Pierre-Richard Dahoo is Professor at the University of Versailles St Quentin (UVSQ), researcher at LATMOS, UMR 8190 CNRS, Manager of the University Institute of Technology of Mantes-en-Yvelines and Program Manager of the Chair "Materials Simulation and Engineering" of the UVSQ in Versailles, France.Azzedine Lakhlifi is Lecturer at the University of Franche-Comté, and researcher at UTINAM Institute, UMR 6213 CNRS, OSU THETA Franche-Comté Bourgogne, University Bourgogne Franche-Comté, Besançon, France.

• Foreword ixPreface xiChapter 1 Symmetry of Triatomic Molecules 11.1. Introduction 11.2. The symmetry group of the Hamiltonian of a triatomic molecule 31.3. Symmetry of the nonlinear triatomic molecule (O3) 61.3.1. The nonlinear asymmetric molecule O3 ( 16O16O18O (668)) 81.3.2. The nonlinear symmetric molecule O3 (16O16O16O (666)) 91.3.3. Symmetry of eigenstates of a nonlinear molecule 111.4. Symmetry of the linear triatomic molecule (CO2) 151.4.1. The linear asymmetric molecule CO2 (16O12C18O (628)) 171.4.2. The linear symmetric molecule CO2 (16O12C16O (626)) 191.5. Selection rules 201.5.1. Symmetry of the eigenstates of a triatomic molecule taking into account the nuclei spins 21Chapter 2 Energy Levels of Triatomic Molecules in Gaseous Phase 252.1. Introduction 262.2. Vibrational–rotational movements of an isolated molecule 272.3. Vibrational movements of an isolated triatomic molecule 342.3.1. Nonlinear triatomic molecules 352.3.2. Linear triatomic molecules 362.3.3. Introduction of the perturbative Hamiltonians H1, H2, H3 372.3.4. Transitions between two vibrational levels: selection rules 382.4. Rotational movement of an isolated rigid molecule 402.4.1. Linear triatomic molecules 422.4.2. Symmetric top molecules 422.4.3. Nonlinear triatomic molecules 432.4.4. Transitions between rotational levels 462.5. Vibrational–rotational energy levels of an isolated triatomic molecule 472.6. Rovibrational transitions: selection rules 482.6.1. Dipole moment in terms of normal coordinates 502.7. Appendices 562.7.1. Rotational matrix 562.7.2. Perturbative Hamiltonians of vibration and vibration–rotation coupling 592.7.3. Components of the angular momentum 602.7.4. Rotational Hamiltonian of a symmetric top 602.7.5. Elements of the rotational matrix 612.7.6. Vibrational anharmonic constants 62Chapter 3 Clathrate Nano-Cages 653.1. Introduction 663.2. Clathrate structures 673.3. Inclusion model of a triatomic molecule in a clathrate nano-cage 693.3.1. Inclusion model 693.3.2. Interaction potential energy 713.4. Thermodynamic model of clathrates 733.4.1. Occupation fractions and Langmuir constants 743.4.2. Determination of the Langmuir constants 743.4.3. Application to triatomic molecules 753.5. Infrared spectrum of a triatomic in clathrate matrix 793.5.1. Infrared absorption coefficient 793.5.2. Hamiltonian of the system and separation of movements 793.5.3. Vibrational motions 833.5.4. Orientational motion 833.5.5. Translational motion 843.5.6. Bar spectra 843.6. Application to the CO2 molecule 863.6.1. Vibrational motions 863.6.2. Orientational motion 883.6.3. Translational motion 943.6.4. Bar spectra 953.7. Appendices 983.7.1. Non-zero orientation matrix elements used to calculate the corrections to first-order perturbation energies 983.7.2. Correction to eigenenergies of the orientation Hamiltonian 993.7.3. Expressions of the vector components derivatives of the dipole moment with respect to the normal vibrational coordinates 1023.7.4. Expressions of the orientational transition elements in the approximation of harmonic librators 102Chapter 4 Nano-Cages of Noble Gas Matrices 1074.1. Introduction 1084.2. The theoretical molecule–matrix model 1104.2.1. Site inclusion model 1104.2.2. 