Convergence of lattice sums and Madelung’s constant

Abstract

The lattice sums involved in the definition of Madelung's constant of an NaCI-type crystal lattice in two or three dimensions are investigated. The fundamental mathematical questions of convergence and uniqueness of the sum of these, not absolutely convergent, series are considered. It is shown that some of the simplest direct sum methods converge and some do not converge. In particular, the very common method of expressing Madelung's constant by a series obtained from expanding spheres does not converge. The concept of analytic continuation of a complex function to provide a basis for an unambiguous mathematical definition of Madelung's constant is introduced. By these means, the simple intuitive direct sum methods and the powerful integral transformation methods, which are based on theta function identities and the Mellin transform, are brought together. A brief analysis of a hexagonal lattice is also given.

Colloquium: Physics of the Riemann hypothesis

“…is divergent which is physically unsatisfactory. The convergence properties of such sums have been extensively investigated (Borwein et al, 1985;Chaba and Pathria, 1975, 1976a,b, 1977. The sum (44) is an alternating and conditionally convergent sum.…”

Convergence analysis of the Wolf method for Coulombic interactions

“…It is interesting to note that summation over regions that may seem to be natural can diverge. As an example, for a 3-dimensional NaCl-type ionic crystal, it was shown that this lattice sum does not converge over expanding spheres [3], expanding ellipsoids, and some specific expanding polygons [4], but it would converge for expanding cubes [3]. The direct summation method is not practical due to the slow rate of convergence.…”

“…Specifically, in what order one should sum the terms of the series (2) to obtain the Ewald's result? Borwein, et al [3] showed that for NaCl-type crystals, summing over cubes would yield the Ewald result. It is interesting to note that ′ n∈Z 3 |n + r| −1 is not a convergent series and since all of the summands are positive, partial sums of this series would be unbounded.…”

Some results on the electrostatic energy of ionic crystals

“…For the simple cubic NaCl structure one can find a much faster and more accurate path to the Madelung constant based on analytical formulae of Borwein et al (7). This is discussed in Sect.…”

“…In the macroscopic limit when N becomes large this quantity will go to minus infinity. If we divide it by the number of (neutral) molecules, N3/2, we get the electrostatic energy per molecule in the form [7] @ = lim qN N-co Note that N should be even if the macroscopic piece is to be neutral; otherwise it has charge minus one, but this does not affect convergence.…”