Calculation of Franck-Condon factors and simulation of photoelectron spectroscopy of D2CO: Including Duschinsky effects

Calculations of Franck-Condon factors are crucial for interpreting vibronic spectra of molecules and studying nonradiative processes. On the base of the closed form expression of the Franck-Condon integrals between arbitrary multidimensional harmonic oscillators under the Duschinsky mixing effects, a more general algebraic expression for the calculation of the three-dimensional four-mode Franck-Condon factors was derived straightforwardly and applied to study the photoelectron spectrum of D2CO+(B∼2A1). Geometry optimization and harmonic vibrational frequency calculations were performed on the X∼1A1 state of D2CO at B3LYP, QCISD, CCSD and CASSCF levels, and the B∼2A1 state of D2CO+ at CIS, TD-B3LYP, and CASSCF levels. Franck-Condon analyses and spectral simulations were carried out on the D2CO+(B∼2A1)-D2CO(X∼1A1) photoionization process. The spectral simulations of vibrational structures based on the computed Franck-Condon factors are in excellent agreement with the observed spectrum. In addition, the equilibrium geometric parameters of the B∼2A1 state of D2CO+ were obtained in the spectral simulations.