Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 4: Problem 8

Determine the irreducible representation for the following: (a) \(\mathrm{C}_{2 v}\) point group (b) Set of \(p\) -orbitals

Short Answer

Expert verified
The irreducible representations for \(\mathrm{C}_{2 v}\) are \(\mathrm{A}_{1}\), \(\mathrm{A}_{2}\), \(\mathrm{B}_{1}\) and \(\mathrm{B}_{2}\). For the set of \(p\)-orbitals; the orbital shapes transform under the \(\mathrm{C}_{2 v}\) point group operations such that they will correspond to one (or more) of these representations. It will depend on the specific characteristics of the \(p\)-orbitals, and how they transform under the operations of the \(\mathrm{C}_{2 v}\) group.

Step by step solution

01

Determine Irreducible representation for \(\mathrm{C}_{2 v}\) group

Irreducible representation for \(\mathrm{C}_{2 v}\) point group can be determined from its character table. For this purpose, we will directly refer to the character table for \(\mathrm{C}_{2 v}\) point group. According to character table, irreducible representations are \(\mathrm{A}_{1}\), \(\mathrm{A}_{2}\), \(\mathrm{B}_{1}\) and \(\mathrm{B}_{2}\).
02

Identify the set of \(p\) -orbitals

The \(p\) -orbitals are three dimensional, one each along the x, y, and z axis, namely the \(p_x\), \(p_y\), and \(p_z\). These three orbitals are treated as one entity under the transformation, and we characterize these representations using the same \(\mathrm{C}_{2 v}\) character table.
03

Determine the characters for the operations on \(p\) orbitals

The transformation of \(p_x\), \(p_y\), and \(p_z\) under the symmetry operations for \(\mathrm{C}_{2 v}\) will be used to determine the characters. The \(p_x\) orbital is not changed by \(E\) or \(\sigma_{x z}\), but changes sign with \(C_2\) and \(\sigma_{y z}\), character can be represented as \(\chi\)=3. Same way characters for \(p_y\) and \(p_z\) orbitals can be determined.
04

Reducible Representation for the set of \(p\) orbitals

Next, combine the characters for the \(p_x\), \(p_y\), and \(p_z\) orbitals under each operation into a reducible representation. This forms a reducible representation for the set of \(p\) -orbitals in the \(\mathrm{C}_{2 v}\) point group. Utilizing characters from step 3, reducible representation \(\Gamma_{red}\) can be constructed.
05

Determine Irreducible representations for p-orbitals

Use projection operator method to reduce the reducible representation of p-orbitals to their irreducible constituents. Each constituent will correspond to either to a \(p_x\), \(p_y\), or \(p_z\) orbital. Compare results with irreducible representations from step 1.

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

Write the multiplication table for \(\mathrm{C}_{2}\), point group.

Discuss the methods for designation of planes of a crystal.

Assign the symmetry point group to the following molecules: (a) Benzene (b) \(\mathrm{H}_{2} \mathrm{O}\) (c) \(\mathrm{C}_{2} \mathrm{H}_{4}\) (d) Acetylene

Determine the number of symmetry operations and assign the symmetry point group to the following molecules: (a) Ruthacene (b) \(\mathrm{N}_{2} \mathrm{O}\) (c) \(\mathrm{PCl}_{5}\) (d) trans \(-\mathrm{N}_{2} \mathrm{~F}_{2}\) (e) \(\mathrm{BF}_{3}\) (f) \(\mathrm{CO}_{2}\) (g) \(\mathrm{N}_{2} \mathrm{O}_{4}\) (h) \(\mathrm{POCl}_{3}\) (i) Acetaldehyde (i) Diborane

What do you mean by Bravais lattices and crystal systems?
See all solutions

Recommended explanations on Chemistry Textbooks

Kinetics

Read Explanation

Ionic and Molecular Compounds

Read Explanation

The Earths Atmosphere

Read Explanation

Physical Chemistry

Read Explanation

Chemical Reactions

Read Explanation

Chemistry Branches

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