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For molecular complexes with a moderate amount of symmetry there are usually vibrational modes that are neither infrared nor Raman active. Sometimes these modes transform as some cubic polynomial in x,y,z. For example the $B_{1u}$ irrep in the $D_{4h}$ group which transforms as $xyz$.

My question is: it possible to measure these kinds of vibrations using some sort of non-linear optics? For example, measuring the absorption of three photons instead of just one like in infrared spectroscopy/reflectivity. In the case of $B_{1u}$ above, this would translate to 3 photons polarized with x,y,z respectively and whose total wavenumber matches a $B_{1u}$ vibration. Or perhaps a more complicated multi-photon version of Raman spectroscopy could do this?

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    $\begingroup$ I think you are at the core of some of the nlo phenomena, perhaps THG or stimulated Raman. Can be suitable for Physics SE, too. $\endgroup$ – Alchimista Oct 8 '19 at 10:00
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The product $xyz$ is an odd function and as such should produce the same selection rules as 1 photon (electric dipole) transitions. Thus CO has a 1 photon transition at 146.7 nm in the vacuum uv that can be more easily reached with a 3-photon transition with a pulsed laser at 440 nm. ( Three photon transitions are also used in bio-imaging where a ir photon is used so as not the excite the sample. Also, as the transition is multiphoton the focus area is smaller than a 1 photon transition and this improves spatial resolution).

As the product $xyz$ can be thought of as $xy$ multiplied by $yz$ then this is like a double Raman transition. Raman involves a change in polarisability. Polarisability is proportional to a volume but the projection for the radiation is 2D hence $xy$ type products in the point group tables. A Raman type transition may occur between 4 levels. Levels 1 and 4 are 'real' and 2 and 3 'virtual'. The steps are 1 up to 3 , 3 down to 2, and 2 up to 4. This has been demonstrated in doppler free geometry between Na $3^2S_{1/2}\to 3^2P_{1/2}$.

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  • $\begingroup$ Ah oops, I guess my question is basically moot since all cubic invariants are odd under parity. I assumed they would be too weak to see because of the higher order, but I was wrong. Is there a similar scheme for higher order multipole vibrations that aren't odd under some parity? $\endgroup$ – user157879 Oct 8 '19 at 13:48
  • $\begingroup$ Say, for example, quadrupoles? $\endgroup$ – user157879 Oct 8 '19 at 13:52

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