Exploring the separability of the three-body Coulomb problem in hyperspherical elliptic coordinates - art. no. 062705

An approximate symmetry of the three-body Coulomb problem which reveals its elf via approximate separability of the hyperspherical adiabatic (HSA) eige nvalue problem in hyperspherical elliptic (HSE) coordinates (eta, xi) and t hus is intimately related to the HSA approximation is discussed. The additi onal integral of motion responsible for this separability is specific to th e Coulomb interaction and generalizes the Laplace-Runge-Lentz vector known from the two-body Coulomb problem and the integral of separation constant o f the two-center Coulomb problem. In the zeroth approximation, this symmetr y leads to a completely separable representation of the three-body wave fun ction, where each state is labeled by a pair of HSE quantum numbers (n(eta) , n(xi)) that generalize the spheroidal quantum numbers for diatomic molecu les, on the one hand, and the Herrick-Lin quantum numbers for two-electron atoms, on the other. This approximation is illustrated by calculations for a number of three-body Coulomb systems for states with zero total angular m omentum. Taking into account the nonadiabatic couplings as well as the nons eparable part of the Coulomb potential by mixing a few separable states per mits one to obtain accurate results for systems with arbitrary masses and c harges of particles and for a wide spectrum below the three-body breakup th reshold.