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Quantum Chemistry
Quantum Chemistry
John P. Lowe, Kirk Peterson
Lowe's new edition assumes little mathematical or physical sophistication and emphasizes an understanding of the techniques and results of quantum chemistry. It can serve as a primary text in quantum chemistry courses, and enables students and researchers to comprehend the current literature. This third edition has been thoroughly updated and includes numerous new exercises to facilitate selfstudy and solutions to selected exercises. * Assumes little initial mathematical or physical sophistication, developing insights and abilities in the context of actual problems * Provides thorough treatment of the simple systems basic to this subject * Emphasizes UNDERSTANDING of the techniques and results of modern quantum chemistry * Treats MO theory from simple Huckel through ab intio methods in current use * Develops perturbation theory through the topics of orbital interaction as well as spectroscopic selection rules * Presents group theory in a context of MO applications * Includes qualitative MO theory of molecular structure, Walsh rules, WoodwardHoffmann rules, frontier orbitals, and organic reactions develops MO theory of periodic systems, with applications to organic polymers.
Categories:
Physics\\Quantum Physics
Year:
2006
Edition:
3rd ed
Publisher:
Elsevier Academic Press
Language:
english
Pages:
726
ISBN 13:
9780124575516
ISBN:
012457551X
File:
PDF, 9.16 MB
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Quantum Chemistry Third Edition 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 Acquisitions Editor: Jeremy Hayhurst Project Manager: A. B. McGee Editorial Assistant: Desiree Marr Marketing Manager: Linda Beattie Cover Designer: Julio Esperas Composition: Integra Software Services Cover Printer: Phoenix Color Interior Printer: MapleVail Book Manufacturing Group Elsevier Academic Press 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, CA 921014495, USA 84 Theobald’s Road, London WC1X 8RR, UK This book is printed on acidfree paper. c 2006, Elsevier Inc. All rights reserved. Copyright 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, email: permissions@elsevier.co.uk. You may also complete your request online via the Elsevier homepage (http://www.elsevier.com), by selecting “Customer Support” and then “Obtaining Permissions.” Library of Congress CataloginginPublication Data Lowe, John P. Quantum chemistry.  3rd ed. / John P. Lowe, Kirk A. Peterson. p. cm. Includes bibliographical references and index. ISBN 012457551X 1. Quantum chemistry. I. Peterson, Kirk A. II. Title. QD462.L69 2005 541'.28dc22 2005019099 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN13: 9780124575516 ISBN10: 012457551X For all information on all Elsevier Academic Press publications visit our Web site at www.books.elsevier.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.elsevier.com  www.bookaid.org  www.sabre.org To Nancy J. L. 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. I yield not.” Men And their computers stand and stare and wonder. W.G. LOWE Contents Preface to the Third Edition xvii Preface to the Second Edition xix Preface to the First Edition xxi 1 Classical Waves and the TimeIndependent Schrödinger Wave Equation 11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 The Classical Wave Equation . . . . . . . . . . . . . . . . . . . . . 14 Standing Waves in a Clamped String . . . . . . . . . . . . . . . . . 15 Light as an Electromagnetic Wave . . . . . . . . . . . . . . . . . . . 16 The Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . 17 The Wave Nature of Matter . . . . . . . . . . . . . . . . . . . . . . 18 A Diffraction Experiment with Electrons . . . . . . . . . . . . . . . 19 Schrödinger’s TimeIndependent Wave Equation . . . . . . . . . . . 110 Conditions on ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Some Insight into the Schrödinger Equation . . . . . . . . . . . . . 112 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Choice Questions . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 4 7 9 10 14 16 19 21 22 23 24 25 26 2 Quantum Mechanics of Some Simple Systems 21 The Particle in a OneDimensional “Box” . . . . . . . . . . . . . 22 Detailed Examination of ParticleinaBox Solutions . . . . . . . 23 The Particle in a OneDimensional “Box” with One Finite Wall . 24 The Particle in an Inﬁnite “Box” with a Finite Central Barrier . . 25 The Free Particle in One Dimension . . . . . . . . . . . . . . . . 26 The Particle in a Ring of Constant Potential . . . . . . . . . . . . 27 The Particle in a ThreeDimensional Box: Separation of Variables 28 The Scattering of Particles in One Dimension . . . . . . . . . . . 29 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Choice Questions . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 27 30 38 44 47 50 53 56 59 60 65 68 . . . . . . . . . . . . . . . . . . . . . . . . ix x Contents 3 The OneDimensional Harmonic Oscillator 31 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Some Characteristics of the Classical OneDimensional Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 The QuantumMechanical Harmonic Oscillator . . . . . . . . . . . . 34 Solution of the Harmonic Oscillator Schrödinger Equation . . . . . . 35 QuantumMechanical Average Value of the Potential Energy . . . . . 36 Vibrations of Diatomic Molecules . . . . . . . . . . . . . . . . . . . 37 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Choice Questions . . . . . . . . . . . . . . . . . . . . . . . 4 The Hydrogenlike Ion, Angular Momentum, and the Rigid Rotor 41 The Schrödinger Equation and the Nature of Its Solutions . . . 42 Separation of Variables . . . . . . . . . . . . . . . . . . . . . 43 Solution of the R, , and Equations . . . . . . . . . . . . . 44 Atomic Units . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Angular Momentum and Spherical Harmonics . . . . . . . . . 46 Angular Momentum and Magnetic Moment . . . . . . . . . . 47 Angular Momentum in Molecular Rotation—The Rigid Rotor 48 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Choice Questions . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 6 . . . . . . . . . . . ManyElectron Atoms 51 The Independent Electron Approximation . . . . . . . . . . . . 52 Simple Products and Electron Exchange Symmetry . . . . . . . 53 Electron Spin and the Exclusion Principle . . . . . . . . . . . . 54 Slater Determinants and the Pauli Principle . . . . . . . . . . . 55 Singlet and Triplet States for the 1s2s Conﬁguration of Helium . 56 The SelfConsistent Field, SlaterType Orbitals, and the Aufbau Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Electron Angular Momentum in Atoms . . . . . . . . . . . . . . 58 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Choice Questions . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postulates and Theorems of Quantum Mechanics 61 Introduction . . . . . . . . . . . . . . . . . . . . . . . 62 The Wavefunction Postulate . . . . . . . . . . . . . . . 63 The Postulate for Constructing Operators . . . . . . . . 64 The TimeDependent Schrödinger Equation Postulate . 65 The Postulate Relating Measured Values to Eigenvalues 66 The Postulate for Average Values . . . . . . . . . . . . 67 Hermitian Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 69 69 72 74 83 84 85 85 88 . . . . . . . . . . . 89 89 105 106 109 110 115 117 119 120 125 126 . . . . . . . . . . . . . . . 127 127 129 132 137 138 . . . . . . . . . . . . . . . . . . 144 149 159 160 164 165 . . . . . . . . . . . . . . . . . . . . . 166 166 166 167 168 169 171 171 xi Contents 68 69 610 611 612 613 614 615 616 617 Proof That Eigenvalues of Hermitian Operators Are Real . . . . . . . Proof That Nondegenerate Eigenfunctions of a Hermitian Operator Form an Orthogonal Set . . . . . . . . . . . . . . . . . . . . . . . . Demonstration That All Eigenfunctions of a Hermitian Operator May Be Expressed as an Orthonormal Set . . . . . . . . . . . . . . . . . Proof That Commuting Operators Have Simultaneous Eigenfunctions Completeness of Eigenfunctions of a Hermitian Operator . . . . . . The Variation Principle . . . . . . . . . . . . . . . . . . . . . . . . The Pauli Exclusion Principle . . . . . . . . . . . . . . . . . . . . . Measurement, Commutators, and Uncertainty . . . . . . . . . . . . TimeDependent States . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Choice Questions . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The Variation Method 71 The Spirit of the Method . . . . . . . . . . . . . . . . . . . . . 72 Nonlinear Variation: The Hydrogen Atom . . . . . . . . . . . 73 Nonlinear Variation: The Helium Atom . . . . . . . . . . . . . 74 Linear Variation: The Polarizability of the Hydrogen Atom . . 75 Linear Combination of Atomic Orbitals: The H+2 Molecule–Ion 76 Molecular Orbitals of Homonuclear Diatomic Molecules . . . . 77 Basis Set Choice and the Variational Wavefunction . . . . . . . 78 Beyond the Orbital Approximation . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Choice Questions . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 The Simple Hückel Method and Applications 81 The Importance of Symmetry . . . . . . . . . . . . . . . . . . 82 The Assumption of σ –π Separability . . . . . . . . . . . . . . 83 The Independent πElectron Assumption . . . . . . . . . . . . 84 Setting up the Hückel Determinant . . . . . . . . . . . . . . . 85 Solving the HMO Determinantal Equation for Orbital Energies 86 Solving for the Molecular Orbitals . . . . . . . . . . . . . . . 87 The Cyclopropenyl System: Handling Degeneracies . . . . . . 88 Charge Distributions from HMOs . . . . . . . . . . . . . . . . 89 Some Simplifying Generalizations . . . . . . . . . . . . . . . 810 HMO Calculations on Some Simple Molecules . . . . . . . . . 811 Summary: The Simple HMO Method for Hydrocarbons . . . . 812 Relation Between Bond Order and Bond Length . . . . . . . . 813 π Electron Densities and Electron Spin Resonance Hyperﬁne Splitting Constants . . . . . . . . . . . . . . . . . . . . . . . . 814 Orbital Energies and OxidationReduction Potentials . . . . . . 815 Orbital Energies and Ionization Energies . . . . . . . . . . . . 816 π Electron Energy and Aromaticity . . . . . . . . . . . . . . . 172 173 174 175 176 178 178 178 180 185 186 189 189 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 190 191 194 197 206 220 231 233 235 241 242 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 244 244 246 247 250 251 253 256 259 263 268 269 . . . . . . . . . . . . 271 275 278 279 xii Contents 817 818 819 820 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 287 289 295 296 305 306 Matrix Formulation of the Linear Variation Method 91 Introduction . . . . . . . . . . . . . . . . . . . . . 92 Matrices and Vectors . . . . . . . . . . . . . . . . 93 Matrix Formulation of the Linear Variation Method 94 Solving the Matrix Equation . . . . . . . . . . . . 95 Summary . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 308 308 315 317 320 320 323 10 The Extended Hückel Method 101 The Extended Hückel Method . . . . . . . . . . . . . 102 Mulliken Populations . . . . . . . . . . . . . . . . . . 103 Extended Hückel Energies and Mulliken Populations . 104 Extended Hückel Energies and Experimental Energies Problems . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 324 335 338 340 342 347 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 348 349 349 350 350 351 352 353 357 357 358 360 . . . . . . . . . . . . . . . . . . . . . 365 367 368 370 384 386 388 9 Extension to Heteroatomic Molecules SelfConsistent Variations of α and β HMO Reaction Indices . . . . . . . . Conclusions . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . Multiple Choice Questions . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 The SCFLCAOMO Method and Extensions 111 Ab Initio Calculations . . . . . . . . . . . . . . . . . . . . . 112 The Molecular Hamiltonian . . . . . . . . . . . . . . . . . . 