net Quantum Chemistry Third Edition www.net Quantum Chemistry Third Edition John P. Lowe Department of Chemistry The Pennsylvania State University University Park, Pennsylvania Kirk A. Peterson Department of Chemistry Washington State University Pullman, Washington Amsterdam • Boston • Heidelberg • London • New York • Oxford Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo www.net Acquisitions Editor: Jeremy Hayhurst Project Manager: A. McGee Editorial Assistant: Desiree Marr Marketing Manager: Linda Beattie Cover Designer: Julio Esperas Composition: Integra Software Services Cover Printer: Phoenix Color Interior Printer: Maple-Vail Book Manufacturing Group Elsevier Academic Press 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA 84 Theobald’s Road, London WC1X 8RR, UK This book is printed on acid-free paper. Copyright c 2006, Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: telephone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: permissions@elsevier. You may also complete your request on-line via the Elsevier homepage (http://www.com), by selecting “Customer Support” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data Lowe, John P. Includes bibliographical references and index.28--dc22 2005019099 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13: 978-0-12-457551-6 ISBN-10: 0-12-457551-X For all information on all Elsevier Academic Press publications visit our Web site at www.com Printed in the United States of America 05 06 07 08 09 10 9 8 7 6 5 4 3 2 1 Working together to grow libraries in developing countries www.net To Nancy -J.net THE MOLECULAR CHALLENGE Sir Ethylene, to scientists fair prey, (Who dig and delve and peek and push and pry, And prove their findings with equations sly) Smoothed out his ruffled orbitals, to say: “I stand in symmetry. Mine is a way Of mystery and magic. Ancient, I Am also deemed immortal. Should I die, Pi would be in the sky, and Judgement Day Would be upon us. For all things must fail, That hold our universe together, when Bonds such as bind me fail, and fall asunder. Hence, stand I firm against the endless hail Of scientific blows.” Men And their computers stand and stare and wonder.net Contents Preface to the Third Edition xvii Preface to the Second Edition xix Preface to the First Edition xxi 1 Classical Waves and the Time-Independent Schrödinger Wave Equation 1 1-1 Introduction . 1 1-3 The Classical Wave Equation . 4 1-4 Standing Waves in a Clamped String . 7 1-5 Light as an Electromagnetic Wave . 9 1-6 The Photoelectric Effect . 10 1-7 The Wave Nature of Matter . 14 1-8 A Diffraction Experiment with Electrons . 16 1-9 Schrödinger’s Time-Independent Wave Equation . 21 1-11 Some Insight into the Schrödinger Equation . 24 Multiple Choice Questions . 26 2 Quantum Mechanics of Some Simple Systems 27 2-1 The Particle in a One-Dimensional “Box” . 27 2-2 Detailed Examination of Particle-in-a-Box Solutions . 30 2-3 The Particle in a One-Dimensional “Box” with One Finite Wall . 38 2-4 The Particle in an Infinite “Box” with a Finite Central Barrier . 44 2-5 The Free Particle in One Dimension . 47 2-6 The Particle in a Ring of Constant Potential . 50 2-7 The Particle in a Three-Dimensional Box: Separation of Variables . 53 2-8 The Scattering of Particles in One Dimension . 60 Multiple Choice Questions .net x Contents 3 The One-Dimensional Harmonic Oscillator 69 3-1 Introduction . 69 3-2 Some Characteristics of the Classical One-Dimensional Harmonic Oscillator . 69 3-3 The Quantum-Mechanical Harmonic Oscillator . 72 3-4 Solution of the Harmonic Oscillator Schrödinger Equation . 74 3-5 Quantum-Mechanical Average Value of the Potential Energy . 83 3-6 Vibrations of Diatomic Molecules . 85 Multiple Choice Questions . 88 4 The Hydrogenlike Ion, Angular Momentum, and the Rigid Rotor 89 4-1 The Schrödinger Equation and the Nature of Its Solutions . 89 4-2 Separation of Variables . 105 4-3 Solution of the R, , and Equations . 