next up previous contents index
Next: Interaction between Molecules Up: Electronic Transitions Previous: Electronic Transitions   Contents   Index

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/.
Image /.automount/brandes/home/brandes/brandes/tex/qm2005//FC1.gif Image /.automount/brandes/home/brandes/brandes/tex/qm2005//FC2.gif

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 \bgroup\color{col1}$ U_\alpha(r)$\egroup and \bgroup\color{col1}$ U_{\alpha'}(r)$\egroup and argues that the electronic transition occurs so fast that the nuclear system has no time to react: before and after the transition, the nuclear coordinate \bgroup\color{col1}$ X$\egroup is the same. This, however, means that the distance \bgroup\color{col1}$ \vert x'\vert\equiv\vert X-r_{\alpha'}\vert$\egroup from the equilibrium position \bgroup\color{col1}$ r_{\alpha'}$\egroup after the transition and the distance \bgroup\color{col1}$ \vert x\vert\equiv \vert X-r_{\alpha}\vert$\egroup from the equilibrium position \bgroup\color{col1}$ r_{\alpha}$\egroup before the transition are not the same: when the nuclei are in equilibrium before the transition ( \bgroup\color{col1}$ X=r_\alpha, x=0$\egroup), their new coordinate \bgroup\color{col1}$ x'$\egroup relative to the new equilibrium \bgroup\color{col1}$ r_{\alpha'}$\egroup is \bgroup\color{col1}$ x'\equiv X-r_{\alpha'}=r_\alpha- r_{\alpha'}\ne 0$\egroup after the transition.

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

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

The transition matrix element in Born-Oppenheimer approximation is ( \bgroup\color{col1}$ \alpha\ne \alpha'$\egroup)
$\displaystyle \langle \alpha' v' \vert {\bf d}_e + {\bf d}_{\rm n} \vert \alpha v\rangle$ $\displaystyle =$ $\displaystyle \int dq dX \psi^*_{\alpha'}(qX) \phi^*_{\alpha',v'}(X) ( {\bf d}_e + {\bf d}_{\rm n})
\phi_{\alpha,v}(X) \psi_{\alpha}(qX)$  
  $\displaystyle =$ $\displaystyle \int dX \phi^*_{\alpha',v'}(X) \left[ \int dq \psi^*_{\alpha'}(qX){\bf d}_e \psi_{\alpha}(qX)\right]
\phi_{\alpha,v}(X)$  
  $\displaystyle +$ $\displaystyle \int dX \phi^*_{\alpha',v'}(X)\phi_{\alpha,v}(X)
{\bf d}_{\rm n} \int dq \psi^*_{\alpha'}(qX) \psi_{\alpha}(qX)$  
  $\displaystyle =$ $\displaystyle \int dX \phi^*_{\alpha',v'}(X) \left[ \int dq \psi^*_{\alpha'}(qX){\bf d}_e \psi_{\alpha}(qX)\right]
\phi_{\alpha,v}(X)+0$  
  $\displaystyle \approx$ $\displaystyle \langle \alpha'\vert {\bf d}_e\vert \alpha\rangle S(v,v'),\quad S(v,v')\equiv \langle v' \vert v \rangle.$ (3.3)

Here it was assumed that the integral
$\displaystyle \int dq \psi^*_{\alpha'}(qX){\bf d}_e \psi_{\alpha}(qX) \approx \langle \alpha'\vert {\bf d}_e\vert \alpha\rangle$     (3.4)

does not depend on the nuclear coordinates \bgroup\color{col1}$ X$\egroup.

The transition between two electronic levels \bgroup\color{col1}$ \alpha$\egroup and \bgroup\color{col1}$ \alpha'$\egroup is therefore determined by the dipole matrix element \bgroup\color{col1}$ \langle \alpha'\vert {\bf d}_e\vert \alpha\rangle$\egroup and the Franck-Condon factors \bgroup\color{col1}$ S(v,v')$\egroup, which are the overlap integrals of the corresponding vibronic states. As these states belong to different electronic states \bgroup\color{col1}$ \alpha$\egroup and \bgroup\color{col1}$ \alpha'$\egroup, the overlaps are not zero, and there is also no selection rule for \bgroup\color{col1}$ \Delta v$\egroup.


next up previous contents index
Next: Interaction between Molecules Up: Electronic Transitions Previous: Electronic Transitions   Contents   Index
Tobias Brandes 2005-04-26