113 The Form of the Wavefunction . . . . . . . . . . . . . . . . . 114 The Nature of the Basis Set . . . . . . . . . . . . . . . . . . 115 The LCAOMOSCF Equation . . . . . . . . . . . . . . . . . 116 Interpretation of the LCAOMOSCF Eigenvalues . . . . . . 117 The SCF Total Electronic Energy . . . . . . . . . . . . . . . 118 Basis Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 The Hartree–Fock Limit . . . . . . . . . . . . . . . . . . . . 1110 Correlation Energy . . . . . . . . . . . . . . . . . . . . . . . 1111 Koopmans’ Theorem . . . . . . . . . . . . . . . . . . . . . . 1112 Conﬁguration Interaction . . . . . . . . . . . . . . . . . . . . 1113 Size Consistency and the Møller–Plesset and Coupled Cluster Treatments of Correlation . . . . . . . . . . . . . . . . . . . 1114 Multideterminant Methods . . . . . . . . . . . . . . . . . . . 1115 Density Functional Theory Methods . . . . . . . . . . . . . . 1116 Examples of Ab Initio Calculations . . . . . . . . . . . . . . 1117 Approximate SCFMO Methods . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Contents 12 TimeIndependent Rayleigh–Schrödinger Perturbation Theory 121 An Introductory Example . . . . . . . . . . . . . . . . . . . . . . . 122 Formal Development of the Theory for Nondegenerate States . . . . 123 A Uniform Electrostatic Perturbation of an Electron in a “Wire” . . 124 The GroundState Energy to FirstOrder of Heliumlike Systems . . 125 Perturbation at an Atom in the Simple Hückel MO Method . . . . . 126 Perturbation Theory for a Degenerate State . . . . . . . . . . . . . 127 Polarizability of the Hydrogen Atom in the n = 2 States . . . . . . . 128 DegenerateLevel Perturbation Theory by Inspection . . . . . . . . 129 Interaction Between Two Orbitals: An Important Chemical Model . 1210 Connection Between TimeIndependent Perturbation Theory and Spectroscopic Selection Rules . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Choice Questions . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 391 391 396 403 406 409 410 412 414 13 Group Theory 131 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 An Elementary Example . . . . . . . . . . . . . . . . . . . . . . . 133 Symmetry Point Groups . . . . . . . . . . . . . . . . . . . . . . . 134 The Concept of Class . . . . . . . . . . . . . . . . . . . . . . . . . 135 Symmetry Elements and Their Notation . . . . . . . . . . . . . . . 136 Identifying the Point Group of a Molecule . . . . . . . . . . . . . . 137 Representations for Groups . . . . . . . . . . . . . . . . . . . . . . 138 Generating Representations from Basis Functions . . . . . . . . . . 139 Labels for Representations . . . . . . . . . . . . . . . . . . . . . . 1310 Some Connections Between the Representation Table and Molecular Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1311 Representations for Cyclic and Related Groups . . . . . . . . . . . 1312 Orthogonality in Irreducible Inequivalent Representations . . . . . 1313 Characters and Character Tables . . . . . . . . . . . . . . . . . . . 1314 Using Characters to Resolve Reducible Representations . . . . . . 1315 Identifying Molecular Orbital Symmetries . . . . . . . . . . . . . . 1316 Determining in Which Molecular Orbital an Atomic Orbital Will Appear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317 Generating Symmetry Orbitals . . . . . . . . . . . . . . . . . . . . 1318 Hybrid Orbitals and Localized Orbitals . . . . . . . . . . . . . . . 1319 Symmetry and Integration . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Choice Questions . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 429 429 431 434 436 441 443 446 451 14 Qualitative Molecular Orbital Theory 141 The Need for a Qualitative Theory . . . . . . . . . . . . . . 142 Hierarchy in Molecular Structure and in Molecular Orbitals 143 H+ 2 Revisited . . . . . . . . . . . . . . . . . . . . . . . . . 144 H2 : Comparisons with H2+ . . . . . . . . . . . . . . . . . . 484 484 484 485 488 . . . . . . . . . . . . . . . . 417 420 427 428 452 453 456 458 462 463 465 467 470 472 476 481 483 xiv Contents 145 146 147 148 149 Rules for Qualitative Molecular Orbital Theory . . . . . . . . . . Application of QMOT Rules to Homonuclear Diatomic Molecules Shapes of Polyatomic Molecules: Walsh Diagrams . . . . . . . . Frontier Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . Qualitative Molecular Orbital Theory of Reactions . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 490 495 505 508 521 524 15 Molecular Orbital Theory of Periodic Systems 151 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 The Free Particle in One Dimension . . . . . . . . . . . . . . . . . 153 The Particle in a Ring . . . . . . . . . . . . . . . . . . . . . . . . . 154 Benzene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 General Form of OneElectron Orbitals in Periodic Potentials— Bloch’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 A Retrospective Pause . . . . . . . . . . . . . . . . . . . . . . . . 157 An Example: Polyacetylene with Uniform Bond Lengths . . . . . . 158 Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . 159 Polyacetylene with Alternating Bond Lengths—Peierls’ Distortion . 1510 Electronic Structure of AllTrans Polyacetylene . . . . . . . . . . . 1511 Comparison of EHMO and SCF Results on Polyacetylene . . . . . 1512 Effects of Chemical Substitution on the π Bands . . . . . . . . . . 1513 PolyParaphenylene—A Ring Polymer . . . . . . . . . . . . . . . 1514 Energy Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 1515 TwoDimensional Periodicity and Vectors in Reciprocal Space . . . 1516 Periodicity in Three Dimensions—Graphite . . . . . . . . . . . . . 1517 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 526 526 529 530 533 537 537 546 547 551 552 554 555 562 562 565 576 578 580 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 xv Contents Appendix 9 Braket 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 Preface to the Third Edition We have attempted to improve and update this text while retaining the features that make it unique, namely, an emphasis on physical understanding, and the ability to estimate, evaluate, and predict results without blind reliance on computers, while still maintaining rigorous connection to the mathematical basis for quantum chemistry. We have inserted into most chapters examples that allow important points to be emphasized, clariﬁed, or extended. This has enabled us to keep intact most of the conceptual development familiar to past users. In addition, many of the chapters now include multiple choice questions that students are invited to solve in their heads. This is not because we think that instructors will be using such questions. Rather it is because we ﬁnd that such questions permit us to highlight some of the deﬁnitions or conclusions that students often ﬁnd most confusing far more quickly and effectively than we can by using traditional problems. Of course, we have also sought to update material on computational methods, since these are changing rapidly as the ﬁeld of quantum chemistry matures. This book is written for courses taught at the ﬁrstyear graduate/senior undergraduate levels, which accounts for its implicit assumption that many readers will be relatively unfamiliar with much of the mathematics and physics underlying the subject. Our experience over the years has supported this assumption; many chemistry majors are exposed to the requisite mathematics and physics, yet arrive at our courses with poor understanding or recall of those subjects. That makes this course an opportunity for such students to experience the satisfaction of ﬁnally seeing how mathematics, physics, and chemistry are intertwined in quantum chemistry. It is for this reason that treatments of the simple and extended Hückel methods continue to appear, even though these are no longer the methods of choice for serious computations. These topics nevertheless form the basis for the way most nontheoretical chemists understand chemical processes, just as we tend to think about gas behavior as “ideal, with corrections.” xvii Preface to the Second Edition The success of the ﬁrst edition has warranted a second. The changes I have made reﬂect my perception that the book has mostly been used as a teaching text in introductory courses. Accordingly, I have removed some of the material in appendixes on mathematical details of solving matrix equations on a computer. Also I have removed computer listings for programs, since these are now commonly available through commercial channels. I have added a new chapter on MO theory of periodic systems—a subject of rapidly growing importance in theoretical chemistry and materials science and one for which chemists still have difﬁculty ﬁnding appropriate textbook treatments. I have augmented discussion in various chapters to give improved coverage of timedependent phenomena and atomic term symbols and have provided better connection to scattering as well as to spectroscopy of molecular rotation and vibration. The discussion on degeneratelevel perturbation theory is clearer, reﬂecting my own improved understanding since writing the ﬁrst edition. There is also a new section on operator methods for treating angular momentum. Some teachers are strong adherents of this approach, while others prefer an approach that avoids the formalism of operator techniques. To permit both teaching methods, I have placed this material in an appendix. Because this edition is more overtly a text than a monograph, I have not attempted to replace older literature references with newer ones, except in cases where there was pedagogical beneﬁt. A strength of this book has been its emphasis on physical argument and analogy (as opposed to pure mathematical development). I continue to be a strong proponent of the view that true understanding comes with being able to “see” a situation so clearly that one can solve problems in one’s head. There are signiﬁcantly more endofchapter problems, a number of them of the “by inspection” type. There are also more questions inviting students to explain their answers. I believe that thinking about such questions, and then reading explanations from the answer section, signiﬁcantly enhances learning. It is the fashion today to focus on stateoftheart methods for just about everything. The impact of this on education has, I feel, been disastrous. Simpler examples are often needed to develop the insight that enables understanding the complexities of the latest techniques, but too often these are abandoned in the rush to get to the “cutting edge.” For this reason I continue to include a substantial treatment of simple Hückel theory. It permits students to recognize the connections between MOs and their energies and bonding properties, and it allows me to present examples and problems that have maximum transparency in later chapters on perturbation theory, group theory, qualitative MO theory, and periodic systems. I ﬁnd simple Hückel theory to be educationally indispensable. xix xx Preface to the Second Edition Much of the new material in this edition results from new insights I have developed in connection with research projects with graduate students. The work of all four of my students since the appearance of the ﬁrst edition is represented, and I am delighted to thank Sherif Kafaﬁ, John LaFemina, Maribel Soto, and Deb Camper for all I have learned from them. Special thanks are due to Professor Terry Carlton, of Oberlin College, who made many suggestions and corrections that have been adopted in the new edition. Doubtless, there are new errors. I would be grateful to learn of them so that future printings of this edition can be made errorfree. Students or teachers with comments, questions, or corrections are more than welcome to contact me, either by mail at the Department of Chemistry, 152 Davey Lab, The Pennsylvania State University, University Park, PA 16802, or by email directed to JL3 at PSUVM.PSU.EDU. Preface to the First Edition My aim in this book is to present a reasonably