Quantum chemistry

704 Pages·2008·80.08 MB·English

by  McQuarrie, Donald A.

QUANTUM CHEMISTRY
SECOND EDITION
QUANTUM CHEMISTRY
SECOND EDITION
Donald A. McQuarrie
DEPARTM ENT O F C HEMISTRY
UNIVERSITY O F CALIFORN IA, DAVIS
UN IVERSITY SCIENCE BOOKS
Mill Va lley, California
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Library of Congress Cataloging-in-Publication Data
McQua1Tie, Donald A. (Donald Allan)
Quantum chemistry I Donald A. McQua1Tie.-2nd ed.
p. cm.
Includes index.
ISBN 978-1-891389-50-4 (alk. paper)
Quantum chemistry. I. Title.
QD462.M4 2007
541'.28-dc22
2007023879
Printed in the United States of America
10 9 8 7 6 5 4 3
Contents
Preface to the Second Edition xi
CHAPTER 1 I The Dawn of the Quantum Theory
1.1 Blackbody Rad iation 2
1 .2 Planck's Quantum Hypothesis 4
1 .3 The Photoelectric Effect 8
1 .4 Vibrations of Atoms in Crystals 11
1 .5 The Hydrogen Atomic Spectrum 12
1 .6 The Rydberg Formula 15
1 .7 Angular Momentum 16
1 .8 Quantized Angular Momentum 1 8
1 .9 Reduced Mass 22
1.10 De Broglie Waves 26
1 .11 The Relation between de Broglie Waves and Quantized Angu lar Momentum 28
1.12 De Broglie Waves Observed 29
1 .13 Two-Sli t Experiments 30
1 .14 The Heisenberg Uncertainty Princip le 34
Problems 36
References 44
MATHCHAPTER A I Complex Numbers 45
Problems 49
CHAPTER 2 I The Classical Wave Equation 53
2.1 The One-Dimensiona l Classical Wave Equation 54
2 .2 Separation of Variables 54
2.3 Oscillatory Solutions to Differential Equat ions 58
2 .4 Superposition of Normal Modes 61
2.5 A Vibrating Membrane 64
2.6 Interference of Waves 68
Problems 72
References 84 v
VI Contents
M ATHCH APTER B I Probabi lity and Statistics 85
Problems 93
CHAPTER 3 I The Schrodinger Equation and a Particle in a Box 97
3.1 The Schrodinger Equation 97
3.2 Linear Operators in Quantum Mechanics 99
3.3 Eigenvalue Problems In Quantum Mechanics 101
3.4 Wave Functions and Their Probabilistic Interpretation 103
3.5 Quantized Energies 105
3.6 Normalized Wave Functions 107
3.7 Average Quantities in Quantum Mechan ics 110
3.8 The Uncertainty Principle and Operators 112
3.9 Particl e in a Three-D imensional Box 114
Problems 120
References 128
M ATHCH APTER C I Vectors 129
Problems 139
CHAPTER 4 I The Postu lates and General Principles of Quantum
Mechanics 143
4.1 State Functions 143
4.2 Quantum-Mechanical Operators and Classical Variables 147
4.3 Observable Quantities and Eigenvalues 149
4.4 Commutators and the Uncerta inty Principle 153
4.5 Hermitian Operators 1 56
4.6 Hermitian Operators and Orthogona I ity 160
4.7 Commuting Operators and Mutua l Eigenfunctions 164
4.8 Probabil ty of a Measurement and Fourier Coefficients 165
4.9 The Time-Dependent Schrodinger Equation 1 70
4.1 0 Quantum Mechanics and the Two-SI it Experiment 175
Problems 179
References 196
MATHC HAPTER 0 I Ser ies and Limits 197
Problems 202
CH APTER 5 I The Harmonic Oscillator and Vibrational
Spectroscopy 207
5.1 Classical Harmon ic Oscillator 207
5.2 Conservation of Energy of a Classical Harmonic Oscillator 210
5.3 Harmonic-Oscillator Model of a Diatom ic Molecu le 213
5.4 The Harmonic-Osci llator Approximation 215
5.5 The Energy Levels of a Quantum-Mechan ical Harmonic Oscillator 218
5.6 Infrared Spectra of D iatomic Molecu les 219
Contents vi i
5 .7 Overtones in Vibrational Spectra 222
5.8 Harmon ic-Oscillator Wave Functions 225
5.9 Parity of Hermite Polynomials 228
5.10 Relations Among Hermite Polynomials 230
5.11 Normal Coord inates 233
5.12 Harmon ic-Oscillator Selection Rule 237
Appendix: Operator Method Solution to the Schrodinger Equation for a Harmon ic
Oscillator 239
Problems 243
References 254
