Polyspherical Parametrization of an N-Atom System: Principles and Applications
Résumé
This study reviews the polyspherical parametrization of a N-atom system. In this general formulation, the N-atom system is described by any set of (N-1) vectors {Ri, (i 1, . . . , N 1)} chosen by the user and characterized by their spherical coordinates (Ri, i, i) in the body fixed (BF) frame. Moreover, we present the method employed to establish the general expression of the kinetic energy operator (KEO) in terms of the total angular momentumJˆ and either the angular momenta {Lˆ i; (i 1, . . . , N 2)} or the conjugate momenta {(pRi, p i, p i); (i 1, . . . , N 2)} associated with the polyspherical coordinates. We propose to adopt a particular definition of the BF frame, but we show how the user can easily modify its definition in order to take into account the specificity of the studied system. Symbolic calculations can be very helpful to determine the general expression of the KEO of a N-atom system. We also demonstrate how this general formalism can be adapted by the user in order to take into account some physical properties of the studied system, e.g., how the symmetry can be introduced, how the system can be separated into two subsystems, and how the polyspherical coordinates can be coupled with an other set of coordinates more adapted to describe the system. We also show that this formulation can be coupled easily and efficiently to numerical methods that solve the time-independent Schro¨dinger equation or that propagate the wave packet using the time-dependent picture. Special emphasis is placed on concrete applications of this parametrization performed very recently by several groups to study the infrared spectroscopy of a large variety of semi-rigid molecules, floppy or van der Waals systems, or scattering processes.