Relation between classical and quantum formulations of the Franck-Condon principle: The generalized r-centroid approximation

The classical transition point $\text{r}{}_{\mathrm{v\prime v″}}{}^1$ for the (v′, v″) band is introduced as the point where the kinetic energy does not change during the transition (classical statement of the Franck-Condon principle). It is shown that when the r-centroid approximation is valid, it implies that the r centroid $\text{r}{}_{\mathrm{v\prime v″}}$ equals $\text{r}{}_{\mathrm{v\prime v″}}{}^1$, allowing a new connection to be made between the classical and quantum statements of the Franck-Condon principle. The r-centroid approximation fails for some classically allowed bands having large Franck-Condon factors. This occurs when the (v′, v″) band has more than one classical transition point. A generalization of the r-centroid approximation permits such bands to be used in determining the variation of the electronic transition moment with internuclear distance.