The Franck-Condon Principle

Figure: Franck-Condon-Principle. Left: Classical picture, right: quantum-mechanical picture. From Prof. Ed Castner's lecture http://rutchem.rutgers.edu/http://rutchem.rutgers.edu/.

Here, a good description is in Atkins/Friedman ch. 11.4.

For simplicity, we leave out the rotations here and just discuss electronic and vibrational transitions. In a classical picture (with respect to the large mass nuclear motion), one considers the two potential curves $U_\alpha(r)$ and $U_{\alpha'}(r)$ and argues that the electronic transition occurs so fast that the nuclear system has no time to react: before and after the transition, the nuclear coordinate $X$ is the same. This, however, means that the distance $\vert x'\vert\equiv\vert X-r_{\alpha'}\vert$ from the equilibrium position $r_{\alpha'}$ after the transition and the distance $\vert x\vert\equiv \vert X-r_{\alpha}\vert$ from the equilibrium position $r_{\alpha}$ before the transition are not the same: when the nuclei are in equilibrium before the transition ( $X=r_\alpha, x=0$), their new coordinate $x'$ relative to the new equilibrium $r_{\alpha'}$ is $x'\equiv X-r_{\alpha'}=r_\alpha- r_{\alpha'}\ne 0$ after the transition.

The total dipole moment operator is a sum of electronic and nuclear dipole moment,

$${\bf d} = -e \sum_i {\bf q}_i + e \sum_s Z_s {\bf X}_s = {\bf d}_e + {\bf d}_{\rm n}.$$ (3.2)

The transition matrix element in Born-Oppenheimer approximation is ( $\alpha\ne \alpha'$)

$$\langle \alpha' v' \vert {\bf d}_e + {\bf d}_{\rm n} \vert \alpha v\rangle = \int dq dX \psi^*_{\alpha'}(qX) \phi^*_{\alpha',v'}(X) ( {\bf d}_e + {\bf d}_{\rm n}) \phi_{\alpha,v}(X) \psi_{\alpha}(qX)$$

$$= \int dX \phi^*_{\alpha',v'}(X) \left[ \int dq \psi^*_{\alpha'}(qX){\bf d}_e \psi_{\alpha}(qX)\right] \phi_{\alpha,v}(X)$$

$$+ \int dX \phi^*_{\alpha',v'}(X)\phi_{\alpha,v}(X) {\bf d}_{\rm n} \int dq \psi^*_{\alpha'}(qX) \psi_{\alpha}(qX)$$

$$= \int dX \phi^*_{\alpha',v'}(X) \left[ \int dq \psi^*_{\alpha'}(qX){\bf d}_e \psi_{\alpha}(qX)\right] \phi_{\alpha,v}(X)+0$$

$$\approx \langle \alpha'\vert {\bf d}_e\vert \alpha\rangle S(v,v'),\quad S(v,v')\equiv \langle v' \vert v \rangle.$$ (3.3)

Here it was assumed that the integral

$$\int dq \psi^*_{\alpha'}(qX){\bf d}_e \psi_{\alpha}(qX) \approx \langle \alpha'\vert {\bf d}_e\vert \alpha\rangle$$ (3.4)

does not depend on the nuclear coordinates $X$.

The transition between two electronic levels $\alpha$ and $\alpha'$ is therefore determined by the dipole matrix element $\langle \alpha'\vert {\bf d}_e\vert \alpha\rangle$ and the Franck-Condon factors $S(v,v')$, which are the overlap integrals of the corresponding vibronic states. As these states belong to different electronic states $\alpha$ and $\alpha'$, the overlaps are not zero, and there is also no selection rule for $\Delta v$.