Infrared Spectroscopy of Symmetric and Spherical Spindles for Space Observation 1

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Pierre-Richard Dahoo. Infrared Spectroscopy of Symmetric and Spherical Spindles for Space Observation 1

Table of Contents

List of Illustrations

List of Tables

Guide

Pages

Infrared Spectroscopy of Symmetric and Spherical Top Molecules for Space Observation 1

Foreword

Preface

1. Group Theory in Infrared Spectroscopy

1.1. Introduction

1.2. The point-symmetry group of a molecule

1.2.1. Symmetry operations and symmetry elements of a molecule

1.2.2. Point symmetry group and laws of composition

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

1.3.1. Irreducible representations

1.3.1.1. Case of the NH3 molecule

1.3.2. Equivalent representations

1.4. Table of characters and fundamental theorems

1.4.1. Tables of characters, classes and irreducible representations

1.4.2. Irreducible representation of group C3v

1.4.3. Schur’s lemma

1.4.4. Orthogonality and normalization theorem

1.4.5. Orthogonality of lines

1.4.6. Orthogonality of columns

1.4.7. Decomposition of a reducible representation on an irreducible basis

1.4.8. Projection operators for irreducible representations

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

1.5. Overall rotation group symmetry of a molecule

1.6. Full symmetry group of the Hamiltonian of a molecule

1.6.1. Permutation operations

1.6.2. Permutation group Sn

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

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

1.6.5. Permutation-inversion group GCNPI

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

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

1.8. Example of group theory applications

1.9. Conclusion

1.10. Appendices: Groups and Lie algebra of SU(2) and SO(3) 1.10.1. Appendix A: Groups SU(2) and SO(3)

1.10.1.1. The relation between the group SU(2) and the group SO(3)

1.10.1.2. Representations of SU(2) and SO(3)

1.10.2. Appendix B: Lie algebra and SO(3) 1.10.2.1. Definition of an algebra

1.10.2.2. Generators of elements of group SO(3)

1.10.2.3. Definition of a Lie algebra

1.10.2.4. Lie algebra of SU(2) and SO(3)

2. Symmetry of Symmetric and Spherical Top Molecules

2.1. Introduction

2.2. Symmetry group of molecular Hamiltonian

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

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

2.3.2. Symmetry group of asymmetric molecular tops NHD2 and NDH2

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

2.4. Symmetry of CH4 and its isotopologues CD4, CHD3, CDH3 and CH2D2. 2.4.1. Symmetry group of spherical tops CH4 and CD4

2.4.2. Symmetry group of symmetric tops CHD3 and CDH3

2.4.3. Symmetry group of the asymmetric top CH2D2

2.5. Symmetry group of the complete CNPI group

2.6. Conclusion

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

3.1. Introduction

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

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

3.4. Transition operator for the selection rules

3.5. Dipole moment operator and line profile

3.6. Irreducible representations of the vibrations of the molecules

3.6.1. Procedure for the decomposition of the reducible representation

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

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

3.6.4. Case of the asymmetric top XY2Z2 (CH2D2)

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

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

3.7. Types of vibrations of irreducible representations. 3.7.1. Case of symmetric tops XY3 and XZY3 (NH3, ND3, CDH3, CHD3)

3.7.2 Case of spherical top XY4 (CH4, CD4)

3.7.3 Case of the asymmetric top XY2Z2 (CD2H2)

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

3.8. Rotation and spin Hamiltonian symmetries

3.8.1 Vibronic degrees of freedom of NH3 and CH4

3.8.2 Rovibronic degrees of freedom

3.8.3 Rotational degrees of freedom

3.8.4 Spin degrees of freedom

3.8.5 IR and Raman selection rules for the rotational levels

3.9. Conclusion

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

4. Energy Levels of Symmetric Tops in the Gas Phase

4.1. Introduction

4.2. Vibrational–rotational motions of an isolated symmetric top

4.3. Vibrational motions of an isolated pyramidal symmetric top. 4.3.1. Kinetic and potential energy functions

4.3.1.1 Cartesian coordinates

4.3.2. Harmonic oscillators – classical approach

4.3.2.1 Internal coordinates

4.3.2.2 Hypothesis of central forces

4.3.2.3 Hypothesis of valence forces

4.3.2.4 Symmetry coordinates

4.3.3. Separation of vibrational modes

4.3.4. Harmonic oscillators – quantum approach

4.3.5. Molecular vibrations beyond the harmonic approximation

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

4.3.7. Transitions between two vibrational levels: selection rules

4.4. Rotational motion of an isolated rigid symmetric top molecule. 4.4.1. Rotational kinetic Hamiltonian and energy level scheme

4.4.2. Transitions between two rotational levels: selection rules

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

4.6. Application to the ammonia NH3 molecule. 4.6.1. Geometric, rotational and vibrational characteristics

4.6.2. Vibrational motions in the harmonic approximation

4.6.3. Vibrational motions beyond the harmonic approximation

4.6.4. Vibration-inversion mode

4.6.5 Dipole moment as a function of normal coordinates

4.7 Appendices. 4.7.1 Appendix A: Rotation matrix

4.7.2. Appendix B: Expressions of force constants

4.7.3. Appendix C: Rotational moments of transition

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

Box 4.1.Values of anharmonic vibrational force constants

5. Spherical Top CH4

5.1. Introduction

5.2. Characteristics of the CH4 molecule in gas phase

5.3. Tensor formalism for the CH4 molecule

5.3.1. Orientation of SO(3) in Td

5.3.2. Vibrational tensor operators

5.3.2.1. Two-dimensional isotropic oscillator

5.3.2.2. Three-dimensional isotropic oscillator

5.3.2.3. Dijon tensor formalism

5.3.3. Rotational tensor operators

5.3.4. Rovibrational tensor operators

5.3.5. Expression of the rovibrational Hamiltonian

5.3.6. Expression of the vibration wave functions

5.3.7. Expression of rotational wave functions

5.3.8. Expression of rovibrational wave functions

5.4. Application to the CH4 molecule

5.4.1. Electric dipole transition moment

5.4.2. Polarizability

5.5. Rotational structure in the degenerate vibrational levels

5.5.1. The degenerate vibrational level v2 = 1

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

5.5.3. Vibration–rotation Coriolis interaction

5.6. Conclusion

5.7. Appendices. 5.7.1. Appendix A: Quantum mechanics review

5.7.2. Appendix B: Creation and annihilation operators

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

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

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

5.7.6. Appendix F: Hamiltonian transformed using the contact method

References

Index

A, B, C

D, E, F

G, H, I

K, L, M

N, O, P

Q, R

S

T

V, W, X

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Infrared Spectroscopy of Symmetric and Spherical

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Ammonia and methane molecules are members of the class of the molecules with a ternary and quaternary axis of symmetry. The first part is organized into two chapters concerning the symmetry and the use of group theory for the study of symmetric and spherical tops, as well as a review of the line profile of molecules that evolve in nanocages or on surfaces. Then the theoretical models elaborated for the study of vibration–rotation spectra of polyatomic molecules in the gas phase are briefly recalled as developed in molecular physics. They allow the study of both cold and hot bands, as well as the combination bands, and have been extensively described in the books devoted to the spectroscopic study in the gas phase.

In this volume, the theoretical description only covers the study of cold bands, whose spectra result from transitions from the fundamental vibration–rotation level. The IR spectroscopic study concerns the libration or hindered rotation movements that result from a strong coupling with the environment and the constrained rotational movements when the coupling with the environment is weak (rotation at low J).

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