A generalized Franck–Condon principle for vibronic bands of diatomic molecules

Abstract

The traditional Franck–Condon principle is limited in usefulness because empirical electronic transition matrix elements are not generally constant across a vibronic molecular transition band. Empirical generalizations that preserve the utility of the Franck–Condon overlap integral account for variations of the electronic matrix elements for diatomic bands in terms of transition frequency, as in a frequency-generalized Franck–Condon approximation [G. W. Robinson and R. A. Auerbach, J. Chem. Phys. 74, 2083 (1981)], or in terms of internuclear distance, as in the well-known r-centroid approximation. This paper illustrates empirically the validity of the Robinson–Auerbach method, and its relationship to the r-centroid approximation, for vibronic bands of some diatomic molecules. Most importantly, the former empirical method is derived from a gauge-invariant theory of matter-radiation interactions developed recently by Brown and Robinson [C. W. Brown and G. W. Robinson, J. Chem. Phys. (submitted)]. Both the empirical method and the gauge-invariant theory utilize continuous variability between dipole length and velocity matrix elements Rμν and Pμν/m, respectively. Using a gauge parameter ξ, the value of which is immaterial with exact molecular eigenstates, this variability is fixed in the empirical method by the ratio (im ωμνRμν/Pμν)ξ, which is unity with exact eigenstates, and in the theory by the difference ξ(im ωμνRμν−Pμν), which vanishes with exact eigenstates. The empirical form of the variability is obtained from the theory when the difference im ωμνRμν −Pμν small. The generalized Franck–Condon principle presented here has never before been successfully derived from theory.