Construction of Tetrahedral Harmonics

Abstract

Explicit, detailed, and convenient algorithms have been developed for constructing tetrahedral harmonics for arbitrary J. These functions are expressed as linear combinations of spherical harmonics and also of symmetric top functions. A projection‐operator technique was used to produce projected functions which transform according to the rows of the irreducible matrix representations of $T_d$ . In general the set of projected functions so constructed may be linearly dependent. The idempotence of the projection operators was invoked to show that all the required orthonormal tetrahedral harmonics can be produced by the diagonalization of the projection‐operator matrices. The elements of these matrices are given explicitly for arbitrary J. In an Appendix, alternative procedures, including the Gram–Schmidt method, for the orthogonalization of the projected functions are treated briefly, with special emphasis on the use of these matrix elements. The “symmetry‐adapted functions” involve the $d_{\mathrm{K\prime K}}^J\text{(π\hspace{0.17em}/\hspace{0.17em}2)}$ , and a recursion relation is presented which facilitates an accurate and rapid calculation of these constants. An important new sum rule useful in calculations of integrated intensities is derived using the idempotence property. Various physical applications are discussed.