long d.a. the raman effect

The Raman Effect: A Unified Treatment of the Theory of Raman Scattering by Molecules. Derek A. Long Copyright  2002 John Wiley & Sons Ltd ISBNs: 0-471-49028-8 (Hardback); 0-470-84576-7 (Electronic) The Raman Effect The Raman Effect A Unified Treatment of the Theory of Raman Scattering by Molecules Derek A. Long Emeritus Professor of Structural Chemistry University of Bradford Bradford, UK Copyright  2002 by John Wiley & Sons Ltd, Baffins Lane, Chichester, West Sussex PO19 1UD, England National 01243 779777 International (+44) 1243 779777 e-mail (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on http://www.wileyeurope.com or http://www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency, 90 Tottenham C ourt Road, London, W1P 9HE, without the permission in writing of the Publisher. Other Wiley Editorial Offices John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, USA Wiley-VCH Verlag GmbH, Pappelallee 3, D-69469 Weinheim, Germany John Wiley Australia, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons (Canada) Ltd, 22 Worcester Road, Rexdale, Ontario, M9W 1L1, Canada Library of Congress Cataloguing in Publication Data Long, D. A. (Derek Albert) The Raman effect : a unified treatment of the theory of Raman scattering by molecules / Derek A. Long. p. cm. Includes bibliographical references and index. ISBN 0-471-49028-8 (acid-free paper) 1. Raman spectroscopy. I. Title. QD96.R34 L66 2001 535.8 0 46—dc21 2001046767 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0 471 49028 8 Typeset in 11/13pt Times by Laserwords Private Limited, Chennai, India Printed and bound in Great Britain by Biddles Ltd, Guildford and King’s Lynn This book is printed on acid-free paper responsibly manufactured from sustainable forestry, in which at least two trees are planted for each one used for paper production. Dedicated to Edward and William Long, grandsons. vii Contents Preface xix Acknowledgements xxiii Part One Theory 1 1 Survey of Light-scattering Phenomena 3 1.1 Introduction 3 1.2 Some Basic Definitions 4 1.3 Rayleigh and Raman Scattering 5 1.3.1 Description 5 1.3.2 Energy transfer model 7 1.4 Hyper-Rayleigh and Hyper-Raman Scattering 10 1.4.1 Description 10 1.4.2 Energy transfer model 10 1.5 Second Hyper-Rayleigh and Second Hyper-Raman Scattering 11 1.5.1 Description 11 1.5.2 Energy transfer model 11 1.6 Coherent anti-Stokes and Coherent Stokes Raman Scattering 11 1.7 Stimulated Raman Gain and Loss Spectroscopy 13 1.8 Typical Spectra 14 1.9 Bases for the Theoretical Treatment of Rayleigh and Raman Scattering 16 1.10 Historical Perspective 16 viii Contents 1.11 Caveat 17 References 17 2 Introduction to Theoretical Treatments of Incoherent Light Scattering 19 2.1 General Considerations 19 2.2 Induced Oscillating Electric Dipoles as Sources of Scattered Radiation 21 2.3 Basis of the Classical Theory of Light Scattering 22 2.4 Basis of the Quantum Mechanical Treatment of Incoherent Light-Scattering Phenomena: Electric Dipole Case 24 2.5 Extension of Quantum Mechanical Treatment of Incoherent Light Scattering to Include Magnetic Dipole and Electric Quadrupole Cases 27 2.6 Comparison of t he Classical and Quantum Mechanical Treatments of Light Scattering 28 2.7 The Way Ahead 29 3 Classical Theory of Rayleigh and Raman Scattering 31 3.1 Introduction 31 3.2 First-order Induced Electric Dipole 31 3.3 Frequency Dependence of the First-order Induced Electric Dipole 34 3.4 Classical Scattering Tensors a Ray and a Ram k 35 3.5 Selection Rules for Fundamental Vibrations 36 3.5.1 General considerations 36 3.5.2 Diatomic molecules 36 3.5.3 Polyatomic molecules 38 3.6 Selection Rules for Overtones and Combinations 43 3.7 Coherence Properties of Rayleigh and Raman Scattering 44 3.8 Limitations of