Knowledge Bank

The multicenter molecular integrals of the overlap, nuclear attraction, kinetic energy and two-electron Coulomb repulsion over the homogeneous solid harmonic spherical Gaussians $r2\mu +1Yirq(\theta \varphi )exp(-\alpha r2)$ are integrated by expanding function from one center to another. The expansion of die regular solid spherical harmonics $riYirq(\theta \phi )$ (and the irregular solid spherical harmonics $r(r+1)Yirq(\theta \phi ))$ about a displaced center is shown lo be ail irreducible tensor coupling of two solid spherical harmonic tensors; one refers to the displaced center and the oilier is made of the displacement vector. The Gaussian exponentials are expanded at the displaced center through the modified plane wave expansion. The overlap integral involving non-homogeneous solid harmonic spherical Gaussians $r(2\mu +1)+1Yirq(\theta \varphi )exp(-\alpha r2)$ has also been integrated. The results obtained are In simple analytical expression. Within these expressions all the magnetic quantum numbers appear only in two places: in the Clebsech-Gordan coefficients and the spherical harmonics of the internuclear vector (referring to an arbitrary France of reference). Molecular geometry of different states can be obtained by optimizing bond distances and bond angles which appear explicitly in each integral expression.