Hyperspherical harmonics as Sturmian orbitals in momentum space: A systematic approach to the few-body Coulomb problem

Abstract

To exploit hyperspherical harmonics (including orthogonal transformations) as basis sets to obtain atomic and molecular orbitals, Fock projection into momentum space for the hydrogen atom is extended to the mathematical d dimensional case, higher than the physical case d = 3. For a system of N particles interacting through Coulomb forces, this method allows us to work both in a d = 3( N - 1) dimensional configuration space (on eigenfunctions expanded on a Sturmian basis) and in momentum space (using a ( d + 1)-dimensional hyperspherical harmonics basis set). Numerical examples for three-body problems are presented. Performances of alternative basis sets corresponding to different coupling schemes for hyperspherical harmonics have also been explicitly obtained for bielectronic atoms and H 2 + (in the latter case, also in the Born-Oppenheimer approximation extending the multicentre technique of Shibuya and Wulfman). Among the various generalizations and applications particularly relevant is the introduction of alternative expansions for multidimensional plane waves, of use for the generalization of Fourier transforms to many-electron multicentre problems. The material presented in this paper provides the starting point for numerical applications, which include various generalizations and hierarchies of approximation schemes, here briefly reviewed.