the Classical Theory 45 3.9 Example of Rayleigh and Raman Scattering 45 3.10 Presentation of Raman Spectra 47 References 48 4 Quantum Mechanical Theory of Rayleigh and Raman Scattering 49 4.1 Introduction 49 4.2 Time-dependent Perturbation Theory and  a fi 50 4.3 Qualitative Discussion of ˛   fi 54 4.3.1 Frequency denominators 55 4.3.2 Transition electric dipole numerators 56 4.3.3 Selection rules 58 4.4 Tensorial Nature of the Transition Polarizability and its Symmetry 58 4.5 Born–Oppenheimer Approximation and the Transition Polarizability Tensor 61 Contents ix 4.6 Simplification of ˛   e f v f :e g v i : General Considerations 64 4.7 Simplification by Radical Approximation: the Placzek Transition Polarizability 65 4.8 Simplification of ˛   e f v f :e i v i by Stages 68 4.8.1 Introduction of Herzberg–Teller vibronic coupling 68 4.8.2 Identification of non-resonance and resonance situations 75 4.9 Normal Electronic (and Vibronic) Raman Scattering 77 4.10 Normal Pure Vibrational Raman Scattering 78 4.11 Electronic (and Vibronic) Resonance Raman Scattering 81 4.12 Vibrational Resonance Raman Scattering 83 4.13 Units and Orders of Magnitude 83 References 84 5 Vibrational Raman Scattering 85 5.1 Introduction 85 5.2 The Placzek Vibrational Transition Polarizability: Recapitulation 86 5.2.1 Cartesian basis 86 5.2.2 The spherical basis 88 5.3 Definition of Illumination–Observation Geometry 89 5.4 Intensity of Scattered Radiation: Some General Considerations 94 5.4.1 Development of a symbol for scattered intensity 94 5.4.2 Scattering cross-section 95 5.5 Intensity Formulae and Polarization Characteristics for a General Vibrational Transition in Various Illumination–Observation Geometries 97 5.5.1 General considerations 97 5.5.2 Linearly polarized incident radiation 98 5.5.3 Natural incident radiation 102 5.5.4 Angular dependence of scattered intensity 103 5.5.5 Circularly polarized incident radiation 106 5.5.6 Symmetry and depolarization ratios, reversal coefficients and degrees of circularity 109 5.6 Stokes Parameters for Scattered Radiation 113 5.7 Specific Vibrational Transitions 116 5.8 Vibrational Selection Rules 120 5.9 Patterns of Vibrational Spectra 123 5.10 Orders of Magnitude 126 5.11 Epilogue 127 References 131 Reference Tables for Chapter 5 132 Reference Table 5.1: Definitions for IÂ; p s ,p i  132 x Contents Reference Table 5.2(a) to 5.2(g): Intensities, Polarization Properties and Stokes Parameters for Vibrational Raman (and Rayleigh) Scattering 132 Reference Table 5.3: Symmetry classes for x, y, z, the rotations R x , R y and R z , and the components of the cartesian basis tensor c a. 145 6 Rotational and Vibration–Rotation Raman Scattering 153 6.1 Introduction 153 6.2 Irreducible Transition Polarizability Components 154 6.3 Symmetric Top 156 6.3.1 Selection rules 156 6.3.2 Placzek invariants  G  j   fi 157 6.3.3 Intensities 167 6.3.4 Subsequent development 169 6.4 Rotational and Vibrational Terms 169 6.5 Statistical Distribution of Molecular Population 171 6.6 Diatomic Molecule 173 6.6.1 Introduction 173 6.6.2 Heteronuclear diatomic molecule: pure rotation 174 6.6.3 Heteronuclear diatomic molecule: vibration–rotation 175 6.6.4 Homonuclear diatomic molecule: nuclear spin degeneracy 179 6.6.5 Intensity distribution 180 6.7 Symmetric Top Molecule 186 6.7.1 Introduction 186 6.7.2 Symmetric top: pure rotation 187 6.7.3 Symmetric top: vibration–rotation 191 6.7.4 Intensities 203 6.8 Linear Molecules 204 6.8.1 Rotation and vibration-rotation Raman spectra 204 6.8.2 Intensities 207 6.9 Contributions from Electronic Orbital and Spin Angular Momenta 208 6.10 Spherical Top Molecules 210 6.11 Asymmetric Top Molecules 211 6.12 Epilogue 211 References 213 Reference Tables for Chapter 6 214 Introduction 214 Reference