The transition polarizability is the essence of the Raman scattering process. A polarizability involves a transition dipole moment from ground state to excited state and a second transition dipole moment from the excited state back to the ground state. We could also call this the square of the transition polarizability. We must carefuly distinguish the transition polarizability from a state polarizability . The state polarizability starts and ends on the same state, both labeled i. The transition polarizability starts on ia nd ends on f. Often we will consider i to be the ground state, but it could be a different state. There is also a polarizability for an excited state. Howeer, this is is also not the same the as the transition polarizability, since the latter involves the a system that starts and ends on differnt states. Clearly, the transition polarizability is needed for Raman spectroscopy since most often Raman starts on the lowest vibrational state <0| and then ends up on the first vibratinoal excited state |1>.
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Distinguishing Franck-Condon from Vibronic modes
Resonance Raman spectroscopy is a vibrational spectroscopy that has a unique capability to distinguish between Franck-Condon and vibronic modes based on studies of the polarization of the Raman signal. The polarization of the Raman scattered line is the crucial observable for this experiment. In the video below we discuss the depolarization ratio, which is the quantity that changes for Raman scattering from totally-symmetric (Franck-Condon active) vibrations and non-totally-symmetric (vibronic) vibrations. Specifically, the depolarization ratio is shown to vary from 0 to 1/3 for Franck-Condon active vibrations and to have a value of 3/4 for vibronic (non-totally-symmetric) modes. The vibronic modes couple electron states because of distortions of the molecule away from its equilibrium symmetry so that higher lying states can participate and mix into a transition. Normally, the transition may be weak or formally forbidden, but this state mixing can give some intensity to the transition. Other methods can infer that there such vibrational modes exist. For example vibronic theory predicts a certain type of absorption line shape that has a "false origin". While this is an experimental handle, the false origin is not as easy to prove as the fact that a vibrational mode is nontotally symmetric because its depolarizatoin ratio has a value of 3/4. Of course, one must be careful to note that not all non-toally-symmmetric modes are vibronically active either. The complete description involves a combination of both absorption and Raman spectroscopy.
The non-resonant Raman paradox
We explain Raman scattering both in terms of the resonant and non-resonant picture. However, why should these two be different? Is it not possible to see non-resonant Raman as basically the same thing as resonant Raman, but with an excitation source tuned far from resonance. In fact, when Albrecht first derived a theory of non-resonant Raman he introduced the idea that non-resonant Raman scattering could only be allowed for non-totally symmetric modes. Perhaps this is a bit of a simplification, but it there is clearly a paradox in the simplest sum-over-states picture of Raman spectroscopy. We show that paradox in the following video. The resolution of the matter can be found in the idea that non-resonant Raman is really just resonant Raman, in which the excitation laser is tuned far from the maximum.
The Raman time correlator
The sum-over-states picture for Raman spectroscopy is very cumbersome. Typical resonant Raman proceses with a number of storngly coupled Franck-Condon active modes have ~1014 possible intermediate state pathways back to the ground state. An alternative computational method is provided by the time-correlator. The time-correlator approach has the advantage that any number of modes can be treated simultaneously. Moreover, the calculation can be carried out at any temperature. This is of great interest in studies of resonance Raman spectroscopy for low frquency mode.
