Multidimensional Franck-Condon integrals and Duschinsky mixing effects

A general method for calculating multidimensional Franck–Condon integrals for polyatomic molecules is given. These integrals are derived by means of a multivariable generating function which incorporates both the transformation of the normal mode coordinates between initial and final electronic states and their frequency changes. The normal mode transformation or mode mixing (Duschinsky effect) scrambles the occupations of the normal modes, leading to unusual distributions, which at certain values of the angles of rotation are confined to some of the modes only or even to a single mode. Mathematically, this selectivity can be described in terms of normal mode displacements generated by the rotation matrix from the potential minima of the ground and excited states. These normal mode displacements, as well as the set of mixed frequency change parameters due to mode mixing, have no counterparts in the parallel mode approximation and are the reason that multidimensional Franck–Condon integrals are of formidable analytical complexity. This complexity is analyzed with particular attention to the symmetry of the integrals with respect to the interchange of quantum occupation numbers. Finally calculation of scattered intensities in the resonance Raman process is made, showing a strong dependence of the corresponding cross section upon the rotation angle.