A new method to calculate Franck-Condon factors of multidimensional harmonic oscillators including the Duschinsky effect.

Calculations of Franck-Condon factors are crucial for interpreting vibronic spectra of molecules and studying nonradiative processes. We have developed a new method for calculating Franck-Condon factors of multidimensional harmonic oscillators including the Duschinsky effect. Closed-form formulas of two-, three-, and four-dimensional Franck-Condon factors were derived straightforwardly by using the properties of Hermite polynomials and Gaussian integrals. This new method was applied to study the photoelectron spectra of H(2)O(+)(B (2)B(2)) and D(2)O(+)(B (2)B(2)), whose equilibrium geometries and harmonic vibrational frequencies were calculated by using the coupled cluster singles and doubles with perturbative triples [CCSD(T)] method together with the basis sets of 6-311++G(3df,2pd) and aug-cc-pVTZ. The adiabatic ionization energies were computed by using the CCSD(T) method extrapolated to the complete basis set limit with aug-cc-pVXZ (X=D,T,Q,5). It was found that the simulated photoelectron spectra were mainly composed of nu(2) progressions and the combination bands of nu(1) and nu(2), whereas pure nu(1) transitions should be too weak to be observable, contrary to the literature reports. It was also found that the first discernible peak in the experimental photoelectron spectra did not correspond to the adiabatic transition. The adiabatic ionization energies of H(2)O(+)(B (2)B(2)) and D(2)O(+)(B (2)B(2)) are proposed to be 16.78 and 16.83 eV, about 0.40 and 0.58 eV lower than the best experimental values, respectively. Conversely, the calculated ionization energies are in agreement with the proposed values within 0.02 eV.