Fundamentals of Molecular Spectroscopy 981990790X, 9789819907908

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Prabal Kumar Mallick

Fundamentals of Molecular Spectroscopy

Fundamentals of Molecular Spectroscopy

Prabal Kumar Mallick

Fundamentals of Molecular Spectroscopy

Prabal Kumar Mallick Former Professor, Department of Physics University of Burdwan Burdwan, India

ISBN 978-981-99-0790-8 ISBN 978-981-99-0791-5 (eBook) https://doi.org/10.1007/978-981-99-0791-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Dedicated To the Memory of My Parents Prokash Chandra Mallick and Monisha Devi

Preface

On several occasions, many postgraduate and Ph.D. students came to me to understand various aspects of molecular spectroscopy, which were not necessarily related to their course of studies. There are a number of good books which cover different areas of molecular spectroscopy. But majority of the books present only those parts of the theories which are absolutely essential to explain the spectral features and some relevant aspects without going into the details. Some of the books of course are there which deal with the theoretical bases very elegantly and critically, but they confine their discussions within limited areas and not in the wider field of molecular spectroscopy. Detailed theoretical backgrounds of different types of spectroscopy are given in scattered ways in different publications. So the readers have to consult different books and journals for getting vivid theoretical pictures of different fields of molecular spectroscopy. Several beautiful theoretical explanations of many observational facts of various aspects of molecular spectroscopy are now available. While reviewing my lecture notes on molecular spectroscopy delivered in different postgraduate classes, an idea came to my mind. It will be very nice to have a book which covers the major fields of molecular spectroscopy stressing on the theoretical background critically. This idea led me to write this book. This book is consisted of 11 chapters starting from Born-Oppenheimer approximation and its relevance to various spectra to some topics on nonlinear spectroscopy through rotational, vibrational, Raman and electronic spectroscopy, group theoretical application, nuclear magnetic resonance, electron spin resonance, nuclear quadrupole resonance and Mossbauer spectroscopy. My intention is to present a good background of the theoretical pictures of the concerned fields which will help the readers to understand the subjects firmly and apply them to their own fields according to their need. In this book, it is presumed that the readers are well acquainted with the fundamentals of the basic subjects of physics, for example, mathematical methods, classical mechanics, quantum mechanics, statistical mechanics and electrodynamics. Another point is to be noted herewith. Since mathematics is a very important tool to explain theoretically many physical processes analytically, I have used it whenever and wherever it is necessary. If someone (reader) feels that some portions are too complex and lengthy, he or she may ignore the detailed mathematical steps and accept only the vii

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Preface

final result for the application in their own fields of interest. For example, the readers, who are not so much inquisitive, may ignore the detailed derivation of the s-vectors for wagging and torsion (which are not available in most of the books), perturbation calculation with Morse potential energy curve, detailed explanation of Ʌ doubling, calculation of susceptibilities of various orders, etc. The purpose of writing is not only to bring a wider field in a single book but also to develop the theories starting from the fundamentals and also from the simple to the final forms through fairly elaborate powerful techniques so that the readers become self-sufficient and apply them accordingly. Hopefully this book will be used both by the students and the teachers and also by the research workers as a textbook and also as a reference book. The author acknowledges deeply the inspiration came from some of my students and friends for writing such a book on molecular spectroscopy. The author is grateful to Prof. J. K. Bhattacharya and Dr. S. K. Sarkar for their kind help and valuable suggestions in publishing this book. The author also acknowledges the help provided by M/S Dhar Brothers for formatting the manuscript. Lastly, the author expresses his sincere thanks to all the members of his family for their patience and extended help all throughout the time of writing this book. Kolkata, India

Prabal Kumar Mallick

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Born–Oppenheimer Approximation . . . . . . . . . . . . . . . . . 1.1.2 Validity of Born–Oppenheimer Approximation . . . . . . . . 1.1.3 Estimation of Different Energies in Molecules . . . . . . . . 1.1.4 Energy-level Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Breakdown of Born–Oppenheimer Approximation . . . . . References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 6 7 10 10 12

2

Rotational Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Nuclear Wave Equation in Diatomic Molecules . . . . . . . . . . . . . . . 2.2 Diatomic Molecule as a Rigid Rotator . . . . . . . . . . . . . . . . . . . . . . . 2.3 Selection Rules and the Spectral Structure . . . . . . . . . . . . . . . . . . . 2.4 Isotopic Effect in Rotational Spectra . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Intensities of Rotational Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Non-rigid Rotator (A Semiclassical Approach) . . . . . . . . . . . . . . . 2.7 Rotational Spectra of Polyatomic Molecules . . . . . . . . . . . . . . . . . 2.7.1 Hamiltonian in Terms of Angular Momentums . . . . . . . . 2.7.2 Different Types of Rotating Molecules . . . . . . . . . . . . . . . 2.8 Stark Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Quadrupole Hyperfine Structure in Molecules . . . . . . . . . . . . . . . . References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 14 15 19 20 21 24 27 30 42 45 53

3

Infrared Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Vibrational Energy Levels of a Diatomic Molecule Considered as a Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . 3.1.1 Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Rotational-Vibrational Spectrum of a Diatomic Molecule with the Potential Function of a Simple Harmonic Oscillator . . . . 3.3 Anharmonic Oscillator and Morse Potential Function . . . . . . . . . .

55 55 57 59 64

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3.4 Dissociation Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 72

4

Raman Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.1 Classical Explanation of Raman Scattering . . . . . . . . . . . . . . . . . . . 74 4.1.1 Polarizability Ellipsoid and Raman Activity . . . . . . . . . . 76 4.2 Quantum Theoretical Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.3 Selection Rules of Rotational Raman Spectra . . . . . . . . . . . . . . . . . 82 4.3.1 Rotational Raman Spectra of Diatomic Molecules . . . . . 82 4.3.2 Rotational Raman Spectra of Polyatomic Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.4 Symmetry Properties of Wave Functions . . . . . . . . . . . . . . . . . . . . . 87 4.4.1 Effect of Nuclear Spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.5 Selection Rules and Characteristics of Vibrational Raman Spectra of Diatomic Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.5.1 Rotational Vibrational Raman Spectra . . . . . . . . . . . . . . . 95 4.6 Raman Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.7 Surface Enhanced Raman Scattering (SERS) . . . . . . . . . . . . . . . . . 97 References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5

Vibrational Spectra of Polyatomic Molecules . . . . . . . . . . . . . . . . . . . . 5.1 Normal Coordinates (Classical Description) . . . . . . . . . . . . . . . . . . 5.2 Normal Coordinates (Quantum Mechanical Description) . . . . . . . 5.3 Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Selection Rules for the Infrared Spectra . . . . . . . . . . . . . . 5.3.2 Selection Rules for the Raman Spectra . . . . . . . . . . . . . . . 5.4 Normal Modes of a Linear Symmetric Triatomic Molecule . . . . . 5.4.1 Parallel Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Perpendicular Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Normal Modes of an Asymmetric Linear Triatomic Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Normal Coordinate Calculation by the Method of Wilson . . . . . . 5.6.1 Internal Coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Determination of s-vectors . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 G-Matrix and Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . 5.6.4 Potential Energy in Terms of Internal Coordinates and the Secular Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Molecular Symmetry and Vibrational Problems . . . . . . . . . . . . . . . 5.8 Fourier Transform Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

105 105 109 111 111 113 114 115 118 121 126 126 127 135 137 140 147 152 159

Electronic Spectra of Diatomic Molecules . . . . . . . . . . . . . . . . . . . . . . . . 161 6.1 Vibrational Coarse Structure of Electronic Bands of a Diatomic Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.2 Isotope Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Contents

Rotational Fine Structure of Vibronic Bands in Diatomic Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Intensity Distribution in the Vibrational Bands of the Electronic Spectra (Franck–Condon Principle) of Diatomic Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Quantum Mechanical Formulation of Franck–Condon Principle . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Quantum Numbers of Electronic States in Diatomic Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Coupling of Angular Momenta . . . . . . . . . . . . . . . . . . . . . 6.5.2 Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Determination of Heat of Dissociation of a Diatomic Molecule from the Observed Electronic Spectra . . . . . . . . . . . . . . 6.7 Predissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Quantum Theory of Valence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.1 Hydrogen Molecule Ion . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.2 Hydrogen Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Electronic Structure of Diatomic Molecules . . . . . . . . . . . . . . . . . . 6.9.1 Homonuclear Diatomic Molecules . . . . . . . . . . . . . . . . . . 6.9.2 Heteronuclear Diatomic Molecules . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

6.3

7

Electronic Spectra of Polyatomic Molecules . . . . . . . . . . . . . . . . . . . . . . 7.1 Hybridization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Conjugated System of Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Free Electron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Molecular Orbital Method . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Relaxation Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Lifetime and Quantum Yield . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Vibronic and Spin–Orbit Interactions and N → π* Transitions in Organic Molecules . . . . . . . . . . . . . . . . 7.3.3 Radiative Sources for T1 → S0 Transition (Phosphorescence Decay) . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Radiation Less Transition . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Molecular Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Charge Transfer Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Hydrogen Bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Excimers and Exciplexes . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.4 Energy Transfer Phenomena . . . . . . . . . . . . . . . . . . . . . . . . 7.4.5 Electron Transfer Phenomena . . . . . . . . . . . . . . . . . . . . . . . 7.5 Photoelectron Spectroscopy (PES) . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Few Examples of Photoelectron Spectra and Their Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167

170 173 176 177 182 184 187 188 189 196 201 201 207 209 213 215 216 219 220 221 225 226 227 237 241 245 245 252 255 262 267 273 275 275

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Appendix 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 8

Application of Group Theory to Molecular Spectroscopy . . . . . . . . . . 8.1 Definition of a Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Multiplication Table and Other Properties of a Group . . . . . . . . . . 8.3 Representations and Their Characteristics . . . . . . . . . . . . . . . . . . . . 8.4 Determination of the Point Group of a Molecule . . . . . . . . . . . . . . 8.5 Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Electronic Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Vibrational Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Symmetry Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Vibronic Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Spin–Orbit Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Depolarization Ratio (ρ) in Raman Spectra . . . . . . . . . . . . . . . . . . References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic Resonance (ESR/EPR) and Nuclear Quadrupole Resonance (NQR) Spectroscopy . . . . . . . . . . . . . . . . . . . . . 9.1 Nuclear Magnetic Resonance (NMR) . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Basic Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Experimental Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.4 Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.5 Chemical Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.6 Spin–Spin Interaction (High-Resolution NMR Spectra) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.7 Solid-State NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.8 Nuclear Magnetic Resonance Imaging (NMRI) . . . . . . . 9.2 Electron Spin or Paramagnetic Resonance (ESR/EPR) . . . . . . . . . 9.2.1 Basic Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 ESR Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Effect of Nuclear Spin (Hyperfine and Super Hyperfine Interactions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.4 Anisotropic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.5 Zero Field Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.6 ESR Spectra of Transition Metal Ion . . . . . . . . . . . . . . . . . 9.3 Nuclear Quadrupole Resonance (NQR) . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Basic Principle of NQR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Axially Symmetric System . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Non-axially Symmetric System . . . . . . . . . . . . . . . . . . . . . Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

289 289 291 293 298 303 304 308 318 335 336 341 346

347 347 348 349 350 355 359 363 369 372 376 376 377 378 385 391 395 400 400 405 406 409 411 413 415

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10 Mossbauer Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Nuclear Recoil and Doppler Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Earlier Experiments on Resonance Absorption . . . . . . . . . . . . . . . . 10.3 Principal of Mossbauer Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 10.4 Experimental Set-up of Mossbauer Spectroscopy . . . . . . . . . . . . . 10.5 Isomer Shift (Chemical Shift) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Nuclear Quadrupole Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 The Effect of Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

417 417 419 420 421 422 424 426 428

11 Some Nonlinear Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Nonlinear Raman Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Normal, Hyper and Second Hyper Rayleigh and Raman Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Stimulated Raman Scattering (SRS) . . . . . . . . . . . . . . . . . 11.1.3 Coherent Antistokes Raman Scattering (CARS) . . . . . . . 11.1.4 Inverse Raman Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.5 Two Photon and Multiphoton Absorption and Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.6 Multiphoton Dissociation and Laser Isotope Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Theoretical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 First-Order Effect and Linear Susceptibility . . . . . . . . . . 11.2.2 Second-Order Effect and Second-Order Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Third-Order Effect and Third-Order Susceptibility . . . . . 11.3 Sum Frequency and Second Harmonics Generation . . . . . . . . . . . 11.4 Stimulated Raman Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Coherent Antistokes Raman Scattering (CARS) . . . . . . . . . . . . . . References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

429 430 430 432 433 435 436 437 440 443 445 449 452 459 466 472

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

List of Figures

Fig. 1.1

Fig. 2.1 Fig. 2.2 Fig. 2.3

Fig. 2.4 Fig. 2.5 Fig. 2.6

Fig. 2.7

Fig. 2.8

‘Energy-level diagram’ showing transitions giving rise to various types of spectra in molecules [E i ’s are different electronic states; V s and J s are the quantum numbers of the respective vibrational and rotational levels of different electronic states. Single- and double-dash superscripts correspond to higher and lower levels in the respective transitions] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotational energy levels of a diatomic molecule considered as a rigid rotator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotational transitions and spectra of a diatomic molecule . . . . . Rotational spectra of a diatomic molecule (continuous line) and its heavier isotope (broken lines). B and B' are the rotational constants of the respective molecules . . . . . . . . . . Relative intensities of the rotational lines . . . . . . . . . . . . . . . . . . Schematic representation of the energy levels and rotational spectra of rigid and non-rigid rotators . . . . . . . . . Bond distances of carbonyl sulphide. ‘o’ is the centre of mass at a distance ‘x’ from the centre (carbon) atom on the right-hand side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic representation of the energy levels and the transitions for the rigid symmetric top molecule: a prolate and b oblate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orientation of the body-fixed axes system (x, y, z) with respect to the space-fixed (X, Y, Z) axes system in terms of the Euler’s angles (ϕ, θ, ψ). a Anticlockwise rotation of the space-fixed axes about Z by an angle ψ, b anticlockwise rotation of the intermediate axes (double dash) about x'' by an angle θ and c anticlockwise rotation of the next axes (single dash) about z' by an angle ϕ to coincide with the body-fixed axes system x, y, z . . . . . . . . . .

10 16 19

20 20 23

32

35

36

xv

xvi

Fig. 2.9

Fig. 2.10 Fig. 2.11 Fig. 2.12

Fig. 3.1

Fig. 3.2 Fig. 4.1 Fig. 4.2

Fig. 4.3

Fig. 4.4

Fig. 4.5 Fig. 4.6

List of Figures

Energy-level diagram of an asymmetric top molecule obtained from a correlation between those of two extreme cases, oblate and prolate symmetric top molecules. (For other details see the text) . . . . . . . . . . . . . . . . . . . . . . . . . . . First-order Stark splitting of a symmetric top molecule for the transition J = 2, k = 1 ← J = 1, k = 1 . . . . . . . . . . . . . Second-order Stark effect of the J = 0 → J = 1 line in linear molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . First-order hyperfine quadrupole splitting of the rotational line arising from the transition J = 0 → J = 1 in CH3 Cl molecule (due to nuclear spin I Cl = 3/2). Note that J = 1, K = ± 1 levels do not take part in these spectral transitions; so they are not shown in the diagram . . . . . . . . . . . . Rotational-vibrational energy levels, transitions and spectral structure around the fundamental vibration of a diatomic molecule. Dashed lines correspond to forbidden transition and absence of a spectral line at the position of the pure vibrational transition . . . . . . . . . . . . . Morse potential energy function and energy levels . . . . . . . . . . . Polarizability ellipsoid of H2 molecule as seen from the direction a across and b along the bond . . . . . . . . . . . . Change of the shape, size and orientation of the polarizability ellipsoid of water molecule during the execution of its three vibrational modes. The central figures correspond to the equilibrium configuration of the molecule and the side ones correspond to the two extreme positions of the respective vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Change in the shape, size and orientation of the polarizability ellipsoid of the carbon dioxide molecule during the execution of its three vibrational modes. The central figures correspond to the equilibrium configuration of the molecule and the side ones correspond to the two extreme positions of the respective vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of the polarizability (α) with the displacement coordinate (ξ ) for the three vibrations of carbon dioxide molecule. [For the three vibrations, see Fig. 4.3.] . . . . . . . . . . . . a Rayleigh’s scattering b Stokes Raman scattering c Antistokes Raman scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . a Rotational Raman transitions and b expected rotational Raman spectra in diatomic molecules, the height and the strength of blackening denoting the relative intensities (see the text). The J-values correspond to those of the lower levels (J '' ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 43 44

53

63 64 75

77

78

79 81

85

List of Figures

Fig. 4.7 Fig. 4.8

Fig. 4.9 Fig. 5.1 Fig. 5.2

Fig. 5.3 Fig. 5.4 Fig. 5.5 Fig. 5.6 Fig. 5.7

Fig. 5.8 Fig. 5.9 Fig. 5.10 Fig. 5.11

Fig. 5.12 Fig. 5.13 Fig. 6.1 Fig. 6.2

Fig. 6.3

Fig. 6.4

xvii

Rotational spectrum of symmetric top molecule . . . . . . . . . . . . . Statistical weights (St.Wts.) of different rotational levels of the lowest vibrational states of the lowest electronic states (1 ∑ g + ) of H2 and N2 /D2 and (3 ∑ g ) of O2 molecules . . . . Charge transfer mechanism in the SERS enhancement . . . . . . . A model linear symmetrical molecule (CO2 ) . . . . . . . . . . . . . . . Longitudinal vibrations of OCO molecule for the three values of λ. a λ = 0; b λ = k/mo and c λ = k/mo {1 + 2mo /mc )} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perpendicular vibrations of OCO molecule for the three values of λ: a λ = 0, b 0 and c 2k θ /(r 2 mo ){1 + 2mo /mc } . . . . . Linear asymmetric molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bond stretching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In-plane bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Top (a) and side (b) views of wagging. The position 1' in b is the location that atom 1 would occupy if the distorted molecule is subjected to rigid rotations and translations which restore the original atoms 2, 3 and 4 to their original plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Torsion angle bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetry coordinates of AB2 molecule (assuming no shift of the centre of mass) . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic principle of Fourier Transform Spectroscopy . . . . . . . . . . Interferogram and spectrum (Fourier transform) of the source emitting two waves of slightly different wavenumbers/wavelengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of truncation (apodization) on the spectral resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectrums and interferograms of a a white source and b a single frequency/wavenumber absorption . . . . . . . . . . . . . . . . v' - and v'' -progressions and sequence associated with an electronic transition of a diatomic molecule . . . . . . . . . Formation of the band heads in the P- and R-branches of the rotational fine structure of the electronic spectra of diatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of Franck–Condon Principle. a equilibrium internuclear distances are equal in both the electronic states. b equilibrium internuclear distance is slightly greater and c much greater in the upper state . . . . . . . . . . . . . . . a Franck–Condon principle in emission and b Condon parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

91 103 115

118 121 121 128 128

131 135 141 148

149 151 152 164

168

171 173

xviii

Fig. 6.5

Fig. 6.6

Fig. 6.7 Fig. 6.8 Fig. 6.9 Fig. 6.10 Fig. 6.11

Fig. 6.12 Fig. 6.13

Fig. 6.14 Fig. 6.15 Fig. 6.16 Fig. 6.17

List of Figures

Energy level diagram for the explanation of the Franck–Condon principle from the standpoint of quantum mechanics. Simple harmonic oscillator type wave functions of different vibrational states of the two electronic states (E 0 and E 1 ) are shown by shaded curves. The upward and downward vertical arrows correspond to the most favoured transitions for absorption and emission, respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Hund’s case (a) of coupling and b the energy levels of the 2 ∏ and 3 Δ states. The dotted lines indicate missing levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Hund’s case (b) and b splitting of 2 ∑ and 3 ∑ states . . . . . . Hund’s coupling cases (c), (d) and (e) . . . . . . . . . . . . . . . . . . . . . Potential energy diagram for the determination of dissociation energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Birge–Sponer extrapolation to determine the dissociation energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Radiationless transition occurs at the point P (of LPM) where the bound state (A) intersects the non-bound state (B) leading to predissociation. b illustration for the appearance of predissociation in the vibrational structure of electronic spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . Elliptic coordinates of the electron (e) in hydrogen molecule ion. (ϕ is measured from the xz-plane.) . . . . . . . . . . . Binding energy of hydrogen ion as a function of internuclear distance for the lowest potential energy curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electron density in hydrogen molecule ion . . . . . . . . . . . . . . . . . Hydrogen molecule with a, b as the two nuclei and 1,2 as the two electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculated Coulombic and total energy of hydrogen molecule as a function of internuclear distance . . . . . . . . . . . . . a Formation of bonding and antibonding orbitals from two 1s orbitals in a homonuclear diatomic molecule. b formation of bonding and antibonding orbitals from a 1s and a 2pz orbitals, z being the direction of the internuclear axis c formation of bonding and antibonding orbitals from two 2pz orbitals, z being the direction of the internuclear axis d formation of bonding and antibonding orbitals from two 2px/y orbitals, x/y being the directions perpendicular to the internuclear axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175

179 181 181 185 186

188 192

195 195 196 200

202

List of Figures

Fig. 6.18

Fig. 6.19 Fig. 6.20 Fig. 7.1 Fig. 7.2 Fig. 7.3 Fig. 7.4 Fig. 7.5

Fig. 7.6 Fig. 7.7 Fig. 7.8 Fig. 7.9 Fig. 7.10

Fig. 7.11 Fig. 7.12 Fig. 7.13

Fig. 7.14 Fig. 7.15 Fig. 7.16

xix

Correlation diagram between the united atom and separated atom states of homonuclear diatomic molecules. Note that the contributions of d-atomic orbitals shown here are not discussed above . . . . . . . . . . . . . . . . Electronic energy levels of HF molecule . . . . . . . . . . . . . . . . . . . Electronic energy levels of carbon monoxide (CO) . . . . . . . . . . Formation of methane (CH4 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . sp3 -hybridization in a methane (CH4 ), b ethane (C2 H6 ), c ammonia (NH3 ) and d water (H2 O) molecules . . . . . . . . . . . . sp2 -hybridization in ethylene (C2 H4 ) . . . . . . . . . . . . . . . . . . . . . . sp-hybridization in acetylene (C2 H2 ) molecule . . . . . . . . . . . . . σ- and π-bondings in octatetraene (C8 H10 ). σ-bonds (indicated by ‘—’) are formed involving the sp2 -hybridized orbitals and the delocalized π-bonds are formed by the remaining 2pz-orbitals of the carbon atoms, oriented perpendicular to the skeletal plane . . . . . . . . . . Energy levels and lowest energetic transition in octatetraene (C8 H10 ) molecule . . . . . . . . . . . . . . . . . . . . . . . . . π-bonding in benzene (C6 H6 ) molecule . . . . . . . . . . . . . . . . . . . Energy levels of the π-molecular orbitals of benzene . . . . . . . . Significant canonical (resonance) structure of Benzene . . . . . . . Fluorescence, internal conversion (IC), phosphorescence, intersystem system crossing (ISC), vibrational relaxation (VR) and their rate constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ground and excited singlet and triplet molecular states . . . . . . . nπ * and π π * states connected by spin–orbit interaction . . . . . . Potential energy curve for the charge transfer spectra. R, I and EA are the internuclear distance, vertical ionization potential of the donor (D) and vertical electron affinity of the acceptor (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular orbital description of the hydrogen bonding system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electronic configurations of M, N, MN and M*····N . . . . . . . . . Potential energy diagram and the related spectral transitions in exciplex (M* N) formation. ΔE 00 corresponds to the 0 ↔ 0 transition between the ground and excited electronic states of M, ΔH ex is the dissociation energy of the exciplex and δErep corresponds to the energy difference between the minimum of the exciplex potential curve and the peak of its emission spectra. (For exciplex M /= N and for excimer M = N) . . . . . . . . . . . . . . . . . . . . . . . . .

205 208 209 216 218 219 219

220 221 222 223 224

226 234 243

247 253 256

257

xx

Fig. 7.17

Fig. 7.18 Fig. 7.19

Fig. 7.20

Fig. 7.21

Fig. 7.22

Fig. 7.23 Fig. 7.24

Fig. 7.25

List of Figures

Concentration dependence of the emission spectrum. The solid lines correspond the monomer (M) and the dashed lines to the exciplex (M*N). The concentrations of M are (C4 > C3 > C2 > C1 ), whereas the concentration of N remains unchanged . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time dependence of fluorescence intensity of the monomer I(t) and the excimer I´(T ) . . . . . . . . . . . . . . . . . Potential energy curves for the initial/reactant state (i = r) and the final/product state (f = p). Crossing point (q0 , v0 ) of the two curves corresponds to activation energy (ΔE). ΔG is the free energy change between the reactants and the products. λ is the reorganization energy. Inset shows the energy splitting at the crossing point . . . . . . . . . . . . . Electronic coupling of the initial and the final states with the virtual, high energy electron transfer states involving the bridge (L) through which the donor and acceptor are connected. The subscripts e and h of H correspond to the electron and hole in the bridge (L), respectively, for the initial (reactant, i) and the final (product, f ) states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of log k el (k el being the electron transfer rate) with driving free energy (−ΔG0 ); a and b correspond to classical (7.125) and quantum mechanical (7.131) curves, respectively. (Remember free energy of spontaneous electron transfer is negative and reorientational energy is positive) . . . . . . . . . . . . . . . . . . . . . Energy level diagrams for a UPS, b XPS and c (AES); in AES, the core (K) hole, created by knocking out the electron from there by the bombardment of the incident X-ray photon, is filled up by an electron from the higher level L I , creating a new hole there and in the level L III by simultaneous ejection of an auger electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental set-up for UV/X-ray spectrophotometer . . . . . . . . He II photoelectron spectra of the inert atoms Ne, Ar, Kr and Xe. a Energy levels of the inert atoms and their ions which has two multiplet levels. b Photoelectron spectral peaks of the corresponding energy of the levels can be obtained by subtracting these from the excitation energy. Energy difference of these peaks is the energy due to the spin–orbit splitting of the valence state of the respective ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nature of photoelectron spectra of H2 . . . . . . . . . . . . . . . . . . . . .

258 260

269

271

273

274 276

277 278

List of Figures

Fig. 7.26

Fig. 7.27 Fig. 7.28

Fig. 7.29 Fig. 7.30 Fig. 7.31 Fig. 8.1 Fig. 8.2 Fig. 8.3 Fig. 8.4 Fig. 8.5 Fig. 8.6 Fig. 8.7 Fig. 8.8 Fig. 8.9 Fig. 8.10 Fig. 8.11 Fig. 8.12 Fig. 8.13

xxi

Photoelectronic spectra of nitrogen molecule (N2 ). Intensity (count rate) is in arbitrary unit and the ionization energy is in ev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (He I Photoelectron spectra of HBr) . . . . . . . . . . . . . . . . . . . . . . XPS of oxygen1s and carbon 1s electrons in a 2: 1 mixture of CO and CO2 gases. (For exact peak positions, see the text) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XPS of a mixture of CO2 and (CH3 )2 CO . . . . . . . . . . . . . . . . . . Carbon 1s chemical shifts observed in the x ray photoelectron spectrum of ethyl trifluoroacetate . . . . . . . . . . . . . Einstein’s coefficients and the rate constants . . . . . . . . . . . . . . . Symmetry elements of an equilateral triangle . . . . . . . . . . . . . . . Methane molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sulphur hexafluoride molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . Essential characteristics of dodecahedrons and icosahedrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flowchart for the determination of the point group of a molecule having at most only one manifold axis . . . . . . . . . Electronic energy levels and their symmetries of a molecule belonging to the point group C 2v . . . . . . . . . . . . . Energy levels and their symmetries in benzene molecule . . . . . Structure of benzene molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . N2 F2 molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ammonia molecule NH3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of porphyrin heterocyclic . . . . . . . . . . . . . . . . . . . . . . . Electronic absorption spectrum of cytochrome c (a model of iron porphyrin) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mixing pathways of 3 B1u /3 B2u states bringing allowed singlet character through the term | 0  0 |   ∂ H  |   0| 0 | | | Tl HS O Sm Sm Q a | Sk0 of Eq. (8.27) . . . . . . . . . . . ∂ Qa a

Fig. 8.14

0

a

Fig. 9.2 Fig. 9.3

282 283 284 285 291 299 300 301 304 305 306 307 316 322 337 337

339

Mixing pathways of 3B1u /3B1u states bringing allowedsinglet character through the term |  | |   0 |  ∂ H0  Q a |Tm0 Tm0 | HS O | Sk0 of Eq. (8.27) . . . . . . . . . . . Tl | ∂ Qa

Fig. 8.15 Fig. 8.16 Fig. 9.1

279 280

0

Choice of axes in naphthalene . . . . . . . . . . . . . . . . . . . . . . . . . . . Excitation scattering geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . Splitting of a nuclear level (with spin I = 1/2) in a magnetic field B0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block diagram of the experimental set-up of NMR spectrometer .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . − → Magnetization M and its components . . . . . . . . . . . . . . . . . . .

340 340 345 348 350 353

xxii

Fig. 9.4 Fig. 9.5 Fig. 9.6

Fig. 9.7 Fig. 9.8

Fig. 9.9

Fig. 9.10 Fig. 9.11 Fig. 9.12 Fig. 9.13 Fig. 9.14 Fig. 9.15 Fig. 9.16 Fig. 9.17 Fig. 9.18

Fig. 9.19 Fig. 9.20 Fig. 9.21

Fig. 9.22

Fig. 9.23 Fig. 9.24 Fig. 9.25

List of Figures

Effect of a 90°–τ –180°–τ –· · · pulse on the magnetization   0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vector M Variation of transverse susceptibilities χ ' and χ '' with (ω0 -ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Energy levels and b absorption in the NMR spectrum of proton in the isolated state and in molecular environment with and without the applied field Bappl . . . . . . . . . Low-resolution NMR spectrum of methanol . . . . . . . . . . . . . . . Proton NMR spectra of two molecules, the elemental formula of each of which is C3 H8 O. The number within each peak indicates relative peak area . . . . . . . . . . . . . . . High-resolution NMR spectrum of methanol, the number within each peak corresponds to respective relative peak area i.e. intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of vicinal coupling with the dihedral angle (Karplus curve) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Long-range coupling in meta (Jmet = 2–3 Hz) and para (Jmet = 0–1 Hz) aromatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . σ –π overlap (hyperconjugation in long range coupling) dihedral angle between the π-bond and the CH bond . . . . . . . . Energy levels of a solid with two identical nuclei with spin ½ a without and b with dipole–dipole interaction . . . Spinning of the sample in MAS NMR . . . . . . . . . . . . . . . . . . . . . Dipolar broadened and high-resolution NMR spectra of solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slice selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phases of the transverse magnetization vectors after the Gx pulse is turned off . . . . . . . . . . . . . . . . . . . . . . . . . . . Splitting of the ground state energy level of hydrogen atom in a magnetic field. E 0 is its energy in the absence of the magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block diagram of the ESR spectrometer . . . . . . . . . . . . . . . . . . . a ESR absorption spectra. b First derivative spectra observed with field modulation . . . . . . . . . . . . . . . . . . . . . . . . . . a Energy levels and the allowed transitions, and b ESR spectrum for an unpaired electron coupled to two non-equivalent nuclei each with spin ½. (a1 > a2 ) . . . . . . . . . . . a Energy levels and the allowed transitions and b ESR spectrum for an unpaired electron coupled to two equivalent nuclei each with spin ½. (a1 = a2 = a) . . . . . . . . . . . ESR spectra of benzene radical anion . . . . . . . . . . . . . . . . . . . . . ESR spectra of pyrazine anion radical anion . . . . . . . . . . . . . . . . ESR spectrum of naphthalene radical anion . . . . . . . . . . . . . . . .

355 359

361 361

362

365 368 369 369 371 371 372 374 374

377 377 378

380

381 382 384 384

List of Figures

Fig. 9.26

Fig. 9.27 Fig. 9.28

Fig. 9.29 Fig. 9.30

Fig. 9.31 Fig. 9.32 Fig. 9.33 Fig. 9.34 Fig. 9.35 Fig. 10.1 Fig. 10.2 Fig. 10.3

Fig. 10.4 Fig. 10.5

Fig. 11.1

Fig. 11.2

xxiii

Angular (θ) dependence of the resonance field (B) and g in an axially symmetric single crystal for rotation about X- (or Y-) axis, θ being the angle between Z-axis and the field direction. (g > g ) . . . . . . . . . . . . . . . . . . . . . . . . . . ESR spectra of an axially symmetric paramagnetic sample in the powder form. (g > g ) . . . . . . . . . . . . . . . . . . . . . Triplet state zero field splitting and Zeeman splitting for the magnetic field applied along the three cartesian axes of the molecule. The full arrows correspond to the selection rule ΔM s = ± 1 and the dashed arrows to ΔM s = ± 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A schematic representation of the angular part of the d-orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Splitting of d-orbitals in octahedral (Oh ) and tetrahedral (T d ) crystal fields. a, c belong to octahedral and b, d to tetrahedral symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Splitting of the states of a d1 ion, b d2 ion, c d3 ion, d d4 ion and e d5 ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Energy levels and transitions in nuclei with spin, a I = 3/2 and b I = 5/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Energy levels and transitions in nuclei with spin (I = 1) . . . . . . Level splitting and transitions in nucleus with spin 3/2 for η /= 0 compared to the case, η = 0 compare to . . . . . . . . . . . Level splitting and transitions in nucleus with spin 1 for η /= 0 compared to the case, η = 0 . . . . . . . . . . . . . . . . . . . . . . . . . Emission and absorption overlaps in atoms (a) and nuclei (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decay scheme leading Mossbauer transition in 57 Fe . . . . . . . . . Experimental arrangements of Mossbauer spectroscopy: a screw thread arrangement and b loudspeaker coil arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quadrupole splitting in a system with I = 1/2 in the ground state and I = 3/2 in the excited state . . . . . . . . . . . Energy level splitting and transitions between the states with I ' = 1/2 and I ' = 3/2. a Free nucleus; b nucleus in a magnetic field only; c nucleus in a magnetic field with quadrupolar interaction. νo is transition frequency in the absence of magnetic and quadrupole interaction (In this illustration, both gn '' and gn ' are taken as positive) . . . . . . . Stimulated Raman scattering. a Energy level diagram. Vi’s (associated with the dotted lines) are virtual states. b Experimental demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . a Diagrammatic description of experimental set-up of CARS and CSRS; b energy level diagram with transitions in CARS and CSRS . . . . . . . . . . . . . . . . . . . . . .

388 390

394 396

397 399 406 406 408 409 418 420

421 425

427

433

434

xxiv

Fig. 11.3 Fig. 11.4

Fig. 11.5

Fig. 11.6 Fig. 11.7

Fig. 11.8

Fig. 11.9

Fig. 11.10

Fig. 11.11

Fig. 11.12 Fig. 11.13 Fig. 11.14 Fig. 11.15 Fig. 11.16

List of Figures

Demonstration of experimental set-up of Inverse Raman Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Raman scattering; b two photon absorption with two photons of equal energy; c two photon absorption with two photons of unequal energies and d three photon absorption with three photons of equal energy. V, V1 and V2 are the intermediate states, here virtual . . . . . . . . . . . . . . Monitoring of double photon absorption spectra of a molecule. a Fluorescence excitation spectra. b Double photon ionization spectra where two absorbed photons (hν) take the molecule from the ground state 1 to the excited state 2, and the third photon ionizes it. V is the virtual intermediate state . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiphoton laser dissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . Energy level diagram for a two photon absorption, b stokes Raman scattering, c antistokes Raman scattering and d two photon emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diagramatic representations of the four processes in the energy level diagram (11.7). a Two photon absorption, b stokes Raman scattering (ω2 > ω1 ), c antistokes Raman (ω2 < ω1 ) scattering and d two photon emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Three photon absorption; b hyper Raman stokes (E g < E n ) and antistokes (E g > E n ) scattering; c sum frequency generation (also hyper Rayleigh scattering) and d three photon emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time ordered graph. a Three photon absorption; b hyper Raman stokes (E g < E n ) and antistokes (E g > E n ) scattering; c sum frequency generation (also hyper Rayleigh scattering) and d three photon emission corresponding to the respective Fig. 11.9a–d . . . . . . . . . . . . . . . a Time ordered graph corresponding to Eq. 11.28. b When E n /= E g , the process is stoke hyper Raman (for E n > E g ) and antistoke hyper Raman (for E n < E g ) scattering. When E n = E g , the process is a sum frequency generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generation of sum frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of output intensity (I 3 ) with (Δk.L/2) curve . . . . . . . . Phase matching of second harmonic generation in negative uniaxial crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Dispersion curve in the normal dispersion region and b perfect phase matching for stokes and antistokes radiations . . . Twelve time ordered graphs for CARS with two incident frequencies ω1 and two scattered photons of frequencies ω2 and ω3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

435

436

438 439

447

447

450

451

452 454 457 458 466

468

List of Figures

Fig. 11.17

Fig. 11.18

xxv

Energy level diagram for non-resonant and resonant CARS. For resonant, CARS è(ω1 −ω2 ) coincides with the energy difference E p –E o of the two rotational-vibrational levels |p > and |o > of the scattering system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase matching in CARS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

471 471

List of Tables

Table 6.1 Table 6.2 Table 7.1 Table 7.2 Table 7.3 Table 7.4 Table 8.1 Table 8.2 Table 8.3 Table 8.4 Table 8.5 Table 8.6 Table 8.7 Table 8.8 Table 8.9 Table 8.10 Table 8.11 Table 9.1 Table 9.2 Table 9.3 Table 9.4

Table 9.5 Table 10.1

Deslandre’s table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electronic configuration of some diatomic molecules . . . . . . . . Effects of the spin operators on the spin part of the wave functions* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . − → Effects of the operations of various components of l on the p-orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matrix elements of H so between singlet and triplet states . . . . . Bond lengths of N2 and N2 + in some excited states . . . . . . . . . . Multiplication table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representation matrices of the point group C 3V . . . . . . . . . . . . . Character table of the group C 2v . . . . . . . . . . . . . . . . . . . . . . . . . Character table for D6h group . . . . . . . . . . . . . . . . . . . . . . . . . . . Character table of the group D4h . . . . . . . . . . . . . . . . . . . . . . . . . Character table for C2h group . . . . . . . . . . . . . . . . . . . . . . . . . . . Character table of the group C 3v . . . . . . . . . . . . . . . . . . . . . . . . . Character table for the point group T d . . . . . . . . . . . . . . . . . . . . Character table for the octahedral group Oh . . . . . . . . . . . . . . . . Transitions, energies and intensities of benzene . . . . . . . . . . . . . Character table of the point group D2h . . . . . . . . . . . . . . . . . . . . Resonance condition for several important nuclei . . . . . . . . . . . Typical values for proton NMR chemical shifts . . . . . . . . . . . . . MI values for different combination of mIi ’s for three equivalent protons (n = 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Number of lines and their relative intensities in the ESR spectra for different number of equivalent protons coupled to a single unpaired electron . . . . . . . . . . . . . . . . . . . . . Ground states and degeneracies of the first transition series in octahedral fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical shift of some Sn119 compounds . . . . . . . . . . . . . . . . .

163 206 232 233 235 279 292 294 305 313 314 315 324 332 334 338 341 351 362 382

382 398 424

xxvii

xxviii

Table 10.2 Table 11.1

List of Tables

Transition frequencies when I ' = 3/2 and I '' = 1/2 (without quadrupole interaction) . . . . . . . . . . . . . . . . . . . . . . . . . Types of phase matching in uniaxial crystal . . . . . . . . . . . . . . . .

427 458

Chapter 1

Introduction

Abstract This chapter discusses clearly how the spectra of molecules differ from those of atoms and how this difference is explained through the introduction of Born– Oppenheimer approximation. This approximation is shown to split the complex molecular wave equation into two parts, one belonging to the electronic motion and the other to the nuclear motion. The nuclear motion is composed of rotations and vibrations of molecules. It is shown that the rotational energy quantum is less than the vibrational energy quantum which again is less than the energy arising from electronic motion. These three motions give rise to three types of spectrum, respectively, lying in the microwave, infrared and visible/ultraviolet regions of electromagnetic radiation. The validity of this approximation and also the condition under which it breaks down are also discussed.

1.1 Introduction In atoms, the valence electrons move in the Coulomb field of the nucleus (or better the atomic core, consisted of the respective nucleus and the closed cell electrons) and also of themselves. In the simplest atom hydrogen, there is only one electron moving in the Coulomb field of the nucleus. There the wave equation can be solved directly by some analytical methods. In fact, during the advent of quantum mechanics, one of the great achievements of this subject came through the solution of the hydrogen atom problem, and since then quantum mechanics has been playing a very vital role in solving many problems in various fields of Physics and Chemistry. But whenever we move on to many electron atoms, this way of solving the wave equation directly, even for the next heavier atom, helium, becomes impossible, and so some approximate methods are utilized. There are various types of approximations used for solving the wave equations and extracting realistic solutions of the specific cases. By realistic, it is meant that the solutions are in conformity with the experimental findings. The cases become further complicated if the electrons are placed in the Coulomb field of several nuclei, that is, in the molecular environment. Not only the motion of the electrons gives rise to electronic energy states, the internal motions of the nuclei also generate nuclear energy states in these systems. The relative motions © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 P. K. Mallick, Fundamentals of Molecular Spectroscopy, https://doi.org/10.1007/978-981-99-0791-5_1

1

2

1 Introduction

of the nuclei in the molecule give rise to both internal vibrations (called normal modes of vibration) and rotational motions of the molecule. Thus, the quantized states of a molecule are composed of electronic, vibrational and rotational energies. It can be shown (to be discussed latter on) that energy-wise, the electronic quanta are greater than the vibrational quanta which in turn are greater than the rotational quanta. Thus, there are several electronic energy states of a molecule, as in the case of any atom. But unlike atoms, each of these electronic states has associated with itself several vibrational levels. Such levels are called vibronic levels of that particular electronic state. Each vibronic level is again associated with several rotational levels. So whenever we consider an electronic transition between two electronic states of a molecule, actually transition takes place between two different sets of vibrational states (i.e. vibronic states) of the respective electronic states, and thus, unlike atoms, this would give rise to the vibrational structure of the electronic spectra. Unlike atoms, it is found that even if an electronic transition is forbidden in a molecule, vibrational structure of the spectrum associated with the respective electronic transition may still be present. Since each vibronic level of a molecule has a number of rotational levels associated with itself, vibrational structure of the electronic spectra in a molecule has also a rotational fine structure. But whenever we use low resolving power instrument for the observation of the spectra, rotational fine structure cannot be resolved, and the vibrational structure appears as bands in the electronic spectra of the molecule due to the lack of proper resolution of the instrument. So the molecular spectra are called band spectra. Such types of bands are also observed in the molecule even if the transitions take place between different vibrational levels of a particular electronic state (the ground electronic state) giving rise to band spectra in the infrared region. All these are critically discussed in the following sections. Remember that the spectra appear as lines in atoms due to the absence of the vibrational and the rotational energy levels, and so the atomic spectra are called line spectra. Spectra arising due to transitions between electronic states are called electronic spectra of molecules. They generally lie in the visible or near ultraviolet region. When the transitions take place between different vibrational levels of the ground electronic state of a molecule (that is, when there is no electronic transition), the relevant spectra are called vibrational spectra, and the corresponding spectral region is infrared. So such spectra are called infrared spectra. Lastly, when there is no electronic and vibrational transition, that is the molecule is in its lowest vibrational level of the ground electronic state, the spectra, arising from the transitions between various rotational levels of that particular vibronic state of the lowest electronic state, are called rotational spectra. Rotational spectra are generally observed in the microwave or in the far infrared region of the electromagnetic spectrum. So they are called microwave spectra. Analyses of molecular spectra in various wavelength regions are very helpful not only to identify molecules but also to study their various properties and characteristics.

1.1 Introduction

3

1.1.1 Born–Oppenheimer Approximation A molecule is a many electron system having a number of atoms. So the respective Hamiltonian can be written as H =− −

h2  2 h2  1 2 ∇i − ∇ 2m e i 2 K MK K

  Z K e2  e2  Z K Z L e2 + + ri,K r RKL i K i< j i, j K to a

16

2 Rotational Spectra

• J

•

TJ

• 20B

4

3

12B

2

6B

1

2B

0

0

Fig. 2.1 Rotational energy levels of a diatomic molecule considered as a rigid rotator

− → final one |f > of a molecule is proportional to the square of the integral < i| M |f > , ) − →( E − called transition dipole moment matrix. Here, M = l ql → rl is the electric dipole moment of the molecule. For a rotational transition, the initial and the final states, |i > and |f > , are the respective rotational states of the molecule. Thus from Eq. (2.9), we get i Miϕ |i( = Y J i Mi = N J i Mi PJMi i (cos θ )e

(2.10a)

and | f ( = YJ f M f = NJ f



PJ f (cos θ )ei M f ϕ Mf

Mf

(2.10b)

− → The three Cartesian components of the electric dipole moment M = i Mx +

j M y + k Mz of the molecule, oriented in the (θ, ϕ) direction in space, are given by Mx = M0 sin θ cos ϕ

(2.11a)

M y = M0 sin θ sin ϕ

(2.11b)

Mz = M0 cos θ

(2.11c)

2.3 Selection Rules and the Spectral Structure

17

M 0 being the dipole moment of the molecule measured in its own reference frame. Let us first determine the z-component of the transition dipole matrix )i|Mz | f (. π 2π (Mz )i f = M0

Y J∗i Mi cosθ Y J f M f sin θ dθ dϕ

(2.12)

θ=0 ϕ=0

Now, we shall use the recurrence relations of spherical harmonics which will help in evaluating this integral. ] (J − M + 1)(J + M + 1) 1/2 Y J +1,M (2J + 1)(2J + 3) ] [ (J − M)( J + M) 1/2 + Y J −1,M (2J − 1)(2J + 1) ] [ (J ± M + 1)(J ± M + 2) 1/2 ±ϕ e sinθ Y J M = ∓ (2J + 1)(2J + 3) ] [ (J ∓ M)(J ∓ M − 1) 1/2 Y J +1,M±1 ± Y J −1,M±1 (2J − 1)(2J + 1) [

cos θ Y J M =

(2.13a)

(2.13b)

Here, the upper and lower signs are evenly corresponded. Since the spherical harmonics are the normalized rotational wave functions (2.10), they constitute a complete set of orthogonal functions. Thus using the first of the recurrence relation (2.13a), the z component of the transition moment matrix (2.12) becomes ⎡) (

)( ) _1/2 Jf − Mf + 1 Jf + Mf + 1 ( )( ) δ J i, J f +1 δ Mi,M f (Mz )i f = M0 ⎣ 2J f + 1 2J f + 3 ⎤ )( )( ) _1/2 Jf − Mf Jf + Mf ( )( ) (2.14) + δ J i,J f −1 δ Mi,M f ⎦ 2J f − 1 2J f + 1 With the form of Mx and My, given in Eq. (2.11), determination of the corresponding transition dipole moment matrices is not easy. So we shall adopt other techniques by taking some linear combinations of them in which case the determination of the transition dipole moment matrices becomes much easier. Instead of using M x and √ shall use M M y , we ξ and M η such that M ξ = (M x + i M y )/ 2 and M η = (M x − i √ M y )/ 2. Then

18

2 Rotational Spectra ( ) ) ( i Mx + j M y = i Mξ + Mη /21/2 + j Mξ − Mη /21/2 i ) ) ( ( = 2−1/2 i −i j Mξ + 2−1/2 i +i j Mη







= l Mξ + m Mη

(2.15)

( ) This means a transformation from one orthogonal system i , j to another ( ) l , m . Anyway, the transition dipole moment evaluated in the new system with the help of the second one of the recurrence relation (2.13b) is given by (

Mξ/η

) if

 

e±iϕ sin θ Y J f M f sin θ dθ dϕ [ ( )( ) Jf ± Mf + 1 Jf ± Mf + 2 ( )( ) δ J i,J f +1 δ Mi,M f ±1 = M0 ∓ 2J f f + 1 2J f + 3 ] ( ) )( Jf ∓ Mf Jf ∓ Mf − 1 )( ) δ J i,J f −1 δ Mi,M f ±1 ± ( (2.16) 2J f − 1 2J f + 1

= M0

Thus from the Eqs. (2.14) and (2.16), we derive the selection rules for rotational spectra, Selection Rules for Rotational Spectra ΔJ = J f − Ji = ±1 and ΔM = M f − Mi = 0, ±1

(2.17)

Since the rotational spectra are studied by absorption phenomenon, so if J i = J, the value of J f = J + 1. So the rotational spectral structure is v = TJ +1 − TJ = B(J + 1)(J + 1) − B J (J + 1) = 2B(J + 1) where J = 0.1, 2, 3, 4, . . .

(2.18)

Here, ν is the wavenumber of the concerned line (corresponding to the transition J → J + 1). Since the energy levels (2.8) are independent of the quantum number (M), the selection rules for M is not necessary here. Thus under rigid rotator approximation, a series of equidistant lines is observed with separation of the successive lines being 2B. The first member of the rotational spectra is observed at the wavenumber 2B, and next are found at 4B, 6B, 8B, 10B, · · · etcetera. This is shown in Fig. 2.2. So from

2.4 Isotopic Effect in Rotational Spectra

19

J 5

TJ 30B

4 20B

3

12B

2 1 0

6B 2B 0

2B 4B

6B

8B 10B

Fig. 2.2 Rotational transitions and spectra of a diatomic molecule

the spectral features, the rotational constant and hence the internuclear distance can be determined.

2.4 Isotopic Effect in Rotational Spectra When an atom is replaced by an isotope, the electronic charge distribution in the relevant chemical bond remains practically unchanged, and so the internuclear distance too also remains unchanged. But as the reduced mass of the diatomic molecule changes, the moment of inertia will also change, and this will change the rotational constant (Eq. 2.8). So in a heavier isotopic molecule, the rotational spectra will be appearing to some extent compressed. Let μ and μ' be the reduced mass of a diatomic molecule and its heavier isotope, respectively. Then relative magnitude of the respective rotational constants can be determined from Eq. (2.8). μ B' 1 = ' = 2 with ρ > 1 B μ ρ

(2.19)

Thus, the wavenumber difference between the corresponding lines in the spectra of the two molecules is given by (Fig. 2.3) ) ( ) ( 1 ' Δν = ν − ν = 2 B − B ' (J + 1) = 2B 1 − 2 ( J + 1) ρ

(2.20)

20

2 Rotational Spectra

2B

2B

4B

4B

6B

6B

8B

8B

10B

10B

Fig. 2.3 Rotational spectra of a diatomic molecule (continuous line) and its heavier isotope (broken lines). B and B' are the rotational constants of the respective molecules

2.5 Intensities of Rotational Lines We have stated earlier that the probability of a transition in a molecule from one rotational level to another mainly depends on the differences between the rotational quantum numbers of the two concerned rotational states (ΔJ), which is ±1. But it hardly depends on the particular set of values of the rotational quantum number, J f and J i . So it is obvious to expect that whenever we are concerned with the spectra of an assemblage of molecules, intensities of different spectral lines depend on the relative population of the initial rotational levels. According to Maxwell Boltzmann distribution law, the population of a rotational level (J) at the absolute temperature To K is N J = N0 (2J + 1)e−

ch B J (J +1) kT

(2.21)

where (2J + 1) is the degeneracy of the rotational level (J), N 0 is the population of the lowest level J = 0, and k is the Boltzmann constant (1.38 × 10–16 ergs per o K). So the intensity (I J ) of a line arising from the transition J → J + 1 is proportional to the population ratio N J /N 0 , and the variation with J is shown in Fig. 2.4. Fig. 2.4 Relative intensities of the rotational lines

NJ / No (IJ)

Imax

JMAX

J

2.6 Non-rigid Rotator (A Semiclassical Approach)

21

The level having maximum population can be determined from the relation ddNJJ = 0 which yields Jmax = (kT /2Bch)1/2 − 1/2

(2.22)

So the wavenumber of the maximum intense band is [ ] ν max = 2B(Jmax + 1) = 2B (kT /2Bch)1/2 + 1/2

(2.23)

In the case when the spectra is not well resolved, from the measurement of the wavenumber of this maximum intense band, the rotational constant and hence the internuclear distance of the molecule can be determined.

