An Algebraic Expression of the Three-Dimensional Franck-Condon Factors and Its Application

Abstract

A more general algebraic expression for the calculation of the four-mode Franck-Condon factors was derived straightforwardly on the basis of the closed form expression of the Franck-Condon integrals between arbitrary multidimensional harmonic oscillators under the Duschinsky mixing effects. This new algebraic expression was applied to study the photoelectron spectra of $D_2CO^+(\widetilde{A} ^2B_1).$ Franck-Condon analyses and spectral simulations were carried out on the $D_2CO^+(\widetilde{A} ^2B_1)$-$D_2CO(\widetilde{X} ^1A_1)$ photoionization processes. The spectral simulations of vibrational structures based on the computed Franck-Condon factors are in excellent agreement with the observed spectra.