109 4-5 Angular Momentum and Spherical Harmonics . 110 4-6 Angular Momentum and Magnetic Moment . 115 4-7 Angular Momentum in Molecular Rotation—The Rigid Rotor . 120 Multiple Choice Questions . 126 5 Many-Electron Atoms 127 5-1 The Independent Electron Approximation . 127 5-2 Simple Products and Electron Exchange Symmetry . 129 5-3 Electron Spin and the Exclusion Principle . 132 5-4 Slater Determinants and the Pauli Principle . 137 5-5 Singlet and Triplet States for the 1s2s Configuration of Helium . 138 5-6 The Self-Consistent Field, Slater-Type Orbitals, and the Aufbau Principle . 144 5-7 Electron Angular Momentum in Atoms . 160 Multiple Choice Questions . 165 6 Postulates and Theorems of Quantum Mechanics 166 6-1 Introduction . 166 6-2 The Wavefunction Postulate . 166 6-3 The Postulate for Constructing Operators . 167 6-4 The Time-Dependent Schrödinger Equation Postulate . 168 6-5 The Postulate Relating Measured Values to Eigenvalues . 169 6-6 The Postulate for Average Values .net Contents xi 6-8 Proof That Eigenvalues of Hermitian Operators Are Real . 172 6-9 Proof That Nondegenerate Eigenfunctions of a Hermitian Operator Form an Orthogonal Set . 173 6-10 Demonstration That All Eigenfunctions of a Hermitian Operator May Be Expressed as an Orthonormal Set . 174 6-11 Proof That Commuting Operators Have Simultaneous Eigenfunctions 175 6-12 Completeness of Eigenfunctions of a Hermitian Operator . 176 6-13 The Variation Principle . 178 6-14 The Pauli Exclusion Principle . 178 6-15 Measurement, Commutators, and Uncertainty . 178 6-16 Time-Dependent States . 186 Multiple Choice Questions . 189 7 The Variation Method 190 7-1 The Spirit of the Method . 190 7-2 Nonlinear Variation: The Hydrogen Atom . 191 7-3 Nonlinear Variation: The Helium Atom . 194 7-4 Linear Variation: The Polarizability of the Hydrogen Atom . 197 7-5 Linear Combination of Atomic Orbitals: The H+2 Molecule–Ion . 206 7-6 Molecular Orbitals of Homonuclear Diatomic Molecules . 220 7-7 Basis Set Choice and the Variational Wavefunction . 231 7-8 Beyond the Orbital Approximation . 235 Multiple Choice Questions . 242 8 The Simple Hückel Method and Applications 244 8-1 The Importance of Symmetry . 244 8-2 The Assumption of σ –π Separability . 244 8-3 The Independent π-Electron Assumption . 246 8-4 Setting up the Hückel Determinant . 247 8-5 Solving the HMO Determinantal Equation for Orbital Energies . 250 8-6 Solving for the Molecular Orbitals . 251 8-7 The Cyclopropenyl System: Handling Degeneracies . 253 8-8 Charge Distributions from HMOs . 256 8-9 Some Simplifying Generalizations . 259 8-10 HMO Calculations on Some Simple Molecules . 263 8-11 Summary: The Simple HMO Method for Hydrocarbons . 268 8-12 Relation Between Bond Order and Bond Length . 269 8-13 π -Electron Densities and Electron Spin Resonance Hyperfine Splitting Constants . 271 8-14 Orbital Energies and Oxidation-Reduction Potentials . 275 8-15 Orbital Energies and Ionization Energies . 278 8-16 π -Electron Energy and Aromaticity .net xii Contents 8-17 Extension to Heteroatomic Molecules . 284 8-18 Self-Consistent Variations of α and β . 287 8-19 HMO Reaction Indices . 296 Multiple Choice Questions . 306 9 Matrix Formulation of the Linear Variation Method 308 9-1 Introduction . 308 9-2 Matrices and Vectors . 308 9-3 Matrix Formulation of the Linear Variation Method . 315 9-4 Solving the Matrix Equation . 323 10 The Extended Hückel Method 324 10-1 The Extended Hückel Method . 335 10-3 Extended Hückel Energies and Mulliken Populations . 338 10-4 Extended Hückel Energies and Experimental Energies . 347 11 The SCF-LCAO-MO Method and Extensions 348 11-1 Ab Initio Calculations . 348 11-2 The Molecular Hamiltonian . 349 11-3 The Form of the Wavefunction . 349 11-4 The Nature of the Basis Set . 350 11-5 The LCAO-MO-SCF Equation . 350 11-6 Interpretation of the LCAO-MO-SCF Eigenvalues . 351 11-7 The SCF Total Electronic Energy . 353 11-9 The Hartree–Fock Limit . 