Quantum dynamics of kinematic invariants in tetra- and polyatomic systems

For the dynamical treatment of polyatomic molecules or clusters as n-body s ystems, coordinates are conveniently broken up into external (or spatial) r otations, 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 spherica l triangle, depending on the number of bodies, upon which angular coordinat es 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 n greater than or equal to 5. Eigenf unctions describing the kinematics of systems with vanishing internal and e xternal angular momentum can be obtained in closed form in terms of Bessel functions of the hyperradius and surface spherical harmonics. They can be u seful as orthonormal expansion basis sets for the hyperspherical treatment of the n-body particle dynamics.