Angular Symmetry and Hylleraas Coordinates in Four-Body Problems

Abstract

Abstract The most accurate studies of few-body Coulomb systems have used wavefunctions of forms that are simple in Hylleraas coordinates (those that explicitly include all the interparticle distances). In most studies the wavefunction has been constructed from Slater-type orbitals about a single center (relative to which the other particles are at positions r i ) by adjoining to each orbital product a polynomial in the other interparticle distances r i j . Matrix elements have then usually been evaluated by expanding the r i j in terms of the r i . This type of wavefunction is not ideal for “nonadiabatic” systems in which all the particles are of comparable (finite) mass; a preferable alternative is to use a wavefunction constructed from exponentials in all the Hylleraas coordinates. It is not practical to use the usual expansion methods for Hylleraas exponential wavefunctions, and this paper considers issues arising when four-body systems are treated directly in the Hylleraas coordinates for states of general angular symmetry. The two problems treated here are (1) convenient and compact expression of the kinetic energy for angular eigenstates, and (2) angular integrations in matrix elements. Both these problems differ from their well-known counterparts in orbital formulations, in part because the r i j are not orthogonal coordinates, and in part because the angles of the r i are not all independent variables.