The Madelung constant of rock salt

The electrostatic energy of an ion in a sodium chloride crystal is set by one number, the Madelung constant $M \approx 1.7476$ . It is a conditionally convergent lattice sum of alternating charges. The two-dimensional analogue has a closed form in terms of Dirichlet series values, but whether the three-dimensional constant can be written in terms of standard functions is unknown.

Problem

Put a unit charge of sign $(-1)^{i+j+k}$ at every point $(i,j,k) \in \mathbb{Z}^3 \setminus \{0\}$ , so the nearest-neighbour distance is 1. The Madelung constant of the rock-salt (NaCl) structure is

M = -\sum_{(i,j,k) \neq (0,0,0)} \frac{(-1)^{i+j+k}}{\sqrt{i^2 + j^2 + k^2}} = 1.74756459463318219063\ldots

The series converges only conditionally. The accepted value comes from summing over expanding cubes $\max(|i|,|j|,|k|) \le N$ with $N \to \infty$ , not over expanding spheres [3, 4]. A rapidly convergent form is Benson's formula [1]:

M = 12\pi \sum_{\substack{i,j \ge 1 \\ i,j \text{ odd}}} \operatorname{sech}^2\Bigl(\frac{\pi}{2}\sqrt{i^2 + j^2}\Bigr).

Find a closed-form expression for $M$ in terms of standard constants and special functions.

Background

Madelung introduced such sums in 1918 to compute the electrostatic energy of ionic crystals. The Born–Landé lattice energy of a crystal such as NaCl is proportional to its Madelung constant [4]. In 1974 Zucker evaluated the analogous alternating sums $M_N(s) = \sum' (-1)^{i_1+\cdots+i_N} (i_1^2 + \cdots + i_N^2)^{-s}$ in dimensions $N = 2, 4, 6, 8$ in terms of the Riemann zeta function and the Dirichlet eta and beta functions [2]. For example, $M_2(s) = -4\beta(s)\eta(s)$ , so the two-dimensional Madelung constant is $4\beta(\tfrac12)\eta(\tfrac12) \approx 1.6155$ in magnitude [4]. No such evaluation is known in three dimensions. Lattice sums of this kind are surveyed in [5].

What is known

$M = 1.7475645946331821906362120355443974034851614366247417581528\ldots$ . HorizonMath gives these 59 significant digits as its reference, and OEIS A085469 lists the same expansion. Benson's formula and other rapidly convergent series yield many more digits [1, 4].
Closed forms exist for the alternating sums in dimensions 1, 2, 4, 6 and 8 [2, 4]. For the three-dimensional constant, Burrows, Cooper and Schwerdtfeger wrote in 2022 that "it is currently not known if the Madelung constant can be expressed in terms of standard functions" [4].

What would count as progress

A closed-form expression that matches $M$ to 20 significant digits, HorizonMath's acceptance threshold. Its evaluator checks this at 100-digit precision and rejects numerical shortcuts such as quadrature, root-finding, infinite series and fitted constants. Benson's formula converges fast enough to test a candidate to hundreds of digits.
A proof that a candidate expression equals $M$ , for example by a reduction of the three-dimensional sum to known Dirichlet series.

References

G. C. Benson, "A simple formula for evaluating the Madelung constant of a NaCl-type crystal", Can. J. Phys. 34 (1956), 888–890. doi:10.1139/p56-095
I. J. Zucker, "Exact results for some lattice sums in 2, 4, 6 and 8 dimensions", J. Phys. A 7 (1974), 1568–1575. doi:10.1088/0305-4470/7/13/011
D. Borwein, J. M. Borwein and K. F. Taylor, "Convergence of lattice sums and Madelung's constant", J. Math. Phys. 26 (1985), 2999–3009. doi:10.1063/1.526675
A. Burrows, S. Cooper and P. Schwerdtfeger, "The Madelung constant in N dimensions", Proc. R. Soc. A 478 (2022), 20220334. doi:10.1098/rspa.2022.0334, arXiv:2202.01392
J. M. Borwein, M. L. Glasser, R. C. McPhedran, J. G. Wan and I. J. Zucker, Lattice Sums Then and Now, Cambridge University Press, 2013. doi:10.1017/CBO9781139626804

Source

Imported from HorizonMath madelung_nacl, curated by Erik Y. Wang, Sumeet R. Motwani et al. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.