Rapid convergence of lattice sums and structural integrals in ordered and disordered systems

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Abstract
We present a new interpretation of the Ewald technique that suggests a simple rule to guide the transformation of any lattice sum into a pair of sums, one over the original lattice and the other over the reciprocal lattice, both converging faster than any power law in the distance from the origin. The rule sets conditions needed in finding an approximation to the long-range part of the function being summed, and it is the Fourier transform of this approximation that then appears in the reciprocal-lattice sum. Once such an approximation is found, the same transformation can be used not only for lattice sums over Bravais lattices or lattices with a basis, but also for the structural integrals associated with noncrystalline systems, including glasses and liquids. The approach can also be applied to quasicrystals, where the transformation takes two different forms depending on whether the sum is viewed in three dimensions or six, for the icosahedral case. Finally, other implications of the interpretation are presented, including a set of conditions under which an alternative planewise summation method can be used.

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