Seventh Edition Quantum Chemistry Ira N.Levine
SEVENTH EDITION Ouantum Chemistry Ira N.Levine Chemistry Department.Brooklyn College,City University of New York PEARSON Boston Columbus Indianapolis New York San Franciso Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singagore Taipei Tokyo
Quantum Chemistry Boston Columbus Indianapolis New York San Franciso Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singagore Taipei Tokyo Se v e nth Edition Ira N. Levine Chemistry Department, Brooklyn College, City University of New York
Contents Preface x Chapter 1 The Schrodinger Equation 1 1.1 Quantum Chemistry 1 1.2 Historical Background of Quantum Mechanics 2 1.3 The Uncertainty Principle 6 1.4 The Time-Dependent Schrodinger Equation 7 1.5 The Time-Independent Schrodinger Equation 11 1.6 Probability 14 1.7 Complex Numbers 16 1.8 Units 17 19 Calculus 18 Chapter2 The Particle in a Box 21 2.1 Differential Equations 21 2.2 Particle in a One-Dimensional Box 22 2.3 The Free Particle in One Dimension 28 2.4 Particle in a Rectangular Well 28 2.5 Tunneling 30 Summary 31 Problems 31 Chapter3 Operators 34 3.1 Operators 34 3.2Eigenfunctions and Eigenvalues 38 3.3Operators and Quantum Mechanics 39 3.4 The Three-Dimensional,Many-Particle Schrodinger Equation 44 3.5 The Particle in a Three-Dimensional Box 47 3.6 Degeneracy 50 3.7 Average Values 51 3.8 Requirements for an Acceptable Wave Function 54 55 iv
iv Preface x Chapter 1 The Schrödinger Equation 1 1.1 Quantum Chemistry 1 1.2 Historical Background of Quantum Mechanics 2 1.3 The Uncertainty Principle 6 1.4 The Time-Dependent Schrödinger Equation 7 1.5 The Time-Independent Schrödinger Equation 11 1.6 Probability 14 1.7 Complex Numbers 16 1.8 Units 17 1.9 Calculus 18 Summary 18 Problems 19 Chapter 2 The Particle in a Box 21 2.1 Differential Equations 21 2.2 Particle in a One-Dimensional Box 22 2.3 The Free Particle in One Dimension 28 2.4 Particle in a Rectangular Well 28 2.5 Tunneling 30 Summary 31 Problems 31 Chapter 3 Operators 34 3.1 Operators 34 3.2 Eigenfunctions and Eigenvalues 38 3.3 Operators and Quantum Mechanics 39 3.4 The Three-Dimensional, Many-Particle Schrödinger Equation 44 3.5 The Particle in a Three-Dimensional Box 47 3.6 Degeneracy 50 3.7 Average Values 51 3.8 Requirements for an Acceptable Wave Function 54 Summary 55 Problems 56 Contents
Chapter4 The Harmonic Oscillator 60 4.1 Power-Series Solution of Differential Equations60 4.2 The One-Dimensional Harmonic Oscillator 62 4.3 Vibration of Diatomic Molecules 71 4.4 Numerical Solution of the One-Dimensional Time-Independent Schrodinger Equation 74 Chapter5 Angular Momentum 90 5.1 Simultaneous Specification of Several Properties 9 5.2 Vectors 94 5.3 Angular Momentum of a One-Particle System 99 5.4The Ladder-Operator Method for Angular Momentum 110 Chapter6 The Hydrogen Atom 118 6.1 The One-Particle Central-Force Problem 118 6.2 Noninteracting Particles and Separation of Variables 120 6.3 Reduction of the Two-Particle Problem to Two One-Particle Problems 121 6.4 The Two-Particle Rigid Rotor 124 6.5 The Hydrogen Atom 128 6.6 The Bound-State Hydrogen-Atom Wave Functions 135 6.7 Hydrogenlike Orbitals 143 6.8 The Zeeman Effect 147 6.9 Numerical Solution of the Radial Schrodinger Equation 149 Chapter 7 Theorems of Quantum Mechanics 155 7.1 Notation 155 7.2 Hermitian Operators 156 7.3 Expansion in Terms of Eigenfunctions 161 7.4 Eigenfunctions of Commuting Operators 167 7.5 Parity 170 7.6 Measurement and the Superposition of States 172 7.7 Position Eigenfunctions 177 78 The Postulates of Quantum Mechanics 180 7Measurement and the Interpretation of Quantum Mechanics 184 7.10 Matrices 187 Summary 191 Problems 191
