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.

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• Foreword ixVincent BOUDONPreface xiChapter 1. Group Theory in Infrared Spectroscopy 11.1. Introduction 11.2. The point-symmetry group of a molecule 21.2.1. Symmetry operations and symmetry elements of a molecule 31.2.2. Point symmetry group and laws of composition 61.3. Representations by square matrices (general linear group of order n on R or C: GLn(R) or GLn(C)) 101.3.1. Irreducible representations 101.3.2. Equivalent representations 131.4. Table of characters and fundamental theorems 141.4.1. Tables of characters, classes and irreducible representations 141.4.2. Irreducible representation of group C3v 161.4.3. Schur’s lemma 171.4.4. Orthogonality and normalization theorem 181.4.5. Orthogonality of lines 181.4.6. Orthogonality of columns 201.4.7. Decomposition of a reducible representation on an irreducible basis 201.4.8. Projection operators for irreducible representations 211.4.9. Characters of irreducible representations of the direct product of two groups 221.5. Overall rotation group symmetry of a molecule 231.6. Full symmetry group of the Hamiltonian of a molecule 261.6.1. Permutation operations 271.6.2. Permutation group Sn 281.6.3. Complete nuclear permutation group (GCNP) of a molecule 301.6.4. Inversion group ε and inversion operations E* and permutation-inversion operations P* 301.6.5. Permutation-inversion group GCNPI 311.6.6. Group SO(3) isomorphic to permutation-inversion group GCNPI 321.7. Correlation between the rotation group and a point-group symmetry of a molecule 341.8. Example of group theory applications 391.9. Conclusion 401.10. Appendices: Groups and Lie algebra of SU(2) and SO(3) 401.10.1. Appendix A: Groups SU(2) and SO(3) 401.10.2. Appendix B: Lie algebra and SO(3) 42Chapter 2. Symmetry of Symmetric and Spherical Top Molecules 452.1. Introduction 462.2. Symmetry group of molecular Hamiltonian 472.3. Symmetry of the NH3 molecule and its isotopologues ND3, NHD2 and NDH2 542.3.1. Symmetry group of the symmetric molecular tops NH3 and ND3 542.3.2. Symmetry group of asymmetric molecular tops NHD2 and NDH2 562.3.3. Symmetry group of the complete group taking into account the inversion 572.4. Symmetry of CH4 and its isotopologues CD4, CHD3, CDH3 and CH2D2 592.4.1. Symmetry group of spherical tops CH4 and CD4 592.4.2. Symmetry group of symmetric tops CHD3 and CDH3 612.4.3. Symmetry group of the asymmetric top CH2D2 622.5. Symmetry group of the complete CNPI group 622.6. Conclusion 66Chapter 3. Line Profiles, Symmetries and Selection Rules According to Group Theory 673.1. Introduction 683.2. Symmetries of the eigenstates of the zeroth-order Hamiltonian 703.3. Intensity of the vibration–rotation lines and bar spectrum 723.4. Transition operator for the selection rules 743.5. Dipole moment operator and line profile 773.6. Irreducible representations of the vibrations of the molecules 823.6.1. Procedure for the decomposition of the reducible representation 823.6.2. Case of symmetric tops XY3 and XZY3 (NH3, ND3, CDH3, CHD3) 843.6.3. Case of spherical top XY4 (CH4, CD4) 883.6.4. Case of the asymmetric top XY2Z2 (CH2D2) 903.6.5. Case of the asymmetric top XY2Z (NDH2 or NHD2) 923.6.6. Case of inversion for NH3, ND3, NDH2 and NHD2 933.7. Types of vibrations of irreducible representations 933.7.1. Case of symmetric tops XY3 and XZY3 (NH3, ND3, CDH3, CHD3) 933.7.2. Case of spherical top XY4 (CH4, CD4) 1003.7.3. Case of the asymmetric top XY2Z2 (CD2H2) 1053.7.4. Case of the asymmetric top XY2Z (NDH2 or NHD2) 1083.8. Rotation and spin Hamiltonian symmetries 1113.8.1. Vibronic degrees of freedom of NH3 and CH4 1113.8.2. Rovibronic degrees of freedom 1143.8.3. Rotational degrees of freedom 1163.8.4. Spin degrees of freedom 1233.8.5. IR and Raman selection rules for the rotational levels 1333.9. Conclusion 1343.10. Appendix: Absorption and emission of a molecule in the gas phase 134Chapter 4. Energy Levels of Symmetric Tops in the Gas Phase 1394.1. Introduction 1404.2. Vibrational–rotational motions of an isolated symmetric top 1414.3. Vibrational motions of an isolated pyramidal symmetric top 1464.3.1. Kinetic and potential energy functions 1464.3.2. Harmonic oscillators – classical approach 1474.3.3. Separation of vibrational modes 1604.3.4. Harmonic oscillators – quantum approach 1614.3.5. Molecular vibrations beyond the harmonic approximation 1654.3.6. Intrinsic inversion phenomenon of certain pyramidal molecules of type XY3 1664.3.7. Transitions between