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

1 Cover

2 Title Page Infrared Spectroscopy of Symmetric and Spherical coordinated by Pierre Richard Dahoo and Azzedine Lakhlifi Volume 3

3 Copyright First published 2021 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK www.iste.co.uk John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA www.wiley.com © ISTE Ltd 2021 The rights of Pierre Richard Dahoo and Azzedine Lakhlifi to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2020951963 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-568-8

4 Foreword

5 Preface

6 1 Group Theory in Infrared Spectroscopy 1.1. Introduction 1.2. The point-symmetry group of a molecule 1.3. Representations by square matrices (general linear group of order non Ror C: GLn(R) or GLn(C)) 1.4. Table of characters and fundamental theorems 1.5. Overall rotation group symmetry of a molecule 1.6. Full symmetry group of the Hamiltonian of a molecule 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)

7 2 Symmetry of Symmetric and Spherical Top Molecules 2.1. Introduction 2.2. Symmetry group of molecular Hamiltonian 2.3. Symmetry of the NH 3molecule and its isotopologues ND 3, NHD 2and NDH 2 2.4. Symmetry of CH 4and its isotopologues CD 4, CHD 3, CDH 3and CH 2D 2 2.5. Symmetry group of the complete CNPI group 2.6. Conclusion

8 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 3.7. Types of vibrations of irreducible representations 3.8. Rotation and spin Hamiltonian symmetries 3.9. Conclusion 3.10. Appendix: Absorption and emission of a molecule in the gas phase

9 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 4.4. Rotational motion of an isolated rigid symmetric top molecule 4.5. Rovibrational energy levels of an isolated symmetric top and selection rules 4.6. Application to the ammonia NH 3molecule 4.7. Appendices

10 5 Spherical Top CH 4 5.1. Introduction 5.2. Characteristics of the CH 4molecule in gas phase 5.3. Tensor formalism for the CH 4molecule 5.4. Application to the CH 4molecule 5.5. Rotational structure in the degenerate vibrational levels 5.6. Conclusion 5.7. Appendices

11 References

12 Index

13 End User License Agreement

1 Chapter 1 Figure 1.1. Symmetry operations and symmetry elements of a molecule Figure 1.2. Equivalent symmetry operations of plane rotation and reflectionover... Figure 1.3. Symmetry elements of the NH3 molecule and its reversed equilibrium c... Figure 1.4. Symmetry operations and flow diagram of point groupsaccording to th... Figure 1.5. The operations equivalent to rotational symmetry (cyclic group of or... Figure 1.6. The effect of operation E* on a CH3D molecule. For a color version o...

2 Chapter 2Figure 2.1. Mobile reference frame and the two equilibrium configurations of the...Figure 2.2. Potential with two minima depending on the distance h between the ni...Figure 2.3. The symmetry elements of the NDH2 molecule and its reverse equilibri...Figure 2.4. Symmetry elements of the CH4 molecule. For a color version of this f...Figure 2.5. The symmetry element of proper rotation C3 and two of three planes o...Figure 2.6. Element of symmetry of proper rotation C2 and the two planes of symm...

3 Chapter 3Figure 3.1. Internal coordinates of NH3 and CDH3. For a color version of this fi...Figure 3.2. Internal coordinates ri of the CH4 molecule inscribed in a cube. For...Figure 3.3. Internal coordinates bi of the CH4 4molecule inscribed in a cube. Fo...Figure 3.4. Internal coordinates of the NDH2 molecule. For a color version of th...

4 Chapter 4Figure 4.1. Geometrical characteristics of a symmetric top molecule of type XY3 ...Figure 4.2. Definition of the fixed direct orthonormal systems of axes: (O, , ...Figure 4.3. Vibration modes of a symmetric top molecule of type XY3. The arrows ...Figure 4.4. System of axes (G, , , ) attached to the equilibrium configuratio...Figure 4.5. Displacement vectors of the nuclei, at an instant t, during a vibrat...Figure 4.6. Definition of the system of symmetry coordinates of a symmetric top ...Figure 4.7. Vibration–inversion mode of a pyramidal symmetric top molecule XY3. ...Figure 4.8. Rotational levels scheme of an oblate symmetric top of type XY3. The...Figure 4.9. Potential energy of the vibration–inversion mode ν 2of an ammonia NH...

5 Chapter 5Figure 5.1. Normal modes of vibration of CH4: a) stretching ν1 of type A1; b) be...Figure 5.2. Spectrum of the degenerate three-dimensional harmonic oscillator [ME...Figure 5.3. a) Angular momentum in the physical space; b) angular momentum in th...Figure 5.4. a) Composition of angular momenta; b) projections of angular momenta Figure 5.5. Possible transitions of a cold band from the fundamental level to th...

1 Chapter 1 Table 1.1. Schoenflies and Hermann–Mauguin notations Table 1.2. (Cayley) group C3v multiplication table Table 1.3. The point groups of 32 classes of symmetry and the equivalence betwee... Table 1.4. Table of characters giving the classes (number of elements hi) of the... Table 1.5. Tables of characters of symmetry operations (Ok), classes of operatio... Table 1.6. Table of characters of the symmetry operations (Ok) for each nucleus ... Table 1.7. Elements of the rotation matrix | for irreducible representations: (a... Table 1.8. Multiplication (Cayley) table of the symmetric group S3 Table 1.9. Table of characters of symmetric groups: a ) for S2 and b ) for S3 Table 1.10. Transformations of Euler angles and components of rovibronic angular...Table 1.11. Characters of the rotation symmetry operations of type C2, C3, C4 an...Table 1.12. Table of correlation and characters of symmetry operations of group ...Table 1.13. Table of correlation and characters of the symmetry operations of gr...Table 1.14. Table of correlation and characters of symmetry operations of group ...Table 1.15. Table of characters of representation of group T generated by the ba...Table 1.16. Direct product of irreducible representations of group C3 vTable 1.17. Direct product of irreducible representations of group D3 hTable 1.18. Direct product of irreducible representations of group Td (for group...