On the symmetries of spherical harmonics

Abstract

It is often necessary to obtain expansions in spherical harmonics that belong to a given irreducible representation of a symmetry group. This is the case, for instance, when the cellular method is applied to investigate the band structure of a metal, where expansions are required that reproduce the symmetry of the group of the k vector (see Bouckaert, Smoluchowski and Wigner(4); von der Lage and Bethe (9)). Another instance where such expansions are necessary appears when hybrid orbitals are obtained for a central atom in a molecule of given symmetry (Kimball (8)). In this case lower order spherical harmonics are considered and tables for them up to l = 2 (functions s, p and d in real form) are given in the literature (cf. for example Eyring, Walter and Kimball (5)). However, interest has recently arisen in hybrids that include f functions (Shirmazan and Dyatkina (12)) and an extension of these tables appears to be desirable.