Quantum Mechanical Foundations of Molecular Spectroscopy

Preface xi

Introduction xv

1 Transition from Classical Physics to Quantum Mechanics 1

1.1 Description of Light as an Electromagnetic Wave 2

1.2 Blackbody Radiation 3

1.3 The Photoelectric Effect 5

1.4 Hydrogen Atom Absorption and Emission Spectra 7

1.5 Molecular Spectroscopy 10

1.6 Summary 12

References 12

Problems 12

2 Principles of Quantum Mechanics 15

2.1 Postulates of Quantum Mechanics 16

2.2 The Potential Energy and Potential Functions 20

2.3 Demonstration of Quantum Mechanical Principles for a Simple, One-Dimensional, One-Electron Model System: The Particle in a Box 21

2.3.1 Definition of the Model System 21

2.3.2 Solution of the Particle-in-a-Box Schrödinger Equation 23

2.3.3 Normalization and Orthogonality of the PiB Wavefunctions 25

2.4 The Particle in a Two-Dimensional Box, the Unbound Particle, and the Particle in a Box with Finite Energy Barriers 27

2.4.1 Particle in a 2D Box 27

2.4.2 The Unbound Particle 28

2.4.3 The Particle in a Box with Finite Energy Barriers 29

2.5 Real-World PiBs: Conjugated Polyenes, Quantum Dots, and Quantum Cascade Lasers 31

2.5.1 Transitions in a Conjugated Polyene 31

2.5.2 Quantum Dots 33

2.5.3 Quantum Cascade Lasers 33

References 34

Problems 35

3 Perturbation of Stationary States by Electromagnetic Radiation 37

3.1 Time-Dependent Perturbation Treatment of Stationary-State Systems by Electromagnetic Radiation 37

3.2 Dipole-Allowed Absorption and Emission Transitions and Selection Rules for the Particle in a Box 40

3.3 Einstein Coefficients for the Absorption and Emission of Light 42

3.4 Lasers 45

References 47

Problems 47

4 The Harmonic Oscillator, a Model System for the Vibrations of Diatomic Molecules 49

4.1 Classical Description of a Vibrating Diatomic Model System 49

4.2 The Harmonic Oscillator Schrödinger Equation, Energy Eigenvalues, and Wavefunctions 51

4.3 The Transition Moment and Selection Rules for Absorption for the Harmonic Oscillator 56

4.4 The Anharmonic Oscillator 59

4.5 Vibrational Spectroscopy of Diatomic Molecules 62

4.6 Summary 65

References 66

Problems 66

5 Vibrational Infrared and Raman Spectroscopy of Polyatomic Molecules 69

5.1 Vibrational Energy of Polyatomic Molecules: Normal Coordinates and Normal Modes of Vibration 69

5.2 Quantum Mechanical Description of Molecular Vibrations in Polyatomic Molecules 73

5.3 Infrared Absorption Spectroscopy 76

5.3.1 Symmetry Considerations for Dipole-Allowed Transitions 76

5.3.2 Line Shapes for Absorption and Anomalous Dispersion 77

5.3.2.1 Line Shapes and Lifetimes 77

5.3.2.2 Anomalous Dispersion 79

5.4 Raman Spectroscopy 81

5.4.1 General Aspects of Raman Spectroscopy 81

5.4.2 Macroscopic Description of Polarizability 81

5.4.3 Quantum Mechanical Description of Polarizability 83

5.5 Selection Rules for IR and Raman Spectroscopy of Polyatomic Molecules 87

5.6 Relationship between Infrared and Raman Spectra: Chloroform 88

5.7 Summary: Molecular Vibrations in Science and Technology 90

References 91

Problems 91

6 Rotation of Molecules and Rotational Spectroscopy 93

6.1 Classical Rotational Energy of Diatomic and Polyatomic Molecules 94

6.2 Quantum Mechanical Description of the Angular Momentum Operator 97

6.3 The Rotational Schrödinger Equation, Eigenfunctions, and Rotational Energy Eigenvalues 99

6.4 Selection Rules for Rotational Transitions 104

6.5 Rotational Absorption (Microwave) Spectra 105

6.5.1 Rigid Diatomic and Linear Molecules 105

6.5.2 Prolate and Oblate Symmetric Top Molecules 108

6.5.3 Asymmetric Top Molecules 110

6.6 Rot–Vibrational Transitions 110

References 113

Problems 113

7 Atomic Structure: The Hydrogen Atom 115

7.1 The Hydrogen Atom Schrödinger Equation 116

7.2 Solutions of the Hydrogen Atom Schrödinger Equation 118

7.3 Dipole Allowed Transitions for the Hydrogen Atom 124

7.4 Discussion of the Hydrogen Atom Results 124

7.5 Electron Spin 126

7.6 Spatial Quantization of Angular Momentum 129

References 130

Problems 130

8 Nuclear Magnetic Resonance (NMR) Spectroscopy 131

8.1 General Remarks 131

8.2 Review of Electron Angular Momentum and Spin Angular Momentum 132

8.3 Nuclear Spin 134

8.4 Selection Rules, Transition Energies, Magnetization, and Spin State Population 137

8.4.1 Electric Dipole Selection Rules for a One-Spin Nuclear System 137