Tables 6.1 to 6.4 216 7 Vibrational Resonance Raman Scattering 221 7.1 Introduction 221 7.2 Vibrational Transition Polarizability Tensor Components in the Resonance Case, Based on Perturbation Theory 222 Contents xi 7.3 Comparison of the A VI ,B VI ,C VI and D VI Terms 224 7.3.1 The A VI term 224 7.3.2 The B VI term 227 7.3.3 The C VI term 229 7.3.4 The D VI term 229 7.3.5 Subsequent developments 230 7.4 A VI Term Raman Scattering from Molecules with Totally Symmetric Modes 231 7.4.1 A VI term Raman scattering from molecules with one totally symmetric mode 231 7.4.2 A VI term Raman scattering from molecules with more than one totally symmetric mode: general considerations 237 7.4.3 A VI term Raman scattering from totally symmetric modes when  k is very small 238 7.5 A VI Term Raman Scattering Involving Non-Totally Symmetric Modes 239 7.5.1 General considerations 239 7.5.2 A VI term scattering involving a change of molecular symmetry of the resonant excited state 239 7.5.3 A VI term scattering involving excited state Jahn–Teller coupling 240 7.5.4 Summary of excited state Jahn–Teller effects in resonance Raman scattering 240 7.6 B VI Term Scattering Involving Vibronic Coupling of the Resonant Excited State to a Second Excited State 241 7.6.1 Introduction 241 7.6.2 B VI term scattering from molecules with non-totally symmetric modes 241 7.6.3 B VI term scattering from molecules with totally symmetric modes 244 7.7 Symmetry, Raman Activity and Depolarization Ratios 246 7.7.1 General symmetry considerations 246 7.7.2 The A VI term 247 7.7.3 The B VI term 250 7.8 Time-Dependent Formulation of Resonance Raman Scattering 262 7.8.1 Introduction 262 7.8.2 Transformation of the A VI term to a time-dependent expression 263 7.8.3 The time-dependent interpretation of resonance Raman scattering 264 7.9 Continuum Resonance Raman Scattering 266 References 270 xii Contents 8 Rotational and Vibration–Rotation Resonance Raman Scattering 271 8.1 Introduction 271 8.2 General Expression for ˛   fi for a Symmetric Top Molecule 272 8.3 General Expression for ˛ j m  fi 274 8.4 Contraction of General Expression for ˛ j m  fi 274 8.5 The Quadratic Term 275 8.6 Selection Rules 276 8.7 Evaluation of j˛ j m  fi j 2 277 8.8 Intensities and Depolarization Ratios 279 8.9 An Illustrative Example 283 8.10 Concluding Remarks 287 Reference 287 9 Normal and Resonance Electronic and Vibronic Raman Scattering 289 9.1 Introduction 289 9.2 Normal Electronic and Vibronic Raman Scattering 289 9.2.1 General considerations 289 9.2.2 A III -term scattering 290 9.2.3 B III C C III -term scattering 291 9.2.4 D III -term scattering 292 9.2.5 Transition tensor symmetry 292 9.3 Resonant Electronic and Vibronic Raman Scattering 292 9.3.1 General considerations 292 9.3.2 A V -term scattering 293 9.3.3 B V -term scattering 296 9.3.4 C V -term scattering 297 9.3.5 D V -term scattering 297 9.4 Selection Rules in Electronic Raman Spectra 297 9.4.1 General symmetry considerations 297 9.5 Intensities and Polarization Properties of Electronic Raman Scattering 301 9.5.1 Intensities: general considerations 301 9.5.2 Excitation profiles 301 9.5.3 Depolarization ratios 302 10 Rayleigh and Raman Scattering by Chiral Systems 303 10.1 Introduction 303 10.2 Outline of the Theoretical Treatment 305 10.3 Intensities of Optically Active Rayleigh Scattering 310 10.3.1 General considerations 310 10.3.2 Intensity formulae 314 10.3.3 Stokes parameters 317 [...]