A Primer on Quantum Chemistry

S. M. Blinder, PhD, is Professor Emeritus of Chemistry and Physics at the University of Michigan, Ann Arbor, USA, and a senior scientist with Wolfram Research in Champaign, Illinois. He has published extensively on quantum chemistry and related fields.

Preface xi

About the Author xiv

About the Companion Website xvi

Mathematica 1

1 The Basic Math Assistant 1

2 Derivatives and Integrals 2

3 Differential Equations 4

4 Symbolic Mathematics 5

5 External Data 5

1 The Old Quantum Theory 8

1.1 Introduction 8

1.2 Blackbody Radiation 8

1.3 The Photoelectric Effect 12

1.4 Line Spectra 13

1.5 Bohr Theory of the Hydrogen Atom 15

1.6 Bohr-Sommerfeld Orbits 19

1.7 The Periodic Structure of the Elements 21

2 The Schrödinger Equation 24

2.1 TheWave-Particle Duality 24

2.2 De Broglie's Hypothesis 26

2.3 Heuristic Derivation of the Schrödinger Equation 29

2.4 Operators and Eigenvalues 31

2.5 TheWavefunction 32

3 Quantum Mechanics of Some Simple Systems 33

3.1 Particle in a Box 33

3.2 Free-Electron Model 37

3.3 Particle in a Ring 39

3.4 Free Electron Model for Aromatic Molecules 40

3.5 Particle in a Three-Dimensional Box 41

3.6 The Free Particle 43

3.7 Deltafunction Normalization 45

3.8 Particle in a Deltafunction PotentialWell 47

4 Principles of Quantum Mechanics 50

4.1 Hermitian Operators 50

4.2 Eigenvalues and Eigenfunctions 51

4.3 Expectation Values 52

4.4 Commutators and Uncertainties 53

4.5 Postulates of Quantum Mechanics 55

4.6 Dirac Bra-Ket Notation 57

4.7 The Variational Method 58

4.8 Perturbation Theory 60

5 The Harmonic Oscillator 64

5.1 Classical Oscillator 64

5.2 Harmonic Oscillator in Old Quantum Theory 66

5.3 Quantum Harmonic Oscillator 67

5.4 Harmonic-Oscillator Eigenfunctions 69

5.5 Operator Formulation of the Harmonic Oscillator 70

5.6 Quantum Theory of Radiation 72

5.7 The Anharmonic Oscillator 74

6 Quantum Theory of Angular Momentum 76

6.1 Rotation in Two Dimensions 76

6.2 Spherical Polar Coordinates 78

6.3 Rotation in Three Dimensions 79

6.4 Spherical Harmonics 81

6.5 Electron Spin 83

6.6 Pauli Spin Algebra 84

6.7 General Theory of Angular Momentum 85

6.8 Addition of Angular Momenta 86

7 Molecular Vibration and Rotation 88

7.1 Molecular Spectroscopy 88

7.2 Vibration of Diatomic Molecules 88

7.3 The Morse Potential 90

7.4 Vibration of Polyatomic Molecules 93

7.5 Normal Modes of a Triatomic Molecule 94

7.6 Rotation of Diatomic Molecules 96

8 The Hydrogen Atom 99

8.1 Schrödinger Equation for Hydrogenlike Atoms 99

8.2 Hydrogen Atom Ground State 101

8.3 Hydrogenic 2s and 3s Orbitals 105

8.4 Solving the Schrödinger Equation 106

8.5 ;;- and ;;-Orbitals 108

8.6 Radial Distribution Functions 110

8.7 Summary on Atomic Orbitals 111

8.8 Connection between Hydrogen Atom and Harmonic Oscillator 111

9 The Helium Atom 114

9.1 Experimental Energies 114

9.2 Schrödinger Equation and Simple Variational Calculation 114

9.3 Improved Computations on the Helium Ground State 117

9.4 The Hydride Ion H− 119

9.5 Spinorbitals and the Exclusion Principle 119

9.6 Excited States of Helium 120

10 Atomic Structure and the Periodic Law 123

10.1 The Periodic Table 123

10.2 Slater Determinants 123

10.3 Self-Consistent Field Theory 126

10.4 Lithium and Beryllium Atoms 127

10.5 Aufbau Principles 131

10.6 Atomic Configurations and Term Symbols 132

10.7 Periodicity of Atomic Properties 135

10.8 Relativistic Effects 137

11 The Chemical Bond 140

11.1 The Hydrogen Molecule 140

11.2 Valence Bond Theory 142

11.3 Hybrid Orbitals and Molecular Geometry 143

11.4 Hypervalent Compounds 146

11.5 Boron Hydrides 148

12 Diatomic Molecules 150

12.1 The Hydrogen Molecule-Ion 150

12.2 The LCAO Approximation 153

12.3 MO Theory of Homonuclear Diatomic Molecules 154

12.4 Variational Computation of Molecular Orbitals 156

12.5 Heteronuclear Molecules 158

13 Polyatomic Molecules and Solids 160

13.1 Hückel Molecular Orbital Theory 160

13.2 Conservation of Orbital Symmetry;Woodward-Hoffmann Rules 163

13.3 Valence-Shell Model 166

13.4 Transition Metal Complexes 168

13.5 The Hydrogen Bond 171

13.6 Proteins and Nucleic Acids 172

13.7 Band Theory of Metals and Semiconductors 175

14 Molecular Symmetry and Group Theory 178

14.1 The Ammonia Molecule 178

14.2 Mathematical Theory of Groups 180

14.3 Group Theory in Quantum Mechanics 181

14.4 Molecular Orbitals for Ammonia 182

14.5 Selection Rules 184

14.6 TheWater Molecule 185

14.7 Walsh Diagrams 186

14.8 Molecular Symmetry Groups 187

14.9 Dipole Moments and Optical Activity 192

15 The Hartree-Fock Method 194

15.1 Hartree Self-Consistent Field Theory 194

15.2 DeterminantalWavefunctions 197

15.3 Hartree-Fock Equations 199

15.4 Hartree-Fock Equations using Second Quantization 203

15.5 Roothaan Equations 206

15.6 Atomic Hartree-Fock Results 210

15.7 Electron Correlation 213

15.8 Post Hartree-Fock Methods 214

16 Density Functional Theory 217

16.1 Thomas-Fermi Model 217

16.2 The Hohenberg-Kohn Theorems 221

16.3 Density Functionals 222

16.4 Slater's X-Alpha Method 223

16.5 The Kohn-Sham Equations 224

16.6 Chemical Potential 225

17 Metaphysical Aspects of the Quantum Theory 227

17.1 Introduction 227

17.2 The Copenhagen Interpretation 228

17.3 Superposition 229

17.4 Schrödinger's Cat 230

17.5 The Einstein-Podolsky-Rosen Experiment 231

17.6 Bell's Theorem 234

17.7 Conclusion 236

18 Quantum Computers 238

18.1 Prospects of Quantum Computation 238

18.2 Qubits 239

18.3 Quantum Gates and Circuits 240

18.4 Simulation of a Stern-Gerlach Experiment 246

18.5 Quantum Fourier Transform 247

18.6 Phase Estimation Algorithm 250

18.7 Many-Electron Systems 252

18.8 Atomic and Molecular Hamiltonians 253

18.9 Time-Evolution of a Quantum System 256

18.10 Trotter Expansions 257

18.11 Simulations of Molecular Structure 258

Bibliography 260

Index 261