Chapter 12 - Selected Topics I: Hyperspherical coordinates

Abstract

The chapter reviews hyperspherical coordinates. A powerful method to cope with many electron systems, including correlations, is based on a description within hyperspherical coordinates. A simplest three-body system, provided by a helium atom, is considered in the chapter. The total eigenfunctions are denoted as hyperspherical harmonics. They are harmonic functions on the hyperspherical surface of a six-dimensional space and are simultaneous eigenfunctions of operators. The surge of applying hyperspherical coordinates owes much of its success to adopting the adiabatic approximation in analogy to the treatment of slow collisions within the molecular orbital model. The coupling terms are smooth functions of single vector within a segment. In this way, each segment is treated separately, and the logarithmic derivatives of wave functions are propagated from one segment to another. In the asymptotic region, the solutions propagated are matched to the desired asymptotic solutions expressed in the appropriate Jacobi coordinates. The hyperspherical close-coupling method is widely applied, usually to low collision energies. A generalization of the method is proposed and demonstrated for charge transfer.