Contents | v Chapter 4 The Harmonic Oscillator 60 4.1 Power-Series Solution of Differential Equations 60 4.2 The One-Dimensional Harmonic Oscillator 62 4.3 Vibration of Diatomic Molecules 71 4.4 Numerical Solution of the One-Dimensional Time-Independent Schrödinger Equation 74 Summary 84 Problems 84 Chapter 5 Angular Momentum 90 5.1 Simultaneous Specification of Several Properties 90 5.2 Vectors 94 5.3 Angular Momentum of a One-Particle System 99 5.4 The Ladder-Operator Method for Angular Momentum 110 Summary 114 Problems 115 Chapter 6 The Hydrogen Atom 118 6.1 The One-Particle Central-Force Problem 118 6.2 Noninteracting Particles and Separation of Variables 120 6.3 Reduction of the Two-Particle Problem to Two One-Particle Problems 121 6.4 The Two-Particle Rigid Rotor 124 6.5 The Hydrogen Atom 128 6.6 The Bound-State Hydrogen-Atom Wave Functions 135 6.7 Hydrogenlike Orbitals 143 6.8 The Zeeman Effect 147 6.9 Numerical Solution of the Radial Schrödinger Equation 149 Summary 150 Problems 151 Chapter 7 Theorems of Quantum Mechanics 155 7.1 Notation 155 7.2 Hermitian Operators 156 7.3 Expansion in Terms of Eigenfunctions 161 7.4 Eigenfunctions of Commuting Operators 167 7.5 Parity 170 7.6 Measurement and the Superposition of States 172 7.7 Position Eigenfunctions 177 7.8 The Postulates of Quantum Mechanics 180 7.9 Measurement and the Interpretation of Quantum Mechanics 184 7.10 Matrices 187 Summary 191 Problems 191
viContents Chapter8 The Variation Method 197 8.1 The Variation Theorem 197 8.2 Extension of the Variation Method 201 8.3 Determinants 202 8.4 Simultaneous Linear Equations 205 8.5 Linear Variation Functions 209 8.6 Matrices,Eigenvalues,and Eigenvectors 215 Chapter9 Perturbation Theory 232 9.1 Perturbation Theory 232 9.2 Nondegenerate Perturbation Theory 233 9.3 Perturbation Treatment of the Helium-Atom Ground State 238 9.4 Variation Treatments of the Ground State of Helium 240 9.5 Perturbaion Theory for a Degnerate Energy Leve 245 9.6 Simplification of the Secular Equation 248 9.7 Perturbation Treatment of the First Excited States of Helium 250 98Time-Dependent Perturbation Theory 256 9.9 Interaction of Radiation and Matter 258 Summary 260 Problems 261 chapter1o Electron Spin and the Spin-Statistics Theorem 265 10.1 Electron Spin 265 10.2 Spin and the Hydrogen Atom 268 10.The Spin-Statistics Theorem 268 10.4 The Helium Atom 271 10.5 The Pauli Exclusion Principle 273 10.6 Slater Determinants 277 10.7 Perturbation Treatment of the Lithium Ground State 278 10.8 Variation Treatments of the Lithium Ground State 279 10.9 Spin Magnetic Moment 280 10.10 Ladder Operators for Electron Spin 283 Summary 285 Problems 285 chapter 11 Many-Electron Atoms 289 11.1 The Hartree-Fock Self-Consistent-Field Method 289 11.2 Orbitals and the Periodic Table 295 1.3 Electron Correlation 298 11.4 Addition of Angular Momenta 300
vi | Contents Chapter 8 The Variation Method 197 8.1 The Variation Theorem 197 8.2 Extension of the Variation Method 201 8.3 Determinants 202 8.4 Simultaneous Linear Equations 205 8.5 Linear Variation Functions 209 8.6 Matrices, Eigenvalues, and Eigenvectors 215 Summary 223 Problems 223 Chapter 9 Perturbation Theory 232 9.1 Perturbation Theory 232 9.2 Nondegenerate Perturbation Theory 233 9.3 Perturbation Treatment of the Helium-Atom Ground State 238 9.4 Variation Treatments of the Ground State of Helium 242 9.5 Perturbation Theory for a Degenerate Energy Level 245 9.6 Simplification of the Secular Equation 248 9.7 