Solvation state selective excitation in resonance Raman spectroscopy. II. Theoretical calculation

We have reported in the previous paper [J. Chem. Phys. 109, 9075 (1998)] that the Raman Stokes shifts of the $\mathrm{C=O}$ and the $\mathrm{C=N}$ stretching modes in the resonance Raman spectra of a solvatochromic dye, phenol blue (PB), appear to depend on the excitation photon energy in polar or protic solvents. The reason was ascribed to the solvation state selective excitation due to the inhomogeneous distribution of the solvation environment. In this paper we have made a theoretical model calculation to demonstrate that the solvation state selective excitation indeed brings about the excitation energy dependence of the resonance Raman Stokes shift in solution. In our model, both the electronic and the vibrational transitions are linearly coupled to the same harmonic bath, to embody the coupling between the fluctuations of the electronic and the vibrational transition energies. The absorption and the resonance Raman cross sections are formulated for this model on the basis of the time dependent path integral method. In the formulation, the finite relaxation time of the bath mode and the vibrational progressions in the absorption spectra are also taken into account. We have calculated the resonance Raman spectra at various excitation photon energies for the model system probable for PB in methanol. The parameters for the calculation are estimated from the analysis of the absorption and the resonance Raman spectra. It is demonstrated that the Raman Stokes shift indeed depends on the excitation energy for this model. It is noted that the vibrational dephasing due to the coupling with the solvent bath mode should be faster than the relaxation time of the bath mode for the excitation energy dependence of the Raman Stokes shift to be observed.