A concise textbook bridging quantum theory and spectroscopy!
Designed as a practical text, Quantum Mechanical Foundations of Molecular Spectroscopy covers the quantum mechanical fundamentals of molecular spectroscopy from the view of a professional spectroscopist, rather than a theoretician. Written by a noted expert on the topic, the book puts the emphasis on the relationship between spectroscopy and quantum mechanics, and provides the background information and derivations of the subjects needed to understand spectroscopy including: stationary energy states, transitions between these states, selection rules, and symmetry.
The phenomenal growth of all forms of spectroscopy over the past eight decades has contributed enormously to our understanding of molecular structure and properties. Today spectroscopy covers a broad field including the modern magnetic resonance techniques, non-linear, laser and fiber-based spectroscopy, surface and surface-enhanced spectroscopy, pico- and femtosecond time resolved spectroscopy, and many more. This up-to-date resource discusses several forms of spectroscopy that are used in many fields of science, such as fluorescence, surface spectroscopies, linear and non-linear Raman spectroscopy and spin spectroscopy. This important text:
- Contains the physics and mathematics needed to understand spectroscopy
- Explores spectroscopic methods the are widely used in chemistry, biophysics, biology, and materials science
- Offers a text written by an experienced lecturer and practitioner of spectroscopic methods
- Includes detailed explanations and worked examples
Written for chemistry, biochemistry, material sciences, and physics students, Quantum Mechanical Foundations of Molecular Spectroscopy provides an accessible text for understanding molecular spectroscopy.
Table of Contents
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