Perturbation Treatment of the First Excited States of Helium 250 9.8 Time-Dependent Perturbation Theory 256 9.9 Interaction of Radiation and Matter 258 Summary 260 Problems 261 Chapter 10 Electron Spin and the Spin–Statistics Theorem 265 10.1 Electron Spin 265 10.2 Spin and the Hydrogen Atom 268 10.3 The Spin–Statistics Theorem 268 10.4 The Helium Atom 271 10.5 The Pauli Exclusion Principle 273 10.6 Slater Determinants 277 10.7 Perturbation Treatment of the Lithium Ground State 278 10.8 Variation Treatments of the Lithium Ground State 279 10.9 Spin Magnetic Moment 280 10.10 Ladder Operators for Electron Spin 283 Summary 285 Problems 285 Chapter 11 Many-Electron Atoms 289 11.1 The Hartree–Fock Self-Consistent-Field Method 289 11.2 Orbitals and the Periodic Table 295 11.3 Electron Correlation 298 11.4 Addition of Angular Momenta 300
Contents vii 11.5 Angular Momentum in Many-Electron Atoms 305 11.6 Spin-Orbit Interaction 316 11.7 The Atomic Hamiltonian 318 11.8 The Condon-Slater Rules 320 Summary 323 Problems324 chapter 12 Molecular Symmetry 328 12.1 Symmetry Elements and Operations 328 12.2 Symmetry Point Groups 335 341 Chapter 13 Electronic Structure of Diatomic Molecules 344 13.1 The Born-Oppenheimer Approximation 344 13.2 Nuclear Motion in Diatomic Molecules 347 13.3 Atomic Units 352 1The Hydrogen Molecule lon 353 13.5 Approximate Treatments of theHGround Electronic State 357 13.6 Molecular Orbitals forH Excited States 365 13.7 MO Configurations of Homonuclear Diatomic Molecules 369 13.8 Electronic Terms of Diatomic Molecules 375 13.9 The Hydrogen Molecule 379 13.10 The Valence-Bond Treatment of H2 382 13.11 Comparison of the MO and VB Theories 384 13.12 MO and VB Wave Functions for Homonuclear Diatomic Molecules 386 13.13 Excited States of H,389 13.14 SCF Wave Functions for Diatomic Molecules 390 13.15 MO Treatment of Heteronuclear Diatomic Molecules 393 13.16VB Treatment of Heteronuclear Diatomic Molecules396 13.17 The Valence-Electron Approximation 396 Summary 397 Problems 398 Chapter14 Theorems of Molecular Quantum Mechanics 402 14.1 Electron Probability Density 402 14.2 Dipole Moments 404 14.3 The Hartree-Fock Method for Molecules 407 14.4 The Virial Theorem 416 14.5 The Virial Theorem and Chemical Bonding 422 14.6 The Hellmann-Feynman Theorem 26 14.7 The Electrostatic Theorem 429 Summary 432 Problems 433
Contents | vii 11.5 Angular Momentum in Many-Electron Atoms 305 11.6 Spin–Orbit Interaction 316 11.7 The Atomic Hamiltonian 318 11.8 The Condon–Slater Rules 320 Summary 323 Problems 324 Chapter 12 Molecular Symmetry 328 12.1 Symmetry Elements and Operations 328 12.2 Symmetry Point Groups 335 Summary 341 Problems 342 Chapter 13 Electronic Structure of Diatomic Molecules 344 13.1 The Born–Oppenheimer Approximation 344 13.2 Nuclear Motion in Diatomic Molecules 347 13.3 Atomic Units 352 13.4 The Hydrogen Molecule Ion 353 13.5 Approximate Treatments of the H+ 2 Ground Electronic State 357 13.6 Molecular Orbitals for H+ 2 Excited States 365 13.7 MO Configurations of Homonuclear Diatomic Molecules 369 13.8 Electronic Terms of Diatomic Molecules 375 13.9 The Hydrogen Molecule 379 13.10 The Valence-Bond Treatment of H2 382 13.11 Comparison of the MO and VB Theories 384 13.12 MO and VB Wave Functions for Homonuclear Diatomic Molecules 386 13.13 Excited States of H2 389 13.14 SCF Wave Functions for Diatomic Molecules 390 13.15 MO Treatment of Heteronuclear Diatomic Molecules 393 13.16 VB Treatment of Heteronuclear Diatomic Molecules 396 13.17 The Valence-Electron Approximation 396 Summary 397 Problems 398 Chapter 14 Theorems of Molecular Quantum Mechanics 402 14.1 Electron Probability Density 402 14.2 Dipole Moments 404 14.3 The Hartree–Fock Method for Molecules 407 14.4 The Virial Theorem 416 14.5 The Virial Theorem and Chemical Bonding 422 14.6 The Hellmann–Feynman Theorem 426 14.7 The Electrostatic Theorem 429 Summary 432 Problems 433
