quantum mechanics - Franck Condon Principle and Born Oppenheimer approximation

My question here is purely fundamental. I am confused with the concept in Franck Condon (FC) principle and Born Oppenheimer (BO) approximation. The FC principle is in accordance with the BO approximation or not? In FC there is a correlation between electronic states and the nuclear motion. So, BO approximation is broken. So, can we say FC is an example of breaking of BO approximation?

Secondly, in the case of mega electron volt ions, is BO approximation valid? In this case, the velocity of electrons are comparable to the velocity of the nucleus!

Answer

You are confused by a slightly misleading aspect of the usual presentation of the Franck-Condon principle.

The FCP does indeed rely on a separation of slow and fast timescales, but now the fast timescale is not that of the electronic motion but that of electronic transitions. The typical setting of single-photon transitions in a weak field is tricky to deal with in the time domain, but the take-home message from the first-order perturbation-theoretic analysis is that you can assume the transition to be instantaneous even if there is a (coherent) probability distribution for when that instant occurs.

Suppose, then, that you know that a transition has occurred. (You can do this by post-selecting the excited molecules, for example.) In that moment there is no correlation between the electronic and nuclear coordinates: wherever the nuclei were, they remain, and the electrons are upgraded to the BO excited state corresponding to those nuclear coordinates.

Right after the transition, then, the electronic potential energy surface changes to that of the excited state. The important thing, though, is that the nuclear wavepacket remains unchanged. It must, because the transition was instantaneous! What does happen, however, is that this wavepacket is no longer an eigenstate of the nuclear hamiltonian, and therefore it has to move. The nuclear wavepacket then begins to slosh around the excited-state potential well until otherwise disturbed.

(If the displacement of the minima is small, then the motion is harmonic and nothing very interesting happens. If the displacement is enough to let the wavepacket "see" the anharmonic edges of the well, on the other hand, then all sorts of interesting TDSE dynamics might happen, like spreading and re-interference.)

So what is all the hullabaloo about Franck-Condon factors/oscillations/so on? As in all TDSE evolutions, one can choose to decompose the initial wavepacket into a superposition of the eigenstates of the (new) potential well. The coefficients will probably oscillate with eigenstate number, but so far these oscillations are purely a mathematical artefact of how we're describing the evolution, and they are not physically measurable.

How then, do we measure the coefficients? Well, that task is really measuring the nuclear energy very precisely, i.e. to a precision greater than the spacing between the vibrational levels. Because of the Uncertainty Principle, this requires a measurement over a time that's longer than the period of the nuclear oscillations. (An example is electronic fluorescence, which happens on a long timescale.) This means you are making your system interact with some measuring device, such as the fluorescent EMR modes, over a long time, and the probability of interaction is a Fourier transform over all the system's degrees of freedom: in particular, the temporal motion of the nuclei gets Fourier transformed to the energy domain, and out you get (of course!) the FC factors.