Franck-Condon Principle

Using the Franck-Condon principle, we are able to calculate the intensities of transitions between vibrational states of the electronic energy levels. Due to their low mass, electron movement is fast compare with the nuclear movement and we can consider these two movements separately. Particularly, the position of nucleus can be considered as fixed during the electron transition. This means that the electron transition can be drawn as a vertical line on the potential curve diagram. However, after the electron transition took place, the nuclear vibrations move the nucleus toward the position of a new equilibrium distance with respect to the new potential energy curve.

In terms of quantum mechanics, the probability of an electron transition is proportional to the squire of the overlap integral between the vibrational wavefunction $\Psi_{v''}(R)$ and $\Psi_{v'}(R$ before and after the electronic transition.

To treat the problem quantitatively, the complete molecular wavefunction of the initial and the final state are needed. Fortunately, it is often possible to present this wavefunction as a product of the electronic wave function $\Psi_{el}$ with depends on the coordinates of all electrons $\textbf{r}_i$ and the vibrational wavefunction $\Psi_{v}(R)$ which is a solution of the Schrödinger equation for nuclei and depend on the internuclear distance $R$. This approach is based on the Born-Oppenheimer approximation which argues with large difference in mass for nuclei and electrons. In the Born-Oppenheimer approximation the probability of a radiative transition is written as (see eq. (3)):

$$W_{k \leftarrow 0} \propto \left\vert\langle k \vert\,\mu_{Z}\vert\rangle \right\vert^2 E^2_z,$$ (70)

where $\langle k \vert\,\mu_{Z}\vert\rangle$ is the matrix element of the transition dipole moment:

$$\langle k \vert\mu_{Z}\vert\rangle = q_{v'' v'}\int {\Psi_{el_k}}^*({\bf r}_i) \mu_z \Psi_{el_0}({\bf r}_i) d{\bf r}_i$$ (71)

$$q_{v'' v'} = \int \Psi^*_{v''}(R)\Psi_{v'}(R) dR$$ (72)

The integral over the electron coordinates ${\bf r}_i$ does not dependent on the vibration of nuclei and it is identical for all pairs of v', v". The integral over $R$ represents overlap of the vibrational wave functions.

The quantities $\vert q_{v'' v'}\vert^2$ are called Franck-Condon factors. No selection rules exist for changes of the vibrational quantum number $v$. This is because the vibrational wave functions of the initial and final states are in general not orthogonal to each other being the subject of Scrödinger equation with two different potentials $V'(R)$ and $V''(R)$. Apart from that, the Franck-Condon-principle allows to calculate the probability of a transition from some vibrational level $v'$ of the initial state to another vibrational level $v''$ of the final state.

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