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Chapter 11: Problem 6

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.

Short Answer

Expert verified
The \(\pi^{*} \leftarrow \pi\) transition in the carbonyl group is allowed because it satisfies the selection rule and its transition dipole moment lies along the \(C=O\) bond, which is along the \(z\)-axis.

Step by step solution

01

Recognize the Perturbation

Determine that the perturbation (light) that induces the transition has \(x\), \(y\), and \(z\) components, meaning that it can interact with the molecular orbitals in those directions.
02

Identify the orbitals

Identify the orbitals involved in the transition. The carbonyl group has a \(\pi\) bonding orbital formed out of the \(p_z\) atomic orbitals and a \(\pi*\) antibonding orbital, also formed from \(p_z\) orbitals.
03

Consider the Symmetry

Identify the symmetry of the orbitals. The \(\pi\) and \(\pi*\) orbitals, formed from \(p_z\) orbitals, have \(A_1\) and \(B_1\) symmetries, respectively, under the \(C_{2 v}\) point group.
04

Apply the Selection Rule

Apply the selection rule for electronic transitions. A transition is allowed if the direct product of the symmetries of the initial state, final state, and the dipole operator is the totally symmetric representation. We can see that \(A_1 \times B_1 \times B_1 = A_1\), the totally symmetric representation of the \(C_{2 v}\), is satisfied.
05

Determine the transition dipole moment direction

Determine that the transition dipole moment lies along the \(C=O\) bond, which is along the \(z\)-axis in our alignment.

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Most popular questions from this chapter

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?

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.

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.

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?

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.
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