Dynamical symmetry of vibronic transitions in polyatomic molecules and the Franck-Condon principle

Abstract

In the present paper the method of analyzing vibronic transitions, worked out in Doktorov, Malkin, and Man'ko, J. Mol. Spectrosc. 56, 1 (1975), for nonlinear molecules of the type AB 2 is generalized to arbitrary polyatomic molecules with N vibrational degrees of freedom. By means of the coherent-state method, general expression for the Franck-Condon factor is obtained in the harmonic oscillator approximation in terms of the Hermite polynomials of 2 N variables. Overlap integrals are shown to be matrix elements of some operator belonging to the Lie group Sp(2 N, R) H( N), which serves as the dynamical symmetry group for the vibrational Hamiltonian. Recurrence relations for the overlap integrals are obtained. These relations are suitable for the effective calculation of the overlap integrals. New sum rules generalizing known Herzberg-Teller formulas for the fractional intensity of the 0-0 transition are derived. The generalization of the iterative method proposed by Coon et. al., J. Mol. Spectrosc. 8, 285 (1962), for the AB 2 molecules is discussed. A new partial analysis method is presented for the investigation of a vibronic structure of the electronic spectra of polyatomic molecules. This method enables us to find the parameters of the Dushinsky transformation from the relative intensities of a few selected vibronic bands measured experimentally. These parameters are then used for calculating relative intensities of the remainder vibronic bands and for finding the excited-state geometry of a molecule. Formulas for the relative intensities of the vibronic bands are also obtained in the case of the electronically forbidden transitions, with due regard for the effect of the second order in vibronic coupling. As an example, the calculation of relative intensities of some bands in the 2600 Å system of the absorption spectrum of benzene is given.