Time-dependent theory of Raman scattering

A time-dependent picture of vibrational Raman scattering in the weak field limit is presented. From this viewpoint we can separate the static effects, due to the coordinate dependence of the electronic transition dipole, from the dynamic effects that arise from wave packet propagation on the Born–Oppenheimer surfaces. Away from resonance, the energy uncertainty relation gives the propagation time necessary to obtain the cross section as being inversely proportional to the mismatch of the excitation frequency with the excited surface. The wave packet, given by the initial vibrational wave function times the transition dipole, hardly moves on the excited surface when the excitation frequency is far from resonance. As the excitation frequency is tuned closer to resonance, the propagated wave packet samples a larger portion of the surface. Using the short time approximation to the propagator, we obtain formulas for the cross section that are applicable for Raman scattering by polyatomics. The short time approximation is expected to be good away from resonance independent of the nature of the surface, and also on resonance with a repulsive surface. For an attractive surface, the approximation gives the average resonant cross section useful in the case when the vibrational structures cannot be observed.