viii Contents chapter15 Molecular Electronic Structure 436 15.1 Ab Initio,Density-Functional,Semiempirical,and Molecular-Mechanics Methods 436 15.2 Electronic Terms of Polyatomic Molecules 437 15.3 The SCF MO Treatment of Polyatomic Molecules 440 15.4 Basis Functions 442 15.5 The SCF MO Treatment of H2O 449 15.6 Population Analysis and Bond Orders 456 15.7The Molecular Electrostatic Potential,Molecular Surfaces,and Atomic Charges 460 15.8 Localized MOs 464 15.9 The SCF MO Treatment of Methane.Ethane.and Ethylene 470 15.10 Molecular Geometry 480 15.11 Conformational Searching 490 15.12 Molecular Vibrational Frequencies 496 15.13 Thermodynamic Properties 498 15.14 Ab Initio Quantum Chemistry Programs 500 15.15 Performing Ab Initio Calculations 501 15.16 Speeding Up Hartree-Fock Calculations 507 15.17 Solvent Effects 510 Problems 518 Chapter 16 Electron-Correlation Methods 525 16.1 Correlation Energy 525 16.2 Configuration Interaction 528 16.3 Moller-Plesset (MP)Perturbation Theory 539 16.4 The Coupled-Cluster Method 546 16.5 Density-Functional Theory 552 16.6 Composite Methods for Energy Calculations 572 16.7 The Diffusion Quantum Monte Carlo Method 575 16.8 Noncovalent Interactions 576 6.9NMR Shielding Constants 78 16.10 Fragmentation Methods 580 16.11 Relativistic Effects 581 16.12 Valence-Bond Treatment of Polyatomic Molecules 582 16.13 The GVB.VBSCF,and BOVB Methods 589 16.14 Chemical Reactions 591 Problems 595 Chapter 17 Semiempirical and Molecular-Mechanics Treatments of Molecules 600 17.1 Semiempirical MOTreatments of Planar Conjugated Molecules 600 17.2 The Huckel MO Method 601 17.3 The Pariser-Parr-Pople Method 619 17.4 General Semiempirical MO and DFT Methods 621
viii | Contents Chapter 15 Molecular Electronic Structure 436 15.1 Ab Initio, Density-Functional, Semiempirical, and Molecular-Mechanics Methods 436 15.2 Electronic Terms of Polyatomic Molecules 437 15.3 The SCF MO Treatment of Polyatomic Molecules 440 15.4 Basis Functions 442 15.5 The SCF MO Treatment of H2O 449 15.6 Population Analysis and Bond Orders 456 15.7 The Molecular Electrostatic Potential, Molecular Surfaces, and Atomic Charges 460 15.8 Localized MOs 464 15.9 The SCF MO Treatment of Methane, Ethane, and Ethylene 470 15.10 Molecular Geometry 480 15.11 Conformational Searching 490 15.12 Molecular Vibrational Frequencies 496 15.13 Thermodynamic Properties 498 15.14 Ab Initio Quantum Chemistry Programs 500 15.15 Performing Ab Initio Calculations 501 15.16 Speeding Up Hartree–Fock Calculations 507 15.17 Solvent Effects 510 Problems 518 Chapter 16 Electron-Correlation Methods 525 16.1 Correlation Energy 525 16.2 Configuration Interaction 528 16.3 Møller–Plesset (MP) Perturbation Theory 539 16.4 The Coupled-Cluster Method 546 16.5 Density-Functional Theory 552 16.6 Composite Methods for Energy Calculations 572 16.7 The Diffusion Quantum Monte Carlo Method 575 16.8 Noncovalent Interactions 576 16.9 NMR Shielding Constants 578 16.10 Fragmentation Methods 580 16.11 Relativistic Effects 581 16.12 Valence-Bond Treatment of Polyatomic Molecules 582 16.13 The GVB, VBSCF, and BOVB Methods 589 16.14 Chemical Reactions 591 Problems 595 Chapter 17 Semiempirical and Molecular-Mechanics Treatments of Molecules 600 17.1 Semiempirical MO Treatments of Planar Conjugated Molecules 600 17.2 The Hückel MO Method 601 17.3 The Pariser–Parr–Pople Method 619 17.4 General Semiempirical MO and DFT Methods 621