rigorous treatment of molecular orbital theory, embracing subjects that are of practical interest to organic and inorganic as well as physical chemists. My approach here has been to rely on physical intuition as much as possible, ﬁrst solving a number of speciﬁc problems in order to develop sufﬁcient insight and familiarity to make the formal treatment of Chapter 6 more palatable. My own experience suggests that most chemists ﬁnd this route the most natural. I have assumed that the reader has at some time learned calculus and elementary physics, but I have not assumed that this material is fresh in his or her mind. Other mathematics is developed as it is needed. The book could be used as a text for undergraduate or graduate students in a half or full year course. The level of rigor of the book is somewhat adjustable. For example, Chapters 3 and 4, on the harmonic oscillator and hydrogen atom, can be truncated if one wishes to know the nature of the solutions, but not the mathematical details of how they are produced. I have made use of appendixes for certain of the more complicated derivations or proofs. This is done in order to avoid having the development of major ideas in the text interrupted or obscured. Certain of the appendixes will interest only the more theoretically inclined student. Also, because I anticipate that some readers may wish to skip certain chapters or parts of chapters, I have occasionally repeated information so that a given chapter will be less dependent on its predecessors. This may seem inelegant at times, but most students will more readily forgive repetition of something they already know than an overly terse presentation. I have avoided early usage of braket notation. I believe that simultaneous introduction of new concepts and unfamiliar notation is poor pedagogy. Braket notation is used only after the ideas have had a change to jell. Problem solving is extremely important in acquiring an understanding of quantum chemistry. I have included a fair number of problems with hints for a few of them in Appendix 14 and answers for almost all of them in Appendix 15.1 It is inevitable that one be selective in choosing topics for a book such as this. This book emphasizes ground state MO theory of molecules more than do most introductory texts, with rather less emphasis on spectroscopy than is usual. Angular momentum is treated at a fairly elementary level at various appropriate places in the text, but it is never given a fullblown formal development using operator commutation relations. Timedependent phenomena are not included. Thus, scattering theory is absent, 1 In this Second Edition, these Appendices are numbered Appendix 12 and 13. xxi xxii Preface to the First Edition although selection rules and the transition dipole are discussed in the chapter on timeindependent perturbation theory. Valencebond theory is completely absent. If I have succeeded in my effort to provide a clear and meaningful treatment of topics relevant to modern molecular orbital theory, it should not be difﬁcult for an instructor to provide for excursions into related topics not covered in the text. Over the years, many colleagues have been kind enough to read sections of the evolving manuscript and provide corrections and advice. I especially thank L. P. Gold and O. H. Crawford, who cheerfully bore the brunt of this task. Finally, I would like to thank my father, Wesley G. Lowe, for allowing me to include his sonnet, “The Molecular Challenge.” Chapter 1 Classical Waves and the TimeIndependent Schrödinger Wave Equation 11 Introduction The application of quantummechanical principles to chemical problems has revolutionized the ﬁeld of chemistry. Today our understanding of chemical bonding, spectral phenomena, molecular reactivities, and various other fundamental chemical problems rests heavily on our knowledge of the detailed behavior of electrons in atoms and molecules. In this book we shall describe in detail some of the basic principles, methods, and results of quantum chemistry that lead to our understanding of electron behavior. In the ﬁrst few chapters we shall discuss some simple, but important, particle systems. This will allow us to introduce many basic concepts and deﬁnitions in a fairly physical way. Thus, some background will be prepared for the more formal general development of Chapter 6. In this ﬁrst chapter, we review brieﬂy some of the concepts of classical physics as well as some early indications that classical physics is not sufﬁcient to explain all phenomena. (Those readers who are already familiar with the physics of classical waves and with early atomic physics may prefer to jump ahead to Section 17.) 12 Waves 12.A Traveling Waves A very simple example of a traveling wave is provided by cracking a whip. A pulse of energy is imparted to the whipcord by a single oscillation of the handle. This results in a wave which travels down the cord, transferring the energy to the popper at the end of the whip. In Fig. 11, an idealization of the process is sketched. The shape of the disturbance in the whip is called the wave proﬁle and is usually symbolized ψ(x). The wave proﬁle for the traveling wave in Fig. 11 shows where the energy is located at a given instant. It also contains the information needed to tell how much energy is being transmitted, because the height and shape of the wave reﬂect the vigor with which the handle was oscillated. 1 2 Chapter 1 Classical Waves and the TimeIndependent Schrödinger Wave Equation Figure 11 Cracking the whip. As time passes, the disturbance moves from left to right along the extended whip cord. Each segment of the cord oscillates up and down as the disturbance passes by, ultimately returning to its equilibrium position. The feature common to all traveling waves in classical physics is that energy is transmitted through a medium. The medium itself undergoes no permanent displacement; it merely undergoes local oscillations as the disturbance passes through. One of the most important kinds of wave in physics is the harmonic wave, for which the wave proﬁle is a sinusoidal function. A harmonic wave, at a particular instant in time, is sketched in Fig. 12. The maximum displacement of the wave from the rest position is the amplitude of the wave, and the wavelength λ is the distance required to enclose one complete oscillation. Such a wave would result from a harmonic1 oscillation at one end of a taut string. Analogous waves would be produced on the surface of a quiet pool by a vibrating bob, or in air by a vibrating tuning fork. At the instant depicted in Fig. 12, the proﬁle is described by the function ψ(x) = A sin(2π x/λ) (11) (ψ = 0 when x = 0, and the argument of the sine function goes from 0 to 2π, encompassing one complete oscillation as x goes from 0 to λ.) Let us suppose that the situation in Fig. 12 pertains at the time t = 0, and let the velocity of the disturbance through the medium be c. Then, after time t, the distance traveled is ct, the proﬁle is shifted to the right by ct and is now given by (x, t) = A sin[(2π/λ)(x − ct)] Figure 12 (12) A harmonic wave at a particular instant in time. A is the amplitude and λ is the wavelength. 1A harmonic oscillation is one whose equation of motion has a sine or cosine dependence on time. 3 Section 12 Waves A capital is used to distinguish the timedependent function (12) from the timeindependent function (11). The frequency ν of a wave is the number of individual repeating wave units passing a point per unit time. For our harmonic wave, this is the distance traveled in unit time c divided by the length of a wave unit λ. Hence, ν = c/λ (13) Note that the wave described by the formula (x, t) = A sin[(2π/λ)(x − ct) + ] (14) is similar to of Eq. (12) except for being displaced. If we compare the two waves at the same instant in time, we ﬁnd to be shifted to the left of by λ/2π. If = π, 3π, . . . , then is shifted by λ/2, 3λ/2, . . . and the two functions are said to be exactly out of phase. If = 2π, 4π, . . . , the shift is by λ, 2λ, . . . , and the two waves are exactly in phase. is the phase factor for relative to . Alternatively, we can compare the two waves at the same point in x, in which case the phase factor causes the two waves to be displaced from each other in time. 12.B Standing Waves In problems of physical interest, the medium is usually subject to constraints. For example, a string will have ends, and these may be clamped, as in a violin, so that they cannot oscillate when the disturbance reaches them. Under such circumstances, the energy pulse is unable to progress further. It cannot be absorbed by the clamping mechanism if it is perfectly rigid, and it has no choice but to travel back along the string in the opposite direction. The reﬂected wave is now moving into the face of the primary wave, and the motion of the string is in response to the demands placed on it by the two simultaneous waves: (x, t) = primary (x, t) + reﬂected (x, t) (15) When the primary and reﬂected waves have the same amplitude and speed, we can write (x, t) = A sin [(2π/λ)(x − ct)] + A sin [(2π/λ)(x + ct)] = 2A sin(2π x/λ) cos(2π ct/λ) (16) This formula describes a standing wave—a wave that does not appear to travel through the medium, but appears to vibrate “in place.” The ﬁrst part of the function depends only on the x variable. Wherever the sine function vanishes, will vanish, regardless of the value of t. This means that there are places where the medium does not ever vibrate. Such places are called nodes. Between the nodes, sin(2π x/λ) is ﬁnite. As time passes, the cosine function oscillates between plus and minus unity. This means that oscillates between plus and minus the value of sin(2π x/λ). We say that the xdependent part of the function gives the maximum displacement of the standing wave, and the tdependent part governs the motion of the medium back and forth between these extremes of maximum displacement. A standing wave with a central node is shown in Fig. 13. 4 Chapter 1 Classical Waves and the TimeIndependent Schrödinger Wave Equation Figure 13 A standing wave in a string clamped at x = 0 and x = L. The wavelength λ is equal to L. Equation (16) is often written as (x, t) = ψ(x) cos(ωt) (17) ω = 2π c/λ (18) where The proﬁle ψ(x) is often called the amplitude function and ω is the frequency factor. Let us consider how the energy is stored in the vibrating string depicted in Fig. 13. The string segments at the central node and at the clamped endpoints of the string do not move. Hence, their kinetic energies are zero at all times. Furthermore, since they are never displaced from their equilibrium positions, their potential energies are likewise always zero. Therefore, the total energy stored at these segments is always zero as long as the string continues to vibrate in the mode shown. The maximum kinetic and potential energies are associated with those segments located at the wave peaks and valleys (called the antinodes) because these segments have the greatest average velocity and displacement from the equilibrium position. A more detailed mathematical treatment would show that the total energy of any string segment is proportional to ψ(x)2 (Problem 17). 