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 Contents Preface to the Third Edition xvii Preface to the Second Edition xix Preface to the First Edition xxi 1 ClassicalWaves and the Time-Independent Schr¨odingerWave Equation 1 1-1 Introduction 1 1-2 Waves . 1 1-3 The ClassicalWave Equation . 4 1-4 StandingWaves in a Clamped String . 7 1-5 Light as an ElectromagneticWave . 9 1-6 The Photoelectric Effect 10 1-7 TheWave Nature of Matter 14 1-8 A Diffraction Experiment with Electrons . 16 1-9 Schr¨odinger’s Time-IndependentWave Equation . 19 1-10 Conditions on ψ 21 1-11 Some Insight into the Schr¨odinger Equation . 22 1-12 Summary 23 Problems 24 Multiple Choice Questions 25 Reference . 26 2 Quantum Mechanics of Some Simple Systems 27 2-1 The Particle in a One-Dimensional “Box” . 27 2-2 Detailed Examination of Particle-in-a-Box Solutions . 30 2-3 The Particle in a One-Dimensional “Box” with One FiniteWall . 38 2-4 The Particle in an Infinite “Box” with a Finite Central Barrier 44 2-5 The Free Particle in One Dimension 47 2-6 The Particle in a Ring of Constant Potential 50 2-7 The Particle in a Three-Dimensional Box: Separation of Variables 53 2-8 The Scattering of Particles in One Dimension . 56 2-9 Summary 59 Problems 60 Multiple Choice Questions 65 References . 68 3 The One-Dimensional Harmonic Oscillator 69 3-1 Introduction . 69 3-2 Some Characteristics of the Classical One-Dimensional Harmonic Oscillator 69 3-3 The Quantum-Mechanical Harmonic Oscillator 72 3-4 Solution of the Harmonic Oscillator Schr¨odinger Equation 74 3-5 Quantum-Mechanical Average Value of the Potential Energy . 83 3-6 Vibrations of Diatomic Molecules . 84 3-7 Summary 85 Problems 85 Multiple Choice Questions . 88 4 The Hydrogenlike Ion, Angular Momentum, and the Rigid Rotor 89 4-1 The Schr¨odinger Equation and the Nature of Its Solutions . 89 4-2 Separation of Variables . 105 4-3 Solution of the R, , and Equations . 106 4-4 Atomic Units 109 4-5 Angular Momentum and Spherical Harmonics . 110 4-6 Angular Momentum and Magnetic Moment 115 4-7 Angular Momentum in Molecular Rotation—The Rigid Rotor 117 4-8 Summary 119 Problems 120 Multiple Choice Questions . 125 References 126 5 Many-Electron Atoms 127 5-1 The Independent Electron Approximation . 127 5-2 Simple Products and Electron Exchange Symmetry 129 5-3 Electron Spin and the Exclusion Principle . 132 5-4 Slater Determinants and the Pauli Principle 137 5-5 Singlet and Triplet States for the 1s2s Configuration of Helium 138 5-6 The Self-Consistent Field, Slater-Type Orbitals, and the Aufbau Principle . 144 5-7 Electron Angular Momentum in Atoms . 149 5-8 Overview 159 Problems 160 Multiple Choice Questions . 164 References 165 6 Postulates and Theorems of Quantum Mechanics 166 6-1 Introduction . 166 6-2 TheWavefunction Postulate . 166 6-3 The Postulate for Constructing Operators 167 6-4 The Time-Dependent Schr¨odinger Equation Postulate . 168 6-5 The Postulate Relating Measured Values to Eigenvalues 169 6-6 The Postulate for Average Values 171 6-7 Hermitian Operators 171 6-8 Proof That Eigenvalues of Hermitian Operators Are Real . 172 6-9 Proof That Nondegenerate Eigenfunctions of a Hermitian Operator Form an Orthogonal Set 173 6-10 Demonstration That All Eigenfunctions of a Hermitian Operator May Be Expressed as an Orthonormal Set . 