2.6 Non-rigid Rotator (A Semiclassical Approach) If the rotational spectra are critically analysed, it is seen that the separation between the successive rotational lines is really not constant but decreases with the increase of wavenumber (i.e. with the increase of J value). However, the amount of decrease is very small, and that is why, apparently this separation appears to remain the same. If for each successive separation, the internuclear distance is calculated, and it is found that with the increase of J value, the internuclear distance also increases. This means that more and more fast is the rotation, more and more is the internuclear distance due to centrifugal distortion. Thus, it is reasonable to assume that the two atoms in a diatomic molecule are fastened at the two ends of a spiral spring rather than a rigid rod. Here, we shall determine the energy levels of such a non-rigid rotator from semiclassical viewpoint. Let the spring constant be k, and r and r o are the instantaneous and equilibrium internuclear distances. If ω is the angular frequency of rotation, then μω2 r = k(r − r0 ) kr0 r= k − μω2 So the rotational energy of this non-rigid rotator becomes μ2 ω4 r 2 1 2 k (I ω)2 I ω + (r − r0 )2 = + 2 2 2I 2k | |2 | |4 | →| | →| |L | |L | = + ' where L→ is the angular momentum 2μr 2 2kμ2 r 6

E rot =

(2.24)

22

2 Rotational Spectra

≈

≈

=

=

| |2 | |4 | →| [ | →| | L | k − μω2 ]2 |L | + k 2μr02 2kμ2 r06 | |2 | |2 | |2 | |4 | →| | →| 2 | →| 2 4 | →| |L | |L | ω |L | μ ω |L | − + + 2 2 2 2 2μr0 kr0 2μr0 k 2kμ2 r06 | |2 | |4 | |6 | |4 | →| | →| | →| | →| |L | |L | |L | |L | − + + 2μr02 kμ2 r06 2μ3 k 2 r010 2kμ2 r06 | |2 | |4 | →| | →| |L | |L | − (neglecting the third term as it is much 2 2μro 2kμ2 r06

smaller in magnitude than the other terms) { | |2 J (J + 1)}2 h4 J (J + 1)h2 | →| − subtituting the eigen value of = |L | 2μr02 2kμ2 r06 = J (J + 1)h2 = EJ

(2.25)

So in wavenumber, the energy levels are Trot = T J = −

h EJ = J (J + 1) ch 8π 2 μr02 c

h3 {J (J + 1)}2 32π 4 kμ2 r06 c

= B J (J + 1) − D{J (J + 1)}2

(2.26)

Here, B is the rotational constant, and D is called the non-rigidity constant. It can be easily shown that 4B 3 h3 = = non-rigidity consant 32π 4 kμ2 r06 c ν 2vib ( )1/2 k 1 where ν vib = 2π c μ = vibrational frequency of the bond.

D=

(2.27)

(2.27a)

It is found that D 0, the zero-order states are the product of a nuclear state (|N > = |ImI > ) and an electronic state (|E > ). The parity operator reverses the signs of the components of a vector. If we consider → r n . Therefore if the nuclear Hamiltonian remains nuclear charges, it is the vector − invariant under space inversion and nuclear states do not have any degeneracy, the nuclear Hamiltonian and the parity operator commute with each other, and in such condition, no contribution of the nuclear electrical multipole moments (in Eqs. 2.91 and 2.92) with odd (m) can exist, because the contributions of the integrand at x n , yn , zn and −x n , −yn , −zn cancel each other. Hence, the leading contribution in the electrostatic interaction energy comes from the term associated with nuclear electrical quadrupole moment (m = 2), and this gives rise to the quadrupole hyperfine structure in the molecular spectra. The contributions from the further higher-order terms are much smaller in magnitude, and we can disregard them. Thus from Eqs. (2.91 and 2.92), we get

46

2 Rotational Spectra





) r 2 (→ − → rn · → dτe dτn ρn (− rn )ρe (→ re re ) n3 P2 − re   ) ρe (→ re ) 1( → = rn2 ρn (− r n ) 3 cos2 θen − 1 dτn dτe 2 re3 re rn ] [   E 3 1 → = ρn (− rn ) xni xn j xei xej − rn2 re2 2 2 τe τn i j

HE2 =

ρe (→ re ) dτe dτn re5 ( ) 1E =− Qi j ∇ E e i j 6 ij

(2.93)

where  Qi j = τn



= τn

) ( ρn (→ rn ) 3xni xn j − δi j rn2 dτn ( ) xni xn j + xn j xni 2 ρn (→ rn ) 3 − δi j rn dτn 2

(2.94a)

and  (∇ E )i j = − e

τe

∂ ∂ ρe (→ re ) ∂ xi ∂ x j

(

) 1 dτe re



) ρe (→ r)( 3xei xej − re2 δi j dτe 5 re τn ( )  ρe (→ r ) xei xej + xej xei 2 3 dτe − r =− δ i j e re5 2 =−

(2.94b)

τn

Here, (Qij ) and (∇ E e )i j are both traceless symmetric tensor, and it is to be noted that the last steps of Eqs. (2.94a, 2.94b) arise since the Cartesian coordinates commute among themselves (i.e. x i x j = x j x i ). Our next task is to determine the quantum mechanical average of the interaction energy. For this, we shall use a theorem (Wigner-Eckart) which we shall not prove, but only state and use it. This can also be applied to all second-rank tensors (as of the form (2.94a, 2.94b)) which (a) are constructed in the same manner from vectors satisfying the same commutation rules with respect to I, (b) are symmetric and (c) have a zero trace. This theorem states that the quantum mechanical matrix elements diagonal in I of all such tensors has the same dependence on the magnetic quantum numbers mI .

2.9 Quadrupole Hyperfine Structure in Molecules

47

Thus according to this theorem, the matrix element of the nuclear quadrupole moment in the 2I + 1-dimensional space of the nuclear ground state is [

< I >i < I > j + < I > j < I >i − δi j < I 2 > < Q i j >= C 3 2

] (2.95)

Here, C is a constant. This constant can be expressed in terms of the scalar quantity Q, which is conventionally called the nuclear quadrupole moment and is defined by 

[ ] (ρn )Im I =I 3z n2 − rn2 dτn | | =< I I |eQ 33 | I I >= C)I I |3I Z2 − I 2 |I I ( ] [ = C 3I 2 − I (I + 1) = C I [2I − 1]

eQ =

(2.96)

Hence, the matrix of the nuclear quadrupole moment tensor (2.95), v after substitution of C from (2.96), is given by eQ I (2I − 1) ] [ < I >i < I > j + < I > j < I >i − δi j < I 2 > × 3 2

< Qi j > =

(2.97)

In a similar manner, we can determine the electronic quadrupole moment matrix elements diagonal in electronic angular momentum quantum number (J). Thus, the matrix elements diagonal in J is ( ) eq J ∇ Ee ij = − J (2J − 1) ] [ (J )i (J ) j + (J ) j (J )i − J 2 δi j × 3 2

(2.98)

where  eq J =

3z e2 − r 2 (ρe ) J m J =J dτe re 5

(2.99)

Therefore, the interaction energy (2.93) becomes E E2 = )HE2 (

] E [ (I )i (I ) j + (I ) j (I )i e2 q J Q 3 − I 2 δi j 6I (2I − 1)J (2J − 1) i j 2 ] [ (J )i (J ) j + (J ) j ( J )i (2.100) − J 2 δi j × 3 2

=

48

2 Rotational Spectra

This equation can be written in an alternative form which is more convenient for the calculation. Since I and J commute with each other, ⎫ ) _⎧ ⎨E ⎬ E E (I )i (I ) j ( J )i (J ) j = (I )i ( J )i (I ) j ( J ) j ⎩ ⎭ ij

i

j

)2 = I→ · J→ (

(2.101)

Likewise, E

(I )i (I ) j δi j J 2 =

ij

E

(J )i (J ) j δi j I 2

ij

= I 2 J 2 and

E

δi j I 2 J 2

ij

= 3I J 2

2

(2.102)

E E The only complicated terms are i j (I )i (I ) j (J ) j (J )i = i j (I ) j (I )i (J )i (J ) j . These terms can be determined by using the commutation relations of different components of angular momentums (I i s) and (J i s). So (I )i (I ) j = (I ) j (I )i + i (I )i× j

(2.103)

Here, the subscript i × j indicates a direction perpendicular to the respective vectors as in the case of vector product. Thus, E

(I )i (I ) j (J ) j (J )i =

ij

E ij

+i

(I ) j (I )i (J ) j ( J )i

E

(I )i× j ( J ) j (J )i

ij

( )2 E = I→ · J→ + i (I )i× j (J ) j ( J )i

(2.104)

ij

Again, (J ) j ( J )i + (J ) j (J )i 2 (J ) j ( J )i + (J )i (J ) j + i (J ) j×i = 2

(J ) j (J )i =

So

(2.105)

2.9 Quadrupole Hyperfine Structure in Molecules

i

E

(I )i× j (J ) j (J )i =

ij

[ ] i E (I )i× j (J ) j (J )i + (J )i (J ) j 2 ij

− =

49

1E (I )i× j ( J ) j×i 2 ij

1E (I )i× j (J )i× j (first term on the right - hand side 2 ij

vanishes since it is antisymmetric in i and j ) 1E 2(I )k ( J )k = I→ · J→ = 2 k

(2.106)

Thus from (2.104 and 2.106), we get E ij

(I )i (I ) j (J ) j (J )i =

E

(I ) j (I )i ( J )i (J ) j

ij

( )2 ( ) = I→ · J→ + I→ · J→

(2.107)

Thus substituting Eqs. (2.102, 2.103 and 2.107) into Eq. (2.100), we get E E2

[ ] 9 → → 2 9[ → → 2 e2 q J Q ( I . J ) + ( I . J ) + ( I→. J→) = 6I (2I − 1)J (2J − 1) 2 2 ] 2 2 2 2 2 2 −3I J − 3I J + 3I J ] [ 3 → → e2 q J Q 2 2 2 → → 3( I . J ) + ( I . J ) − I J = 2I (2I − 1)J (2J − 1) 2

(2.108)

For atomic cases, this formula can be applied as such. But it is less convenient for quadrupole moments of 1 ∑ linear molecules rotating with different rotational angular momentum quantum numbers J. This is true because qJ depends on the magnitude of J. Hence in molecules, the entity qJ is determined in the molecular frame of reference instead of the space-fixed frame of reference, since such a quantity will be the same for molecules which differs only in mJ and J. First, we shall determine this value for a linear molecule in terms of the molecule-fixed axis instead of the space-fixed axis. Let θ e and ϕ e be the spherical polar coordinates of a point in the molecule relative to the z-axis of the space-fixed coordinate system, while θe' and ϕe' are the corresponding entities with respect to the molecular axis of symmetry z0 . Let θ ˝ and ϕ˝ be the angular orientation of the z0 axis with respect to the space-fixed z-axis. If we apply addition theorem of spherical harmonics to Eq. (2.99), we get

50

2 Rotational Spectra

) )( 1( 3 cos2 θ '' − 1 3 cos2 θ ' e−1 2 + other terms involving eiϕ

3 cos2 θ e−1 =

(2.109)

(see also Appendix 9.1). The rotational wave function (2.9) for a linear molecule in the rotational state (JmJ = J) is / ψ J J = (−1)

J

) (2J + 1) J ( '' PJ cos θ '' ei J φ 4π

= YJ J

(2.110)

Here, Y JJ is the normalized spherical harmonics. Thus, the average value of qJ in this state is given by | ( (q J )average = J J |q J ||J J (  1 3z e2 − r 2 = (ρe ) J m J =J dτe e re 5  1 3 cos2 θe − 1 = (ρe ) J m J =J dτe e re 3  ( ) 1 = Y J∗J 3 cos2 θ // − 1 Y J J sin θ // dθ // dϕ '' 2e  / 3 cos θe − 1 × (ρe )dτe/ r3

(2.111)

Applying the recursion relation of spherical harmonics, the first integral (of 2.111) can be evaluated to be − 2J2J+ 3 . So Eq. (2.111) becomes (q J )average

 / J 1 3 cos θe − 1 =− (ρe )dτe/ 2J + 3 e r3 J 1 ∂2V e =− 2J + 3 e ∂z 02

(2.112)

where V e is the potential at the centre of the concerned nucleus arising from all the electronic and other nuclear charges outside a small sphere surrounding the concerned nucleus and is given by  Ve =

1 ρe dτe/ r

(2.113)

Thus from Eqs. (2.108 and 2.112), we get the final expression for the energy arising from the quadrupole interaction (2.108) as

2.9 Quadrupole Hyperfine Structure in Molecules

E E2

[ ] eQ ∂ 2 V e /∂z 02 =− 2I (2I − 1)(2J + 3)(2J − 1) [ ] 3 → → 2 2 2 → → 3( I . J ) + ( I . J ) − I J 2

51

(2.114)

− → Let the total angular momentum of the molecule be F which is equal to J→ + I→ . Then the diagonal elements for the operators I 2 , J 2 and I→. J→ are I(I + 1), J(J + 1) and C/2, respectively, where C = F(F + 1) − J (J + 1) − I (I + 1)

(2.115)

Thus, Eq. (2.114) for the quadrupole interaction energy for a linear molecule becomes ] [ eQ ∂ 2 V e /∂ z 02 E E2 = − 2I (2I − 1)(2J + 3)(2J − 1) [ ] 3 C(C + 1) − I (I + 1) J (J + 1) (2.116) 4 This theory can also be extended to a symmetric top molecule. For a symmetric top molecule, the angular momentum is not exactly perpendicular to the molecular − → axis. There is a component of J along the molecular (symmetry) axis z0 , and the rotational wave function is represented by the ket | J K M( unlike | J M( in linear molecule where both J and K take up values 0, ± 1, 2, ± 3, …. Then the analogous to the above Eq. (2.111) gives 

( ) ψ J∗ K M=J 3 cos2 θ '' − 1 ψ J K M=J sin'' dθ '' ] [ 2J 3K 2 −1 = J (J + 1) 2J + 3

(2.117)

and hence | ( (q J )average = J k J |q J ||J k J (  1 = ψ J∗k M=J (3 cos2 θ '' − 1)ψ J k M=J sin θ '' dθ '' 2e  / 3 cos θe − 1 × (ρe )dτe/ r3 [ ]  / K2 1 J 3 cos θe − 1 1−3 =− (ρe )dτe/ 2J + 3 J (J + 1) e r3 [ ] 2 e K2 1∂ V J 1−3 =− (2.118) 2J + 3 J (J + 1) e ∂z 02

52

2 Rotational Spectra

With this value of (qJ )av , the quadrupole interaction energy of a symmetric top molecule becomes ] [ ][ 2 eQ ∂ 2 V e /∂z 02 1 − 3 J (JK+1) E E2 = − 2I (2I − 1)(2J + 3)(2J − 1) [ ] 3 C(C + 1) − I (I + 1) J (J + 1) (2.119) 4 This formula becomes identical to that of (2.116) of a linear molecule for states with K = 0. From this stand point, linear molecule can be looked upon as a special case of symmetric top molecule. Another thing to be kept in mind is that in a symmetric top molecule, the electric field is assumed to be symmetrical about the molecular axis (z0 ), i.e. ∂ 2 V e /∂ 2 x 0 = ∂ 2 V e /∂ 2 y0 . This is true for all cases when a nucleus lies on the molecular axis since symmetric arrangements of atoms are needed to make the moments of inertia of the molecule about x0 and y0 equal. Otherwise if the nucleus does not lie on the molecular axis, the theory becomes more complex because there may be other nuclei with quadrupole coupling. The above formula also does not hold good for accidentally nearly symmetric top molecule. As an example, consider a prolate symmetric top molecule CH3 Cl having the nucleus Cl with nuclear spin I = 3/2 lying on the molecular symmetry axis. We shall confine our discussion related to the rotational transition J = 0 → J = 1. For J = 0 level, K = 0 and F = I = 3/2. For J = 1, K = 0, ± 1 and F = 5/2, 3/2 and 1/2 with C = 3, −2 and −5, respectively. The selection rules for hyperfine quadrupole transition are ΔJ = ±1, ΔK = 0 and ΔF = 0, ±1

(2.120)

Without the quadrupole interaction, the above transition J = 0 → J = 1 with K = 0 gives a line at the wavenumber 2B, where B is the rotational constant corresponding to rotation about an axis perpendicular to the molecular axis. But with the quadrupole interaction taken into consideration, this line splits into three components with wavenumbers ⎫ ν 1 = 2B − 0.25 eQ(∂ 2 V /∂z o2 )/ch ⎬ (2.119) ν 2 = 2B − 0.05 eQ(∂ 2 V /∂z o2 )/ch ⎭ 2 2 ν 3 = 2B + 0.20 eQ(∂ V /∂z o )/ch and this is shown in Fig. 2.12: Quadrupole moment (Q) of a nucleus can be determined from the microwave spectra if (∂ 2 V e /∂z0 2 ) is determined by some other method. The value of this field gradient (∂ 2 V e /∂z0 2 ) can be estimated from the wave mechanical consideration of the nature of bond orbital hybridization and the type of resonance between the different molecular structures. For many atoms, it can be found from the experimentally

References and Suggested Reading

53

F=3/2 J = 1, K = 0 5/2

0

1/2

2B

)/ch 1

2

3

)/ch

J = 0, K = 0

3/2

)/ch

Fig. 2.12 First-order hyperfine quadrupole splitting of the rotational line arising from the transition J = 0 → J = 1 in CH3 Cl molecule (due to nuclear spin I Cl = 3/2). Note that J = 1, K = ± 1 levels do not take part in these spectral transitions; so they are not shown in the diagram

measured fine structure of the atom, since qJ or (∂ 2 V e /∂z0 2 ) also depends on the mean value of 1/r e 3 . However, Q can also be determined from the atomic beam experiment. Hence if the quadrupole coupling constant is determined from microwave spectroscopy, then it is possible to obtain the value of (∂ 2 V e /∂z0 2 ) which depends on the surroundings of the nucleus under consideration, and hence, it gives valuable information of the nature of the bonding electrons in that surroundings. Since s-electrons and completed inner shells have spherical symmetry and d- and f-electrons do not in general move near the nucleus, only bonding p-electrons contributes significantly to (∂ 2 V e /∂z0 2 ).

References and Suggested Reading 1. E.B. Wilson, J.C. Decius, P.C. Cross, Molecular Vibrations: The Theory of Infrared and Raman Spectra. (McGraw Hill Book Company, New York. 1955) 2. C.H. Townes, A.L. Schawlow, Microwave Spectroscopy (Dover Publication, New Tork, 1975) 3. I.I. Gol’dman, V.D. Krivchenkov, Problems in Quantum Mechanics. (Dover Publication, New York, 1993) 4. L. Pauling, E.B. Wilson, Jr, Introduction to Quantum Mechanics with applications to Chemistry (McGraw Hill Book Co. London, New York, 1935) 5. G. Aruldhas, Molecular Structure and Spectroscopy (Prentice Hall of India Pvt. Ltd., New Delhi, 2001) 6. G.M. Barrow, Introduction to Molecular Spectroscopy (McGraw Hill, New York, 1962) 7. N.F. Ramsay, Nuclear Moments. (Wiley, 1953) 8. G. Herzberg, Molecular Spectra and Molecular Structure; II. Infrared and Raman Spectra of Polyatomic Molecules (D. Van Nostrand Company, Inc, Princeton, New Jersey, New York, USA, 1945)

Chapter 3

Infrared Spectra

Abstract Vibrational energy levels of a diatomic molecule are found out by solving the wave equation using a simple harmonic oscillator potential function. The selection rules have been derived, and the nature of the spectra has been discussed along with calculation of force constants. Pointing out the drawbacks of simple harmonic model, the complete wave equation has been solved using the anharmonic oscillator potential function introduced by Morse, and hence, the energy levels and the spectral structure are determined. The effect of coupling between the rotational and the vibrational motions on the spectral features has also been discussed. Lastly, the dissociation energy of a diatomic molecule is determined in terms of vibrational constants.

In the last chapter, we have discussed about the rotational motion in diatomic molecules. There we have not considered the associated vibrational motion. In the present chapter, we shall consider the spectral characteristics of diatomic molecules in the infrared spectral region which arises from the internal vibrations of the said system. Nowadays studies on the infrared spectra are very common as these are of great help in extracting various information related to the molecules such as molecular structure, symmetry, bond strength, intra- and intermolecular interaction. First, we shall consider the nature of pure vibrational spectra of diatomic molecules on the basis of simple harmonic oscillator model and then extend our discussion to various other cases of physical reality including rotational fine structure.

3.1 Vibrational Energy Levels of a Diatomic Molecule Considered as a Simple Harmonic Oscillator Consider that the two atoms in the diatomic molecule are fastened at the two ends of a spiral spring of force constant (k) instead of a rigid rod. If we consider pure vibrational motion, i.e. molecule having no rotational motion, the Schrödinger equation becomes (see Eqs. 2.5, 2.6 and 2.7)

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 P. K. Mallick, Fundamentals of Molecular Spectroscopy, https://doi.org/10.1007/978-981-99-0791-5_3

55

56

3 Infrared Spectra

  1 d 2 dR(r ) h2 (r ) + {E el (r ) − E} = 0 − 2μ R(r )r 2 dr dr

(3.1)

where all the symbols are explained in Eq. 2.5. Unlike rigid rotator, here the internuclear distance is not constant but varies with the execution of the internal vibration. First, we shall consider a simple model and assume that the vibration to be a simple harmonic one. Then the potential energy E el (r) is given by E el =

1 k(r − re ) 2

(3.2)

Here, r is the instantaneous internuclear distance, and r e is the equilibrium internuclear distance, that is, the distance when both the atoms of the molecule are at their respective equilibrium positions. Then the above Eq. (3.1) becomes −

1 h2 1 d 2 dR(r ) (r ) + k(r − re )2 R(r ) = ER(r ) 2 2μ r dr dr 2 Putting

R(r ) = S(r )/r

(3.3) (3.4)

in Eq. (3.3), we get −

h 2 d2 S(r ) 1 + k(r − re )2 S(r ) = ES(r ) 2μ dr 2 2

(3.5)

Since r e is a constan t quantity, let us put ρ = r−r e , which is a measure of the instantaneous displacement of the inter nuclear distance from its equlibrium value. Then the above Eq. (3.5) becomes −

h2 d2 S(ρ) 1 2 + kρ S(ρ) = ES(ρ) 2μ dρ2 2

(3.6)

This is the standard quantum mechanical equation of a linear harmonic oscillator. Its solution is very well known and is given below:   EV 1 E = = V+ hν e τ= ch ch 2

(3.7a)

/ where ν e = 2π1 c μk and V is the vibrational quantum number which can take up values 0, 1, 2, 3… and S(ρ) = SV (ρ) = N V eαρ where

α=

2

/2

HV

√

μω 2π vμ = h h

αρ



(3.7b) (3.8a)

3.1 Vibrational Energy Levels of a Diatomic Molecule Considered …

57

and H V is the Hermite polynomial of degree V, and N V is the normalization constant given by  NV =

√

α √ V 2 (V !) π

1/2 (3.8b)

Thus, the successive energy levels are equally separated, and E 0 = 21 hve = 21 hc ve is called the zero-point energy.

3.1.1 Selection Rules As in the case of rotational spectra, here also the intensity of radiation arising from a spectral transition from an initial vibrational state |i > to a final one |f > is proportional  f . For pure vibrational spectra, to the square of the transition dipole moment i| M| both |i > and |f > are the wave functions (eigen kets) of the respective vibrational states. During execution of small oscillations of the two atoms in the molecule about their respective equilibrium positions along the inter nuclear axis, the electric dipole  too will be changing accordingly. So expanding M  in a Teller’s series moment, M, about the equilibrium positions of the two atoms (that is, about the equilibrium inter nuclear distance), the transition dipole moment becomes  f  = i| M 0 + i| M|

 ∂M ∂ρ

0

 1 ∂2 M dρ + (dρ)2 2 ∂ρ 2 0

+ . . . higher order terms(hot)| f 



2   ∂ M ∂ 1 M  0 i | f  + i|ρ| f  + i|ρ 2 | f  + hot =M ∂ρ 2 ∂ρ 2 0

(3.9)

0

The subscript zeros correspond to equilibrium nuclear configuration. Here, we have put dρ = ρ(t)−ρ(t = 0) = ρ(t) = instantaneous (t) value of the inter nuclear distance, ρ. Since the vibrational wave functions constitute a complete set of orthonormal wave functions, so the first term on the right-hand side of Eq. (3.9) will be non-vanishing only when i /= f which corresponds to no vibrational transition. This term will only contribute to the intensities of rotational lines. Again applying the recursion relation of Hermite polynomials x Hn (x) = Hn+1 (x)/2 + n Hn−1 (x)

(3.10)

58

3 Infrared Spectra

we get, √ 1/2 √ √ α 1 2 e 2 αρ HV +1 ( α ρ)/2 + V HV −1 ( α ρ) √ V 2 π (V !) / / V +1 V |V + 1 + |V − 1 (using 3.8b) (3.11) = 2 2

√ αρ|V  =



Using the relation (3.11) and orthonormalization condition of the vibrational wave functions, we see that the second term on the right-hand side of Eq. (3.9) is nonvanishing only when ΔV = ± 1. This term is therefore responsible for the appearance of the fundamental vibration in the infrared spectra, the intensity of which is propor 2  tional to the square of the first gradient of the dipole moment, i.e. ∂∂ρM . Similarly 0 it can be shown that the next term is non-vanishing only when ΔV = 0, ± 2. The selection rule ΔV = ± 2 gives rise to the second harmonic vibration whose intensity  2 2  is proportional to ∂∂ρM2 . Similarly, further higher harmonics of the vibration appear 0 from the next higher-order terms. Since the magnitude of the dipole moment gradient decreases with the increase of its order, the intensity of the fundamental vibration will be maximum and those of the higher harmonics will gradually decrease with the increase of the order of the harmonics. One very important thing to be kept in  = 0 and mind is that for a homonuclear diatomic molecule, the dipole moment M dipole moment gradients of various orders are also zero. So homonuclear diatomic molecules do not show any vibrational, i.e. infrared spectra. Infrared spectra of only heteronuclear diatomic molecules are observed as their dipole moments and various order dipole moment gradients are nonzero. The spectral structure of these molecules is given by ν V = V νe

(3.12)

With V = 1, 2, 3, ….., the spectra are consisted of a number of equidistant bands with intensities gradually decreasing towards higher harmonics. The wavenumber of the first band and successive band separation is both ν e cm−1 . (Note here that the bands are observed due to poor resolution of the instrument; otherwise, rotational fine structure would be observed in the form of lines with high resolving power instrument.)

3.2 Rotational-Vibrational Spectrum of a Diatomic Molecule …

59

3.2 Rotational-Vibrational Spectrum of a Diatomic Molecule with the Potential Function of a Simple Harmonic Oscillator If we do not disregard the rotational motion in the molecule, the modified form of the Eq. (3.6) becomes −

h 2 d 2 S(ρ) J (J + 1)h2 1 + + k ρ 2 S(ρ) = E S(ρ) 2μ dρ 2 2μr 2 2

(3.13)

Unlike rigid rotator, here the internuclear distance (r) does no longer remain constant. Replacing r by r e + ρ, we get   2ρ 1 1 1 3ρ 2 1 − = = + + (hot) r2 (re + ρ)2 re2 re re2

(3.14)

Since ρ < r e , we have expanded 1/r 2 in terms of Teller’s series. Thus, Eq. (3.13) becomes   h 2 d 2 S(ρ) J (J + 1)h2 3ρ 2 1 2ρ − + + + (hot) + k ρ 2 S(ρ) = ES(ρ) 1 − 2 2 2 2μ dρ 2μre re re 2     2 2 3σ h d S(ρ) σ 1 2σ k + 4 ρ 2 S(ρ) = ES(ρ) i.e. − + 2 − 3ρ+ 2μ dρ 2 re re 2 re +1)h where σ = J ( J2μ and the terms in the Teller series higher than the second order are neglected. 2

 2  3σ σ/re3 1 h 2 d2 S(ρ) J (J + 1)h2 k+ 4 ρ− i.e. − + S(ρ) + 2μ dρ 2 2μre2 2 re (k/2) + 3σ/re4 S(ρ) − i.e − −

σ 2 /re6 S(ρ) = E S(ρ) (k/2) + 3σ/re4

h 2 d2 S(ξ ) J (J + 1)h2 1 + S(ξ ) + k ' ξ 2 S(ξ ) 2 2μ dξ 2μre2 2

h4 /4μ2 re6 {J (J + 1)}2 S(ξ ) = ES(ξ ) (k/2) + 3σ/re4

(3.15)

Here, we have put 3σ 3J (J + 1)h2 1 ' 1 1 k = k + 4 = k + 2 2 re 2 2μre4

(3.16a)

60

3 Infrared Spectra

⎧ ⎨ σ/re3 re + ξ =ρ− = r − ⎩ (k/2) + 3σ/re4

h2 /2μre3 k 2

+

3J ( J +1)h2 2μre4

J (J + 1)

⎫ ⎬ ⎭

= r − r0

(3.16b)

Assuming that 3σ /r e 4 is much smaller than k/2, which is very often the case, the solution for energies of the Eq. (3.15) becomes E = (V + 1/2)hνe' +

J (J + 1)h2 h4 {J (J + 1)}2 − 2 2μre 2kμ2 re6

(3.17)

where / /  /  1/2  ' 6σ 3σ k k k 1 1 1 ' 1+ 4 1+ 4 νe = ≈ = 2π μ 2π μ kre 2π μ kre   3J (J + 1)h2 = νe 1 + 2μkre4

(3.18)

Substitution of Eq. (3.18) in (3.17), we get E = E VJ = (V + 1/2)hνe +

J (J + 1)h2 2μre2

3J (J + 1)h2 hνe h4 {J (J + 1)}2 (V + 1/2) − 2μkre4 2kμ2 re6   h2 3h 3 = (V + 1/2)hνe + + (V + 1/2) J (J + 1) 8π 2 μre2 32π 4 μ2 νe re4 +

−

h4 {J ( J + 1)}2 32π 4 kμ2 re6

(3.19)

and the corresponding term values are EV J = (V + 1/2)¯ν e T = TVJ = ch   3h 2 h + + (V + 1/2) J (J + 1) 8π 2 cμre2 32π 4 cμ2 νe re4 −

h3 32π 4 ckμ2 re6

{J (J + 1)}2

= (V + 1/2)¯ν e + {Be + αe (V + 1/2)}J (J + 1) − D{J (J + 1)}2 = (V + 1/2)¯ν e + BV J (J + 1) − D{J (J + 1)}2

(3.20)

3.2 Rotational-Vibrational Spectrum of a Diatomic Molecule …

61

where ν e = νe /c, Be =

h 8π 2 cμre2

, αe =

BV = {Be + αe (V + 1/2)} =

3h 3 32π 4 cμ2 νe re4

,D =

h3 32π 4 ckμ2 re6

h 8π 2 cμr V2

and (3.21)

Here, Be is the rotational constant corresponding to the equilibrium internuclear distance r e , D is called the non-rigidity constant, and BV is defined as the rotational constant for the V th vibrational state with r V being defined as the average internuclear distance in the vibrational state V. The rotational fine structure of the vibrational spectra is given by ν V = (V ' − V '' )ν e + BV ' J ' (J ' + 1) − BV '' J '' (J '' + 1)   2 2 − D '' J ' (J ' + 1) + D '' J '' (J '' + 1)

(3.22)

where the superscripts (‘) and (“) correspond to the higher (V ' , J ' )and lower(V '' , J '' ) rotational—vibrational states of the associated transition. We know that the vibrational spectra is commonly studied in the infrared region by absorption technique. So by applying the selection rules ΔV = V ' −V '' = + 1, + 2, + 3,… and ΔJ = J ' −J '' = ± 1, we see that we get two systems of lines, one arising from the selection rule ΔJ = −1 is called P-branch and the other arising from the selection rule ΔJ = + 1 is called R-branch. So the P- and R-branches of the vth harmonic for the transition V'' = 0, J '' = J → V ' = V, J ' = J ± 1 are given by v V (P) = V ve − (BV ' + BV ' )J + (BV ' − BV ' )J 2 − D ' {(J − 1)J }2 + D '' {J ( J + 1)}2 with J = 1, 2, 3, . . .

(3.23)

and v V (R) = V v e + 2BV ' + (3BV ' − BV '' )J + (BV ' − BV '' ) J 2 − D ' {(J + 1)( J + 2)}2 + D '' {J (J + 1)}2 with J = 0, 1, 2, 3, . . .

(3.24)

If we disregard the variation of the non-rigidity term (i.e. D' ≈ D'' = D, say), these two equations can be represented by a single equation, v V (m) = V ve + (BV ' + BV '' )m + (BV ' − BV '' )m 2 − 4Dm 3

(3.25)

where m = J + 1 correspond to R − branch and m = −J correspond to P − branch. If we also ignore the small variation of the rotational constants and assume that BV ' ≈ BV ” = B, Eq. (3.25) becomes (ignoring also the non-rigidity constant)

62

3 Infrared Spectra

ν V (m) = V ν e + 2Bm

(3.26)

This gives a series of nearly equidistant lines both in the P- and in the R-branches with successive line separation 2B (Fig. 3.1). Thus analysing the spectra, both the rotational constant and hence the internuclear distance of the molecule can be determined. One very important point becomes noteworthy in this connection. Due to the selection rule ΔJ = ± 1 for the two branches, no line is observed at the wavenumber V ν e which corresponds to the pure vibrational transition. This is called the zero line. Again the wavenumbers of the first members of the P- and R-branches are V ν e − 2B and V ν e + 2B. Since the zero line lies in the region between these two lines, this region is called the zero line gap. The width of the zero line gap is 4B which is double the successive separation of lines in the P- and R-branches. So by examining the rotational-vibrational spectra (mainly around the fundamental vibration and few of its harmonics after which the intensities become too small to be observed experimentally), the respective zero lines are determined from which the force constant of the bond, which too / is very important to the molecular scientists, 1 may be determined. Since ν e = 2π c μk , for heavier isotope, i.e. with higher value of μ, the zero line is shifted towards lower wavenumber region. Moreover, due to the decrease in the rotational constant in the case of heavier isotope, the entire spectra will appear contacted. The variation in the probabilities of transitions from different rotational levels of the ground vibrational state (V = 0) to those of higher vibrational levels is very small, so the intensity of a rotational line in the vibrational spectra is determined by the population of the initial rotational level of the ground vibrational state from where the transition starts. An apprehended variation of intensities of the rotational lines in the vibrational spectrum is shown in Fig. 3.1. Since quantum number of the rotational level of the ground vibrational state for which the population is maximum, according to Eq. (2.22), is Jmax = (kT /2Bch)1/2 − 1/2

(3.27)

the wavenumbers of the most intense lines in the P- and R-branches of the fundamental vibration are given by 1/2 ν max − 1/2} fund (P) = ν e − 2B{(kT /2B ch)

(3.28a)

1/2 ν max + 1/2} fund (R) = ν e + 2B{(kT /2B ch)

(3.28b)

So the wave number difference between the maximum intense lines (or peak heights in the case of unresolved spectra) in the P—and R—branches is given by 1/2 Δν max = (8B kT/ch)1/2 fund (RP) = 4B(kT/2B ch)

(3.29)

3.2 Rotational-Vibrational Spectrum of a Diatomic Molecule …

ΔJ = J - J = -1 P – Branch

63

ΔJ = J - J = +1 R - Branch

4 J

3 V =1

2 1 0

4 J

3 V =0

2 1 0

Zero Line

Intensity

e

2B

2B Zero Line Gap 4B

Fig. 3.1 Rotational-vibrational energy levels, transitions and spectral structure around the fundamental vibration of a diatomic molecule. Dashed lines correspond to forbidden transition and absence of a spectral line at the position of the pure vibrational transition

Thus by knowing the temperature (T ) and Δν max fund (RP) from experimental observation, the rotational constant and hence the internuclear distance of the molecule can also be determined.

64

3 Infrared Spectra

3.3 Anharmonic Oscillator and Morse Potential Function Simple harmonic oscillator model for a diatomic molecule has some serious drawbacks. Critical analysis of the vibrational spectra shows that the wavenumber separation of the successive harmonics is not constant but decreases towards higher wavenumber region, i.e. with the increase of the order of the harmonics. But according to this model (SHO), the average internuclear distance decreases with the increase of vibrational quantum number (due to the rotational-vibrational coupling, Eqs. 3.20 and 3.21) which goes against the common idea of structural behaviour of molecules expected in their excited states. Moreover, the potential function (Eq. 3.2) indicates that the potential energy and hence the restoring force increase with the increase of the internuclear distance (r). So this model is unable to explain dissociation of molecules. So it is expected that an anharmonic oscillator model could be a better representative for the potential function. In fact a function, more realistic, although empirically introduced by P.M. Morse, is found to be a good approximation for the potential function yielding results which comply with those found from experimental observations. This potential function, named after Morse, is given by  2 V (r ) = E el (r ) = De 1 − e−β(r −re )

(3.30)

where De is the dissociation energy measured with respect to the potential energy at the equilibrium internuclear position (r e ) and β is a small positive constant which varies from molecule to molecule. Note that Do (= De −E o ) is the dissociation energy of the molecule measured with respect to the zero-point energy, E 0 (Fig. 3.2). With simple harmonic oscillator potential function, the wave Eq. (3.1) after replacing R(r) by S(r)/r reduces to Eq. (3.13). Here, the wave equation will be same as Eq. (3.13), but the potential function (1/2)kρ 2 will be replaced by the Morse function (3.30). Thus, we get Fig. 3.2 Morse potential energy function and energy levels

V(r)

Continuum

D0 4 3 2 1 0

De

3.3 Anharmonic Oscillator and Morse Potential Function

−

65

 2 h 2 d 2 S(ρ) J ( J + 1)h2 + S(ρ) + De 1 − e−β ρ S(ρ) = ES(ρ) 2μ dρ 2 2μr 2

(3.31)

Expanding both r −2 in Taylor’s series as in Eq. (3.14) and also the exponential function keeping terms up to fourth order in ρ, we get   h 2 d2 S(ρ) J ( J + 1)h2 3ρ 2 4ρ 3 5ρ 4 2ρ − + + 2 − 3 + 4 S(ρ) 1− 2μ dρ 2 2μre2 re re re re   2 β2ρ2 β 3ρ3 + S(ρ) = ES(ρ), + De βρ − 2 6    3σ σ 2σ h 2 d2 S(ρ) 2 + 2 S(ρ) + − 3 ρ + + De β ρ 2 i.e. − 2μ dρ 2 re re re4     4σ 5σ 7 3 3 4 De β ρ 4 S(ρ) = ES(ρ), − + De β ρ + + re5 re6 12 here σ = i.e. −

h 2 d2 S(ρ) σ + 2 S(ρ) + 2μ dρ 2 re



J (J + 1)h2 2μ 3σ 1 k+ 4 2 re

 ρ−

σ/re3 (k/2) + 3σ/re4

2

σ 2 /re6 S(ρ) − S(ρ) (k/2) + 3σ/re4     4σ 5σ 7 3 3 4 De β ρ 4 S(ρ) − + De β ρ S(ρ) + + re5 re6 12 = ES(ρ), where k = 2De β 2   1 σ h 2 d2 S(ρ) σ 2 /re6 S(ρ) + ko ξ 2 S(ρ) i.e. − + 2− 2 4 2μ dρ re (k/2) + 3σ/re 2   3 4 + Aρ + Bρ S(ρ) = ES(ρ) where ξ = ρ − a=

(3.32)

σ/re3 = ρ − a = r − (re + a), (k/2) + 3σ/re4

σ/re3 , (k/2) + 3σ/re4 3σ 1 1 ko = k + 4 2 2 re   4σ 3 A = − 5 + De β re

(3.33a) (3.33b) (3.33c)

66

3 Infrared Spectra

and B =

5σ 7 De β 4 + re6 12

(3.33d)

Now, put ρ = ξ + a and assume ρ 3 ≈ ξ 3 + 3ξ 2 a and ρ 4 ≈ ξ 4 + 4ξ 3 a, neglecting second- and higher-order terms in ‘a’ (as ‘a’ is very small in magnitude). Thus, the above Eq. (3.32) becomes     σ 1 h 2 d 2 S(ρ) σ 2 /re6 ko + 3a A − + 2− S(ρ) + 2μ dρ 2 re (k/2) + 3σ/re4 2 ξ 2 S(ρ) + (A + 4a B)ξ 3 S(ρ) + Bξ 4 S(ρ) = ES(ρ)   h 2 d2 S(ξ ) σ σ 2 /re6 i.e. − + − 2μ dξ 2 re2 (k/2) + 3σ/re4 1 S(ξ ) + k1 ξ 2 S(ξ ) + A1 ξ 3 S(ξ ) + Bξ 4 S(ξ ) = ES(ρ) 2 where k1 = ko + 6a A and A1 = A + 4a B

(3.34) (3.35)

Since both A (hence A1 ) and B are small quantities, we can apply perturbation theory to solve Eq. (3.34). Here the total Hamiltonian is H = Ho + H1 + H 2

(3.36a)

wher e the unpertur bed part o f the H amiltonian is Ho = −

  1 h2 d 2 σ σ 2 /re6 + k1 ξ 2 + − 2μ dξ 2 re2 (k/2) + 3σ/re4 2

(3.36b)

and the perturbed part consists of two terms H1 = A1 ξ 3 and H2 = Bξ 4

(3.36c)

The solutions of the unperturbed equation Ho So (ξ ) = E o So (ξ ) or Ho |V  = E o |V 

(3.37)

are 

E o = E VJ

σ σ 2 /re6 = (V + 1/2)hνe1 + 2 − re (k/2) + 3σ/re4

≈ (V + 1/2)hνe1 +



h 2 J (J + 1) h4 − {J (J + 1)}2 8π 2 μre2 32π 4 kμ2 re6

(3.38a)

3.3 Anharmonic Oscillator and Morse Potential Function

67

/ 1 k1 where, νe1 = 2π μ  √ 1/2 α1 √ 1 2 and So (ξ ) = |V  = e 2 α1 ξ HV ( α1 ξ ) √ 2V (V !) π

(3.38b)

(3.39a)

where HV is the Hermite Polynomial of degree V and α1 =

√ μk1 h

(3.39b)

Now using the recurrence relation (3.10) of the Hermite polynomial H V (ξ) and of the vibrational wave functions (3.11), it is found that √ 2 α1 ξ |V  =

/

/   1 (V + 1)(V + 2) V (V − 1) |V + 2 + V + |V  + |V − 2 4 2 4 (3.40)

This equation helps us in doing the perturbation calculation. It can be easily understood that no first-order correction to energy is obtained from H1 , which only gives nonzero second-order correction and the first-order correction is obtained from the perturbed Hamiltonian H 2 . First-order perturbation calculation The first-order correction to the energy is |  |   | B | E 2'(1) = V |H2 |V  = V | Bξ 4 |V = 2 V |α12 ξ 4 |V α1    B = 2 V |α1 ξ 2 α1 ξ 2 |V  α1     1 2 V (V − 1) B (V + 1)(V + 2) + V+ + = 2 4 2 4 α1    2 1 3 B 3 V+ + = 2 2 8 α1 2 !2     7    3 5σ + 12 De β 4 re6 1 2 1 1 2 1 7 D e β 4 h2 V+ V+ = + + ≈ 2μk1 2 4 8 μk1 2 4 (3.41) neglecting the term

5σ 7 as it is much less than the term De β 4 . re 12

68

3 Infrared Spectra

Second-order perturbation calculation The second-order correction to the energy, as calculated from the perturbed part H1 of the Hamiltonian, is E 1'(2)

|    | " V |H1 |V ' V ' |H1 |V " | V |H1 |V ' |2 = = EV J − EV ' J EV J − EV ' J V ' /=V V ' /=V | | | | " | V | A 1 ξ 3 | V ' |2 = (V − V ' )hvel V ' /=V |/ |√ 3 || ' \||2 | | V α ξ | | |V | " 1 A1 = 3 ' (V − V )hvel α1 V ' /=V

(3.42)

where in the unperturbed part (3.36), rotational energy is treated as constan t. Using the relations (3.11) and (3.40), we get √

3 |  α1 ξ |V ' =

/

 (V ' + 1)(V ' + 2)(V ' + 3) || ' V +3 8 |   3 √ ' 3 √ | + √ (V + 1)3 |V ' + 1 + √ V '3 |V ' − 1 2 2 2 2 / ' ' '  V (V − 1)(V − 2) || ' V −3 + 8

(3.43)

Thus, Eq. (3.42) becomes 

 V (V − 1)(V − 2) (V + 1)(V + 2)(V + 3) 9 3 9 3 − − + V (V + 1) 8×3 8 8 8×3 α13 hve1   2   A21 9V + 9V + 6 =− 3 + 9 3V 2 + 3V + 1 3 8α1 hve1    ' A21 & 15 A21 7 1 2 2 =− 3 30V + 30V + 11 = − + V+ 4 α13 hve1 2 60 8α1 hvel    2 2 6 3 7 15 De β (h/2π ) 1 + ≈ −√ V+ √ 2 16 (μk1 )3 (h/2π ) k1 /μ 4     De2 β 6 (h/2π )2 15 1 2 7 =− V+ + (3.44) 4 2 16 μk12

E 1' (2) =

A21

Here, we have put A1 ≈ A ≈ −De β 3 (according to Eq. (3.33c, 3.35) and used Eqs. (3.38b and 3.39b). So, by approximating k 1 ≈ 2De β 2 as shown below, we get

3.3 Anharmonic Oscillator and Morse Potential Function '

E =

E 2'(1)

+

E 1'(2)

  1 2 1 β 2h2 V+ ≈− 2 μ 2

69

(3.45)

From Eqs. (3.32, 3.33 and 3.35), we get,   4σ 3σ 1 1 1 σ/re3 3 k1 = k0 + 3a A = k + 4 − 3 + De β 2 2 2 re (k/2) + 3σ/re4 re5 3σ 3σ ≈ De β 2 + 4 − 3 β, putting k /2 ≈ De β 2 re r   e 3σ 1 2 (3.46) = De β − 3 β 1 − re re β Thus, the unperturbed energy (3.38) becomes /

h 2 J (J + 1) k1 h4 (V + 1/2) + − {J (J + 1)}2 2 2 4 μ 8π μre 32π kμ2 re6 /   3σ h 2De β 2 h 2 J (J + 1) 1− = (1 − 1/βr ) (V + 1/2) + e 2π μ β De re3 8π 2 μre2

h E0 = 2π

h4 {J (J + 1)}2 32π 4 kμ2 re6 /   3 h 2De β 2 h 1− = (1 − 1/βre ) 2 2 J (J + 1) (V + 1/2) 2π μ β De re3 8π μre −

h4 h 2 J (J + 1) − {J (J + 1)}2 8π 2 μre2 c 32π 4 kμ2 re6 / k h (V + 1/2) − αe ch(V + 1/2){J (J + 1)} = 2π μ +

h 2 J (J + 1) h4 − {J (J + 1)}2 2 2 4 8π μre 32π kμ2 re6   h2 = hνe (V + 1/2) + − αe ch(V + 1/2) {J (J + 1)} 8π 2 μre2 +

−

h4 {J (J + 1)}2 32π 4 kμ2 re6

= hνe (V + 1/2) + ch BV J (J + 1) − ch D{J (J + 1)}2 / 1 where νe = 2π

k h , BV = − αe (V + 1/2) 2 μ 8π μre2 c

(3.47)

70

3 Infrared Spectra

h3 32π 4 kμ2 re6 c 3νe h (1 − 1/βre ) 2 2 and αe = β De re3 8π μre c

= Be − αe (V + 1/2), D =

(3.47a)

Thus, the total energy becomes E = E VJ = E 0 + E ' = hνe (V + 1/2) −

1 β 2h2 (V + 1/2)2 2 μ

+ ch BV J (J + 1) − ch D{J (J + 1)}2

(3.48)

and the term value is T = TVJ = E VJ /ch = ν¯ e (V + 1/2) − ν¯ e xe (V + 1/2)2 + BV J (J + 1) − D{J ( J + 1)}2

(3.48a)

1 β 2h2 2 μ ch = anharmonicity constant(sometimes xe is called anharmonicity constant.) (3.49)

where ν e = νe /c and ν e xe =

Here also applying the same selection rules, ΔV = 0, ± 1, ± 2, ± 3,….etc. and ΔJ = + 1 and −1 for P- and R-branches and neglecting small variation of the rotational and non-rigidity (or centrifugal stretching) constants in different vibrational states, the P- and R-branches of the vth harmonics are given by the formulas (from Eq. 3.25) v¯ V (m) = V v¯ e − v¯ e xe V (V + 1) + 2Bm − 4Dm 3

(3.50)

v¯ V (m) = V v¯ e − v¯ e xe V (V + 1) + 2Bm

(3.50a)

which reduces to

if the centrifugal stretching constant is neglected. As before m = J + 1 and J—for R and P—branches respectively with the respective J values starts from 0 and 1. Thus, for pure vibrational spectra (m = 0) ⎫ v fundamentd = v e − 2v e xe ⎪ ⎪ ⎪ ⎪ ⎪ v 2nd harmonicsl = 2v e − 6v e xe ⎪ ⎪ ⎪ ⎪ v 3rd harmonicsl = 3v e − 12v e xe ⎬ v 4th harmonicsl = 4v e − 20v e xe ⎪ ⎪ ⎪ .............................. ⎪ ⎪ ⎪ ⎪ .............................. ⎪ ⎪ ⎭ v V th harmonics = V v e − v e xe V (V + 1)

(3.51)

3.4 Dissociation Energy

71

Thus, analysing the vibrational spectrum, the vibrational constants of the molecule can be determined. This, in turn, is helpful in determining the dissociation energy which is discussed in the following section.

3.4 Dissociation Energy From Eq. (3.51), we see that the wavenumber separation between successive vibrational levels decreases with the increase of the vibrational quantum number. At dissociation level, this difference vanishes, and beyond this vibrational state, the energy becomes continuous. The pure vibrational level is given by (3.48a) TV = E V /ch = v e (V + 1/2) − v e xe (V + 1/2)2

(3.52)

The maximum value of the vibrational quantum number (V max ) up to which the energy is discrete can be found as follows. At the beginning of the continuum, TV +1 − TV = 0 i.e. v e − 2v e xe (V + 1) = 0 1 i.e. V = Vmax = −1 2xe

(3.53)

Hence, the dissociation energy as measured from the minimum of Morse potential energy (Fig. 3.2) is De = TVmax = v¯ e (Vmax + 1/2) − v¯ e xe (Vmax + 1/2)2     1 1 1 1 2 − v¯ e xe = v¯ e − − 2xe 2 2xe 2 v¯ e v¯ e xe = − 4xe 4

(3.54)

The dissociation energy as measured from the lowest vibrational level is therefore D0 = De − T0 = =

v¯ e v¯ e − 4xe 2

v¯ e v¯ e xe v¯ e v¯ e xe − + − 4xe 4 2 4 (3.55)

Thus knowing the vibrational constants, the dissociation energies can be determined.

72

3 Infrared Spectra

References and Suggested Reading 1. H.A. Bethe, R.W. Jackiew, Intermeduate Quantum Mechanics. (W.A. Benjamin. New York, USA, 1968) 2. H. Eyring, J. Walter, G.E. Kimball, Quantum Chemistry (Wiley, New York, London, 1944) 3. J. Michael Hollas, Modern Spectroscopy (Wiley, Chichester, England, 2004). 4. G. Aruldhas, Molecular Structure and Spectroscopy (Prentice Hall India Pvt. Ltd., New Delhi, 2001) 5. G.M. Barrow, Introduction to Molecular Spectroscopy (McGraw Hill Book Company Inc., New York, USA, 1962) 6. C.N. Banwell, Fundamentals of Molecular spectroscopy (Tata McGraw Hill, New Delhi, India, 1972)

Chapter 4

Raman Spectroscopy

Abstract Raman effect is explained from classical, semiclassical and quantum mechanical standpoints. Selection rules for rotational and vibratioanal Raman spectra are determined and the characteristic features of rotational, vibrational and rotationalvibrational Raman spectra have been discussed including Raman and the infrared activities and polarization properties of Raman bands along with their intensities. The effect of nuclear spin on the spectral behaviour is also presented. Surface enhanced Raman scattering has been discussed and the relevant theoretical background of this process is given.