two vibrational levels: selection rules 1694.4. Rotational motion of an isolated rigid symmetric top molecule 1714.4.1. Rotational kinetic Hamiltonian and energy level scheme 1714.4.2. Transitions between two rotational levels: selection rules 1734.5. Rovibrational energy levels of an isolated symmetric top and selection rules 1754.6. Application to the ammonia NH3 molecule 1774.6.1. Geometric, rotational and vibrational characteristics 1774.6.2. Vibrational motions in the harmonic approximation 1784.6.3. Vibrational motions beyond the harmonic approximation 1814.6.4. Vibration-inversion mode 1824.6.5. Dipole moment as a function of normal coordinates 1844.7. Appendices 1844.7.1. Appendix A: Rotation matrix 1844.7.2. Appendix B: Expressions of force constants 1854.7.3. Appendix C: Rotational moments of transition 1894.7.4. Appendix D: Values of non-zero anharmonic vibrational force constants and corrected eigenvectors 192Chapter 5. Spherical Top CH4 1955.1. Introduction 1955.2. Characteristics of the CH4 molecule in gas phase 1995.3. Tensor formalism for the CH4 molecule 2025.3.1. Orientation of SO(3) in Td 2045.3.2. Vibrational tensor operators 2075.3.3. Rotational tensor operators 2155.3.4. Rovibrational tensor operators 2185.3.5. Expression of the rovibrational Hamiltonian 2185.3.6. Expression of the vibration wave functions 2195.3.7. Expression of rotational wave functions 2195.3.8. Expression of rovibrational wave functions 2215.4. Application to the CH4 molecule 2215.4.1. Electric dipole transition moment 2225.4.2. Polarizability 2255.5. Rotational structure in the degenerate vibrational levels 2285.5.1. The degenerate vibrational level v2 = 1 2295.5.2. The degenerate vibrational level vs = 1 (s = 3 or 4) 2305.5.3. Vibration–rotation Coriolis interaction 2335.6. Conclusion 2365.7. Appendices 2365.7.1. Appendix A: Quantum mechanics review 2365.7.2. Appendix B: Creation and annihilation operators 2395.7.3. Appendix C: Clebsch–Gordan coefficients and Wigner 3j-symbols 2415.7.4. Appendix D: Tensor operators and the Wigner–Eckart theorem 2425.7.5. Appendix E: Hamiltonian as a function of dimensionless normal coordinates up to fourth order 2435.7.6. Appendix F: Hamiltonian transformed using the contact method 245References 249Index 259

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.

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• Foreword ixVincent BOUDONPreface xiChapter 1. Group Theory in Infrared Spectroscopy 11.1. Introduction 11.2. The point-symmetry group of a molecule 21.2.1. Symmetry operations and symmetry elements of a molecule 31.2.2. Point symmetry group and laws of composition 61.3. Representations by square matrices (general linear group of order n on R or C: GLn(R) or GLn(C)) 101.3.1. Irreducible representations 101.3.2. Equivalent representations 131.4. Table of characters and fundamental theorems 141.4.1. Tables of characters, classes and irreducible representations 141.4.2. Irreducible representation of group C3v 161.4.3. Schur’s lemma 171.4.4. Orthogonality and normalization theorem 181.4.5. Orthogonality of lines 181.4.6. Orthogonality of columns 201.4.7. Decomposition of a reducible representation on an irreducible basis 201.4.8. Projection operators for irreducible representations 211.4.9. Characters of irreducible representations of the direct product of two groups 221.5. Overall rotation group symmetry of a molecule 231.6. Full symmetry group of the Hamiltonian of a molecule 261.6.1. Permutation operations 271.6.2. Permutation group Sn 281.6.3. Complete nuclear permutation group (GCNP) of a molecule 301.6.4. Inversion group ε and inversion operations E* and permutation-inversion operations P* 301.6.5. Permutation-inversion group GCNPI 311.6.6. Group SO(3) isomorphic to permutation-inversion group GCNPI 321.7. Correlation between the rotation group and a point-group symmetry of a molecule 341.8. Example of group theory applications 391.9. Conclusion 401.10. Appendices: Groups and Lie algebra of SU(2) and SO(3) 401.10.1. Appendix A: Groups SU(2) and SO(3) 401.10.2. Appendix B: Lie algebra and SO(3) 42Chapter 2. Symmetry of Symmetric and Spherical Top Molecules 452.1. Introduction 462.2. Symmetry group of molecular Hamiltonian 472.3. Symmetry of the NH3 molecule and its isotopologues ND3, NHD2 and NDH2 542.3.1. Symmetry group of the symmetric molecular tops NH3 and ND3 542.3.2. Symmetry group of asymmetric molecular tops NHD2 and NDH2 562.3.3. Symmetry group of the complete group taking into account the inversion 572.4. Symmetry of