Hyperspherical harmonics for polyatomic systems: basis sets for kinematic rotations

In a symmetrical hyperspherical framework, the internal coordinates for the treatment of N-body systems are conveniently broken up into kinematic invariants and kinematic rotations. Kinematic rotations describe motions that leave unaltered the moments of the inertia of the N-body system and perform the permutation of particles. This article considers the corresponding expansions of the wave function in terms of hyperspherical harmonics giving explicit examples for the four-body case, for which the space of kinematic rotations (the "kinetic cube") is the space SO(3)/V 4 and then the related eigenfunctions will provide a basis on such manifold, as well as be symmetrical with respect to the exchange of identical particles (if any). V 4 is also denoted as D 2. The eigenfunctions are obtained studying the action of projection operators for V 4 on Wigner D-functions. When n of the particles are identical, the exchange symmetry can be obtained using the projection operators for the S n group. This eigenfunction expansion basis set for kinematic rotations can be also of interest for the mapping of the potential energy surfaces.