# 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!

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I would suggest branching off your second point into a question of its own. What exactly do you mean by MeV ions? If that is the energy of the center-of-mass motion, then the BOA stays unchanged as a simple change of frame will put the ion to rest, and unless it collides with something else there is nothing that can couple to the COM motion. If, on the other hand, you have > 1 MeV in the internal nuclear motion of your molecule, it wil very quickly tear itself apart. – Emilio Pisanty May 10 at 23:22

## 3 Answers

The Franck-Condon principle is a direct consequence of the Born-Oppenheimer approximation stating that since nuclei are so much slower then electrons, they cannot move during the electronic excitation. There is no violation because the is no energy exchange between the electron and the nuclei - they both get necessary energy from the photon.

I'm not sure what you mean by megavolt ions. If the ion is just moving very fast, but nuclei and electron cloud travel together, then I don't see why the BO approximation should be violated.

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If we assume direct dissociation of molecules (when molecules are irradiated with laser) and pre dissociation, then why the predissociation is said to be non adiabatic (means Born Oppenheimer approximation is broken)? In both direct and predissociation, electrons are excited and cause the molecules to brake (electro-vibronic coupling). – albedo Mar 11 at 11:55
The Franc-Condon factor influences the process of electronic excitation, which is very vast. Predissociation happens much later, when nuclei move around sufficiently. Hence, FC factor has no direct relation to predissociation. – gigacyan Mar 12 at 7:37

First, one can derive the Franck-Condon principle from purely quantum mechanical considerations. In that case, the separation of the electronic and vibrational motion of the nucleus comes from the Born-Oppenheimer approximation which allows to treat separately the electronic and nuclear degrees of freedom.

Second, in the Born-Oppenheimer approximation the first step is to neglect the kinetic energy of the nucleus. This is justified by the assumption that the nucleus is heavy and moves slowly while the electrons move much faster so that we can consider that the adiabatic hypothesis applies. This will make sense if one assumes that the momentum of the nucleus and the electrons are of the same order of magnitude $p_N \simeq p_e$ so that the kinetic energy $E_N \ll E_e$ since we assume $m_N \gg m_e$. From this it follows that if the velocity of the nucleus and the electron are similar, the Born-Oppenheimer approximation still holds.

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The electronic transitions are non adiabatic. Electronic transitions are accompanied by vibrational transitions and hence electronic transition is coupled with nuclear motion. So, Born-Oppenheimer approximation is can not be valid! – albedo Mar 11 at 5:53
but if velocities of the nucleus and the electron are similar, then kinetic energy of the nucleus is three orders of magnitude larger than that of the electron! – gigacyan Mar 11 at 7:35
I think, it is not the kinetic energy but the velocity which is important here. The BO approximation is valid if the electrons can adjust quickly with the nuclear motion. In the normal cases, the electrons adjust very quickly with the nucleus (adiabatic process). But, for eg., if an electromagnetic wave interact with an ion, the ion oscillate faster, then the motion is non adiabatic. – albedo Mar 11 at 7:57
What the Franck-Condon principle is telling you is that since the electronic transitions are virtually instantaneous (since we assume the Born-Oppenheimer approximation) compared to the time scale of nuclear motion, this sets constraints to the vibrational transitions during the electronic transition since the new vibrational state should be instantaneously compatible with the nuclear positions and momentum in which the molecule was before. Born-Oppenheimer approximation does not decouple electronic and vibrational motion and indeed the energy of the electrons depend on the nuclear position. – DaniH Mar 11 at 17:56
@DaniH: I think you didn't get my point! I was saying about the non adiabatic case. The Born Oppenheimer approximation neglects the non adiabatic effects. For example, when there is a crossing between two Born Oppenheimer surfaces or conical intersections. What you said is true in the case of vertical transitions. Am I right? – albedo Mar 12 at 9:46
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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.

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