MATHCHAPTER E I Spherical Coordinates 255
Problems 263
CHAPTER 6 I The Rigid Rotator and Rotational Spectroscopy 267
6.1 The Energy Levels of a Rigid Rotator 267
6.2 The Rigid Rotator Model of a D iatomic Molecule 272
6 .3 Rotation-Vibrational Spectra 2 75
6.4 Rotation- Vibration Interaction 278
6.5 A Nonrigid Rotator 281
6.6 Spherical Harmon ics 282
6.7 Rigid-Rotator Selection Rule 288
6 .8 Angular Momentum and Measurements 290
Append ix: Determ ination of the Eigenvalues of L2 and i., by Operator Methods 296
Problems 300
References 308
MATHCHAPTER F I Determ inants 309
Problems 317
CHAPTER 7 I The Hydrogen Atom 321
7.1 The Schrod inger Equation for a Hydrogen Atom 321
7 .2 s Orbitals 32 7
7.3 pOrbita ls 334
7.4 The Zeeman Effect 339
7.5 ElectronSpin 344
7 .6 Spin-Orbit Interaction 349
7.7 Hydrogen Atomic Term Symbols 353
7 .8 The Zeeman Effect Revisited 357
7 .9 The Schrod inger Equation for a Helium Atom 359
Problems 360
References 366
MATHCHAPTER G I Matrices 367
Problems 376
viii Contents
CHAPTER 8 I Approx imation Methods 381
8.1 The Variationa l Method 381
8.2 Trial Functions That Depend Linearly on Variational Parameters 387
8.3 Trial Functions That Depend Non linearly on Variational Parameters 395
8.4 Introduction to Perturbation Theory 396
8.5 First-Order Pertubation Theory 399
8.6 Selection Rules and Time-Dependent Perturbation Theory 404
Problems 410
References 426
MATHCHAPTER H I Matrix Eigenvalue Problems 427
Problems 432
CHAPTER 9 I Many-Electron Atoms 435
9.1 Atomic U nits 435
9.2 Classic Ca lcu lations on a Helium Atom 439
9.3 Hartree-Fock Equations for a Helium Atom 444
9.4 Antisymmetry of Electronic Wave Functions 447
9.5 Slater Determ inants 450
9.6 The Hartree-Fock- Roothaan Method 453
9.7 Hartree-Fock- Roothaan Results for A toms 458
9.8 Correlation Energy 463
9.9 Atomic Term Symbols 466
9.10 Addition of Angu lar Momenta 470
9.11 Hund's Rules 474
9.12 Atomic Term Symbols and Atomic Spectra 475
9.13 Russell- Saunders Coupling 479
Appendix: An SCF Calculation of a Helium Atom 482
Problems 489
References 497
CHAPTER 10 I The Chemical Bond : One- and Two-Electron
Molecu les 499
10.1 The Born- Oppenheimer Approximation 500
10.2 The Hydrogen Molecular Ion, Hi 501
10.3 Molecular Orbita ls Constructed from a Linear Combination of Atom ic Orbitals 506
10.4 Bonding and Antibonding Orbitals 513
10.5 Molecular Orbital Theory and the Vir ial Theorem 515
10.6 Polarization Terms in Basis Sets 521
10.7 The Sc hrodinger Equation for H2 523
10.8 Molecular Orbita l Theory Resu lts for H2 526
10.9 Configuration Interaction 531
10.10 An SCF Calcu lation on H2 537
Appendix: Molecular Orbital Theory of H2 543
Problems 547
References 557
Contents ix
CH APTER 11 I Qual itative Theory of Chemical Bonding 559
11.1 Molecu lar Orbitals 560
11 .2 Molecu lar Electron Configurations 564
11.3 Molecu lar Orbital Theory and Heteronuclear Diatom ic Molcu les 570
11 .4 Molecu lar Term Symbols 573
11 .5 Molecu lar Term Symbols and Symmetry Properties 577
11 .6 The rr -Electron Approximation 581
11 .7 HOckel Molecular Orbital Theory and Bond Orders 588
11.8 HOckel Molecular Orbita l Theory in Matr ix Notation 595
Problems 597
References 605
CHAPTER 12 I The Hartree-Fock-Roothaan Method 607
12.1 The Hartree-Fock- Roothaan Equations 608
12.2 M inimal Gaussian Basis Sets 614
12.3 Extended Gaussian Basis Sets 621
12.4 Basis Sets w ith Orbita l Polarization Terms 626
12.5 Using Gaussian 03 and WebMO 631
12.6 Hartree- Fock- Roothaan Results 636
12.7 Post-Hartree-Fock Methods 643
Problems 653
References 660
References for Post-Hartree-Fock Methods 660
Answers to the Numerical Problems 663
Index 683
Illustration Credits 690