# What is the Franck-Condon principle?

by physics love
Tags: franckcondon, principle
 P: 5 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!