Diagrammatic theory of temperature-dependent resonance Raman scattering from polyatomic systems with general harmonic potential surfaces

A detailed derivation and discussion is given of our diagrammatic theory of temperature-dependent resonance Raman (RR) scattering and the optical absorption for multimode systems having general quadratic plus linear electron-vibrational coupling. By combining the time-correlator reformulation of RR scattering with suitably developed nonzero temperature many-body diagrammatic techniques, we obtain the RR excitation profiles and the absorption as one-dimensional Fourier transforms of analytic expressions involving just the model parameters and the temperature. The expressions are very convenient for explicit multimode model calculations. In addition, the theory brings out in a natural way the relation between RR profiles and the absorption, such that within well-defined special cases useful ‘‘transform’’ techniques can be developed for computing profiles directly from the observed temperature-dependent absorption. The many practical advantages of the theory for the analysis of experimental data have been demonstrated in earlier papers dealing with specific systems. In this paper we provide a comprehensive discussion of the theoretical details, which have not been given previously. The theory applies for any number of normal modes, and for arbitrary normal coordinate mixing, mode frequency shifts, and atomic equilibrium position shifts under electronic excitaiton. It involves products of phonon operators having both positive and negative time ordering, necessitating specialized combinatorial arguments. The use of an appropriate linked cluster expansion is shown to lead very naturally to a separation of the RR scattering into ‘‘orders,’’ which is the essential component producing the important general features listed above.Detailed derivations are given of the exact expressions for the first-order RR profiles in the most general model, and for first- , second- , and third-order profiles in the frequency-shift limit of no mode mixing. The latter formulas are recast into their absorption→profile ‘‘transform’’ versions, and these are simplified to more useful approximate forms for the practically important special case of small frequency shifts. Renormalizations of the linear electron-vibrational coupling parameters due both to mode mixing and to frequency shifts are also briefly discussed.