Doubly resonant SFG and DFG spectroscopies: An analytic model for data analysis including distorted and rotated vibronic levels. I. Theory.

We present an analytic description of doubly resonant infrared-visible sum (SFG) and difference frequency generation (DFG) spectroscopies. Within the Born-Oppenheimer and Condon approximations for harmonic oscillators, we extend the usual theory, limited to linear electron-vibration coupling, and introduce the quadratic coupling phenomena (mode distortion and mode mixing) in the excited state. The excitation spectra of vibrations in SFG and DFG experiments are calculated in integral form for arbitrary mode distortions and small amplitude mode mixing between pairs of modes. Mode distortion modifies all orders of vibronic coupling including the fundamental process, whereas mode mixing appears as a perturbation added to the distorted mode case. For small quadratic coupling amplitudes, the results may be recast in simple analytic forms after the introduction of the overlap spectral function and developed in sums and products of Lorentzian functions.