Infrared Spectroscopy of Symmetric and Spherical Spindles for Space Observation 1

Pierre-Richard Dahoo is Professor and Holder of the Chair Materials Simulation and Engineering at the University of Versailles Saint-Quentin in France. He is Director of Institut des Sciences et Techniques des Yvelines and a specialist in modeling and spectroscopy at the LATMOS laboratory of CNRS. Azzedine Lakhlifi is Senior Lecturer at the University of Franche-Comté and a researcher, specializing in modeling and spectroscopy at UTINAM Institute, UMR 6213 CNRS, OSU THETA Franche-Comté Bourgogne, University Bourgogne Franche-Comté, Besançon, France.

Foreword ix
Vincent BOUDON

Preface xi

Chapter 1. Group Theory in Infrared Spectroscopy 1

1.1. Introduction 1

1.2. The point-symmetry group of a molecule 2

1.2.1. Symmetry operations and symmetry elements of a molecule 3

1.2.2. Point symmetry group and laws of composition 6

1.3. Representations by square matrices (general linear group of order n on R or C: GLn(R) or GLn(C)) 10

1.3.1. Irreducible representations 10

1.3.2. Equivalent representations 13

1.4. Table of characters and fundamental theorems 14

1.4.1. Tables of characters, classes and irreducible representations 14

1.4.2. Irreducible representation of group C3v 16

1.4.3. Schur’s lemma 17

1.4.4. Orthogonality and normalization theorem 18

1.4.5. Orthogonality of lines 18

1.4.6. Orthogonality of columns 20

1.4.7. Decomposition of a reducible representation on an irreducible basis 20

1.4.8. Projection operators for irreducible representations 21

1.4.9. Characters of irreducible representations of the direct product of two groups 22

1.5. Overall rotation group symmetry of a molecule 23

1.6. Full symmetry group of the Hamiltonian of a molecule 26

1.6.1. Permutation operations 27

1.6.2. Permutation group Sn 28

1.6.3. Complete nuclear permutation group (GCNP) of a molecule 30

1.6.4. Inversion group ε and inversion operations E* and permutation-inversion operations P* 30

1.6.5. Permutation-inversion group GCNPI 31

1.6.6. Group SO(3) isomorphic to permutation-inversion group GCNPI 32

1.7. Correlation between the rotation group and a point-group symmetry of a molecule 34

1.8. Example of group theory applications 39

1.9. Conclusion 40

1.10. Appendices: Groups and Lie algebra of SU(2) and SO(3) 40

1.10.1. Appendix A: Groups SU(2) and SO(3) 40

1.10.2. Appendix B: Lie algebra and SO(3) 42

Chapter 2. Symmetry of Symmetric and Spherical Top Molecules 45

2.1. Introduction 46

2.2. Symmetry group of molecular Hamiltonian 47

2.3. Symmetry of the NH3 molecule and its isotopologues ND3, NHD2 and NDH2 54

2.3.1. Symmetry group of the symmetric molecular tops NH3 and ND3 54

2.3.2. Symmetry group of asymmetric molecular tops NHD2 and NDH2 56

2.3.3. Symmetry group of the complete group taking into account the inversion 57

2.4. Symmetry of CH4 and its isotopologues CD4, CHD3, CDH3 and CH2D2 59

2.4.1. Symmetry group of spherical tops CH4 and CD4 59

2.4.2. Symmetry group of symmetric tops CHD3 and CDH3 61

2.4.3. Symmetry group of the asymmetric top CH2D2 62

2.5. Symmetry group of the complete CNPI group 62

2.6. Conclusion 66

Chapter 3. Line Profiles, Symmetries and Selection Rules According to Group Theory 67

