Quantum dynamics of kinematic invariants in tetra- and polyatomic systems

Abstract

For the dynamical treatment of polyatomic molecules or clusters as n-body systems, coordinates are conveniently broken up into external (or spatial) rotations, kinematic invariants, and internal (or kinematic) rotations. The kinematic invariants are related to the three principal moments of inertia of the system. At a fixed value of the hyperradius (a measure of the total moment of inertia), the space of kinematic invariants is a certain spherical triangle, depending on the number of bodies, upon which angular coordinates can be imposed. It is shown that this triangle provides the 24-element (group O) octahedral tesselation of the sphere for n=4 and the 48-element (group O_h) octahedral tesselation for n5. Eigenfunctions describing the kinematics of systems with vanishing internal and external angular momentum can be obtained in closed form in terms of Bessel functions of the hyperradius and surface spherical harmonics. They can be useful as orthonormal expansion basis sets for the hyperspherical treatment of the n-body particle dynamics.