13 The Classical Wave Equation It is one thing to draw a picture of a wave and describe its properties, and quite another to predict what sort of wave will result from disturbing a particular system. To make such predictions, we must consider the physical laws that the medium must obey. One condition is that the medium must obey Newton’s laws of motion. For example, any segment of string of mass m subjected to a force F must undergo an acceleration of F /m in accord with Newton’s second law. In this regard, wave motion is perfectly consistent with ordinary particle motion. Another condition, however, peculiar to waves, is that each segment of the medium is “attached” to the neighboring segments so that, as it is displaced, it drags along its neighbor, which in turn drags along its neighbor, 5 Section 13 The Classical Wave Equation Figure 14 A segment of string under tension T . The forces at each end of the segment are decomposed into forces perpendicular and parallel to x. etc. This provides the mechanism whereby the disturbance is propagated along the medium.2 Let us consider a string under a tensile force T . When the string is displaced from its equilibrium position, this tension is responsible for exerting a restoring force. For example, observe the string segment associated with the region x to x + dx in Fig. 14. Note that the tension exerted at either end of this segment can be decomposed into components parallel and perpendicular to the x axis. The parallel component tends to stretch the string (which, however, we assume to be unstretchable), the perpendicular component acts to accelerate the segment toward or away from the rest position. At the right end of the segment, the perpendicular component F divided by the horizontal component gives the slope of T . However, for small deviations of the string from equilibrium (that is, for small angle α) the horizontal component is nearly equal in length to the vector T . This means that it is a good approximation to write slope of vector T = F /T at x + dx (19) But the slope is also given by the derivative of , and so we can write Fx+dx = T (∂/∂x)x+dx (110) At the other end of the segment the tensile force acts in the opposite direction, and we have Fx = −T (∂/∂x)x (111) The net perpendicular force on our string segment is the resultant of these two: (112) F = T (∂/∂x)x+dx − (∂/∂x)x The difference in slope at two inﬁnitesimally separated points, divided by dx, is by deﬁnition the second derivative of a function. Therefore, F = T ∂ 2 /∂x 2 dx (113) 2 Fluids are of relatively low viscosity, so the tendency of one segment to drag along its neighbor is weak. For this reason ﬂuids are poor transmitters of transverse waves (waves in which the medium oscillates in a direction perpendicular to the direction of propagation). In compression waves, one segment displaces the next by pushing it. Here the requirement is that the medium possess elasticity for compression. Solids and ﬂuids often meet this requirement well enough to transmit compression waves. The ability of rigid solids to transmit both wave types while ﬂuids transmit only one type is the basis for using earthquakeinduced waves to determine how deep the solid part of the earth’s mantle extends. 6 Chapter 1 Classical Waves and the TimeIndependent Schrödinger Wave Equation Equation (113) gives the force on our string segment. If the string has mass m per unit length, then the segment has mass m dx, and Newton’s equation F = ma may be written T ∂ 2 /∂x 2 = m ∂ 2 /∂t 2 (114) where we recall that acceleration is the second derivative of position with respect to time. Equation (114) is the wave equation for motion in a string of uniform density under tension T . It should be evident that its derivation involves nothing fundamental beyond Newton’s second law and the fact that the two ends of the segment are linked to each other and to a common tensile force. Generalizing this equation to waves in threedimensional media gives 2 ∂2 ∂2 ∂ 2 (x, y, z, t) ∂ + + (115) y, z, t) = β (x, ∂x 2 ∂y 2 ∂z2 ∂t 2 where β is a composite of physical quantities (analogous to m/T ) for the particular system. Returning to our string example, we have in Eq. (114) a timedependent differential equation. Suppose we wish to limit our consideration to standing waves that can be separated into a spacedependent amplitude function and a harmonic timedependent function. Then (x, t) = ψ(x) cos(ωt) (116) and the differential equation becomes cos(ωt) d 2 cos(ωt) m d 2 ψ (x) m = ψ(x) = − ψ(x)ω2 cos(ωt) dx 2 T dt 2 T (117) or, dividing by cos(ωt), d 2 ψ(x)/dx 2 = −(ω2 m/T )ψ(x) (118) This is the classical timeindependent wave equation for a string. We can see by inspection what kind of function ψ(x) must be to satisfy Eq. (118). ψ is a function that, when twice differentiated, is reproduced with a coefﬁcient of −ω2 m/T . One solution is (119) ψ = A sin ω m/T x This illustrates that Eq. (118) has sinusoidally varying solutions such as those discussed √ in Section 12. Comparing Eq. (119) with (11) indicates that 2π/λ = ω m/T . Substituting this relation into Eq. (118) gives d 2 ψ(x)/dx 2 = −(2π/λ)2 ψ(x) (120) which is a more useful form for our purposes. For threedimensional systems, the classical timeindependent wave equation for an isotropic and uniform medium is (∂ 2 /∂x 2 + ∂ 2 /∂y 2 + ∂ 2 /∂z2 )ψ(x, y, z) = −(2π/λ)2 ψ(x, y, z) (121) Section 14 Standing Waves in a Clamped String 7 where λ depends on the elasticity of the medium. The combination of partial derivatives on the lefthand side of Eq. (121) is called the Laplacian, and is often given the shorthand symbol ∇ 2 (del squared). This would give for Eq. (121) ∇ 2 ψ(x, y, z) = −(2π/λ)2 ψ(x, y, z) (122) 14 Standing Waves in a Clamped String We now demonstrate how Eq. (120) can be used to predict the nature of standing waves in a string. Suppose that the string is clamped at x = 0 and L. This means that the string cannot oscillate at these points. Mathematically this means that ψ(0) = ψ(L) = 0 (123) Conditions such as these are called boundary conditions. Our question is, “What functions ψ satisfy Eq. (120) and also Eq. (123)?” We begin by trying to ﬁnd the most general equation that can satisfy Eq. (120). We have already seen that A sin(2π x/λ) is a solution, but it is easy to show that A cos(2π x/λ) is also a solution. More general than either of these is the linear combination3 ψ(x) = A sin(2π x/λ) + B cos(2π x/λ) (124) By varying A and B, we can get different functions ψ. There are two remarks to be made at this point. First, some readers will have noticed that other functions exist that √ satisfy Eq. (120). These are A exp(2π ix/λ) and A exp(−2πix/λ), where i = −1. The reason we have not included these in the general function (124) is that these two exponential functions are mathematically equivalent to the trigonometric functions. The relationship is exp(±ikx) = cos(kx) ± i sin(kx). (125) This means that any trigonometric function may be expressed in terms of such exponentials and vice versa. Hence, the set of trigonometric functions and the set of exponentials is redundant, and no additional ﬂexibility would result by including exponentials in Eq. (124) (see Problem 11). The two sets of functions are linearly dependent.4 The second remark is that for a given A and B the function described by Eq. (124) is a single sinusoidal wave with wavelength λ. By altering the ratio of A to B, we cause the wave to shift to the left or right with respect to the origin. If A = 1 and B = 0, the wave has a node at x = 0. If A = 0 and B = 1, the wave has an antinode at x = 0. We now proceed by letting the boundary conditions determine the constants A and B. The condition at x = 0 gives ψ(0) = A sin(0) + B cos(0) = 0 (126) 3 Given functions f , f , f . . . . A linear combination of these functions is c f + c f + c f + · · · , where 1 2 3 1 1 2 2 3 3 c1 , c2 , c3 , . . . are numbers (which need not be real). 4 If one member of a set of functions (f , f , f , . . . ) can be expressed as a linear combination of the remaining 1 2 3 functions (i.e., if f1 = c2 f2 + c3 f3 + · · · ), the set of functions is said to be linearly dependent. Otherwise, they are linearly independent. 8 Chapter 1 Classical Waves and the TimeIndependent Schrödinger Wave Equation However, since sin(0) = 0 and cos(0) = 1, this gives B =0 (127) Therefore, our ﬁrst boundary condition forces B to be zero and leaves us with ψ(x) = A sin(2π x/λ) (128) Our second boundary condition, at x = L, gives ψ(L) = A sin(2π L/λ) = 0 (129) One solution is provided by setting A equal to zero. This gives ψ = 0, which corresponds to no wave at all in the string. This is possible, but not very interesting. The other possibility is for 2πL/λ to be equal to 0, ±π, ±2π, . . . , ±nπ, . . . since the sine function vanishes then. This gives the relation 2πL/λ = nπ, n = 0, ±1, ±2, . . . (130) or λ = 2L/n, n = 0, ±1, ±2, . . . (131) Substituting this expression for λ into Eq. (128) gives ψ(x) = A sin(nπ x/L), n = 0, ±1, ±2, . . . (132) Some of these solutions are sketched in Fig. 15. The solution for n = 0 is again the uninteresting ψ = 0 case. Furthermore, since sin(−x) equals −sin(x), it is clear that the set of functions produced by positive integers n is not physically different from the set produced by negative n, so we may arbitrarily restrict our attention to solutions with positive n. (The two sets are linearly dependent.) The constant A is still undetermined. It affects the amplitude of the wave. To determine A would require knowing how much energy is stored in the wave, that is, how hard the string was plucked. It is evident that there are an inﬁnite number of acceptable solutions, each one corresponding to a different number of halfwaves ﬁtting between 0 and L. But an even larger inﬁnity of waves has been excluded by the boundary conditions—namely, all waves having wavelengths not divisible into 2L an integral number of times. The result Figure 15 Solutions for the timeindependent wave equation in one dimension with boundary conditions ψ(0) = ψ(L) = 0. Section 15 Light as an Electromagnetic Wave 9 of applying boundary conditions has been to restrict the allowed wavelengths to certain discrete values. As we shall see, this behavior is closely related to the quantization of energies in quantum mechanics. The example worked out above is an extremely simple one. Nevertheless, it demonstrates how a differential equation and boundary conditions are used to deﬁne the allowed states for a system. One could have arrived at solutions for this case by simple physical argument, but this is usually not possible in more complicated cases. The differential equation provides a systematic approach for ﬁnding solutions when physical intuition is not enough. 15 Light as an Electromagnetic Wave Suppose a charged particle is caused to oscillate harmonically on the z axis. If there is another charged particle some distance away and initially at rest in the xy plane, this second particle will commence oscillating harmonically too. Thus, energy is being transferred from the ﬁrst particle to the second, which indicates that there is an oscillating electric ﬁeld emanating from the ﬁrst particle. We can plot the magnitude of this electric ﬁeld at a given instant as it would be felt by a series of imaginary test charges stationed along a line emanating from the source and perpendicular to the axis of vibration (Fig. 16). If there are some magnetic compasses in the neighborhood of the oscillating charge, these will be found to swing back and forth in response to the disturbance. This means that an oscillating magnetic ﬁeld is produced by the charge too. Varying the placement of the compasses will show that this ﬁeld oscillates in a plane perpendicular to the axis of vibration of the charged particle. The combined electric and magnetic ﬁelds traveling along one ray in the xy plane appear in Fig. 17. The changes in electric and magnetic ﬁelds propagate outward with a characteristic velocity c, and are describable as a traveling wave, called an electromagnetic wave. Its frequency ν is the same as the oscillation frequency of the vibrating charge. Its wavelength is λ = c/ν. Visible light, infrared radiation, radio waves, microwaves, ultraviolet radiation, X rays, and γ rays are all forms of electromagnetic radiation, their only difference being their frequencies ν. We shall continue the discussion in the context of light, understanding that it applies to all forms of electromagnetic radiation. Figure 16 A harmonic electricﬁeld wave emanating from a vibrating electric charge. The wave magnitude is proportional to the force felt by the test charges. The charges are only imaginary; if they actually existed, they would possess mass and under acceleration would absorb energy from the wave, causing it to attenuate. 