CH4 and its isotopologues CD4, CHD3, CDH3 and CH2D2 592.4.1. Symmetry group of spherical tops CH4 and CD4 592.4.2. Symmetry group of symmetric tops CHD3 and CDH3 612.4.3. Symmetry group of the asymmetric top CH2D2 622.5. Symmetry group of the complete CNPI group 622.6. Conclusion 66Chapter 3. Line Profiles, Symmetries and Selection Rules According to Group Theory 673.1. Introduction 683.2. Symmetries of the eigenstates of the zeroth-order Hamiltonian 703.3. Intensity of the vibration–rotation lines and bar spectrum 723.4. Transition operator for the selection rules 743.5. Dipole moment operator and line profile 773.6. Irreducible representations of the vibrations of the molecules 823.6.1. Procedure for the decomposition of the reducible representation 823.6.2. Case of symmetric tops XY3 and XZY3 (NH3, ND3, CDH3, CHD3) 843.6.3. Case of spherical top XY4 (CH4, CD4) 883.6.4. Case of the asymmetric top XY2Z2 (CH2D2) 903.6.5. Case of the asymmetric top XY2Z (NDH2 or NHD2) 923.6.6. Case of inversion for NH3, ND3, NDH2 and NHD2 933.7. Types of vibrations of irreducible representations 933.7.1. Case of symmetric tops XY3 and XZY3 (NH3, ND3, CDH3, CHD3) 933.7.2. Case of spherical top XY4 (CH4, CD4) 1003.7.3. Case of the asymmetric top XY2Z2 (CD2H2) 1053.7.4. Case of the asymmetric top XY2Z (NDH2 or NHD2) 1083.8. Rotation and spin Hamiltonian symmetries 1113.8.1. Vibronic degrees of freedom of NH3 and CH4 1113.8.2. Rovibronic degrees of freedom 1143.8.3. Rotational degrees of freedom 1163.8.4. Spin degrees of freedom 1233.8.5. IR and Raman selection rules for the rotational levels 1333.9. Conclusion 1343.10. Appendix: Absorption and emission of a molecule in the gas phase 134Chapter 4. Energy Levels of Symmetric Tops in the Gas Phase 1394.1. Introduction 1404.2. Vibrational–rotational motions of an isolated symmetric top 1414.3. Vibrational motions of an isolated pyramidal symmetric top 1464.3.1. Kinetic and potential energy functions 1464.3.2. Harmonic oscillators – classical approach 1474.3.3. Separation of vibrational modes 1604.3.4. Harmonic oscillators – quantum approach 1614.3.5. Molecular vibrations beyond the harmonic approximation 1654.3.6. Intrinsic inversion phenomenon of certain pyramidal molecules of type XY3 1664.3.7. Transitions between two vibrational levels: selection rules 1694.4. Rotational motion of an isolated rigid symmetric top molecule 1714.4.1. Rotational kinetic Hamiltonian and energy level scheme 1714.4.2. Transitions between two rotational levels: selection rules 1734.5. Rovibrational energy levels of an isolated symmetric top and selection rules 1754.6. Application to the ammonia NH3 molecule 1774.6.1. Geometric, rotational and vibrational characteristics 1774.6.2. Vibrational motions in the harmonic approximation 1784.6.3. Vibrational motions beyond the harmonic approximation 1814.6.4. Vibration-inversion mode 1824.6.5. Dipole moment as a function of normal coordinates 1844.7. Appendices 1844.7.1. Appendix A: Rotation matrix 1844.7.2. Appendix B: Expressions of force constants 1854.7.3. Appendix C: Rotational moments of transition 1894.7.4. Appendix D: Values of non-zero anharmonic vibrational force constants and corrected eigenvectors 192Chapter 5. Spherical Top CH4 1955.1. Introduction 1955.2. Characteristics of the CH4 molecule in gas phase 1995.3. Tensor formalism for the CH4 molecule 2025.3.1. Orientation of SO(3) in Td 2045.3.2. Vibrational tensor operators 2075.3.3. Rotational tensor operators 2155.3.4. Rovibrational tensor operators 2185.3.5. Expression of the rovibrational Hamiltonian 2185.3.6. Expression of the vibration wave functions 2195.3.7. Expression of rotational wave functions 2195.3.8. Expression of rovibrational wave functions 2215.4. Application to the CH4 molecule 2215.4.1. Electric dipole transition moment 2225.4.2. Polarizability 2255.5. Rotational structure in the degenerate vibrational levels 2285.5.1. The degenerate vibrational level v2 = 1 2295.5.2. The degenerate vibrational level vs = 1 (s = 3 or 4) 2305.5.3. Vibration–rotation Coriolis interaction 2335.6. Conclusion 2365.7. Appendices 2365.7.1. Appendix A: Quantum mechanics review 2365.7.2. Appendix B: Creation and annihilation operators 2395.7.3. Appendix C: Clebsch–Gordan coefficients and Wigner 3j-symbols 2415.7.4. Appendix D: Tensor operators and the Wigner–Eckart theorem 2425.7.5. Appendix E: Hamiltonian as a function of dimensionless normal coordinates up to fourth order 2435.7.6. Appendix F: Hamiltonian transformed using the contact method 245References 249Index 259