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Chapter 11: Problem 3

The Franck-Condon principle and the BornOppenheimer approximation have an important qualitative feature in common. What feature do they share that to a large extent justifies their usefulness?

Short Answer

Expert verified
Both the Franck-Condon Principle and the Born-Oppenheimer approximation share the qualitative feature that they separate electronic and nuclear motions. In other words, they both assume that electronic transitions occur while the nuclei are essentially stationary.

Step by step solution

01

Understand the Franck-Condon Principle

This principle states that an electronic transition is most likely to occur without any change in the position of the nuclei within the molecule. This is because the time scale of nuclear motion (10^-12 to 10^-9 seconds) is much slower than the typical timescale of an electronic transition (10^-15 seconds), making the nuclei appear stationary during that transition.
02

Understand the Born-Oppenheimer Approximation

The approximation named after Max Born and J. Robert Oppenheimer is also based on the fact that nuclear motion is much slower than electronic motion. It allows us to separate the wavefunction of a molecule into its electronic and nuclear components, where the electronic motion is considered in the nuclear clamped locations. This means the electrons move in the field of fixed nuclei.
03

Identify the Common Feature

Both the Franck-Condon Principle and the Born-Oppenheimer approximation reduce the complexity of multi-electronic systems by considering that the motion of the nuclei is much slower than that of the electrons. Thus, they both assume that during an electronic transition, the nuclear configuration remains unchanged, allowing separation of nuclear and electronic motions.

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Most popular questions from this chapter

Consider a case in which two electronic states have the same force constant but in which the equilibrium bond lengths differ by \(\Delta R\). Find an expression for the relative intensity of the \(0-1\) transition \((v=1\) is the upper vibrational state \()\) as a function of \(\Delta R .\) Hint: Follow Example \(11.1 ;\) use mathematical software to evaluate the integral numerically.

In an aromatic molecule of \(D_{2 \mathrm{h}}\) symmetry the lowest triplet term was identified as \(^{3} \mathrm{B}_{1 \mathrm{u}} .\) What is the polarization of its phosphorescence? Hint. Decide which singlet terms can mix with \(^{3} \mathrm{B}_{1 \mathrm{u}}\) and assess the polarization of the light involved in the return of that state to the \(^{1} \mathrm{A}_{\mathrm{g}}\) ground state.

Assess the polarization of the \(^{1} \mathrm{A}_{2} \leftarrow^{1} \mathrm{A}_{1}\) transition in \(\mathrm{H}_{2} \mathrm{CO}\) and of the \(^{1} \mathrm{B}_{2 \mathrm{u}} \leftarrow^{1} \mathrm{A}_{\mathrm{g}}\) transition in \(\mathrm{CH}_{2}=\mathrm{CH}_{2} .\) Hint. Use \(C_{2 v}\) and \(D_{2 \mathrm{h}}\) respectively; consider the role of vibrational coupling.

Show that in the carbonyl group the \(\pi^{*} \leftarrow \pi\) transition is allowed, its transition dipole moment lying along the bond. Hint. Consider the carbonyl group to be of \(C_{2 v}\) symmetry with the \(C=O\) bond along the \(z\) -axis.

In a diamagnetic octahedral complex of \(\mathrm{Co}^{3+}\), two transitions can be assigned to \(^{1} \mathrm{T}_{1 \mathrm{g}} \leftarrow^{1} \mathrm{A}_{1 \mathrm{g}}\) and \(^{1} \mathrm{T}_{2 \mathrm{g}} \leftarrow^{1} \mathrm{A}_{1 \mathrm{g}}\) Are these transitions forbidden? If they are forbidden, what symmetries of vibrations would provide intensity? Can the intensities be ascribed to the admixture of configurations involving p-orbitals?
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