10 Chapter 1 Classical Waves and the TimeIndependent Schrödinger Wave Equation Figure 17 A harmonic electromagnetic ﬁeld produced by an oscillating electric charge. The arrows without attached charges show the direction in which the north pole of a magnet would be attracted. The magnetic ﬁeld is oriented perpendicular to the electric ﬁeld. If a beam of light is produced so that the orientation of the electric ﬁeld wave is always in the same plane, the light is said to be plane (or linearly) polarized. The planepolarized light shown in Fig. 17 is said to be z polarized. If the plane of orientation of the electric ﬁeld wave rotates clockwise or counterclockwise about the axis of travel (i.e., if the electric ﬁeld wave “corkscrews” through space), the light is said to be right or left circularly polarized. If the light is a composite of waves having random ﬁeld orientations so that there is no resultant orientation, the light is unpolarized. Experiments with light in the nineteenth century and earlier were consistent with the view that light is a wave phenomenon. One of the more obvious experimental veriﬁcations of this is provided by the interference pattern produced when light from a point source is allowed to pass through a pair of slits and then to fall on a screen. The resulting interference patterns are understandable only in terms of the constructive and destructive interference of waves. The differential equations of Maxwell, which provided the connection between electromagnetic radiation and the basic laws of physics, also indicated that light is a wave. But there remained several problems that prevented physicists from closing the book on this subject. One was the inability of classical physical theory to explain the intensity and wavelength characteristics of light emitted by a glowing “blackbody.” This problem was studied by Planck, who was forced to conclude that the vibrating charged particles producing the light can exist only in certain discrete (separated) energy states. We shall not discuss this problem. Another problem had to do with the interpretation of a phenomenon discovered in the late 1800s, called the photoelectric effect. 16 The Photoelectric Effect This phenomenon occurs when the exposure of some material to light causes it to eject electrons. Many metals do this quite readily. A simple apparatus that could be used to study this behavior is drawn schematically in Fig. 18. Incident light strikes the metal dish in the evacuated chamber. If electrons are ejected, some of them will strike the collecting wire, giving rise to a deﬂection of the galvanometer. In this apparatus, one can vary the potential difference between the metal dish and the collecting wire, and also the intensity and frequency of the incident light. Suppose that the potential difference is set at zero and a current is detected when light of a certain intensity and frequency strikes the dish. This means that electrons Section 16 The Photoelectric Effect Figure 18 11 A phototube. are being emitted from the dish with ﬁnite kinetic energy, enabling them to travel to the wire. If a retarding potential is now applied, electrons that are emitted with only a small kinetic energy will have insufﬁcient energy to overcome the retarding potential and will not travel to the wire. Hence, the current being detected will decrease. The retarding potential can be increased gradually until ﬁnally even the most energetic photoelectrons cannot make it to the collecting wire. This enables one to calculate the maximum kinetic energy for photoelectrons produced by the incident light on the metal in question. The observations from experiments of this sort can be summarized as follows: 1. Below a certain cutoff frequency of incident light, no photoelectrons are ejected, no matter how intense the light. 2. Above the cutoff frequency, the number of photoelectrons is directly proportional to the intensity of the light. 3. As the frequency of the incident light is increased, the maximum kinetic energy of the photoelectrons increases. 4. In cases where the radiation intensity is extremely low (but frequency is above the cutoff value) photoelectrons are emitted from the metal without any time lag. Some of these results are summarized graphically in Fig. 19. Apparently, the kinetic energy of the photoelectron is given by kinetic energy = h(ν − ν0 ) (133) where h is a constant. The cutoff frequency ν0 depends on the metal being studied (and also its temperature), but the slope h is the same for all substances. We can also write the kinetic energy as kinetic energy = energy of light − energy needed to escape surface (134) 12 Chapter 1 Classical Waves and the TimeIndependent Schrödinger Wave Equation Figure 19 Maximum kinetic energy of photoelectrons as a function of incident light frequency, where ν0 is the minimum frequency for which photoelectrons are ejected from the metal in the absence of any retarding or accelerating potential. The last quantity in Eq. (134) is often referred to as the work function W of the metal. Equating Eq. (133) with (134) gives energy of light − W = hν − hν0 (135) The materialdependent term W is identiﬁed with the materialdependent term hν0 , yielding energy of light ≡ E = hν (136) where the value of h has been determined to be 6.626176 × 10−34 J sec. (See Appendix 10 for units and conversion factors.) Physicists found it difﬁcult to reconcile these observations with the classical electromagnetic ﬁeld theory of light. For example, if light of a certain frequency and intensity causes emission of electrons having a certain maximum kinetic energy, one would expect increased light intensity (corresponding classically to a greater electromagnetic ﬁeld amplitude and hence greater energy density) to produce photoelectrons of higher kinetic energy. However, it only produces more photoelectrons and does not affect their energies. Again, if light is a wave, the energy is distributed over the entire wavefront and this means that a low light intensity would impart energy at a very low rate to an area of surface occupied by one atom. One can calculate that it would take years for an individual atom to collect sufﬁcient energy to eject an electron under such conditions. No such induction period is observed. An explanation for these results was suggested in 1905 by Einstein, who proposed that the incident light be viewed as being comprised of discrete units of energy. Each such unit, or photon, would have an associated energy of hν,where ν is the frequency of the oscillating emitter. Increasing the intensity of the light would correspond to increasing the number of photons, whereas increasing the frequency of the light would increase the energy of the photons. If we envision each emitted photoelectron as resulting from a photon striking the surface of the metal, it is quite easy to see that Einstein’s proposal accords with observation. But it creates a new problem: If we are to visualize light as a stream of photons, how can we explain the wave properties of light, such as the doubleslit diffraction pattern? What is the physical meaning of the electromagnetic wave? Section 16 The Photoelectric Effect 13 Essentially, the problem is that, in the classical view, the square of the electromagnetic wave at any point in space is a measure of the energy density at that point. Now the square of the electromagnetic wave is a continuous and smoothly varying function, and if energy is continuous and inﬁnitely divisible, there is no problem with this theory. But if the energy cannot be divided into amounts smaller than a photon—if it has a particulate rather than a continuous nature—then the classical interpretation cannot apply, for it is not possible to produce a smoothly varying energy distribution from energy particles any more than it is possible to produce, at the microscopic level, a smooth density distribution in gas made from atoms of matter. Einstein suggested that the square of the electromagnetic wave at some point (that is, the sum of the squares of the electric and magnetic ﬁeld magnitudes) be taken as the probability density for ﬁnding a photon in the volume element around that point. The greater the square of the wave in some region, the greater is the probability for ﬁnding the photon in that region. Thus, the classical notion of energy having a deﬁnite and smoothly varying distribution is replaced by the idea of a smoothly varying probability density for ﬁnding an atomistic packet of energy. Let us explore this probabilistic interpretation within the context of the twoslit interference experiment. We know that the pattern of light and darkness observed on the screen agrees with the classical picture of interference of waves. Suppose we carry out the experiment in the usual way, except we use a light source (of frequency ν) so weak that only hν units of energy per second pass through the apparatus and strike the screen. According to the classical picture, this tiny amount of energy should strike the screen in a delocalized manner, producing an extremely faint image of the entire diffraction pattern. Over a period of many seconds, this pattern could be accumulated (on a photographic plate, say) and would become more intense. According to Einstein’s view, our experiment corresponds to transmission of one photon per second and each photon strikes the screen at a localized point. Each photon strikes a new spot (not to imply the same spot cannot be struck more than once) and, over a long period of time, they build up the observed diffraction pattern. If we wish to state in advance where the next photon will appear, we are unable to do so. The best we can do is to say that the next photon is more likely to strike in one area than in another, the relative probabilities being quantitatively described by the square of the electromagnetic wave. The interpretation of electromagnetic waves as probability waves often leaves one with some feelings of unreality. If the wave only tells us relative probabilities for ﬁnding a photon at one point or another, one is entitled to ask whether the wave has “physical reality,” or if it is merely a mathematical device which allows us to analyze photon distribution, the photons being the “physical reality.” We will defer discussion of this question until a later section on electron diffraction. EXAMPLE 11 A retarding potential of 2.38 volts just sufﬁces to stop photoelectrons emitted from potassium by light of frequency 1.13 × 1015 s−1 . What is the work function, W , of potassium? SOLUTION Elight = hν = W + KEelectron , W = hν − KEelectron = (4.136 × 10−15 eV s) (1.13 × 1015 s−1 ) − 2.38 eV = 4.67 eV − 2.38 eV = 2.29 eV [Note convenience of using h in units of eV s for this problem. See Appendix 10 for data.] 14 Chapter 1 Classical Waves and the TimeIndependent Schrödinger Wave Equation EXAMPLE 12 Spectroscopists often express E for a transition between states in wavenumbers , e.g., m−1 , or cm−1 , rather than in energy units like J or eV. (Usually cm−1 is favored, so we will proceed with that choice.) a) What is the physical meaning of the term wavenumber? b) What is the connection between wavenumber and energy? c) What wavenumber applies to an energy of 1.000 J? of 1.000 eV? SOLUTION a) Wavenumber is the number of waves that ﬁt into a unit of distance (usually of one centimeter). It is sometimes symbolized ν̃. ν̃ = 1/λ, where λ is the wavelength in centimeters. b) Wavenumber characterizes the light that has photons of the designated energy. E = hν = hc/λ = hcν̃. (where c is given in cm/s). c) E = 1.000 J = hcν̃; ν̃ = 1.000 J/ hc = 1.000 J /[(6.626 × 10−34 J s)(2.998 × 1010 cm/s)] = 5.034 × 1022 cm−1 . Clearly, this is light of an extremely short wavelength since more than 1022 wavelengths ﬁt into 1 cm. For 1.000 eV, the above equation is repeated using h in eV s. This gives ν̃ = 8065 cm−1 . 