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This book is dedicated to the description and application of various different theoretical models to identify the near and mid-infrared spectra of symmetric and spherical top molecules in their gaseous form.
Theoretical models based on the use of group theory are applied to rigid and non-rigid molecules, characterized by the phenomenon of tunneling and large amplitude motions. The calculation of vibration-rotation energy levels and the analysis of infrared transitions are applied to molecules of ammonia (NH3) and methane (CH4). The applications show how interactions at the molecular scale modify the near and mid-infrared spectra of isolated molecules, under the influence of the pressure of a nano-cage (the substitution site of a rare gas matrix, clathrate, fullerene or zeolite) or a surface, and allow us to identify the characteristics of the perturbing environment.
This book provides valuable support for teachers and researchers but is also intended for engineering students, working research engineers and Master?s and doctorate students.
List of contents
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 (G¯CNP) of a molecule 30
1.6.4. Inversion group epsilon and inversion operations E¯* and permutation-inversion operations P¯* 30
1.6.5. Permutation-inversion group G¯CNPI 31
1.6.6. Group SO(3) isomorphic to permutation-inversion group G¯CNPI 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. Cas
About the author
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.