3.1. Introduction 68

3.2. Symmetries of the eigenstates of the zeroth-order Hamiltonian 70

3.3. Intensity of the vibration–rotation lines and bar spectrum 72

3.4. Transition operator for the selection rules 74

3.5. Dipole moment operator and line profile 77

3.6. Irreducible representations of the vibrations of the molecules 82

3.6.1. Procedure for the decomposition of the reducible representation 82

3.6.2. Case of symmetric tops XY3 and XZY3 (NH3, ND3, CDH3, CHD3) 84

3.6.3. Case of spherical top XY4 (CH4, CD4) 88

3.6.4. Case of the asymmetric top XY2Z2 (CH2D2) 90

3.6.5. Case of the asymmetric top XY2Z (NDH2 or NHD2) 92

3.6.6. Case of inversion for NH3, ND3, NDH2 and NHD2 93

3.7. Types of vibrations of irreducible representations 93

3.7.1. Case of symmetric tops XY3 and XZY3 (NH3, ND3, CDH3, CHD3) 93

3.7.2. Case of spherical top XY4 (CH4, CD4) 100

3.7.3. Case of the asymmetric top XY2Z2 (CD2H2) 105

3.7.4. Case of the asymmetric top XY2Z (NDH2 or NHD2) 108

3.8. Rotation and spin Hamiltonian symmetries 111

3.8.1. Vibronic degrees of freedom of NH3 and CH4 111

3.8.2. Rovibronic degrees of freedom 114

3.8.3. Rotational degrees of freedom 116

3.8.4. Spin degrees of freedom 123

3.8.5. IR and Raman selection rules for the rotational levels 133

3.9. Conclusion 134

3.10. Appendix: Absorption and emission of a molecule in the gas phase 134

Chapter 4. Energy Levels of Symmetric Tops in the Gas Phase 139

4.1. Introduction 140

4.2. Vibrational–rotational motions of an isolated symmetric top 141

4.3. Vibrational motions of an isolated pyramidal symmetric top 146

4.3.1. Kinetic and potential energy functions 146

4.3.2. Harmonic oscillators – classical approach 147

4.3.3. Separation of vibrational modes 160

4.3.4. Harmonic oscillators – quantum approach 161

4.3.5. Molecular vibrations beyond the harmonic approximation 165

4.3.6. Intrinsic inversion phenomenon of certain pyramidal molecules of type XY3 166

4.3.7. Transitions between two vibrational levels: selection rules 169

4.4. Rotational motion of an isolated rigid symmetric top molecule 171

4.4.1. Rotational kinetic Hamiltonian and energy level scheme 171

4.4.2. Transitions between two rotational levels: selection rules 173

4.5. Rovibrational energy levels of an isolated symmetric top and selection rules 175

4.6. Application to the ammonia NH3 molecule 177

4.6.1. Geometric, rotational and vibrational characteristics 177

4.6.2. Vibrational motions in the harmonic approximation 178

4.6.3. Vibrational motions beyond the harmonic approximation 181

4.6.4. Vibration-inversion mode 182

4.6.5. Dipole moment as a function of normal coordinates 184

4.7. Appendices 184

4.7.1. Appendix A: Rotation matrix 184

4.7.2. Appendix B: Expressions of force constants 185

4.7.3. Appendix C: Rotational moments of transition 189

4.7.4. Appendix D: Values of non-zero anharmonic vibrational force constants and corrected eigenvectors 192

Chapter 5. Spherical Top CH4 195

5.1. Introduction 195

5.2. Characteristics of the CH4 molecule in gas phase 199

5.3. Tensor formalism for the CH4 molecule 202

5.3.1. Orientation of SO(3) in Td 204

5.3.2. Vibrational tensor operators 207

5.3.3. Rotational tensor operators 215

5.3.4. Rovibrational tensor operators 218

5.3.5. Expression of the rovibrational Hamiltonian 218

5.3.6. Expression of the vibration wave functions 219

5.3.7. Expression of rotational wave functions 219

5.3.8. Expression of rovibrational wave functions 221

5.4. Application to the CH4 molecule 221

5.4.1. Electric dipole transition moment 222

5.4.2. Polarizability 225

5.5. Rotational structure in the degenerate vibrational levels 228

5.5.1. The degenerate vibrational level v2 = 1 229

5.5.2. The degenerate vibrational level vs = 1 (s = 3 or 4) 230

5.5.3. Vibration–rotation Coriolis interaction 233

5.6. Conclusion 236

5.7. Appendices 236

5.7.1. Appendix A: Quantum mechanics review 236

5.7.2. Appendix B: Creation and annihilation operators 239

5.7.3. Appendix C: Clebsch–Gordan coefficients and Wigner 3j-symbols 241

5.7.4. Appendix D: Tensor operators and the Wigner–Eckart theorem 242

5.7.5. Appendix E: Hamiltonian as a function of dimensionless normal coordinates up to fourth order 243

5.7.6. Appendix F: Hamiltonian transformed using the contact method 245

References 249

Index 259