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Addition theorems as three‐dimensional Taylor expansions. II. B functions and other exponentially decaying functions
Author(s) -
Weniger Ernst Joachim
Publication year - 2002
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.948
Subject(s) - eigenfunction , operator (biology) , mathematical physics , simple (philosophy) , laplace operator , laguerre polynomials , mathematics , translation (biology) , nabla symbol , taylor series , tensor (intrinsic definition) , physics , pure mathematics , mathematical analysis , quantum mechanics , omega , chemistry , biochemistry , eigenvalues and eigenvectors , philosophy , epistemology , repressor , messenger rna , transcription factor , gene
Addition theorems can be constructed by doing three‐dimensional Taylor expansions according to f ( r + r ′)=exp ( r ′⋅∇) f ( 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 ′∂/∂ x )exp ( y ′∂/∂ y )exp ( z ′∂/∂ z ) would lead to enormous technical problems. A better alternative consists in using a series expansion for the translation operator exp ( r ′⋅∇) involving powers of the Laplacian ∇ 2 and spherical tensor gradient operators \documentclass{article}\pagestyle{empty}\begin{document}$\mathcal{Y}^{m}_{\ell}(\nabla)$\end{document} , which are irreducible spherical tensors of ranks zero and ℓ, respectively (Santos, F. D. Nucl Phys A 1973, 212, 341). In this way, it is indeed possible to derive addition theorems by doing three‐dimensional Taylor expansions (Weniger, E. J. Int J Quantum Chem 2000, 76, 280). 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. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002