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

Abstract

In the present paper the Franck-Condon factor for two-dimensional harmonic oscillator wave functions is expressed in terms of the Hermite polynomials of several variables. The results are applied to allowed electronic transitions in polyatomic molecules. Recurrence relations for overlap integrals are obtained. The coefficients of these relations depend on a few parameters which are found from experimental data. If there is no shift of origin of the potential function with the electronic transition, the overlap integrals are expressed in terms of Wigner's D-function. For this case a number of new sum rules for Franck-Condon factors are obtained which can be considered as a criterion for the applicability of the harmonic approximation. The appearance of these sum rules is a consequence of the possibility of representing the overlap integral as a matrix element of some operator of a dynamical group representation. A method is proposed for obtaining the potential function of the excited electronic state. The Herzberg-Teller formula for the fractional intensity of the 0-0 band is generalized to two-dimensional vibrations. As examples, some simple polyatomic molecules of the type AB 2 are considered, for which geometrical structures of excited states are found and the band intensities predicted.