When a beam of monochromatic radiation is incident on a molecular system, three cases arise. It may be absorbed if the energy of the radiation quantum is equal to the difference of energies between two quantum states of the system. If not absorbed, the beam may be transmitted. There is a third option, that is, the beam may be scattered by the molecular system. While experimenting on this scattering phenomena, Lord Rayleigh in 1871 found that the intensity of the scattered radiation (I s ) is inversely proportional to the fourth power of the wavelength (λ) of the incident radiation, i.e. I s = (constant).λ−4 . Thus, he explained that small wavelength portion of the visible radiation is scattered by the particles in the atmosphere to the maximum extent and so the clear sky appears blue. Later on, when experimenting on the scattering of light by liquids like benzene, chloroform, carbon tetrachloride, etc., C. V. Raman (in 1928) found through spectroscopic analyses that the scattered radiation not only contains the incident (i.e. unchanged wavelength) radiation, called Rayleigh’s radiation, but also some weak radiation of wavelengths, both greater and smaller than that of the incident radiation. Earlier such possibility was predicted by A. Smekal in 1923. Out of these weak radiations, those which lie on the lower wavelength side (higher energy side) of the Rayleigh radiation are found to be much weaker than those lying on the higher wavelength side (lower energy side). Since then this scattering process was named after the discoverer, that is, Raman scattering. The weak radiations with frequencies greater and less than that of the incident radiation are called Raman radiations or Raman lines, lines in the sense that they are very sharp. Lines lying on the lower frequency (or higher wavelength) side of the incident radiation are called stokes and © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 P. K. Mallick, Fundamentals of Molecular Spectroscopy, https://doi.org/10.1007/978-981-99-0791-5_4

73

74

4 Raman Spectroscopy

those lying on the higher frequency (or lower wavelength) side are called antistokes Raman lines. At first, Raman thought that this may be an optical analogue of Compton scattering. This idea was discarded on the basis that the wavelengths of these weak radiations for a particular scattering system remain unchanged even if the angle of scattering is changed. During the course of understanding of the origin of the phenomenon, once it was thought to be some kind of luminescence phenomenon. This idea was also discarded on the basis of the following observations. For excitation of luminescence from a molecular system, excitation radiation was found to have a minimum frequency, below which no luminescence could be observed. Raman scattering was found to be excited with radiation of any frequency, no such minimum frequency radiation was necessary. Besides this, luminescence frequencies were found to be the characteristics of the system. That means they remain constant for a particular molecular system independent of the frequency of the exciting radiation. However in the case of Raman scattering, the frequency differences between the Raman lines and that of the incident radiation (very often called Raman frequencies or Raman shifts) and not the actual frequencies of the Raman lines were found to be the characteristics of the scatterer. Moreover unlike Raman lines, the luminescence spectra are found to appear as bands broader than those of the Raman lines. At this juncture, a striking observation opened the door of understanding of the origin of the phenomenon. Raman frequencies or Raman shifts were found to be the same as those of the infrared frequencies of the same molecular system. This proved that the origin of the phenomenon must be molecular in nature.

4.1 Classical Explanation of Raman Scattering Whenever a radiation is incident on a molecule, the electric field associated with the electromagnetic radiation induces an electric dipole moment in the molecule, given by → μ → =α·E

(4.1)

→ is the electric field of the incident radiation and α is called the polarizability where E of the molecule, which is a tensor of rank two ⎤ αxx αxy αxz α = ⎣ αyx αyy αyz ⎦ αzx αzy αzz ⎡

(4.2)

For a diatomic molecule, say H2 , the polarizability is anisotropic. Because of the fact that the electrons forming the bond are more easily displaced by electric fields applied along the bond axis than across the bond direction. The polarizabilities in

4.1 Classical Explanation of Raman Scattering Fig. 4.1 Polarizability ellipsoid of H2 molecule as seen from the direction a across and b along the bond

75

H

(a) H

H

(b) H

various directions are most conveniently represented by polarizability ellipsoids. This ellipsoid is a three-dimensional surface whose distance from the electrical centre √ is proportional to 1/ αi where αi is the polarizability along the line joining the electrical centre with the point ‘i’ on the ellipsoid. This is similar to the moment of inertia ellipsoid. This is shown in Fig. 4.1. → oscillating with a frequency equal to that of the incident The electric field E, radiation νo , is → =E → o cos(2π νo t) E

(4.3)

The molecule has some internal motions, rotations or vibrations which change the polarizability periodically with a period of the molecular frequency. First consider the rotational motion of the molecule. As can be understood from Fig. 4.1, if the molecule rotates with an angular frequency 2π ν r , the corresponding polarizability ellipsoid rotates twice as fast. This change in polarizability during rotation causes a change in the induced dipole moment given by → o cos(2π νo t) μ → = (α or + α 1r ) · E → o cos(2π νo t) = α or · E 1 → o [cos{2π (νo + 2νr )t} + cos{2π(νo − 2νr )t}] + α 1r · E 2

(4.4)

This oscillating dipole moment thus radiates three radiations of frequencies: ν o (Rayleigh), ν o + 2ν r (antistokes Raman) and ν o − 2ν r (stokes Raman). Here, α or is the average polarizability and α 1r is the amplitude of the change in polarizability during rotation. Now consider the vibrational motion in the molecule. For the execution of an internal vibrational mode in the molecule, Q = Qo cos (2π ν V t), with a frequency, say, ν V , the polarizability changes as: ( α = (α o ) +

∂α ∂Q

) Q 0

(4.5)

76

4 Raman Spectroscopy

Neglecting the higher-order terms, the induced dipole moment becomes ) ] [ ( ∂α μ → = (α o ) + Q E 0 cos(2π νo t) ∂Q 0 → o cos(2π νo t) = αoE ( ) 1 ∂α → o cos[{2π (νo + νV )t} + cos{2π (νo − νV )t}] + Qo E 2 ∂Q 0

(4.6)

Oscillating induced dipole moment (μ) → in Eq. (4.6) therefore emits radiation of three distinct frequencies. (i) Rayleigh line with frequency ν = ν o (ii) Antistokes Raman line with frequency ν = ν o + νV (iii) Stokes Raman line with frequency ν = ν o − νV . ) ) ∂α It may be noted that if ∂Q is zero, no Raman line will be observed. So the prime 0 criteria of observing a vibrational Raman line is that α must be varying with the vibrational coordinate Q with nonzero gradient at the equilibrium nuclear configuration, designated by the subscript ‘o’.

4.1.1 Polarizability Ellipsoid and Raman Activity From examination of the variation of the polarizability ellipsoid (in size, shape and orientation), during the execution of the corresponding vibration, Raman activities of different normal modes can be understood. First consider the case of the simple molecule H2 O (which is an asymmetric top molecule). These are illustrated in Fig. 4.2a–c. For each mode, the central figures correspond to the equilibrium configuration of the molecule and the side figures correspond to two extreme positions of the respective vibrations. The approximate shapes of the polarizability ellipsoids are shown below the respective vibrations. Note that when a bond is stretched, the electrons forming this bond are less strongly held by the nuclei and hence the bond becomes more easily polarizable which results in the decrease of the length of the polarizability ellipsoid in this direction. Reverse is the case when the bond is compressed. Thus, when the molecule executes the symmetric stretching vibration (ν1 ), the polarizability ellipsoid decreases or increases in size when the two bonds are simultaneously stretched or compressed. However, the shape of the polarizability ellipsoid remains more or less unchanged. When the bending mode (ν2 ) is executed, the molecule in the two extreme positions of the vibration approaches towards linear (left side) and towards diatomic molecular (right side) configurations. Following the previous arguments, the shapes and sizes of the polarizability ellipsoids become as shown in the Fig. 4.2b. Finally for the asymmetric stretching vibration (ν3 ), the shape and size of the polarizability ellipsoid remain more or less unchanged, but the direction of the major axis changes drastically

4.1 Classical Explanation of Raman Scattering

O

77

O

O H

H

H

H (a)

H

Symmetric Stretching vibration (ν1)

O

O

H

H

H

O

H

H

H

H

(b) Scissoring (bending) vibration (ν2) O

O

H H

(c)

H

O H

H H

Asymmetric stretching vibration (ν3)

Fig. 4.2 Change of the shape, size and orientation of the polarizability ellipsoid of water molecule during the execution of its three vibrational modes. The central figures correspond to the equilibrium configuration of the molecule and the side ones correspond to the two extreme positions of the respective vibrations

as shown Fig. 4.2c. Thus in all three cases, the polarizability ellipsoid changes at least in one aspect and hence all the three vibrations are Raman active. Now consider the linear symmetric triatomic molecule CO2 . This molecule possesses three normal vibrations, just like H2 O molecule. Only difference is that, since the molecule CO2 is linear, the bending vibrations are doubly degenerate, because with respect to the internuclear axis, two mutually perpendicular directions, both perpendicular to the internuclear axis, are identical. However, the shape and size of the polarizability ellipsoids are as shown in Fig. 4.3 which can be easily understood from the arguments, similar to those made in the case of H2 O molecule, above. Thus one may expect that all the three modes are also Raman active in this case also. However, the fact is that it is not so; the symmetric stretching mode is

78

4 Raman Spectroscopy

only Raman active and the other two are Raman inactive. In order to understand the reason, we shall explore the case more critically. Let us now study the change in polarizability with the change of the displacement coordinate (ξ ) which is extension (positive) or compression (negative) of the bond under consideration for the stretching vibration and displacement of the bond angle from its equilibrium value for the bending mode. For the bending vibration, (ξ ) is positive or negative according to the two different orientations with respect to the linear axis of the molecule in the bent form. Since polarizability ellipsoid measures the reciprocal of α, so α increases when the bond is stretched and decreases when it is contracted. Thus, the Fig. 4.4 is sketched. The details of the curve are not necessary since we are concerned only with the variation near the origin ξ = 0. For the symmetric stretching mode (ν1 ), the gradient at the origin, (∂α/∂ξ )0 /= 0, as can be easily understood from Fig. 4.3, so this vibration is Raman active. But for the asymmetric stretching vibration (ν3 ), the polarizability decreases for both positive and negative values of (ξ ) with respect to its value at the origin (see Fig. 4.3). So the polarizability gradient (∂α/∂ξ )0 = 0 and hence the vibration becomes Raman inactive. For the angle bending mode (ν2 ), it can

C

O

C

C

O

C

O

C

C

(a) Symmetric Stretching vibration (v1) C

C

C

O

C

O

O C

C

(b) Angle bending vibration (ν2) C

O

C

C

O

C

C

O

C

(c) Asymmetric stretching vibration (ν3)

Fig. 4.3 Change in the shape, size and orientation of the polarizability ellipsoid of the carbon dioxide molecule during the execution of its three vibrational modes. The central figures correspond to the equilibrium configuration of the molecule and the side ones correspond to the two extreme positions of the respective vibrations

4.2 Quantum Theoretical Explanation

α

79

α

α

ξ Fig. 4.4 Variation of the polarizability (α) with the displacement coordinate (ξ ) for the three vibrations of carbon dioxide molecule. [For the three vibrations, see Fig. 4.3.]

be understood, in a similar way, that the polarizability increases on both sides of the origin (ξ = 0) in the same way and this gives rise to a minimum of the polarizability curve at the origin. Thus, this mode also becomes Raman inactive. In the same way, we can establish the Raman activities of the three modes of vibration of water molecule. For them the polarizability variation can be represented by curves similar to Fig. 4.4. For each of them, the polarizability gradient (∂α/∂ξ )0 is nonzero near the origin and hence each vibration is Raman active. Since the intensity of a Raman band depends on the square of the gradient of the polarizability at the origin, so in order to know the relative intensities of these three modes, we need to know the relative values of these three polarizability gradients. It is found that this slope is large for the symmetric stretching vibration and small for the other two modes. So although all the three modes are Raman active, symmetric stretching vibration appears very strongly in the Raman spectra but the intensities of the other two modes are so small that they are not generally observed. Even for the overtone or combination vibrations of small molecules or the fundamental vibrations of more and more complex molecules, determination of the Raman activities of different vibrations of the respective molecules is not possible if we follow this procedure. We can see later on that quantum mechanics and group theory can be applied more elegantly to determine the Raman and infrared activities of different vibrations of molecules possessing certain symmetry properties.

4.2 Quantum Theoretical Explanation When an incident radiation interacts with a molecule, the electric dipole moment → of the incident radiation is, induced by the electric field (E)

80

4 Raman Spectroscopy

→ μ → = α·E

(4.1)

where ‘α’ is the polarizability tensor of rank two ⎛

⎞ αxx αxy αxz ( ) α = ⎝ αyx αyy αyz ⎠ = αij ; i, j = x, y, z. αzx αzy αzz

(4.7)

If the interaction brings a change of the molecular system from an eigen state |m( to another eigen state |n(, then the intensity of the scattered radiation of frequency ν is given by (following the quantum mechanical analogue of electric dipole radiation), Is =

1 4πεo

|(μ → mn (|2 (2π ν)4 3C 3

(4.8)

/ \ → where μ the induced transition dipole moment (oscillating with a frequency ν) mn

between the initial and final molecular eigen states |m( and |n( respectively is given by, → (μ → mn ( = (m|μ|n( → = (m|α E|n(

(4.9)

The initial and final molecular states are given by, |m( = e

−2 π iEm t h |Um (→r )(

(4.10)

−2 π iEn t h |Un (→r )(

(4.11)

|n( = e

Let the electric field associated with the incident radiation is given by, [ ] → =E → o cos(2π νo t) = 1 E → ei2πνo t + e−i2πνo t E 2 o

(4.12)

νo being the frequency of the incident radiation. (→ ) μ mn becomes, Thus, the induced transition dipole moment − μ → mn

) ) ] ( ( [ E −E m → o · i2π νo − En −E t −i2π νo + n m t E h h (Um (→r )|α|Un (→r )( = +e e 2

(4.13)

Equation (4.13) indicates that the scattered radiation will contain frequencies (ν o − ν mn ) and (ν o + ν mn ), where [ν mn = (E n − E m )/h]. Note, as will be shown below, that the second term of the right-hand side of Eq. (4.13), yields some unrealistic results, and henceforth will not be considered. This equation can be physically interpreted from the photonic views of radiation. Consider three specific cases.

4.2 Quantum Theoretical Explanation

81

Case I When the incident radiation is scattered elastically leaving the molecule in the same state as the initial one (i.e. |n( = |m(), a radiation of frequency (ν o ) is observed. This corresponds to Rayleigh scattering. This can be explained as follows. By absorbing the incident photon (of energy hν o ), the molecule gets excited to a virtual state |s( (whose energy is hν o above that of the initial state |m(), wherefrom it is de-excited again to the initial state by emitting a radiation of frequency ν o . This is shown in Fig. 4.5a. Here |n( = |m( and so En = Em . Case II Unlike case I, if the molecule scatters the incident radiation inelastically and is excited from a lower energy initial state |m( to a higher energy final state m |n((En (Em ), a radiation of frequency νStokes = νo − En −E is found. This is shown h in Fig. 4.5b. By absorbing the incident photon, the molecule, being initially in the

s

(a)

h

o

n h

o

m

(b)

s h

Stokes

=h

o

(En Em)

n h

o

m

(c)

s h h

o

anti-Stokes

=h

o

+ (Em En)

m n

Fig. 4.5 a Rayleigh’s scattering b Stokes Raman scattering c Antistokes Raman scattering

82

4 Raman Spectroscopy

lower energy state |m(, is excited to a virtual state |s(, and there from returns to the another molecular state |n((En (Em ) by the emission of radiation frequency νStokes = m . This radiation is called stokes Raman radiation. This picture contradicts νo − En −E h the process arising from the second term of the right-hand side of Eq. (4.13) and so that term is not considered. Case III Here also the molecule scatters inelastically the incident radiation as shown in Fig. 4.5c. The molecule was initially| in) the state |m(. By absorbing the incident radiation, it is excited to the virtual state |s' wherefrom it comes to the state |n((En (Em ) by the emission of radiation, known as antistokes Raman radiation. The frequency of m . This is in compliance with the first term on this radiation is νanti Stokes = νo − En −E h right-hand side of Eq. (4.13), as before, but in contradiction with that arises from the second term of the same equation. So again that term is not taken into consideration.

4.3 Selection Rules of Rotational Raman Spectra The integral (Um (→r )|α|Un (→r )( = (m|α |n( in equation (4.13) actually determines the selection rule for Raman spectra which in turn determines the relevant spectral characteristics. First, we shall derive the selection rules for rotational Raman spectroscopy.

4.3.1 Rotational Raman Spectra of Diatomic Molecules Here |m( and |n( are the initial and final rotational levels of the molecule, considered as a rigid rotator and they are given by |m( = NJm Mm PJm Mm (cos θ ) eiMm φ = YJm Mm and

|n( = NJn Mn PJn Mn (cos θ ) eiMn φ = YJMn

(4.14a) (4.14b)

where Ns’ and Ps, are normalization constants and associated Legendre’s polynomials and Ys’ are normalized spherical harmonics, as discussed in Chap. 2 (see Eq. 2.10). Now consider the molecule with principal axes (X, Y, Z), i.e. the axes fixed in the molecule are rotating in the laboratory frame (x, y, z), i.e. in space. The molecule is irradiated with a plane polarized light for which the electric fields are E x = 0 = E y , E z = E. Then, the z-component of the dipole moment induced in the molecule is μz = αzz Ez = μX cos(X , z) + μY cos(Y , z) + μz cos(Z, z)

4.3 Selection Rules of Rotational Raman Spectra

= αXX E X cos(X , z) + αYY EY cos(Y , z) + αZZ EZ cos(Z, z) [ ] sin ce μL = αLL EL for L = X , Y and Z(the principal axes) [ ] = αXX cos2 (X , z) + αYY cos2 (Y , z) + αZZ cos2 (Z, z) Ez

83

(4.15)

Let the axis of the diatomic molecule be along z-direction, so α XX = α YY . Thus αzz = αXX cos2 (X , z) + αYY cos2 (Y , z) + αZZ cos2 (Z, z) = αXX + (αZZ − αXX ) cos2 (Z, z) = αXX + (αZZ − αXX ) cos2 θ Since cos2 (X , z) + cos2 (Y , z) + cos2 (Z, z) = 1

(4.16) (4.16a)

Here θ is the polar angle, i.e. the angle, the linear axis (Z) of the diatomic molecule makes with the space fixed axis (z). Hence for this component (zz) of the polarizability tensor (α zz ), the integral or matrix element (m|αzz |n( between the rotational states |m( and |n( becomes non-vanishing, only when ⎫ ΔJ = Jn − Jm = 0, ±2 ⎬ and ⎭ ΔM = Mn − Mm = 0

(4.17)

(following the orthogonality condition and the recursion relations of the spherical harmonics). In a similar way, for the other components of the polarizability tensor (say, α xy , α xz and α yz , etc.), we get the same selection rule for J same, but the selection rules for ΔM becomes 0, ± 1. Thus, the ultimate selection rules for rotational Raman spectra are ⎫ ΔJ = Jn − Jm = 0, ±2 ⎬ (4.17a) and ⎭ ΔM = Mn − Mm = 0, ±1 Since rotational term value for the diatomic molecule is τ J = BJ (J + 1), the selection rule concerned with the quantum number M is not considered in determining the structure of the rotational Raman spectra. Ignoring the selection rule ΔJ = 0, which gives rise only to the Rayleigh’s radiation, we see that the wavenumbers of different Raman lines, arising from the selection rules ΔJ = ± 2, are: ν = ν stokes = ν o − (τJ +2 − τJ ) = ν o − 4B(J + 3/2), for ΔJ = +2(S - branch)and J = 0, 1, 2, 3, 4, . . .

(4.18a)

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4 Raman Spectroscopy

and ν = ν antistokes = ν o + (τJ − τJ −2 ) = ν o + 4B(J − 1/2), for ΔJ = −2(O - branch)and J = 2, 3, 4, ...

(4.18b)

Here, ν o is the wavenumber of the exciting radiation. Thus on both sides of the Rayleigh’s line (of wavenumber ν o ), a series of equidistant lines are observed with separation of successive lines being 4B (Fig. 4.6). The lines on the lower wavenumber side of the Rayleigh’s line are different members of stokes line. These lines are arising from the selection rules ΔJ = + 2 and are called S-branch. Similarly, the group of antistokes lines is called O-branch and it arises from the selection rule ΔJ = −2. The wavenumbers of the first members of the S- and O-branches are ν o − 6B and ν o + 6B which correspond to the first members of the stokes and antistokes lines. Thus, the wavenumber separation between these two lines is 12B. Thus Wave number separation between the first members of the antistokes and stokes lines wave number separation between the successive lines in the S- or O-branch wns(anti-stokes − stokes)1st = wnssl 3 (4.19) = 1 ( ) Since the transition polarizability matrix (m|α|n( = J ' M ' |α|J '' M '' depends very weakly on the J (= J m ) value, so the intensity of each Raman line is proportional to the population of the initial level (J = J m ) NJ = N0 (2J + 1)e−

chBJ (J +1) kT

(4.20)

Thus, the intensities of different members of the stokes and antistokes Raman lines vary accordingly and they are shown in the Fig. 4.6. Relative intensities of different Raman lines are roughly indicated by their heights and the strength of blackening of the respective lines in Fig. 4.6. The rotational quantum number (J = J max ) for which the population maximum is J = Jmax = (kT /2Bch)1/2 − 1/2

(4.21)

Thus, the wavenumber separation of the maximum intense lines in the stokes and antistokes regions is 1/2 Δν max st−anti = 8B(Jmax + 1/2) = 8B(kT /2Bch)

(4.22)

4.3 Selection Rules of Rotational Raman Spectra

85

Stokes Transitions in Molecules

Antistokes Transitions in Molecules J = J" = 7

6

5

4 3 2 1 0 (a) 4B

4B 6B

J=J"= 5 4 3 2 1 0 Stokes lines (O – branch) (b)

6B o

0 1 2 3 4 5 Antistokes lines (S – branch)

Fig. 4.6 a Rotational Raman transitions and b expected rotational Raman spectra in diatomic molecules, the height and the strength of blackening denoting the relative intensities (see the text). The J-values correspond to those of the lower levels (J '' )

86

4 Raman Spectroscopy

4.3.2 Rotational Raman Spectra of Polyatomic Molecules (Case a). Linear molecules Here, the energy levels, selection rules and hence the spectral structure are same as those for the diatomic molecules. Unlike diatomic molecules, only the rotational constant (B), in the present case, is expressed by a much complex relation involving various bond lengths of the molecules. (Case b). Symmetric Top Molecules For such molecules, the energy levels are given by TJ = BJ (J + 1) + (A − B)K 2 , with J = 1, 2, 3, 4, . . . and K = 0, ±1, ±2, ±3, ±4, . . . ± J

(4.23)

The selection rules are ΔJ = 0, ±1, ±2, ΔK = 0; but for Km = Kn = 0, ΔJ = ±2.

(4.24)

Here also ΔJ = 0 gives rise to Rayleigh’s line. The wavenumbers of the stokes Raman lines are ν = ν stokes = ν o − (Tn − Tm ) = ν o − 2B(J + 1), J = 1, 2, 3, . . . ; ΔJ = +1(R - branch)

(4.25a)

ν = ν stokes = ν o − (Tn − Tm ) = ν o − 4B(J + 3/2), J = 0, 1, 2, 3, . . . ; ΔJ = +2(S - branch)

(4.25b)

Note that for J = 0 state, K = 0. So according to the selection rules (4.24), J = 0 is excluded in (4.25a) for the R-branch. Similarly, the wavenumbers of the antistokes Raman lines are ν = ν antistokes = ν o − (Tn − Tm ) = ν o + 2BJ , J = 1, 2, 3, . . . ; ΔJ = −1(P - branch)

(4.25c)

ν = ν antistokes = ν o − (Tn − Tm ) = ν o + 4B(J − 1/2), J = 2, 3, . . . ; ΔJ = −2(O - branch)

(4.25d)

The spectral characteristics are shown in Fig. 4.7. In the complete spectrum, every alternate R- (and P-) line is overlapped by a S- (and O-) line. This gives rise to the

4.4 Symmetry Properties of Wave Functions

Stokes

87

Antistokes

P/R Branch

O/S Branch

P /R +O/S Branch

ν0

ν0

Fig. 4.7 Rotational spectrum of symmetric top molecule

alteration of intensities of successive lines in the complete spectrum. More intensities of the overlapped lines are shown in the Fig. 4.7 by more blackening. (Case c). Spherical Top and Asymmetric Top Molecules For the spherical top molecules (methane (CH4) , carbon tetrachloride (CCl4 ), silane (SiH4 ), etc.), the polarizability ellipsoids are spherical and so rotation cannot produce any effect on these ellipsoids. Thus, pure Raman spectra is completely inactive in these molecules. However, information about rotational motion of such molecules can be obtained from the rotational fine structure of the respective infrared and vibrational Raman spectra. For asymmetric top molecules, all rotations are Raman active, however, the Raman spectra are very complicated. They are analysed and interpreted by considering the molecules as intermediate between the prolate and oblate symmetric top.

4.4 Symmetry Properties of Wave Functions According to Born-Oppenheimer approximation, the wave function of a molecule is given by (without considering the spin of the nuclei, see Eq. (1.13)) ψ(r, R) = ψmol (r, R) = ψelec (r, R) · ψvib (R) · ψrot (θ, ϕ)

(4.26)

where r and R refer to the coordinates of the electrons and the nuclei, respectively, and θ and ϕ are the polar and azimuthal angles designating the orientation of the

88

4 Raman Spectroscopy

molecule in space. If a molecule possesses a centre of inversion, then the inversion operation at this centre transforms a right handed coordinate system to a left handed one. The total Hamiltonian of the molecule remains unchanged under this operation and so the total wave function (4.26) must either remain unchanged or change sign only. This can also be shown as follows: In a coordinate system, consider an operator O operates on a function ψ. Let R be another operator which acts on the coordinate system and generates a new one in which the former operator and the function become O' and ψ ' , respectively. Then O' ψ ' = O' Rψ = R(Oψ) This gives R−1 O' R = O or O' = ROR−1 Here, R−1 is the inverse of the operation R, so RR−1 = R−1 R = E (identity operation). Thus, we see that the operator in the transformed coordinate system (O' ) is related to that in the initial coordinate system (O) by a similarity transformation. In the present case, let R be the inversion operation I and O be the Hamiltonian (H) of the molecule such that H = H ' . Therefore IH ψ=I Eψ i.e., (I H I −1 )(Iψ) = E Iψ (since I −1 I = E, identity operation) i.e., H ' Iψ = E Iψ = H Iψ. Thus, we see that both ψ and Iψ are the eigen functions of H. So, if we choose Iψ = ψ ' , we can say both ψ ' and Iψ ' are eigen functions of H. Since Iψ ' = I 2 ψ, so we can say that all the three functions ψ, Iψ and I 2 ψ have the same eigen value of the Hamiltonian, H. This is possible only when the eigen value of I is either + 1 or − 1. We shall now consider the effect of inversion on the wave function in Eq. (4.26) of a diatomic (or linear) molecule. This operation is equivalent to a rotation by 180° about an axis (say, y-), perpendicular to the linear axis (say, z-), of the molecule followed by a reflection in the xz-plane. The first operation cannot change the electronic wave function since it leaves the relative positions of the electrons and the nuclei unchanged. The second operation reflects the electronic wave function on the above (xz) plane designated by σ v . This effect depends on the nature on the electronic wave function. If it (electronic wave function) remains unchanged, then the wave function is designated by a superscript(‘+’) and if it changes sign only, then it is designated by the superscript(‘−’). Therefore + + − σV ψelec = ψelec and σV ψelec− = −ψelec

(4.27)

The vibrational wave function ψ vib (R) of a diatomic molecule does not change under the inversion operation. However for a polyatomic molecule, the matter is not

4.4 Symmetry Properties of Wave Functions

89

so simple. For a symmetric vibration of such a molecule, the inversion operation does not change the sign of the vibrational wave function but for an antisymmetric vibration, the inversion operation changes the sign of the wave function (of the first excited vibrational state).The rotational wave function of a diatomic molecule is a function of the angular coordinates only. Under rigid rotator approximation ψrot (θ, ϕ) = PJM (cos θ )eiM ϕ ( 2 )M /2 )J d J +M 1 ( 1 − cos2 θ cos θ − 1 eiM ϕ = J J +M 2 (J !) d (cos θ )

(4.28)

Inversion operation sends θ and ϕ to π − θ and π + ϕ, respectively. So I ψrd (θ, ϕ) = (−1)J ψrot (θ, ϕ) = ±ψrot (θ, ϕ)

(4.29)

which means inversion operation changes only the signs of the wave functions of the odd J-levels and leaves the even J-levels unchanged. A rotational level is said to be positive if the molecular wave function remain unchanged under inversion operation and if it changes the sign only then the rotational level is said to be negative. Thus for a ψ elec + (r, R) state, all even J-levels are positive and all odd J-levels are negative. Similarly for a ψ elec − (r, R) state, the reverse is the case, i.e. all even J-levels are negative and all odd J-levels are positive. The selection rules are ⎧ ⎫ ⎪ ⎨ Positive ← = → Negative ⎪ ⎬ Positive ← /= → Positive ⎪ ⎪ ⎩ ⎭ Negative ← /= → Negative

(4.30)

These rules can be easily derived. These are consistent with the selection rules ΔJ = ± 1 (in the rotational spectra).

4.4.1 Effect of Nuclear Spins First we shall consider the effect of exchange of two identical nuclei (designated by the operator P) on the total wave function of the molecule (Eq. 4.26). This operation is equivalent to two operations, (a) An inversion of the total wave function of the molecule, ψ(r, R), at the centre of inversion. (b) An inversion of the electronic part of the total wave function ψ elec (r, R) at the centre of inversion. The effect of the first operation has been discussed above. If the second operation changes the sign of the electronic wave function only, then it is called an ‘Ungerade’

90

4 Raman Spectroscopy

(i.e. odd) state (u) and if the wave function remains unchanged under this inversion operation, then the state is called a ‘Gerade’ (i.e. even) state(g). Thus I ψelec(g) = ψelec(g) and I ψelec(u) = −ψelec(u)

(4.31)

If nuclear spins are included, then the total molecular wave function becomes ψtotal (r, R) = ψmol (r, R) · ψns = ψelec (r, R) · ψvib (R) · ψrot (θ, ϕ) · ψns

(4.32)

Here also the exchange of two identical nuclei does not change the total Hamiltonian of the system. So, as shown above, the total wave function ψ total (r, R) will either remain unchanged or change sign only under this operation. Thus, P is an eigen operator with eigen value ± 1 and eigen function ψ total (r, R). Thus Pψtotal (r, R) = I σv [ψelec (r, R)]ψvib (R) · I ψtot (θ, ϕ) · Pψns = I σv [ψelec (r, R)] · ψvib (R) · (−1)J ψrot (θ, ϕ) · Pψns

(4.33)

Thus, the spin of the nuclei that undergo exchange is the index number which will determine whether the sign of the total wave function [ψ total (r, R)] of the molecule will change sign or not under this operation. If the spin of two identical nuclei, which undergo exchange operation (P), is even, i.e. they are bosons, then this operation does not change the sign of the total wave function, ψ total (r, R). However, if the spin is odd, i.e. if the interchanged nuclei are fermions, then the effect of P on ψ total (r, R) is to change its sign. Since both proton and neutron have nuclear spin ½, they are fermions. Therefore, if a nucleus has an even number of nucleons (even mass number), then it can be considered as a boson, and if it contains an odd number of nucleons (odd mass number), then it can be considered as a fermion. Thus we see that along with the degeneracy (2 J + 1) of each rotational level, there is weight factor, called statistical weight, associated with each rotational level which will determine the relative intensities of the successive lines in rotational spectra. If I is the nuclear spin of the exchanged nuclei, then this ratio is (I + 1)/I. This will be understood from the following examples which too will clearly explain the effect of nuclear spin on the spectral features of molecules. Example 1 Hydrogen molecule (H2 ) Hydrogen nucleus (or proton) is a spin-1/2 particle (fermion). So the net effect of the operation of P on ψ total (r, R) is to change its sign only, i.e. to −ψ total (r, R). The ground state electronic wave function of hydrogen molecule is 1 ∑ g + (i.e. ψ elec (r, R) = ψ e(g) + ). So Pψtotal (r, R) = P[ψelec (r, R) · ψvib (R) · ψrot (θ, ϕ) · Pψns ] = P[ψe · ψv · ψr ] · Pψns ) ) + · ψv · (−1)J ψr · Pψns = I σv ψe(g)

4.4 Symmetry Properties of Wave Functions

91

= (−1)J ψtotal (r, R) · Pψns = −ψtotal (r, R)

(4.34)

The spin of each of the hydrogen nuclei can orient itself in such a way that its component along the internuclear axis is| either so)the corresponding | ) + 1/2 or −1/2; nuclear states are specified by |α( = | 21 21 and |β( = | 21 − 21 . The nuclear spin function of the molecule thus formed is ⎫ ⎪ α(1)α(2) ⎬ 1 β(1)β(2) = ψns(symmetric) and √ [α(1)β(2) − α(2)β(1)] ⎪ 2 √1 [α(1)β(2) + α(2)β(1)] ⎭ 2 = ψns(anti symmetric) The symmetric and antisymmetric subscripts correspond to no change or change of signs of the respective functions under the exchange operation (P) on the identical nuclei. Thus, we see from Eq. (4.34) that all even J-levels will combine with the antisymmetric nuclear spin function and all odd J-levels will combine with the symmetric nuclear spin function. Thus, the statistical weights of the respective states are one and three along with their respective (2J + 1) degeneracies (Fig. 4.8). The first type of molecules is called para hydrogen and the second type of molecules are called ortho hydrogen. Thus, the ratio of intensities of successive lines in the rotational Raman spectra is 1:3 which is compatible with the general formula I/(I + 1) for I = ½ of hydrogen nucleus.

J 5

St.Wt. 3

J 5

1 4

4

St.Wt. 1

0

J 5

St.Wt. 3

4

6

3

3

3

1

3

3

2 1 0

1 3 1

2 1 0

0 1 0

2 1 0

6 3 6

H2

O2

N2/D2

Fig. 4.8 Statistical weights (St.Wts.) of different rotational levels of the lowest vibrational states of the lowest electronic states (1 ∑ g + ) of H2 and N2 /D2 and (3 ∑ g ) of O2 molecules

92

4 Raman Spectroscopy

Example 2 D2 molecule Here also, like H2 molecule, the ground electronic state is 1 ∑ g + (i.e. ψ elec (r, R) = ψ e(g) + ). So Iσ v (ψ e(g) + ) = (ψ e(g) + ). Thus, all even Jlevels are positive and all odd J-levels are negative states. Unlike proton (hydrogen nucleus), the deuterium nucleus has spin one. Thus, the spin states of this nucleus are designated by |γ ( = |11(, |δ( = |10(and|λ( = |1 − 1(. Therefore, the nuclear spin functions of the molecule are ⎫ γ (1)γ (2), δ(1)δ(2), λ(1)λ(2), ⎪ ⎪ ⎪ ⎬ √1 [γ (1)δ(2) + γ (2)δ(1], 2 √1 [δ(1)λ(2) + δ(2)λ(1)], ⎪ ⎪ 2[ ⎪ ] ⎭ 1 √ λ(1)γ (2) + λ(2)γ (1) 2 [ ] ⎫ √1 γ (1)δ(2) − γ (2)δ(1) , ⎪ ⎬ 2 √1 [δ(1)λ(2) − δ(2)λ(1)], 2[ ]⎪ ⎭ √1 λ(1)γ (2) − λ(2)γ (1) 2

= ψns(symmetric) and

= ψns (anti symmetric)

Since deuterium nucleus is a boson, so Pψ total (r, R) = ψ total (r, R). So the positive rotational levels will combine with the symmetric nuclear spin function and the negative rotational levels will combined with the antisymmetric nuclear spin function. Therefore, all the even levels will have statistical weight six and all the odd rotational levels will have statistical weight three. These are shown in Fig. 4.8. Thus, the intensities of the successive lines in the rotational Raman spectra are in the ratio 2:1. Since nitrogen nucleus has spin one and nitrogen molecule has the ground electronic state 1 ∑ g + , the case is identical to D2 molecule. In fact in the early days, the measured intensity ratio of the successive lines in the rotational Raman spectra was found to be 2:1 from which the spin of the nitrogen nucleus was predicted to be one. Example 3 Oxygen molecule (O2 ) ) ) − − The ground electronic state of oxygen molecule is 3 ∑ g i.e.ψelec (r, R) = ψe(g) . So Iσ v (ψ e(g) − ) = (−ψ e(g) − ). Thus, all odd rotational levels are positive and all even rotational levels are negative. Since the nuclear spin of oxygen is zero, it possesses only one spin state |η( = |00(. So only possible nuclear spin function of oxygen molecule is η(1)η(2) which is symmetric with respect to the exchange (P) of the two oxygen nuclei. So all the odd rotational levels have statistical weight one and all the even rotational levels have zero statistical weight. Thus, all the even rotational levels will be missing. So no rotational Raman transition can originate from the even rotational levels. So the alternate lines in the rotational Raman spectra will be missing. This cannot be understood from the direct intensity measurement of the successive rotational lines due to the absence of intensity alteration of successive rotational lines. However, from the critical analysis of the spectra, the missing of alternate lines can be understood and this is shown below. Had there been no missing of lines, we could get [from Eq. (4.19) and Fig. 4.6]

4.5 Selection Rules and Characteristics of Vibrational Raman Spectra …

93

Wave number separation between the first members of the stokes and the antistokes lines wave number separation between the successive lines in the S- or O-branch 3 wns(st-antist)1st = (4.19) = wnssl 1

If the alternate lines (odd members, i.e. first, third, fifth ……., etc., members) are found to absent in the spectra, the above ratio becomes Wave number separation between the first members of the stokes and and antistokes lines wave number separation between the successive lines in the S- or O-branch 20 5 wns(st-antist)1st = = = (4.35a) wnssl 8 2

The observed experimental result is closer to 5/2 which establishes the prediction of zero nuclear spin of oxygen. Such prediction can also be made from the observation of the stokes spectra only. For no missing of lines (from Eq. 4.19, Fig. 4.6) 6B 3 Raman shift of first rotational line = = Wavenumber difference of successive lines 4B 2

(4.35b)

For missing of all odd members of the spectra, the above ratio becomes 10B 5 Raman shift of first rotational line = = Wavenumber difference of successive lines 8B 4

(4.35c)

Actual value of the above ratio is found to be 5:4 which predicts zero spin of the oxygen nucleus.

4.5 Selection Rules and Characteristics of Vibrational Raman Spectra of Diatomic Molecules For vibrational Raman spectra, both before and after irradiation with the exciting radiation, the molecule under consideration is in one of its vibrational states; for stokes scattering, the initial state |m( is a lower energy vibrational state, and for antistokes Raman scattering, |m( is the higher energy vibrational state. As before, the integral which determines the selection rule is (Um (→r )|α|Un (→r )( = (m|α |n(. For vibrational Raman spectroscopy, the states |m( and |n( are the vibrational states of the molecule. Since in a molecule, the atoms are not static, they are executing small oscillations about their respective equilibrium positions with amplitudes depending on the ambient temperature. Thus expanding the polarizability in a Taylor series about the equilibrium nuclear configuration of the molecule (designated by a subscript ‘0’), we can expand the transition polarizability matrix as

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4 Raman Spectroscopy

) ) ( ∂α 1 ∂ 2α Q+ Q2 + . . . |n( ∂Q o 2 ∂Q2 o ) ) ( ( ∂α 1 ∂ 2α (m|Q|n( + (m|Q2 |n( = (α)o (m | n( + ∂Q o 2 ∂Q2 o (

(m|α|n( = (m|(α)o +

(4.36)

Here, |m( and |n( are the vibrational states associated with the vibrational quantum numbers V m and V n of the molecule. Assuming the states to be of simple harmonic type and following the recursion relation 3.11 described in Sect. 3.1.1 and the reasoning given therein, we get the selection rules ΔV = Vn − Vm = 0, ±1, ±2, ±3

(4.37)

ΔV = 0 corresponds to the Rayleigh line. + and − signs correspond to the stokes and antistokes lines. |ΔV | = 1 corresponds to the fundamental mode, |ΔV | = 2 to the first overtone or second harmonic, |ΔV | = 3 to the second overtone or third harmonic, etc. If the wavenumber of the exciting radiation is ν o , following the above selection rules, the wavenumber of the Vth harmonic, under anharmonic oscillator model, becomes (see Eq. 3.51) ν V th harmonic = ν o ± (τV − τo ) } { = ν o ± [ ν e (V + 1/2) − ν e xe (V + 1/2)2 − {ν e /2 − ν e xe /4}] = ν o ± [V ν e − V (V + 1)ν e xe ] (4.38) where ν e is the wavenumber of the vibration for equilibrium configuration and ν e xe is the anharmonicity. Thus ⎫ ν fundamenta(stokes) = ν o − [ν e − 2ν e xe ] ⎪ ⎪ ⎬ ν 2nd harmonics(stokes) = ν o − [2ν e − 6ν e xe ] ν 3rd harmonics(stokes) = ν o − [3ν e − 12ν e xe ] ⎪ ⎪ ⎭ ν 4th harmonics(stokes) = ν o − [4ν e − 20ν e xe ]

(4.39a)

⎫ ν fundamenta(anti stokes) = ν o + [ν e − 2ν e xe ] ⎪ ⎪ ⎬ ν 2nd harmonics(anti stokes) = ν o + [2ν e − 6ν e xe ] ν 3rd harmonics(anti stokes) = ν o + [3ν e − 12ν e xe ] ⎪ ⎪ ⎭ ν 4th harmonics(anti stokes) = ν o + [4ν e − 20ν e xe ]

(4.39b)

and

4.5 Selection Rules and Characteristics of Vibrational Raman Spectra …

95

4.5.1 Rotational Vibrational Raman Spectra Each vibrational Raman band possesses a fine structure which arises due to Raman transitions from different rotational levels of one vibrational state to those of another vibrational state, all being in the ground electronic state of the molecule. Assuming anharmonic oscillator and rigid rotator model, the term values of the rotational vibrational levels are given by (see Eq. 3.48a), T = TV J = EV J /ch = v e (V + 1/2) − v e xe (V + 1/2)2 + BV J (J + 1)

(4.40)

Using the selection rules for rotational Raman spectra, ΔJ = 0(Q-branch), ΔJ = −2(O-branch) and ΔJ = + 2(S-branch), the fine structure of the Vth mode becomes, ν V th(Q - branch) = ν o ± (TV J =0 − T0J =0 ) = ν o ± [V ν e − V (V + 1)ν e xe ]

(4.40a)

v Vih(0 - branch) = v o ± (TV J '' −2 − T0J '' ) = v o ± [{V v e − V (V + 1)v e xe } ( )( ) )}] { ( + BV J '' − 2 J '' − 1 − Bo J '' J '' + 1 = v o ± [{V v e − V (V + 1)v e xe } }] { + 2Bv − (3BV + Bo ) + (BV − Bo )J ''2 , J '' = 2, 3, 4, . . . . = v o ± [{V v e − V (V + 1)v e xe } − 4B(J + 3/2)] ( ) assuming BV ≈ Bo = B, J = J '' − 2 = 0, 1, 2, 3, . . . (4.40b) ν V th(S - branch) = ν o ± (TV J +2 − T0J ) = ν o ± [{V ν e − V (V + 1)ν e xe } +{BV (J + 2)(J + 3) − Bo J (J + 1)}] = ν o ± [{V ν e − V (V + 1)ν e xe } }] { + 6BV + (5BV − Bo )J + (BV − Bo )J 2 = ν o ± [{V ν e − V (V + 1)ν e xe } + 4B(J + 3/2)], (assuming BV ≈ Bo = B, J = 0, 1, 2, 3, . . .)

(4.40c)

Here also, ‘ − ’ sign corresponds to stokes and ‘ + ’ sign corresponds toantistokes Raman lines. Intensities of different rotational lines are proportional to the population of the initial level (V,J) 2 NoJ = N0 (2J + 1)e−ch[{ve (V+1/2)−vex (V+1/2) }+BJ(J+1)]/kT

(4.41)

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4 Raman Spectroscopy

Thus, stokes Raman intensities will be more than the intensities of the corresponding vibrations in the antistoke region. The fundamental vibration will be of maximum intensity and the intensities of the higher and higher harmonics decrease with the increase of the order of the harmonic. Again around each antistoke vibrational band, S-branch will be situated on the higher energy side of the corresponding Q-branch and O-branch will be situated on the lower energy side of that Q-branch. Reverse will be the case for the stoke Raman vibrational band. Around each stoke Raman vibration, S-branch will appear on the lower energy side of the corresponding Q-branch and O-branch will appear on the higher energy side of the Q-branch.

4.6 Raman Intensities In the Raman process, an incident photon is destroyed, a scattered photon is created and the molecule undergoes a transition from an initial state to a final state. The process is coherent and therefore these three events are not independent and resolvable, i.e. we cannot view this scattering process as a sequential absorption and then emission of a photon but instead look it as one in which all three events occur more or less simultaneously. Irrespective of the nature of the excitation, that is whether the excitation is in the resonance or in the off-resonance region with an electronic band of the molecule, the same theoretical treatment is supposed to be applicable in all the cases. Scattering intensity I s of a Raman line of frequencyνs , for a molecular transition from the initial state |G( to a final state |F( is given by, Is = Kvs4 I0

∑ |( ) | | αρσ |2 ρσ

IF

(4.42)

where ( ) K is a constant, I0 is the intensity of the incident radiation (of frequencyν0 ), αρσ GF is the polarizability (scattering) tensor for the molecular transition from the initial state |G( to the final state |F( with the incident and scattered polarizations being indicated by ρ and σ (ρ, σ = x, y, z), respectively. Here, averaging over all orientations has been considered. For molecules with ( no ) absorption band in the visible or near ultraviolet region, the scattering tensor αρσ GF is nearly a constant and so Raman intensities in these molecules depend more or less on the fourth power of the scattered frequencies (ν s 4 ), just like Raleigh scattering.( But) for molecules having absorption bands in the visible or near ultraviolet region, αρσ GF no longer remains constant but depends on various properties of the excited electronic states. Since the scattering intensity is proportional to the square of the scattering tensor, so the study of intensity pattern of different Raman bands and their variation with the change of excitation frequency (called Raman excitation profiles, i.e. REPs) may unravel many intricate characteristics of the molecule in various excited electronic states. With the help of time-dependent perturbation theory [Kramers-Heisenberg-Dirac (KHD)], it can be shown that (ρ, σ )-th component of the polarizability matrix

4.7 Surface Enhanced Raman Scattering (SERS)

97

between the initial and final states is, (

αρσ

) GF

( | | )] [ (F|Pσ |I ( I |Pρ |G 1 ∑ (F|Pρ |I ((I |Pσ |G( = + h I νIG − ν0 − i[I νIF + ν0 − i[I

(4.43)

Here, |I ( is an intermediate state of the molecule. [I is the damping term reflecting the homogeneous width of the state |I (. For simplicity, we have not included the effect of damping of the initial and final states |G( and |F(. The states |G( and |F( are the rotational vibrational states, both of the ground electronic states of the molecule. Pρ /σ is the ρ/σ the component of the electric dipole moment operator. The first term in the square bracket on the right-hand side of Eq. (4.43) is called the resonant term and the second one the non-resonant term. For exciting frequency lying much below the frequency of the lowest electronic band, contributions of both the terms are effective. Such kind of scattering is called normal Raman scattering (NRS). But for resonant excitation with a particular electronic band, only the contribution of the resonant term from that particular electronic state is mostly effective. This time the scattering is called resonance Raman scattering (RRS). In the case of resonance excitation, the intensities of the Raman bands increase to several orders of the normal Raman intensities. In this case, long progression of vibrations may be observed. In the case of resonance Raman scattering, sometimes the electronic states lying in the close neighbourhood of the resonance state may also contribute to the scattering process. When the excitation lies in the region of discrete vibrational states of the resonating electronic state, the scattering is called discrete resonance Raman scattering. For such scattering process, the intensity pattern of the Raman spectra changes very haphazardly with the change of excitation frequency. But when the excitation lies in the region of continuums of the vibrational states of the resonating electronic state, the scattering is called continuous resonance Raman scattering. The intensities of the overtones change in this case in a more systematic manner rather than in a haphazard manner as in the case of discrete resonance Raman scattering.

4.7 Surface Enhanced Raman Scattering (SERS) In general, intensities of Raman bands are weakin normal Raman scattering. But in the seventies of the last century, some investigators found that when molecules are adsorbed on electrochemically roughened coinage metal surface, the intensity of Raman bands increase by several orders. In fact in 1974, Fleishman and others found that Raman signal from pyridine molecules increases by several orders when adsorbed on electrochemically roughened silver metal. In fact the average enhancement observed was of about 106 times. Since then this process is named as surface enhanced Raman scattering (SERS). Later, it was found that not only by adsorption of molecules on roughened surface but in aqueous solutions of coinage metal colloids, similar enhancement occurred. So it was understood that this effect is not

98

4 Raman Spectroscopy

only a surface effect but a nanostructure effect. In this section, we shall explain the basic principles which are responsible for this enhancement. There are two main factors, as seen from Eq. (4.42), which control the intensities of the Raman bands. One is the field intensity (E) on the square of which depends the intensity of the incident radiation (I0 ) and the other is the polarizability (α). The enhancement occurring due to the increase of the electric field strength near the metal surface is called electromagnetic enhancement and the other, a short range effect appears due to the increase of the polarizability (α) is called chemical effect. The two effects will be explained one by one as follows. Electromagnetic enhancement Free electrons in metals can be considered as electron gas or electron plasma. Each of these electrons oscillates with respect to one of the ions considered to be fixed and constitute a plasma oscillation. The quasiparticles appearing from the quantization of these plasma oscillations are called plasmons. The roughened metal surface may be considered to have a nanostructure. So in order to explain the enhancement effect both in the case of roughened surface and in aqueous solution of metal colloid, let us assume that the molecule is adsorbed on the surface of a nanometal sphere of radius (a) which is much smaller in size than the wavelength λ (a/λ < 1) of the exciting radiation such that the electric field near the metal surface can be considered to be uniform. The electric field associated with the exciting radiation is able to sustain the dipole oscillation of the localized surface plasmons (LSP) of the metal sphere under the condition of resonant excitation to be discussed below. At resonance, the electric field intensity is enhanced. Let the enhancement factor be denoted by g. So if E 0 be the incident electric field intensity, the molecule adsorbed on the metal surface will experience an enhanced field gE 0 . This enhanced field will generate an oscillation of the stokes/antistokes Raman field of the molecule which will be similarly enhanced in the metal sphere, this time by a factor g' , since the frequency of the incident radiation undergoes a Raman shift. So the net field experienced by the metal sphere is gg' E 0 and the oscillating dipole in the metal at the Raman shifted frequency will radiate whose intensity is proportional to (gg' )2 E 0 2 . So the intensity of the stokes/antistokes Raman frequency will be increases by a factor of (gg' )2 . In silver, g ~ g' ~ 30 for excitation around 400 nm. So the enhancement factor is ~ 106 . Now let us see how the resonance condition is fulfilled. We know from the electromagnetic theory that the polarizability of the small metal sphere of radius (a) of dielectric constant εin embedded in a medium of dielectric constant εout is α = a3

εin − εout = ga3 (say) εin + 2εout

(4.44)

→0 → 0 of the incident radiation will induce a dipole moment μ The electric field E → = αE in the metal sphere. The dielectric constant of the metal is a function of the wavelength of the incident radiation i.e. εin = εin (λ). This polarizability becomes very large when the denominator of the Eq. (4.44) is very small (nearly zero), i.e. when the real part of εin (λ) is equal to −2, since the dielectric constant of the surrounding medium is

4.7 Surface Enhanced Raman Scattering (SERS)

99

very nearly equal to one (εout ~ 1). This condition is called the resonant excitation of the surface plasmon. If the adsorbed molecule is at a distance →r from the centre of the → mol ) experienced by the adsorbed molecule will be the metal sphere, the total field (E → 0 plus the field produced by the induced dipole moment sum of the incident field E μ → of the metal sphere at the position of the molecule. Thus, it can be shown that the electric field at the molecule is ] [ → · →r )→r μ → (μ → → − 3 Emol = E0 + 3 r5 r ) ) ⎡ ⎤ → 0 · →r →r E →0 E → 0 + α ⎣3 − 3⎦ (4.45) =E r5 r → 0 = E0 zˆ , then this Let the incident field be polarized in the z-direction, i.e. E equation becomes ) ) → → mol = 1 − α E → 0 + 3α z E0 − r E 3 r r5

(4.46)

So in magnitude, this field is inversely proportional to the cube of the distance (r) i.e. Emol ∞ r13 . On the surface of the sphere r = a, the square of the absolute value of this field becomes | | )] [ ( | → |2 → ∗ → mol = |1 − g|2 + 3 cos2 θ 2greal + |g|2 E02 (4.47) |Emol | = Emol · E θ being the angular orientation of the position vector of the adsorbed molecule (→r ) | | | → |2 with respect to the incident field direction zˆ . Thus, we see that |E mol | is proportional to cos2 θ. For large value of |g|, E mol 2 = E 0 2 |g|2 (1 + 3 cos2 θ).This means that the field E mol is maximum when the molecule lies on the direction of the incident field, and 2 ∼ 4g 2 E02 . The ratio between the maximum to minimum therefore for large g, Emol intensity on the metal surface is 4:1 and the radially averaged intensity is E mol 2 = 2 E 0 2 |g|2 . This field will create a Raman field at the Raman shifted frequency in the molecule, and this will be further enhanced in the metal sphere in a similar manner, only then g will be changed to g' to correspond to the shifted frequency. Thus, the maximum enhancement factor is EF = (assuming εout = 1).

