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

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 ∇ and spherical tensor gradient operators Y l (∇), which are irreducible spherical tensors of ranks zero and l, respectively [F.D. Santos, Nucl. Phys. A 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. 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 Slatertype 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.