Symmetry-adapted functions for the point groups

Abstract

Abstract The aim of this chapter is to derive the linear combinations of spherical harmonics that belong to the various rows of the matrix representations of the crystallographic point groups, and to give tables of these functions. Such functions as these have been given the name, by Melvin (1956), of symmetry-adaptedfunctions because they have the correct properties required by the representations under transformation by the group elements. Work on the determination of symmetry-adapted functions has been done by Altmann (1956, I 957), Bell (1954), Bethe (1929), Betts (1959), Callen and Callen (1963), Cohan (1958), Cornwell (1969), Flodmark (1963), Flower, March, and Murray (1960), McIntosh (1960a, 1963), Melvin (1956), Meyer (1954), Nesbet (1961), Schiff (1955, 1956), and von der Lage and Bethe (1947). This chapter follows quite closely the treatment given by Altmann (1957), Altmann and Bradley (1963a, b, 1965), and Altmann and Cracknell (1965). Given a particular matrix representation of a group the problem is to find all possible bases that are symmetry-adapted to that representation. Then any function that can be expanded in terms of spherical harmonics and is to have the transformation properties of a given row of that representation, will include only those linear combinations of spherical harmonics that are symmetry-adapted to have the same transformation properties. We use the term surface harmonic for such a linear combination of spherical harmonics that is symmetry-adapted to one row of a point-group representation.