A general analytical expression for evaluation of an arbitrary n-dimensional Franck-Condon overlap integral including the Duschinsky effect

[Display omitted] •A more general analytical expression for calculation of the n-dimensional Franck-Condon integral of harmonic oscillators was derived.•Presenting a kind of method for determining the coefficient parameter matrices by matrix manipulations.•Simulating and elucidating the vibrational structure of the photodetachment spectra of SeO2‾.•Employing the iterative Franck-Condon analysis procedure, the geometries of the ground state of anion SeO2‾ are obtained. Evaluations of the multidimensional Frank-Condon (FC) overlapping integrals are needed theoretically to investigate the mechanism of radiative and non-radiative processes among the different electronic states of polyatomic molecules. We extended our recently reported approach to calculate an arbitrary n-dimensional Franck–Condon overlapping integrals based on harmonic oscillator approximation and taking into account the Duschinsky effects. An analytical formula was derived straightforwardly based on the Hermite polynomials expanding properties and directly evaluating Gaussian integrals. A method to determine the coefficient parameter matrices of four-dimensional Franck-Condon integrals by matrix operations was presented. In addition, as a practical application, we simulated and elucidated the vibrationally resolved peaks of the photodetachment spectra of SeO2‾. Moreover, in the spectral simulations, employing our iterative Franck-Condon analysis procedure, fairly reliable geometries of the ground state X~2B1 of the negative ion SeO2‾ are obtained.