion energy is fast and no accelerated-convergence treatment is normally required.


Glasser, M. L. & Zucker, I. J. (1980). Lattice sums. Theor. Chem. Adv. Perspect. 5, 67–139.

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