12-6 L-J potential 1124.2.3. Site distortion 1164.2.4. Coupling of the molecule–matrix system 1184.2.5. Vibrational frequency displacements 1194.2.6. The calculation of the orientational modes 1234.2.7. Bar spectra and spectral profiles 1244.3. Application to triatomic molecules 1264.3.1. The triatomic molecule C3 1264.3.2. The nonlinear triatomic molecule O3 1354.4. Appendix: Program for determining the equilibrium configuration of an O3 molecule in a noble gas matrix nano-cage 140Chapter 5 Effect of Nano-Cages on Vibration 1455.1. Introduction 1455.2. The theoretical molecule–matrix model 1465.3. Calculation of the shift of vibrational frequencies 1475.3.1. Calculation principle 1475.3.2. Application of the MAPLE program 1515.4. Application to linear triatomic molecules 1555.4.1. Experimental study of linear triatomic molecules (CO2, N2O) 1555.4.2. Frequency shift calculation for degenerate mode ν2 1565.4.3. Calculation results for linear triatomic molecules (CO2, N2O) 1585.5. Appendices 1635.5.1. Transition from Cartesian coordinates to normal coordinates 1635.5.2. MAPLE program for displacement/shifts of vibrational frequency modes of a CO2 molecule in a noble gas nano-cage matrix 166Chapter 6 Adsorption on a Graphite Substrate 1736.1. Molecule adsorbed on a graphite substrate (1000) at low temperature 1736.1.1. Astrophysical context 1736.1.2. Molecule adsorbed onto a graphite substrate 1756.1.3. Graphite substrate–molecule interaction energy 1766.2. Adsorption observables at low temperature 1786.2.1. Equilibrium configuration and potential energy surface 1786.2.2. Adsorption energy 1816.2.3. Diffusion constant 1816.3. Interaction energy between two molecules 1836.3.1. Electrostatic contribution 1846.3.2. Induction contribution 1876.3.3. Dispersion–repulsion contribution 1886.4. Appendices 1886.4.1. Expressions of action tensors 1886.4.2. Multipolar moments and dipolar polarizability of a molecule relative to the fixed (absolute) reference frame 1916.4.3. Code in the FORTRAN language for the calculation of the interaction potential energy between two molecules 191Bibliography 203Index 211

Pierre-Richard Dahoo is Professor at the University of Versailles St Quentin (UVSQ), researcher at LATMOS, UMR 8190 CNRS, Manager of the University Institute of Technology of Mantes-en-Yvelines and Program Manager of the Chair "Materials Simulation and Engineering" of the UVSQ in Versailles, France.Azzedine Lakhlifi is Lecturer at the University of Franche-Comté, and researcher at UTINAM Institute, UMR 6213 CNRS, OSU THETA Franche-Comté Bourgogne, University Bourgogne Franche-Comté, Besançon, France.

• Foreword ixPreface xiChapter 1 Symmetry of Triatomic Molecules 11.1. Introduction 11.2. The symmetry group of the Hamiltonian of a triatomic molecule 31.3. Symmetry of the nonlinear triatomic molecule (O3) 61.3.1. The nonlinear asymmetric molecule O3 ( 16O16O18O (668)) 81.3.2. The nonlinear symmetric molecule O3 (16O16O16O (666)) 91.3.3. Symmetry of eigenstates of a nonlinear molecule 111.4. Symmetry of the linear triatomic molecule (CO2) 151.4.1. The linear asymmetric molecule CO2 (16O12C18O (628)) 171.4.2. The linear symmetric molecule CO2 (16O12C16O (626)) 191.5. Selection rules 201.5.1. Symmetry of the eigenstates of a triatomic molecule taking into account the nuclei spins 21Chapter 2 Energy Levels of Triatomic Molecules in Gaseous Phase 252.1. Introduction 262.2. Vibrational–rotational movements of an isolated molecule 272.3. Vibrational movements of an isolated triatomic molecule 342.3.1. Nonlinear triatomic molecules 352.3.2. Linear triatomic molecules 362.3.3. Introduction of the perturbative Hamiltonians H1, H2, H3 372.3.4. Transitions between two vibrational levels: selection rules 382.4. Rotational movement of an isolated rigid molecule 402.4.1. Linear triatomic molecules 422.4.2. Symmetric top molecules 422.4.3. Nonlinear triatomic molecules 432.4.4. Transitions between rotational levels 462.5. Vibrational–rotational energy levels of an isolated triatomic molecule 472.6. Rovibrational transitions: selection rules 482.6.1. Dipole moment in terms of normal coordinates 502.7. Appendices 562.7.1. Rotational matrix 562.7.2. Perturbative Hamiltonians of vibration and vibration–rotation coupling 592.7.3. Components of the angular momentum 602.7.4. Rotational Hamiltonian of a symmetric top 602.7.5. Elements of the rotational matrix 612.7.6. Vibrational