2 2 ' · Emetd −1 Emol εin − 1 εin ' · = 4gg = 4 ' 4 εin + 2 εin + 2 E0

(4.48)

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4 Raman Spectroscopy

In order to get a better insight into the resonance condition, we shall discuss about the dispersion relation of the metal sphere. First we shall apply Drude-Sommerfeld model to free electrons. The incident → 0 ) will displace the electron by (→r ) from the relevant ion which gives electric field (E → = e→r associated with the particular pair of electron and rise to a dipole moment μ ion. The cumulative effect of these dipole moments in unit volume will result in the n ∑ → = nμ macroscopic polarization, given by P → = e →ri . → r , t), Thus, the electric displacement D(→

i=1

→ = ε0 εE → → r , t) = ε0 E →o + P D(→ i.e. ε = 1 +

→ P → ε0 E

(4.49)

Here, ε = εin = εin (λ) is the dielectric constant of the metal. So in order to determine the dielectric constant, the equation of motion of the electrons has to be solved. The equation of motion of the free electrons in the conduction band can be expressed under Drude-Sommerfled model as m

d 2 →r d→r → 0 e−iωt = eE + mγ 2 dt dt

(4.50)

→ 0 and ω are the electric Here, e and m are the charge and mass of the electrons, E field and the angular frequency of the incident radiation. Here, the damping term is assumed to be proportional to the velocity and the damping constant γ ~ vf /l, where vf is the Fermi velocity and l is the mean free path of the electrons. Let the steady state solution of this equation be →r = →r0 e−iωt . Substituting this in (4.50), we get →r0 = −

e/m → E0 ω2 + iωγ

(4.51)

which gives 2 → = ne→r = ne→r0 e−iωt = − ne /m E → 0 e−iωt P 2 ω + iωγ

(4.51a)

Substituting this equation in (4.49), we get εDruck = 1 − Here, ωP = electrons).

√

ωp2 γ ωp2 ne2 /mε0 ) ( = 1 − + i ω2 + iωγ ω2 + γ 2 ω ω2 + γ 2

(4.52)

ne2 /mε0 is the plasmon frequency (n being the number density of the

4.7 Surface Enhanced Raman Scattering (SERS)

101

Drude-Sommerfeld method gives good results for the optical properties of metals in the infrared region. But for excitation in the visible and the higher energy regions of the electromagnetic radiation, contribution of valence electrons becomes important, because the metal to molecule and/or molecule to metal charge transfer states lie in these regions. So the excitation of the valence electron to the conduction band (called interband transition) modifies the condition of plasmon resonance. If me is the effective mass of the bound electrons, γ is the damping constant describing the radiation damping of the bound electrons and α is the force constants of the bound electrons, and the equation of motion of the bound electrons becomes me

d 2 →r d→r → 0 e−iωt + α→r = eE + me γ dt 2 dt

(4.53)

The solution of this equation can be determined in the same manner as described before. √ e/me → 0 e−iωt , where ω0 = α/me ) →r0 = ( 2 E 2 ω0 − ω − iωγ

(4.54)

Thus, the polarization of the bound electrons is 2 e → = nb e→r = nb e→r0 e−iωt = ( nb e /m → 0 e−iωt ) P E 2 ω0 − ω2 − iωγ

(4.55)

where nb is the number density of the bound electrons. Substituting this expression in Eq. (4.49), the dielectric constant arising from the bound electrons (εb ) becomes ω2p nb e2 /me ε0 ) ( 2 ) εb = 1 + ( 2 = 1 + ω0 −ω2 − iωγ ω0 −ω2 − iωγ ( ) ω2p ω02 −ω2 γ ωω2p =1+ ( + i ) ( )2 2 ω02 −ω2 + γ 2 ω2 ω02 −ω2 + γ 2 ω2

(4.56)

where ωp = nb e2 /me e0 is the plasmon frequency of the bound electrons. Since both the conduction electrons and the bound electrons (specially for excitation in the region of the interband transition) contribute to the plasmons, Drude model can be used by adding an extra term εb to the middle part of Eq. (4.52) and thus we get ε = εb + 1 − Substitution of this, Eq. (4.44) gives

ω2p ω2 + iωγ

(4.57)

102

4 Raman Spectroscopy

)

) εb ω2 − ωp2 + iωγ εb ] α = a3 [ (εb + 3)ω2 − ωp2 + iωγ (εb + 3)

(4.58)

Both the real and the imaginary parts of the polarization have a pole at the √ frequency ω = ωR = ωp / (εb + 3) and width of that resonance is γ (εb + 3). When γ is large, i.e. in the case of metals having poor conductivity, the peak height diminishes and so also the SERS enhancement. Similarly when the contribution of the interband transition is large in the region of the exciting wavelength, i.e. εb (generally a function of the wavelength) is large, the band width increases and hence the SERS enhancement also diminishes. This explains why, all things remaining same, SERS enhancement is found to be more in silver than in gold which again is more than in copper. The contribution of the interband transition to the dielectric function of these metals increases in that order, i.e. from silver to gold and from gold to copper. Most of the transition metals have poor SERS enhancement ability because for them both γ is large (i.e. conductivity is low) and the participation of the interband transition in the dielectric function is high (i.e. εb is large). In fact the electromagnetic enhancement is small for spherical particles, because for them the plasmon frequency lies in the higher frequency side and the band width is large due to greater contribution of the interband transition. For prolate spheroidal objects, plasmon frequency is shifted to the red side relative to the spherical objects and so the band width is less which results in the greater enhancement. What we have discussed till now has a closed relation with Mie scattering (which is the scattering from particles of dimension of the order of the exciting wavelength) from spherical objects. Mie’s scattering theory can be solved analytically if the object is spherical. For deviation from spherical geometry, this theory can be extended to objects of higher aspect (i.e. width to height) ratio by changing the dielectric resonance condition (in the denominator of the expression for g in Eq. 4.44) from εin + 2εout = 0 to εin + χεout = 0, where χ is a measure of the aspect ratio. Since the real part of εin is a function of wavelength, i.e. εin (λ), the resonance condition is met in the visible region of the electromagnetic spectrum for the metal nanoparticles such as gold and silver with high aspect ratio (large χ). So in these metal nanoparticles, electromagnetic Raman enhancement is observed for excitation in the visible region. In the long wavelength limit of the Mie scattering (a/λ interferogram ….

Spectrum …. . ν

….

….. ν

ν Fig. 5.11 Interferogram and spectrum (Fourier transform) of the source emitting two waves of slightly different wavenumbers/wavelengths

150

5 Vibrational Spectra of Polyatomic Molecules

plate, the waves at the point of centre burst are not in phase. So there is an asymmetry in the centre burst. If we move away from this centre burst in either direction, the multitudinous cosine waves become out of phase and the signal intensity dies off rapidly to lower intensity oscillation. The less the spectral structure, more rapid the oscillations die out. Thus, we can say that higher resolution in the spectra is contained in the wings of the interferogram, i.e. in the region of higher values of the path difference (δ). Another point is worth noting in this regard. In order to make Fourier transform, one has to do the integration varying the retardation (δ) from − ∞ to +∞. But practically, this is not possible. There is a practical limit of the range in which (δ) can vary. So a compromise is made which results in the diminution of the spectral resolution. There is a theoretical limit of the resolution. If δ max is the maximum value of the retardation, the best possible resolution is Δν = 1/δmax

(5.135)

But this theoretical limit is never attained due to truncation (or apodization) of the interferogram. The movable mirror cannot be displaced to vary the path difference (δ) from −∞ to +∞. Let its movement be confined within a certain region for which δ varies from –Δ to +Δ. Within this region, the interferogram is thus truncated and the interferogram is convoluted within this range by some mathematical function. Let one such function f (δ) be f (δ) = 0, when − Δ > δ > +Δ and = 1, when − Δ < δ < +Δ

(5.136)

This type of truncation (apodization) is called boxcar truncation (apodization). The Fourier transform of this function is a sinc function (sin x/x) which has a central peak whose full width at half height (FWHH) is 0.605/Δ. On either side of this peak, there is a series of lobes with gradually diminishing undulating positive and negative intensities. This is called ringing. The first negative peak on either side of the central peak is of height 22% of the central peak height. So if there is a spectral line within this region, that will be lost, i.e. is absent in the spectra. This is the price to be paid for this truncation. This is shown in Fig. 5.12. Another important truncation is triangular apodization. In this apodization, the peak heights in the interferogram are not restricted to unit values within the range –Δ < δ < +Δ, but a mathematical function is used which reduces linearly the heights from unity at the peak position to zero at the two extreme ends (±Δ). Here, the Fourier transform becomes a sinc2 function (sin2 x/x 2 ). So there will be no ringing but the spectral resolution decreases to FWHH value 0.805/Δ compared to that in boxcar truncation (Fig. 5.12). Other than these two types of truncation, there are several other mathematical functions which are offered for the purpose of apodization to the users of the FTIR spectrophotometers according to their specific needs. FTIR spectra are generally obtained by absorption technique. So the sample is placed at a position so that the interfered beams transmit through it before entering

5.8 Fourier Transform Spectroscopy

151

Fig. 5.12 Effect of truncation (apodization) on the spectral resolution

the detector (Fig. 5.10). If no sample is inserted in the sample chamber, the interferogram and the Fourier transform of this which corresponds to the spectrum (called background) in the wavenumber domain are observed as shown in Fig. 5.13.In this case, the spectrum shows a continuum throughout the spectral region. If a sample having absorption at a single wavenumber (ν a ) is inserted, this wavenumber will not take part in cancelling the signal outside the central burst region, so an absorption is observed on the continuous transmission at that wavenumber as shown in Fig. 5.13. Extending this idea to a sample having absorptions at several wavenumbers, a complicated spectral structure (which is the FTIR spectrum of the sample) is observed. In most of the cases, the background spectrum is continuous but not of constant intensities. In the spectrophotometer, the FT data of both the spectrum and the background are stored in the computer memory and their ratio gives the required FTIR spectrum of the sample. To increase the signal-to-noise ratio, average of a number of scans is taken which reduces the noise proportional to the square root of the number of scans. Multiple scanning of the spectra is also helpful in observing the weak bands.

152

5 Vibrational Spectra of Polyatomic Molecules Fourier transform ------- >

Spectrum

interferogram White light

ν

I (ν) δ

I(δ)

Single frequency absorption ν

Fig. 5.13 Spectrums and interferograms of a a white source and b a single frequency/wavenumber absorption

Appendix s-Vectors for Torsional Vibrations Analytically, the dihedral (torsion) angle τ (see Fig. 5.8) can be defined as cos τ =

(→e12 × e→23 ) · (→e23 × e→34 ) sin φ2 sin φ3

(5.137)

×→e23 ) e34 ) where (→e12 and (→e23sin×→ are the unit vectors perpendicular to the planes 1, 2, 3 sin φ2 φ3 and 2, 3, 4, respectively. By differentiation of Eq. (5.137), small variation of τ is obtained.

(→e12 × Δ→e23 ) · (→e23 × e→34 ) (Δ→e12 × e→23 ) · (→e23 × e→34 ) + sin φ2 sin φ3 sin φ2 sin φ3 (→e12 × e→23 ) · (Δ→e23 × e→34 ) + sin φ2 sin φ3

− sin τ Δτ =

s-Vectors for Torsional Vibrations

153

(→e12 × e→23 ) · (→e23 × Δ→e34 ) sin φ2 sin φ3 [ ] (→e12 × e→23 ) · (→e23 × e→34 ) cos φ2 Δφ2 cos φ3 Δφ3 − + sin φ2 sin φ3 sin φ2 sin φ3 = T1 + T2 + T3 + T4 + T5 + T 6 (5.138) +

where Δτ =

4 ∑

s→tα · ρ→α

(5.138a)

α=1

In deriving the expressions for the s→ vectors, the terms (T i s, i = 1–6) are needed to be determined. (Δ→e12 × e→23 ) · (→e23 × e→34 ) sin φ2 sin φ3 ⎧ ⎫ r12 Δ→ r12 − (Δr12 )→ r12 1 e23 × e→34 ), from (5.67) = × e → 23 · (→ 2 sin φ2 sin φ3 r12 1 = [{(ρ→2 − ρ→1 ) r12 sin φ2 sin φ3 − {→e12 · (ρ→2 − ρ→1 )} e→12 } × e→23 ] · (→e23 × e→34 ), from (5.69) and (5.71) [ ] {−ρ→1 × e→23 + (→e12 · ρ→1 )(→e12 × e→23 )} · (→e23 × e→34 ) 1 = r12 sin φ2 sin φ3 +{ρ→2 × e→23 − (→e12 · ρ→2 )(→e12 × e→23 )} · (→e23 × e→34 ) ⎡ ⎤ ρ→1 · { − e→23 × (→e23 × e→34 ) ⎢ +→e (→e × e→ ) · (→e × e→ )}⎥ 1 23 23 34 ⎥ ⎢ 12 12 (5.139a) = ⎢ ⎥ ⎦ r12 sin φ2 sin φ3 ⎣ + ρ→2 · { e→23 × (→e23 × e→34 )

T1 =

−→e12 (→e12 × e→23 ) · (→e23 × e→34 )} Similarly, the next three terms of T 2 , T 3 and T 4 can be determined as ⎡ T2 =

1 r23 sin φ2 sin φ3

⎤ ρ→2 · { e→12 × (→e23 × e→34 ) ⎢ +→e (→e × e→ ) · (→e × e→ )}⎥ 23 23 34 ⎥ ⎢ 23 12 ⎢ ⎥ ⎣ + ρ→3 · { − e→12 × (→e23 × e→34 ) ⎦

(5.139b)

−→e23 (→e12 × e→23 ) · (→e23 × e→34 )} ⎡ T3 =

1 r23 sin φ2 sin φ3

⎤ ρ→2 · { ( e→12 × e→23 ) × e→34 ⎢ +→e (→e × e→ ) · (→e × e→ )}⎥ 23 23 34 ⎥ ⎢ 23 12 ⎢ ⎥ ⎣ +ρ→3 · { −(→e12 × e→23 ) × e→34 ⎦ −→e23 (→e12 × e→23 ) · (→e23 × e→34 )}

(5.139c)

154

5 Vibrational Spectra of Polyatomic Molecules

⎡ T4 =

1 r34 sin φ2 sin φ3

⎤ ρ→3 · {− ( e→12 × e→23 ) × e→23 + ⎢ +→e (→e × e→ ) · (→e × e→ )}⎥ 23 23 34 ⎥ ⎢ 34 12 ⎢ ⎥ ⎣ + ρ→4 · { (→e12 × e→23 ) × e→23 ⎦

(5.139d)

−→e34 (→e12 × e→23 ) · (→e23 × e→34 )} T 5 and T 6 may be determined in the following manner. (→e12 × e→23 ) · (→e23 × e→34 ) cos φ2 Δφ2 sin φ2 sin φ3 sin φ2 ⎡ − cos φ e→ − e→ ⎤ 2 12 23 · ρ→1 ⎢ ⎥ r12 sin φ2 ⎢ ⎥ ⎢ cos φ2 e→23 + e→12 ⎥ ⎢ ⎥ · ρ→3 (→e12 × e→23 ) · (→e23 × e→34 ) ⎢+ ⎥ r sin φ =− cos φ 23 2 ⎢ ⎥ 2 2 ⎢ ⎥ sin φ2 sin φ3 ⎢ (r12 − r23 cos φ2 )(−→e12 ) ⎥ ⎢ ⎥ ⎣ + (r23 − r12 cos φ2 )→e23 ⎦ + · ρ→2 r12 r23 sin φ2 (from Eq. 5.63) ⎡ (→e12 · e→23 ) e→12 − e→23 · ρ→1 ⎢ (→e12 × e→23 ) · (→e23 × e→34 ) r12 sin φ2 ⎢ =− cos φ2 ⎣ · ρ→3 −(→e12 · e→23 ) e→23 + e→12 sin2 φ2 sin φ3 + r23 sin φ2 ⎤ ⎛ ⎞ 1 {→e23 − (→e12 · e→23 ) e→12 } ⎥ ⎜ r12 sin φ2 ⎟ ⎟ · ρ→2 ⎥ + ⎜ ⎦ ⎝ ⎠ 1 {−→e12 + (→e12 · e→23 ) e→23 } + r23 sin φ2 ⎡ e→12 × (→e12 × e→23 ) · ρ→1 ⎢ (→e12 × e→23 ) · (→e23 × e→34 ) r12 sin φ2 ⎢ cos φ2 ⎣ · ρ→3 =− (→e12 × e→23 ) × e→23 sin2 φ2 sin φ3 − r23 sin φ2 ) ] ( e→12 × (→e12 × e→23 ) (→e12 × e→23 ) × e→23 · ρ→2 (5.140a) + − + r12 sin φ2 r23 sin φ2

T5 = −

Similarly, it can be shown that ⎡

e→23 × (→e23 × e→34 ) · ρ→2 r23 sin φ3 · ρ→4 (→e23 × e→34 ) × e→34 − r34 sin φ3 ) ] (→e23 × e→34 ) × e→34 · ρ→3 (5.140b) + r34 sin φ3

⎢ (→e12 × e→23 ) · (→e23 × e→34 ) T6 = − cos φ3 ⎢ 2 ⎣ sin φ2 sin φ3 +

( e→23 × (→e23 × e→34 ) − r23 sin φ3

s-Vectors for Torsional Vibrations

155

The s→t1 vector, which is the scalar product multiplier of ρ→1 (5.138a), is found to have contributions only from the terms T 1 and T 5 . The contributions of these terms are. { − e→23 × (→e23 × e→34 ) + e→12 (→e12 × e→23 ) · (→e23 × e→34 )} r12 sin φ2 sin φ3 (→e12 × e→23 ) · (→e23 × e→34 ) − cos φ2 {→e12 × (→e12 × e→23 )} r12 sin3 φ2 sin φ3 1 [ { − e→23 × (→e23 × e→34 ) = r12 sin3 φ2 sin φ3 + e→12 (→e12 × e→23 ) · (→e23 × e→34 )}(→e12 × e→23 ) · (→e12 × e→23 ) + (→e12 × e→23 ) · (→e23 × e→34 ){(→e12 · e→23 )2 e→12 − (→e12 · e→23 )→e23 } ], [since cos φ2 = −(→e12 · e→23 ) and sin2 φ2 = (→e12 × e→23 ) · (→e12 × e→23 ) ] 1 = [ { − e→23 × (→e23 × e→34 ) + e→12 (→e12 × e→23 ) r12 sin3 φ2 sin φ3 · (→e23 × e→34 )}(→e12 × e→23 ) · (→e12 × e→23 ) + (→e12 × e→23 ) · (→e23 × e→34 ){(1 − (→e12 × e→23 )2 )→e12 − (→e12 · e→23 )→e23 } ] [since cos2 φ2 = {−(→e12 · e→23 ) }2 = 1 − sin2 φ2 = 1 − (→e12 × e→23 )2 ] 1 = [ { − e→23 × (→e23 × e→34 ) }(→e12 × e→23 ) · (→e12 × e→23 ) r12 sin3 φ2 sin φ3 + (→e12 × e→23 ) · (→e23 × e→34 ){(→e12 − (→e12 · e→23 )→e23 } ] 1 = [ { − e→23 × (→e23 × e→34 ) }(→e12 × e→23 ) · (→e12 × e→23 ) 3 r12 sin φ2 sin φ3 + (→e12 × e→23 ) · (→e23 × e→34 ){→e23 × (→e12 × e→23 } ] e→23 × [− (→e23 × e→34 ) (→e12 × e→23 ) · (→e12 × e→23 ) = r12 sin3 φ2 sin φ3 + (→e12 × e→23 ) · (→e23 × e→34 )(→e12 × e→23 )}] e→23 × [(→e12 × e→23 ) × {(→e12 × e→23 ) × (→e23 × e→34 )}] = r12 sin3 φ2 sin φ3 e→23 · {(→e12 × e→23 ) × (→e23 × e→34 )} = (→e12 × e→23 ) r12 sin3 φ2 sin φ3 [since e→23 · (→e12 × e→23 ) = 0] sin τ = (→e12 × e→23 ) r12 sin2 φ2 e→23 · {(→e12 × e→23 ) × (→e23 × e→34 ) [since sin τ = }] (5.141) sin φ2 sin φ3 which, from Eq. (5.138a), gives

156

5 Vibrational Spectra of Polyatomic Molecules

s→t1 = −

(→e12 × e→23 ) r12 sin2 φ2

(5.142)

Similarly, the s→t2 vector is the scalar product multiplier of ρ→2 (5.138) and this term will get contributions from the terms T 1 , T 2 , T 3 , T 5 and T 6 . First consider the contributions from the terms T 3 and T 6 { ( e→12 × e→23 ) × e→34 + e→23 (→e12 × e→23 ) · (→e23 × e→34 )} r23 sin φ2 sin φ3 e→23 × (→e23 × e→34 ) (→e12 × e→23 ) · (→e23 × e→34 ) − cos φ3 2 r23 sin φ3 sin φ2 sin φ3 { ( e→12 · e→34 )→e23 − ( e→23 · e→34 )→e12 + e→23 [→e12 · {→e23 × (→e23 × e→34 )}] = r23 sin φ2 sin φ3 e→23 × (→e23 × e→34 ) (→e12 × e→23 ) · (→e23 × e→34 ) − cos φ3 r23 sin φ3 sin φ2 sin2 φ3 { ( e→12 · e→34 )→e23 − ( e→23 · e→34 )→e12 + e→23 [→e12 · {→e23 (→e23 · e→34 ) − e→34 }] = r23 sin φ2 sin φ3 e→23 × (→e23 × e→34 ) (→e12 × e→23 ) · (→e23 × e→34 ) − cos φ3 2 r23 sin φ3 sin φ2 sin φ3 −( e→23 · e→34 ){→e23 × (→e12 × e→23 )} 2 = sin φ3 r23 sin φ2 sin3 φ3 e→23 × (→e23 × e→34 ) (→e12 × e→23 ) · (→e23 × e→34 ) − cos φ3 2 r23 sin φ3 sin φ2 sin φ3 cos φ3 = [ (→e23 × e→34 ) · (→e23 × e→34 ){→e23 × (→e12 × e→23 )} r23 sin φ2 sin3 φ3 − (→e12 × e→23 ) · (→e23 × e→34 ){→e23 × (→e23 × e→34 )}], since cos φ3 = −( e→23 · e→34 ) and sin2 φ3 = (→e23 × e→34 ) · (→e23 × e→34 ) cos φ3 = e→23 × [(→e23 × e→34 ) · (→e23 × e→34 )(→e12 × e→23 ) r23 sin φ2 sin3 φ3 − (→e23 × e→34 )(→e12 × e→23 ) · (→e23 × e→34 )] cos φ3 = e→23 × [(→e23 × e→34 ) × {(→e12 × e→23 ) × (→e23 × e→34 )}] r23 sin φ2 sin3 φ3 cos φ3 (→e23 × e→34 ) = e→23 · {(→e12 × e→23 ) × (→e23 × e→34 )}, r23 sin φ2 sin3 φ3 since e→23 · (→e23 × e→34 ) = 0 cos φ3 (→e23 × e→34 ) , see Eq. (5.141) (5.143a) = sin τ r23 sin2 φ3 Next consider the contributions from T 2 and the second term of the scalar multiplier of ρ→2 in τ5 (the first term will be considered along with T 1 ).

s-Vectors for Torsional Vibrations

157

{ e→12 × (→e23 × e→34 ) + e→23 (→e12 × e→23 ) · (→e23 × e→34 )} r23 sin φ2 sin φ3 (→e12 × e→23 ) × e→23 (→e12 × e→23 ) · (→e23 × e→34 ) − cos φ2 2 r23 sin φ2 sin φ2 sin φ3 { (→e12 · e→34 )→e23 − (→e12 · e→23 )→e34 } + e→23 [→e12 · {→e23 (→e23 · e→34 ) − e→34 }] = r23 sin φ2 sin φ3 (→e12 × e→23 ) × e→23 (→e12 × e→23 ) · (→e23 × e→34 ) − cos φ2 2 r23 sin φ2 sin φ2 sin φ3 −(→e12 · e→23 )→e34 + e→23 (→e12 · e→23 )(→e23 · e→34 ) = r23 sin φ2 sin φ3 (→e12 × e→23 ) × e→23 (→e12 × e→23 ) · (→e23 × e→34 ) − cos φ2 r23 sin φ2 sin2 φ2 sin φ3 cos φ2 {→e23 × (→e23 × e→34 )} 2 = − sin φ2 r23 sin3 φ2 sin φ3 (→e12 × e→23 ) × e→23 (→e12 × e→23 ) · (→e23 × e→34 ) − cos φ2 r23 sin φ2 sin2 φ2 sin φ3 since cos φ2 = −(→e12 · e→23 ) cos φ2 e→23 × { (→e23 × e→34 )(→e12 × e→23 ) · (→e12 × e→23 ) =− r23 sin3 φ2 sin φ3 − (→e12 × e→23 )(→e12 × e→23 ) · (→e23 × e→34 )} sin2 φ2 = (→e12 × e→23 ) · (→e12 × e→23 ) cos φ2 e→23 × [(→e12 × e→23 ) × {(→e12 × e→23 ) × (→e23 × e→34 )}] = r23 sin3 φ2 sin φ3 cos φ2 (→e12 × e→23 ) = [→e23 · {(→e12 × e→23 ) × (→e23 × e→34 )}] r23 sin3 φ2 sin φ3 cos φ2 (→e12 × e→23 ) = sin τ, see Eq. (5.141) r23 sin2 φ2

since

(5.143b)

The remaining contributions come from T 1 and the first term of the scalar multiplier of ρ→2 in T 5 . { e→23 × (→e23 × e→34 ) − e→12 (→e12 × e→23 ) · (→e23 × e→34 ) r12 sin φ2 sin φ3 e→12 × (→e12 × e→23 ) (→e12 × e→23 ) · (→e23 × e→34 ) + cos φ2 r12 sin φ2 sin2 φ2 sin φ3 e→23 × (→e23 × e→34 ) = (→e12 × e→23 ) · (→e12 × e→23 ) r12 sin3 φ2 sin φ3

158

5 Vibrational Spectra of Polyatomic Molecules

+

−→e12 (→e12 × e→23 ) · (→e23 × e→34 )(→e12 × e→23 ) · (→e12 × e→23 ) −(→e12 · e→23 )(→e12 × e→23 ) · (→e23 × e→34 ){→e12 × (→e12 × e→23 )}

, r12 sin3 φ2 sin φ3 since cos φ2 = − (→e12 · e→23 ) and sin2 φ2 = (→e12 × e→23 ) · (→e12 × e→23 ) e→23 × (→e23 × e→34 ) (→e12 × e→23 ) · (→e12 × e→23 ) = r12 sin3 φ2 sin φ3 (→e12 × e→23 ) · (→e23 × e→34 ) + [ e→12 {→e23 · {→e12 × (→e12 × e→23 )}} r12 sin3 φ2 sin φ3 − (→e12 · e→23 ){→e12 × (→e12 × e→23 )}] e→23 × (→e23 × e→34 ) (→e12 × e→23 ) · (→e12 × e→23 ) = r12 sin3 φ2 sin φ3 (→e12 × e→23 ) · (→e23 × e→34 ) + e→23 × [→e12 × {→e12 × (→e12 × e→23 )}] r12 sin3 φ2 sin φ3 e→23 × (→e23 × e→34 ) = (→e12 × e→23 ) · (→e12 × e→23 ) r12 sin3 φ2 sin φ3 (→e12 × e→23 ) · (→e23 × e→34 ) − {→e23 × (→e12 × e→23 )} r12 sin3 φ2 sin φ3 since e→12 · (→e12 × e→23 ) = 0 e→23 × [(→e23 × e→34 )(→e12 × e→23 ) · (→e12 × e→23 ) = r12 sin3 φ2 sin φ3 − (→e12 × e→23 )(→e12 × e→23 ) · (→e23 × e→34 )] e→23 × [{(→e12 × e→23 ) × {(→e12 × e→23 ) × (→e23 × e→34 )}] =− r12 sin3 φ2 sin φ3 e→23 · {(→e12 × e→23 ) × (→e23 × e→34 )} =− (→e12 × e→23 ), since e→23 · (→e12 × e→23 ) = 0 r12 sin3 φ2 sin φ3 (→e12 × e→23 ) =− sin τ, see Eq. (5.141) (5.143c) r12 sin2 φ2 Thus substituting the terms from Eqs. (5.87, a, b and c) into the Eq. (5.82a), we get s→t2 =

r23 − r12 cosφ2 e→12 × e→23 cosφ3 e→23 × e→34 − r12 r23 sinφ2 sinφ2 r23 sinφ3 sinφ3

(5.144)

In a similar fashion, s→t3 and s→t4 can be determined. But without entering into the detailed calculation, the expressions for the respective vectors can be obtained by exchanging the indexes (1,4) and (2,3) of s→t2 and s→t1 , respectively. The s-vectors thus found are,

References and Suggested Reading

159

s→t1 = − s→t2 =

e→12 × e→23 r12 sin2 φ2

r23 − r12 cosφ2 e→12 × e→23 cosφ3 e→23 × e→32 − r12 r23 sinφ2 sinφ2 r23 sinφ3 sinφ3

(5.145a) (5.145b)

s→t3 = [(14)(23)]→st2

(5.145c)

s→t4 = [(14)(23)]→st1

(5.145d)

where the expressions in brackets (14) and (23) indicate that the latter vectors can be obtained by permutation of the atom subscripts by 1 & 4, and 2 & 3 in the expressions for the first two vectors.

References and Suggested Reading 1. E.B. Wilson, J.C. Decius, P.C. Cross, Molecular Vibrations (The Theory of Infrared and Raman Spectra) (McGraw Hill Book Company, New York. 1955) 2. L.A. Woodward, Introduction to the Theory of Molecular Vibrational Spectroscopy (Clarendan Press, Oxford, 1972) 3. G.M. Barrow, Introduction to Molecular spectroscopy (McGraw Hill Book company Inc., New York, 1962) 4. J.M. Hollas, Modern Spectroscopy, 4th edn. (Wiley, Chichester, London, 1986) 5. W.D. Perkins, Fourier transform infrared spectroscopy. J. Chem. Educ. 63 (1986)

Chapter 6

Electronic Spectra of Diatomic Molecules

Abstract In this chapter, electronic spectra of diatomic molecules have been discussed extensively. Vibrational structure of the electronic spectra and isotopic effect on these spectra and their fine structure, including formation of band heads and shading of bands, have been discussed. A critical description of the Franck– Condon principle in relation to intensity of the vibrational bands of the electronic spectra has been given. Different types of coupling between molecular rotation and electronic motions have been discussed to investigate the rotational fine structure of electronic spectra. The idea of molecular orbitals has been introduced by solving the problem fof the simplest molecule, hydrogen molecule ion. Valence bond method has been presented for hydrogen molecule. Electronic structures of different diatomic molecules have been described. A theoretical discussion on the Ʌ doubling is also given.

In the earlier chapters, we have discussed about pure rotational, pure vibrational and rotational-vibrational spectra of molecules. Molecules also have several electronic states, and the transitions amongst these levels generate the electronic spectra of the molecules. But there is a characteristics difference between the electronic spectra of the atoms and molecules. In the case of the atoms, there is a single nucleus around which the electrons move and the spectra appear as lines. However in the case of molecules, there are several nuclei, and so, the electronic transitions will be accompanied by changes in the vibrational and rotational levels of the relevant electronic states. As we have seen earlier that energy wise E elec > E vib > E rot , so if we disregard the contribution of the rotational energy, we get vibrational structure of the electronic spectra (vibronic bands). If the electronic spectra are examined by high resolution spectrograph, rotational fine structures of the vibronic bands are observed. Anyway proper analyzes of the vibrational and rotational fine structures of the electronic spectra help to determine many interesting features of the molecule in the ground and excited electronic states. In this chapter, we shall first discussed the essential features of the vibrational structure and then the rotational fine structures of the electronic spectra of diatomic molecules. Latter, we shall extend our discussion to polyatomic molecules in the

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 P. K. Mallick, Fundamentals of Molecular Spectroscopy, https://doi.org/10.1007/978-981-99-0791-5_6

161

162

6 Electronic Spectra of Diatomic Molecules

following chapter because the treatment of the spectral features of the electronic spectra of polyatomic molecules is different.

6.1 Vibrational Coarse Structure of Electronic Bands of a Diatomic Molecule Ignoring the rotational energy, the energy of a vibronic level of a diatomic molecule becomes E mol = E ev = E elec + E vib

(in ergs/joules )

Tmol = Tev = Telec + Tvib (in wavenumbes , cm−1 )

(6.1a) (6.1b)

Assuming the molecule to be an an harmonic oscillator, the vibrational component of the energy level is ( Tvib =

( ) ) 1 1 2 v+ νe − v + ν e xe 2 2

(6.2)

? ) ( where v is the vibrational quantum number (0, 1, 2, 3,…), ν e = 2π1 c kμe (k e being the equilibrium force constant and μ being the reduced mass) is the equilibrium vibrational wavenumber and ν e x e is the anharmonicity constant of the molecule. So for a transition between two vibronic levels, e' v' (higher) and e'' v'' (lower), the wavenumber of the spectral line can be written as ) ( ) ( ) ( v = v v'' v'' = Te' v' − Te'' v'' = T'e + Tv' − T''e + Tv'' = T'e − T''e + (Tv' − Tv'' ) (( ) ) ) ( 1 ' 1 2 ' ' ' ' v + v − v + ve xe = v e' e'' + 2 e 2 (( ) ) )2 ( 1 1 v'' + v '' − v'' + v ''e xe'' − 2 e 2 [ ] ) 1( ) 1( = v e' e'' + v'e − v''e − v 'e xe' − v'e xe' 2 4 ) ) ] ( ) ( '' [( ' ' ' ' + v e − v e xe v − ve − v 'e xe' v'' − v 'e xe' v'2 − v 'e xe'' v''2 ) ( ) ( = v oo + v 'o v' − v ''o v'' − v 'o xo' v'2 − v''o xo' vˆ ''2 (6.3) where

6.1 Vibrational Coarse Structure of Electronic Bands of a Diatomic Molecule

) 1( ) 1( ' v e − v 'e − v 'e xe' − v'e xe' , 2 4 v 'o = v'e − v 'e x ' , v ''o = ve'' − v 'e xe''

163

voo = ve' e'' +

(6.3a)

and

v 'o xo' = v 'e xe' and

(6.3b)

v'o x '' = v 'e xe''

The change in vibrational quantum number accompanying an electronic transition is not governed by any selection rule, so any transition v' ↔ v'' may occur but with some definite probability. In some cases when it is large, the intensity of the corresponding vibronic band is large; when this probability is small, the band appears with weak intensity. In this section, only vibrational structure of the electronic band will be discussed, and the intensities of the vibrational bands will be discussed in a latter section. The series of lines arising from the transitions from a particular vibrational level (v'' ) of the lower electronic state (e'' ) to different vibrational levels (v' ) of the excited electronic state (e' ) (for absorption) or those arising out of transitions '' from different vibrational v' to a particular vibrational level v (for emission) are ' called v -progression. Similarly, the transitions originating from or terminating to a particular vibrational level v' give rise to a series of lines called v'' -progression. The series of lines arising from the transitions for which Δv = v' –v'' = constant is called sequence. ν oo is called the 0–0 (zero–zero) band which arises from the transition between the lowest vibrational states of the two electronic states. For a particular electronic transition, the wavenumbers of all the vibrational bands are arranged in a Table 6.1, called Deslandre’s table. Table. 6.1 Deslandre’s table v' → 0 v'' ↓

1

2

3

4

5

6

7

8

9

10

• • • • • • •

0

ν 00

ν101

ν 02

ν 03

ν 04

ν 05

ν 06

ν 07

ν 08

ν 09

ν 0,10

• • • • • • •

1

ν 10

ν 11

ν 12

ν 13

ν 14

ν 15

ν 16

ν 17

ν 18

ν 19

ν 1,10

• • • • • • •

2

ν 20

ν 21

ν 22

ν 23

ν 24

ν 25

ν 26

ν 27

ν 28

ν 29

ν 2,10

• • • • • • •

3

ν 30

ν 31

ν 32

ν 33

ν 34

ν 35

ν 36

ν 37

ν 38

ν 39

ν 3,10

• • • • • • •

4

ν 40

ν 41

ν 42

ν 43

ν 44

ν 45

ν 46

ν 47

ν 48

ν 49

ν 4,10

• • • • • • •

5

ν 50

ν 51

ν 52

ν 53

ν 54

ν 55

ν 56

ν 57

ν 58

ν 59

ν 5,10

• • • • • • •

6

•

•

•

•

•

•

•

•

•

•

•

• • • • • • •

7

•

•

•

•

•

•

•

•

•

•

•

• • • • • • •

8

•

•

•

•

•

•

•

•

•

•

•

• • • • • • •

164

6 Electronic Spectra of Diatomic Molecules

In this table, each row corresponds to a v' -progression, and the respective v' value increases from the left to the right. Similarly, each column corresponds to a v'' progression, and the v'' -value increases from the top to the bottom. The main diagonal of this table corresponds to the sequence, Δv = 0, and all the other sequences are parallel to it, those lying above the main diagonal correspond to Δv = positive and those lying below correspond to Δv = negative. The first and the second differences of the horizontal (v' -progression) and the vertical (v'' -progression) columns are very helpful in determining the vibrational constants of the molecule in the two electronic states (Fig. 6.1). The first differences are

Fig. 6.1 v' - and v'' -progressions and sequence associated with an electronic transition of a diatomic molecule

6.1 Vibrational Coarse Structure of Electronic Bands of a Diatomic Molecule

165

( ) Δ(1) Tv'' + 21 = v v' ,v'' +1 − v v' ,v'' = Tv'' +1 − Tv'' = v e'' − 2v e'' xe'' v'' + 1

(6.4a)

( ) Δ(1) Tv' + 21 = vv' +1,v'' − v v' v'' = Tv' +1 − Tv' = v e' − 2v e' xe' v' + 1

(6.4b)

and

The second differences are Δ(2) Tv'' +1 = Δ(1) Tv'' + 23 − Δ(1) Tv'' + 21 = −2v e'' xe''

(6.5a)

Δ(2) Tv' +1 = Δ(1) Tv' + 23 − Δ(1) Tv' + 21 = −2ν e' xe'

(6.5b)

and

Thus, we see that the second differences are constants, and therefore from the second differences, the anharmonicities (i.e. anharmonicity constants) of the respective electronic states can be determined. Then from the first difference, the equilibrium vibrational wavenumber from the knowledge of the anharmonicity and hence the equilibrium force constant of each of the two electronic states can be determined. Remember in determining the first and the second differences, the average values of the said differences of all the rows and columns are to be taken. At normal temperature, most of the molecules are in the lowest vibrational states of the ground electronic states. So in the absorption spectra of molecules, a v' progression is observed (with v'' = 0), ν abs = ν 00 + ν '0 v' − ν '0 x0' v'2

(6.6)

In this progression, the wavenumber difference of the successive lines gradually decreases with the increase of the v' -value, and ultimately, this difference becomes zero. Thus, the discrete bands ultimately join to a continuum. Actually, such kind of continuum joining to a set of discrete bands is observed in the absorption spectrum of iodine. With the increase of temperature, the higher and higher vibrational levels of the ground electronic states will get more and more populated giving rise to v' progressions originating from higher and higher values of v'' (1, 2, 3, 4…). These are called satellite bands or hot bands lying very close to the respective bands originating from v'' = 0. The wavenumbers of these bands can be determined from Eq. (6.3) by proper substitution of the v'' -values. However following Maxwell-Boltzman distribution law, the intensities of the hot bands gradually decrease with the gradual increase of the v'' -values from which the absorption originates. In the case of emission, all the vibrational levels of the excited electronic states are evenly populated, so different v'' -progressions are found to originate from different values of v' (Eq. 6.3). If the molecule is excited with light of proper wavenumber, the molecule first absorbs this radiation and gets excited from the level v'' = 0

166

6 Electronic Spectra of Diatomic Molecules

to a particular v' -level of the excited electronic state and there from emits a v'' progression. In this progression, all the bands are stokes bands excepting one which brings the molecule down to the level v'' = 0, and this is called the resonance band. Such emission is called resonance fluorescence. However if absorption originates from higher values of v'' , both stokes and antistokes bands are observed. Obviously, the antistokes bands are weak as they are satellite bands. For sequence, Δv = v' −v'' = constant; so we can put v' = Δv + v'' in Eq. (6.3) to determine the wavenumbers of different members of the sequence v seq = v 00 + v'0 Δv − v '0 x0' (Δv)2 ( ) + v'0 − v ''0 − 2v '0 x0' Δv v'' ) ( − v'0 x0' − v ''0 x0'' v''2

(6.7)

where Δv can take up values 0, 1, 2, 3,… for various sequences.

6.2 Isotope Effect For a diatomic molecule ν e =

1 2π c

?

ke , μ

where μ =

mass of the molecule. For isotopic substitution, heavier isotope and ke remains unchanged. Thus, νe = ν iso e

?

μ μiso

[

+

1 m1

=

1 ρ

1 m2

]−1

is the reduced

(say), where ρ › 1, for

μiso = ρ μ

(6.8)

Since the electronic energy levels remain unaffected by isotopic substitution, ν 00 remains unchanged, and so from Eq. (6.3), we get

v v' − viso w'

) ⎡( ( ) 1( ' ' ) ⎤ 1 ' '' ' ' ⎢ 2 ν e − ve − 4 ν e xe − ve xe ⎥ ⎢ ) ) }⎥ ( '' = (1 − 1/ρ)⎢ {( ' ' ' ' ' ' '' ⎥ ⎣ + v e − v e xe v − ve − ve xe v ⎦ ( ) − v 'e xe' v'2 − v 'e xe'' v''2

(6.9)

For the 0–0 band, ] [ ) 1( ' ' ) 1( ' '' '' ' − v 00 − v iso = (1 − 1/ρ) v − v v x − v x 00 e e e 2 e 4 e e

(6.10)

Thus, an isotopic shift of the 0–0 band is expected in the electronic spectra. Such kind of shift is observed experimentally, and this proves the existence of zero-point energy of the oscillatory motion.

6.3 Rotational Fine Structure of Vibronic Bands in Diatomic Molecules

167

6.3 Rotational Fine Structure of Vibronic Bands in Diatomic Molecules In the previous sections, discussions have been confined to the vibrational structure of the electronic bands. If these vibronic bands are examined with instruments of high resolving power, each band is found to consist of a number of lines the resolution which depends on the resolving power of the instrument. This kind of fine structure arises due to rotational motion of the molecule. For a particular vibronic level, the total energy in term value is given by Tevj = Telec + Tvib + Trot = Te + Tv + TJ = TevJ = Tev + TJ

(6.11)

Thus for a vibronic band (corresponding to a transition e' v' ↔ e'' v'' ), the rotational structure is ) ( ) ( v = Te' v' J ' − Te'' V '' J '' = Te' V ' − Te'' V '' + TJ ' − TJ '' ( ) = v0 + B ' J ' J ' + 1 ( ) − B '' J '' J '' + 1 (under rigid rotator approximation )

(6.12)

ν 0 is called the band origin or zero line which arises from transition (J ' = 0)↔ (J = 0). The selection rules for rotational transitions in the electronic spectra of diatomic molecule are ''

ΔJ = J ' − J '' = 0 (Q−branch ), Ʌ /= 0, for at least one state ΔJ = +1 (R − branch ) = −1 (P − branch ) and

(

) ( ) J ' = 0 ← × → J '' = 0 i.e. transition between states, J ' , J '' = 0

is forbidden.

(6.13)

For an electronic transition, the upper and lower energy states may have different orbital angular momenta about the internuclear axis, and this momentum is Ʌè (where Ʌ = 0, 1, 2, 3, 4,…). The electronic state with Ʌ = 0 is called a ∑-state. For a ∑ → ∑ electronic transition, ΔJ = 0 is forbidden, i.e. Q-branch is absent. Otherwise, all the three branches are observed. The wavenumbers of the spectral lines in the three branches are (writing J '' = J) ν = ν 0 + (B ' − B '' )J + (B ' − B '' ) J 2 = ν Q ( f or Q − branch, J = 1, 2, 3, ...)

(6.14a)

168

6 Electronic Spectra of Diatomic Molecules

ν = ν 0 − (B ' + B '' ) J + (B ' − B '' ) J 2 = ν P ( f or P − branch, J = 1, 2, 3, ...)

(6.14b)

ν = ν 0 + 2B ' + (3B ' − B '' ) J + (B ' − B '' )J 2 = ν R ( f or R − branch, J = 0, 1, 2, 3, ...)

(6.14c)

All the branches are represented by three parabolic equations. The P- and R-branch can be represented by a single equation ν = ν 0 + (B ' + B '' )m + (B ' − B '' )m 2 = ν m (where m = −J for P − branch and ( J + 1) f or R − branch)

(6.15)

If the wavenumbers (ν) are plotted along x-axis and rotational quantum numbers (J/m) are plotted along y-axis, the three branches represent three parabolas, which are called Fortrat parabolas, and such diagram is called Fortrat diagram (Fig. 6.2). In general, two cases arise depending on the relative values of the rotational constants (B' and B'' ), i.e. the internuclear distances (r ' and r '' ) of the two electronic states. Case (A) B' > B'' or r ' < r '' From Eq. (6.14a, b), we see that in the P-branch, the magnitude of the coefficient of the linear term (whose sign is negative) is greater than that of the quadratic term (which is positive). So for smaller values of J, the contribution of the linear term is greater than that of the quadratic term. So the wavenumbers of the respective lines and also the wavenumber differences between the successive lines decrease with

Fig. 6.2 Formation of the band heads in the P- and R-branches of the rotational fine structure of the electronic spectra of diatomic molecules

6.3 Rotational Fine Structure of Vibronic Bands in Diatomic Molecules

169

the increase of J-value. This means that the lines in the P-branch come closer and closer as J-value increases. So there exists a certain value of J where this difference vanishes. This corresponds to an equal contribution from the linear and the quadratic terms. With the further increase of J-value, the wavenumber increases, and the lines become more and more separated. Thus, we can say that a band head is formed in the P-branch at this extremum value of J (J head ), and the bands are shaded towards violet. For the other two branches, the coefficients of the both the linear and the quadratic terms are positive. So no band heads are formed in these branches and the wavenumbers, and the wavenumber separation of the successive lines increases with the increase on J-value. Case (B) B' < B'' or r ' > r '' In this case, it can be shown in the similar way that a band head is formed in the R-branch and the bands are shaded towards red. In this case, the lines are more and more separated and spread towards red as J-value increases. When the internuclear distances in the two electronic states are nearly equal, then B' ≈ B'' . In such case, the band heads are formed at large values of (J head ). Besides, a band head is appeared to be formed at the beginning of the Q-branch. So under such circumstances, two heads appear to be formed, i.e. in the P- and Q-branches or in the R- and Q-branches depending on the relative values of B' and B'' . Since the rotational constants of different vibrational states of a particular electronic state are related by the equation. ((Bv = Be − αe (v + 1/2), where Be is the rotational constant for the equilibrium nuclear configuration of the electronic state (e) and α e is small positive constant much smaller, in magnitude, than Be ), so when B' ≈ B'' , shading of bands may change direction from band to band, depending on B' , greater or less than B'' . dν The position of the band heads can be determined from the relation dm = 0 which gives m head = −

(B ' + B '' ) (B ' + B '' )2 and ν − ν = − head 0 2(B ' − B '' ) 4(B ' − B '' )

(6.16)

Thus, we see that when B' > B'' , both mhead and ν head − ν 0 are negative. These mean that a band head is formed in the P-branch (6.15) and the bands are shaded towards violet. Similarly when B' < B'' , both mhead and ν head − ν 0 are positive. So in such a case, a band head is formed in the R-branch, and the bands are shaded towards red. These are in compliance with the former discussion. Isotopic effect in the rotational fine structure can be determined in the following manner. Since BBisoe = μμiso = ρ 2 (> 1, for heavier isotope), from Eq. (6.14a, b), we see that v − v0 = ρ2 ν iso − ν iso 0

(6.17)

170

6 Electronic Spectra of Diatomic Molecules

So the bands are contracted by a factor of 1/ρ 2 in the heavier isotope. Note that the zero lines in the two ? molecules (natural and the isotope) are different due to the

variation of ν e = 2π1 c kμe in the two cases. For homonuclear diatomic molecule, the rotational levels are either symmetric or antisymmetric with respect to exchange of the nuclei. The symmetric and antisymmetric nature of the rotational levels depend on the (+, −) and (g, u) characters of the electronic states and rotational quantum number (J).We know that the state of a nucleus with spin I has (2I + 1) degeneracy. So in a homo nuclear diatomic molecule, the number of asymmetric states is 2I+1 C 2 = I(2I + 1), and the number of symmetric states is (2I + 1) + 2I+1 C 2 = (I + 1)(2I + 1), where I is the spin of each nucleus. Since, according to the selection rule, transitions are only possible between two symmetric (s) or two antisymmetric (a) sates, so the intensity ratio of the successive rotational lines in the electronic spectra is about (I + 1)/I. This has been extensively discussed in Chap. 4.