358 11-12 Configuration Interaction . 360 11-13 Size Consistency and the Møller–Plesset and Coupled Cluster Treatments of Correlation . 367 11-15 Density Functional Theory Methods . 368 11-16 Examples of Ab Initio Calculations . 370 11-17 Approximate SCF-MO Methods .net Contents xiii 12 Time-Independent Rayleigh–Schrödinger Perturbation Theory 391 12-1 An Introductory Example . 391 12-2 Formal Development of the Theory for Nondegenerate States . 391 12-3 A Uniform Electrostatic Perturbation of an Electron in a “Wire” . 396 12-4 The Ground-State Energy to First-Order of Heliumlike Systems . 403 12-5 Perturbation at an Atom in the Simple Hückel MO Method . 406 12-6 Perturbation Theory for a Degenerate State . 409 12-7 Polarizability of the Hydrogen Atom in the n = 2 States . 410 12-8 Degenerate-Level Perturbation Theory by Inspection . 412 12-9 Interaction Between Two Orbitals: An Important Chemical Model . 414 12-10 Connection Between Time-Independent Perturbation Theory and Spectroscopic Selection Rules . 420 Multiple Choice Questions . 428 13 Group Theory 429 13-1 Introduction . 429 13-2 An Elementary Example . 429 13-3 Symmetry Point Groups . 431 13-4 The Concept of Class . 434 13-5 Symmetry Elements and Their Notation . 436 13-6 Identifying the Point Group of a Molecule . 441 13-7 Representations for Groups . 443 13-8 Generating Representations from Basis Functions . 446 13-9 Labels for Representations . 451 13-10 Some Connections Between the Representation Table and Molecular Orbitals . 452 13-11 Representations for Cyclic and Related Groups . 453 13-12 Orthogonality in Irreducible Inequivalent Representations . 456 13-13 Characters and Character Tables . 458 13-14 Using Characters to Resolve Reducible Representations . 462 13-15 Identifying Molecular Orbital Symmetries . 463 13-16 Determining in Which Molecular Orbital an Atomic Orbital Will Appear . 465 13-17 Generating Symmetry Orbitals . 467 13-18 Hybrid Orbitals and Localized Orbitals . 470 13-19 Symmetry and Integration . 476 Multiple Choice Questions . 483 14 Qualitative Molecular Orbital Theory 484 14-1 The Need for a Qualitative Theory . 484 14-2 Hierarchy in Molecular Structure and in Molecular Orbitals .net xiv Contents 14-5 Rules for Qualitative Molecular Orbital Theory . 490 14-6 Application of QMOT Rules to Homonuclear Diatomic Molecules . 490 14-7 Shapes of Polyatomic Molecules: Walsh Diagrams . 505 14-9 Qualitative Molecular Orbital Theory of Reactions . 524 15 Molecular Orbital Theory of Periodic Systems 526 15-1 Introduction . 526 15-2 The Free Particle in One Dimension . 526 15-3 The Particle in a Ring . 530 15-5 General Form of One-Electron Orbitals in Periodic Potentials— Bloch’s Theorem . 537 15-7 An Example: Polyacetylene with Uniform Bond Lengths . 546 15-9 Polyacetylene with Alternating Bond Lengths—Peierls’ Distortion . 547 15-10 Electronic Structure of All-Trans Polyacetylene . 551 15-11 Comparison of EHMO and SCF Results on Polyacetylene . 552 15-12 Effects of Chemical Substitution on the π Bands . 554 15-13 Poly-Paraphenylene—A Ring Polymer . 562 15-15 Two-Dimensional Periodicity and Vectors in Reciprocal Space . 562 15-16 Periodicity in Three Dimensions—Graphite . 580 Appendix 1 Useful Integrals 582 Appendix 2 Determinants 584 Appendix 3 Evaluation of the Coulomb Repulsion Integral Over 1s AOs 587 Appendix 4 Angular Momentum Rules 591 Appendix 5 The Pairing Theorem 601 Appendix 6 Hückel Molecular Orbital Energies, Coefficients, Electron Densities, and Bond Orders for Some Simple Molecules 605 Appendix 7 Derivation of the Hartree–Fock Equation 614 Appendix 8 The Virial Theorem for Atoms and Diatomic Molecules 624 www.net Contents xv Appendix 9 Bra-ket Notation 629 Appendix 10 Values of Some Useful Constants and Conversion Factors 631 Appendix 11 Group Theoretical Charts and Tables 636 Appendix 12 Hints for Solving Selected Problems 651 Appendix 13 Answers to Problems 654 Index 691 www.
