Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 11: Problem 7

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.

Short Answer

Expert verified
For the \(^{1}\mathrm{A}_{\mathrm{2}} \leftarrow ^{1}\mathrm{A}_{\mathrm{1}}\) transition in formaldehyde, the polarization is in the \(y\) direction, while for the \(^{1}\mathrm{B}_{\mathrm{u}} \leftarrow ^{1}\mathrm{A}_{\mathrm{g}}\) transition in ethene, the polarization can occur in the \(x\) or \(y\) direction, indicating a doubly degenerate transition.

Step by step solution

01

Identifying Molecular Symmetry

First, identify the point groups of the molecules. Formaldehyde belongs to the \(C_{2v}\) point group and ethene belongs to the \(D_{2h}\) point group.
02

Listing Symmetry of Vibrational Modes

List all of the vibrational modes for each molecule and identify their symmetries. In \(C_{2v}\), \(^{1}\mathrm{A}_{1}\) and \(^{1}\mathrm{A}_{2}\) represent vibrational modes. In \(D_{2h}\), \(^{1}\mathrm{A}_{\mathrm{g}}\) and \(^{1}\mathrm{B}_{\mathrm{u}}\) represent vibrational modes.
03

Identifying Allowed Transitions

In order for a transition to be allowed, the product of the symmetries of the initial and final states and the transition dipole moment must contain the totally symmetric representation. The transition dipole moment has \(x\), \(y\), and \(z\) components. For \(C_{2v}\), the symmetries of the \(x\), \(y\), and \(z\) components are \(^{1}\mathrm{B}_{\mathrm{1}}\), \(^{1}\mathrm{B}_{\mathrm{2}}\), and \(^{1}\mathrm{A}_{\mathrm{1}}\) respectively.
04

Determining the Polarization

Calculate the direct product of the symmetries of the initial and final states for each transition. For formaldehyde, for the transition \(^{1} \mathrm{A}_{\mathrm{1}}\) to \(^{1}\mathrm{A}_{\mathrm{2}}\), the polarization can occur in the \(y\) direction. For ethene, for the transition \(^{1}\mathrm{A}_{\mathrm{g}}\) to \(^{1}\mathrm{B}_{\mathrm{u}}\), the polarization can occur in the \(x\) or \(y\) direction, indicating a doubly degenerate transition.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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?

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.

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.

The Bixon-Jortner approach to radiationless transitions was sketched in a very simplified form in Example \(11.4 .\) The following is a slightly more elaborate version. Let \(\psi,\) an eigenstate of the system hamiltonian \(H^{(\mathrm{sys})}\) with eigenvalue \(E,\) be the state populated initially, and let \(\varphi_{n}\) an eigenstate of the bath hamiltonian \(H^{( \text {bath) } }\) with eigenvalue \(E_{n},\) be a state of the bath. Let \(\Psi=a \psi+\Sigma_{n} b_{n} \varphi_{n}\) be an eigenstate of the true hamiltonian \(H\) with energy \(\mathcal{E}\). Let \(\left\langle\psi | \varphi_{n}\right\rangle=0\) and \(H^{\prime}=H-H^{(\mathrm{sys})}-H^{(\text {bath })}\) have constant matrix elements \(\left\langle\varphi_{n}\left|H^{\prime}\right| \psi\right\rangle=V\) for all \(n .\) Show that \(H \Psi=\mathcal{E} \Psi\) leads to \(V a+\left(E_{n}-\mathcal{E}\right) b_{n}=0\) and \((E-\mathcal{E}) a+V \Sigma_{n} b_{n}=0 .\) Hence find an expression relating \(a\) and \(b_{n^{*}}\) Letting \(\mathcal{E}-E_{n}=(\gamma-n) \varepsilon\) and using \(\Sigma_{n=-\infty}^{\infty} 1 /(\gamma-n)=-\pi \cot \pi \gamma\) and \(\rho=1 / \varepsilon,\) show that \(E-\mathcal{E}-\pi \rho V^{2} \cot \pi \gamma=0,\) an equation for \(\mathcal{E} .\) Go on to show on the basis that \(a^{2}+\Sigma_{n} b_{n}^{2}=1,\) that \(a^{2}=V^{2} /\left\\{(E-\mathcal{E})^{2}+V^{2}\right.\) \(\left.+\left(\pi V^{2} \rho\right)^{2}\right\\}\)
See all solutions

Recommended explanations on Chemistry Textbooks

Chemical Analysis

Read Explanation

Making Measurements

Read Explanation

Physical Chemistry

Read Explanation

Chemical Reactions

Read Explanation

Organic Chemistry

Read Explanation

Kinetics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free