The Spherical Tensor Gradient Operator

Abstract

The spherical tensor gradient operator ${\mathcal{Y}}_{\ell}^{m} (\nabla)$, which is obtained by replacing the Cartesian components of $\bm{r}$ by the Cartesian components of $\nabla$ in the regular solid harmonic ${\mathcal{Y}}_{\ell}^{m} (\bm{r})$, is an irreducible spherical tensor of rank $\ell$. Accordingly, its application to a scalar function produces an irreducible spherical tensor of rank $\ell$. 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 ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ can be understood easily with the help of an old theorem on differentiation by Hobson [Proc. London Math. Soc. {\bf 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 ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ is applied to them. Fourier transformation is very helpful to understand the properties of ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ since it produces ${\mathcal{Y}}_{\ell}^{m} (-\mathrm{i} \bm{p})$. It is also possible to apply ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ to generalized functions, yielding for instance the spherical delta function $\delta_{\ell}^{m} (\bm{r})$. The differential operator ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ 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 $r^{\nu} {\mathcal{Y}_{\ell}^{m}} (\bm{r})$ with $\nu \in \mathbb{R}$.