17 The Wave Nature of Matter Evidently light has wave and particle aspects, and we can describe it in terms of photons, which are associated with waves of frequency ν = E/ h. Now photons are rather peculiar particles in that they have zero rest mass. In fact, they can exist only when traveling at the speed of light. The more normal particles in our experience have nonzero rest masses and can exist at any velocity up to the speedoflight limit. Are there also waves associated with such normal particles? Imagine a particle having a ﬁnite rest mass that somehow can be made lighter and lighter, approaching zero in a continuous way. It seems reasonable that the existence of a wave associated with the motion of the particle should become more and more apparent, rather than the wave coming into existence abruptly when m = 0. De Broglie proposed that all material particles are associated with waves, which he called “matter waves,” but that the existence of these waves is likely to be observable only in the behaviors of extremely light particles. De Broglie’s relation can be reached as follows. Einstein’s relation for photons is E = hν (137) But a photon carrying energy E has a relativistic mass given by E = mc2 (138) E = mc2 = hν = hc/λ (139) mc = h/λ (140) Equating these two equations gives or 15 Section 17 The Wave Nature of Matter A normal particle, with nonzero rest mass, travels at a velocity v. If we regard Eq. (140) as merely the highvelocity limit of a more general expression, we arrive at an equation relating particle momentum p and associated wavelength λ: mv = p = h/λ (141) λ = h/p (142) or Here, m refers to the rest mass of the particle plus the relativistic correction, but the latter is usually negligible in comparison to the former. This relation, proposed by de Broglie in 1922, was demonstrated to be correct shortly thereafter when Davisson and Germer showed that a beam of electrons impinging on a nickel target produced the scattering patterns one expects from interfering waves. These “electron waves” were observed to have wavelengths related to electron momentum in just the manner proposed by de Broglie. Equation (142) relates the de Broglie wavelength λ of a matter wave to the momentum p of the particle. A higher momentum corresponds to a shorter wavelength. Since kinetic energy T = mv 2 = (1/2m)(m2 v 2 ) = p2 /2m (143) it follows that p= √ 2mT (144) Furthermore, Since E = T + V , where E is the total energy and V is the potential energy, we can rewrite the de Broglie wavelength as h λ= √ 2m(E − V ) (145) Equation (145) is useful for understanding the way in which λ will change for a particle moving with constant total energy in a varying potential. For example, if the particle enters a region where its potential energy increases (e.g., an electron approaches a negatively charged plate), E − V decreases and λ increases (i.e., the particle slows down, so its momentum decreases and its associated wavelength increases). We shall see examples of this behavior in future chapters. Observe that if E ≥ V , λ as given by Eq. (145) is real. However, if E < V , λ becomes imaginary. Classically, we never encounter such a situation, but we will ﬁnd it is necessary to consider this possibility in quantum mechanics. EXAMPLE 13 A He2+ ion is accelerated from rest through a voltage drop of 1.000 kilovolts. What is its ﬁnal deBroglie wavelength? Would the wavelike properties be very apparent? SOLUTION Since a charge of two electronic units has passed through a voltage drop of 1.000 × 103 volts, the ﬁnal kinetic energy of the ion is 2.000 × 103 eV. To calculate λ, we ﬁrst 16 Chapter 1 Classical Waves and the TimeIndependent Schrödinger Wave Equation convert from eV to joules: KE ≡ p2 /2m = (2.000 × 103 eV)(1.60219 × 10−19 J/eV) = 3.204 −16 J. m −3 kg/g)(1 mol/6.022 × 1023 atoms) = 6.65 × 10−27 kg; × 10√ H e = (4.003 g/mol)(10 −27 p = 2mH e · KE = [2(6.65 × 10 kg)(3.204 × 10−16 J)]1/2 = 2.1 × 10−21 kg m/s. λ = h/p = (6.626 × 10−34 Js)/(2.1 × 10−21 kg m/s) = 3.2 × 10−13 m = 0.32 pm. This wavelength is on the order of 1% of the radius of a hydrogen atom–too short to produce observable interference results when interacting with atomsize scatterers. For most purposes, we can treat this ion as simply a highspeed particle. 18 A Diffraction Experiment with Electrons In order to gain a better understanding of the meaning of matter waves, we now consider a set of simple experiments. Suppose that we have a source of a beam of monoenergetic electrons and a pair of slits, as indicated schematically in Fig. 110. Any electron arriving at the phosphorescent screen produces a ﬂash of light, just as in a television set. For the moment we ignore the light source near the slits (assume that it is turned off) and inquire as to the nature of the image on the phosphorescent screen when the electron beam is directed at the slits. The observation, consistent with the observations of Davisson and Germer already mentioned, is that there are alternating bands of light and dark, indicating that the electron beam is being diffracted by the slits. Furthermore, the distance separating the bands is consistent with the de Broglie wavelength corresponding to the energy of the electrons. The variation in light intensity observed on the screen is depicted in Fig. 111a. Evidently, the electrons in this experiment are displaying wave behavior. Does this mean that the electrons are spread out like waves when they are detected at the screen? We test this by reducing our beam intensity to let only one electron per second through the apparatus and observe that each electron gives a localized pinpoint of light, the entire diffraction pattern building up gradually by the accumulation of many points. Thus, the square of de Broglie’s matter wave has the same kind of statistical signiﬁcance that Einstein proposed for electromagnetic waves and photons, and the electrons really are localized particles, at least when they are detected at the screen. However, if they are really particles, it is hard to see how they can be diffracted. Consider what happens when slit b is closed. Then all the electrons striking the screen must have come through slit a. We observe the result to be a single area of light on the screen (Fig. 111b). Closing slit a and opening b gives a similar (but displaced) Figure 110 The electron source produces a beam of electrons, some of which pass through slits a and/or b to be detected as ﬂashes of light on the phosphorescent screen. Section 18 A Diffraction Experiment with Electrons 17 Figure 111 Light intensity at phosphorescent screen under various conditions: (a) a and b open, light off; (b) a open, b closed, light off; (c) a closed, b open, light off; (d) a and b open, light on, λ short; (e) a and b open, light on, λ longer. light area, as shown in Fig. 111c. These patterns are just what we would expect for particles. Now, with both slits open, we expect half the particles to pass through slit a and half through slit b, the resulting pattern being the sum of the results just described. Instead we obtain the diffraction pattern (Fig. 111a). How can this happen? It seems that, somehow, an electron passing through the apparatus can sense whether one or both slits are open, even though as a particle it can explore only one slit or the other. One might suppose that we are seeing the result of simultaneous traversal of the two slits by two electrons, the path of each electron being affected by the presence of an electron in the other slit. This would explain how an electron passing through slit a would “know” whether slit b was open or closed. But the fact that the pattern builds up even when electrons pass through at the rate of one per second indicates that this argument will not do. Could an electron be coming through both slits at once? To test this question, we need to have detailed information about the positions of the electrons as they pass through the slits. We can get such data by turning on the light source and aiming a microscope at the slits. Then photons will bounce off each electron as it passes the slits and will be observed through the microscope. The observer thus can tell through which slit each electron has passed, and also record its ﬁnal position on the phosphorescent screen. In this experiment, it is necessary to use light having a wavelength short in comparison to the interslit distance; otherwise the microscope cannot resolve a ﬂash well enough to tell which slit it is nearest. When this experiment is performed, we indeed detect each electron as coming through one slit or the other, and not both, but we also ﬁnd that the diffraction pattern on the screen has been lost and that we have the broad, featureless distribution shown in Fig. 111d, which is basically the sum of the singleslit experiments. What has happened is that the photons from our light source, in bouncing off the electrons as they emerge from the slits, have affected the momenta of the electrons and changed their paths from what they were in the absence of light. We can try to counteract this by using photons with lower momentum; but this means using photons of lower E, hence longer λ. As a result, the images of the electrons in the microscope get broader, and it becomes more and more ambiguous as to which slit a given electron has passed through or that it really passed through only one slit. As we become more and more uncertain about the path 18 Chapter 1 Classical Waves and the TimeIndependent Schrödinger Wave Equation of each electron as it moves past the slits, the accumulating diffraction pattern becomes more and more pronounced (Fig. 111e). (Since this is a “thought experiment,” we can ignore the inconvenient fact that our “light” source must produce X rays or γ rays in order to have a wavelength short in comparison to the appropriate interslit distance.) This conceptual experiment illustrates a basic feature of microscopic systems—we cannot measure properties of the system without affecting the future development of the system in a nontrivial way. The system with the light turned off is signiﬁcantly different from the system with the light turned on (with short λ), and so the electrons arrive at the screen with different distributions. No matter how cleverly one devises the experiment, there is some minimum necessary disturbance involved in any measurement. In this example with the light off, the problem is that we know the momentum of each electron quite accurately (since the beam is monoenergetic and collimated), but we do not know anything about the way the electrons traverse the slits. With the light on, we obtain information about electron position just beyond the slits but we change the momentum of each electron in an unknown way. The measurement of particle position leads to loss of knowledge about particle momentum. This is an example of the uncertainty principle of Heisenberg, who stated that the product of the simultaneous uncertainties in “conjugate variables,” a and b, can never be smaller than the value of Planck’s constant h divided by 4π : a · b ≥ h/4π (146) Here, a is a measure of the uncertainty in the variable a, etc. (The easiest way to recognize conjugate variables is to note that their dimensions must multiply to joule seconds. Linear momentum and linear position satisﬁes this requirement. Two other important pairs of conjugate variables are energy–time and angular momentum–angular position.) In this example with the light off, our uncertainty in momentum is small and our uncertainty in position is unacceptably large, since we cannot say which slit each electron traverses. With the light on, we reduce our uncertainty in position to an acceptable size, but subsequent to the position of each electron being observed, we have much greater uncertainty in momentum. Thus, we see that the appearance of an electron (or a photon) as a particle or a wave depends on our experiment. Because any observation on so small a particle involves a signiﬁcant perturbation of its state, it is proper to think of the electron plus apparatus as a single system. The question, “Is the electron a particle or a wave?” becomes meaningful only when the apparatus is deﬁned on which we plan a measurement. In some experiments, the apparatus and electrons interact in a way suggestive of the electron being a wave, in others, a particle. The question, “What is the electron when were not looking?,” cannot be answered experimentally, since an experiment is a “look” at the electron. In recent years experiments of this sort have been carried out using single atoms.5 EXAMPLE 14 The lifetime of an excited state of a molecule is 2 × 10−9 s. What is the uncertainty in its energy in J? In cm−1 ? How would this manifest itself experimentally? 