Cancellation effects in Franck-Condon integrals

Cancellation effects in Franck-Condon integrals, which lead to the absence of certain (totally symmetric) progression members in electronic spectra, are discussed. In an approximate theory, where the upper- and lower-state vibrational frequencies are assumed to be equal, it is shown that the positions of the vibrational bands of least intensity are closely related to the zeros of the associated Laguerre polynomials. In particular, it is found that: (i) if the ratio of the intensities of the (1, 0) and (0, 0) bands equals m, the band (m, 1) will be absent from the spectrum; (ii) if the ratio of the intensities of the (1, 0) and (0, 0) bands is equal to one of the n zeros of the Laguerre polynomial L_n(x), the sequence member (n, n) will be absent from the spectrum. (Here (m, n) refers to the transition m <- n, where m and n are vibrational quantum numbers.)

These results appear to be approximately valid for many real spectra. The 2500 Angstrom system of PN, the second positive system of nitrogen, and the 2600 Angstrom system of benzene are considered as examples, and the expected effects are found. Corresponding effects in non-totally symmetric vibrations are considered, and shown to be of small importance. A note is included on the abnormal prominence of the (1, 1) sequence band which occurs when the change in geometry accompanying an electronic transition is large.