anharmonic constants 62Chapter 3 Clathrate Nano-Cages 653.1. Introduction 663.2. Clathrate structures 673.3. Inclusion model of a triatomic molecule in a clathrate nano-cage 693.3.1. Inclusion model 693.3.2. Interaction potential energy 713.4. Thermodynamic model of clathrates 733.4.1. Occupation fractions and Langmuir constants 743.4.2. Determination of the Langmuir constants 743.4.3. Application to triatomic molecules 753.5. Infrared spectrum of a triatomic in clathrate matrix 793.5.1. Infrared absorption coefficient 793.5.2. Hamiltonian of the system and separation of movements 793.5.3. Vibrational motions 833.5.4. Orientational motion 833.5.5. Translational motion 843.5.6. Bar spectra 843.6. Application to the CO2 molecule 863.6.1. Vibrational motions 863.6.2. Orientational motion 883.6.3. Translational motion 943.6.4. Bar spectra 953.7. Appendices 983.7.1. Non-zero orientation matrix elements used to calculate the corrections to first-order perturbation energies 983.7.2. Correction to eigenenergies of the orientation Hamiltonian 993.7.3. Expressions of the vector components derivatives of the dipole moment with respect to the normal vibrational coordinates 1023.7.4. Expressions of the orientational transition elements in the approximation of harmonic librators 102Chapter 4 Nano-Cages of Noble Gas Matrices 1074.1. Introduction 1084.2. The theoretical molecule–matrix model 1104.2.1. Site inclusion model 1104.2.2. 12-6 L-J potential 1124.2.3. Site distortion 1164.2.4. Coupling of the molecule–matrix system 1184.2.5. Vibrational frequency displacements 1194.2.6. The calculation of the orientational modes 1234.2.7. Bar spectra and spectral profiles 1244.3. Application to triatomic molecules 1264.3.1. The triatomic molecule C3 1264.3.2. The nonlinear triatomic molecule O3 1354.4. Appendix: Program for determining the equilibrium configuration of an O3 molecule in a noble gas matrix nano-cage 140Chapter 5 Effect of Nano-Cages on Vibration 1455.1. Introduction 1455.2. The theoretical molecule–matrix model 1465.3. Calculation of the shift of vibrational frequencies 1475.3.1. Calculation principle 1475.3.2. Application of the MAPLE program 1515.4. Application to linear triatomic molecules 1555.4.1. Experimental study of linear triatomic molecules (CO2, N2O) 1555.4.2. Frequency shift calculation for degenerate mode ν2 1565.4.3. Calculation results for linear triatomic molecules (CO2, N2O) 1585.5. Appendices 1635.5.1. Transition from Cartesian coordinates to normal coordinates 1635.5.2. MAPLE program for displacement/shifts of vibrational frequency modes of a CO2 molecule in a noble gas nano-cage matrix 166Chapter 6 Adsorption on a Graphite Substrate 1736.1. Molecule adsorbed on a graphite substrate (1000) at low temperature 1736.1.1. Astrophysical context 1736.1.2. Molecule adsorbed onto a graphite substrate 1756.1.3. Graphite substrate–molecule interaction energy 1766.2. Adsorption observables at low temperature 1786.2.1. Equilibrium configuration and potential energy surface 1786.2.2. Adsorption energy 1816.2.3. Diffusion constant 1816.3. Interaction energy between two molecules 1836.3.1. Electrostatic contribution 1846.3.2. Induction contribution 1876.3.3. Dispersion–repulsion contribution 1886.4. Appendices 1886.4.1. Expressions of action tensors 1886.4.2. Multipolar moments and dipolar polarizability of a molecule relative to the fixed (absolute) reference frame 1916.4.3. Code in the FORTRAN language for the calculation of the interaction potential energy between two molecules 191Bibliography 203Index 211

Pierre-Richard Dahoo is Professor at the University of Versailles St Quentin (UVSQ), researcher at LATMOS, UMR 8190 CNRS, Manager of the University Institute of Technology of Mantes-en-Yvelines and Program Manager of the Chair "Materials Simulation and Engineering" of the UVSQ in Versailles, France.Azzedine Lakhlifi is Lecturer at the University of Franche-Comté, and researcher at UTINAM Institute, UMR 6213 CNRS, OSU THETA Franche-Comté Bourgogne, University Bourgogne Franche-Comté, Besançon, France.