5 See F. Flam [1]. Section 19 Schrödinger’s TimeIndependent Wave Equation 19 SOLUTION The Heisenberg uncertainty principle gives, for minimum uncertainty E · t = h/4π. E = (6.626 × 10−34 J s)/[(4π )(2 × 10−9 s)] = 2.6 × 10−26 J (2.6 × 10−26 J) (5.03 × 1022 cm−1 J−1 ) = 0.001 cm−1 (See Appendix 10 for data.) Larger uncertainty in E shows up as greater linewidth in emission spectra. 19 Schrödinger’s TimeIndependent Wave Equation Earlier we saw that we needed a wave equation in order to solve for the standing waves pertaining to a particular classical system and its set of boundary conditions. The same need exists for a wave equation to solve for matter waves. Schrödinger obtained such an equation by taking the classical timeindependent wave equation and substituting de Broglie’s relation for λ. Thus, if ∇ 2 ψ = −(2π/λ)2 ψ (147) h λ= √ 2m(E − V ) (148) −(h2 /8π 2 m)∇ 2 + V (x, y, z) ψ(x, y, z) = Eψ(x, y, z) (149) and then Equation (149) is Schrödinger’s timeindependent wave equation for a single particle of mass m moving in the threedimensional potential ﬁeld V . In classical mechanics we have separate equations for wave motion and particle motion, whereas in quantum mechanics, in which the distinction between particles and waves is not clearcut, we have a single equation—the Schrödinger equation. We have seen that the link between the Schrödinger equation and the classical wave equation is the de Broglie relation. Let us now compare Schrödinger’s equation with the classical equation for particle motion. Classically, for a particle moving in three dimensions, the total energy is the sum of kinetic and potential energies: (1/2m)(px2 + py2 + pz2 ) + V = E (150) where px is the momentum in the x coordinate, etc. We have just seen that the analogous Schrödinger equation is [writing out Eq. (149)] 2 ∂ ∂2 ∂2 −h2 + + + V (x, y, z) ψ(x, y, z) = Eψ(x, y, z) (151) 8π 2 m ∂x 2 ∂y 2 ∂z2 It is easily seen that Eq. (150) is linked to the quantity in brackets of Eq. (151) by a relation associating classical momentum with a partial differential operator: px ←→ (h/2π i)(∂/∂x) (152) and similarly for py and pz . The relations (152) will be seen later to be an important postulate in a formal development of quantum mechanics. 20 Chapter 1 Classical Waves and the TimeIndependent Schrödinger Wave Equation The lefthand side of Eq. (150) is called the hamiltonian for the system. For this reason the operator in square brackets on the LHS of Eq. (151) is called the hamiltonian operator6 H . For a given system, we shall see that the construction of H is not difﬁcult. The difﬁculty comes in solving Schrödinger’s equation, often written as H ψ = Eψ (153) The classical and quantummechanical wave equations that we have discussed are members of a special class of equations called eigenvalue equations. Such equations have the format Op f = cf (154) where Op is an operator, f is a function, and c is a constant. Thus, eigenvalue equations have the property that operating on a function regenerates the same function times a constant. The function f that satisﬁes Eq. (154) is called an eigenfunction of the operator. The constant c is called the eigenvalue associated with the eigenfunction f . Often, an operator will have a large number of eigenfunctions and eigenvalues of interest associated with it, and so an index is necessary to keep them sorted, viz. Op fi = ci fi (155) We have already seen an example of this sort of equation, Eq. (119) being an eigenfunction for Eq. (118), with eigenvalue −ω2 m/T . The solutions ψ for Schrödinger’s equation (153), are referred to as eigenfunctions, wavefunctions, or state functions. EXAMPLE 15 a) Show that sin(3.63x) is not an eigenfunction of the operator d/dx. b) Show that exp(−3.63ix) is an eigenfunction of the operator d/dx. What is its eigenvalue? c) Show that π1 sin(3.63x) is an eigenfunction of the operator ((−h2 /8π 2 m)d 2 /dx 2 ). What is its eigenvalue? d sin(3.63x) = 3.63 cos(3.63x) = constant times sin(3.63x). SOLUTION a) dx d b) dx exp(−3.63ix) = −3.63i exp(−3.63ix) = constant times exp(−3.63ix). Eigenvalue = −3.63i. d cos(3.63x) c) ((−h2 /8π 2 m)d 2 /dx 2 ) π1 sin(3.63x) = (−h2 /8π 2 m)(1/π)(3.63) dx = [(3.63)2 h2 /8π 2 m] · (1/π) sin(3.63x) = constant times (1/π) sin(3.63x). Eigenvalue = (3.63)2 h2 /8π 2 m. 6An operator is a symbol telling us to carry out a certain mathematical operation. Thus, d/dx is a differential operator telling us to differentiate anything following it with respect to x. The function 1/x may be viewed as a multiplicative operator. Any function on which it operates gets multiplied by 1/x. 21 Section 110 Conditions on ψ 110 Conditions on ψ We have already indicated that the square of the electromagnetic wave is interpreted as the probability density function for ﬁnding photons at various places in space. We now attribute an analogous meaning to ψ 2 for matter waves. Thus, in a onedimensional problem (for example, a particle constrained to move on a line), the probability that the particle will be found in the interval dx around the point x1 is taken to be ψ 2 (x1 ) dx. If ψ is a complex function, then the absolute square, ψ2 ≡ ψ*ψ is used instead of ψ 2 .7 This makes it mathematically impossible for the average mass distribution to be negative in any region. If an eigenfunction ψ has been found for Eq. (153), it is easy to see that cψ will also be an eigenfunction, for any constant c. This is due to the fact that a multiplicative constant commutes8 with the operator H , that is, H (cψ) = cH ψ = cEψ = E(cψ) (156) The equality of the ﬁrst and last terms is a statement of the fact that cψ is an eigenfunction of H . The question of which constant to use for the wavefunction is resolved by appeal to the probability interpretation of ψ2 . For a particle moving on the x axis, the probability that the particle is between x = −∞ and x = +∞ is unity, that is, a certainty. This probability is also equal to the sum of the probabilities for ﬁnding the particle in each and every inﬁnitesimal interval along x, so this sum (an integral) must equal unity: +∞ c*c −∞ ψ*(x)ψ (x) dx = 1 (157) If the selection of the constant multiplier c is made so that Eq. (157) is satisﬁed, the wavefunction ψ = cψ is said to be normalized. For a threedimensional function, cψ(x, y, z), the normalization requirement is +∞ c*c −∞ +∞ −∞ +∞ −∞ ψ*(x, y, z)ψ(x, y, z) dx dy dz ≡ c2 ψ2 dv = 1 all space (158) As a result of our physical interpretation of ψ2 plus the fact that ψ must be an eigenfunction of the hamiltonian operator H , we can reach some general conclusions about what sort of mathematical properties ψ can or cannot have. First, we require that ψ be a singlevalued function because we want ψ2 to give an unambiguous probability for ﬁnding a particle in a given region (see Fig. 112). Also, we reject functions that are inﬁnite in any region of space because such an inﬁnity will always be inﬁnitely greater than any ﬁnite region, and ψ2 will be useless as a measure of comparative probabilities.9 In order for H ψ to be deﬁned everywhere, it is necessary that the second derivative of ψ be deﬁned everywhere. This requires that the ﬁrst derivative of ψ be piecewise continuous and that ψ itself be continuous as in Fig.1d. (We shall see an example of this shortly.) 7 If f = u + iv, then f *, the complex conjugate of f , is given by u − iv, where u and v are real functions. 8 a and b are said to commute if ab = ba. 9 There are cases, particularly in relativistic treatments, where ψ is inﬁnite at single points of zero measure, so that ψ2 dx remains ﬁnite. Normally we do not encounter such situations in quantum chemistry. 22 Chapter 1 Classical Waves and the TimeIndependent Schrödinger Wave Equation Figure 112 (a) ψ is triple valued at x0 . (b) ψ is discontinuous at x0 . (c) ψ grows without limit as x approaches +∞ (i.e., ψ “blows up,” or “explodes”). (d) ψ is continuous and has a “cusp” at x0 . Hence, ﬁrst derivative of ψ is discontinuous at x0 and is only piecewise continuous. This does not prevent ψ from being acceptable. Functions that are singlevalued, continuous, nowhere inﬁnite, and have piecewise continuous ﬁrst derivatives will be referred to as acceptable functions. The meanings of these terms are illustrated by some sample functions in Fig. 112. In most cases, there is one more general restriction we place on ψ, namely, that it be a normalizable function. This means that the integral of ψ2 over all space must not be equal to zero or inﬁnity. A function satisfying this condition is said to be squareintegrable. 111 Some Insight into the Schrödinger Equation There is a fairly simple way to view the physical meaning of the Schrödinger equation (149). The equation essentially states that E in H ψ = Eψ depends on two things, V and the second derivatives of ψ. Since V is the potential energy, the second derivatives of ψ must be related to the kinetic energy. Now the second derivative of ψ with respect to a given direction is a measure of the rate of change of slope (i.e., the curvature, or “wiggliness”) of ψ in that direction. Hence, we see that a more wiggly wavefunction leads, through the Schrödinger equation, to a higher kinetic energy. This is in accord with the spirit of de Broglie’s relation, since a shorter wavelength function is a more wiggly function. But the Schrödinger equation is more generally applicable because we can take second derivatives of any acceptable function, whereas wavelength is deﬁned Section 112 Summary 23 (a) Since V = 0, E = T . For higher T , ψ is more wiggly, which means that λ is shorter. (Since ψ is periodic for a free particle, λ is deﬁned.) (b) As V increases from left to right, ψ becomes less wiggly. (c)–(d) ψ is most wiggly where V is lowest and T is greatest. Figure 113 only for periodic functions. Since E is a constant, the solutions of the Schrödinger equation must be more wiggly in regions where V is low and less wiggly where V is high. Examples for some onedimensional cases are shown in Fig. 113. In the next chapter we use some fairly simple examples to illustrate the ideas that we have already introduced and to bring out some additional points. 112 Summary In closing this chapter, we collect and summarize the major points to be used in future discussions. 1. Associated with any particle is√a wavefunction having wavelength related to particle momentum by λ = h/p = h/ 2m(E − V ). 2. The wavefunction has the following physical meaning; its absolute square is proportional to the probability density for ﬁnding the particle. If the wavefunction is normalized, its square is equal to the probability density. 3. The wavefunctions ψ for timeindependent states are eigenfunctions of Schrödinger’s equation, which can be constructed from the classical wave equation by requir√ ing λ = h/ 2m(E − V ), or from the classical particle equation by replacing pk with (h/2πi)∂/∂k, k = x, y, z. 24 Chapter 1 Classical Waves and the TimeIndependent Schrödinger Wave Equation 4. For ψ to be acceptable, it must be singlevalued, continuous, nowhere inﬁnite, with a piecewise continuous ﬁrst derivative. For most situations, we also require ψ to be squareintegrable. 5. The wavefunction for a particle in a varying potential oscillates most rapidly where V is low, giving a high T in this region. The low V plus high T equals E. In another region, where V is high, the wavefunction oscillates more slowly, giving a low T , which, with the high V , equals the same E as in the ﬁrst region. 