8.4.2 Transition Energies 138

8.4.3 Magnetization 138

8.4.4 Spin State Population Analysis 139

8.5 Chemical Shift 140

8.6 Multispin Systems 141

8.6.1 Noninteracting Spins 141

8.6.2 Interacting Spins: Spin–Spin Coupling 143

8.6.3 Interaction of Multiple Spins 144

8.7 Pulse FT NMR Spectroscopy 146

8.7.1 General Comments 146

8.7.2 Description of NMR Event in Terms of the “Net Magnetization” 147

References 148

Problems 149

9 Atomic Structure: Multi-electron Systems 151

9.1 The Two-electron Hamiltonian, Shielding, and Effective Nuclear Charge 151

9.2 The Pauli Principle 152

9.3 The Aufbau Principle 153

9.4 Periodic Properties of Elements 155

9.5 Atomic Energy Levels 156

9.5.1 Good and Bad Quantum Numbers and Term Symbols 156

9.5.2 Selection Rules for Transitions in Atomic Species 159

9.6 Atomic Spectroscopy 160

9.7 Atomic Spectroscopy in Analytical Chemistry 161

References 162

Problems 162

10 Electronic States and Spectroscopy of Polyatomic Molecules 163

10.1 Molecular Orbitals and Chemical Bonding in the H2 ⁺ Molecular Ion 163

10.2 Molecular Orbital Theory for Homonuclear Diatomic Molecules 168

10.3 Term Symbols and Selection Rules for Homonuclear Diatomic Molecules 171

10.4 Electronic Spectra of Diatomic Molecules 173

10.4.1 The Vibronic Absorption Spectrum of Oxygen 173

10.4.2 Vibronic Transitions and the Franck–Condon Principle 175

10.5 Qualitative Description of Electronic Spectra of Polyatomic Molecules 177

10.5.1 Selection Rules for Electronic Transitions 178

10.5.2 Common Electronic Chromophores 178

10.5.2.1 Carbonyl Chromophore 178

10.5.2.2 Olefins 179

10.5.2.3 Benzene 180

10.5.2.4 Other Aromatic Molecules 180

10.5.2.5 Transition Metals in the Electrostatic Field of Ligands 181

10.6 Fluorescence Spectroscopy 181

10.6.1 Fluorescence Energy Level (Jablonski) Diagram 182

10.6.2 Intersystem Crossing and Phosphorescence 183

10.6.3 Two-Photon Fluorescence 183

10.6.4 Summary of Mechanisms for Raman, Resonance Raman, and Fluorescence Spectroscopies 184

10.7 Optical Activity: Electronic Circular Dichroism and Optical Rotation 185

10.7.1 Circularly Polarized Light and Chirality 185

10.7.2 Manifestation of Optical Activity: Optical Rotation, Optical Rotatory Dispersion and Circular Dichroism 187

10.7.3 Optical Activity of Asymmetric Molecules: The Magnetic Transition Moment 188

10.7.4 Optical Activity of Dissymmetric Molecules: Transition Coupling and the Exciton Model 191

10.7.5 Vibrational Optical Activity 192

References 193

Problems 194

11 Group Theory and Symmetry 199

11.1 Symmetry Operations and Symmetry Groups 200

11.2 Group Representations 204

11.3 Symmetry Representations of Molecular Vibrations 211

11.4 Symmetry-Based Selection Rules for Dipole-Allowed Processes 214

11.5 Selection Rules for Raman Scattering 217

11.6 Character Tables of a Few Common Point Groups 218

References 219

Problems 219

Appendix 1 Constants and Conversion Factors 221

Appendix 2 Approximative Methods: Variation and Perturbation Theory 223

A2.1 General Remarks 223

A2.2 Variation Method 224

A2.3 Time-independent Perturbation Theory for Nondegenerate Systems 225

A2.4 Detailed Example of Time-independent Perturbation: The Particle in a Box with a Sloped Potential Function 226

A2.5 Time-dependent Perturbation of Molecular Systems by Electromagnetic

Radiation 230

Reference 231

Appendix 3 Nonlinear Spectroscopic Techniques 233

A3.1 General Formulation of Nonlinear Effects 233

A3.2 Noncoherent Nonlinear Effects: Hyper-Raman Spectroscopy 234

A3.3 Coherent Nonlinear Effects 235

A3.3.1 Second Harmonic Generation 236

A3.3.2 Coherent Anti-Stokes Raman Scattering (CARS) 237

A3.3.3 Stimulated Raman Scattering (SRS) and Femtosecond Stimulated Raman Scattering (FSRS) 240

A3.4 Epilogue 242

References 242

Appendix 4 Fourier Transform (FT) Methodology 243

A4.1 Introduction to Fourier Transform Spectroscopy 243

A4.2 Data Representation in Different Domains 244

A4.3 Fourier Series 244

A4.4 Fourier Transform 247

A4.5 Discrete and Fast Fourier Transform Algorithms 248

A4.6 FT Implementation in EXCEL or MATLAB 249

References 251

Appendix 5 Description of Spin Wavefunctions by Pauli Spin Matrices 253

A5.1 The Formulation of Spin Eigenfunctions 𝛼 and 𝛽 as Vectors 254

A5.2 Form of the Pauli Spin Matrices 255

A5.3 Eigenvalues of the Spin Matrices 256

Reference 257

Index 259