Properties of Cartesian-spherical transformation coefficients

Abstract

In a previous paper (see Molec. Phys., vol.29, p.1461 (1975)), a standard transformation was proposed for expressing Cartesian tensors and tensor expressions in spherical form, and vice versa. Some properties of the transformation coefficients are derived. A recursion formula is given, which provides a simple means of generating the coefficients from those of the next rank below. The coefficients ( alpha ₁,... alpha _n mod j₁...j_n; m) are tabulated for nn=0. Another recursion formula gives the coefficients for m not=0 in terms of those for m=0. A graphical method is used to derive formulae for manipulating the coefficients, and formulae are given for handling products of tensors, contracted or uncontracted.