112.A Problems10 11. Express A cos(kx) + B sin(kx) + C exp(ikx) + D exp(−ikx) purely in terms of cos(kx) and sin(kx). 12. Repeat the standingwaveinastring problem worked out in Section 14, but clamp the string at x = +L/2 and −L/2 instead of at 0 and L. 13. Find the condition that must be satisﬁed by α and β in order that ψ (x) = A sin(αx) + B cos(βx) satisfy Eq. (120). 14. The apparatus sketched in Fig. 18 is used with a dish plated with zinc and also with a dish plated with cesium. The wavelengths of the incident light and the corresponding retarding potentials needed to just prevent the photoelectrons from reaching the collecting wire are given in Table P14. Plot incident light frequency versus retarding potential for these two metals. Evaluate their work functions (in eV) and the proportionality constant h (in eV s). TABLE P14 Retarding potential (V) λ(Å) Cs Zn 6000 3000 2000 1500 1200 0.167 2.235 4.302 6.369 8.436 — 0.435 2.502 1.567 6.636 15. Calculate the de Broglie wavelength in nanometers for each of the following: a) An electron that has been accelerated from rest through a potential change of 500 V. b) A bullet weighing 5 gm and traveling at 400 m s−1 . 16. Arguing from Eq. (17), what is the time needed for a standing wave to go through one complete cycle? 10 Hints for a few problems may be found in Appendix 12 and answers for almost all of them appear in Appendix 13. 25 Section 112 Summary 17. The equation for a standing wave in a string has the form (x, t) = ψ(x) cos(ωt) a) Calculate the timeaveraged potential energy (PE) for this motion. [Hint: Use PE = − F d; F = ma; a = ∂ 2 /∂t 2 .] b) Calculate the timeaveraged kinetic energy (KE) for this motion. [Hint: Use KE = 1/2mv 2 and v = ∂/∂t.] c) Show that this harmonically vibrating string stores its energy on the average half as kinetic and half as potential energy, and that E(x)av αψ 2 (x). 18. Indicate which of the following functions are “acceptable.” If one is not, give a reason. a) b) c) d) e) ψ =x ψ = x2 ψ = sin x ψ = exp(−x) ψ = exp(−x 2 ) 19. An acceptable function is never inﬁnite. Does this mean that an acceptable function must be square integrable? If you think these are not the same, try to ﬁnd an example of a function (other than zero) that is never inﬁnite but is not square integrable. 110. Explain why the fact that sin(x) = −sin(−x) means that we can restrict Eq. (132) to nonnegative n without loss of physical content. 111. Which of the following are eigenfunctions for d/dx? a) b) c) d) e) f) x2 exp(−3.4x 2 ) 37 exp(x) sin(ax) cos(4x) + i sin(4x) 112. Calculate the minimum de Broglie wavelength for a photoelectron that is produced when light of wavelength 140.0 nm strikes zinc metal. (Workfunction of zinc = 3.63 eV.) Multiple Choice Questions (Intended to be answered without use of pencil and paper.) 1. A particle satisfying the timeindependent Schrödinger equation must have a) an eigenfunction that is normalized. b) a potential energy that is independent of location. c) a de Broglie wavelength that is independent of location. 26 Chapter 1 Classical Waves and the TimeIndependent Schrödinger Wave Equation d) a total energy that is independent of location. e) None of the above is a true statement. 2. When one operates with d 2 /dx 2 on the function 6 sin(4x), one ﬁnds that a) b) c) d) e) the function is an eigenfunction with eigenvalue −96. the function is an eigenfunction with eigenvalue 16. the function is an eigenfunction with eigenvalue −16. the function is not an eigenfunction. None of the above is a true statement. 3. Which one of the following concepts did Einstein propose in order to explain the photoelectric effect? a) A particle of rest mass m and velocity v has an associated wavelength λ given by λ = h/mv. b) Doubling the intensity of light doubles the energy of each photon. c) Increasing the wavelength of light increases the energy of each photon. d) The photoelectron is a particle. e) None of the above is a concept proposed by Einstein to explain the photoelectric effect. 4. Light of frequency ν strikes a metal and causes photoelectrons to be emitted having maximum kinetic energy of 0.90 hν. From this we can say that a) light of frequency ν/2 will not produce any photoelectrons. b) light of frequency 2ν will produce photoelectrons having maximum kinetic energy of 1.80 hν. c) doubling the intensity of light of frequency ν will produce photoelectrons having maximum kinetic energy of 1.80 hν. d) the work function of the metal is 0.90 hν. e) None of the above statements is correct. 5. The reason for normalizing a wavefunction ψ is a) b) c) d) e) to guarantee that ψ is squareintegrable. to make ψ*ψ equal to the probability distribution function for the particle. to make ψ an eigenfunction for the Hamiltonian operator. to make ψ satisfy the boundary conditions for the problem. to make ψ display the proper symmetry characteristics. Reference [1] F. Flam, Making Waves with Interfering Atoms. Physics Today, 921–922 (1991). Chapter 2 Quantum Mechanics of Some Simple Systems 21 The Particle in a OneDimensional “Box” Imagine that a particle of mass m is free to move along the x axis between x = 0 and x = L, with no change in potential (set V = 0 for 0 < x < L). At x = 0 and L and at all points beyond these limits the particle encounters an inﬁnitely repulsive barrier (V = ∞ for x ≤ 0, x ≥ L). The situation is illustrated in Fig. 21. Because of the shape of this potential, this problem is often referred to as a particle in a square well or a particle in a box problem. It is well to bear in mind, however, that the situation is really like that of a particle conﬁned to movement along a ﬁnite length of wire. When the potential is discontinuous, as it is here, it is convenient to write a wave equation for each region. For the two regions beyond the ends of the box −h2 d 2 ψ + ∞ψ = Eψ, 8π 2 m dx 2 x ≤ 0, x ≥ L (21) Within the box, ψ must satisfy the equation −h2 d 2 ψ = Eψ, 8π 2 m dx 2 0<x <L (22) It should be realized that E must take on the same values for both of these equations; the eigenvalue E pertains to the entire range of the particle and is not inﬂuenced by divisions we make for mathematical convenience. Let us examine Eq. (21) ﬁrst. Suppose that, at some point within the inﬁnite barrier, say x = L + dx, ψ is ﬁnite. Then the second term on the lefthand side of Eq. (21) will be inﬁnite. If the ﬁrst term on the lefthand side is ﬁnite or zero, it follows immediately that E is inﬁnite at the point L + dx (and hence everywhere in the system). Is it possible that a solution exists such that E is ﬁnite? One possibility is that ψ = 0 at all points where V = ∞. The other possibility is that the ﬁrst term on the lefthand side of Eq. (21) can be made to cancel the inﬁnite second term. This might happen if the second derivative of the wavefunction is inﬁnite at all points where V = ∞ and ψ = 0. For the second derivative to be inﬁnite, the ﬁrst derivative must be discontinuous, and so ψ itself must be nonsmooth (i.e., it must have a sharp corner; see Fig. 22). Thus, we see that it may be possible to obtain a ﬁnite value for both E and ψ at x = L + dx, provided that ψ is nonsmooth there. But what about the next point, x = L + 2 dx, and all the other points outside the box? If we try to use the same device, we end up with the requirement that ψ be nonsmooth at every point where V = ∞. A function that is 27 28 Figure 21 Chapter 2 Quantum Mechanics of Some Simple Systems The potential felt by a particle as a function of its x coordinate. continuous but which has a pointwise discontinuous ﬁrst derivative is a contradiction in terms (i.e., a continuous f cannot be 100% corners. To have recognizable corners, we must have some (continuous) edges. We say that the ﬁrst derivative of ψ must be piecewise continuous.) Hence, if V = ∞ at a single point, we might ﬁnd a solution ψ which is ﬁnite at that point, with ﬁnite energy. If V = ∞ over a ﬁnite range of connected points, however, either E for the system is inﬁnite, and ψ is ﬁnite over this region or E is not inﬁnite (but is indeterminate) and ψ is zero over this region. We are not concerned with particles of inﬁnite energy, and so we will say that the solution to Eq. (21) is ψ = 0.1 Turning now to Eq. (22) we ask what solutions ψ exist in the box having associated eigenvalues E that are ﬁnite and positive. Any function that, when twice differentiated, yields a negative constant times the selfsame function is a possible candidate for ψ. Such functions are sin(kx), cos(kx), and exp(±ikx). But these functions are not all independent since, as we noted in Chapter 1, exp(±ikx) = cos(kx) ± i sin(kx) (23) We thus are free to express ψ in terms of exp(±ikx) or else in terms of sin(kx) and cos(kx). We choose the latter because of their greater familiarity, although the ﬁnal answer must be independent of this choice. The most general form for the solution is ψ (x) = A sin(kx) + B cos(kx) (24) where A, B, and k remain to be determined. As we have already shown, ψ is zero at x ≤ 0, x ≥ L and so we have as boundary conditions ψ(0) = 0 ψ(L) = 0 (25) (26) Mathematically, this is precisely the same problem we have already solved in Chapter 1 for the standing waves in a clamped string. The solutions are ψ(x) = A sin(nπ x/L), n = 1, 2, . . . , ψ(x) = 0, x ≤ 0, x ≥ L 0<x <L (27) 1 Thus, the particle never gets into these regions. It is meaningless to talk of the energy of the particle in such regions, and our earlier statement that E must be identical in Eqs. (21) and (22) must be modiﬁed; E is constant in all regions where ψ is ﬁnite. Section 21 The Particle in a OneDimensional “Box” 29 Figure 22 As the function f (x) approaches being nonsmooth, δ approaches zero (the width of one point) and n approaches inﬁnity. One difference between Eq. (27) and the string solutions is that we have rejected the n = 0 solution in Eq. (27). For the string, this solution was for no vibration at all— a physically realizable circumstance. For the particleinabox problem, this solution is rejected because it is not squareintegrable. (It gives ψ = 0, which means no particle on the x axis, contradicting our starting premise. One could also reject this solution for the classical case since it means no energy in the string, which might contradict a starting premise depending on how the problem is worded.) Let us check to be sure these functions satisfy Schrödinger’s equation: −h2 d 2 [A sin(nπ x/L)] 8π 2 m dx 2 nπ x 2 2 n π2 −h −A 2 sin = 8π 2 m L L 2 2 nπ x n h = Eψ (x) A sin = 8mL2 L H ψ(x) = (28) This shows that the functions (27) are indeed eigenfunctions of H . We note in passing that these functions are acceptable in the sense of Chapter 1. 30 Chapter 2 Quantum Mechanics of Some Simple Systems The only remaining parameter is the constant A. We set this to make the probability of ﬁnding the particle in the well equal to unity: L L 2 2 ψ (x)dx = A sin2 (nπ x/L)dx = 1 (29) 0 0 This leads to (Problem 22) A = 2/L (210) which completes the solving of Schrödinger’s timeindependent equation for the problem. Our results are the normalized eigenfunctions ψn (x) = (2/L) sin(nπ x/L), n = 1, 2, 3, . . . (211) and the corresponding eigenvalues, from Eq. (28), En = n2 h2 /8mL2 , n = 1, 2, 3, . . . (212) Each different value of n corresponds to a different stationary state of this system. 22 Detailed Examination of ParticleinaBox Solutions Having solved the Schrödinger equation for the particle in the inﬁnitely deep squarewell potential, we now examine the results in more detail. 22.A Energies The most obvious feature of the energies is that, as we move through the allowed states (n = 1, 2, 3, . . . ), E skips from one discrete, wellseparated value to another (1, 4, 9 in units of h2 /8mL2 ). Thus, the particle can have only certain discrete energies—the energy is quantized. This situation is normally indicated by sketching the allowed energy levels as horizontal lines superimposed on the potential energy sketch, as in