Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Problem 13

The symmetry of the ground electronic state of the water molecule is \(\mathrm{A}_{1}\). (a) An electric field, (b) a magnetic field is applied perpendicular to the molecular plane. What symmetry species of excited states may be mixed into the ground state by the perturbations? Hint. The electric interaction has the form \(H^{(1)}=a x ;\) the magnetic interaction has the form \(H^{(1)}=b l_{x^{*}}\)

Short Answer

Expert verified
The excited states that may be mixed into the ground state by the electric and magnetic field perturbations are \(B_{1}\) and \(B_{2}\), respectively.

Step by step solution

01

Identify the Symmetry of the Perturbing Operators

The ground state of the water molecule is \(A_{1}\). For the electric field, \(H^{(1)}=a x\) is the perturbing operator. The symmetry of \(x\) is \(B_{1}\) for the \(C_{2v}\) point group. For the magnetic field, \(H^{(1)}=b l_{x^{*}}\) is the perturbing operator. The \(l_{x^{*}}\) operator has symmetry \(B_{2}\) for the \(C_{2v}\) point group.
02

Determine Possible Symmetries of Excited States

An interaction can only mix states of the same symmetry. The excited states that can be mixed into the ground state by the perturbations must be product with \(A_{1}\) (our ground state) to give the symmetry of the perturbing operator.
03

Electric Field Case

The product of the symmetries of states that can be mixed by the electric field must result in \(B_1\). From group theory, we know that \(A_{1} \times B_{1} = B_{1}\), which means the symmetry species of the excited state may be \(B_{1}\).
04

Magnetic Field Case

For the magnetic field, the product must result in \(B_2\). So, \(A_{1}\times B_{2} = B_{2}\). Therefore, the symmetry species of the excited state may be \(B_{2}\).

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

Repeat the last problem but set \(H_{\mathrm{s}, \mathrm{s}}=\gamma\) and \(S_{\mathrm{s}^{\prime}} \neq 0\) Evaluate the overlap integrals between 1 s-orbitals on centres separated by \(R ;\) use $$S=\left\\{1+\frac{R}{a_{0}}+\frac{1}{3}\left(\frac{R}{a_{0}}\right)^{2}\right\\} \mathrm{e}^{-R / a_{0}}$$ Suppose that \(\beta / \gamma=S_{\mathrm{s}, \mathrm{s}_{2}} / S_{\mathrm{s}, \mathrm{s}} .\) For a numerical result, take \(R=80 \mathrm{pm}, a_{0}=53 \mathrm{pm}\)

Calculate the second-order energy correction to the ground state of a particle in a one-dimensional box for a perturbation of the form \(H^{(1)}=-\varepsilon \sin (\pi x / L)\) by using the closure approximation. Infer a value of \(\Delta E\) by comparison with the numerical calculation in Example \(6.4 .\) These two problems \((6.14 \text { and } 6.15)\) show that the parameter \(\Delta E\) depends on the perturbation and is not simply a characteristic of the system itself.

A hydrogen atom in a \(2 \mathrm{s}^{1}\) configuration passes into a region where it experiences an electric field in the \(z\) -direction for a time \(\tau .\) What is its electric dipole moment during its exposure and after it emerges? Hint. Use eqn 6.62 with \(\omega_{21}=0 ;\) the dipole moment is the expectation value of \(-e z\) \\[\text { use } \int \psi_{2 \mathrm{s}} z \psi_{2 \mathrm{p}} \mathrm{d} \tau=3 a_{0}\\]

Show group-theoretically that when a perturbation of the form \(H^{(1)}=a z\) is applied to a hydrogen atom, the 1 s-orbital is contaminated by the admixture of \(n \mathrm{p}_{z^{-}}\) orbitals. Deduce which orbitals mix into (a) \(2 p_{x}\) -orbitals, (b) \(2 \mathrm{p}_{z}\) -orbitals (c) \(3 d_{x y}\) -orbitals.

Calculate the first-order correction to the energy of a ground-state harmonic oscillator subject to an anharmonic potential of the form \(a x^{3}+b x^{4}\) where \(a\) and \(b\) are small (anharmonicity) constants. Consider the three cases in which the anharmonic perturbation is present (a) during bond expansion \((x \geq 0)\) and compression \((x \leq 0)\) (b) during expansion only, (c) during compression only.
See all solutions

Recommended explanations on Chemistry Textbooks

Inorganic Chemistry

Read Explanation

Physical Chemistry

Read Explanation

Nuclear Chemistry

Read Explanation

The Earths Atmosphere

Read Explanation

Kinetics

Read Explanation

Chemical Analysis

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