The Spherical Tensor Gradient Operator

The spherical tensor gradient operator Ylm(∇), which is obtained by replacing the Cartesian components of r by the Cartesian components of ∇ in the regular solid harmonic Ylm(r), is an irreducible spherical tensor of rank l. Accordingly, its application to a scalar function produces an irreducible spherical tensor of rank l. Thus, it is in principle sufficient to consider only multicenter integrals of scalar functions: Higher angular momentum states can be generated by differentiation with respect to the nuclear coordinates. Many of the properties of Ylm(∇) can be understood easily with the help of an old theorem on differentiation by Hobson [Proc. Math. London Soc. 24, 54 (1892)]. It follows from Hobson's theorem that some scalar functions of considerable relevance as for example the Coulomb potential, Gaussian functions, or reduced Bessel functions produce particularly compact results if Ylm(∇) is applied to them. Fourier transformation is very helpful in understanding the properties of Ylm(∇) since it produces Ylm(-ip). It is also possible to apply Ylm(∇) to generalized functions, yielding for instance the spherical delta function δlm(r). The differential operator Ylm(∇) can also be used for the derivation of pointwise convergent addition theorems. The feasibility of this approach is demonstrated by deriving the addition theorem of rvYlm(r) with v ∈ πR.