174 6-11 Proof That Commuting Operators Have Simultaneous Eigenfunctions 175 6-12 Completeness of Eigenfunctions of a Hermitian Operator 176 6-13 The Variation Principle 178 6-14 The Pauli Exclusion Principle . 178 6-15 Measurement, Commutators, and Uncertainty 178 6-16 Time-Dependent States 180 6-17 Summary 185 Problems 186 Multiple Choice Questions 189 References . 189 7 The Variation Method 190 7-1 The Spirit of the Method 190 7-2 Nonlinear Variation: The Hydrogen Atom 191 7-3 Nonlinear Variation: The Helium Atom 194 7-4 Linear Variation: The Polarizability of the Hydrogen Atom . 197 7-5 Linear Combination of Atomic Orbitals: The H+2 Molecule–Ion . 206 7-6 Molecular Orbitals of Homonuclear Diatomic Molecules . 220 7-7 Basis Set Choice and the VariationalWavefunction 231 7-8 Beyond the Orbital Approximation 233 Problems 235 Multiple Choice Questions 241 References . 242 8 The Simple H¨uckel Method and Applications 244 8-1 The Importance of Symmetry . 244 8-2 The Assumption of σ–π Separability . 244 8-3 The Independent π-Electron Assumption . 246 8-4 Setting up the H¨uckel Determinant 247 8-5 Solving the HMO Determinantal Equation for Orbital Energies . 250 8-6 Solving for the Molecular Orbitals 251 8-7 The Cyclopropenyl System: Handling Degeneracies . 253 8-8 Charge Distributions from HMOs . 256 8-9 Some Simplifying Generalizations 259 8-10 HMO Calculations on Some Simple Molecules 263 8-11 Summary: The Simple HMO Method for Hydrocarbons . 268 8-12 Relation Between Bond Order and Bond Length . 269 8-13 π-Electron Densities and Electron Spin Resonance Hyperfine Splitting Constants . 271 8-14 Orbital Energies and Oxidation-Reduction Potentials . 275 8-15 Orbital Energies and Ionization Energies . 278 8-16 π-Electron Energy and Aromaticity 279 8-17 Extension to Heteroatomic Molecules 284 8-18 Self-Consistent Variations of α and β 287 8-19 HMO Reaction Indices 289 8-20 Conclusions 295 Problems . 296 Multiple Choice Questions 305 References 306 9 Matrix Formulation of the Linear Variation Method 308 9-1 Introduction 308 9-2 Matrices and Vectors . 308 9-3 Matrix Formulation of the Linear Variation Method . 315 9-4 Solving the Matrix Equation . 317 9-5 Summary . 320 Problems . 320 References 323 10 The Extended H¨uckel Method 324 10-1 The Extended H¨uckel Method 324 10-2 Mulliken Populations . 335 10-3 Extended H¨uckel Energies and Mulliken Populations 338 10-4 Extended H¨uckel Energies and Experimental Energies . 340 Problems . 342 References 347 11 The SCF-LCAO-MO Method and Extensions 348 11-1 Ab Initio Calculations 348 11-2 The Molecular Hamiltonian . 349 11-3 The Form of theWavefunction 349 11-4 The Nature of the Basis Set . 350 11-5 The LCAO-MO-SCF Equation 350 11-6 Interpretation of the LCAO-MO-SCF Eigenvalues . 351 11-7 The SCF Total Electronic Energy 352 11-8 Basis Sets . 353 11-9 The Hartree–Fock Limit . 357 11-10 Correlation Energy 357 11-11 Koopmans’ Theorem . 358 11-12 Configuration Interaction . 360 11-13 Size Consistency and the Møller–Plesset and Coupled Cluster Treatments of Correlation 365 11-14 Multideterminant Methods 367 11-15 Density Functional Theory Methods . 368 11-16 Examples of Ab Initio Calculations . 370 11-17 Approximate SCF-MO Methods . 384 Problems . 386 References 388