Hyperspherical coordinates for molecular dynamics by the method of trees and the mapping of potential energy surfaces for triatomic systems

Some results on hyperspherical coordinates and harmonics for the representation of the many-body problem are presented, extensive use being made of the method of trees. Properties of these trees are examined: a lemma on the simplification of trees possessing a particular symmetry is proven, and used to discuss the internal coordinates for a system of three particles and the mapping of potential energy surfaces. A framework is provided for relating different couplings of particles by rotations on hyperspheres and alternative hyperangular parametrizations by orthogonal basis transformations. Extensions to nonzero angular momentum or to more than three particles are shown not to be trivial, and the possible role of developments of the tree method, leading to more general hyperspherical coordinates, is briefly considered.