Analytical methods for fast converging lattice sums for cubic and hexagonal close-packed structures

Fast convergent series are presented for lattice sums associated with the simple cubic, face-centered cubic, body-centered cubic, and hexagonal close-packed structures for interactions described by an inverse power expansion in terms of the distances between the lattice points, such as the extended Lennard-Jones potential. These lattice sums belong to a class of slowly convergent series, and their exact evaluation is related to the well-known number-theoretical problem of finding the number of representations of an integer as a sum of three squares. We review and analyze this field in some detail and use various techniques such as the decomposition of the Epstein zeta function introduced by Terras or the van der Hoff-Benson expansion to evaluate lattice sums in three dimensions to computer precision.