This book is dedicated to the application of the different theoretical models described in Volume 1 to identify the near-, mid- and far-infrared spectra of linear and nonlinear triatomic molecules in gaseous phase or subjected to environmental constraints, useful for the study of environmental sciences, planetology and astrophysics.
The Van Vleck contact transformation method, described in Volume 1, is applied in the calculation and analysis of IR transitions between vibration-rotation energy levels. The extended Lakhlifi-Dahoo substitution model is used in the framework of Liouville’s formalism and the line profiles of triatomic molecules and their isotopologues subjected to environmental constraints are calculated by applying the cumulant expansion.
The applications presented in this book show how interactions at the molecular level modify the infrared spectra of triatomics trapped in a nano-cage (substitution site of a rare gas matrix, clathrate, fullerene, zeolite) or adsorbed on a surface, and how these interactions may be used to identify the characteristics of the perturbing environment.
Table of Contents
Foreword ix
Preface xi
Chapter 1 Symmetry of Triatomic Molecules 1
1.1. Introduction 1
1.2. The symmetry group of the Hamiltonian of a triatomic molecule 3
1.3. Symmetry of the nonlinear triatomic molecule (O3) 6
1.3.1. The nonlinear asymmetric molecule O3 ( 16O16O18O (668)) 8
1.3.2. The nonlinear symmetric molecule O3 (16O16O16O (666)) 9
1.3.3. Symmetry of eigenstates of a nonlinear molecule 11
1.4. Symmetry of the linear triatomic molecule (CO2) 15
1.4.1. The linear asymmetric molecule CO2 (16O12C18O (628)) 17
1.4.2. The linear symmetric molecule CO2 (16O12C16O (626)) 19
1.5. Selection rules 20
1.5.1. Symmetry of the eigenstates of a triatomic molecule taking into account the nuclei spins 21
Chapter 2 Energy Levels of Triatomic Molecules in Gaseous Phase 25
2.1. Introduction 26
2.2. Vibrational-rotational movements of an isolated molecule 27
2.3. Vibrational movements of an isolated triatomic molecule 34
2.3.1. Nonlinear triatomic molecules 35
2.3.2. Linear triatomic molecules 36
2.3.3. Introduction of the perturbative Hamiltonians H1, H2, H3 37
2.3.4. Transitions between two vibrational levels: selection rules 38
2.4. Rotational movement of an isolated rigid molecule 40
2.4.1. Linear triatomic molecules 42
2.4.2. Symmetric top molecules 42
2.4.3. Nonlinear triatomic molecules 43
2.4.4. Transitions between rotational levels 46
2.5. Vibrational-rotational energy levels of an isolated triatomic molecule 47
2.6. Rovibrational transitions: selection rules 48
2.6.1. Dipole moment in terms of normal coordinates 50
2.7. Appendices 56
2.7.1. Rotational matrix 56
2.7.2. Perturbative Hamiltonians of vibration and vibration-rotation coupling 59
2.7.3. Components of the angular momentum 60
2.7.4. Rotational Hamiltonian of a symmetric top 60
2.7.5. Elements of the rotational matrix 61
2.7.6. Vibrational anharmonic constants 62
Chapter 3 Clathrate Nano-Cages 65
3.1. Introduction 66
3.2. Clathrate structures 67
3.3. Inclusion model of a triatomic molecule in a clathrate nano-cage 69
3.3.1. Inclusion model 69
3.3.2. Interaction potential energy 71
3.4. Thermodynamic model of clathrates 73
3.4.1. Occupation fractions and Langmuir constants 74
3.4.2. Determination of the Langmuir constants 74
3.4.3. Application to triatomic molecules 75
3.5. Infrared spectrum of a triatomic in clathrate matrix 79
3.5.1. Infrared absorption coefficient 79
3.5.2. Hamiltonian of the system and separation of movements 79
3.5.3. Vibrational motions 83
3.5.4. Orientational motion 83
3.5.5. Translational motion 84
3.5.6. Bar spectra 84
3.6. Application to the CO2 molecule 86
3.6.1. Vibrational motions 86
3.6.2. Orientational motion 88
3.6.3. Translational motion 94
3.6.4. Bar spectra 95
3.7. Appendices 98
3.7.1. Non-zero orientation matrix elements used to calculate the corrections to first-order perturbation energies 98
3.7.2. Correction to eigenenergies of the orientation Hamiltonian 99
3.7.3. Expressions of the vector components derivatives of the dipole moment with respect to the normal vibrational coordinates 102
3.7.4. Expressions of the orientational transition elements in the approximation of harmonic librators 102
Chapter 4 Nano-Cages of Noble Gas Matrices 107
4.1. Introduction 108
4.2. The theoretical molecule-matrix model 110
4.2.1. Site inclusion model 110
4.2.2. 12-6 L-J potential 112
4.2.3. Site distortion 116
4.2.4. Coupling of the molecule-matrix system 118
4.2.5. Vibrational frequency displacements 119
4.2.6. The calculation of the orientational modes 123
4.2.7. Bar spectra and spectral profiles 124
4.3. Application to triatomic molecules 126
4.3.1. The triatomic molecule C3 126
4.3.2. The nonlinear triatomic molecule O3 135
4.4. Appendix: Program for determining the equilibrium configuration of an O3 molecule in a noble gas matrix nano-cage 140
Chapter 5 Effect of Nano-Cages on Vibration 145
5.1. Introduction 145
5.2. The theoretical molecule-matrix model 146
5.3. Calculation of the shift of vibrational frequencies 147
5.3.1. Calculation principle 147
5.3.2. Application of the MAPLE program 151
5.4. Application to linear triatomic molecules 155
5.4.1. Experimental study of linear triatomic molecules (CO2, N2O) 155
5.4.2. Frequency shift calculation for degenerate mode ν2 156
5.4.3. Calculation results for linear triatomic molecules (CO2, N2O) 158
5.5. Appendices 163
5.5.1. Transition from Cartesian coordinates to normal coordinates 163
5.5.2. MAPLE program for displacement/shifts of vibrational frequency modes of a CO2 molecule in a noble gas nano-cage matrix 166
Chapter 6 Adsorption on a Graphite Substrate 173
6.1. Molecule adsorbed on a graphite substrate (1000) at low temperature 173
6.1.1. Astrophysical context 173
6.1.2. Molecule adsorbed onto a graphite substrate 175
6.1.3. Graphite substrate-molecule interaction energy 176
6.2. Adsorption observables at low temperature 178
6.2.1. Equilibrium configuration and potential energy surface 178
6.2.2. Adsorption energy 181
6.2.3. Diffusion constant 181
6.3. Interaction energy between two molecules 183
6.3.1. Electrostatic contribution 184
6.3.2. Induction contribution 187
6.3.3. Dispersion-repulsion contribution 188
6.4. Appendices 188
6.4.1. Expressions of action tensors 188
6.4.2. Multipolar moments and dipolar polarizability of a molecule relative to the fixed (absolute) reference frame 191
6.4.3. Code in the FORTRAN language for the calculation of the interaction potential energy between two molecules 191
Bibliography 203
Index 211
