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Addition Theorems as Three-Dimensional Taylor Expansions. II. $B$ Functions and Other Exponentially Decaying Functions | | Ernst Joachim Weniger
; | | Date: |
12 Sep 2001 | | Subject: | Mathematical Physics | math-ph math.MP | | Abstract: | Addition theorems can be constructed by doing three-dimensional Taylor expansions according to $f (mathbf{r} + mathbf{r}’) = exp (mathbf{r}’ cdot mathbf{
abla}) f (mathbf{r})$. 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 $exp (mathbf{r}’ cdot mathbf{
abla})$ involving powers of the Laplacian $mathbf{
abla}^2$ and spherical tensor gradient operators $mathcal{Y}_{ell}^{m} (
abla)$, which are irreducible spherical tensors of ranks zero and $ell$, respectively [F.D. Santos, Nucl. Phys. A {f 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. {f 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. | | Source: | arXiv, math-ph/0109011 | | Services: | Forum | Review | PDF | Favorites | |
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