Contents ix 17.5 The Molecular-Mechanics Method 634 17.6 Empirical and Semiempirical Treatments of Solvent Effects 648 17.7 Chemical Reactions 652 17.8 The Future of Quantum Chemistry 655 Problems 656 Appendix 661 Bibliography 665 Answers to Selected Problems 667 Index 679
Contents | ix 17.5 The Molecular-Mechanics Method 634 17.6 Empirical and Semiempirical Treatments of Solvent Effects 648 17.7 Chemical Reactions 652 17.8 The Future of Quantum Chemistry 655 Problems 656 Appendix 661 Bibliography 665 Answers to Selected Problems 667 Index 679
Preface This book or rt-yernda nced undergraduate courses in quantun ch mis 76: text pro with an i nd of principles.T nath included merivations ar nted in full ster p-bysep detail hat students at all le can easily follow and understand.A rich variety of homework problems (both quantitative and conceptual)is given for each chapter. NEW TO THIS EDITION The following improvements were made to the seventh edition .Thorough latest chemistry research and methods of computational chemistry,includin many new liter re refe New problems have been added to most chapters,including additional computational problems in Chapters 15 and 16. Explanations have been revised in areas where students had difficulty. Color has been added to figures to increase the visual appeal of the book. The computer programs in the Solutions Manual and the text were changed from BASIC to C++ The text is enlivened by references to modern research in quantum mechanics such as the Ozawa reformulation of the uncertainty principle and the observation of interference effects with very large molecules. New and expanded material in the seventh edition includes New theoretical and ental wo ork on the uncertainty pri nciple (Section 5.1) eld-I charges (c 15.7) 161 Expanded tr tof extra n to the mplete-basis-set (CBS)limit (Sections 15 5 16 1 and 164) Use of the two-electron reduced density matrix(Section 16.2). The DFT-D3 method (Section 16.5). The VV10 correlation functional for dispersion(Section 16.5). The WI-F12 and W2-F12 methods (Section 16.6). Dispersion(stacking)interactions in DNA(Section 16.8). The MP2.5.MP2.X.SCS(MI)-CCSD,and SCS(MI)-MP2 methods(Section 16.8). An expanded discussion of calculation of NMR shielding constants and spin-spin coupling constants including linear scaling(Section 16.9). The PM-H d PM metods (7) 0 Resources:Optional Spartan Student Edition molecular provide eling pac arge nal fet in e sy- pter pro oblem the bo ed.c
x This book is intended for first-year graduate and advanced undergraduate courses in quantum chemistry. This text provides students with an in-depth treatment of quantum chemistry, and enables them to understand the basic principles. The limited mathematics background of many chemistry students is taken into account, and reviews of necessary mathematics (such as complex numbers, differential equations, operators, and vectors) are included. Derivations are presented in full, step-by-step detail so that students at all levels can easily follow and understand. A rich variety of homework problems (both quantitative and conceptual) is given for each chapter. New to this Edition The following improvements were made to the seventh edition: • Thorough updates reflect the latest quantum chemistry research and methods of computational chemistry, including many new literature references. • New problems have been added to most