Addition Theorems as Three-Dimensional Taylor Expansions. II. $B$ Functions and Other Exponentially Decaying Functions

TL;DR

This paper develops addition theorems for exponentially decaying functions like B functions using three-dimensional Taylor expansions and spherical tensor operators, simplifying their derivation and enabling immediate application to related functions.

Contribution

It introduces a method to derive addition theorems for B functions and related functions using spherical tensor expansions, simplifying previous complex Cartesian approaches.

Findings

Addition theorems for B functions are derived with a compact structure.

Functions based on Laguerre polynomials can be expressed as finite sums of B functions.

The method simplifies the derivation of addition theorems for exponentially decaying functions.

Abstract

Addition theorems can be constructed by doing three-dimensional Taylor expansions according to $f (\mathbf{r} + \mathbf{r}') = \exp (\mathbf{r}' \cdot \mathbf{\nabla}) f (\mathbf{r})$. Since, however, one is normally interested in addition theorems of irreducible spherical tensors, the application of the translation operator in its Cartesian form $\exp (x' \partial /\partial x) \exp (y' \partial /\partial y) \exp (z' \partial /\partial z)$ would lead to enormous technical problems. A better alternative consists in using a series expansion for the translation operator $\exp (\mathbf{r}' \cdot \mathbf{\nabla})$ involving powers of the Laplacian $\mathbf{\nabla}^2$ and spherical tensor gradient operators $\mathcal{Y}_{\ell}^{m} (\nabla)$, which are irreducible spherical tensors of ranks zero and $\ell$, respectively [F.D.\ Santos, Nucl. Phys. A {\bf 212}, 341 (1973)]. In this way, it is indeed possible to derive addition theorems by doing three-dimensional Taylor expansions [E.J. Weniger, Int. J. Quantum Chem. {\bf 76}, 280 (2000)]. The application of the translation operator in its spherical form is particularly simple in the case of $B$ functions and leads to an addition theorem with a comparatively compact structure. Since other exponentially decaying functions like Slater-type functions, bound-state hydrogenic eigenfunctions, and other functions based on generalized Laguerre polynomials can be expressed by simple finite sums of $B$ functions, the addition theorems for these functions can be written down immediately.