• Foreword ixPreface xiChapter 1 Symmetry of Triatomic Molecules 11.1. Introduction 11.2. The symmetry group of the Hamiltonian of a triatomic molecule 31.3. Symmetry of the nonlinear triatomic molecule (O3) 61.3.1. The nonlinear asymmetric molecule O3 ( 16O16O18O (668)) 81.3.2. The nonlinear symmetric molecule O3 (16O16O16O (666)) 91.3.3. Symmetry of eigenstates of a nonlinear molecule 111.4. Symmetry of the linear triatomic molecule (CO2) 151.4.1. The linear asymmetric molecule CO2 (16O12C18O (628)) 171.4.2. The linear symmetric molecule CO2 (16O12C16O (626)) 191.5. Selection rules 201.5.1. Symmetry of the eigenstates of a triatomic molecule taking into account the nuclei spins 21Chapter 2 Energy Levels of Triatomic Molecules in Gaseous Phase 252.1. Introduction 262.2. Vibrational–rotational movements of an isolated molecule 272.3. Vibrational movements of an isolated triatomic molecule 342.3.1. Nonlinear triatomic molecules 352.3.2. Linear triatomic molecules 362.3.3. Introduction of the perturbative Hamiltonians H1, H2, H3 372.3.4. Transitions between two vibrational levels: selection rules 382.4. Rotational movement of an isolated rigid molecule 402.4.1. Linear triatomic molecules 422.4.2. Symmetric top molecules 422.4.3. Nonlinear triatomic molecules 432.4.4. Transitions between rotational levels 462.5. Vibrational–rotational energy levels of an isolated triatomic molecule 472.6. Rovibrational transitions: selection rules 482.6.1. Dipole moment in terms of normal coordinates 502.7. Appendices 562.7.1. Rotational matrix 562.7.2. Perturbative Hamiltonians of vibration and vibration–rotation coupling 592.7.3. Components of the angular momentum 602.7.4. Rotational Hamiltonian of a symmetric top 602.7.5. Elements of the rotational matrix 612.7.6. Vibrational anharmonic constants 62Chapter 3 Clathrate Nano-Cages 653.1. Introduction 663.2. Clathrate structures 673.3. Inclusion model of a triatomic molecule in a clathrate nano-cage 693.3.1. Inclusion model 693.3.2. Interaction potential energy 713.4. Thermodynamic model of clathrates 733.4.1. Occupation fractions and Langmuir constants 743.4.2. Determination of the Langmuir constants 743.4.3. Application to triatomic molecules 753.5. Infrared spectrum of a triatomic in clathrate matrix 793.5.1. Infrared absorption coefficient 793.5.2. Hamiltonian of the system and separation of movements 793.5.3. Vibrational motions 833.5.4. Orientational motion 833.5.5. Translational motion 843.5.6. Bar spectra 843.6. Application to the CO2 molecule 863.6.1. Vibrational motions 863.6.2. Orientational motion 883.6.3. Translational motion 943.6.4. Bar spectra 953.7. Appendices 983.7.1. Non-zero orientation matrix elements used to calculate the corrections to first-order perturbation energies 983.7.2. Correction to eigenenergies of the orientation Hamiltonian 993.7.3. Expressions of the vector components derivatives of the dipole moment with respect to the normal vibrational coordinates 1023.7.4. Expressions of the orientational transition elements in the approximation of harmonic librators 102Chapter 4 Nano-Cages of Noble Gas Matrices 1074.1. Introduction 1084.2. The theoretical molecule–matrix model 1104.2.1. Site inclusion model 1104.2.2. 12-6 L-J potential 1124.2.3. Site distortion 1164.2.4. Coupling of the molecule–matrix system 1184.2.5. Vibrational frequency displacements 1194.2.6. The calculation of the orientational modes 1234.2.7. Bar spectra and spectral profiles 1244.3. Application to triatomic molecules 1264.3.1. The triatomic molecule C3 1264.3.2. The nonlinear triatomic molecule O3 1354.4. Appendix: Program for determining the equilibrium configuration of an O3 molecule in a noble gas matrix nano-cage 140Chapter 5 Effect of Nano-Cages on Vibration 1455.1. Introduction 1455.2. The theoretical molecule–matrix model 1465.3. Calculation of the shift of vibrational frequencies 1475.3.1. Calculation principle 1475.3.2. Application of the MAPLE program 1515.4. Application to linear triatomic molecules 1555.4.1. Experimental study of linear triatomic molecules (CO2, N2O) 1555.4.2. Frequency shift calculation for degenerate mode ν2 1565.4.3. Calculation results for linear triatomic molecules (CO2, N2O) 1585.5. Appendices 1635.5.1. Transition from Cartesian coordinates to normal coordinates 1635.5.2. MAPLE program for displacement/shifts of vibrational frequency modes of a CO2 molecule in a noble gas nano-cage matrix 166Chapter 6 Adsorption on a Graphite Substrate 1736.1. Molecule adsorbed on a graphite substrate (1000) at low temperature 1736.1.1. Astrophysical context 1736.1.2. Molecule adsorbed onto a graphite substrate 1756.1.3. Graphite substrate–molecule interaction energy 1766.2. Adsorption observables at low temperature 1786.2.1. Equilibrium configuration and potential energy surface 1786.2.2. Adsorption energy 1816.2.3. Diffusion constant 1816.3. Interaction energy between two molecules 1836.3.1. Electrostatic contribution 1846.3.2. Induction contribution 1876.3.3. Dispersion–repulsion contribution 1886.4. Appendices 1886.4.1. Expressions of action tensors 1886.4.2. Multipolar moments and dipolar polarizability of a molecule relative to the fixed (absolute) reference frame 1916.4.3. Code in the FORTRAN language for the calculation of the interaction potential energy between two molecules 191Bibliography 203Index 211