Hyperspherical harmonics for polyatomic systems: Basis set 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.