Madelung Constants

Madelung constants are the quantities obtained from cubic, hexagonal, etc., lattice sums, with the exponent and summation convention specified.

For cubic lattice sums, define

where is a positive integer and the prime excludes the origin. The sum converges absolutely for . Closed forms for at even indices include those summarized in the following table, where is the Dirichlet beta function and is the Dirichlet eta function.

To obtain the closed form for , separate the two coordinate axes from the four quadrants. The symmetry under sign changes gives

The second of these sums can be done analytically as

for . In the case , this reduces to

The first sum is more difficult, but in the case can be written

Combining these then gives the original sum as

For a positive integer , the Coulomb values have the convergent integral

where is a Jacobi theta function. This integral specifies the summation convention when the underlying lattice sum is not absolutely convergent. In this convention, low-dimensional values are

(Finch 2003, pp. 76-77), where is the Riemann zeta function and is the Dirichlet beta function.

For , the lattice sum is interpreted using expanding cubes or analytic continuation (Tyagi 2004). It is given by Benson's formula (Borwein and Bailey 2003, p. 24)

(OEIS A085469), where the prime indicates that summation over (0, 0, 0) is excluded.

is sometimes called "the" Madelung constant and corresponds to the Madelung constant for a three-dimensional NaCl crystal. Crandall (1999) gave the expression

Similar results were found by Tyagi (2004),

the last of which converges rapidly. Averaging (17) and (14) then gives the beautiful equation

which agrees with to nine decimal places when the sum is omitted (Tyagi 2004).

However, no closed form for is known (Bailey et al. 2006).

For hexagonal lattice sums, is expressible in closed form as

(OEIS A086055).