Nuclear coordinate dependence of electronic matrix elements for radiationless transitions

Abstract

The nuclear coordinate dependence of the electronic matrix elements for radiationless transitions (in the weak coupling limit) is investigated by the use of a Q-centroid approximation. This approach bears a similarity to the familiar r-centroid method in diatomic spectroscopy, but has a wholy different physical character. Because the Q-centroid for electronic relaxation is obtained as an average with density of states weighted Franck—Condon factors, it is not restricted to geometries near the equilibrium position of the initial electronic state as it is in the case of radiative transitions (in the weak coupling limit). For totally symmetric vibrations, it is shown that the Q-centroids for poor accepting modes are in the vicinity of the equilibrium positions for this vibration, while those for good accepting modes tend towards the surface crossing along those vibrational modes. Thus, in the case of dominant accepting modes, the electronic matrix element reflects a Teller surface crossing mechanism for electronic relaxation, even though the density of states weighted Franck—Condon factors reflect a tunnelling mechanism. For non-totally symmetric vibrations, Q-centroids may be large or small independent of their accepting mode capabilities. Thus, coupling mechanisms, which are “forbidden” at the equilibrium geometry in aromatic hydrocarbons, may become allowed and even dominant because of very distorted Q-centroid configurations. This leads to another possible reason for the absence of observation of a vibration that is clearly assignable as a promoting mode in the single vibronic level fluorescence studies of benzene-like molecules. The results underscore previous warnings as to the enormous errors incurred by using the Condon approximation for the nuclear coordinate dependent energy denominators that appear in the electronic matrix elements.