—   crystal structure

For any $\alpha>0$, a neutral unit cell with volume $V$, formal charges $q_i$ at positions $r_i$, and $N_{\rm f.u.}$ formula units satisfies $M=-\frac{r_0}{N_{\rm f.u.}}\left[\frac12\sum_{i,j}\sum_n' q_iq_j\,\operatorname{erfc}(\alpha|r_i-r_j+n|)/|r_i-r_j+n| + \frac{2\pi}{V}\sum_{k\ne0}|\sum_i q_i e^{ik\cdot r_i}|^2\,e^{-|k|^2/(4\alpha^2)}/|k|^2 - \alpha\pi^{-1/2}\sum_i q_i^2\right]$, where $n$ runs over lattice translations, $k$ over nonzero reciprocal-lattice vectors, and the prime omits the term with $i=j$ and $n=0$ [1].

The rock-salt row also satisfies Benson's identity $M=12\pi\sum_{m,n\geq1,\ m,n\ {\rm odd}} \operatorname{sech}^2(\frac{\pi}{2}\sqrt{m^2+n^2})$ [3].

The sign convention puts the minus sign in the energy formula and stores $M>0$. The charges in the Ewald sum are the formal ionic charges, so the fluorite, antifluorite and cuprite rows include charge products with magnitudes larger than one.

The normalising distance is the nearest cation-anion distance. In terms of a cubic lattice parameter $a$, it is $a/2$ for rock salt, $\sqrt3 a/2$ for caesium chloride, and $\sqrt3 a/4$ for zincblende, fluorite, antifluorite and cuprite.

The wurtzite row is the ideal wurtzite structure with $c/a=\sqrt{8/3}$ and internal coordinate $u=3/8$, so $r_0=a\sqrt{3/8}$. Rutile, anatase and corundum are not included because their Madelung constants depend on structural parameters whose values vary from compound to compound, whereas ideal wurtzite has a conventional parameter choice.

P. P. Ewald, Die Berechnung optischer und elektrostatischer Gitterpotentiale, Annalen der Physik 369 (1921), no. 3, 253-287. (doi)

Wikipedia: Madelung constant

MathWorld: Madelung Constants

LibreTexts: Madelung Constants

Watson integrals of the cubic lattices —   another lattice-sum table; its sums are random-walk return sums on cubic lattices, while these are Coulomb sums over periodic charged arrangements

Packing densities and Hermite numbers of the classical lattices —   records geometric invariants of the classical lattices that underlie several of these crystal structures

Kissing numbers $\tau_n$ —   both use nearest-neighbour geometry of lattice arrangements; here the nearest cation-anion distance normalises an all-pair Coulomb sum

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds rock salt, caesium chloride, zincblende, ideal wurtzite, fluorite, antifluorite and cuprite, not every ionic crystal structure)

How they were obtained:

The generator computes the neutral-cell Ewald sum in arb ball arithmetic with $\alpha=4$, including explicit upper bounds for the omitted real-space and reciprocal-space tails. It writes $100$ digits.

more

The rock-salt row was checked against Benson's rapidly convergent $\operatorname{sech}^2$ series [3]. The rows for rock salt, caesium chloride, zincblende, fluorite, cuprite and ideal wurtzite were checked against published prefixes in [3] and [4].