N-atom molecular systems: bunch of relative position vectors, local coordinates and quantum-mechanical kinetic energy operators

Abstract

After elimination of the centre of mass translation, a vector parametrization for an N-atom molecular system, consisting of N − 1 relative position vectors, is used. Sets of 3 N − 6 local coordinates well suited for describing the system are introduced: they are all made up of the N − 1 vector lengths, N − 2 planar angles of vector pairs and N − 3 dihedral angles for vector triplets. In addition, three Euler angles describe the orientation of the body-fixed frame: the first two angles allow one to orient the z-BF axis, either parallel to one vector or perpendicular to the plane of two vectors; the third angle is for rotation around z, completing the link between the molecule (i.e. the vectors) and the BF axes. The three Euler angles, together with the 3 N − 6 local coordinates, make up a set of N − 1 triplets of spherical coordinates for the relative position vectors, with respect to various frames. This property is used to derive exact expressions of general quantum mechanical kinetic energy operators, and also to propose a polyspherical-harmonics representation in which the kinetic energy matrix may take a relatively simple form (i.e. prediagonalized to a large extent)