A general formula of two-dimensional Franck-Condon integral and the photoelectron spectroscopy of sulfur dioxide

We derived a general formula of Franck-Condon integral for two-dimensional harmonic oscillators (〈v1v2|v1′v2′〉) taking into account the Duschinsky effect and applied it to study the photoelectron spectroscopy of SO2 and SO2-. The equilibrium geometries and harmonic vibrational frequencies of SO2+, SO2 and SO2- were calculated by using the density functional theory (B3LYP functional) and the coupled cluster singles and doubles with perturbative triples [CCSD(T)] methods with various basis sets up to 6-311+G(3df) and aug-cc-pVTZ. The adiabatic ionization energy and electron affinity were computed by using the CCSD(T) method extrapolated to the complete basis set limit with aug-cc-pVXZ (X = D, T, Q, 5). The simulated photoelectron spectra of both SO2 and SO2- are in accord with the experiment. While the Duschinsky effect plays a role for some weak transitions of SO2, it can be neglected for SO2-. A splitting observed in the experimental photoelectron spectrum of SO2 is interpreted as contributing from hot bands and combination bands of ν1 and ν2, rather than arising from perturbation of a potential barrier as previous researchers proposed. The calculated adiabatic ionization energy and electron affinity are in agreement with the experiment within 0.027 and 0.040 eV, respectively.