### Abstract

It is well known how to expand in spherical harmonics the gradient of a radial function in turn multiplied by a spherical harmonic. This expansion involves the use of the Wigner–Eckart theorem for the familiar O(3) ⊆O(2) chain of groups, and leads to Wigner coefficients in the formula together with reduced matrix elements which are simple first order differential operators in the radial variable. In the present paper we extend the above analysis to the application of the momentum operator π_{ m } to functions of the collective coordinates α_{ m }, *m*=2,1,0,−1,−2 associated with quadrupole vibrations. The spherical harmonics are now replaced by the complete but nonorthonormal set of functions χ^{λ} _{ s L M }, characterized by the irreducible representations λ,*L*,*M* of the O(5) ⊆O(3) ⊆O(2) chain of groups as well as by an extra labelling index *s*, that were derived in a previous publication. The application of the gradient to a product of a function *F* (β), β^{2}=Σ_{ m }α_{ m }α^{ m }, by χ^{λ} _{ s L M } requires an extension of the Wigner–Eckart theorem for the nonorthonormal basis. Results similar to the ones mentioned in the previous paragraph are obtained, though, of course, now we will have Wigner coefficients in the O(5) ⊆ (3) ⊆O(2) chain which have already been derived and programmed. With the help of the gradient formula we discuss the effect of the operators [π×π]^{ L } _{ m }, *L*=0,2,4, [α×π]^{ L } _{ m }, L=1,3 on basis of the O(5) ⊆O(3) chain of groups and indicate some of their applications.

Commenting has been disabled for this content