... earlier book, the copyright of which has been assigned to me by the original publishers D A Long The Raman Effect: A Unified Treatment of the Theory of Raman Scattering by Molecules Derek A Long Copyright  2002 John Wiley & Sons Ltd ISBNs: 0-471-49028-8 (Hardback); 0-470-84576-7 (Electronic) 1 Part One THEORY A unified treatment of the theory of Raman scattering by molecules The Raman Effect: A Unified... substantial population of the final state f It should be noted that as n2 tends to zero the stimulated Raman gain process tends towards the normal Raman process Consequently the theoretical treatment of the stimulated Raman effect can embrace the normal Raman process by treating the latter as a limiting case The interaction of the radiation with the material system can also result in the creation ¯ of a photon... observing Q M If this were the case only Rayleigh ‡ Chapter 3, Section 3.9 discusses the presentation of experimentally observed Rayleigh and Raman spectra 16 The Raman Effect and Raman scattering would normally find practical application because the equipment necessary for other types of light scattering is considerably more expensive However, knowledge of the theory underlying these phenomena will reveal... (vii) Energy level diagram Second hyper -Raman scattering (Stokes) Second hyper-Rayleigh scattering Hyper -Raman scattering (Stokes) (viii) Name Survey of Light-scattering Phenomena 9 10 The Raman Effect say nothing about the mechanisms of the photon–molecule interactions or the probability of their occurrence Raman scattering is inherently incoherent and as a result the intensity of scattering from a material... and both the intensity and polarization of the scattered radiation depend on the direction of observation 1.4.2 Energy transfer model Here again the model is based on the photon description of electromagnetic radiation ¯ We consider that before the interaction there are n1 photons each of energy hω1 and the molecule has energy Ei The interaction of the radiation with a molecule leads to the successive... electronic transition energy The theoretical treatment of CARS and CSRS involves concepts that lie outside the scope of this book Consideration of the interaction of the waves of frequencies ω1 and ω2 involves the bulk or macroscopic properties of the material system which must then be related to the individual or microscopic properties of the molecules From such considerations emerge the special properties... Stokes hyper -Raman scattering, and stimulated hyper -Raman spectroscopy More than 25 types of Raman spectroscopies are now known! 4 The Raman Effect This book is concerned almost entirely with aspects of Rayleigh and Raman scattering To set the scene for the detailed developments in subsequent chapters we now give some basic definitions and present a description of these two processes, together with a... after its discoverer C.V Raman (Plate 1.1) In the spectrum of the scattered radiation, the new frequencies are termed Raman lines, or bands, and collectively are said to constitute a Raman spectrum Raman bands at frequencies less than the incident frequency (i.e of the type ω1 ωM ) are referred to as Stokes bands, and those at frequencies greater than the incident frequency (i.e of the type ω1 C ωM ) as... Phenomena 7 1.3.2 Energy transfer model This model is based on the photon description of electromagnetic radiation We consider that before the interaction of the radiation with the system there are n1 photons each of energy hω1 , where ω1 is the frequency of the incident monochromatic radiation, and ¯ the molecule has energy Ei The interaction of the radiation with a molecule leads to ¯ annihilation of... sense, as there is ¯ no conservation of energy in this stage of the process The role of the incident radiation is rather to perturb the molecule and open the possibility of spectroscopic transitions other than direct absorption If hω1 approaches an electronic transition energy, enhancement of ¯ the scattered intensity is observed For classification purposes in this and subsequent applications of the energy