chapters, including additional computational problems in Chapters 15 and 16. • Explanations have been revised in areas where students had difficulty. • Color has been added to figures to increase the visual appeal of the book. • The computer programs in the Solutions Manual and the text were changed from BASIC to C++. • The text is enlivened by references to modern research in quantum mechanics such as the Ozawa reformulation of the uncertainty principle and the observation of interference effects with very large molecules. New and expanded material in the seventh edition includes • New theoretical and experimental work on the uncertainty principle (Section 5.1). • The CM5 and Hirshfeld-I methods for atomic charges (Section 15.7). • Static and dynamic correlation (Section 16.1). • Expanded treatment of extrapolation to the complete-basis-set (CBS) limit (Sections 15.5, 16.1 and 16.4). • Use of the two-electron reduced density matrix (Section 16.2). • The DFT-D3 method (Section 16.5). • The VV10 correlation functional for dispersion (Section 16.5). • The W1-F12 and W2-F12 methods (Section 16.6). • Dispersion (stacking) interactions in DNA (Section 16.8). • The MP2.5, MP2.X, SCS(MI)-CCSD, and SCS(MI)-MP2 methods (Section 16.8). • An expanded discussion of calculation of NMR shielding constants and spin-spin coupling constants including linear scaling (Section 16.9). • Fragmentation methods (Section 16.10). • The PM6-D3H4 and PM7 methods (Section 17.4). Resources: Optional Spartan Student Edition molecular modeling software provides access to a sophisticated molecular modeling package that combines an easy-to-use graphical interface with a targeted set of computational functions. A solutions manual for the end-of-chapter problems in the book is available at http://www.pearsonhighered.com/ advchemistry. Preface
Preface xi hemishiy desirable for all chemistry studentd on,and thi nis goal in m ve tr explar nd whe e points ns clear and co are given v the houg I to mak The aim is to ve students a solid understandi f the of quantum mechanics and molecular structure.The tobe useful to students in all branches of chemistry,not just future quantum chemists.Howeve the presentation is such that those who do go on in quantum chemistry will have a good foundation and will not be hampered by misconceptions. An obstacle faced by many chemistry students in learning quantum mechanics is their unfamiliarity with much of the required mathematics.In this text I have included detailed treatments of the needed mathematics.Rather than putting all the mathematics in an introductory chapter or a series of appendices,I have integrated the mathematics with the physics and chemistry.Immediate application of the mathematics to solving a tcs more r student udy of the m make the mather have als kept in mir limited physics a nany ch g top in physics Leland Auen.N.Colin Ba k have beneft 6 en.David F Gordon A.Gallup.Daniel Gerrity.David Goldbe Robert Griffin.Tracy Hamilto Sharon Hamr Ja s Harrison John Head.Warren Hehre, Robert Hinde Hans Jaffe,Miklos Kertes z Neil Kestne er,Harry King,Peter Kollman,Anna Krylov,Mel Levy,Errol Lewars,Joel Liebman,Tien-Sung Tom Lin,Ryan McLaughlin,Frank Meeks, Robert Metzger,Charles Millner,John H.Moore,Pedro Muifio,William Palke,Sharon Palmer,Kirk Peterson,Gary Pfeiffer,Russell Pitzer.Oleg Prezhdo.Frank Rioux,Kenneth Sando,Harrison Shull,James J.P.Stewart,Richard Stratt,Fu-Ming Tao,Ronald Terry, Alexander Van Hook,Arieh Warshel,Peter Weber,John S.Winn,and Michael Zerner. Reviewers for the seventh edition were ohn Asbury.State University Mu-Hyun ufts of Science and Technology versity of Ca anta Barbara Mo Tach Ruben Parra.DePaul University Michael Wedlock,Gettysburg College I wish to thank all these people and several anonymous reviewers for their helpful suggestions. I would greatly appreciate receiving any suggestions that readers may have for improving the book. Ira N.Levine INLevine@brooklyn.cuny.edu