6.4 Intensity Distribution in the Vibrational Bands of the Electronic Spectra (Franck–Condon Principle) of Diatomic Molecule In the previous Sect. 6.1, we have discussed about the vibrational coarse structure of the electronic spectra without referring anything related to their intensity distribution. There is no selection rule for the vibrational transition, but the intensities are found to vary from band to band in a systematic manner. In this section, we shall study the variation in intensity of the vibrational bands in the electronic spectra. There are three distinct types of intensity distribution, observed in the electronic absorption spectra of diatomic molecules (Fig. 6.3). In the first type, the intensity ' '' of the 0–0 band is maximum and joins onto it a v -progression (with v = 0) with ' gradually decreasing intensity towards the bands of higher v values. This type of intensity distribution is observed in the red part of the solar spectrum of atmospheric oxygen. In the second type, the intensity gradually increases from the 0–0 band and ' ' attains a maximum at a particular value of v . Above this value of v , intensities of the ' bands are gradually found to decrease towards higher members of the v -progression. This pattern of intensity distribution is observed in the absorption spectra of carbon monoxide (CO) molecule. In the last type, the intensities of the first few members of ' the v -progression are so weak that they are not observed. The first band is observed ' ' at a higher v -value, and then, the intensities of the further higher members of the v progression go on increasing attaining a maximum very near to or in the continuum. This type is observed in the visible absorption spectrum of iodine molecule. Intensity variation of the vibrational bands in the electronic spectra can be explained with the help of Franck–Condon (FC) principle. Actually, Franck gave the basic idea, and later, Condon gave a mathematical basis of this principle. This principal states that ‘an electronic transition in a molecule takes place so rapidly

6.4 Intensity Distribution in the Vibrational Bands of the Electronic Spectra …

171

Fig. 6.3 Illustration of Franck–Condon Principle. a equilibrium internuclear distances are equal in both the electronic states. b equilibrium internuclear distance is slightly greater and c much greater in the upper state

that the nuclei cannot compete with it, and during the transition, the nuclear coordinates (i.e. the internuclear distance) and momentums (i.e. velocities) remain unchanged’. This principle is illustrated in Fig. 6.3. At normal tempera'' ture, most of the molecules are accommodated in the lowest vibrational state (v = '' 0) of the lower electronic state (e ) from which the transition (for absorption) starts. First consider the case where the equilibrium internuclear distances of the two states are nearly equal (r e '' ≈ r e ' ). If we disregard the zero-point energy, the transition will '' start from the point A (minimum of the electronic state, e ). For the transition to the point B (0–0 band), the violation of FC principle is minimum, so this will be the most probable transition, and hence, the corresponding intensity will be maximum. For transition to any other v' (>0) state, say CED, violation of the FC principle is greater. Because if the transition takes place to any one of the classical turning points C or D, the internuclear distance changes. On the other hand for the transition to terminate at the point E (where r e '' ≈ r e ' ), the velocity undergoes a change which corresponds to the change in kinetic energy of amount EB. For transition to any other point between C and D, both internuclear distance and the nuclear velocity ' change. The more the state CD is away from the level v = 0, more is the violation

172

6 Electronic Spectra of Diatomic Molecules

of FC principle, and less is the intensity. This is illustrated in Fig. 6.3a). When r e '' is slightly less than r e ' , 0–0 transition will not be the most probable transition, because for this transition, the internuclear distance changes. But for the transition to the level CD (v' > 0), the violation is minimum, and hence, the corresponding band intensity will be maximum. For the higher members of the v' -progression, the intensities will go on decreasing again as in the case (a). This is illustrated in Fig. 6.3b. When r e '' is significantly greater than r e ' , the transition probabilities and hence the respective band intensities in the beginning of the v' -progression are too weak to be observed. In fact, the first line in the v' -progression, in such case, is observed for a high value of v' , and for further higher values of v' , the intensities go on increasing as before and the maximum is observed near to or within the continuum. This is illustrated in Fig. 6.3c. In the case of emission, the molecule is initially in the excited electronic state, and all the vibrational levels of this state are evenly populated as the Maxwell-Boltzman law is not valid in the excited electronic state. So the downward transitions are possible from all the vibrational levels (v' ), generating different v'' -progressions. The v' -progression in emission which originates from v'' = 0 is similar in nature to the v' -progression originating from v'' -0 in the case of absorption. In the case of other v'' - progressions (with v' > 0), the process is something different. Consider any such state AB (with v' > 0, Fig. 6.4). Any of the two classical turning points (A or B) is the most favourable points for the downward transitions to begin. Because here, the nuclear velocities are zero, and so, the nuclei spend maximum time at these points. So if the transition starts from the point A, out of all the transitions, the transition to the vibrational state (EF) of the lower electronic state (e'' ) is most probable. Because very near to E, the left classical turning point, there is a point on the lower potential energy curve where the vertical line drawn from the point A intersects. Here, the violation of the FC principle is minimum. Similarly out of all the downward transitions originating from the right classical turning point B of the electronic state (e' ), transition to the point D (or very near to it) of the vibrational level CD of (e'' ) is most favourable. Hence, all the v'' -progressions (with v' /= 0) exhibit two maxima. If all such maximum intense bands of the v'' -progressions are marked in the Deslandre’s table, they are found to form a parabola. This parabola is called Condon parabola. The axis of this parabola is parallel to the main diagonal of the Deslandre’s table. The width of this parabola depends on the difference (r e ' ~ r e '' ) of the potential minima of the two electronic states. When this difference is large, the parabola is more open, and when this difference is small, the width of the parabola is small, and in the extreme case, the parabola ceases to be a straight line coinciding the main diagonal of the Deslandre’s table for the case re' ≈ re'' .

6.4 Intensity Distribution in the Vibrational Bands of the Electronic Spectra …

173

Fig. 6.4 a Franck–Condon principle in emission and b Condon parabola

6.4.1 Quantum Mechanical Formulation of Franck–Condon Principle This formulation is based on Condon approximation. According to this approximation, the electronic transition occurs on a time scale short compared to the nuclear motion so that the transition probability can be calculated at a fixed nuclear position. Note that this is a more restrictive approximation than the Born–Oppenheimer approximation. The B-O approximation states that nuclear and electronic motions can be separated but does not demand that the nuclear positions are frozen at a fixed nuclear configuration. We know that the intensity of a spectral line is proportional to | the\ square of the transition dipole moment between the initial (|ψi ⟨) and final (|ψ f ) states of the concerned transition, i.e. | | | \|2 | \|2 | → n |ψ f || → |ψ f || = ||⟩ψi | M →e + M Iif ∝ |⟩ψi | M

(6.18)

where subscripts e and n correspond to the contributions from the electronic and nuclear charges of the molecule. Considering each state as a Born–Oppenheimer state, we have

174

6 Electronic Spectra of Diatomic Molecules

| \ |ψi ⟨ ≡ |ψi (q, Q)⟨ = |ψie (q, Q) . ψvi (Q)

(6.19a)

| | \ \ |ψ f ≡ |ψi (q, Q)⟨ = |ψ e (q, Q) . ψv (Q) f f

(6.19b)

and

Here, according to |Born–Oppenheimer approximation, the electronic wave functions, \ | e \ |ψ (q, Q) and ||ψ e (q, Q) , , are functions of the electronic coordinates (q’s) i f having parametric| dependence on| the nuclear \ \ coordinates (Q’s), and the vibrational wave functions, ( |ψvi (Q) and |ψv f i (Q) ), are functions of the nuclear coordinates only. Thus, the transition dipole moment becomes | \ | \ → i f ) = ⟩ψi | M → n |ψ f → |ψ f = ⟩ψi | M →e + M (M | | ⟨ \ →e + M → n |ψ ef (q, Q).ψv f (Q) = ψie (q, Q).ψvi (Q)| M | [⟨ | | e ⟨ \] | \ → e |ψ f (q, Q) |ψv f (Q) = ψvi (Q) | ψie (q, Q)| M | | ⟨ ⟨ \| \ → n ψie (q, Q) | ψ ef (q, Q) |ψv f (Q) + ψvi (Q) | M | | | | ⟨ \ ⟨ \ → n δif |ψv f (Q) = ψvi (Q) | ( R→e (Q))if |ψv f (Q) + ψvi (Q) | M (due to orthogonal properties of the electronic wave functions)

(6.20)

The integral ( R→e (Q))if over the electronic space is a slowly varying function of the nuclear coordinates (according to Condon approximation) and can be expanded about the equilibrium internuclear configuration (designated by the subscript ‘0’) in terms |⟨ | \ |2 ⟨ | \ of Taylor’s series as vi | v f is called the vibrational overlap factor, and | vi | v f | is called the Franck–Condon (FC) factor on which the intensity of the band vf ↔ vi depends. So the vibrational overlap factor (and hence the FC factor) plays vital roles in determining the intensities of the vibrational bands of the electronic spectra (Fig. 6.5). (

) →e (Q))if δ( R ( R→e (Q))if = ( R→e (Q 0 ))if + (Q − Q 0 ) δQ 0 ) ( 1 δ 2 ( R→e (Q))if + (Q − Q 0 )2 + · · · 2 δ Q2

(6.21)

0

⟨ | \ → if ) = ( R→e (Q 0 ))if . vi | v f (M

(6.22a)

| | | |2 |⟨ | \ | 2 | → |2 | → | | | ( R vf | Iif ≈ |( M ) = (Q )) | | if e 0 if | . vi

(6.22b)

and hence the intensity is

6.4 Intensity Distribution in the Vibrational Bands of the Electronic Spectra …

175

Fig. 6.5 Energy level diagram for the explanation of the Franck–Condon principle from the standpoint of quantum mechanics. Simple harmonic oscillator type wave functions of different vibrational states of the two electronic states (E 0 and E 1 ) are shown by shaded curves. The upward and downward vertical arrows correspond to the most favoured transitions for absorption and emission, respectively

| \ |⟨ | \ |2 vi | v f is called the vibrational overlap factor, and | vi | v f | is called the Franck-Condon (FC) factor on which the intensity of the band v f ↔ vi depends. So the vibrational overlap factor (and hence the FC factor) plays vital roles in determining the intensities of the vibrational bands of the electronic spectra. This is demonstrated in Fig. 6.5. Wave functions of different vibrational states of the two electronic states (E 0 and E 1 ) are shown by shaded curves. These curves are assumed to correspond to the wave functions of simple harmonic oscillator. For such wave functions, other than the lowest state, all other vibrational states have two broad and large maxima (or minima) near their classical turning points. In the region between the classical turning points, there are other smaller maxima (or minima). The heights of these intermediate maxima (or minima) decrease with the increase of the vibrational quantum number. However for the lowest vibrational state, there exist only one maximum at the equilibrium position. So in evaluating the overlap factor in a qualitative manner, we can disregard the contributions of these intermediate maxima (or minima), and the chief contribution comes from the peaks near the classical turning points. So for absorption at normal temperature (i.e. transitions originating from the level v'' = 0), maximum band intensity occurs at such a v' -value for which there is a classical turning point at or near to r ' = r e '' (i.e. nearly vertically above the minimum of the lower potential curve). So this v' -progression exhibits only one maximum. Similarly in the case of emission, originating from a level (v' /= 0), the overlap (hence FC) factor is maximum in magnitude for that v'' -state for which the large and broad peak near any of its classical turning points lie vertically below any of the similar peaks of the originating (v' ) state. Since for each state (v' /= 0), there ⟨

176

6 Electronic Spectra of Diatomic Molecules

are two broad maximum or minimum near the classical turning points, two intensity maxima are observed in the v'' -progression. Again since there is only one peak near the equilibrium position (r e ' ) for the state v' = 0, the nature of the v'' -progression originating from this v' -level is same as that of absorption discussed above. Thus, we arrive at the same conclusion by quantum mechanical approach as that of the semiclassical one.

6.5 Quantum Numbers of Electronic States in Diatomic Molecules In atomic spectra, several quantum numbers are necessary to specify a particular electronic state. They are principal quantum number (n), orbital angular quantum number (l), electron and nuclear spin quantum numbers (s and I), or some combination of these. In addition to these, several other quantum numbers are necessary to specify the states when the atom is placed in electric or magnetic field. In atoms, the electrons move in spherically symmetric Coulomb field of the nucleus, but in the case of a diatomic molecule, there exist an electric field along the internuclear axis of the molecule. So to specify the electronic states of such a molecule, different sets of quantum numbers are necessary. (a) Orbital angular momentum In a diatomic (or in any linear polyatomic) molecule, there exists an electric field along the internuclear axis (z) of the molecule. So the total (orbital) angular momentum → loses its meaning, but its component along the internuclear axis (Ʌ) → remains ( L) constant as L→ precesses about the said axis. If E z is the electric field along the axis (z), the interaction energy is proportional to E z |Ʌ|, where the quantum number Ʌ can take up values 0, ±1, ±2, ±3, . . . , ±L. . So the levels with |Ʌ| /= 0 are all doubly degenerate, whereas the level Ʌ = 0 is non-degenerate. Thus, the overall effect of this interaction is splitting of the level into (L + 1) components. The levels with |Ʌ| = 0, 1, 2, 3, 4, … are, respectively, designated as ∑, ∏, Δ, Φ, ..., etc. These are the analogous classifications to those employed in atoms, where they are termed as S, P, D, F, … states when L = 0, 1, 2, 3, 4,…. (b) Spin angular momentum → of the electrons precesses about the magnetic Total spin angular momentum ( S) field along the internuclear axis produced by the orbital angular momentum (for states other than ∑). So the component of S→ along z-axis is conserved, and the corresponding quantum number is designated by ∑ (which should not be confused → ∑ can take up values –S, –S + 1, –S with the electronic state ∑). For a particular S, + 2,…,S–2, S–1, S.

6.5 Quantum Numbers of Electronic States in Diatomic Molecules

177

(c) Total angular momentum → obtained The component of total angular momentum along the internuclear axis (Ω), → and ∑, → is conserved. So Ω is a good quantum number by combining Ʌ Ω=Ʌ+∑ So, for the state (Ʌ /= 0), the associated quantum number Ω can take up 2S + 1 values: Ʌ–S, Ʌ–(S − 1), Ʌ–(S−2),…, Ʌ + (S–1), Ʌ + S. 2S + 1 is called the multiplicity of the state. So full multiplicity is possible only for the sate (Ʌ /= 0). The electronic state of a diatomic molecule is designated by (2S+1) ɅΩ , analogous to 2S+1 LJ , in the case of an atom. Thus, the 3 Δ level splits into three states as shown below: 3

Δ3

3

Δ ⇒ 3 Δ2

3

Δ1 .

This splitting arises due to spin–orbit interaction. So the multiplet splitting of an electronic state of term value (T 0 ) is given approximately by Te = T0 + AɅ ∑

(6.23)

Here, T 0 is the term value in the absence of the spin–orbit interaction. A is known as the spin–orbit interaction constant. For example, the Ω-values of the split levels of the 4 ∏ state are: 5/2, 3/2, 1/2 and −1/2. Note that Ω can take up negative values, and state with Ω = Ʌ + ∑ is degenerate with the state −Ω = −Ʌ − ∑. However, the states with same Ω formed by different combination of Ʌ, ∑ are not degenerate. So in the above example of 4 ∏—state, the levels with the set of (Ʌ, ∑) values(3/2, − 1) and (−3/2, 1) are degenerate, but they are not degenerate with another degenerate level(1/2, −1) or (−1/2, 1).When the spin–orbit interaction constant (A) is positive, the multiplet is called regular. For such cases, the energy of the sublevel increases with the increase of Ω-value. Similarly for negative values of A, the energy of the sublevel increases with the decrease of Ω-value. Such multiplets are called inverted.

6.5.1 Coupling of Angular Momenta So far we have considered various types of electronic angular momenta of the diatomic molecules. However, the rotation and the rotational energy of the molecule may be influenced by the electronic motion. So in this section, we shall consider various cases arising from coupling of the angular momentums of the electrons and

178

6 Electronic Spectra of Diatomic Molecules

the nuclei in the molecule. Here, we shall disregard the contribution of the nuclear spin. In a diatomic molecule, there exists an electrostatic field along the molecular axis. This field interacts with the electric dipole moment and hence causes the orbital angular momentum of the electrons to precess about this axis. As a result of this precession, a magnetic field is set along the axis about which the electronic spin angular momentum precess. In addition to this, a magnetic field is also generated due to rotation of the entire molecule and acts in the direction of the nuclear (rotational) angular momentum, i.e. perpendicular to the axis of the molecule. For the cases where Ʌ > 0 and S > 0, the interactions between the nuclear momentum and orbital and spin angular momenta of the electrons are magnetic in nature. Hund was the first to show how these various sources of angular momenta could be coupled together to form a resultant angular momentum. Different cases of these coupling are discussed below. Hund’s case (a) This applies to diatomic molecules where internuclear distance is small enough to produce a strong electric field along the axis. This prevents the angular momenta L→ and S→ to couple directly. These two vectors precess independently about the → and ∑ → which are internuclear axis and produce the component angular momenta Ʌ → → + ∑. → constants of motion. These two vectors then form the resultant vector Ω = Ʌ → → Finally, a weak interaction between the angular momentum of rotation ( N ) and (Ω) forms a resultant angular momentum ( J→), where the quantum number J can take up values Ω, Ω + 1, Ω + 2, Ω + 3,….The precession of various vectors is shown in Fig. 6.6a. The rotational energy is given by

E rot = Bv ch [ J ( J + 1) − Ω2 ]

(6.24)

where Bv is the rotational constant of the vibrational level (v). The Hund’s coupling case (a) is applicable when Ʌ, S > 0 (i.e. for non-singlet state). The ∑ (singlet)-state with Ʌ = 0 comes under the purview of case (b) which is discussed below. Hund’s case (b) When Ʌ = 0 and S /= 0, the spin vector is not coupled with the internuclear axis. Sometimes also for light molecules, this thing may happen even when Ʌ /= 0 because → is not defined. the magnetic field along the internuclear axis is weak. In such case, Ω → form a resultant vector K→ , where the quantum Here, nuclear rotation N→ and Ʌ number K can take up values. K = Ʌ, Ʌ + 1, Ʌ + 2, Ʌ + 3, . . . For ∑ (i.e. Ʌ = 0)-states, therefore, K can take up all positive integral values including zero. The energy of each level, designated by the quantum number K, is given by

6.5 Quantum Numbers of Electronic States in Diatomic Molecules

179

Fig. 6.6 a Hund’s case (a) of coupling and b the energy levels of the 2 ∏ and 3 Δ states. The dotted lines indicate missing levels

Er ot = Bv ch [ K (K + 1) − Ʌ2 ]

(6.25)

Finally, K→ and S→ form the resultant angular momentum J→ where the quantum number can take up values K = K + S, K + S − 1, K + S − 2, K + S − 3, . . . , |K − S| Thus except for the case K < S, J can take up 2S + 1 values i.e. each level specified by a given K splits into 2S + 1 (multiplicity) components. Here, the coupling between ← K→ and S is magnetic in nature. Because the rotation produces a magnetic field ← − along K→ and its interaction with the magnetic moment along the spin S generates ←

the resultant angular momentum J→. But the precessions of K→ and S→ about J are → about K→ . much slower than the nutation of the figure axis, i.e. Ʌ It can be shown that the term values of the rotational levels of the state 2 ∑ are given by

180

6 Electronic Spectra of Diatomic Molecules

F1 (K ) = Bv K (K + 1) +

1 1 γ K , for J = K + 2 2

and

(6.26a)

1 1 F2 (K ) = Bv K (K + 1) − γ (K + 1), for J = K − 2 2 where γ is generally, but not necessarily, positive and is very small in magnitude with respect to the rotational constant Bv . For the triplet state 3 ∑, the multiplet components are F1 (K ) = Bv K (K + 1) + (2K + 3)Bv − λ ? + (2K + 3)2 Bv2 + λ2 − 2λBv + γ (K + 1), for J = K + 1 F2 (K ) = Bv K (K + 1), for J = K F1 (K ) = Bv K (K + 1) − (2K − 1)Bv − λ ? + (2K − 1)2 Bv2 + λ2 − 2λBv − γ K , for J = K − 1

(6.26b)

So the multiplet splitting of the singlet and the triplet states is different. Naturally for the singlet state (S = 0), there is no distinction between case (a) and case (b). For such case, Ω = Ʌ and K = J. The precessional motion of the angular momenta and the splitting of the levels, 2 ∑ and 3 ∑, under Hund’s coupling case (b) is shown in Fig. 6.7. Amongst the five cases of Hund’s coupling, cases (a) and (b) are the most important and are mostly used to explain the observed splitting. Hund’s cases (c), (d) and (e). When the internuclear distance is sufficiently large, the electric field along the internuclear axis may not be adequate to couple the orbital and spin angular momenta with the axis. In such cases, specially for certain heavy molecules, the orbital and spin angular momenta form a resultant J→a = L + S→ which precesses about the → along it (where the quantum number Ω internuclear axis to form a component Ω → couples with can take up values J a , J a −1, J a −2, J a −3,…1/2 or 0.). Finally, this Ω nuclear rotation N→ to form a resultant angular momentum J→. The rotational energy is given by E rot = Bv ch [ J (J + 1) − Ω2 ]

(6.27)

This corresponds to Hund’s coupling case (c) and is shown in Fig. 6.8a. In Hund’s coupling (d), the interaction between L→ and the internuclear axis is − → weak. L is coupled with nuclear rotation N→ to form a resultant vector K→ , where the quantum number K can take up values: K = L + N, L + N−1, L + N−2, L + N−3,…. Finally, this K→ interacts with S→ to form the resultant J→. But most of the time they are so weak that formation of J→ is neglected. The energy levels are

6.5 Quantum Numbers of Electronic States in Diatomic Molecules

Fig. 6.7 a Hund’s case (b) and b splitting of 2 ∑ and

Fig. 6.8 Hund’s coupling cases (c), (d) and (e)

3∑

states

181

182

6 Electronic Spectra of Diatomic Molecules

Er ot = Bv ch [ N (N + 1) ]

(6.28)

where N can take up the positive integral values including zero. In Hund’s coupling case (e), the coupling between L→ and S→ is so strong that they form a resultant J→a as in the case (c). Finally, this J→a is coupled with the nuclear r otation N→ to form the resultant vector J→. Practically, no case is found which comes under this category. Lamda doubling (Ʌ doubling) Hund’s coupling cases are idealized cases, because some appreciable interactions amongst various angular momenta have not been considered in these idealized cases. In Hund’s coupling cases (a) and (b), the interaction between nuclear rotation N→ and electronic angular momentum L→ has been neglected. It is found that for larger speed of rotation, this interaction becomes significant and splits each J-level into two sublevels (if Ʌ /= 0). The splitting increases with the increase of J-value. This is called Ʌ doubling, because it removes Ʌ degeneracy for states with nonzero Ʌ. How does this thing occur? Each state with Ʌ /= 0 is doubly degenerate, and these two components have opposite parity. Strictly speaking, this degeneracy is maintained for non-rotating molecule (J = 0) for states with Ω > 0 [Hund’s case (a)]. The rotational levels (J /= 0) of the degenerate electronic state are mixed with the corresponding levels of the nearby ∑-state. The ∑-state is non-degenerate, so it can interact only with that component of the degenerate state which has the same parity with its own. Thus, the degeneracy is lifted, and it leads to Ʌ doubling. The amount of splitting increases with the increase of J–value. It can be shown that Ʌ doubling effect is largest for molecules in the ∏–electronic states. If the reader is interested, he/she can see appendix for further details.

6.5.2 Selection Rules General Rules For absorption and emission, the general selection rules for transitions between various electronic and rotational levels are ⎫ ΔJ = 0, ±1, J = 0 ←+→ J = 0;⎪ ⎪ ⎪ ⎪ ⎪ + ←+→ −, + ↔ +, − ↔ −; ⎪ ⎬ s ←+→ a, s ↔ s, a ↔ a; ⎪ ⎪ ⎪ g ←+→ g, u ←+→ u, g ↔ u . ⎪ ⎪ ⎪ ⎭

(6.29)

6.5 Quantum Numbers of Electronic States in Diatomic Molecules

Δ Ʌ = 0, ± 1.

183

(6.30)

[Note that + and – signs correspond to symmetric and antisymmetric behaviour of the wave function with respect to reflection on a plane of symmetry. Similarly, s and a correspond to symmetric and antisymmetric character with respect to exchange of two identical nuclei, and g and u correspond to symmetrical and antisymmetrical property with respect to inversion.] The selection rule (6.30) thus predicts that the following transitions are allowed.: ∑ ↔ ∑, ∑ ↔ ∏, ∏ ↔ ∏, ∏ ↔ Δ etc., but the following transitions are not: ∑ ←+→ Δ, ∏ ←+→ Φ, ∑ ←+→ Φ. Also from (6.29) and (6.30), we see that + + − + ∑ ↔ + ∑, − ∑ ↔ − ∑, + ∑g ↔ ∑ ∑u , − ∑ g ↔ ∑− ↔ ∑ ∑u+, ∑∑ ↔ ∏,+ ∑ + − ←+→ and ∏, etc. are all allowed transitions but g ←+→ g are not The spin selection rule is ΔS = 0.

(6.31)

Δ∑ = 0.

(6.32)

Selection rules for cases (a)

This means 2 ∏1/2 ↔ 2 Δ3/2 , 2 ∏1/2 ↔ 2 ∏1/2 , 2 ∏3/2 ↔ 2 Δ5/2 transitions are allowed, but the following ones are forbidden: 2 ∏1/2 ←/=→ 2 ∏3/2 , 2 ∏3/2 ←/=→ 2 Δ3/2 , etc. Again ΔΩ = 0, ±1.

(6.33) '

''

Δ J = 0 is forbidden for Ω = Ω = 0

(6.33a)

Apart from these selection rules, certain rules can be framed from the calculation of band intensity. A transition with ΔΩ /= 0 is accompanied by a strong Q-branch, but for ΔΩ = 0, the Q-branch is weak in intensity, and the intensity decreases with ' '' the ∑ increase ∑ of J-value. In addition if Ω = Ω = 0, no Q-branch is observed (as in ↔ transition). Selection rules for cases (b) Here ΔK = 0, ±1 (ΔK = 0 forbidden for Ʌ' = Ʌ'' = 0 i.e. for

∑

(6.34) ↔

∑

transition).

184

6 Electronic Spectra of Diatomic Molecules

For ΔɅ = 0, the Q-branch (ΔK = 0) decreases very rapidly in intensity with increasing K, and therefore, very often are not observed. Similarly in branches with ΔJ /= ΔK, the intensity falls rapidly with increasing K. These are called satellite bands, because they lie very close to those with ΔJ = ΔK. The intensities of the satellite bands are small in compared to the main branches as long as case (b) is applied to both the states. Selection rules for cases (c) Here ⎫ ΔΩ = 0, ±1 ⎬ ΔJ = 0, ±1 )⎭ is forbidden for Ω' = Ω'' = 0

(6.35)

Selection rules for cases (d) ΔK = 0, ±1 ΔL = 0, ±1 (The coupling between the electronic motion and nuclear

⎫ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎭ rotation is so weak that practically no change in N can occur)

(6.36)

6.6 Determination of Heat of Dissociation of a Diatomic Molecule from the Observed Electronic Spectra We know that the potential energy of an electronic state of a diatomic molecule is best represented by the Morse potential [ ]2 V (r ) = De 1 − e−β(r − re )

(6.37)

where De is the dissociation energy with respect to the minimum of the potential energy curve and β is a small positive constant. [Note that the dissociation energy measured from the lowest vibrational state of the molecule is represented by D0 . Experimentally, this entity is first determined, then De is determined from the knowledge of the zero-point energy]. Dissociation energy (D00 ) is the energy required to dissociate the molecule from the lowest rotational level of the lowest vibrational state of the ground electronic state into two normal atoms. This is very often defined as the heat of dissociation

6.6 Determination of Heat of Dissociation of a Diatomic Molecule …

185

ΔH00 arising from the chemical reaction AB → A + B occurring at 0° C under ideal gas condition. Dissociation energy can be determined from the vibrational structure of the experimentally observed electronic absorption spectrum of a diatomicmolecule (Fig. 6.9). The term value of a vibrational level (under anharmonic oscillator model) is given by Tv = ν e (v + 1/2) − ν e xe (v + 1/2)2

(6.38)

At normal temperature, most of the molecules are accumulated in the lowest vibrational state v'' = 0, and the transitions in the absorption occur from this states to different vibrational levels (v' ) of the excited electronic state giving rise to a v' progression. The wavenumber difference between the term values of the successive bands in this progression is given by ΔTv' +1/2 = ν v' +1 − ν v' = Tv' +1 − Tv' = ν e' − 2ν e' xe' (v' + 1)

(6.39)

Where v v' = T0'' v' − T0'' 0' So if ΔT v' +1/2 is plotted against v' , a line with negative gradient is expected. But, in fact, a nonlinear curve is observed as shown in Fig. 6.10.This may be the result of non-inclusion of the higher-order anharmonicity terms in Eq. (6.39) and may also be due to some other reasons. Observed spectra exhibit a finite number of bands, and ΔT v' +1/2 are plotted for a finite number of v' Fig. 6.9 Potential energy diagram for the determination of dissociation energy

186

6 Electronic Spectra of Diatomic Molecules

values (say v' = o to v' max ). The transition v'' = 0 → v' = v' max , in general, does not correspond to dissociation. So Birge and Sponer suggested to extrapolate this curve up to v' = v' c where it cuts the v' -axis, which corresponds to dissociation of the molecule in its upper electronic state (the dashed portion in Fig. 6.10). I ex (in Fig. 6.9) is the wavenumber difference between the starting of the continuums of the two electronic states of the molecule. So I ex corresponds to the excitation wavenumber of one (or rarely both) of the atoms produced on dissociation. So, if A is the area under ∑v ' this curve (Fig. 6.10) in the first quadrant, then 0c ΔTv' +1/2 = A. So we have ν 0' 0'' + A = D01 = D00 + Iex

(6.40)

From the experiment, ν 0' 0'' and A are obtained, and I ex is available from the previously determined excitation wavenumbers of the atoms. So knowing all the other entities in Eq. (6.40), the dissociation energy D00 can be determined. In some cases, for example, in iodine molecule, the excited state potential minimum is so much shifted with respect to that of the ground state (i.e. re' ∼ re'' is so large), ' that not only the 0–0 band, but some first few members of the v -progression are too weak to be observed. If v' 1st be the v' -value of the first observed band of the ' v' -progression, then ν 0' 0'' is to be replaced by ν v1st 0'' in Eq. (6.40) and A becomes ∑vc' equal to v' ΔTv' +1/2 . 1st Dissociation energy may also be determined by emission (fluorescence) method. A molecule after absorbing a radiation gets excited into the continuum of the upper electronic state and then gets dissociated into a normal atom and another in its excited state. If the excited state of the latter atom is not a metastable state, then photo dissociation is accompanied by fluorescence emission from the excited atom. By varying the frequency of the exciting radiation, lowest frequency causing such atomic fluorescence is determined. From the frequency difference between this minimum energy Fig. 6.10 Birge–Sponer extrapolation to determine the dissociation energy

6.7 Predissociation

187

exciting radiations which yield this atomic fluorescence and the atomic fluorescence line, at least an upper limit of the dissociation energy of the molecule can be obtained. This method has been applied to the halides of the alkali metals. To demonstrate this, consider the case of sodium iodide. It is a polar molecule. After absorbing a radiation of energy chν, the molecule dissociates into a normal iodine atom and an excited sodium atom (in the 32 P-state). NaI + chν → (NaI )∗ → Na∗ (32 P) + I (52 P3/2 ) Na∗ (32 P) → Na∗ (32 S) + chν D The excited sodium atom falls to the ground state (32 S) by emitting the D-line radiation of wavenumber ν D . If ν is the minimum frequency observed to yield luminescence from the photo dissociated excited atom, the dissociation energy is given by D00 = ν − ν D

(6.41)

The value of dissociation energy thus obtained experimentally may not be the true dissociation energy, but at least an upper limit of it. In fact, the values thus obtained for the alkali halide molecules are very near to those obtained by other methods.

6.7 Predissociation Sometimes, it is found that in the v' -progression, the rotational fine structure is clear and distinct for high and low values of the vibrational quantum number (v' ), but it is either blurred or completely continuous for a band with a particular v' -value and sometimes also above this band. Appearance of such kind of diffuse structure or complete continuum in the region well below the actual dissociation limit of the molecule is called predissociation. This occurs if the excited state (having a definite minimum, i.e. a bound state) either intersects or having another state in the closed vicinity with no minimum (repulsive or dissociative state). This is illustrated in the Fig. 6.11. The bound state A intersects a dissociative state B at the point P (of the curve LPM) which is the right classical turning point of a vibrational level of the state A. By absorption whenever the molecule goes to this level, there is a possibility of crossover from the state A to B. This type of transfer is called radiationless transition (because the energy remains unchanged). The time period of molecular vibration is ~10–14 s, whereas the rotational time period is ~10–10 s. Hence, a molecule may vibrate several times before completing a single rotation. Thus during the transfer, the molecule no longer remains in the discrete rotational quantized state, and hence, the rotational fine structure becomes diffuse and so also becomes the corresponding vibrational band. However, other members of the progression above and below this level (whose right

188

6 Electronic Spectra of Diatomic Molecules

Fig. 6.11 a Radiationless transition occurs at the point P (of LPM) where the bound state (A) intersects the non-bound state (B) leading to predissociation. b illustration for the appearance of predissociation in the vibrational structure of electronic spectra

turning point is P) exhibit sharp rotational fine structure. Such kind of predissociation was first observed in the electronic absorption spectra of S 2 .

6.8 Quantum Theory of Valence How the atoms are held together in a molecule was a basic question to the chemist for a long time. This mechanism was understood after the advent of quantum mechanics. Two methods were developed by quantum mechanics both of which successfully solved the problem by describing the electronic structure of molecules. They are

6.8 Quantum Theory of Valence

189

(a) Molecular orbital method and (b) Valence bond method. Molecular orbital method was developed by Hund, Mulliken and Hückel and the valence bond method by Heitler and London. In atoms, the electrons are present in different atomic orbitals. Likewise, in the molecular orbital method, the electrons in molecules are supposed to be distributed in different molecular orbitals. So the main problem lies in the construction of molecular orbitals. There are several ways of doing so. If linear combinations of atomic orbitals of all the atoms in the molecule are used to form molecular orbitals, they are found to be efficient for calculation. For a localized bond, the molecular orbital is localized between the two constituent atoms, and for a non-localized bond, the molecular orbital is polycentric and extended over a number of nuclei. Once the molecular orbitals are formed, the electrons are placed in these orbitals just like atoms following Pauli’s exclusion principle. In the valence bond method, the atoms with their valence electrons come closer to each other, and their valence electrons are allowed to interact to form the bonds and hence the molecule. This method was successfully applied by Heitler and London to hydrogen molecule in 1927. Later, this idea was extended by Slater and Pauling to solve the problem of electronic structure of complex molecules.

6.8.1 Hydrogen Molecule Ion In the theory of electronic structure of molecules, the simplest molecule, hydrogen molecule ion (H2 + ), plays the same role as hydrogen atom plays in the study of electronic structure of complex atoms. So at first, we shall discuss the problem of this simplest molecule through the knowledge of molecular orbitals. In this molecule, the lone electron is supposed to be moving in the potential field of the two nuclei (i.e. protons), designated by a and b. Then, the Schrödinger equation for the electronic motion becomes ] [ 1 1 1 1 ψ = EΨ (6.42) H Ψ = − ∇2 − − + 2 ra rb R Here, r a and r b are the distance of the electron from the nuclei a and b, respectively, and R is the distance between the two nuclei. For large distance (R), the normal state of the system is an atomic hydrogen ion (proton) and a hydrogen atom in the ground state (1 s), the electron belonging to any one of the nuclei. So we can construct a reasonable molecular orbital Ψ taking a linear combination of the two 1 s-orbitals of the normal hydrogen atoms:

190

6 Electronic Spectra of Diatomic Molecules

Ψ = ca ψa + cb ψb = ca (1s)a + cb (1s)b

(6.43a)

where (in atomic units) 1 ψa = (1s)a = √ e−ra π

(6.43b)

1 ψb = (1s)b = √ e−rb π

(6.43c)

and

We now apply the variation principle to determine the constants c and d and also the energy E after normalizing the wave function Ψ. The secular equation thus obtained is | | | Haa − E Hab − ES | | | (6.44) | Hab − ES Hbb − E | = 0 where ( Haa =

(

( ψ ∗a H ψa dτ ,

(

Hbb =

ψ ∗b H ψb dτ ;

Hab = ψ ∗a H ψb dτ = ψ ∗b H ψa dτ = Hba , and ( ( S = ψ ∗a ψb dτ = ψ ∗b ψa dτ = overlap integral.

(6.45)

The solution of this determinantal equation is E1 =

Haa + Hab 1+S

and E 2 =

Haa − Hab 1−S

(6.46)

(since both the atomic orbitals are 1 s, so H aa = H bb ). The coefficients, c and d, satisfy the following simultaneous equations (Haa − E) ca + (Hab − ES) cb = 0 (Hab − ES) ca + (Hbb − ES) cb = 0

(6.47)

Also from the normalization condition ⟩Ψ | Ψ⟨ of the wave function Ψ, we get ca2 + cb2 + 2ca cb S = 1 Thus from Eqs. (6.47) and (6.48), we get

(6.48)

6.8 Quantum Theory of Valence

191

/ ca = cb

=

and ca = −cb

1 for E 1 2(1 + S) / 1 = for E 2 2(1 − S)

(6.49)

Thus, the wave functions and their energies are ψa + ψb Haa + Hab with E 1 = Ψ1 = √ 1 + S 2(1 + S) and ψa − ψb Haa − Hab with E 2 = Ψ2 = √ 1 − S 2(1 − S)

(6.50)

If E H is the energy of the hydrogen atom, then the matrix elements of the energy become 1 1 − εaa , Hbb = E H + R R ) ( 1 S − εab − εbb and Hab = E H + R ( ( ψb∗ ψb ψa∗ ψa where εaa = dτ = dτ = εbb rb ra ( ( ψa∗ ψb ψa∗ ψb and εab = dτ = = dτ etc. rb ra

Haa = E H +

(6.51)

So the energies are E1 = E H +

εaa + εab 1 − R 1 + S

and E2 = E H

(6.52) εaa − εab 1 − + R 1 − S

It is convenient to use elliptical coordinates (Fig. 6.12) to evaluate the integrals S, εaa and εab : ra − rb ra + rb , ν = , φ; R = 2a R R R3 2 (μ − ν 2 ) dμ dν dφ; dτ = 8 1 ≤ μ ≤ ∝, −1 ≤ ν ≤ 1, 0 ≤ φ ≤ 2π.

μ =

(6.53)

192

6 Electronic Spectra of Diatomic Molecules

Fig. 6.12 Elliptic coordinates of the electron (e) in hydrogen molecule ion. (ϕ is measured from the xz-plane.)

For the overlap integral, we have ( S=

(∝

R3 = 8π

e−Rμ dμ

1

=

1 π (1

ψ ∗a ψb dτ =

3

(∝

3

(∝

R 4

[

˚

R3 2 (μ − ν 2 ) dμ dν dφ; 8 (2π 2 2 (μ − ν ) dν dφ

−1

e−(ra + rb )

0

e−Rμ 2μ2 −

]

2 dμ 3

1

R3 μ e dμ − 6 1 [ ] R2 1 + R + 3

R = 2

= e−R

2 −Rμ

(∝

e−Rμ dμ

1

(6.54)

Using the formula (∝

x n e−ax d x =

1

The other integrals are

n n ! e−a ∑ a k = An (a) an + 1 k = 0 k !

(say)

(6.54a)

6.8 Quantum Theory of Valence

εaa

1 = π =

(

R2 2

193

˚ −R (μ + ν) 3 R 1 2 e−2ra e (μ2 − ν 2 ) dμ dν dφ; dτ = rb π R (μ − ν) 8 ⎤ ⎡ ∝ ( (1 (∝ (1 ⎣ μ e−Rμ dμ e−Rν dν + e−Rμ dμ ν e−Rν dν ⎦ −1

1

−1

1

} 1 { = 1 − e−2 R (1 + R) R

(6.55a) and ˚ 1 2 e−(ra + rb ) e−Rμ R 3 2 (μ − ν 2 ) dμ dν dφ dτ = ra π R (μ + ν) 8 ˚ 1 R2 = e−Rμ (μ − ν) dμ dν dφ π 4 ⎤ ⎡ ∝ ( (1 (∝ (1 R2 ⎣ = μ e−Rμ dμ dν − e−Rμ dμ ν dν ⎦ 2

1 εab = π

(

−1

1

=e

−R

1

−1

(1 + R)

(6.55b)

[Note that in evaluating the integrals in ν, one may also use the relation (1

x n e−ax d x = (−1)n+1 An (−a) − An (a)]

(6.55c)

−1

With the substitution of Eqs. (6.54) and (6.55a–c) in Eq. (6.52), the total energy becomes { } 1 − e−2 R (1 + R) ± R e− R (1 + R) 1 { }] [ − E 1,2 = E H + (6.56) 2 R R 1 ± e−R 1 + R + R 3

where + and – signs correspond to the subscripts 1 and 2 on the left hand side. In order to test the goodness of the approximation, let us consider two extreme cases. For large R (i.e. R → ∞), H aa = E H (= −1/2, in atomic units), H ab = 0, so that E 1 = E 2 = E H, i.e. the energy of the hydrogen atom, as per expectation. For R → 0, S = 1 and H aa = E H + 1/R−1 = H ab . So, neglecting the nuclear repulsion 1/R, E ' 1 = E 1 (R → 0) = 3E H and E ' 2 = E 2 (R → 0) = 0. This is against the expectation that for R → 0, both the energies and the wave functions should approach the corresponding entities of the ground state of singly ionized helium atom. In fact, in the above ´ approximation, it is found that De = |E 1 (r e ) ~ E H (1 s)| = 1.76 eV and r e = 1.32 Å ´ The which differ from the corresponding experimental values 2.791 eV and 1.06Å.

194

6 Electronic Spectra of Diatomic Molecules

error may be due to the fact that for R → 0, the atomic orbitals no longer remain as those of pure hydrogenic atom. These errors can be reduced by choosing atomic orbitals of variable charge α, such as ( ψa =

α3 π

)1/2

e−α ra and ψb =

(

α3 π

)1/2

e−α rb

(6.57)

By varying the parameter α, the energy is minimized, and the results are found to ´ improve. The improved results are De = 2.25 eV and re = 1.06Å.With the inclusion of the 2p-orbitals, the results are found to improve further, and the above entities are ´ respectively, more closer to the observed values. The found to be 2.71 eV and 1.06Å, value of De can be further improved by including more and more hydrogen orbitals into the initial atomic wave functions. Figure 6.13 shows the variation of E 1,2 –E 1S with the internuclear distance (r). The curve for E 1 exhibits a minimum which leads to the formation of a stable molecular ion. So the corresponding orbital is called a bonding orbital. On the other hand, the energy state E 2 exhibits no minimum. This is a repulsive state, and no stable molecule can be formed in this state. So the corresponding orbital is called an antibonding orbital. Figure 6.13 also shows that |E 1 –E 1S | < |E 2 –E 1S |which means that depletion of energy of the bonding orbital is less than the increase of that of the antibonding orbital with respect to the energy of the system in the dissociation state (i.e. one hydrogen atom and a proton). This can also be understood from Eqs. 6.52 and 6.56. For simplicity, if the overlap integral (S) is taken to be negligible with \ respect to unity, the upshift of the state ψ2 is by an amount (1 + R) (e−R + e−2R R) with respect to the hydrogen atom and \ the proton whilst the state ψ1 becomes stable by an amount (1 + R) (e−R − e−2R R). Thus, the state ψ2 becomes more unstable than the stability of the state ψ1 .This can be shown in another way. The electron densities of the states ψ1 and ψ2 are (Fig. 6.14). ρ1 = ψ1∗ ψ1 =

1 (ψ 2 + ψb2 + 2ψa ψb ) 2 + 2S a

and

(6.58)

1 ρ2 = ψ2∗ ψ2 = (ψ 2 + ψb2 − 2ψa ψb ) 2 − 2S a So the electronic charge density between the two nuclei is more for the state ψ1 than ψ2 . At the midpoint between the two nuclei, they are ρ1 =

4 ψ2 2 + 2S a

and

ρ2 = 0.

(6.59)

The attraction between this accumulated charge and the two protons is considered as the source of stability of the state ψ1 . So more and more is the accumulation of electronic charge in between the two nuclei, more and more is the stability of the state.

6.8 Quantum Theory of Valence

195

Fig. 6.13 Binding energy of hydrogen ion as a function of internuclear distance for the lowest potential energy curves Fig. 6.14 Electron density in hydrogen molecule ion

196

6 Electronic Spectra of Diatomic Molecules

However, this charge density is maximum for a particular value of the internuclear distance, which corresponds to the equilibrium internuclear distance (r e ) of that electronic state. This distance, in general, varies from one electronic state to other.

6.8.2 Hydrogen Molecule For the hydrogen molecule (Fig. 6.15), the Schrödinger equation is ] [ 1 1 1 1 1 1 1 1 − − − + + Ψ H Ψ = − ∇12 − ∇22 − 2 2 ra1 ra2 rb1 rb2 r12 Rab = EΨ (6.60) A trial function for Ψ can be constructed from the molecular orbital of hydrogen molecule ion (6.50) using Pauli’s exclusion principle, as follows: ⎫ Ψ ∼ [ψa (1) + ψb (1)][ψa (2) + ψb (2)] ⎪ ⎪ ⎪ = [ψa (1)ψa (2) + ψb (1)ψb (2)] + [ ψa (1)ψb (2)⎬ ⎪ + ψa (2) ψb (1)] ⎪ ⎪ ⎭ = Ψion + Ψcov

(6.61)

where ψa , ψb are the atomic orbitals of the two atoms a, b in their 1S state. Ψion corresponds to that part of the molecular orbital where both the electrons lie on either of the two nuclei (protons), which means a negatively charged hydrogen atom H and a positively charged nucleus of hydrogen H+ (i.e. proton). So this state is called an ionic state. In the other part of the molecular orbital (Ψcov ), the contributions of both the nuclei are considered in forming a covalent bond. So( this part is called the Ψ∗H ψ dτ covalent form of the molecular orbital. The energy E = ( Ψ∗ ψ dτ of hydrogen molecule was approximately calculated by Hellmann using the wave function Ψ given by Eq. (6.61). He found the equilibrium internuclear distance (r e ) to be 1.6ao Fig. 6.15 Hydrogen molecule with a, b as the two nuclei and 1,2 as the two electrons

6.8 Quantum Theory of Valence

197

´ and dissociation energy to be 2.65 eV which are well different from the (~0.85Å) ´ and 4.72 eV. corresponding experimental values 1.4ao (~0.74Å) Since the electron affinity (~0.75 eV) of hydrogen atom is much less than its ionization potential (13.6 eV), so the ionic form is less probable, and this term can be dropped out. Accordingly, Heitler and London choose the covalent form of the wave function to explain the bonding characteristics of the molecule. Since both the forms ψa (1)ψb (2) and ψa (2)ψb (1) are equally strong for the choice of the wave function, let us proceed by choosing a normalized trial wave function as Ψ = c ψa (1)ψb (2) + d ψa (2)ψb (1) Applying variational method

( Ψ∗H ψ dτ ( , Ψ∗ ψ dτ

(∂E ∂c

= 0 =

∂E ∂d

)

(6.62)

to minimize the energy E

=

the simultaneous equations thus found are

c [⟩ψa (1)ψb (2)| H |ψa (1)ψb (2)⟨ − [ + d ⟩ψa (1)ψb (2)| H |ψa (2)ψb (1)⟨ [ c ⟩ψa (1)ψb (2)| H |ψa (2)ψb (1)⟨ − + d [⟩ψa (2)ψb (1)| H |ψa (2)ψb (1)⟨

⎫ ⎪ ⎪ ⎪ ] ⎪ 2 − E S = 0 (a)⎬ ] ⎪ E S2 ⎪ ⎪ ⎪ ⎭ − E] = 0 (b) E]

For non-trivial solutions, the secular determinant must be zero, i.e. | | | [⟩ψa (1)ψb (2)| H |ψa (1)ψb (2)⟨ − E] | | [ ]|| | 2 | + ⟩ψa (1)ψb (2)| H |ψa (2)ψb (1)⟨ − E S | |[ ] || = 0 | | ⟩ψa (1)ψb (2)| H |ψa (2)ψb (1)⟨ − E S 2 | | | | +[⟩ψa (2)ψb (1)| H |ψa (2)ψb (1)⟨ − E] |

(6.63)

(6.64)

The solutions of these equations are E± =

Hab,ab ± Hab,ba 1 ± S2

(6.65)

where Hab,ab = ⟩ψa (i )ψb ( j )| H |ψa (i )ψb ( j )⟨,

(6.66a)

Hab,ba = ⟩ψa (i )ψb ( j )| H |ψb (i )ψa ( j )⟨ and

(6.66b)

S = ⟩ψa (i ) | ψb (i )⟨ = ⟩ψb ( j ) | ψa ( j )⟨ = overlap factor (i, j = 1, 2)

(6.66c)

198

6 Electronic Spectra of Diatomic Molecules

With this set of solution, the constants c and d can be determined from the simultaneous Eq. (6.63) and the normalization condition ⟩Ψ | Ψ⟨ = 1. Thus, we get Ψ+ =

ψa (1)ψb (2) + ψa (2)ψb (1) √ 2 (1 + S 2 )

f or E = E +

(6.67a)

Ψ− =

ψa (1)ψb (2) − ψa (2)ψb (1) √ for E = E + 2 (1 − S 2 )

(6.67b)

and

H ab,ab and H ab,ba are determined as follows: 1 1 Hab,ab = ⟩ψa (1)ψb (2) | − ∇12 − ∇22 2 2 1 1 1 1 1 1 |ψa (1)ψb (2) ⟨ − − − − + + ra1 ra2 rb1 rb2 r12 Rab = 2E H + ⟩ψa (1)ψb (2)| 1 1 1 1 |ψa (1)ψb (2)⟨ − + + (6.68a) − ra2 rb1 r12 Rab and 1 1 Hab,ba = ⟩ψa (1)ψb (2) | − ∇12 − ∇22 2 2 1 1 1 1 1 1 |ψb (1)ψa (2) ⟨ − − − − + + ra1 ra2 rb1 rb2 r12 Rab = 2E H S 2 + ⟩ψa (1)ψb (2)| 1 1 1 1 |ψb (1)ψa (2)⟨ − + + (6.68b) − ra2 rb1 r12 Rab Thus, E ± = 2E H +

1 J ± K + Rab 1 ± S2

Here, J = ⟩ψa (1)ψb (2)| − +

1 ra2

1 |ψa (1)ψb (2)⟨ r12

−

1 rb1

(6.69)

6.8 Quantum Theory of Valence

199

= −2εaa + ⟩ψa (1)ψb (2)|

1 |ψa (1)ψb (2)⟨ r12

(6.70a)

and K = ⟩ψa (1)ψb (2)| − −

1 ra2

1 1 |ψb (1)ψa (2)⟨ + rb1 r12

= −2εab S + ⟩ψa (1)ψb (2)|

1 |ψb (1)ψa (2)⟨ r12

(6.70b)

where S, εaa and εab can be obtained from Eqs. (6.54) and (6.55a–c). The integral J is called Coulomb integral, and it represent the interaction between the charge densities |ψa (i )|2 and |ψb (2)|2 . Actually in the expression for J in Eq. (6.70a), the last integral gives the Coulombic repulsion between the electron clouds of the two atoms a and b. Likewise, the integral K is called the exchange integral. The results ´ and D = 3.14 eV. Though these values thus obtained were r e = 1.64ao (i.e. ~0.87Å) e are only slightly better than the results obtained from molecular orbital calculation, yet they are not near to the corresponding experimental values, however, this method is easier to handle than the molecular orbital method. It is found that K is negative, so E + < E which means Ψ + state lies below the Ψ state. It is also found that the greater part of the binding energy comes from the exchange term (about 85–90%), and this is shown in Fig. 6.16. One point to be noted here is that the exchange integral K is roughly proportional to the overlap factor S. So in order to have strong bonding, good amount of overlap of the two atomic orbitals (i.e. large S) is required. The results may be improved by adding an extra term λΨ ion (representing ionic bonding) to the Heitler London covalent form of wave function and then apply variational principle to determine the constant λ. Thus, the equilibrium internuclear ´ and 0.147 au distance and the dissociation energy are found to be 1.42a.u (0.75Å) (4.00 eV) which are closer to the corresponding experimental values compared to the results obtained from molecular orbital and valence bond (Heitler London) method. Since the Hamiltonian (6.60) does not contain any term depending on spins of the electrons, so the total wave function must be a product of the space dependent and the spin dependent parts. As in the case of two electron system, the spin part is consisted of a singlet and a triplet state: 1 χ00 (1, 2) = √ α(1)β(2) − α(2)β(1) = singlet state; 2 ⎫ χ11 (1, 2) = α(1)α(2) ⎪ ⎪ ⎪ ⎬ χ1−1 (1, 2) = β(1)β(2) = triplet state. ⎪ 1 ⎪ χ10 (1, 2) = √ α(1)β(2) + α(2)β(1) ⎪ ⎭ 2

(6.71)

200

6 Electronic Spectra of Diatomic Molecules

Fig. 6.16 Calculated Coulombic and total energy of hydrogen molecule as a function of internuclear distance

[| \] [| \] where α and β are the up | 21 21 and down | 21 − 21 spin functions of an electron. As the electron is a fermion, the total molecular wave functions will be antisymmetric with respect to the interchange of the two electrons. The total wave functions, thus obtained, are ) ∑g (singlet state) = Ψ+ · χ00 (1, 2) [ψa (1)ψb (2) + ψa (2)ψb (1)] √ = [α(1)β(2) − α(2)β(1)] 2 (1 + S 2 ) with M S = 0 (3 ) Ψ ∑u (triplet state) [ψa (1)ψb (2) − ψa (2)ψb (1)] √ = Ψ− · χ11 (1, 2) = α(1)α(2) 2(1 − S 2 ) with M S = +1 Ψ

(1

= Ψ− · χ1−1 (1, 2) = with M S = − 1,

[ψa (1)ψb (2) − ψa (2)ψb (1)] √ β(1)β(2) 2(1 − S 2 )

(6.71a)

(6.71b)

(6.71c)

6.9 Electronic Structure of Diatomic Molecules

[ ] [ψa (1)ψb (2) − ψa (2)ψb (1)] α(1)β(2) √ = Ψ− · χ10 (1, 2) = +α(2)β(1) 2 (1 − S 2 )

201

(6.71d)

with M S = 1. ( ) The lower state 1(∑g ,)having a potential minimum, is a singlet and a bound state. But the higher state 3 ∑u , having no potential minimum, is a triplet, and it is called a repulsive state. If the molecule is excited to this state, it will dissociate into two fragments of normal hydrogen atoms.

