Homework 12

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Q1

Franck-Condon factors are usually difficult to calculate, but often are approximated by simple harmonic oscillator models. Imagine a diatomic with two electronic states that are identically harmonic except for the location of their minima. Imagine one has a minimum displacement at a distance \(\delta\) from each other. We will consider only the \(v=0\) levels of each state so that vibrational wavefunction are just the zero-point harmonic oscillator functions:

\[ | v =0 \rangle^{g} = \dfrac{1}{\pi^{1/4}} e ^{-q_1^2/2} = \dfrac{1}{\pi^{1/4}} \exp\left[-\dfrac{k}{2\hbar \omega} (R-R_{eq})^2\right]\]

and

\[ | v =0 \rangle^{e} = \dfrac{1}{\pi^{1/4}} e ^{-q_2^2/2} = \dfrac{1}{\pi^{1/4}} \exp\left[-\dfrac{k}{2\hbar \omega} (R-R_{eq}-\delta)^2\right]\]

Where \(R_{eq}\) is the bond length of ground electronic state and \(R_{eq} + δ\) is the bond length of the excited electronic state.

1. Show that the integral for the Franck-Condon overlap \[S_{00}= \sqrt{\dfrac{\alpha}{\pi}} e^{-\alpha(R_{eq} - R_{eq}-\delta )^2/4} \int_{-\infty}^{\infty} e ^{-\alpha\{R - 1/2(R_e + R_{eq} -\delta )\}^2} dR\] where \[\alpha = \dfrac{\sqrt{mk}}{\hbar}\]
2. This is a Gaussian Integral and is known in standard integral tables. Solve for the analytical value of \(S_{00}\) for the 0-0 vibrational transition.
3. What value of \(\delta\) gives \(S=1\) for the 0-0 transition?
4. How does \(S\) depends on the absolute positions (i.e., electronic energies\) of the two electronic states?

Q2

Calculate the Frank-Condon factors for the 0-0 transition using the model in Q1 with the following displacements:

• \(δ=0.0 \; Å\)
• \(δ=0.05 \; Å\)
• \(δ=0.1 \; Å\)
• \(δ=0.5\; Å\)

for the vibration \(\dfrac{k}{\hbar \omega} =200 \; Å^{-2}\). These numbers should convince you of the importance of the Franck-Condon factor in controlling one’s access to vibration levels of excited electronic states.