An algebraic formula to calculate the three-dimensional Franck–Condon factors including the Duschinsky effect

Abstract

Based on the contour integral expression of Hermite polynomials, a closed form for the Franck–Condon integrals between multidimensional harmonic oscillators has been presented under the Duschinsky mixing effects. Furthermore, an exact analytical expression for calculation of the three-dimensional six-mode Franck–Condon integral has been derived. In addition, a general explicit formula to calculate the three-dimensional Franck–Condon factors has also been given and applied to study the intensity distribution in the photoelectron spectrum of the H₂CO⁺( ² B ₁)–H₂CO( ¹ A ₁) transition of formaldehyde. The spectral simulations of vibrational structures based on the computed Franck–Condon factors are in excellent agreement with the observed spectrum.

Keywords:

• Franck–Condon integral
• contour integral
• Franck–Condon factors

Acknowledgement

This work was supported by the National Nature Science Foundation of China (No. 10674002), the Nature Science Foundation of the Education Committee of Anhui (No. KJ2009A131), the Program for Innovative Research Team in Anhui Normal University, and the Doctoral Research Foundation of Anhui Normal University (No. 750706).