VECTOR PARAMETRIZATION OF THE N-BODY PROBLEM IN QUANTUM-MECHANICS - POLYSPHERICAL COORDINATES

The configuration of an N-body system can be entirely represented by N - 1 relative position vectors after separation of the center-of-mass motion. Many of the sets of coordinates that are commonly used for describing molecular configurations can be viewed as spherical coordinates, for the various vectors, collected together. The spherical angles are local; i.e., they are defined for frames that change from one vector to another. Each particular set of coordinates of that nature (polyspherical coordinates) consists of three Euler angles for the overall rotation of the body-fixed frame and 3N - 6 internal coordinates: the N - 1 vector lengths, N - 2 planar angles between pairs of vectors, and N - 3 dihedral angles between two vectors around a third one. This article aims at developing an example of this type of parametrization, where the body-fixed-frame z axis is parallel to one vector. The quantum-mechanical kinetic-energy operator for the system so described is derived. The operator action on the angular part of the functional basis set is studied (Wigner rotation matrix elements for the Euler angles and spherical harmonics for the internal angles), and the structure of the matrix representing the kinetic-energy operator is described in detail. The advantages and drawbacks of the present vector parametrization and the polyspherical coordinates are discussed. The principal advantage is in numerically calculating the matrix elements of the kinetic-energy operator: The integration over all angles turns out to be analytically achieved, so that the numerical effort is to be concentrated only on the N - 1 radial coordinates. Radial basis functions are to be selected according to the physical context (collisional or vibrational, or any other). Thus the angular basis set proposed constitutes an adequate finite-basis representation for the kinetic-energy operator and, combined with a discrete-variable representation for the potential energy, is likely to provide an efficient collocation framework for the dynamical study of more-than-three particle systems.