The Spherical Tensor Gradient Operator

The spherical tensor gradient operator Y l (∇), which is obtained by replacing the Cartesian components of r by the Cartesian components of ∇ in the regular solid harmonic Y l (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 Y l (∇) can be understood easily with the help of an old theorem on differentiation by Hobson [Proc. London Math. Soc. 24, 54 67 (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 Y l (∇) is applied to them. Fourier transformation is very helpful to understand the properties of Y l (∇) since it produces Y l (−ip). It is also possible to apply Y l (∇) to generalized functions, yielding for instance the