6.9 Electronic Structure of Diatomic Molecules In the above Sect. 6.8.1, we have seen how the molecular orbitals are formed from the linear combinations of two atomic orbitals (LCAO). In more complicated diatomic molecules, only those orbitals which correspond to outer shell electrons of the atoms are considered for the formation of molecular orbitals. These contributing electrons are called valence electrons.

6.9.1 Homonuclear Diatomic Molecules As in the case of hydrogen molecule ion, any two s–electrons give rise to a bonding (designated as nsσg ) and an antibonding orbital (nsσu *). It is known that the atomic orbitals of the s-electrons are spherically symmetric, and the sign of the wave function is either positive or negative everywhere. So when two such orbitals form a bonding molecular orbital (nsσg ), it is ellipsoidal in nature, and symmetric about the internuclear axis and the sign of the wave function is same everywhere. Building up of electronic charge between the nuclei acts as a sort of gum which holds the two nuclei together. Similarly when two ns-atomic orbitals are negatively combined (means the wave functions of the two orbitals are opposite in sign) to form an antibonding orbital (nsσu *), the electrons avoid the position between the two nuclei (the electronic charge density being zero at the midpoint). But instead, they prefer to concentrate around the nuclei in such a manner that the molecular orbital (Ψ MO ) becomes positive around one nucleus and negative around the other. The antibonding orbital has also an axial (cylindrical) symmetry as the bonding one. But in the case of the bonding one, Ψ MO is symmetric, and for the case of the antibonding orbital, Ψ MO is asymmetric with respect to inversion about the midpoint of the two nuclei. These are shown in Fig. 6.17a. The p-electrons of the atoms have three possible atomic orbitals. Each of them looks like a dumbbell, and the two lobes of this dumbbell are opposite in sign, the positive lobe is directed towards the positive direction of the corresponding Cartesian axis.

202

6 Electronic Spectra of Diatomic Molecules

a

b

c

Fig. 6.17 a Formation of bonding and antibonding orbitals from two 1s orbitals in a homonuclear diatomic molecule. b formation of bonding and antibonding orbitals from a 1s and a 2pz orbitals, z being the direction of the internuclear axis c formation of bonding and antibonding orbitals from two 2pz orbitals, z being the direction of the internuclear axis d formation of bonding and antibonding orbitals from two 2px/y orbitals, x/y being the directions perpendicular to the internuclear axis

6.9 Electronic Structure of Diatomic Molecules

203

d

Fig. 6.17 (continued)

Thus, these three dumbbells are oriented along the three Cartesian axes, just like the three components of a linear vector. (Remember that the pZ -orbital corresponds to the wave function Ψ nlm = Ψ n10 . But the wave functions Ψ nlm = Ψ n1±1 are of complex forms. The proper linear combinations of these two wave functions are taken to generate two real orbitals px and py having the dumbbell forms, oriented along the respective, x- and y-directions, mentioned above). When an s-atomic orbital of an atom approaches a p-atomic orbital along the z-direction (taken as the molecular axis), a bonding molecular orbital is formed when the charge clouds of the s-orbital and the overlapping lobe of the p-orbital have same sign, i.e. Ψ MO (spσ) = Ψ a (ns)– Ψ b (npz )], and the antibonding orbital is formed in the other way, i.e. Ψ MO (spσ*) = Ψa(ns) + Ψ(npz ) (Fig. 6.17b). As before, here also, there is an increase of electronic charge density in the case of the bonding orbital (σ) and depletion of charge density in the antibonding orbital(σ*) in the region between the two atoms. In fact, no molecular orbital is formed by the combination of s-and px/y atomic orbitals, since the overlap factor is zero. When two p-orbitals approach each other along the head on direction (z), bonding and antibonding molecular orbitals are formed by the linear combinations of atomic orbitals, Ψ MO (npz σg ) = Ψ a (npz )–Ψ b (npz ) and Ψ MO (npz σu *) = Ψ a (npz ) + Ψ b (npz ), respectively, where the subscripts ‘g’ and ‘u’ correspond to symmetric and antisymmetric nature of the molecular orbitals under inversion about the centre of inversion (midpoint of the bond) of the molecule. Both these molecular orbitals are axially symmetric. But when two p-atomic orbitals approach each other sidewise (i.e. either two px or two py -orbitals approach each other), bonding and antibonding orbitals are formed, each having a nodal plane passing through the axis of the molecule. For the bonding orbital, Ψ MO (npx/y πu ) = Ψ a (npx ) + Ψ b (npx ), and for the antibonding orbital, Ψ MO (npx πg *) = Ψ a (npx )—Ψ b (npx/y ). The bonding orbital is antisymmetric, and

204

6 Electronic Spectra of Diatomic Molecules

the antibonding orbital is symmetric with respect to inversion about the centre of inversion. Formation of various types of σ– and π–bonds from s- and p- atomic orbitals is shown in the Figs. 6.17a–d. For the same value of the total quantum number (n) of the atom, the molecular orbitals arising from the linear combination of the x and y components of the atomic p-orbitals are degenerate. The bonding formed by end on combination of atomic orbitals is called σ-bonding, and those formed by sideways combination is called π-bonding. Generally, the σ-bonding is stronger than the π-bonding. It may be noted that the relative energies of the molecular orbitals arising from 2p-atomic orbitals may change with the atomic number. For molecules lighter than O2 , the orbitals 2pπu lie above the 2pσg orbitals. However, for molecules heavier than O2 , the orbitals 2pπu lie below the 2pσig orbitals. Above the second row diatomic molecules, these arrangements are somewhat uncertain. For the determination of electronic configuration of molecules, the MOs are arranged in order of increasing energy as the atomic orbitals in the case of atoms, and the electrons are inserted in these orbitals one by one following Pauli’s exclusion principle. In order to know how the internuclear distance affects the energies of the MOs, it is useful to study the correlation diagram (Fig. 6.18) which gives an idea about the relative positions of the MOs as the internuclear distance changes. This diagram determines a one to one correspondence amongst the MOs for different internuclear distance, starting from the united atom (R = 0) to the separated atom limit (R = ∞) through the equilibrium configuration (R = Re ). In the Fig. 6.18, on the left side, various states of the united atoms and the possible molecular orbitals into which they can split are given. On the right-hand side, the various states of the separated atoms and the molecular orbitals formed from the linear combinations of these atomic orbitals are shown. In drawing this correlation diagram, three following points are to be kept in mind: 1. The component of the orbital angular momentum along the internuclear axis (Ʌ) is to be kept constant throughout the path of the internuclear distance (i.e. to follow the conservation principle of the component of orbital angular momentum along the internuclear axis). 2. The parity of the wave function (g or u) is preserved throughout the path (i.e. R = 0 to R = ∞). 3. Non-crossing rule (of Neumann-Wigner) must be obeyed, i.e. potential energy curves corresponding to orbitals having the same symmetry do not cross anywhere for any values of R from o to ∞. In determining the electronic configuration of a molecule, just like atoms, the electrons are placed in the above molecular orbitals (MOs) in order of increasing energies following Pauli’s exclusion principle. In the early days, following Lewis, it was thought that a normal chemical bond was formed by sharing two electrons with opposite spins between two atoms. Such a bond is known as a single bond with bond order one. Later, it was felt that a new definition was needed in order to define the bond order between two atoms in a complex molecule. Now, the bond order (B.O.) is defined as

6.9 Electronic Structure of Diatomic Molecules

205

Fig. 6.18 Correlation diagram between the united atom and separated atom states of homonuclear diatomic molecules. Note that the contributions of d-atomic orbitals shown here are not discussed above

[Number of electrons in the bonding orbitals− −Number of electrons in the antibonding orbitals] B.O = 2

(6.72)

H2 and He2 : Thus, we can say that the electronic configuration of hydrogen molecule is (1sσg )2 with B.O. 1. The same for normal helium molecule is (1sσg )2 (1sσu *)2

206

6 Electronic Spectra of Diatomic Molecules

with zero B.O. Thus, a stable normal helium molecule cannot be formed. In fact, the ground electronic state of the helium molecule (He2 ) is a repulsive state which means that whenever the possibility of forming such a state arises, it dissociates into its constituent normal He atoms. However, an excited electronic configuration of the molecule (say, (1sσg )2 (1sσu *)1 (2sσg )1 ) is possible whose B.O is 1. This state is a bound state. When a He2 * molecule in the above excited state decays to the ground state, it gives rise to an intense continuous radiation in the wavelength range of 60–100 nm which is used as a continuous ultraviolet source of radiation. Li2 : In lithium molecule, the electronic configuration is (1sσg )2 (1sσu *)2 (2sσg )2 . Two electrons in each of the bonding (1sσg ) and antibonding (1sσu *) orbitals lead to cancellation of bonding between the two Li atoms. The entire bonding comes from the two electrons in the remaining MO (2sσg )2 . The energy difference between the MO (1sσg ) or the MO (1sσu *) and the respective atomic orbital (1s) is not much. In fact, more and more the two atoms are heavy, less and less is the above energy difference which means, in heavier atoms, the core atomic orbitals have little contribution in the formation of molecule. Thus, the electronic configuration of Li2 molecule can be written as (KK)4 (2sσg )2 . In this way, we can determine the electronic configurations of different homonuclear diatomic molecules in their respective ground states, some of which are shown in Table 6.2 For B2 molecule, (2pσg ) and (2pπu ) molecular orbitals have nearly the same energy, so the valence electron configuration is (2pσg )1 (2pπu )1 which lead to the observed paramagnetism. Note that we have not included C2 molecule in this discussion, because bonding in molecules with carbon atoms has some novelty, and these have been discussed in Chap. 7. It should be noted that a molecular orbital can be labelled either by separated or united atom limits (shown in Fig. 6.18). For example, the valence electron of O2 is Table. 6.2 Electronic configuration of some diatomic molecules Molecule Electronic configuration

Bond Magnetic order behaviour

(H2 )1

(1s σg )1

1/2

Paramagnetic

(H2 )2

(1s σg )2

1

Diamagnetic

(He2 )3

(1s σg )2 (1s σu *)1

1/2

Paramagnetic

(He2 )4 (Li2 )6 (Be2 )8 (B2 )10

(1s σg )2 (1s σu *)2 (KK)4 (2sσg )2 (KK)4 (2sσg )2 (2sσu * )2 (KK)4 (2sσg )2 (2sσu * )2 [(2pσg )2 /(2pσg ) (2pπu )]

0

–

1

Diamagnetic

0

–

1

Paramagnetic (see text)

(C2 )12

(KK)4 (2sσg )2 (2sσu * )2 (2pσg )2 (2pπu )2

2

(See Chap. 7)

(N2 )14 (O2 )16 (F2 )18 (Ne2 )20

(KK)4 (2sσg )2 (2sσu * )2 (2pσg )2 (2pπu )4

3

Diamagnetic

(KK)4 (2sσg )2 (2sσu * )2 (2pπu )4 (2pσg )2 (2pπg * )2

2

Paramagnetic

(KK)4 (2sσg )2 (2sσu * )2 (2pπu )4 (2pσg )2 (2pπg * )4 1 (KK)4 (LL)16 [(2sσg )2 (2sσu * )2 (2pπu )4 (2pσg )2 (2pπg * )4 (2pσu *)2 ] 0

Diamagnetic –

6.9 Electronic Structure of Diatomic Molecules

207

labelled as 2pπg * in the separated atom limit and 3dπg * in the united atom limit. The antibonding orbitals are represented with asterisk marks in the superscript.

6.9.2 Heteronuclear Diatomic Molecules When the two atoms a and b are different, a molecular orbital formed out of the two atomic orbitals ψa and ψb is ΨMO = ca ψa + cb ψb

(6.73)

Unlike homo nuclear diatomic molecule ca /= cb . Using variation principle, constants ca and cb are determined that makes the energy minimum. The more unlike are the energies of the atomic orbitals ψa and ψb , more unlike are ca and cb . It can be shown that unless the energies of the respective molecular orbitals are fairly close to one another, no true molecular orbitals can be formed, and the electrons remain confine to their respective atomic orbitals. We shall discuss few examples. Example 1. Hydrogen fluoride (HF). Here, 1S- and 2S-orbital energies of fluorine atom are so different from the energy of the atomic orbital 1S of hydrogen atom that no true molecular orbitals can be formed out of these atomic orbitals. The energy of the 2P-atomic orbital of fluorine atom is close to the atomic orbital 1S of the hydrogen atom. So the atomic orbital 1SH will combine with the atomic orbital 2pZ of fluorine (z being the internuclear axis) to form bonding and antibonding σ-orbitals as shown in Fig. 6.19. The remaining two p-atomic orbitals (px and py ) will remain more or less unaffected in the molecule, and they are called non-bonding orbitals (nπx and nπy ). The electrons in these orbitals do not take part in the bonding phenomena. So the electronic configuration of HF molecule is ( )4 (HF) : (1S F )2 (2S F )2 (σ 2 pz )2 nπx nπ y and the bond order is one. The lowest energetic band is expected to arise from an n → σ* transition. Example 2. Carbon monoxide (CO) In this molecule, the situation is similar but more complicated. The energies of the 1S-orbitals of both the atoms are so different that practically, they do not take part in forming molecular orbitals. So these orbitals of the two atoms remain more or less unaffected in the molecule. Since no orbital of the carbon atom lies in the neighbourhood of the 2S atomic orbital of oxygen, the latter atomic orbital remains more or less unaffected in the molecule. The two electrons in this orbital (2SO ) remain as lone pair electrons on the oxygen atom. The energy of the 2pO -orbital on the oxygen atom is close to that of the 2sC- and 2pC orbitals on the carbon atom. This results in the formation of σ and σ* type molecular orbitals from (2pO )z and 2sC atomic orbitals (z-direction being along the internuclear

208

6 Electronic Spectra of Diatomic Molecules

Fig. 6.19 Electronic energy levels of HF molecule

axis) and π and π* type molecular orbitals from the combinations of the 2px - and 2py -orbitals of the two atoms. The atomic orbital (2pC )z remains unaffected and the two electrons in this orbital stay as two non-bonding electrons in the molecule (Fig. 6.20). Thus, the electronic configuration of the molecule is (CO) : (KK)4 (2SO)2 (σ )2 (π )4 (n)2 The bond order of this molecule is three, and the lowest energetic band arises from an n → π* transition.

Appendix

209

Fig. 6.20 Electronic energy levels of carbon monoxide (CO)

Appendix Lamda doubling (Ʌ doubling) If we consider the coupling of the perpendicular components of the orbital and spin angular momentums of the electrons directly with the nuclear rotation as a small perturbation in the Hund’s coupling case (a), the total angular momentum of the → and hence, the kinetic energy of nuclear molecule can be written as J→ = N→ + L→ + S, rotation of the diatomic molecule becomes ⎤ 2 2 + + S J (L ) ⊥ ⊥ 1 2 1⎣ )]⎦ [ ( N = = 2I 2I +Jz2 − 2 J→ · J→z + L→ ⊥ + S→⊥ ⎤ ⎡ 2 2 + + S J(J + 1) − Ω (L ) ⊥ ⊥ 1⎣ ) ( ⎦ = 2I −2 J→⊥ · L→ ⊥ + S→⊥ ⎡

E rot

(6.74)

210

6 Electronic Spectra of Diatomic Molecules

Taking the terms {−Ω2 + (L ⊥ + S⊥ )2 } with the electronic (vibronic) energy, the energy of nuclear rotation of the molecule becomes )] [ ( E rot = B J(J + 1) − 2 J→⊥ · L→ ⊥ + S→⊥ = BJ(J + 1) + H'

(6.75)

of which the first term corresponds to the unperturbed energy of rotation, and the second term, considered as a small perturbation arising due to interaction between the rotational and the perpendicular component of the electronic angular momentums, is ) ( (6.76) H ' = −2B J→⊥ · L→ ⊥ + S→⊥ (Here, we have considered B in energy unit and è = 1). The Eigen function/ket in Hund’s case (a) is |ψ⟨ = |J S∑ɅΩ⟨ . . The first-order correction to the energy is E (1) = H '

| )| ( 1 | | J S∑ɅΩ|2 J→⊥ · L→ ⊥ + S→⊥ |S∑ɅΩ 2I | | | (J L + J L ) | 1 + − | | − + = − J S∑ɅΩ| | J S∑ɅΩ = 0 | +(J− S+ + J+ S− )| 2I

=−

(6.77)

(considering the nuclear integration in space-fixed coordinate system and the electronic one in the molecule-fixed coordinate system and using the properties of angular momentum). So no first-order correction exists. We have to look for the second-order correction. It is found that for states with Ʌ > 1, even no second-order correction exists, and for these cases, further higher-order correction only lifts the Ʌ degeneracy. Here, we shall consider only the ∏–states (Ʌ = 1) and see how the Ʌ degeneracy is lifted in the second order for the singlet electronic states. Before going to this job, we shall discuss some properties of the energy eigen states. For a diatomic molecule, the parity (inversion) operator (P) is an important operator which inverts all the electronic and nuclear coordinates in a space fixed coordinate system. In Hund’s coupling case (a), the molecular wave function (eigen ket) is |ψ = |n J SɅ∑ , n being the other quantum specification of the electronic state than S, Ʌ and ∑, and the effect of the parity operation on it is P|n J SɅ∑ = (−1) J −S+sn |n J S − Ʌ − ∑

(6.78)

In fact, the parity operator itself is the product of two operations, C 2 (rotation about an axis perpendicular to the molecular axis by an angle 180°) and σv (reflection on a plane passing through the internuclear (say-z) axis and perpendicular to the C 2 axis). Here, sn is related to the other symmetry operation, the plane of reflection σ v , such

Appendix

211

that σv |n J SɅ∑ = (−1)sn |n J S − Ʌ − ∑

(6.78a)

where sn = 0 for states with Ʌ /= 0 and for the state with Ʌ = 0, it is 0 for ∑ + and 1 for ∑ − states. Moreover [P, H e ] = [σv , H e ] = 0, which means that P, σv and H e have simultaneous eigen functions. So the states with Ʌ /= 0 are doubly degenerate, and hence, we can form linear combinations of the two basis functions which are eigen kets of P (and hence of H e ), i.e. ] 1 [ |ψ⟨ = √ |n J SɅ∑⟨ + (−1)s |n J S − Ʌ − ∑⟨ 2

(6.79)

with eigen values (of the parity operator) (−1) J −S+sn +s . Here, s = 0 and s = 1 represent two different parities for given values of J, S and sn . Here, Ʌ ≥ 0, and ∑ ≥ 0 for Ʌ = 0. Since ∑ states (Ʌ = 0) are non-degenerate, they are either symmetric or antisymmetric with respect to σv, and they are represented by ∑ + and ∑ − , respectively. But all the other states with Ʌ ≥ 1 are doubly degenerate, and they have parity (−1) J −S+sn +s . Thus if we take |ψ as a ∏–state, the matrix element of H ' (6.76) in the representation of |ψ (6.79) is ⎡/

⎤ \' ' n J SɅ∑|H ' |n ' J SɅ ∑ ⎢ ⎥ / \ ⎢ ⎥ ⎢ +(−1)s ' n J SɅ∑|H ' |n ' S J − Ʌ' − ∑ ' ⎥ ⎢ ⎥ ⟨ \ 1 ' ' ⎢ ⎥ ψ|H |ψ = ⎢ / \ ⎥ ' 2⎢ s ' ' ' ⎥ ⎢ +(−1) n J S − Ʌ − ∑|H |n J SɅ ∑ ⎥ ⎣ / \⎦ ' ' ' +(−1)s+s n J S − Ʌ − ∑|H ' |n ' J S − Ʌ − ∑

(6.80)

⟨ \ Notice that the integral ψ|H ' |ψ ' vanishes if ψandψ ' belong to different parities. For diagonal elements in n and Ʌ, this integral becomes ⟨

\ ⟨ \ ψ|H ' |ψ ' = n J SɅ∑|H ' |n J SɅ∑ ' ⟨ \ + (−1)s n J SɅ∑|H ' |n J S − Ʌ − ∑ '

(6.81)

For states with Ʌ = 0, there of non-degeneracy) ⟨ is no s-dependent term (because \ and for states with Ʌ ≥ 1, n J Ʌ∑|H ' |n J − Ʌ − ∑ ' = 0 for the reason of different symmetries of the bra and ket vectors in the Ʌ-space. So there exists no first-order correction, and first-order perturbation does not lift the Ʌ-degeneracy. Now, we shall go to the second-order correction to the energy of the singlet state with Ʌ = 1 (1 ∏),

212

6 Electronic Spectra of Diatomic Molecules

E

(2)

=

|⟨ \| ∑ | ψ|H ' |ψ ' | 2 ψ/=ψ '

Eψ − Eψ '

(6.82)

For this state, the wave function (eigen ket) is |n J Ʌ in Hund’s case (a).It is found that the primed state of the matrix element is nonzero only for Ʌ' = 0. So the second-order correction is given by

(2) E∏

|⟨ \ | n J Ʌ|H ' |n ' J Ʌ' = Ʌ − 1 = 0 ⟨ \|2 1 ∑ + (−1)s n J − Ʌ|H ' |n ' J Ʌ' = Ʌ − 1 = 0 | = 2 n' En − En'

(6.83)

In determining the integral in (6.83), it is to be kept in mind that →j refers to spacefixed axis and L→ refers to molecule-fixed axis. Taking H ' from (6.76 and 6.77) and using the properties of angular momentum, we see that \ \ | | ( )' | | \ | | n J Ʌ| H ' |n ' J Ʌ − 1 = n J Ʌ|−2B J→⊥ | · L→ ⊥ + S→⊥ J Ʌ − 1 \ ⟨ = −B n J Ʌ|(J− L + + J+ L − )|n ' J Ʌ − 1 ⟨ \ = −B nɅ|L + |n ' Ʌ − 1 [(J + Ʌ)( J − Ʌ + 1)]1/2

⟨

(6.84)

Substituting this in Eq. (6.83) with Ʌ = 1, we get (2) E∏

|⟨ \| 2 ] ∑ | nɅ|B L + |n ' 0 | [ s = 1 + (−1) J (J + 1) En − En' n' =

] 1[ 1 + (−1)s q J (J + 1) 2

(6.85)

Since Ʌ' = Ʌ–1 = 0 state is non-degenerate with overall parity (–1)J (which includes the contribution of J), it will interact with that component of the Ʌ state having same parity. So the energy of one of the Ʌ = 1 state will change and that of the other one remains unchanged. So this second-order correction to the energy lifts the degeneracy of the Ʌ = 1 state, and this breaking of degeneracy is called Ʌ doubling. Note that although the Ʌ = 0 state is non-degenerate, there will be a second-order correction to the energy of this level, and proceeding in a similar way, it can be shown that this correction is |⟨ \| ∑ | nɅ|B L + |n ' Ʌ' = 1 | 2 (2) E∑ = 2 J ( J + 1) En − En' n' = q ∗ J (J + 1)

(6.86)

References and Suggested Reading

213

Note that in 6.85 and 6.86, q and q* can be looked upon as entities containing contributions of the electrons to the moment of inertia of the molecule from the classical viewpoint. For states with ∑ > 0 (non-singlet) and Ʌ > 1, the problem is too involved and complicated and is not discussed here.

References and Suggested Reading 1. B.H. Bransden, C.J. Joachin, Physics of Atoms and Molecules. (Pearson Education Ltd. 2003) 2. H. Eyring, J. Walter, G.E. Kimball, Quantum Chemistry (Wiley, NY, USA, 1944) 3. G. Aruldhas, Molecular Structure and Spectroscopy (Prentice Hall India Pvt. Ltd., New Delhi, 2001) 4. L. Veseth, Lecture Notes on the Spectroscopy of Diatomic Molecules. Department of Physics, University of Oslo, Norway. folk.uio.no/leifve/molspec.pdf. (For Ʌ–doubling) 5. J.M. Hollas, Modern Spectroscopy (Wiley, Chichester, UK, 2004) 6. B.P. Straughan, S. Walker, Spectroscopy, Vol. 3 (Chapman and Hall, London, UK, 1976)

Chapter 7

Electronic Spectra of Polyatomic Molecules

Abstract This chapter is concerned with the electronic spectra of polyatomic molecules. Hybridizations of the orbitals are presented. A short introduction to the free electron model related to molecular spectra is given. Molecular orbital method has been discussed in relation to some aromatic molecules. Whenever a molecule is excited to a higher energy state, it comes down to the ground state by various relaxation mechanisms. Among such processes, fluorescence and phosphorescence are very important. Related to these luminescence, various radiative and non-radiative mechanisms have been discussed extensively on the basis of vibronic and spin– orbit coupling. Polarization, life time and the characteristics of the spectra arising from π → π* and n → π* transitions have also been discussed. The theoretical backgrounds of different molecular interactions, such as charge transfer, hydrogen bonding, formation of excimer and exciplex, energy transfer and electron transfer, are given.

In polyatomic molecules, the electrons move in the field of several (say, N-) nuclei. Obviously, the problems for these molecules are more complex than diatomic molecules. As in the case of diatomic molecules, both valence bond and molecular orbital methods may be applied to study the electronic structure of polyatomic molecules. In the valence bond method, all the bonds are considered to be two centre bonds and each single bond is shared by two electrons with opposite spins. A polyatomic molecule is supposed to have a number of pairs of atoms each constituting a bond, and there exist several such combinations of bonds of the molecule, called resonance structures. The total wave function is expressed as a linear combination of wave functions of all such linearly independent structures each of which corresponds to a chemically meaningful structure, thus introducing the idea of delocalization of bonds. But in the molecular orbital method, the molecular orbitals are extended all over the molecule which means that the electrons in these orbitals are delocalized. In some of these orbitals, the wave functions may be very significant only in some parts and insignificant elsewhere in the molecule. Thus, localized bonding is implicit within the MO theory. This chapter will discuss about the electronic structures, spectral characteristics and other electronic properties of polyatomic molecules.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 P. K. Mallick, Fundamentals of Molecular Spectroscopy, https://doi.org/10.1007/978-981-99-0791-5_7

215

216

7 Electronic Spectra of Polyatomic Molecules

7.1 Hybridization The electronic configuration of normal carbon atom is 1s2 2s2 2p2 . So it is expected that it should form CH2 molecule by attaching the electron of each of the two hydrogen atoms with one of the two unpaired electrons in the valence cell 2p of carbon atom. Each CH bond is thus expected to have a pair of electrons with opposite spins and the two bonds are mutually perpendicular to each other. CH2 group in molecule (methylene) is found to exist in two classes: singlet and triplet. The respective bond angles are found to be around 102° and 125–140° , i.e. the bond angle is not 90° as predicted by simple valence bond theory. Why this is so? Here comes the idea of hybridization. We know that CH4 is a very stable molecule. If a 2 s electron of the carbon atom is excited to the 2p state, then it gets a favourable environment to form a CH4 molecule which is very stable (Fig. 7.1). Here, one 2 s-orbital and all the three 2p-orbitals lose their identities and they mixed up together to form four sp3 -hybridized orbitals. Actually, the stabilization energy (25.31 eV) is much greater than the amount of energy (8.26 eV) required to excite a 2 s electron to the 2p state to form four sp3 -hybridized orbitals as shown in Fig. 7.1. Each of these four hybridized orbitals forms a valence bond with the 1 s-orbital hydrogen atom. There are three types of spn -hybridization, namely sp3 , sp2 and sp which will be discussed in the following section. sp3 -hybridization The four atomic orbitals in the shell n = 2 can be expressed as Ψ2 s ∼ R2 s (r) Ψ2 px ∼ R2p (r) sin θ cos ϕ Ψ2 py ∼ R2p (r) sin θ cos ϕ Ψ2 pz ∼ R2p (r) cos θ

(7.1)

The angular part of the wave functions indicates that Ψ 2px , Ψ 2py and Ψ 2pz have some directional properties, i.e. their values are maximum in the direction along the three respective Cartesian axes. Taking linear combinations of these four atomic orbitals, four linearly independent, orthonormal orbitals are formed. These are the four sp3 -hybridized orbitals, given by Fig. 7.1 Formation of methane (CH4 )

(sp3) (8.26 3

C

P0 (25.31ev) CH

7.1 Hybridization

217

Ψ1 =

1 [ψ2s + ψ2px + ψ2py + ψ2pz ] 2

(7.2a)

Ψ2 =

1 [ψ2s + ψ2px − ψ2py −ψ2pz ] 2

(7.2b)

Ψ3 =

1 [ψ2s − ψ2px + ψ2py −ψ2pz ] 2

(7.2c)

Ψ4 =

1 [ψ2s − ψ2px − ψ2py + ψ2pz ] 2

(7.2d)

Each member of this set has its maximum value along one of the lines from the carbon atom to a corner of a regular tetrahedron with carbon atom at the centre. That is, if we consider a cube each side of which is of length two units, then the carbon atom is at the centre (0,0,0) and the maximum values of the wave functions are directed along the line drawn from the origin (position of the carbon atom) to the four points (1,1,1), (1,−1,−1), (−1,1,−1) and (−1,−1,1). The angle between any two of these four directions can be determined in the following manner. If we disregard the contribution of ψ2s (since it is spherically symmetric) and consider ψ2px , ψ2py and ψ2pz as unit vectors directed along the three respective Cartesian directions, then the angle between any pair of the sp3 -hybridized orbitals is given by cos θ = −1/3, which gives θ = 109° 28' . With each of these hybridized orbitals, a σ-bond is formed by combining with the 1 s-orbital of a hydrogen atom and thus four σ-bonds (CH) are formed in methane (CH4 ). Now let us consider the structure of ethane (C2 H6 ) molecule. Here, also the four sp3 -hybridized orbitals of each carbon atom form four σ-bonds, three with the 1 s-orbitals of three hydrogen atoms and the fourth one with another sp3 -hybridized orbital of the other carbon atom. The molecule has a cylindrical symmetry about the CC(σ)-bond. Any rotation of the molecule about this axis does not change the electronic charge density along this bond. Thus, there exists a possibility of revolution of the atoms around the CC bond which forms different rotational isomers (CIS, TRANS or GAUCHE) in ethane derivatives and also in molecules having single bond between carbon atom. Similar kind of promotion of 2 s electron to 2p state may also take place in other atoms. The promotion is complete in the case of carbon, but not so in the case of nitrogen and oxygen atoms, because the 2p state has more electrons in the latter atoms than carbon. Moreover, the promotional energy increases from carbon to nitrogen and then to oxygen atoms. Moreover, this promotion does not increase the valence of the nitrogen and oxygen atoms. So, although tetrahedral sp3 -orbitals are formed in all the cases, they are not absolutely as pure as those in carbon atoms. The angles between any pair of these (slightly lesser extent) tetrahedral sp3 -hybridized orbitals are 107° 18' and 104° 30' in nitrogen and oxygen atoms, respectively. In ammonia, three such orbitals (of nitrogen atom) form three σ-bonds by combining with three 1 s- orbitals of three hydrogen atoms. One sp3 -hybridized orbital remains filled with two lone pair (with opposite spins) electrons which cannot form bond with any atom.

218

7 Electronic Spectra of Polyatomic Molecules

H 109.20

H C

H

H

111.20

H H 109.2

H

(a)

0

C

H

C

H

H

H

••

104.30

N

O

H H

H

107.10

(c)

(b)

• •• •

H

(d)

Fig. 7.2 sp3 -hybridization in a methane (CH4 ), b ethane (C2 H6 ), c ammonia (NH3 ) and d water (H2 O) molecules

Similarly, two such sp3 -hybridized orbitals form two bonds with two hydrogen atoms through their 1 s-orbitals in H2 O molecule and two such other orbitals are occupied by lone pair electrons. These are shown in Fig. 7.2. Sp2 -hybridization In this hybridization, the 2 s-orbital is mixed up with two of the three 2p (1px , 2py , 2pz )-orbitals leaving the remaining 2p-orbital unchanged. The normalized orthogonal set of these hybridized orbitals is 1 Ψ1 = √ ψ2s + 3

/

2 ψ2 px 3

(7.3a)

1 1 1 Ψ2 = √ ψ2s − √ ψ2px + √ ψ2py 3 6 2

(7.3b)

1 1 1 Ψ3 = √ ψ2s − √ ψ2px − √ ψ2py 3 6 2

(7.3c)

The ethylene molecule exhibits sp2 -hybridization. The three sp2 -hybridized orbitals of each carbon atom form three σ-bonds by combining with the 1 s-orbitals of two hydrogen atoms and one of the three sp2 -hybridized orbital of the other carbon atom, the maximum value of which is lying along the CC bond. The maximum extent of overlaps between the remaining unaffected two 2p-orbitals of each of the two carbon atoms occurs when they are so oriented with respect to each other that two strong π-bonds are formed between the two carbon atoms. This gives the molecule a planar structure. This is shown in Fig. 7.3. All the sp2 -hybridized bond angles in this molecule are expected to be 120° , but in the observed structure they are found to be slightly different (specially the HCC angle). Sp-hybridization Here, a s-orbital is mixed up with a pz -orbital of a carbon atom to give rise two sp2 -hybridized orbitals.

7.2 Conjugated System of Molecules Fig. 7.3 sp2 -hybridization in ethylene (C2 H4 )

219

112.50

H

π

H

σ 117.40

σ σ

C

C

σ

σ

H

H

π

π

Fig. 7.4 sp-hybridization in acetylene (C2 H2 ) molecule

H

σ

π σ

C π

C

σ

H

π

] 1 [ Ψ1 = √ ψ2s + ψ2pz 2

(7.4a)

] 1 [ Ψ2 = √ ψ2s − ψ2pz 2

(7.4b)

In acetylene molecule, ψ1 of one carbon atom is mixed up with the ψ2 of the other carbon atom to form a σ-bond between the two carbon atoms. Other ψ2 and ψ1 of the respective carbon atoms form two σ-bonds by combining with two 1 s-orbitals of the two hydrogen atoms. The remaining two 2px - and two 2py- orbitals of the two carbon atoms form two π-bonds between the two carbon atoms. Thus, the bond order of the CC bond in acetylene molecule is three, in comparison with two and one in ethylene and ethane molecules, respectively. This is shown in Fig. 7.4.

7.2 Conjugated System of Molecules A conjugated system is a system in which the p-orbitals of the adjacent atoms overlap across the intervening sigma bonds. This allows delocalization of the π-electrons across the adjacent atoms of the molecules having alternating single and double bonds. Such system of molecules may be cyclic, acyclic, linear or mixed. We shall consider only two types of molecules containing conjugated double bonds. The first type comprised of molecules with open systems like butadiene (C4 H6 ), octatetraene (C8 H10 ), etc., and the other group consists of closed planar systems like aromatic molecules, benzene, naphthalene, pyridine, etc. In both types, delocalization of the electrons becomes important and they spread all over the molecular skeleton. The

220

7 Electronic Spectra of Polyatomic Molecules

electronic transitions among the states of these delocalized electrons give rise to the spectra in the visible or in the ultraviolet regions. In the following sections, we shall discuss two methods which are often used (specially the latter one) to study the spectral characteristics of such molecules.

7.2.1 Free Electron Model It is very well known that more and more are the conjugated double bonds, longer and longer are the wavelengths of the absorption bands of the conjugated system. Let us take a specific case of molecules, octatetraene (C8 H10 ). Here, each of the eight skeletal carbon forms three σ-bonds with the nearby hydrogen and other carbon atoms with its three sp2 -hybridized orbitals. Remaining eight 2pz-orbitals (perpendicular to the skeletal plane) form delocalized π-bonds spread over the entire molecule. In the free electron model, it is assumed that the eight 2pz electrons are free to move in the molecular skeleton with a constant potential and the potential is infinite outside the molecule. Thus, we can apply the free electron or particle in a box model to solve the Schrödinger equation. One problem arises as to the length of the box. It is generally assumed that it is better to take the length of the box as the linear distance between the end carbon atoms than the distance along the zig-zag path of the chain. The solutions are available in the books of quantum mechanics, and we shall use the results. The wave functions for the particle in a box model are (Fig. 7.5) ψn (x) = A sin

nπx , a

(7.5a)

and En =

n2 h2 , n = 1, 2, 3, 4, . . . . . . , 8ma2

(7.5b)

9.5Å

H

+ C

H

-

H + C -

+

+ C H

+

C -

C -

H

H

H

+ +

C -

C -

H

+ C -

H H

Fig. 7.5 σ- and π-bondings in octatetraene (C8 H10 ). σ-bonds (indicated by ‘—’) are formed involving the sp2 -hybridized orbitals and the delocalized π-bonds are formed by the remaining 2pz-orbitals of the carbon atoms, oriented perpendicular to the skeletal plane

7.2 Conjugated System of Molecules

221 n

Fig. 7.6 Energy levels and lowest energetic transition in octatetraene (C8 H10 ) molecule

5

4

3 2

9.5 Å

‘m’ being the electronic mass and ‘a’ being the linear dimension of the box. Energy levels are shown in Fig. 7.6. According to Pauli’s exclusion principle, two electrons are to be accommodated in each level. Thus, the four lowest levels are filled with eight electrons. So the lowest energy transition occurs from n = 4 to n = 5. The wavenumber of the radiation, as estimated from Eq. 7.5(b), is (with a = 9.5 Å) [ ] ν = 52 − 42

h = 30227 cm−1 8mca2

which is in good agreement with the experimentally observed band maximum at 33,000 cm−1 . However, the method is oversimplified.

7.2.2 Molecular Orbital Method In the valence bond method, all the bonds are considered to be two centred bonds. But in the molecular orbital (MO) method, orbitals of all the atoms are utilized to generate a set of orbitals which extend all over the molecule. In some of these molecular orbitals, the wave functions are significant in certain parts of the molecule and negligible elsewhere. Thus, localized bonding, as considered in the valence bond approach, in one sense is a special case of molecular orbital theory. So MO theory is more difficult to visualize but it is a very effective and powerful model to predict the spectral properties of polyatomic molecules. As an example, we shall consider the method as applied to the benzene molecule. In benzene (C6 H6 ) molecule, all the three sp2 -hybridized orbitals of each carbon atom form three σ-bonds with two such orbitals of the adjacent carbon atoms and a 1 s-orbital of the nearest hydrogen atom. This explains the planar structure of the regular hexagonal ring (Fig. 7.7). Let us consider the plane of the molecule to be

222

7 Electronic Spectra of Polyatomic Molecules

xy-plane. Then, linear combinations of the remaining 2pz -orbitals of the six carbon atoms generate a set of six orthonormal MOs (called π-orbitals). The kth π-molecular orbital is ∑ ψk = cki ϕi , i, k = 1, 2, . . . 6 (7.6) i

Here, cki ’s are constant coefficients and ϕ i is the 2pz -orbital of the ith carbon atom such that (ϕi |ϕi ) = 1. Also due to orthonormality condition, (ψk |ψl ) = δkl . The energy levels are determined by solving the Schrödinger equation Hψ = Eψ, where the subscript ‘k’ of the MO (ψ k ) has been dropped and H is the Hamiltonian associated with the ∑ six 2pz electrons. E is determined by solving the equation, E = ∑ | ci ϕi ) ( ∑ci ϕi |H∑ (ψ|H |ψ) (ψ|ψ) = ( c ϕ | c ϕ ) , by variation method. The set of equation thus found are i i

i i

6 ∑

[Hij − ESij ]cj = 0, i = 1, 2, . . . , 6

(7.7)

j=1

For non-trivial solution, the required secular equation is || || ||Hij − ESij || = 0

(7.8)

( ( ) ) where Hij = ϕi |H |ϕj and Sij = ϕi |ϕj . So in order to solve this secular equation of order six, we shall introduce Huckel’s approximation:S ij = δ ij ; H ii = α, H ij = β for i = j ± 1 and zero otherwise. Thus, the secular equation becomes

H

H

+ C

+ C

H

-

+

C

H Fig. 7.7 π-bonding in benzene (C6 H6 ) molecule

+ C

+

C

+

H

-

C

H

7.2 Conjugated System of Molecules

223

|| || || || (α − E) β 0 0 0 β || || || || β β 0 0 0 (α − E) || || || || 0 β β 0 0 (α − E) || || || = 0 || || || 0 0 β β 0 (α − E) || || || || 0 0 0 β − E) β (α || || || β 0 0 0 β (α − E) ||

(7.9)

The solution of this equation for the energy (E) is α ± 2β and α ± β, the last pair being doubly degenerate. The normalized molecular orbitals of the corresponding energy levels, as found from Eq. (7.7), are (Fig. 7.8) 1 ψ1 = √ [ϕ1 + ϕ2 + ϕ3 + ϕ4 + ϕ5 + ϕ6 ], E1 = α + 2β. 6

(7.10a)

1 ψ2a = √ [2ϕ1 + ϕ2 − ϕ3 − 2ϕ4 − ϕ5 + ϕ6 ], E2 = α + β. 12

(7.10b)

ψ2b =

1 [ϕ2 + ϕ3 − ϕ5 − ϕ6 ], E2 = α + β. 2

1 ψ3a = √ [2ϕ1 − ϕ2 − ϕ3 + 2ϕ4 − ϕ5 − ϕ6 ], E3 = α − β. 12 ψ3b =

1 [ϕ2 − ϕ3 + ϕ5 − ϕ6 ], E3 = α − β. 2

1 ψ4 = √ [ϕ1 − ϕ2 + ϕ3 − ϕ4 + ϕ5 − ϕ6 ], E4 = α − 2β. 6

(7.10b) (7.10c) (7.10c) (7.10d)

Same results can also be obtained from the standpoint of group theory, which will be discussed in the following chapter. In the ground (lowest energy) state, each of the three lowest orbitals are filled up with pair of electrons having opposite spins. The total energy of the six π-MOs in the ground state is thus 6α + 8β. If we disregard interaction between the atoms (i.e., β = 0), the total energy becomes 6α. Since α and β (called resonance integral) are both negative, the π-bonding has stabilized the Fig. 7.8 Energy levels of the π-molecular orbitals of benzene

E4 = α-2β

(ψ4)

E3(a,b) = α- β

(ψ3(a,b))

E E2(a,b) = α+ β (ψ2(a,b)) E1 = α+2β

(ψ1)

224

7 Electronic Spectra of Polyatomic Molecules

(Kekule structure)

(Dewar Structure)

Fig. 7.9 Significant canonical (resonance) structure of Benzene

molecule by an amount 8β. So the ψ 1 and ψ 2 are called bonding π-orbitals, and the other two, ψ 3 and ψ 4 , are antibonding orbitals designated by π *-MOs. Next, we shall determine the delocalization or resonance energy. This energy is defined as the difference between energy of the most stable canonical structure and the actual energy. There are five such structures (two equivalent Kekule structures and three equivalent Dewar structures) which are sufficiently low in energy (Fig. 7.9): Assuming that one of the Kekule structure to be the most stable structure, the corresponding secular equation is || || || || (α − E) β 0 0 0 β || || || || β 0 0 0 0 (α − E) || || || || 0 0 β 0 0 (α − E) || || || = 0 || || || 0 0 β 0 0 (α − E) || || || || 0 0 0 0 β (α − E) || || || 0 0 0 0 β (α − E) ||

(7.11)

Placing two electrons in each of the three localized π-bonds of one of the said Kekule structure, the energy of the six pz -electrons is 6(α + β). So the resonance or delocalization energy is 2β. It is difficult to estimate the value β directly. However, there is an indirect way of estimating the delocalization energy experimentally in the following manner. The energy of one of the Kekule structure can be estimated from the experimental values of bond energies of CC, C = C and CH bonds found in other molecules like ethane and ethylene. Again, the actual enthalpy of formation of benzene can be determined by thermochemical measurements. The difference of these two energies is the experimental value of the resonance or delocalization energy of benzene. Experimental value of β, thus obtained for benzene, is 18–20 kcal/mol. Same value of β is also obtained in other aromatic molecules, like naphthalene and anthracene. The lowest energy transition (ψ2 → ψ3 ), in benzene, is a π → π * transition. This transition generates four singlet excited states of which one is doubly degenerate. The transition from the ground state to the doubly degenerate state is strongly allowed, and the other two transitions to the two non-degenerate states are forbidden. Actually, benzene absorption spectrum is composed of three bands of which two are weak and one is strong. The weak bands are observed around 2600 Å and 2100 Å (first one being less intense but structured) and the strong one around 1800 Å. All these

7.3 Relaxation Mechanism

225

are discussed in much detail in the chapter on the application of group theory to molecular spectroscopy.

7.3 Relaxation Mechanism On absorption, a molecule goes from the ground state (S 0 ) to different vibrational levels of higher excited electronic states (S 1 , S 2 , S 3 , S 4 , etc., designated in order of increasing energy) following Franck–Condon principle. In the excited states, the molecule may or may not preserve its identity. If it fails to preserve its chemical identity, the process is called photochemical process, and if the identity is preserved, the process is called photophysical process. We shall confine our discussion to photophysical processes. From the higher excited states (say, S 2 , S 3 etc.), the molecule drops down very rapidly (10–12 to 10–13 s) to the state S 1 by releasing energy to the environment. This process is called vibrational relaxation. In a similar way, the molecule goes down from different excited vibrational states to the lowest vibrational state of the same electronic state S 1 . In most cases, the lowest vibrational level of S 1 is the most important state from which all deactivation starts in a molecule. If we designate the singlet states by ‘S’ and triplet states by ‘T ’, then according to Kasha’s rule, in condensed media of organic molecules, the deactivating levels are the lowest excited levels of respective multiplicities (i.e., S 1 for the singlet state and T1 for the triplet state). [However, exception to this rule was found in the case of azulene, where the radiative decay occurs from the transition S 2 ~~~ > S 0 and not from S 1 ~~~> S 0 .] Further deactivation from these states can occur in two ways, radiative and nonradiative. The radiative pathways are called luminescence. For deactivation from the state S1 , the radiative process is called fluorescence. The radiation less transition, directly back to the ground state S0 , is called internal conversion (IC). There is another pathway for non-radiative transition, i.e. S1 ~~~> T1 . Any such transition S i ~~~> T j is called intersystem crossing (ISC), which is a comparatively slow process (Fig. 7.10). So the intersystem crossing (ISC) are responsible for populating the levels T i ’s. Anyway, if by any intersystem crossing mechanism, the molecule is excited to the triplet state, say, T i , it will rapidly decay down to the lowest state (T 1 ) of this triplet manifold by IC mechanism. Radiative deactivation from this (T 1 ) state to the ground state (S 0 ) (S 1 ~~~ > S 0 ) is called phosphorescence. Although in the zeroth order, singlet ↔ triplet transition is spin forbidden, it becomes allowed in the higher order through spin–orbit interaction. In the higher order, the triplet state no longer remains pure triplet, but becomes a mixture of singlet and triplet states due to spin–orbit interaction. Thus, in general, fluorescence is much stronger in intensity than phosphorescence. The non-radiative decay from the lowest triplet to the ground state (T 1 ~~~> S 0 ) is also achieved by ISC mechanism. An idea about the life times of different decay mechanisms can be found from the rates of different decay processes, shown in Fig. (7.10). Generally for organic molecules, the fluorescence life times are found in the range of nanosecond to microsecond (~10–9 to 10–6 s) and phosphorescence life times in the range of millisecond to several seconds (~10–4 to

226

7 Electronic Spectra of Polyatomic Molecules

VR > 1012/sec

IC > 1012/sec

S2

T2

ISC ~ 104-1012/sec

VR

Fluorescence 106-109/sec

Absorption 1015-1016 /sec

S1

T1

ISC ~ 10−-1-105/sec Phosphoresence 10−2 – 104 /sec

T2 absorption

S0

T1 absorption

T1

S0

S0

Fig. 7.10 Fluorescence, internal conversion (IC), phosphorescence, intersystem system crossing (ISC), vibrational relaxation (VR) and their rate constants

102 s). In rigid glassy matrix, all deactivating collisions are generally reduced and molecules are able to exhibit phosphorescence. In almost all cases of fluorescence and phosphorescence transitions in molecules, we are concerned with the spontaneous emission probabilities. For fluorescence radiative transition, the rate constant is given by, Ks =

|2 64π 4 nν 3 || − → | (S | ) |S M | | 1 0 3hc3

(7.12)

− → ∑ → where M = i e− ri is the electric dipole moment operator, n is refractive index of the medium, ν is the frequency of radiation of the concerned transition and the other terms have usual meanings. We can calculate Ks if we know the initial and final wave functions. Similarly for phosphorescence, the radiative transition rate is, Kp =

|2 64π 4 nν 3 || − → | (T | ) |S M | | 1 0 3hc3

(7.13)

7.3.1 Lifetime and Quantum Yield The radiative life time τ0 of an excited state is defined as the time for the radiation intensity to decay to 1/e times the initial value. If Ke is the spontaneous emission, rate then we can write τ0 = K1e . Even if a molecule is in isolation in vacuum, there exist several ‘radiation less’ transitions due to the vibronic interactions, spin–orbit

7.3 Relaxation Mechanism

227

interactions, etc., among the states of the molecule itself. But when the molecule is embedded in such media as fluid or solid environments, the situation becomes quite complicated. Therefore, the observed decay time for any radiative emission transition 1 , which is shorter than the radiative lifetime τ0 . Ki represents is given by, τ = Ke +K i transition rate arising from various non-radiative pathways. Therefore, fluorescence life time is, τF =

1 Ks + KISC + Kns

(7.14)

where Ks is the decay rate for S 1 → S 0 fluorescence, KISC is the decay rate for S 1 ~~~> T 1 intersystem crossing and Kns is the decay rate for S 1 ~~~> S 0 internal conversion. Similarly, phosphorescence life time can be written as, τp =

1 Kp + Knp

(7.15)

where Kp and Knp are the radiative (T 1 → S 0 ) and non-radiative (T 1 ~~~> S 0 intersystem crossing) phosphorescence decay rates. The ground (singlet) state, lowest energy excited singlet and triplet states are generally the most important ones which determine the photophysical behaviours of most of the molecules. A two-state quantum yield φ II is defined as the number of molecules in an excited state taking part in the particular photophysical process divided by the number of quanta absorbed in going from the ground state to the same excited state. A three-state quantum yield φ III is defined as the number of molecules undergoing a particular photophysical process in a particular excited state divided by the number of quanta absorbed in going from the ground to some other higher energy excited state. Therefore, the fluorescence(φ F ) and phosphorescence (φ P ) quantum yields are given by, Ks . Ks + KISC + Kns

(7.16a)

KISC Kp ). ( (Ks + KISC + Kns ) Kp + Knp

(7.16b)

φF = φP =

7.3.2 Vibronic and Spin–Orbit Interactions and N → π* Transitions in Organic Molecules In aromatic and hetero-aromatic (like benzene, pyridine, aromatic carbonyl etc.) molecules, there are certain low lying excited states, which play very important roles in determining their photophysical behaviour. These low lying excited states arise from the transitions among the highest occupies π-MOs, n-orbitals (which are,

228

7 Electronic Spectra of Polyatomic Molecules

as said earlier, the atomic orbitals which remain unaffected in molecules) and the lowest unoccupied antibonding MOs, (π *). Any excited state arising from an n → π * transition is called an nπ*-state and that arising from a π → π * transition is called a π π * state. Generally, the band arising from the n → π * transition is much weaker in intensity than that arising from π → π * transition. For singlet → singlet transition, intensification of both n → π * and weak π → π * bands occur due to borrowing of intensity from a strongly allowed band lying in their near vicinity through vibronic coupling. On the other hand, a singlet ↔ triplet transition is in general forbidden, but becomes allowed due to spin–orbit interaction between a singlet and a triplet states. This spin–orbit interaction is very weak between two π π *- or between two nπ*-states, but strong between a π π * and a nπ * states of different multiplicities. So we shall first discuss the vibronic and spin–orbit coupling mechanisms. Vibronic coupling Total electronic Hamiltonian in a molecule (say, a π-electronic system, such as benzene, pyridine and aromatic ketone) is given by H = Ho + H '

(7.17)

where H o is the unperturbed part (i.e., the part which remains unaffected due to vibrational motion) and the perturbed part (according to Herzberg and Teller) is given by H ' = HV =

3N∑ −6(5) ( a=1

∂H ∂Qa

) Qa

(7.18)

o

where Qa is the displacement of the ‘a’ th normal coordinate from its equilibrium value, the subscript ‘0’ corresponds to equilibrium nuclear configuration and other symbols have their usual meaning. This term is determined from the Taylor series expansion of the electronic Hamiltonian (H) about the equilibrium nuclear configuration. Since the oscillations of the nuclei are small, here we have kept only the linear term and all other higher-order terms are neglected. | ) So, according to the first-order perturbation theory, the excited singlet states |Si' s and the ground state |S0 ) are given by \ |( )| \ | ∂H | | \ Sj | ∂Q |Si ∑ ∑ | 0) a | | ( ) Qa |Sj0 |Si ) = |Si (q, Q) ) = Si + 0 0 Ei − Ej j\=i a

(7.19a)

and \ |( )| \ | ∂H | | \ | ) Sj | ∂Q |S0 ∑ ∑ | 0) a | | ( ) Qa |Sj0 ≈ |S00 |S0 ) = |S0 (q, Q)) = S0 + 0 0 E0 − Ej j\=i a

(7.19b)

7.3 Relaxation Mechanism

229

( ) because of comparatively large value of the denominator E00 − Ej0 , q and Q being the electronic and nuclear coordinates, respectively. Here, the superscript ‘0’ of the states corresponds to respective unperturbed states. The corresponding vibronic or Born–Oppenheimer (BO) states are given by |i, vi ) = |Si (q, Q) ) · |vi (Q) )

(7.20a)

| ) | ) | ) |g, vg = |Sg (q, Q) · |vg (Q)

(7.20b)

vi ’s being the vibrational states of the Ith electronic state. So the transition moment for the transition from ground ( |g ) = |S0 )) to the excited state |i ) (i.e., |g ) → |i )), on the square of which the intensity of the corresponding bands depend, is given by \ |( )| \ | ∂H | ( ) ( Sj | ∂Q |Si ( ) ( ∑ ∑ ) ) a → → jg · vi |Qa |vg → ( ) M Mig = Mig · vi |vg + 0 0 0 0 Ei − Ej j\=i a

(7.21)

The first term on the right-hand side of Eq. (7.21) is of electronic origin and the intensities ( )of the bands arising from this term are controlled by the Franck–Condon term vi |vg which is the overlap integral. The second term constitutes the forbidden part of the spectra which arises due to borrowing of intensity from a nearby allowed electronic transition |g ) → |j ) through vibronic coupling. This transition moment is responsible for fluorescence emission provided the state |i ) is a singlet state. Spin–Orbit Coupling For spin–orbit coupling, the perturbation part of the Hamiltonian (in Eq. 7.17) is given by H ' = Hso = = =

N n ) 1 ∑ ∑ 1 ∂ V (riK ) (→ l · → s i i 2m2 c2 K=1 i=1 rik ∂riK

N n ) 1 ∑ ∑ 1 ∂ V (riK ) ( lxi sxi + lyi syi + lzi szi 2 2 2m c K=1 i=1 rik ∂riK n ∑

( ) Ai lxi sxi + lyi syi + lzi szi = Hx' + Hy' + Hz'

i=1

= (Hso )x + (Hso )y + (Hso )z

(7.22)

where

Ai =

1 ∂ V (riK ) 1 ∑N K=1 rik 2m2 c2 ∂riK

(7.23)

230

7 Electronic Spectra of Polyatomic Molecules

where i and K correspond to electrons and nuclei. Let us now consider one component of Hso , say Hx' and write it in the following form, (Hso )x = Hx' =

n ∑ i=1

+

1 4

1 ∑∑ (Ai lxi + Aj lxj )(sxi + sxj ) 4 i=1 j=1 n

Ai lxi sxi =

n ∑ n ∑

n

(Ai lxi − Aj lxj )(sxi − sxj )

(7.24)

i=1 j=1

Both the terms on the right-hand side of Eq. (7.24) are symmetric with respect to exchange of electrons. So the operation of (H so )x on any wave function preserves the necessary antisymmetry. However, if we consider only the spin part of the operator (H so )x , the spin operator (sxi + sxj ) is symmetric and (sxi − sxj ) is antisymmetric with respect to electron exchange. Thus, we see that (sxi + sxj ) preserves the electron exchange characteristic of the spin part of a molecular wave function. Hence, the first term on the right-hand side of Eq. (7.24) preserves the multiplicity of the molecular wave function and it leads to multiplet splitting of the concerned level. However, the operation of (sxi − sxj ) on the spin part of a molecular wave function yields a function where the characteristic of the previous wave function under electron exchange is no longer preserved. Since the symmetric and antisymmetric characteristics of the spin part of a wave function under the operation of electron exchange determines the multiplicity of a state, so the second term on the right-hand side of Eq. (7.24) leads to a mixing of states of different multiplicities which makes a singlet ↔ triplet transition allowed by borrowing intensity from a nearby spin allowed (singlet ↔ singlet) transition. More specifically, it can be shown (see below) that Hx' and Hy' mix states for which ΔM s = ± 1 and Hz' mixes states for which ΔM s = 0. One-electron spin operators, operating on the one electron spin functions ( |α )and |β )), have the following properties: (α|α) = (β|β) = 1; (α|β) = (β|α) = 0 1 1 sz |α) = h|α); sz |β) = − h|β) 2 2 1 1 sx |β) = h|α) sx |α) = h|β); 2 2 1 1 sy |β) = −i h|α) sy |α) = i h|β); 2 2 3 2 3 2 2 2 s |β) = h |β) s |α) = h |α); 4 4

(7.25)

Let us now consider two molecular electronic states, one triplet and one singlet, arising from two molecular orbitals ‘a’ and ‘b’. Furthermore, consider ‘a’ as an n or π (HOMO)-orbital and ‘b’ as a π * (LUMO)-orbital so that the following states would arise from the excited electronic configuration nπ * or π π *.

