# What is the Franck-Condon principle?

1. Feb 9, 2013

### physics love

hi guys

I want to know what is the Franck-Condon principle?.... please in details

thanks for all

2. Feb 10, 2013

### DrDu

3. Feb 11, 2013

### schafspelz

From the BO approximation, we have the product of the electronic $\varphi_i(r,R)$ and the nuclear $\eta_i(R)$ wavefunction. For the transition, we use Fermi's Golden rule, where the Dipole-Operator $\mu$ "initiates" the transition. So we end up in
$r_{i\rightarrow j}=\left\langle \eta_i(R) \varphi_i(r,R) |\mu| \varphi_j(r,R) \eta_j(R) \right\rangle$.
Here we have an inner integral over the electron coordinates $r$ and an outer integral over the nuclei coordinates $R$. It is important to note here that the inner integral $\left\langle \varphi_i(r,R) |\mu| \varphi_j(r,R) \right\rangle$ is a function of $R$. The approximation is now that this inner integral is taken out of the outer interal, even though the former one is dependent of $R$ - which is the integration variable of the outer integral. Now the above equation looks like this:
$r_{i\rightarrow j}= \left\langle\varphi_i(r,R) |\mu| \varphi_j(r,R) \right\rangle \cdot\left\langle \eta_i(R)|\eta_j(R) \right\rangle$
So actually the electronic integral is handled independently of the nuclei integral. The former one is a usual transition (with an operator for the transition according to Fermi's Golden rule), while the latter one is only an overlap of wavefunctions any more!

4. Feb 11, 2013

### DrDu

Schafspelz, the approximation you made can be justified further by using diabatic electronic states $\eta_j$ instead of the adiabatic electronic wavefunctions. The diabatic states depend only very little on R.
A second step in the Franck-Condon approximation is to replace the dipole integral by a semiclassical expression so that only the neighbourhoods of the turning points of the nuclear motion contribute to the integrand.