# Do Franck-Condon oscillations have natural lineshapes?

I recently found a paper (for the curious, this one) that talks about observing the motion of a nuclear wavepacket in H2O, as initiated by tunnel ionization. This wavepacket should be thought of as a superposition of the different vibronic states of the molecule after ionization, and is therefore described by a Franck-Condon oscillation: the oscillation of the overlap between the ground-state nuclear wavefunction and the different eigenstates of the nuclear degrees of freedom.

That paper reproduces a figure from a 1975 paper (L. Karlsson et al. Isotopic and vibronic coupling effects in the valence electron spectra of H216O, H218O, and D216O. J. Chem. Phys. 62 no. 12 (1975), p. 4745) that experimentally observed the Franck-Condon oscillations in energy-resolved photoelectron spectra:

The measurement clearly resolves the different peaks in the spectrum, their roughly equal spacing, and even a slight displacement for the different isotopic combinations. However, it is clear to me that the shape of the lines, particularly in such an old paper, must come from inhomogeneous broadening mechanisms. (I would by default blame the thermal velocity spread of the molecules prior to ionization, but I can't tell for sure.) I would therefore like to pose the question:

• if one were to remove all sources of inhomogeneous broadening, can one observe natural lineshapes for Franck-Condon oscillations such as these?

• If so, what are the physical mechanisms behind them and what theoretical tools and models exist to study them?

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Regarding your 2 questions I would give the following answers:

1. The thermal Doppler broadening is evaluated to be $25$ meV at $T=293~K$, this broadening will reflect itself by a spectral band being split into two bands, one corresponding to an electron being ejected parallely to the velocity vector of the molecule and the other one being ejected antiparallely. This means that you need at least a resolution of $12$ meV to be able to observe this broadening. In the case where this broadening is removed you are able to observe the oscillations in the Franck-Condon factors in case the total instrumental broadening is smaller than the separation between two consecutive vibrational energy levels of the excited state being probed by ionization. Also I would also add that the lifetime broadening should be small to observe such oscillations, a case almost always fulfilled in case of valence ionization, on the other hand core-ionization induces a very large lifetime broadening which could forbid one from observing such oscillations.
2. The oscillations in the Franck-Condon factors are due to the overlap between the vibrational wavefunction of level $v=0$ of the Ground State and the different vibrational levels of the inonized state. I would recommend that you see the theoretical description of the Franck-Condon factors in the book by Atkins entitled Molecular Quantum Mechanics.

Best regards

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I'm sorry, but I don't see how this is useful. The first question is how can one remove the inhomogeneous broadening, not what the obvious sources of that are. The fact that a sharp electronic linewidth is required to observe the Franck-Condon oscillations is obvious and has little to do with the lineshapes of the latter. –  Emilio Pisanty Sep 9 '14 at 11:50
On the second question, I am perfectly aware of the origin of the oscillations, but those overlaps predict delta-function spectra which are never observed in the real world where experiments take finite time; the question is what the shape is once one removes the inhomogeneous broadening and what physics governs that. I will look at Atkins, though. –  Emilio Pisanty Sep 9 '14 at 11:51
Suppose you remove the in-homogeneous broadening, you are still left with 3 broadening factors: the incoming photon's broadening, the excited state's lifetime broadening and the broadening of your electronic spectrometer. These 3 factors prevent you from observing discrete line shapes in the spectra as theory predicts. All the above-mentioned broadening factors are homogeneous since the lifetime broadening is associated to a Lorentzian function and the other two are associated to a gaussian function. –  Elie Kawerk Sep 9 '14 at 20:55
For the physics governing the lifetime broadening i suggest that you look at Sakurai's book 'Advanced Quantum Mechanics', chapter 2 where he treats the so called lifetime broadening as being the depletion of an excited state due to the interaction of the EM field with the electron. For the instrumental broadening the gaussian distribution aims to mimick experimental curves and usually it does this well. –  Elie Kawerk Sep 9 '14 at 20:58
For your question about how to remove the inhomogeneous linewidth i would propose by cooling the studied systems to very low temperature, where translational and rotational motion does not play any considerable role in absorption through the Doppler effect. This way you 're left with the instrumental and lifetime homogeneous broadening factors. –  Elie Kawerk Sep 9 '14 at 21:01