A Comment On Recent Proposals for the Calculation of Vibration-rotation Energies in More-than-3 Atom Molecules

In a recent paper (Molec. Phys. 1991, 73, 1183), Bramley et al. analysed the physical and mathematical context in which the vibrational rotational Hamiltonian operator for a molecule is expressed in terms of Euler angles for the rotation and internal curvilinear coordinates for the vibration. In the case of a generic tetra-atomic molecule, they derived a few rules which determine the choice of the basis functions, in order (i) to ensure continuity and single-valuedness of the eigenfunctions and (ii) to cancel out the singularities (infinite integrals) that unavoidably crop up in the matrix representation of the vibration-rotation kinetic energy operator. In another recent paper (Phys. Rev. A, 1992, 45, 6217), Chapuisat and lung considered the vibration-rotation of a polyatomic molecule as resulting from the rotational-vibrational motion of coupled relative vectors; with the help of a standard representation for the angular motion of each vector viewed in the moving frame resulting from the first two Euler rotations, i.e. for the whole molecule a direct product of spherical harmonics, they have shown that it is possible to get rid of the singularities and to derive, after completing the third Euler rotation, analytical expressions of the kinetic energy operator matrix elements in the usual bending-torsion-rotation basis. In implementing this approach, the rules derived for tetra-atomics by Bramley et al. are confirmed and are next generalized to the case of the N-atom molecule. The restrictions to the direct product representation which must be explicitly taken into account are established on the basis of a purely physical analysis of the coupled vector motion.