7.3 Relaxation Mechanism

231

] / [/ / 1 1 1 1 |aα1 bβ2 |− |aβ1 bα2 | = (a1 b2 + a2 b1 )(α1 β2 − α2 β1 ) S1 = 2 2 2 2 / / 1 1 |aα1 bα2 | = T1+1 = (a1 b2 − a2 b1 )α1 α2 2 2 [ ] / / / 1 1 1 1 |aα1 bβ2 | + |aβ1 bα2 | = (a1 b2 − a2 b1 )(α1 β2 + α2 β1 ) T10 = 2 2 2 2 / / 1 1 −1 |aβ1 bβ2 | = T1 = (7.26) (a1 b2 − a2 b1 )β1 β2 2 2 Here, we have disregarded the closed shell electrons as they are not taking part in the electronic transitions. Now, we shall see the effect of the spin operators on these functions. First, let us try with the operator (sy1 ± sy2 ). /

(sy1 ±

sy2 )T1+1

1 |aα1 bα2 | = (sy1 ± sy2 ) 2 / 1 (sy ± sy2 ){aα1 bα2 − aα2 bα1 } = 2 1 / 1 h i [{aβ1 bα2 − aα2 bβ1 } ± {aα1 bβ2 − aβ2 bα1 }] = 2 2

(7.27)

Thus, (sy1 +

sy2 )T1+1

( ) 23 1 = i h[(aβ1 bα2 − aβ2 bα1 ) + (aα1 bβ2 − aα2 bβ1 )] 2 1 1 1 = √ i h [|aβ1 bα2 | + |aα1 bβ2 |] = √ ihT10 (7.28a) 2 2 2

and (sy1 −

sy2 )T1+1

( ) 23 1 = i h[(aβ1 bα2 + aβ2 bα1 ) − (aα1 bβ2 + aα2 bβ1 )] 2 ] 1 1 1[ = √ i h |aβ1 bα2 |+ − |aα1 bβ2 |+ = − √ i hS1+ (7.28b) 2 2 2

where |aβ1 bα2 |+ = (aβ1 bα2 + aβ2 bα1 ) is a fully symmetric determinant, called the permanent of |aβ1 bα2 |, etc., and the electron number of the spatial part of the wave function is indicated by the subscript of the adjacent spin function on the right-hand side. In this way, we can determine the effects of all the symmetric and antisymmetric spin operators on the singlet and triplet states (7.26) and the results are given in the following Table 7.1.

232

7 Electronic Spectra of Polyatomic Molecules

Table 7.1 Effects of the spin operators on the spin part of the wave functions* ΔS

ΔMS

T 01

0

±1

0

±1

0

−iT 01 √ −1 2T 1

0

0

−(S1 )+

0

(S1 )+

1

±1

Sy1 − Sy2

+ T −1 1 ( )+ i T +1 1

−i(S1 )+

0

−i(S1 )+

1

±1

Sz1 − Sz2

)+ ( +i T −1 1 √ ( 0 )+ 2 T1

0

√ 2(S1 )+

0

1

0

Spin operators

Wave functions S1

T +1 1

T 01

T −1 1

0

T 01

−1 T +1 1 + T1

Sy1 + Sy2

0

iT 01

−iT +1 1

Sz1 + Sz2

0

Sx1 + Sx2

Sx1 − Sx2

)+ ( − T +1 1 (

* Permanents √1 h 2

√ +1 2T 1

+ iT −1 1

)+

are designated by ‘ + ’ sign in the superscript on the right side of a term. The factor

is omitted throughout

Now, we shall consider the effect of the spatial part (7.24) of the operator H so , (Ai lxi − Aj lxj ), on the spatial part of the wave function (7.26). Since the coefficients → Ai s are functions of the radial coordinates − ri s, it will not take part in the above operation, so we shall investigate only the effects of the operations of lxi , lyi and lzi s on the above wave function. In aromatic molecules, the bonding, antibonding and non-bonding orbitals taking part in electronic transitions are generated from s- and p-atomic orbitals. Since the s-orbitals are only functions of the radial distances, so we shall consider only the operations of the above orbital angular momentum operators lx' i set c. on the various components of p-orbitals, px , py and pz , the forms of which are given below. px = ξ (r) sin θ cos ϕ py = ξ (r) sin θ sin ϕ pz = ξ (r) cos θ

(7.29)

Various components of the orbital angular momentum operator, (→l), in spherical polar coordinates are (from the knowledge of angular momentum operators in quantum mechanics) ) ( ∂ ∂ + cot θ cos ϕ lx = i sin ϕ ∂θ ∂ϕ ) ( ∂ ∂ + cot θ sin ϕ ly = ih − cos ϕ ∂θ ∂ϕ

7.3 Relaxation Mechanism

233

lz = −ih

∂ ∂ϕ

(7.30)

Let us operate on the p-orbitals (7.29) with the operator lx : ) ( ∂ ∂ + cot θ cos ϕ (ξ (r) sin θ cos ϕ) lx px = ih sin ϕ ∂θ ∂ϕ = ihξ (r)(sin ϕ cos ϕ cos θ − sin ϕ cos ϕ cos θ ) = 0

(7.31a)

) ( ∂ ∂ + cot θ cos ϕ (ξ (r) sin θ sin ϕ) lx py = ih sin ϕ ∂θ ∂ϕ ( 2 ) = ihξ (r) sin ϕ cos θ + cos2 ϕ cos θ = ihpz

(7.31b)

) ( ∂ ∂ (ξ (r) cos θ lx pz = ih sin ϕ + cot θ cos ϕ ∂θ ∂ϕ = ihξ (r)(− sin ϕ sin θ ) = −ihpy

(7.31c)

In a similar way, we can determine the results for the other components of →l and they are shown in the following Table 7.2. ( ) Let us now determine the spin–orbit coupling matrix, ψi |Hso |ψf , between two molecular states of different multiplicities, i.e. singlet state ψ i and triplet state ψ f . As before, we shall disregard the contributions of the electrons in the filled molecular orbitals and consider only those occupied by the unpaired electrons. From the knowledge of the form of the operator H so and the modes of operation of the spin and orbital angular momentums on the respective wave functions (Tables 7.1 and 7.2), it is understood that a nonzero value of the spin–orbit coupling matrix element is found only when the two molecular states have different multiplicities and the two molecular states must differ by not more than one molecular orbital. Let the HOMO of the molecule be a, and b and c are the two excited molecular orbitals. These are all n- or π / π *-orbitals made of s- and/or p-orbitals of the constituent atoms. The ground and the excited singlet and triplet states are formed from the electronic configurations as shown in Fig. (7.11). The singlet and triplet wave functions are Table 7.2 Effects of the operations of various − → components of l on the p-orbitals

Operators

Atomic orbitals px

py

pz

lx

0

ièpz

−ièpy

ly

−ièpz

0

ièpx

lz

ièpy

−ièpx

0

234

7 Electronic Spectra of Polyatomic Molecules

] / [/ / 1 1 1 |aα1 bβ2 | − |aβ1 bα2 | |Si ) = |ψi ) = 2 2 2 | \ | \ /1 | +1 | +1 |aα1 cα2 | |Tf = |ψf = 2 [ ] / | \ | \ /1/1 1 | 0 | 0 |aα1 cβ2 | + |aβ1 cα2 | |Tf = |ψf = 2 2 2 | \ | \ /1 | −1 | −1 |aβ1 cβ2 | |Tf = |ψf = 2 \ \ Let us first determine the matrix element Si |Hso |Tf+1 : \

Si |Hso |Tf+1

\

(7.32)

(7.33a)

(7.33b)

(7.33c)

( )3/2 1 = [(aα1 bβ2 |Hso (1) + Hso (2)|aα1 cα2 ) 2 − (aα1 bβ2 |Hso (1) + Hso (2)|aα2 cα1 ) − (aα2 bβ1 |Hso (1) + Hso (2)|aα1 cα2 ) + (aα2 bβ1 |Hso (1) + Hso (2)|aα2 cα1 ) − (aβ1 bα2 |Hso (1) + Hso (2)|aα1 cα2 ) + (aβ1 bα2 |Hso (1) + Hso (2)|aα2 cα1 ) + (aβ2 bα1 |Hso (1) + Hso (2)|aα1 cα2 ) − (aβ2 bα1 |Hso (1) + Hso (2)|aα2 cα1 )]

(7.34)

Using orthogonality relations of spin and orbital wave functions and remembering that the two particles are identical, the above matrix become b

b c a Filled MOs Ground state

c

b c

a

a

Filled MOs Ψi (Excited singlet state)

Fig. 7.11 Ground and excited singlet and triplet molecular states

Filled MOs Ψf (Excited triplet state)

7.3 Relaxation Mechanism

235

Table 7.3 Matrix elements of H so between singlet and triplet states \ \ \ \ Components of H so Si |H so |T +1 Si |H so |T 0f f √ 1/2(b|(H so )lx |c) 0 (H so )x | ) ( | √ 1/2i b|(H so )ly |c 0 (H so )y | | 0 −b|(H so )lz |c (H so )z

\

Si |Hso |Tf+1

\

\ \ Si |H so |T −1 f √ − 1/2(b|(H so )lx |c) | ) ( | √ − 1/2i b|(H so )ly |c 0

/

1 (aα1 |aα1 ) (bβ2 |Hso (2)|cα2 ) 2 / | ) 1 ( || bβ2 (Hso )x (2) + (Hso )y (2) + (Hso )z (2)|cα2 = 2 / ]| ) 1 ( || [ = bβ2 A2 lx (2)sx (2) + ly (2)sy (2) + lz (2)sz (2) |cα2 2 / [ ] ( | | )h h 1 (b|Alx |c) + i b|Aly |c = (7.35) 2 2 2

=

In the last line, the electron subscription is omitted, since the two electrons are identical. The other matrix elements can be determined in the same way and the results are shown in Table 7.3. In the above table (following Eqs. 7.22, 7.23 and 7.35), we have used (Hso )li =

N N ∑ h h e2 ∑ ZK Ali = l = (Hso (K))li , i = x, y, z i 2 4m2 c2 K=1 rK3 K=1

(7.36)

According to the results of Table 7.3, determination of the spin–orbit matrix element between the singlet and the triplet states ultimately reduces to that between two MOs, b and c. The molecular orbitals b and c can be written as linear combinations of the atomic orbitals χ μ ’s. So writing b and c as b=

N ∑

bμ χμ and c =

N ∑

μ=1

cν χν

(7.37)

ν=1

Thus, we get, (b|(Hso )li |c) =

N ∑ μ,ν=1

bμ cν

N ∑ (

χμ |(Hso (K))li |χν

)

(7.38)

K=1

(assuming all the coefficients to be real). We see that nonzero value of the matrix (b|(Hso (K))li |c) is possible only when there exists at least one nucleus (K) for which the contributions of the p-orbitals in

236

7 Electronic Spectra of Polyatomic Molecules

χ μ and χ ν are different in accordance with the results given in Table 7.2. This is possible only in the case of an n → π * transition. Let us take the example of monoazinearomatic, pyridine. Here, the non-bonding / | | ) ) (sp3 )-orbital (n) associated with the N-atom, say, is |χμ = |n ) = 23 |pyN + / 1 |s ) and χν = pzN , where y-direction is along the rotational axis (C2 ) of the 3 N molecule and z-direction is perpendicular to the pyridyl ring. Let us neglect all the multicentred integrals which are expected to be small with respect to the one-centred integral.( Then, only the atom (N) will contribute to spin–orbit coupling since the ) integral χμ |(Hso (N ))li |χν is nonzero (Table 7.2) only when i = x as shown below: (

//

)

χμ |(Hso (N ))li |χν = /

2 pyN + 3

/

| | \ 1 || h || sN Alx pzN 3 |2 |

) 2 h( pyN |A| − ihpyN 32 / | \ / | | he2 ZK | 2h | = −i pyN | 2 2 3 ||pyN , (Table 7.2) 32 2m c rK / | | \ 2 2 |1| 1 h e = − √ i 2 2 ZN pyN || 3 ||pyN r 6 2m c =

N

ZN3 (eff) 1 h2 e 2 = − √ i 2 2 ZN 3 3 a0 n l(l + 1/2)(l + 1) 6 2m c 1 = − √ iξN , 6 where ξN =

ZN3 (eff) h2 e 2 in energy unit Z N 2m2 c2 a03 n3 l(l + 1/2)(l + 1)

(7.39)

Here Z N is the Slater charge, Z N (eff) is the effective nuclear charge for the spin–orbit coupling purposes of the nitrogen atom and a0 is the Bohr radius. Z N for 2p electron of nitrogen atom is 3.75 and if we assume Z N (eff) = 3.9, then it is found that ξ N = 56 cm−1 . From the above Eq. (7.39), we see that more and more the non-bonding electron contributing atom is heavy, more and more is the spin–orbit interaction and hence more and more strong will be the n → π * transitions.

7.3 Relaxation Mechanism

237

7.3.3 Radiative Sources for T1 → S0 Transition (Phosphorescence Decay) The total Hamiltonian of the molecule can be written as ∑ ( ∂HSO ) ∑ ( ∂H0 ) H = H0 (Q0 ) + HSO (Q0 ) + Qa + Qa ∂Qa 0 ∂Qa 0 a a = H0 (Q0 ) + HSO (Q0 ) + HV + HSV

(7.40)

where H0 (Q0 ) is the Hamiltonian at equilibrium nuclear configuration (Q0 ), HSO (Q0 ) is the spin–orbit interaction energy at equilibrium nuclear configuration, and HV and HSV correspond to vibronic and spin-vibronic interaction, respectively. Thus, the transition dipole moment MT1 →S0 between the lowest triplet state (T 1 ) and the ground state (S 0 ) can be written as, − → − → M T1 →S0 = (T1 | M |S0 ) | ) ( |− [( | 1 →| ) ∑ ( 0 ) T10 | HSO |Sk0 ( 0) = T10 | M |S00 + E T1 − E Sk k ( ) | ) ( | ∑ ∂HSO Qa |Sk0 + T10 | ∂Qa 0 a | 0 )( 0 | | 0) ( 0 | ∑ ( ∂H0 ) | | | | ∑ T1 a ∂Qa 0 Qa Tl Tl HSO Sk ( ) ( ) + E T10 − E Tl0 l\=1 | )( | ∑ ( ∂H ) | ) ( 0| 0 |HSO |S 0 S 0 | T Q |S 0 m m ∂Qa 0 a k ∑ 1 ( |− →| ) ( 0) a ( ) + ] × Sk0 | M |S00 0 E T1 − E Sm m\=l

(7.41)

In Eq. (7.41), the first two terms inside the parenthesis represent the transition induced by direct spin–orbit and spin-vibronic couplings between T10 and Sk0 , while the third and fourth terms represent the transitions induced by higher-order mecha∑ ( ∂H0 ) Qa ) perturbations. nisms involving both spin–orbit and vibronic (HV ≡ ∂Qa a

Q0

So, neglecting the spin-vibronic term which is relatively small, the spin-forbidden singlet–triplet transition becomes allowed in the first and higher orders, respectively, through spin–orbit and spin–orbit and vibronic interactions taken together. Vibronic structure and polarization characteristics of the phosphorescence spectra, arising from different mechanisms, are discussed below.

238

7 Electronic Spectra of Polyatomic Molecules

Mechanism I: 3 3

ππ

∗

1

S0 =

ππ ∗ H SO 1ππ ∗

( ππ ) − E ( ππ ) ∗

3

∗

1

ππ ∗

1

1

S0

Polarization character determining factor HSO 1

ππ ∗

3

ππ ∗

1

S0

Mechanism II: 3 3

ππ ∗

1

S0 = `

ππ ∗

SO

1

nπ ∗

( ππ ) − ( nπ ) ∗

3

1

∗

1

π∗

1

S0

Polarization character determining factor HSO 1

nπ

∗

3

ππ ∗

1

S0

Mechanism III: 3 3

ππ ∗

1

S0 =

ππ ∗ H SO 1σπ ∗

( ππ ) − E ( σπ ) 3

∗

1

∗

1

σπ ∗

1

S0

Polarization character determining factor. HSO 1

σπ ∗

3

ππ ∗

1

S0

7.3 Relaxation Mechanism

239

Mechanism IV: 3 3

ππ ∗

1

S0 =

ππ ∗

( ππ ) − ( ∗

3

∗

3

V

π ∗)

3

π ∗ H SO

3

∗

1

( ππ ) − E ( ππ ) ∗

3

1

1

∗

ππ ∗

1

S0

Polarization characterdetermining factor HSO 1

ππ

HV

∗

3

nπ ∗ 1

3

ππ

∗

S0

Mechanism V: 3 3

ππ ∗

1

=

0

ππ ∗

( ππ ) − ( ∗

3

∗

1

SO

1

∗

1

π ∗)

V

1

ππ ∗

( ππ ) − ( ππ ) ∗

3

1

1

∗

ππ ∗

Polarization character determining factor HSO

HV 1

ππ

∗

1

nπ

∗

1

3

ππ

∗

S0

Mechanism VI: π∗

3 3

π∗

1

0

=

(

3

π

SO

∗

1

ππ ∗

) − ( ππ ) ∗

1

1

ππ ∗

Polarization character determining factor

1

ππ

∗

HSO

1

S0

3

nπ ∗

1

0

1

0

240

7 Electronic Spectra of Polyatomic Molecules

Mechanism VII: π∗

3 3

π∗

1

0

=

(

3

V

π ∗) −

3

ππ ∗

3

ππ ∗

( ππ ) ( 3

∗

3

SO

∗

π∗

1

)− (

1

∗

)

1

π∗

1

0

Polarization character determining factor HV.

HSO 1

nπ

∗

3

ππ 1

∗

3

nπ ∗

S0

In planar aromatic molecules, the direct spin–orbit coupling between 1 π π ∗ and π π ∗ states is extremely small (since all one-centre integrals of the form (π |HSO |π ) are zero). So the mechanism I is not an important source of phosphorescence intensity despite the fact that 1 π π ∗ → S0 transitions are of low energy (therefore, small energy denominator) and large value of transition moment. Mechanism II involves the spin–orbit interactions between the lowest energy 3 π π ∗ state and 1 nπ ∗ state which have large one-centre contributions of the form (n|HSO |π ( ). Because ) ( of this ) and the fact that the energy gap between the interacting states, E 1 nπ ∗ − E 3 π π ∗ , is usually small, this mechanism is recognized as an important intensity source for T 1 (1 π π ∗ ) → S0 transitions in nitrogen heterocyclic and aromatic carbonyl compounds. In mechanism III, the T 1 (3 π π ∗ ) → S0 radiative transition is induced by spin–orbit perturbation of the lowest triplet state by singlet states of σ π ∗ or π σ ∗ character, which is effective through the non-vanishing one-centre contributions of the type (σ |HSO |π ). However, since the perturbing σ → π ∗ (or π → σ ∗ ) singlet–singlet transitions are of high energy (therefore, large energy denominator) and low radiative power, this mechanism is not expected to be as important as mechanism II. Although the mechanisms IV and V are higher-order mechanisms requiring vibronic as well as spin–orbit coupling, they can be very important sources of phosphorescence intensity since the perturbing singlet–singlet transitions are low-energy π → π ∗ transitions which carry high oscillator strength (which is a measure of intensity of a spectral band, proportional to the square of the transition dipole matrix). Thus, mechanisms II, IV and V are possible sources for T1 (3 π π ∗ ) → S0 phosphorescence. Similarly, mechanisms VI and VII are the possible radiative sources for T1 (3 nπ ∗ ) → S0 transition. Although all the mechanisms discussed above are more or less important to induce a radiative phosphorescence transition, mechanisms II and VI are the most important for T 1 (3 π π ∗ ) → S0 and T 1 (3 nπ ∗ ) → S0 transitions, respectively. 3

7.3 Relaxation Mechanism

241

Polarization characteristics of the phosphorescence spectra The order of the triplet state lifetime τP is one of the measures which helps to recognize the nature of the triplet state. Generally, a 3 π π ∗ state has a life time of the order of seconds and a 3 nπ ∗ one has a life time of the order of milliseconds. Besides, polarization characteristics of the phosphorescence bands are also helpful to identify the phosphorescence state (3 π π ∗ or 3 nπ ∗ ). According to mechanisms IVII, the transition moment of T 1 (3 π π * or 3 nπ*)→ S0 is determined by the transition moment associated with the singlet–singlet transition through which the borrowing of intensity is conducted. Thus, the 0–0 band of the phosphorescence spectra are inplane polarized through mechanisms I, IV, V and VI, whereas out-of-plane polarized for the mechanisms II, III and VII. As stated before, the mechanisms II and VI are the most important of the radiative processes, so 0–0 band of the phosphorescence spectra in most cases is out-ofplane polarized when the lowest triplet state is of π π ∗ type and is plane polarized when lowest triplet state is of nπ ∗ type in N-heterocyclic and aromatic carbonyl compounds. Vibronic structure of the phosphorescence spectra Vibrational structure of the phosphorescence spectra is also helpful to determine the nature of the phosphorescence state. For (3 π π ∗ (T1 ) → S0 ) phosphorescence, bands of electronic origin are out-of-plane polarized and those of vibronic origin are in-plane polarized. Reverse is the case for (3 nπ ∗ (T1 ) → S0 )) phosphorescence. Generally, it has been found that in the phosphorescence emission from the 3 nπ ∗ state, the C = O stretching vibration is very active in aromatic carbonyls and in many cases progression of this vibration is also found to appear. On the other hand, ring frequencies are found active in the phosphorescence from the 3 π π ∗ state.

7.3.4 Radiation Less Transition Radiation less transition in the condensed phase involves conversion of electronic energy of the initial state into the vibrational energy of the final electronic state. From time-dependent perturbation theory in quantum mechanics, it can be shown that the rate constant for a radiation less transition |i) → |f ) is given by, Knr (l) =

|2 2π ∑ || (f |H ' |i)| δ(El − Es ) h

(7.42)

l

where ‘δ’ is the Dirac delta function. H ' is the perturbation Hamiltonian which is HSO for intersystem crossing and HV for internal conversion. Let us consider the case of radiation less transition from the lowest triplet state (T1 ) to the ground state (S0 ). So according to Born–Oppenheimer approximation,

242

7 Electronic Spectra of Polyatomic Molecules

|i) = Ψ1n (q, Q) = ψ1 (q, Q)ϕ1n (Q)

(7.43)

(where ψ’s and ϕ’s are electronic and vibrational wave functions, n signifies the vibrational state of the respective electronic state). First, consider the lowest triplet state to be a 3 π π ∗ -state. So the radiation less transition from the ground vibrational level of this triplet state to a vibrationally excited ground electronic state, occurring through spin–orbit cum vibronic coupling, involving a promoting mode Qp , between these two states becomes, | ) ( | (Ψ0n |HSO |Ψ10 ) = (ϕ0n | ψ00 |HSO |ψ10 |ϕ10 ) | )( |( ∂H ) | 0 ) ( 0| |ψ ψ0 |HSO |ψ20 ψ20 | ∂Q 1 p 0 (ϕ0n |Qp |ϕ10 ) = 0 0 E1 − E2

(7.44)

Note that in the initial state i = 1, the molecule was in the ground vibrational level n = 0 and the mixing of the ψ10 - (for being a 3 ππ*-state) with a 1 nπ*-state via spin– orbit coupling is not taken into consideration because of large energy denominator. Here, we have assumed that the final (ground) electronic state, which is a 1 π π * state, remains unperturbed, i.e.ψ0 = ψ00 . So the intervening state ψ20 must be triplet nπ *it can be shown from state (i.e., 3 nπ *). Since the electronic state ψ10 is a 3 π π(*- state, ( 0 | ∂H ) | 0 ) |ψ , is non-vanishing group theory (Chap. 8), that the vibronic integral, ψ | 2

∂Qp

0

1

only when Qp (the promoting mode) is an out-of-plane mode of the hetero-aromatic molecule. Again the vibrational wave functions in the respective electronic states can be expressed as products of wave functions of different normal modes, ϕ0n =

∏ l

1 ϕ10 = χp0

χln0 l ∏

1 χk0

(7.45)

k\=p

Here, nl is the vibrational quantum number associated with the wavefunctions of the 1 is lth normal mode designated by χln0 l of the ground electronic state. Similarly, χk0 the vibrational wavefunction of the kth normal mode in the ground vibrational state of the respective electronic state. As discussed earlier, spin–orbit matrix is nonzero only for n → π* transition. So it can be easily understood from Fig. (7.12) that |( ∂H ) | 0 ) ( 0 | ∂ || 0 )) ( 0 | ∂ || 0 )) |ψ ψ20 | ∂Q ψ2 | ∂Qp H ||ψ1 − ψ2 |H ∂Qp ||ψ1 1 ∂ p 0 ( 0 ) ( 0 ) |π )0 = ≈ (n| ∂Qp E1 − E20 E1 − E20

(

where the states ψ10 and ψ20 have π ∗ -orbital in common.

(7.46)

7.3 Relaxation Mechanism

243

n

π

π∗

π∗

π∗

n

n

π

π

Fig. 7.12 nπ * and π π * states connected by spin–orbit interaction

Substituting Eqs. (7.45 and 7.46) in Eq. (7.44), we get | | \ \ | ) ∂ | 1 0 | |π )0 χpn (Ψon |HSO |Ψ10 ) = (n|HSO |π ∗ 0 (n| |Qp |χp0 p ∂Qp ∏( | ) × χ0 | χ1 knk

k0

(7.47)

k\=p

Thus, the rate constant (7.42) for radiation less decay becomes, 2π ∑ |(Ψ0n |HSO |Ψ10 )|2 δ(E10 − E0n ) h n | |2 | | ∗ )|2 | ∂ 2π || | | | (n| (n|HSO π |π )0 || = | h ∂Qp ⎧ ⎫ | | \|2 ∏ |( ⎬ ∑ ⎨||\ | )| 2 | | | 0 1 | χ 0 | χ 1 | δ(E10 − E0n ) Q χ × | | | | χpn p p0 kn k0 k p ⎩ ⎭ n

Knr =

(7.48)

k\=p

Note that the normal mode Qp appears both in the electronic integral (n| ∂Q∂ p |π ) | | \ \ | 1 0 | and also in the vibrational integral χpn . Thus, vibrationally active out-of|Qp |χp0 p plane mode Qp acts both as a promoting mode (which makes the electronic factor of the radiation less process non-vanishing) and an accepting mode (which contributes to the vibrational factor) for these radiation less processes. The intersystem crossing between nπ ∗ and π π ∗ states of different spin multiplicities or between the nπ ∗ -phosphorescence state and the ground state does not require any out-of-plane vibration acting as a promoting mode for the process to be

244

7 Electronic Spectra of Polyatomic Molecules

conducted. In these processes and in many S1 → S0 internal conversion, the vibrationally active out-of-plane mode p enters only in the vibrational part of the rate expression. Thus, the non-radiative rate for transition from nπ ∗ triplet state to the ground state 1 ( ππ*) becomes, 2π ∑ |(Ψ0n |HSO |Ψ10 )|2 δ(E10 − E0n ) h n | )|2 2π ∑ ||( 0 || ψ0 HSO |ψ10 | |(φ0n | φ10 )|2 δ(E10 − E0n ) = h n | )|2 ∏ |( 0 | 1 )|2 2π ∑ ||( 0 || | χ | χ | δ(E10 − E0n ) ψ0 HSO |ψ10 | = lnl l0 h n

Knr =

(7.49)

l

For internal conversion from Ψ10 → Ψ0n , the non-radiative rate expression is, | | 2π ∑ ||( 0 || ∂H || 0 )||2 2 ψ ψ 1 | |(ϕon |Qk |ϕ10 )| δ(E10 − E0n ) | 0 ∂Q h k k | | ∑ |( 0 | ∂H | 0 )|2 |( 0 | | 1 )|2 ∏ |( 0 | 1 )|2 2π | | | | | | | | | χ | χ | δ(E10 − E0n ) = lnl l0 | ψ0 ∂Q ψ1 | χknk Qk χk0 h k

Knr =

k

l\=k

(7.50) The vibrational factors for the radiation less transitions, in general, depend upon the frequency of vibration in the final state, equilibrium displacement and frequency change on electronic excitation and vibrational anharmonicity. For large electronic energy gap, the vibrational overlap integral decreases rapidly with increasing vibrational quanta. Thus, for radiation less transition in aromatic hydrocarbons involving large energy gaps, the vibrational overlap factors are dominated by CH stretching vibrations having high frequencies of vibration. The dramatic reduction of T1 → S0 intersystem crossing rate observed in deuterated aromatic hydrocarbons supports such conjecture. For small energy gap, the CC stretching modes become important for non-radiative transitions as the frequencies and bond lengths undergo changes on electronic excitations which are not small. But from the same standpoint, out-ofplane modes are not that important in non-radiative transitions in aromatic molecules. But in hetero-aromatic and carbonyl molecules, out-of-plane modes play significant roles in the non-radiative transitions. In such molecules, sometimes the energy gap between the lowest nπ * and π π * states is found to be so small that they may not maintain their pure identities, they become admixtures of the two. In such cases, the promoting mode(s) may be strongly distorted (Δνp \= 0) and displaced ΔQp \= 0 which give rise to large increase in the overlap factor for the non-radiative transitions.

7.4 Molecular Interaction

245

7.4 Molecular Interaction In the previous section, we have seen how the excitation of a molecule to a higher state is lost and the molecule returns to the ground state by various radiative and non-radiative pathways. However, in the presence of other molecule(s) in its neighbourhood some interesting processes arise due to molecular interaction and in the present section we shall discuss about these processes.

7.4.1 Charge Transfer Process When two molecules are mixed in a solvent, interaction between them may result in the formation of a molecular complex. In the usual case, this interaction is electrostatic in nature, i.e. dipole–dipole interaction, dipole-induced dipole interaction, dispersive force and stark effect. This type of interaction is not strong and produces small changes in the spectral characteristics of the molecules. However, in some cases this interaction is not so small and strong bands are observed at positions far from the wavelengths of the bands of the individual molecules. For example, when iodine is mixed with starch a strong blue band is observed. A strong colour appears when aromatic hydrocarbons and aromatic nitro or quinines such as nitrobenzene and p-benzoquinone are mixed in a solution. These kinds of colour change actually arise due to the formation of a molecular complex called charge transfer complex, and this process is named as charge transfer process. In the process of forming this charge transfer complex, a fraction of electronic charge is transferred from one molecule to another one (intermolecular) or from one part of a molecule to another part (intramolecular) of the same molecule. The electrostatic attractive force between the transferred fraction of electronic charge and that of the hole stabilizes the complex. The molecule which transfers electronic charge is called the donor molecule and which receives it is called the acceptor molecule. However, this attractive force is much weaker than the energy of a covalent bond. In such molecular complexes, electronic transitions may take place from a lower to an excited electronic state resulting in an absorption band in the optical region of the electromagnetic spectrum. Thus, optical spectroscopy plays a powerful technique for studying the characteristics of charge transfer complexes. A theoretical explanation was first given by R. S. Mulliken to explain the formation of charge transfer complex. Let us consider that a donor molecule (D) weakly interacts with an acceptor molecule (A) to form a molecular complex (D–A). We shall consider the simple case of a 1: 1 complex (D − A) in which both D and A are neutral having closed shell structure. We also assume that the donor molecule (D) can easily release an electron from the HOMO to form a cation (D+ ) and the acceptor molecule (A) can easily take this electron and place it in the LUMO to form an anion (A– ). Now, these two oppositely charged ions interact and form a complex (D+ - A– ) in a singlet state by pairing the electrons on D+ and A– in their respective HOMO and LUMO

246

7 Electronic Spectra of Polyatomic Molecules

orbitals. We represent the normalized wave function of the charge transfer structure by ψ 1 = ψ(D+ - A– ) and that of the non-ionic structure arising from interactions such as dispersive force by ψ 0 = ψ((D–A). Due to the above interactions, the ground (ψ N ) and excited (ψ E ) electronic states of the complex will no longer remain ψ 0 and ψ 1 , respectively, but become some appropriate combinations of them. ψ = ψN /E = aψ0 + bψ1

(7.51)

Since these functions are ortho normal, so we get (ψ|ψ) = a2 + b2 + 2abS01

(7.52)

(ψN |ψE ) = 0

(7.53)

and

Here, S 01 is the overlap integral (ψo |ψ1 ). The energy values can be determined from the secular equation after applying the variational method | | | E0 − E H01 − S01 E | | | | H01 − S01 E E1 − E | = 0

(7.54)

where E0 = (ψ0 |H |ψ0 ), E1 = (ψ1 |H |ψ1 ) and H01 = (ψ0 |H |ψ1 ). The solutions of this Eq. (7.54) are 1 ) [(E1 + E0 ) − 2S01 H01 E= ( 2 2 1 − S01 # $ ] 4(H01 − S01 E1 )(H01 − S01 E0 ) 1/2 ±(E1 − E0 ) 1 + (E1 − E0 )2 2 In a weak complex, S01 >> 1 and solutions become

4(H01 −S01 E1 )(H01 −S01 E0 ) (E1 −E0 )2

(7.55)

>> 1. So the approximate

E = EN ≈ E0 −

(H01 − S01 E0 )2 (E1 − E0 )

(7.56)

E = EE ≈ E1 +

(H01 − S01 E1 )2 (E1 − E0 )

(7.57)

and the corresponding approximate ratios of the two coefficients for the two solutions are b (H01 − S01 E0 ) , for E = EN ≈− a (E1 − E0 )

(7.58)

7.4 Molecular Interaction

247

a (H01 − S01 E1 ) ≈− , for E = EE b (E1 − E0 )

(7.59)

The potential energy curves of the two states are shown in Fig. 7.13. Due to 2 01 −S01 E0 ) and the molecular interaction, the ground state is stabilized by an amount (H(E 1 −E0 ) upper or the excited state is uplifted (i.e., destabilized) by an amount the excitation energy of the charge transfer complex is hνCT = EE − EN = E1 − E0 +

(H01 −S01 E1 )2 . (E1 −E0 )

β02 + β12 E1 − E0

So

(7.60)

where β 0 = H 01 - S 01 E 0 and β 1 = H 01 - S 01 E 1 . The energies E 1 and E 0 can be approximately written as E1 = E∞ + IP − EA − G1

(7.61)

E0 = E∞ − G0

(7.62)

Here, E ∞ , I P , E A , G1 and G0 correspond to the energy of the complex at infinite separation of D and A, (vertical, Franck–Condon) ionization potential of the donor, (vertical, Franck–Condon) electron affinity of the acceptor, an electrostatic energy Fig. 7.13 Potential energy curve for the charge transfer spectra. R, I and EA are the internuclear distance, vertical ionization potential of the donor (D) and vertical electron affinity of the acceptor (A)

EA E1

IP EE

Energy

E1

E0 ( E0 EN

R

R=

)

248

7 Electronic Spectra of Polyatomic Molecules

sum between D+ and A– and the sum of several electrostatic interaction energies including dipole–dipole interaction, van der Waals interactions, etc., between D and A at ground state. The sign of G0 depends on the nature of the complex. In hydrogen bonding, it may be positive. In the strong charge transfer complex, usually it is found that D-A distance is much shorter than the estimated van der Waals distance and G0 is negative due to exchange repulsive force (Fig. 7.13). If no approximation is made, then from Eq. (7.55), we can get [

E1 − E0 (hνCT ) = (EE − EN ) = 2 1 − S01 2

2

]2 [ 1+

4β0 β1 (E1 − E0 )2

] (7.63)

For a series of complex in which the acceptor (A) is the same molecule and the donor (D) varies but having the same kind of active centre, E A + G1 –G0 remains more or less a constant (C). Then, E 1 –E 0 = I P —C (from Eqs. 7.61 and 7.62) and the above Eq. (7.63) becomes [ (hνCT )2 = (EE − EN )2 =

IP − C 2 1 − S01

]2 [ 1+

4β0 β1 (IP − C)2

] (7.64)

This equation was successfully applied to the amine-iodine complexes. Now, we shall discuss about the intensities of the charge transfer spectra. The transition moment, on the square of which the intensity depends, for the charge transfer transition is % % ∑ →ri ψE dτ μ → CT = ψN μψ → E dτ = −e ψN (7.65) Taking ψN = aψ0 + bψ1 and ψE = cψ0 + d ψ1

(7.66)

where the constants a, b, c and d satisfy the conditions specified by the Eqs. (7.52, 7.53, 7.58 and 7.59), we get μ → CT = acμ → 00 + bd μ → 11 + (ad + bc)μ → 01

(7.67)

Here, % μ → ij = −e

ψi

∑

→rl ψj dτ, i, j = 0

(7.68)

Since ψ N and ψ E are orthonormal, then (ac + bd) = −(ad + bc)S01

(7.69)

7.4 Molecular Interaction

249

S01 being the overlap factor (ψ0 |ψ1 ). Thus, the Eq. (7.67) becomes μ → CT = bd(μ → 11 − μ → 00 ) + (ad + bc)(μ → 00 ) → 01 − S01 μ

(7.70)

Let us now consider that the charge transfer occurs from HOMO (ψ d ) of the donor to the LUMO (ψ a ) of the acceptor. Since the electrons in the closed shells of both the donor and the acceptor, excepting (ψ d ), are not taking part in the charge transfer mechanism, we can exclude them from the formation of the molecular wave functions ψ 0 and ψ 1 . We consider here only the singlet states. 1 ψ0 = ψd (1)ψd (2) · √ [α(1)β(2) − α(2)β(1)] 2

(7.71)

1 1 ψ1 = / ( ) [ψd (1)ψa (2) + ψd (2)ψa (1)] · √2 [α(1)β(2) − α(2)β(1) 2 2 1 + Sda (7.72) Here, ψ 1 has been generated in the manner similar to the formation of an antisymmetric MO from two atomic orbitals (see hydrogen molecule in Chap. 6). The overlap integral S 01 is √

2Sda S01 = (ψ0 |ψ1 ) = ( ) 2 1/2 1 + sda

(7.73)

Now, we shall proceed to determine the transition moments, μ → 01 and μ → 11 % μ → 01 = −e

ψ0

∑

→ri ψ1 dτ % ) [ψd (1)ψd (2)] · [→r1 + →r2 ] · [ψd (1)ψa (2) + ψd (2)ψa (1)]dτ

−e =/ ( 2 2 1 + Sda √ % % − 2e ∑ = /( [ ψ (i) · → r · ψ (i)dτ + S ψd (i) · →ri · ψd (i)dτi ] d i a i da ) 2 1 + Sda

(7.74) Similarly, we can obtain μ → 11

% ∑ −e ) =( [2Sda ψd (i)(→ri )ψa (i)dτi 2 1 + Sda % % + ψd (i)(→ri )ψd (i)dτi + ψa (i)(→ri )ψa (i)dτi ]

(7.75)

250

7 Electronic Spectra of Polyatomic Molecules

& ∑ We make the integral ψd i (→ri )ψd dτ = →rd which represents the average position of the electron in the orbital ψd .Now, introduce an approximation (called Mulliken approximation) ∑%

ψd (→ri )ψa dτ ≈ Sda

→rda 2

(7.76)

& ∑& Again, we can write ψd (i)(→ri )ψd d τ + ψa (i)(→ri )ψa d τ ∼ →rda (the distance between D and A) at least in its order of magnitude. Thus, we get, from Eqs. (7.74) and (7.75) μ → 01

√ ] ] [ [ −e 2 Sda →rda →rda = /( ) →rd + 2 = − es01 →rd + 2 2 1 + Sda

(7.77)

and

μ → 11 ≈ −e→rda

(7.78)

This is reasonable since ψ1 correspond to the dative structure D+ —A– . For convenience, we can choose the origin of the coordinate system at →rd (i.e., at → 00 . the centre of gravity of the electrons in the molecular orbital ψ D ) and neglect μ From Eqs. (7.70), (7.77) and (7.78), we get μ → CT = bdμ → 11 + (ad + bc)μ → 01 S01 → 11 ≈ bd μ = bd μ → 11 + (ad + bc) μ → 11 2

(7.79)

since for weak complex S01 is not large, so it is approximately taken as zero. Intensity of a spectral line is expressed by an entity called oscillator strength (f ) which is theoretically expressed for the present case as f theo = 4.704 × 10−7 × ν max × | μ → CT | 2

(7.80)

Here, μCT is the transition moment of the charge transition band expressed in Debye and ν max is the wavenumber of the band maximum. The corresponding experimental value is calculated from the observed spectra according to the formula f

expt

= 4.319 × 10

−9

% ×

] [ d ν = 4.319 × 10−9 × max Δν 1/2

(7.81)

where max is the molar extinction coefficient (expressed in per mole per cm) at the peak intensity of the band and Δν 1/2 is the band half width in cm−1 at the position of the band maximum. Thus, from the last two equations we get

7.4 Molecular Interaction

251

[

max Δν 1/2 |μ → CT | = 0.0958 × ν max

]1/2 (7.82)

From the normalization conditions of the wave functions ψN and ψE , using equations like (7.79) and (7.82), etc., and introducing certain logistic approximations, the constants a, b, c and d can be determined in different ways. Using Eqs. (7.58), (7.59) and (7.66), we get, βo2

( )2 b = (H01 − S01 E0 ) = (E1 − E0 )2 a 2

(7.83)

and β12 = (H01 − S01 E1 )2 =

( c )2 d

(E1 − E0 )2

(7.84)

Thus from Eqs. (7.56) and (7.57), we get EN = E0 −

( )2 b (E1 − E0 ) a

(7.85)

and EE = E1 +

( c )2 d

(E1 − E0 )

(7.86)

Therefore, ( )2 ( ) ] c 2 b = (EE − EN ) = (E1 − E0 ) 1 + + a d [

hνCT

(7.87)

E N –E 0 may be equated to the heat of formation (ΔH) of the complex (i.e. in the gaseous phase or in non-polar solvent) if the term G0 is neglected in Eq. (7.62). Thus, ( )2 b ΔH ≈ − (E1 − E0 ) a

(7.88)

In weak complexes a >> b and d >> c, therefore hν CT ≈ E 1 –E 0 . Thus, ( )2 ΔH b ≈− hνCT a Again taking μ → 00 as zero, we can write

(7.89)

252

7 Electronic Spectra of Polyatomic Molecules

% μ →N =

ψN · μ → · ψN dτ = a2 μ → 00 + b2 μ → 11 + 2abμ → 01 ≈ b2 μ → 11 + 2abμ → 01

(7.90)

Since μ → N and ΔH can be determined experimentally, the constants a and b are determined from the last two equations (using Eqs. 7.77 and 7.78). In a similar way, the other constants c and d can also be determined. More and more strong is the complex, the larger is the contribution from the dative structure, i.e. b becomes larger. Hence from Eqs. (7.79), (7.80) and (7.89), → CT | , and hence, the oscillator strength f CT increases with we can say that ΔH, | μ the increase of the value of the constant b, i.e. stronger is the complex, larger is the intensity of the charge transfer band at par with the expectation. All these theoretical predictions are confirmed in the benzene (and its methyl derivative)—iodine charge transfer complexes.

7.4.2 Hydrogen Bonding Hydrogen bonding is the bonding between a hydrogen atom (attached to another atom through a covalent bond) with a more electronegative atom. Generally, the electronegative atoms, like oxygen (O), fluorine (F) and nitrogen (N), etc., in the said bond, due to their higher electron affinity attract electron from the hydrogen atom, leaving it slightly positively charged. Due to the small size of the hydrogen atom with respect to other atoms, not only the positive charge density on it is high but also it comes close to another proton acceptor. Then, an electrostatic force between the hydrogen atom and the proton acceptor form a kind of bond, which is not as strong as a covalent bond but stronger than that for van der Waals interaction. Thus, the hydrogen bond length is greater than a covalent bond length but less than the van der Waals radii. The hydrogen bonding is described as electrostatic dipole–dipole interaction. There may be also some covalent character. Let us represent the hydrogen bonding in a molecular complex as X–H······Y, where X and Y are two atoms, more electronegative than H. A typical example for such a complex is –O–H······O < . Typical examples of different resonance structures of (–O–H······O < ) system are: −O − H O < −O− H+ O < −O− H − O+ < The first one (1) corresponds to no bond structure. The second one (2) corresponds to an ionic nature of the OH bond causing an electrostatic interaction with the proton acceptor. Structure (3) is a charge transfer structure where a covalent bond is formed

7.4 Molecular Interaction Fig. 7.14 Molecular orbital description of the hydrogen bonding system

253

(antibonding) ψσ* ○○

Ψlone

(lone pair) acceptor

ψσ

○○ (bonding) donor, O―H

between hydrogen and proton acceptor atom. The last structure plays an active role in the covalent character of hydrogen bond. There are other two resonance structures −O+ H− O < −OH− O+ < But their contributions seem to be much small since chemical and physicochemical evidence shows that the polarization form of O–H to be Oδ– –Hδ+ and not as shown in the last two structures. In the molecular orbital picture, the structure 3 can be explained as follows. Here comes into consideration the σ (O–H) bond and the lone pair orbital of the proton acceptor (O). In this structure, the two lone pair electrons are distributed in the covalent bond formed between the lone pair orbital of the acceptor and the antibonding orbital of the donor (Fig. 7.14). This will definitely weaken the donor bond, and as a consequence, the donor bond length increases. Not only this, a new and long bond between the proton and proton acceptor (here in the present case, O–H+δ , δ < 1) is also formed. If the wave functions ψ1 , ψ2 and ψ3 are attached to the three resonance structures 1, 2 and 3, respectively, then the wave functions of the system can be represented by Ψ = C1 ψ1 + C2 ψ2 + C3 ψ3

(7.91)

where C1 2 + C2 2 + C3 2 = 1. So the percentage of the ith structure is Ci2 wi × 100 = × 100 2 2 2 w1 + w2 + w3 C1 + C2 + C3

(7.92)

Many works have been done on the percentage calculations of different resonance structures in different hydrogen bonding systems. For the O–H O < system, it has been found that the major contributions come from the resonance structures 1 and 2

254

7 Electronic Spectra of Polyatomic Molecules

for the equilibrium position of hydrogen. The covalent bond structure contribution is less than 10%. However, it has also been found that: When the distance between the O atom of the O–H bond and acceptor O atom decreases, hydrogen bond becomes stronger and the contribution of w3 becomes larger. When the hydrogen atom moves towards the acceptor O atom, then also the contribution of w3 increases, i.e. the stronger the hydrogen bonding system, the larger is the contribution of covalent structure to hydrogen bonding process. However, the percentage contributions from the electrostatic interactions (structures 1 and 2) remain always much above that from the covalent structure (3). In general when the hydrogen bond becomes stronger, the relevant bond distance becomes smaller. The strength of hydrogen bond is 21 kJ/mole for O–H······O and 161.5 kJ/mole for F–H······F. Thus, O–H bond distance is less than H······O in H2 O, whereas F–H bond length are nearly same as H······F in HF, i.e. the proton lies midways between the two fluorine ions in the latter case. Another interesting thing has been obtained from theoretical calculation. Covalent contribution is found to increase with the use of sp3 -hybridized orbital than pure 2p-orbital of the oxygen atom, and this can be attributed to the greater overlap power or ability of the former orbital. Electronegativities of carbon and hydrogen are nearly equal, so generally this bond (CH) is not supposed to form hydrogen bond. However, the presence of neighbouring electronegative atom(s) may activate this group to form effective hydrogen bond. For example, hydrogen bonds are found in HCN, HCF3 and the like compounds. Simple example of hydrogen bond is water molecule. In isolated water molecule, there are one hydrogen atom and two lone pairs of electrons attached to each of the two oxygen atoms. So each water molecule acts as both proton donor and acceptor, and at most, it can form four hydrogen bonds. Owing to difficulty in breaking these bonds, liquid water has a high boiling point relative to its low molecular mass. In fact boiling points of NH3 , H2 O and HF are found to be higher than the corresponding heavier analogues PH3 , H2 S and HCl. Increase in the melting point, boiling point, solubility and viscosity can be explained from the viewpoint of hydrogen bonding. Besides lowering of frequencies of OH (or similar), stretching modes are found in the vibrational spectra on hydrogen bonding in their respective compounds. In the n → π * electronic transition in a molecule, the initial state is found to be depleted but the final state is not found to be much affected on account of hydrogen bonding. So such type of band generally exhibits blue shift due to hydrogen bonding. Similar effect is also found when the molecule undergoes change from less to more polar environment.

