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Using the words of the wikipedia article Raman Scattering:

The Raman effect corresponds, in perturbation theory, to the absorption and subsequent emission of a photon via an intermediate vibrational state, having a virtual energy level.

From this, we can derive the Kramers-Heisenberg Formula for the differential cross section, which takes account of this intermediate state, but, from what I understood, excludes the absortion of more than one photon.

Well, my question is: the others processes where more than a photon is absorbed, like Hyper Raman Scattering or second Hyper Raman Scattering, are described by the Kramers-Heisenberg Formula as well?

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Most likely yes, since K-H formula comes from causality principle (which is very general). Also, some papers seemed to use Kramers-Heseinberg when describing Hyper-raman scattering e.g. sciencedirect.com/science/article/pii/000926149385050X –  pcr Nov 13 '11 at 1:03

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