Hyperspherical harmonics for polyatomic systems: basis set for collective motions

As a continuation of our previous work for the construction of expansion basis sets for the quantum mechanical treatment of N-body problems of interest for intermolecular and intramolecular and reactive dynamics of polyatomic molecules and clusters, we develop here a group-theoretical procedure which allows us to obtain explicitly the hyperspherical harmonics for the description of the collective motions. The coordinates involved are related to the invariants of the N-body system, which are referred to the moments of inertia. Although this work is limited to the case where both external and internal (kinematic) rotations are zero, and the example of N=4 is explicitly worked out, this method, which gives hyperspherical harmonics as linear combinations of ordinary spherical harmonics, can be extended to cover the general N-body case.