Preface | xi The extraordinary expansion of quantum chemistry calculations into all areas of chemistry makes it highly desirable for all chemistry students to understand modern methods of electronic structure calculation, and this book has been written with this goal in mind. I have tried to make explanations clear and complete, without glossing over difficult or subtle points. Derivations are given with enough detail to make them easy to follow, and wherever possible I avoid resorting to the frustrating phrase “it can be shown that.” The aim is to give students a solid understanding of the physical and mathematical aspects of quantum mechanics and molecular electronic structure. The book is designed to be useful to students in all branches of chemistry, not just future quantum chemists. However, the presentation is such that those who do go on in quantum chemistry will have a good foundation and will not be hampered by misconceptions. An obstacle faced by many chemistry students in learning quantum mechanics is their unfamiliarity with much of the required mathematics. In this text I have included detailed treatments of the needed mathematics. Rather than putting all the mathematics in an introductory chapter or a series of appendices, I have integrated the mathematics with the physics and chemistry. Immediate application of the mathematics to solving a quantum-mechanical problem will make the mathematics more meaningful to students than would separate study of the mathematics. I have also kept in mind the limited physics background of many chemistry students by reviewing topics in physics. Previous editions of this book have benefited from the reviews and suggestions of Leland Allen, N. Colin Baird, Steven Bernasek, James Bolton, W. David Chandler, Donald Chesnut, R. James Cross, Gary DeBoer, Douglas Doren, David Farrelly, Melvyn Feinberg, Gordon A. Gallup, Daniel Gerrity, David Goldberg, Robert Griffin, Tracy Hamilton, Sharon Hammes-Schiffer, James Harrison, John Head, Warren Hehre, Robert Hinde, Hans Jaffé, Miklos Kertesz, Neil Kestner, Harry King, Peter Kollman, Anna Krylov, Mel Levy, Errol Lewars, Joel Liebman, Tien-Sung Tom Lin, Ryan McLaughlin, Frank Meeks, Robert Metzger, Charles Millner, John H. Moore, Pedro Muiño, William Palke, Sharon Palmer, Kirk Peterson, Gary Pfeiffer, Russell Pitzer, Oleg Prezhdo, Frank Rioux, Kenneth Sando, Harrison Shull, James J. P. Stewart, Richard Stratt, Fu-Ming Tao, Ronald Terry, Alexander Van Hook, Arieh Warshel, Peter Weber, John S. Winn, and Michael Zerner. Reviewers for the seventh edition were John Asbury, Pennsylvania State University Mu-Hyun Baik, Indiana University Lynne Batchelder, Tufts University Richard Dawes, Missouri University of Science and Technology Kalju Kahn, University of California, Santa Barbara Scott Kirkby, East Tennessee State University Jorge Morales, Texas Technical University Ruben Parra, DePaul University Michael Wedlock, Gettysburg College I wish to thank all these people and several anonymous reviewers for their helpful suggestions. I would greatly appreciate receiving any suggestions that readers may have for improving the book. Ira N. Levine INLevine@brooklyn.cuny.edu