Addition Theorems as Three-Dimensional Taylor Expansions. II. $B$ Functions and Other Exponentially Decaying Functions
| dc.creator | Weniger, Ernst Joachim | |
| dc.date | 2001-09-11 | |
| dc.date.accessioned | 2026-07-07T04:28:38Z | |
| dc.date.available | 2026-07-07T04:28:38Z | |
| dc.description | 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. | |
| dc.description | 17 pages, LaTeX2e, 0 figures. Submitted to the Per-Olof Löwdin Honorary Volume, International Journal of Quantum Chemistry | |
| dc.identifier | https://arxiv.org/abs/math-ph/0109011 | |
| dc.identifier | http://arxiv.org/abs/math-ph/0109011 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/56863 | |
| dc.subject | Mathematical Physics | |
| dc.title | Addition Theorems as Three-Dimensional Taylor Expansions. II. $B$ Functions and Other Exponentially Decaying Functions | |
| dc.type | text |