Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 24

Find the symmetry-adapted linear combinations of (a) \(\sigma\) -orbitals, (b) \(\pi\) -orbitals on the ligands of an octahedral complex. Hint. Set Cartesian axes on each ligand site, with \(z\) pointing towards the central ion, determine how the orbitals are transformed under the operations of the group \(\mathrm{O},\) and use the procedures for establishing symmetry-adapted orbitals as described in Chapter 5.

Short Answer

Expert verified
The sigma orbitals combine into SALCs that transform as \(a_1g\) under the operations of the octahedral group. On the other hand, the pi orbitals combine into SALCs that transform as \(e_g\). These SALCs describe the bonding, nonbonding, and antibonding molecular orbitals of the octahedral complex.

Step by step solution

01

Setup Cartesian axes

Start by placing Cartesian axes at each ligand site. The z-axis should be pointing towards the central ion. The x and y axes can point in any direction perpendicular to z and each other.
02

Identify symmetry operations of Octahedral group

Identify the symmetry operations of the octahedral group (O_h). These include identity, rotations about the x, y, and z axis, and inversions through the center of the molecule. Note how each of these operations transforms the orbitals.
03

Construct symmetry-adapted orbitals for sigma orbitals

Start with the sigma orbitals. Sigma orbitals have cylindrical symmetry around the bond axis, in this case, the z-axis. Under the operations of O_h group, the sigma orbitals are nondegenerate, transforming as a1g. Use the methods from Chapter 5 to combine the sigma orbitals into SALCs. Each SALC will belong to a different irreducible representation of the O_h group.
04

Construct symmetry-adapted orbitals for pi orbitals

Repeat the procedure for the pi orbitals. Unlike sigma orbitals, pi orbitals have a nodal plane containing the bond axis. Pi orbitals are doubly-degenerate, transforming as e_g. Again, use the methods from Chapter 5 to combine the pi orbitals into SALCs. These SALCs will form the bonding, nonbonding, and antibonding molecular orbitals of the octahedral complex.

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

Predict the ground configuration of (a) CO, (b) NO. For each species, decide which term lies lowest in energy, compute the bond order and identify the HOMO and LUMO.

Use a minimal basis set for the \(\mathrm{M} \mathrm{O}\) description of the molecule \(\mathrm{H}_{2} \mathrm{O}\) to show that the secular determinant factorizes into \((1 \times 1),(2 \times 2),\) and \((3 \times 3)\) determinants. Set up the secular determinant, denoting the Coulomb integrals \(\alpha_{\mathrm{H}}, \alpha_{\mathrm{O}}^{\prime},\) and \(\alpha_{\mathrm{O}}\) for \(\mathrm{H} 1 \mathrm{s}, \mathrm{O} 2 \mathrm{s},\) and \(\mathrm{O} 2 \mathrm{p},\) respectively, and writing the \((\mathrm{O} 2 \mathrm{p}, \mathrm{H} 1 \mathrm{s})\) and \((\mathrm{O} 2 \mathrm{s}, \mathrm{H} 1 \mathrm{s})\) resonance integrals as \(\beta\) and \(\beta^{\prime},\) respectively. Neglect overlap. First, neglect the \(2 s\) -orbital, and find expressions for the energies of the molecular orbitals for a bond angle of \(90^{\circ}\).

Heterocyclic molecules may be incorporated into the Hückel scheme by modifying the Coulomb integral of the atom concerned and the resonance integrals to which it contributes. Consider pyridine, \(\mathrm{C}_{5} \mathrm{H}_{5} \mathrm{N}\) (symmetry group \(\left.C_{2 v}\right) .\) Construct and solve the Hückel secular determinant with \(\beta_{\mathrm{CC}} \approx \beta_{\mathrm{CN}} \approx \beta\) and \(\alpha_{\mathrm{N}}=\alpha_{\mathrm{C}}+^{1 / 2} \beta .\) Estimate the electron energy and the delocalization energy. Hint. The roots of the determinants are best found on a computer.

Explore the role of p-orbital overlap in \(\pi\) -electron calculations. Take the cyclobutadiene secular determinant, but construct it without neglect of overlap between neighbouring atoms. Show that in place of \(x=(\alpha-E) / \beta\) and 1 the elements of the determinant become \(\omega=(\alpha-E) /\) \((\beta-E S)\) and \(1,\) respectively. Hence the roots in terms of \(\omega\) are the same as the roots in terms of \(x .\) Solve for \(E .\) Typically \(S=0.25\).

Set up and solve the secular determinants for (a) hexatriene, (b) the cyclopentadienyl radical in the Hückel \(\pi\) -electron scheme; find the energy levels and molecular orbitals, and estimate the delocalization energy.
See all solutions

Recommended explanations on Chemistry Textbooks

The Earths Atmosphere

Read Explanation

Kinetics

Read Explanation

Chemical Analysis

Read Explanation

Ionic and Molecular Compounds

Read Explanation

Making Measurements

Read Explanation

Organic Chemistry

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