Inverse Gaussian transforms: General properties and application to Slater-type orbitals with noninteger and integer n in the coordinate and momentum representations

Publisher Website
Google Scholar
Abstract
The use of Gaussian‐type orbitals (GTO) facilitates the evaluation of the multicenter integrals encountered in quantum chemistry by reducing all integrals of more than two centers to two‐center integrals. On the other hand Slater‐type orbitals (STO), while leading to more time‐consuming integral evaluations, provide a better approximation to variationally determined atomic orbitals. Thus, for a basis set of given size, STO’s generally give better accuracy than GTO’s. Kikuchi proposed the representation of STO’s as integral Gaussian transforms, or in effect by continuous expansions in GTO’s, and Shavitt, Karplus, and Kern have applied this technique to the evaluation of multicenter integrals over STO’s. If these procedures are to be extended, it is desirable to develop a more systematic approach to the representation of a given basis function, ψ (r), as a Gaussian transform, ψ (r) =G[f (t);r] =F∞0f (t)exp(−r2t) dt; what this reduces to is the problem of calculating the inverse Gaussian transform, f (t) =G−1[ψ (r);t]. In the present investigation it is pointed out that f (t) =L−1[ψ (s1/2);t], where L−1 represents the inverse Laplace transformation. On this basis conditions on ψ (r) necessary for the existence of a unique continuous Gaussian inverse, f (t), are formulated, and general rules for the manipulation of inverse Gaussian transforms are developed. Finally, the formulas for the inverse Gaussian transforms of STO’s obtained previously by Kikuchi and Wright are generalized to noninteger principal quantum number, and angle‐dependent STO’s, in both the coordinate and momentum representations.