# Homework 12 (Due 7/19/16 @ 10:00 a.m.)

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Section: _____________________________

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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.