Application of Houston's Method to the Sum of Plane Waves over the Brillouin Zone. I. Simple-Cubic and Face-Centered-Cubic Lattices

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Abstract
In certain problems in solid‐state physics, the radial functions gj(r) in the expansion χ(r) = Σj=0∞ gj(r)Kj(θ, φ), where χ(r) is a known function and the Kj's are Kubic Harmonics, are of interest. This paper deals with the functions χ(r) ≡ N−1 Σk eik·r, where the sum runs over the first Brillouin Zone of a crystal. In particular, the functions χ(r) for simple cubic and face‐centered cubic lattices are expanded into series of Kubic Harmonics and the radial functions gj(r) for several values of j are found using Houston's method, in which the expansion into series of Kubic Harmonics contains only a finite number of terms with lowest j's. g0(r) is calculated using 3, 6, and 9‐term expansion, g2(r) and g3(r) using only 3 and 6‐term expansion. Comparing gj(r) obtained from the formulas with different numbers of terms it is established that for r in the region 〈0, 2a〉, where a is the lattice constant, the 6‐term approximation is very good. In practice, the functions gj(r) usually occur in integrands, together with atomic orbitals, and the tabulated results are expected to be particularly useful in the study of Wannier functions in the OPW scheme.