Group Theoretical Analysis of Vibronic Interactions in Molecules

Abstract

The canonical transformation methods previously developed for the solution of problems involving dynamical vibronic interactions in molecules are examined by means of group theory. It is found that ${\mathrm{SU}}_2$ and the quantum mechanical group (the group generated by the position and momentum operators) afford physical insight into the reasons for the success of the transformation methods. In each method, the Hamiltonian is closely related to and can be written in terms of the elements of the groups. In these methods we use the fact that the basis functions of the representations of the quantum mechanical group are the harmonic oscillator functions, the natural functions for the problem. Products of representation matrices are in fact the matrix elements of the Hamiltonian. We also make use of the fact that the representation matrices are themselves the wavefunctions of the two‐dimensional isotropic harmonic oscillator. In addition to the physical insight, the group theoretic methods simplify the calculations.