Chemical Applications of Group Theory

Retains the easy-to-read format and informal flavor of the previous editions, and includes new material on the symmetric properties of extended arrays (crystals), projection operators, LCAO molecular orbitals, and electron counting rules. Also contains many new exercises and illustrations.

F. ALBERT COTTON, PhD, is a professor in the Department of Chemistry at Texas A&M University.

Part I. Principles

Introduction

3

(14)

Definitions and Theorems of Group Theory

The Defining Properties of a Group

6

(2)

Some Examples of Groups

8

(4)

Subgroups

12

(1)

Classes

13

(4)

Exercises

16

(1)

Molecular Symmetry and the Symmetry Groups

17

(51)

General Remarks

17

(1)

Symmetry Elements and Operations

18

(1)

Symmetry Planes and Reflections

18

(4)

The Inversion Center

22

(1)

Proper Axes and Proper Rotations

22

(5)

Improper Axes and Improper Rotations

27

(2)

Products of Symmetry Operations

29

(3)

Equivalent Symmetry Elements and Equivalent Atoms

32

(1)

General Relations Among Symmetry Elements and Operations

33

(1)

Symmetry Elements and Optical Isomerism

34

(5)

The Symmetry Point Groups

39

(5)

Symmetries with Multiple High-Order Axes

44

(6)

Classes of Symmetry Operations

50

(4)

A Systematic Procedure for Symmetry Classification of Molecules

54

(2)

Illustrative Examples

56

(12)

Exercises

61

(7)

Representations of Groups

68

(32)

Prefactory Comments on Matrices and Vectors

68

(10)

Representations of Groups

78

(3)

The ``Great Orthogonality Theorem'' and Its Consequences

81

(9)

Character Tables

90

(5)

Representations for Cyclic Groups

95

(5)

Exercises

99

(1)

Group Theory and Quantum Mechanics

100

(14)

Wave Functions as Bases for Irreducible Representations

100

(5)

The Direct Product

105

(4)

Identifying Nonzero Matrix Elements

109

(5)

Exercises

113

(1)

Symmetry-Adapted Linear Combinations

114

(19)

Introductory Remarks

114

(1)

Derivation of Projection Operators

115

(4)

Using Projection Operators to Construct SALCs

119

(14)

Exercises

129

(4)

Part II. Applications

Molecular Orbital Theory and Its Applications in Organic Chemistry

133

(71)

General Principles

133

(7)

Symmetry Factoring of Secular Equations

140

(2)

Carbocyclic Systems

142

(17)

More General Cases of LCAO--MO π Bonding

159

(13)

A Worked Example: Naphthalene

172

(4)

Electronic Excitations of Naphthalene: Selection Rules and Configuration Interaction

176

(4)

Three-Center Bonding

180

(8)

Symmetry-Based ``Selection Rules'' for Cyclization Reactions

188

(16)

Exercises

201

(3)

Molecular Orbital Theory for Inorganic and Organometallic Compounds

204

(49)

Introduction

204

(1)

Transformation Properties of Atomic Orbitals

205

(4)

Molecular Orbitals for σ Bonding in ABn Molecules: The Tetrahedral AB4 Case

209

(5)

Molecular Orbitals for σ Bonding in Other ABn Molecules

214

(8)

Hybrid Orbitals

222

(5)

Molecular Orbitals for π Bonding in ABn Molecules

227

(3)

Cage and Cluster Compounds

230

(10)

Molecular Orbitals for Metal Sandwich Compounds

240

(13)

Exercises

251

(2)

Ligand Field Theory

253

(51)

Introductory Remarks

253

(1)

Electronic Structures of Free Atoms and Ions

254

(6)

Splitting of Levels and Terms in a Chemical Environment

260

(5)

Construction of Energy Level Diagrams

265

(16)

Estimation of Orbital Energies

281

(8)

Selection Rules and Polarizations

289

(8)

Double Groups

297

(7)

Exercises

302

(2)

Molecular Vibrations

304

(44)

Introductory Remarks

304

(1)

The Symmetry of Normal Vibrations

305

(4)

Determining the Symmetry Types of the Normal Modes

309

(6)

Contributions of Particular Internal Coordinates to Normal Modes

315

(2)

How to Calculate Force Constants: The F and G Matrix Method

317

(7)

Selection Rules for Fundamental Vibrational Transitions

324

(4)

Illustrative Examples

328

(10)

Some Important Special Effects

338

(10)

Exercises

346

(2)

Crystallographic Symmetry

348

(109)

Introduction

348

(2)

The Concept of a Lattice---In Two Dimensions

350

(8)

Two-Dimensional Space Symmetries

358

(10)

Three-Dimensional Lattices and Their Symmetries

368

(7)

Crystal Symmetry: The 32 Crystallographic Point Groups

375

(5)

Interrelating Lattice Symmetry, Crystal Symmetry, and Diffraction Symmetry

380

(4)

Additional Symmetry Elements and Operations: Glide Planes and Screw Axes

384

(4)

The 230 Three-Dimensional Space Groups

388

(12)

Space Groups and X-Ray Crystallography

400

(17)

Exercises

410

(7)

Part III. Appendices

I. Matrix Algebra

417

(9)

IIA. Character Tables for Chemically Important Symmetry Groups

426

(11)

IIB. Correlation Table for Group Oh

437

(1)

III. Some Remarks About the Resonance Integral β

438

(3)

IV. The Shapes of f Orbitals

441

(2)

V. Character Tables for Some Double Groups

443

(1)

VI. Elements of The g Matrix

444

(4)

VII. Stereographic Projections for the 32 Crystallographic Point Groups

448

(2)

VIII. The 230 Space Groups

450

(2)

IX. Reading List

452

(5)

Index

457

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Chemical Applications of Group Theory

Summary

Author Biography

Table of Contents

Supplemental Materials