Translational and rotational expansion of spherical Gaussian wave functions for multicenter molecular integrals

Abstract

The expansion of the regular solid spherical harmonics rlYlm(θφ) [and the irregular solid spherical harmonics r−(l+1)Ylm(θφ)] about a displaced center is shown to be an irreducible tensor coupling of two solid spherical harmonic tensors—one refers to the displaced center and the other is made of the displacement vector. The Gaussian exponentials are expanded at the displaced center through the modified plane wave expansion. Combining these two expansions, the multicenter molecular integrals of the overlap, nuclear attraction, kinetic energy, and two-electron Coulomb repulsion over the homogeneous solid harmonic spherical Gaussians r2n+lYlm(θφ)exp(−αr2) are integrated straightforwardly in spherical coordinates. The overlap integral involving nonhomogeneous solid harmonic spherical Gaussians r(2n+1)+lYlm(θφ)exp(−αr2) has also been integrated. The results obtained are in simple analytical expressions. Within these expressions, all the magnetic quantum numbers appear only in two places—in the Clebsch–Gordan coefficients and in the spherical harmonics of the displacement vector (referring to an arbitrary frame of reference). The general analytical expression for each integral is similar to that obtained through the Talmi transformation. They become identical when explicit expressions for Talmi coefficients and numerical values for quantum numbers n and l are used.