7.4 Molecular Interaction

255

7.4.3 Excimers and Exciplexes When two molecules, M and N, in their ground states, come close to each other, in general they do not form any composite configuration. Only very weak dispersive force arises at a relatively long intermolecular distance. However when one of them, say, M is excited (M*) and the other one (N) in its ground state is polar or polarizable, some charge transfer interaction may take place between them which results in a stabilization of energy and a molecular complex M*N is formed. Such a complex has a longer lifetime than the corresponding ground state collision complex MN (if at all is formed) and such excited state (M*N) is metastable. This excited state complex exhibits the properties which are different from those of the individual molecules. Such a complex is called exciplex when M and N are different and excimer when both are same: M∗ + N → M∗ N (exciplex)

(7.93a)

M∗ + M → M∗ M (excimer)

(7.93b)

N∗ + N → N∗ N (excimer)

(7.93c)

A simple explanation of the formation of such complex can be given from the standpoint of molecular orbital theory as follows. Let us consider that the highest occupied molecular orbitals (HOMO) of both the molecules are completely filled with two electrons with opposite spins in their respective ground states and let them be designated by ϕ hm and ϕ hn of the molecules M and N, respectively. Their respective lowest unoccupied molecular orbitals (LUMO) are ϕ lm and ϕ ln . From the basic principle of the formation of molecular orbitals, we can say that here the two orbitals ϕ hm and ϕ hn will form two orbitals of the complex, namely ψ h (bonding) and ψ h * (antibonding). In a similar way, the two LUMOs ϕ lm and ϕ ln will form two orbitals of the complex (Fig. 7.15), namely ψ l (bonding) and ψ l * (antibonding). So the electronic configurations of (MN) and (M*····N) are ψ h 2 ψ h *2 and ψ h 2 ψ h *1 ψ l 1 , respectively, and the corresponding bond orders are zero and one. So in the ground state, the complex (MN) does not form any bonding configuration, i.e. it gives rise to a repulsive state which leads to dissociation into the individual species M and N. But the (M*····N) gives rise to a bonding configuration with a minimum in the potential energy curve. This is illustrated in Fig. 7.16. Since no complex is formed in the ground state, so no evidence of the formation of exciplex can be found from the absorption spectra. However whenever a downward transition occurs, it starts from the potential minimum of the excited state (of the exciplex). Since the lower state is a repulsive

256

7 Electronic Spectra of Polyatomic Molecules

Fig. 7.15 Electronic configurations of M, N, MN and M*····N

one, so the emission spectra of the exciplex do not exhibit any vibrational structure. It exhibit a broad band, the peak of which corresponds to the vertical transition according to Franck–Condon principle. Both singlet and triplet exciplexes are observed, but the latter one is less often observed by their emission. The formation of the exciplex can be proved if the fluorescence intensity of the monomer (M) is found to decrease with the increase of relative concentration of M with respect to N and a new band system is formed on the longer wavelength side with gradual increase in intensity. This is illustrated in Fig. 7.17. In fact, fluorescence excimer was observed by Th. Föster and K. Kasperof Pyrene in solution. They found decrease of monomer fluorescence and appearance of a new band system on the longer wavelength side of the monomer band with increasing intensity with the increase of concentration. The exciplex M*····N possesses all the properties of any electronically excited state, i.e. emission, radiation less transition and photochemistry.

7.4 Molecular Interaction M*N

257

MN

M

N

Monomer discrete excited Vibrational states ΔHex

ΔE00

ΔEex- ΔHex

Monomer discrete ground Vibrational states

rMN

ΔErep

I Eex Exciplex emission

E00 Monomer (M) emission absorption

Fig. 7.16 Potential energy diagram and the related spectral transitions in exciplex (M* N) formation. ΔE 00 corresponds to the 0 ↔ 0 transition between the ground and excited electronic states of M, ΔH ex is the dissociation energy of the exciplex and δErep corresponds to the energy difference between the minimum of the exciplex potential curve and the peak of its emission spectra. (For exciplex M \= N and for excimer M = N)

Let us now consider various mechanisms associated with the excimer formation and decomposition and see how the quantum yields and the various decay rates are determined from the experimental observations. Reactions M + hva → M∗ M∗ → M + hvf M∗ → M M∗ + M → M∗2 M∗2 → M + M∗ M∗2 → M + M + hvf' M∗2 → M + M the asterisk denoting the excited state. So the rate equations are given by

Reactions rates Iabs[ ] kf[ M∗] ki M∗ [ ] ∗ k1 [M] [ ∗ ]M k2[ M2] k'f[ M∗2] k'i M∗2

(7.94)

258

7 Electronic Spectra of Polyatomic Molecules

Emission intensity

C4

C1

C3

C2 C2

C3

C1

C4

λ Fig. 7.17 Concentration dependence of the emission spectrum. The solid lines correspond the monomer (M) and the dashed lines to the exciplex (M*N). The concentrations of M are (C4 > C3 > C2 > C1 ), whereas the concentration of N remains unchanged

) ( [ ] [ ∗] d[M ∗ ] 1 = Iabs + k2 M2 − + k1 [M ] M ∗ dt τ [ ] ) ( [ ∗] [ ] d M2∗ 1 M2 = k1 [M ] M ∗ − + k 2 ' dt τ

(7.95a)

(7.95b)

) ( ) ( Here, τ = 1/ ki + kf and τ ' = 1/ ki' + kf' . Quantum yield of a particular outlet from an excited state is a very important entity and is defined by the fraction of that particular outlet out of all the possible decay processes from that excited state. So the fluorescence quantum yields of the monomer (ηf ) and the excimer (η' f ) emissions are kf [M ∗ ]

ηf = (kf + ki

)[M ∗ ]

+

( ) kf' +ki' ∗ ( k1 [M ][M ] ' ' ) kf +ki +k2

ηfm kf τ ( ( ) )=( ) ) =( k1 τ k1 τ 1 + 1+k 1 + [M ] [M ] ' ' 1+k2 τ 2τ and ηf'

[ ] kf' M2∗ )[ ] =( [ ] kf' + ki' M2∗ + k2 M2∗

(kf +ki ) (kf +ki +k1 [M ])

(7.96a)

7.4 Molecular Interaction

=(

259 '

kf' τ '

ηfm

)=( ) ' 2τ 1 + k1+k 1 τ [M ] ( ) ' ' k τ m 1 ηf 1+k2 τ ' [M ] ηfm k1 τ [M ] ( ( ) ) = = k1 τ (k1 τ [M ] + 1 + k2 τ ' ) 1 + 1+k ' [M ] τ 2 1+

k2 τ ' 1+k1 τ [M ]

' since for photostationary state (1

τ'

k2 )= + k2

(1 τ

)=( 1+

'

ηfm

d[M ∗ ] dt

k2 τ ' 1+k1 τ [M ]

(7.96b)

( d M∗ = 0 = [ dt2 ] ,

) + k1 [M ] 1 + k2 τ ' k2 τ ' = i.e. k1 [M ] 1 + k1 τ [M ] k1 τ [M ]

(7.97)

'

putting Iabs = 0 in Eq. (7.95). Here, ηfm = kf τ and ηfm = kf' τ ' . Thus, we see from Eq. (7.96) that ηf = ηfm and ηf' = 0 for low [M], i.e. when, [M] → 0 and ηf = 0 and ' ηf' = ηfm for high [M], which are usually expected.. If one excites M by applying a light pulse of very short duration (of the order of nanosecond) and studying the formation and destruction of both the monomer and the excimer from the time dependence of the fluorescence of the monomer and the excimer, it is possible to determine the various rate constants associated with the monomer and the excimer which are helpful for getting greater insight into the photophysical behaviour of the system. Let us assume that the pulse width is much smaller than the emission life time. Then, the time-dependent emission intensity of the monomer and the excimer can be determined by solving the Eq. (7.95) under non-photostationary condition. Putting Iabs = 0 and using the initial condition, I(0) = Kf [M*(0)], I' (0) = 0, I(∞) = 0 = I' (∞) = 0, the solution of the simultaneous differential Eq. (7.95) gives [ ] I (t) = kf M ∗ (t) # ∗ $ [M (0)] [(β − μ) exp(−αt) = kf (β − α) + (μ − α) exp(−βt)] [ ] I ' (t) = kf' M2∗ (t) # ∗ $ [ ] ' [M (0)]K1 [M ] exp(−αt) − exp(−βt) = kf (β − α)

(7.98a)

(7.98b)

where α β

$ =

) *1/2 ( 1' (μ + ν) ∓ (ν − μ)2 + 4k1 k2 [M ] 2

(7.99a)

260

7 Electronic Spectra of Polyatomic Molecules

Fig. 7.18 Time dependence of fluorescence intensity of the monomer I(t) and the excimer I´(T )

I´(t)

I(t)

t

μ=

1 + k1 [M ] τ

(7.99b)

1 + k2 τ'

(7.99c)

ν=

Here, it is assumed that [M] remains practically unchanged. It can also be seen that I(0) = k f [M*(0)], I ' (0) = 0, I(∞) = I ' (∞) = 0 and the time dependence of the intensities (7.98) are shown in Fig. (7.18). Let us now study the concentration dependence of the decay constants, α and β. When [M] → 0, 1 τ

(7.100a)

1 + k2 τ'

(7.100b)

α→ β→

( ) τ τ ' k2 ∂α → k1 1 − δ[M ] τ − τ ' + τ τ ' k2

(7.100c)

τ τ ' k1 k2 ∂β → δ[M ] (τ − τ ' + τ τ ' k2 )

(7.100d)

When [M] → ∞, [(

]1/2 )2 1 1 − + k2 − k1 [M ] + 4k1 k2 [M ] τ' τ +( ( )2 ) 1 1 1 1 = − + k2 − 2k1 [M ] ' − + k2 − 2k2 τ' τ τ τ

) *1 (ν − μ)2 + 4k1 k2 [M ] 2 =

7.4 Molecular Interaction

261

)] [ ( 1 1 − k = k1 [M ] − − 2 τ' τ '( ⎡ )2 ( 1 )2 ( ⎤ 21 1 1 1 − + k − − − k 2 2 τ' τ τ' τ ⎣1 + ⎦ [ )]2 (1 1 k1 [M ] − τ ' − τ − k2 )] ( [ 1 1 (7.101) ≈ k1 [M ] − − − k2 τ' τ + k12 [M ]2

*1/2

Thus, we see that under this condition ([M] → ∞), 1 τ'

(7.101a)

1 + k2 + k1 [M ] τ

(7.101b)

∂α →0 δ[M ]

(7.101c)

∂β → k1 δ[M ]

(7.101d)

α→ β→

Moreover for all values of [M] α+β =μ+ν =

1 1 + ' + k2 + k1 [M ] τ τ

(7.102)

Therefore from the measurements of the decay constants, α and β, at various ' concentration and quantum yields, ηfm and ηfm , one can determine all the rate constants ' ' K f , K i , K f , K i , K 1 and K 2 from the above equations. If k 1 and k 2 values are measured at various temperatures, it may be possible to determine the activation energy for excimer formation and decomposition reactions, i.e. k1 = k10 exp{−E1 /KT} and k2 = k20 exp{−E2 /KT}

(7.103)

From their difference, the binding energy (B = E 2 –s1 ) between the monomers in the excimer can be obtained. The activation energy can also be determined in another way. From Eq. (7.96) ηf' ηf

'

=

ηfm

ηfm

(

0 1 ) kf' k1 [M ] k1 τ [M ] = 1 + k2 τ ' kf kf' + ki' + k2

(7.104) '

At low temperatures k2 may be assumed to be very small compared to k f + ' ' k i .Assuming that k f , k f and k i have small dependence on temperature, it is seen '

262

7 Electronic Spectra of Polyatomic Molecules

from Eq. (7.104) that ηf'

≈

ηf

kf' [M ] ( ) k10 exp(−E1 /KT ) ' ' kf kf + ki

(7.105)

'

'

If we assume that at high temperature k2 is large compared with K f + K f , then Eq. (7.104) becomes ηf'

kf'

Kf' k10 k0 = [M ] 0 exp{(E2 − E1 )/KT} = [M ] 10 exp{B/KT} ηf Kf Kf k2 k2

(7.106)

If at high and low temperatures, linear relations are obtained between log(ηf' /ηf ) and 1/T for constant [M], then from Eqs. (7.105) and (7.106), E1 and B can be obtained. In fact, such linear relations were obtained and Birks and coworkers measured the binding energy for the formation of excimer of pyrene in different solvents and these were found not to depend largely on the solvent.

7.4.4 Energy Transfer Phenomena When two molecules D and A come close to each other, the excitation of energy of one may be transferred to the other non-radiatively and sometimes electron may also be transferred which will not be discussed here. In the energy transfer process, the transfer occurs from one (called, donor molecule, D) to the other (called acceptor molecule, A). So we can say the system makes a radiation less transition D*A → A*D. In that case, if the acceptor concentration is increased (without changing the concentration of the donor), it is found that the donor emission decreases and the acceptor luminescence increases. The rate of the probability of such transition is given by Fermi’s Golden rule %

)|2 ( ) ( ) 2π ||( ψi |V |ψf | ρf Ef δ Ei − Ef dEf h )|2 2π ||( ψi |V |ψf | ρf (Ei ) = h

PD→A =

(7.107)

where ψi and ψf are the initial and final state wave functions of the system (with energies E i and E f , respectively), ρ f (energy−1 ) is the final state density and V is the perturbing Hamiltonian. We shall assume that the excitation transfer is much slower than the vibrational relaxation process, so the donor and acceptor molecules may exist in any of the respective vibrational levels before and after the process. Hence, the initial and the final states may be taken as summation over i and f implying summation over the vibrational states of the respective electronic states. Actually, the perturbing Hamiltonian is

7.4 Molecular Interaction

263

V = V (i, a) + V (j, d ) + V (i, j) + V (d , a)

(7.108)

where i and j represent the electrons belonging to the donor (d) and the acceptor (a) molecules in the separated system. Note that V (I, d) and V (j, a) are included in the unperturbed part of the Hamiltonian. If the distance between the donor and the acceptor molecules is large (large in the sense that one can neglect the overlap of the donor–acceptor wave functions, i.e. one can neglect the delocalization of electrons between them) then the interaction (or perturbing) Hamiltonian may be expanded into a series of multipole interactions, i.e. dipole–dipole, dipole-quadrupole, quadrupole– quadrupole, etc., of which the dipole–dipole interaction is the strongest. So we shall consider only the most important dipole–dipole interaction which is of longest range and is given by V =

[ )( )] 1 3( → → μ → μ → · μ → − · R μ → · R D A D A n2 R3 R2

(7.109)

where R is the distance between D and A. Thus, we get, |Vif |2 =

→ D )|2 |(μ → A )|2 K 2 |(μ n4 R6

(7.109a)

where n is the refractive index of the medium and K is a constant depending on the → D ) and (μ → A ) of the donor and orientation of the two transition dipole moments (μ acceptor molecules, respectively, given by, K = u→D · u→A −

)( ) 3( → → − → u→D · R uA · R 2 R

(7.109b)

u→D and u→A being the unit vectors in the directions of the respective dipole moments. In the present case, we shall assume that the distance between D (or d) and A (or a) is so large that the electronic interaction energy is much smaller than the vibronic level width. In other words, the rate of excitation transfer is much smaller than the frequency of molecular vibration. So D and A can be regarded as two completely isolated systems and the wave functions are considered as the product of the electronic and vibrational wave functions of the respective molecules according → D ) and (μ → A ) are replaced by the to Born–Oppenheimer approximation. In that case, (μ → D ) and (μ → A ) and the Franck–Condon product of the electronic transition moment (μ factor weighted by the Boltzmann distribution over the vibronic levels of the respective final states of the molecules. Substitution of Eqs. (7.109) in Eq. (7.107), we get (reminding that the entities within the square brackets are functions of frequencies) by summing over the vibronic levels PD→A =

|( |( )|2 )|2 2π K 2 ρf ∑ [|μ → D |2 gv' (D)| v' |v'' | ] · [|μ → A |2 gv'' (A)| v' |v'' | ] 4 6 h n R v' ,v''

(7.110)

264

7 Electronic Spectra of Polyatomic Molecules

The first square bracket determines the shape of the emission spectrum of the donor molecule and the second square bracket determines the shape of the absorption spectrum of the acceptor molecule. For complex organic molecules in condensed media, the energy levels are i.e. Boltzmann distribution functions (g’s) ( continuous, ) and the overlap integrals ( v' |v'' ’s are continuous functions of frequencies. So the Eq. (7.110), in the continuous energy spectrum becomes PD→A

2π K 2 ρf = h n4 R6

% ' |( |( )|2 (' )|2 ( |μ → A |2 gv'' (A)| v' |v'' | dν (7.111) → D |2 gv' (D)| v' |v'' | |μ

Thus, Eq. (7.111) is proportional to the overlap integral of the fluorescence spectrum of the donor and absorption spectrum of the acceptor molecules. The transition probability can be easily calculated using the spectral data using Einstein’s relations: (ν) =

|( )|2 → A |2 23 π 3 N ν|μ gv'' (A)| v' |v'' | 3000hc(ln10)

(7.112a)

f (ν) =

|( )|2 → D |2 26 π 4 ν 3 n3 |μ gv' (D)| v' |v'' | 3 3hc

(7.112b)

and

where p(ν) is the molar extinction coefficient for the absorption spectrum of the acceptor, f (ν) is the fluorescence quantum spectrum of the donor and N is the Avogadro’s constant. Thus, the above Eq. (7.111) becomes, PD→A ≈

9000K 2 ln(10)ke dν ∫ f (ν) · (ν) 4 7 5 4 6 2 π Nn R ν

(7.113)

where ke is the radiative transition rate of the donor. (For details see Appendix 7.1). This energy transfer probability may also be written as, PD→A

1 = τDf

(

R0 R

)6 (7.114)

where τDf is the fluorescence decay time of the donor in the absence of the acceptor and [

9000(ln 10)K 2 ϕf dν ∫ f (ν) · (ν) 4 Ro = 5 4 128π n N ν

]1/6 (7.115)

ϕf being the fluorescence quantum yield of the donor. Ro is called the ‘Critical transfer distance’, where the rate of energy transfer becomes equal to the rate of decay of the excited state of the donor. So when R is less than the critical transfer distance

7.4 Molecular Interaction

265

(Ro ), energy transfer mechanism dominates and the decay (of the donor) mechanism dominates for the reverse case. For aromatic molecules, Ro is estimated to be about 10–100 Å in its order of magnitude. So far we have considered the case when the donor acceptor distance is large (i.e., the interaction is a long range) for which the initial and the final states in Eq. (7.107) correspond to the system D*A and DA*. However, when the distance between the donor and the acceptor is small, electron exchange is possible. So we can write down the wave functions of the initial and final sates, Ψ i and Ψ f , in terms of the normalized antisymmetric forms as follow: 1 |Ψi ) = √ [ψD' (1)ψA (2) − ψD' (2)ψA (1)] 2

(7.116a)

| ) |Ψf = √1 [ψD (1)ψ ' (2) − ψD (2)ψ ' (1)] A A 2

(7.116b)

where the dashed and the undashed superscripts indicate excited and ground states of the respective species occupied by the electrons (1 and 2). Thus, when these forms are inserted in Eq. (7.107) taking electron spin into consideration, we get %

ψD' ∗ (1)ψA (2)V ψD (1)ψA' (2)d τ1 d τ2 × σD' (1)σA (2)σD (1)σA' (2) % − ψD' ∗ (1)ψA (2)V ψD (2)ψA' (1)d τ1 d τ2 × σD' (1)σA (2)σD (2)σa' (1) (7.117)

(Vif ) =

where σ’s are the spin functions of the marked electrons associated with the donor/acceptor in the ground or excited states as indicated in the subscripts. The first integral of (Vif ) in Eq. (7.117) is the Coulomb integral which vanishes unless σ D ' = σ D and σ A = σ A ' . The discussion so far has been made is based on the Coulombic interaction. In this mechanism, the electrons initially on D* stay on D and the electrons initially on A stay on A*. This is a long range mechanism appearing from the electromagnetic interaction between the two species, donor and acceptor that does not require physical contact between them. Since all the ground states of molecules are singlet, only singlet → singlet energy transfer is possible through the Coulomb term. So the coulombic transfer is expected to be efficient when the when the transitions between the ground and the excited states of each partner have high oscillator strength. The second term of (Vif ) is the exchange integral with V ≡ V (i,j) = e2 /kr12 , k being the dielectric constant of the medium. It represents the electrostatic repulsion ' ' between two charge clouds Q' (1) = eψ D (1)ψ A (1) and Q(2) = eψ D (2)ψ A (2). Since each ψ dies off exponentially with distance from D or A, so the product Q' or Q will be very small throughout all space unless the separation between D and A is very small. So if the separation is small, both functions (Q' and Q) will be sizeable & 2 in the same region of space where r12 is small. Thus, the integral Q' (1) · re12 · Q(2)dτ1 dτ2 may be sizeable (for small R) even though the overlap integrals are

266

7 Electronic Spectra of Polyatomic Molecules

not that sizeable. The exchange integral vanishes unless σ D ' = σ A ' and σ A = σ D . However, σ' is not necessarily equal to σ, so the spins of the donor and acceptor remain unchanged separately in their ground and excited states. So if the excitation of the donor appears from a spin-forbidden transition, the energy transfer mechanism preserves this forbidden character. Thus, triplet → triplet energy transfer may occur through this exchange term but not through the Coulomb term. If the contribution of the vibrational wave functions are introduced in the exchange term of < V if > of Eq. (7.117) as in Eq. (7.111 and 7.112), the transfer probability, appearing from this part, becomes ex PD→A =

2π 2 Z h

% fD (ν)FA (ν)dν =

2π 2 Z JDA h

(7.118)

where fD (ν) and FA (ν) are the normalized line shape (i.e. aand ∫ FA (ν)dν = 1) of the emission and absorption bands of the donor and acceptor, respectively, J DA is the spectral overlap factor and | |2 ∑ |% | 1 ' | | Z = | Q (1) kr Q(2)dτ1 dτ2 | 2

i,f

12

(7.119)

i,f

The separation and concentration dependence is hidden in Z 2 which is expected to fall off exponentially as the distance between the electron and the nucleus is increased. Thus, the rate constants proposed by Dexter takes the form ] [ RDA ex PD→A = CJDA exp −2 L

(7.120)

where C is related to the specific orbital interactions and L is the contact distance between the donor and the acceptor molecule. This process of energy transfer occurring through the exchange term needs overlap of electron clouds and was explained by D. L. Dexter, and hence, it is called Dexter process. This is possible when the donor and the acceptor come closer to each other. This closeness is comparable to the classical process of collision. So sometimes this process is also called collisional energy transfer. The short distance that makes this energy transfer happen is very near to the collisional diameter. Actually, this exchange transfer occurs for donor– acceptor distance less than 10 Å (or better between 5 and 10 Å). In this connection, it is to be pointed out that the other process which occurs through the coulomb (also called resonance) term (Eq. 7.114) by dipole–dipole interaction was theoretically explained by T. Förster and so it is named after him, i.e. Förster mechanism. This process works when the distance between the donor and the acceptor is large.

7.4 Molecular Interaction

267

7.4.5 Electron Transfer Phenomena Electron transfer is a very significant and universal process in chemistry and biology. This mechanism is involved in many chemical and biochemical processes. The way we get energy from food and oxygen (in respiration) and the way the plants make the food and oxygen we consume (in the photosynthesis) depends on the electron transfer reactions between the cofactors of proteins (which help the proteins to carry out their functions). In general, this is a very fast process (occurring within few picoseconds) which pushes electrons and protons around, and in doing so, it converts energy from one form to another. A theoretical explanation of this process was given by R. A. Marcus which will be discussed below in a nut shell. A Classical Picture Most often, the electron transfer occurs in polar environments. The interactions occur in the electron transfer mechanisms are: A∗ + B → A+ + B− (oxidative electron transfer)

(7.121a)

A∗ + B → A− + B+ (reductive electron transfer)

(7.121b)

As the energy transfer process, the electron transfer process also (i) quenches an electronically excited state and prevents its luminescence and/or reactivity and (ii) sensitizes other species, for example to cause luminescence, that do not absorb light. Here, the transferable electron is located on the donor A* and B is the acceptor. The asymmetric charge distribution (A*/A− − B/B+ ) of the reactants is solvated by a polarized solvent. The whole system can be looked upon as having the inner shell (which is comprised of the bond lengths and angles of the reactants) and the outer shell comprised of the solvent molecules. The solvent molecules, in close proximity, may be coordinated to the donor and the acceptor and form the inner shell and the rest of the solvent molecules form the outer shell of the system. The charge distribution of the redox pair will polarize both the inner and the outer shells. The polarization of the inner shell produces a geometrical distortion of it, and the other case is associated with a dipolar field of the surrounding solvent that might be described by dielectric continuum theory. In either case, a favourable solvation energy is obtained when the polar environment organizes around an asymmetric charge distribute on accordingly. The loss of asymmetry in the charge distribution will therefore be associated with the loss of favourable solvation energy. So the energy of the state where the electronic charge is located on one redox centre is less than that when the charge is distributed over the two redox centres. The latter state is a resonating state, and it is also the transition state. So some energy is required to reach this state from the former state, and this energy is called the activation energy for the electron transfer process. Once this state is attained, the system can move back to where it started or to the state where the electron is transferred to the other species of the redox pair, i.e. A+ + B– . The latter

268

7 Electronic Spectra of Polyatomic Molecules

process is called the electron transfer process. Now, the question is how to overcome the barrier of the activation energy. For instantaneous electron transfer to occur when one of the redox states is solvated, this barrier may be crossed without any external supply through photoexcitation. For it to happen in the dark, the energy required for electron transfer comes from the fluctuation of the molecular environment. This fluctuation will generate a temporal situation for creating a transition state where the two redox states are degenerate so that the electron resonates between the two sites. The electron will then localize only after the environment moves towards one of the two solvated configurations. In order to get an analytical picture of this process, consider the system is taking a transition from the reactant state to the product state. Let us represent each state by a potential energy curve. Let these curves be of simple harmonic type representing the parabolic dependence of energy of the system on a special type of coordinate called reaction coordinate. This reaction coordinate includes all the changes of the nuclei as we go from the reactant to the product system, i.e. it includes the changes of all the nuclear coordinates of the reactant and product species and the molecules of the surrounding medium. It is really a line in a many dimensional space representing fluctuations of all the atoms of the reacting species including the molecules of the surrounding medium. For electron transfer, these fluctuations lead the spatial configuration of all the atoms suitable for the electronic structure of the reactants to that suited for the products. The scale of the axes in Fig. (7.19) is chosen in such a way that the initial parabola be represented by the equation y = x 2 . Assume the final parabola to be the same as the initial one in shape but is only displaced. Let the coordinates of the minima of the two parabolas be (0,0) and (c, d) for the reactant (i = r) and the product (f = p), respectively. Then vr = qr2

(7.122a)

(vp − d ) = (qp − c)2

(7.122b)

and so at the crossing point Q(qo , vo ), we have vo = qo2

(7.122c)

vo = (qo − c)2 + d

(7.122d)

and

From 7.122 (c and d), we get the crossing point ( ) d + c2 qo = 2c

(7.123a)

Energy (v)

7.4 Molecular Interaction

269

b

a

a

b

2Hel

I=r

λ

Q(q0, v0)

ΔE ΔG

f=p

A B

Reaction coordinate (q) Fig. 7.19 Potential energy curves for the initial/reactant state (i = r) and the final/product state (f = p). Crossing point (q0 , v0 ) of the two curves corresponds to activation energy (ΔE). ΔG is the free energy change between the reactants and the products. λ is the reorganization energy. Inset shows the energy splitting at the crossing point

and vo = qo2 =

(d + c2 )2 4c2

(7.123b)

From the Fig. (7.19), we see that at the point qi = c on the reactant curve, c2 = = vi = λ, what is called the reorganization energy. The reorganization energy is thus the amount of energy required to distort the nuclear configuration of the reactants from their own nuclear configurations to those of the products without the electron being actually transferred. This energy thus increases with the increase of the separation of the minima of the two parabolas and also with the increase of the force constant of the initial (reactant) state. The figure also demonstrates that the free energy of the reaction, which is the energy difference between the minima of the two parabolas, is ΔG = d. Thus, the activation energy (i.e., the energy required to cross the barrier to go from the reactant to the product state) is qi2

ΔE = vo =

(d + c2 )2 (ΔG + λ)2 = 4c2 4λ

The rate constant of the electron transfer process can be written as

(7.124)

270

7 Electronic Spectra of Polyatomic Molecules ΔE

k = Ae− kT = Ae−

(ΔG+λ)2 4λkT

(7.125)

Here, A is a constant. This constant can be expressed as an exponential decay function of the distance (r) between the donor and the acceptor of the electron and thus the rate constant becomes k = Ae−

(ΔG+λ)2 4λkT

= Be−βr e−

(ΔG+λ)2 4λkT

(7.125a)

β is a constant depending on the medium between the donor and acceptor. In fact, some medium is found to be more efficient than others for the electron transfer process. B is another constant. The reorganization energy λ can be expressed as the sum of two independent contributions: one is called inner reorganization energy (λi ) which arises from the nuclear modes constituting the bond lengths and bond angles within the two reaction partners and the other is called outer reorganization energy (λo ) which arises from the nuclear modes associated with the solvent reorientation around the reacting pair, i.e. λ = λi + λo . In most of the cases, the outer reorientation energy is predominant in the electrontransfer process and to a first approximation it can be expressed by the relation ( )( ) 1 1 1 1 1 2 (7.126) λo = e − + − εop εs 2rA 2rB rAB where εop and εs are optical and static dielectric constants of the solvent, r A and r B are the radii of the reactants and r AB is the centre to centre distance of the reactants. Equation (7.126) shows that λo is particularly large for polar solvents and for the distance between the reaction partners is large. The expression for the vibrational term λi is given by λi =

1∑ p kj (Qjr − Qj )2 2 j

(7.127)

where Qj r and Qj p are the equilibrium positions of the jth normal coordinate Q associated with a reduced force constant k j which was approximated by Mercus as 2 k j r k j p /(k j r + k j p ), k j r and k j p being the force constant of the jth mode of the reactants and the products, respectively. Quantum Mechanical Picture The rate constant for the electron transfer process, according to Fermi’s Golden rule, is given by kel =

4π 2 ||( el )||2 (FC) H h

(7.128)

7.4 Molecular Interaction

271

where H el corresponds to that part of the Hamiltonian which arises due to electronic interaction between the reactant and the product and FC is the Franck–Condon factor between the corresponding electronic states. In the absence of any intervening medium (through space mechanism), the electronic Hamiltonian is supposed to have the following form $ # β el H = H (0) exp − (rAB − ro ) 2 el

el

(7.129)

where r AB is the donor-acceptor distance, H el (o) is the interaction Hamiltonian for the contact distance r o and β el is an appropriate attenuation parameter. The ½ factor in the argument of the exponential is introduced because β el was initially introduced as an attenuation parameter of the rate constant which depends on the square of the electronic Hamiltonian H el (7.128). When the donor and the acceptor are separated by matter (say, a bridge L), the electron transfer process can be mediated by the bridge. If the electron is temporarily localized on the bridge, an intermediate is produced and the process takes place by a sequential or hopping process. On the other hand, the electronic coupling can occur by mixing the initial and final states of the system with the virtual, high energy electron transfer states involving the intervening medium as illustrated in Fig. (7.20). The FC term of Eq. (7.128) is the thermally averaged Franck–Condon factor between the initial and final states. It contains a sum of the overlap integrals between

E

Hih

A

L+

B−

A+

L−

B

Hie HfeHfh

A*

L

B A+

L

B−

Fig. 7.20 Electronic coupling of the initial and the final states with the virtual, high energy electron transfer states involving the bridge (L) through which the donor and acceptor are connected. The subscripts e and h of H correspond to the electron and hole in the bridge (L), respectively, for the initial (reactant, i) and the final (product, f ) states

272

7 Electronic Spectra of Polyatomic Molecules

the nuclear wave functions of the initial and final states of same energy, involving both inner and outer (solvent) modes. Determination and general expression of FC is very complicated. It can be shown that for high temperature (hν < kT), this factor can be approximately expressed by [11] ( FC =

1 4π λkT

)1/2

[ ( )2 ] ΔG 0 + λ exp − 4λkT

(7.130)

Thus, substitution of Eqs. (7.129) and (7,130) in Eq. (7.128), the rate constant for the electron transfer process becomes 4π 2 · kel = h

(

1 4π λkT

)1/2

[ ( )2 ] 0 |( el )|2 + λ ΔG | H (0) | exp{−β el (rAB − ro )} exp − 4λkT (7.131)

Here, λ is the sum of inner (λi ) and outer (λo ) reorganization energies. The exponential term is same as that predicted by Marcus from classical standpoint (7.125). This expression (7.131) shows a nonlinear relationship of the rate constant with the free energy ΔG0 of the reaction and this variation is shown in Fig. (7.21) for a homogeneous series of reactions (i.e., the reactions, for which λ and H el (0) are same). This variation can be categorized in three regimes specified below. In the normal regime for small driving force (−λ < ΔG0 < 0), where the process is thermally activated, ln (k el ) increases with the increase of the driving energy (−ΔG) until a maximum is reached at ΔG0 = -λ. (ii). In the activation less regime where (−λ ≈ ΔG0 ), the reaction rate does not change to a large extent for the change in the driving force. (iii). In the regime where ΔG0 < −λ (called Marcus inverted region), the rate constant decreases with the increase of the driving energy (−ΔG0 ). Inverted behaviour means that extra vibrational excitation is needed to reach the crossing point as the acceptor well is lowered. Note that in this region, the quantum mechanical curve is practically linear rather than parabolic (broken curve in Fig. 7.21). (i).

Initially, Marcus inverted region was not observed experimentally because all the initial experiments were confined to bimolecular redox reactions where for high driving energy, diffusion of reaction reagents, rather than electron transfer, was mainly responsible for determining the reaction rate and this obscured the electron transfer effect. But later the existence of the inverted region was proved when the donor and acceptor were kept firmly separated by some stiff spacer. But it took about thirty years to prove it by Miller et al. [16].

7.5 Photoelectron Spectroscopy (PES)

−λ σOH . Therefore, the applied magnetic fields at resonance are different, Ba > Bb . The total magnetic energy is ΔE = −g p μn Ba (1 − σa )M I a − g p μn Bb (1 − σb )Mlb + h Jab M I a M I b

(9.35)

where σa/b are the shielding factors around the respective protons, M Ia/b are the total magnetic quantum numbers of the respective protons and hJab is the spin–spin coupling constant between the two types of proton in energy unit. Here we have made use of the fact that same type of protons do not have this spin–spin coupling. Since proton has a spin ½, M Ia can have the following values with the relative alignments of the spins of the individual protons: 1 2 1 2 1 − 2 1 − 2

1 2 1 + 2 1 − 2 1 − 2 +

1 2 1 − 2 1 + 2 1 − 2 +

3 2 1 = 2 =

1 2 3 =− 2 =−

There are three transitions associated with the ‘a–type’ NMR peak, namely 23 → → − 21 and − 21 → − 23 (because of the absorption selection rule ΔM Ia = −1). All the three transitions give rise to the same frequency which appears from the first term on the right-hand side of Eq. (9.35). This peak is split into two components due to the contribution from the last term of the Eq. (9.35) corresponding to two values of M Ib (±½). Similarly, the NMR absorption of the ‘b-type’ proton arises from the transition 21 → − 21 and the corresponding transition frequency appears from the second term of Eq. (9.35) (the selection rule being ΔM Ib = −1). This term also exhibits splitting corresponding to four values of M Ia (±3/2, ± 1/2). The frequencies of the split components at a- and b-resonance are 1 1 , 2 2

(for a-resonance)

νa1 =

g p μn (1 − σa )Ba 1 − Jab h 2

(9.36a)

and νa2 =

g p μn (1 − σa )Ba 1 + Jab h 2

(for a-resonance)

νb1 =

g p μn (1 − σb )Bb 3 − Jab h 2

(9.36b) (9.37a)

9.1 Nuclear Magnetic Resonance (NMR)

365

νb2 =

g p μn (1 − σb )Bb 1 − Jab h 2

(9.37b)

νb3 =

g p μn (1 − σb )Bb 1 + Jab h 2

(9.37c)

νb4 =

g p μn (1 − σb )Bb 3 + Jab h 2

(9.37d)

and

Splitting between the adjacent lines within any of the envelopes of the two peaks is J ab , the spin–spin coupling constant. The relative intensities of the four lines in the b-envelop are determined by the statistical weight of each spin state of the a-nuclei. Since the statistical weight is three for the state with M Ia = ± ½ and one for that with M Ia = ± 3/2, so the relative intensities of the successive quartet in Eq. (9.37a, 9.37b, 9.37c, 9.37d) are 1: 3: 3:1. Similarly, the two lines in the a-envelope are equally intense since the statistical weight for both the states, M Ib = ± ½, is same (one). Moreover the relative peak areas for the a- and b-protons are 3: 1. So the relative intensities, within the a- and b-envelopes, of the spin–spin split components are shown in the Fig. 9.8, and they are in the ratio 12:12 and 1:3:3:1. Same kind of splitting is also observed in acetaldehyde [(CH3 )CHO) (Fig. 9.9). This type of splitting due to spin–spin interaction is observed not only between non-equivalent nuclei but also between two different nuclei. Let us illustrate this with the example of HD molecule. Since magnetic moments of proton and deuteron are different, they resonate at two different magnetic fields. But nuclear spin of

Transmissio

CH3 (a)

OH (b)

1

3

3

1

12

12

Field Fig. 9.9 High-resolution NMR spectrum of methanol, the number within each peak corresponds to respective relative peak area i.e. intensity

366

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

deuteron is 1. So the proton peak will split into three equally intense components corresponding to three different values of the magnetic quantum number M ID = + 1, 0 and −1, whereas the deuteron peak will split into two components of equal area, corresponding to two values of M IP = + ½ and -½, due to spin–spin interaction. There are two types of spin–spin coupling: (a) dipolar coupling, and (b) scalar coupling. Dipolar coupling occurs between the two dipoles. So this is not observed in isotropic phases (i.e. in gases and liquids). Because of molecular tumbling in liquid, this effect averages out to zero. However, this effect exists in solids or in anisotropic liquid (liquid crystal). Scalar coupling between two nucleons is transmitted through the bonding electrons. So this interaction decreases with the increase on the intervening bonds. Spin– spin interaction between nucleons separated by more than three bonds is generally small and therefore neglected. This coupling is best characterized by Fermi contact interaction. We know that only the s-electrons of an atom have finite probability of being at the nucleus. So the s-electron around one nucleus transmits nuclear information to the other nucleus through other bonding s-electrons. This interaction can be described as follows when the two nuclei are directly coupled. Electron, that has a nonzero electron density at the nucleus, has its favourable spin orientation (energetically) antiparallel to the spin of the nucleus. The spin of the other electron in the bonding molecular orbital is antiparallel to the former one according to Pauli’s exclusion principle. The second nucleus can have its spin either parallel or antiparallel to that of the latter electron. The antiparallel spin is energetically favourable and that gives a signal of lower frequency, whereas for the parallel spin the signal frequency is higher. This coupling constant is generally found to be positive. From theoretical consideration, the magnitude of this coupling between two nuclei A and B is found to be J = J AB = (2μ0 ge μ B /3)2 γ A γ B |ψ A (0)|2 |ψ B (0)|2 C 2A C B2 (1/ΔT )

(9.38)

where μ0 is the permeability of free space, ge is the g-value of electron, μB is Bohr magneton, γ A and γ B are gyromagnetic ratios of nuclei A and B, |ψA (0)|2 and |ψB (0)|2 are electron densities at nuclei A and B, C A and C B are the coefficients at A and B respectively in the molecular orbital and ΔT is the energy difference between the singlet and triplet states. When one of the hydrogen nucleus in H 2 is replaced by deuteron, the coupling constant changes to J H D = J H H (γ D /γ H ) = J H H (g D /g P ) = J H H (0.86/5.58) = 0.154 J H H

9.1 Nuclear Magnetic Resonance (NMR)

367

Such coupling constants may be compared by introducing the idea of reduced coupling constant, K AB , which is isotope independent. The reduced coupling constant between the nuclei A and B is given by K AB = (J AB / h)(2π/γ A )(2π/γ B )

(9.39)

The spin–spin coupling between non-directly bonded nuclei can be categorized into three types: 2 J HH (geminal, i.e. when the two protons are attached to the same atom), 3 J HH (vicinal, i.e. when the two protons are separated by three bonds) and n J HH (long range, i.e. the two protons are separated by n-bonds, where n is greater than 3). Geminal coupling (2 JHH ) depends on three factors: (a) H–C–H bond angle, (b) hybridization of the carbon atom, and (c) substituents. More is the bond angle, less is the magnitude of 2 J HH . More is the s-character of the carbon (or the relevant) atom, greater is the magnitude of 2 J HH . These are shown in the examples below. H

H

φ

H2C φ

H H2C

C

H

H

H

Φ = 1090

Φ = 1200

Φ = 1200

2

2

2

JHH = -12.4 Hz

JHH = -4.3 Hz

JHH = +2.5 Hz

In order to observe separate proton frequencies due to this coupling, the germinal bonded protons need to be diastereotopic. Vicinal coupling (3 JHH ) depends on the following factors: (a) (b) (c) (d)

dihedral angles, substituents, HC−CH bond distance, and H–C–C angle.

Dependence of the vicinal coupling constant (3 JHH ) between protons with the dihedral angle ϕ is given by the relation 3

JHH = 7 − cos ϕ + 5 cos 2ϕ

(9.40a)

This is shown in the above Fig. 9.10. The coupling is minimum for the dihedral angle ϕ = 90°, i.e. when the two bonds, holding the two protons H A and H B are mutually perpendicular. The scalar coupling between the protons, conducted through the

368

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic … 3

Fig. 9.10 Variation of vicinal coupling with the dihedral angle (Karplus curve)

JHH (Hz)

φ

HB HA

φ/rad

electrons in the intervening molecular orbitals, is affected for this relative orientation of the two bonds holding the protons. For 1 H, 13 C couplings, the relation is 3

J = 3.81 − 0.9 cos ϕ + 3.83 cos 2ϕ

(9.40b)

These equations are called Karplus relations. The general form of this relation is 3

JHH = A + B cos ϕ + C cos 2ϕ

(9.41)

These are very helpful for determining the stereochemistry of the compounds. Vicinal coupling (3 JHH ) is found to decrease with the increase of the electronegativity of the substituents, HCC bond angle and C − C bond length. Since electronegative substituent withdraws electrons, so due to the decrease of electron density in the bonds through which the coupling is conducted, the coupling strength (3 JHH ) decreases. Increase of the HCC bond angle and C−C bond length also lessen the coupling efficiency. Long range coupling (4 JHH and 5 JHH ), in meta and para aromatics, may be transmitted through the intervening σ-bonds. In general, scalar coupling cannot be conducted through π-electrons, because they have nodes at the positions of the nuclei, i.e. C−H bond is orthogonal to the π-molecular orbitals. But these long-range couplings take place through the combined effect of geminal and vicinal couplings as shown in Fig. 9.11. But in certain cases there exists some overlap between σ- and π-bonds and the long-range coupling is conducted through these bonds (hyperconjugation). These are demonstrated in Fig. 9.12. In allylic coupling (4 JHH , Fig. 9.12), the strength is negative and lies in the range from −2 to −4 Hz. The magnitude of 4 JHH in allylic

9.1 Nuclear Magnetic Resonance (NMR)

369

Ha

Ha 1

1

Meta

Para

2 4

2 3

3 HB HB Fig. 9.11 Long-range coupling in meta (Jmet = 2–3 Hz) and para (Jmet = 0–1 Hz) aromatics

H C

C

C

H

H

C

C C

C

H (a). Allylic coupling (4JHH)

(b). Homoallylic coupling (5JHH)

Fig. 9.12 σ –π overlap (hyperconjugation in long range coupling) dihedral angle between the π-bond and the CH bond

coupling varies as cos2 ϕ, where ϕ is the dihedral angle between the π-bond and the CH bond. On the other hand, in homoallylic coupling (5 JHH , Fig. 9.12), the strength is positive and lies in the range from 0 to + 4 Hz. Here 5 JHH arises from the hyperconjugative overlap of the CH bonds and the π-bonding molecular orbital and so its magnitude varies as cos2 ϕ cos2 ϕ ' . Note that if any CH bond is perpendicular to the intervening π –MO, the overlap is zero and in that case, no coupling occurs through such mechanism.

9.1.7 Solid-State NMR We have seen that dipole–dipole interaction is averaged out to zero in isotropic medium (say, liquid) due to molecular tumbling. But this is not so in the solid medium. When a magnetic field is applied, these nuclear dipoles are thermally distributed to different mI values. These dipoles interact and give rise to an interaction energy → → μ 1 and − μ2 between a pair of nuclear dipoles −

370

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

Hint =

μ0 4π

(

μ →1 · μ →2 → 2 · r→) → 1 · r→)(μ (μ −3 r3 r5

) (9.42)

→ where μo is the permeability of free space and − r is the internuclear separation vector. Let us consider the case of two identical nuclei, each with nuclear spin I = ½. The total Hamiltonian of the system is Htot = −gμn Ba m I +

μ0 4π

(

μ →1 · μ →2 → 2 · r→) → 1 · r→)(μ (μ −3 r3 r5

= H0 + H (1)

)

(9.43)

The second term on the right-hand side of Eq. (9.43) is much smaller than the first term, so the first-order perturbation theory can be applied to find the first-order correction to energy. For two ½ spin nuclei, the dipolar mI can take up values + 1, 0 and −1. Thus, the statistical weights of the three unperturbed states, | 1 + 1) , | 10) and| 1 − 1) are 1, 2 and 1, respectively. The first-order perturbation energies can be easily found out, and they are (1) E +1 =

( ) μ0 (gμn )2 (1/2)2 1 − 3cos2 θ , 4π r 3

E 0(1) = −2 and

( ) μ0 (gμn )2 (1/2)2 1 − 3cos2 θ 3 4π r

(1) E −1 =

( ) μ0 (gμn )2 (1/2)2 1 − 3cos2 θ . 3 4π r

(9.44a)

(9.44b)

(9.44c)

where θ is the angle between the field direction and the separation vector and the dipole moments are taken to be oriented along or against the direction of the applied field. Thus, the frequencies of two transitions arising from the selection rules ΔmI = ± 1 are ν1 = and

) gμn Ba (gμn )2 ( −3 1 − 3cos2 θ 3 h r

ν2 =

) gμn Ba (gμn )2 ( +3 1 − 3cos2 θ h r3

(9.45a)

(9.45b)

respectively. This is demonstrated in Fig. 9.13. In the absence of dipole–dipole coupling, only one line was expected due to the selection rule ΔmI = ± 1 because the energy difference of the three successive levels (mI = + 1, 0 and −1) are equal. In the proton NMR experiment with the single crystal of gypsum (CaSO4 , 2H2 O) two doublets are found. Each doublet appears due to the dipole–dipole coupling of the

9.1 Nuclear Magnetic Resonance (NMR)

371

mI

Fig. 9.13 Energy levels of a solid with two identical nuclei with spin ½ a without and b with dipole–dipole interaction

-1 ν2 0 ν1 +1

(b)

(a)

two equivalent protons in each water molecule. Thus, the presence of two doublets can be considered as the proof of the distinctness of the two water molecules.

9.1.7.1

Magic Angle Spinning (MAS) NMR

Actually in solid sample the angle θ is not constant, the molecules are thermally distributed over various θ values. Along with this, chemical shifts are not isotropic as in the case of liquids. There is a chemical shift anisotropy (CSA) in solids. Both these two effects broaden the spectral line widths of solids significantly. If we can suppress this dipole–dipole coupling, resolution of the spectra can be increased. In liquids and solutions, (1 − 3 cos2 θ ) average out to zero and there the resolution of the spectra is high. The question is how to remove this dipole–dipole coupling in the case of solids? It is seen from Eq. (9.45a, 9.45b) that dipole–dipole interaction vanishes when θ = 54.74°. But this cannot be made possible experimentally. Experimentally the thing what is done is described below. The sample, in a polycrystalline state, is taken in a cylindrical drum which spins very rapidly (30–50 kHz) about its axis S that makes an angle α with the field direction as shown in Fig. 9.14. → r will thus generate a cone. Let the semi The rotation of the separation vector − vertical angle of the cone is β. It can be shown that the average value of (1 − 3cos2 θ ) = −1/2 (1 − 3 cos2 β) (1 − 3 cos2 α). (See Appendix 1 at the end of this chapter). Fig. 9.14 Spinning of the sample in MAS NMR

Ba(z)

S α θ β

372

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic … Dipolar broadened NMR spectra

Fig. 9.15 Dipolar broadened and high-resolution NMR spectra of solids B0

High resolution NMR spectra 54.7 0

We have no control over the angle β. But experimentally we can set α to any desired value. When α is set equal to 54.74°, the average value of (1–3 cos2 θ ) becomes zero which means that dipole–dipole effect vanishes. This angle is called magic angle, and the technique is called magic angle spinning (MAS) NMR. Rotation at this magic angle not only removes the dipole–dipole splitting but also averages the chemical shift to a nonzero value resulting in the reduction of the spectral linewidth. That is MAS NMR average the chemical shift anisotropy to achieve good sensitivity and resolution. Although high-resolution solid-state NMR spectra of 13 C is achievable with moderate spinning frequency, but for obtaining such high-resolution solid-state proton NMR spectra, spinning frequency required is found to be above 25 kHz. This is due to the strong multispin dipolar-coupling network among the abundant protons, which is also responsible for the smearing of the typical features of powder spectra in the static limit (Fig. 9.15).

9.1.8 Nuclear Magnetic Resonance Imaging (NMRI) Nuclear magnetic resonance imaging (or simply magnetic resonance imaging, MRI) is a medical technique for the acquisition of high-quality images of inside of human bodies. MRI diagnoses a variety of diseases, such as strokes, tumours, spinal cord injuries, multiple sclerosis, eye or inner ear problems. It is used in the research of brain structure and function among other things. The risk factor is considerably less with respect to CT scan (X-ray computerized tomography) because it does not use any ionizing radiation. However, the patient under MRI scan must be free from pacemakers, surgical clips and implanted materials made of ferromagnetic materials such as, iron, cobalt, nickel. Human bodies are mostly composed of water which is found in tissues and other parts. In water, each hydrogen nucleus or proton possesses

9.1 Nuclear Magnetic Resonance (NMR)

373

a magnetic moment associated with its spin. So when the human body is placed in a magnetic field, an equilibrium magnetization is induced. By the application of appropriate 90° radio frequency pulse and recording the signal (FID), it is possible to point out the position of the signal emitting proton in the human body and get an image of the region from which the signal is coming out. Other than protons, nuclei, such as 14 N, 19 F, 23 Na, 31 P, can also be used for the same purpose. We shall present here the basic principle of this imaging technique. This technique is mainly comprised of three steps, namely slice selection, phase encoding and frequency encoding. We shall discuss them one by one sequentially. Slice selection Let us discuss this with reference to a region in a small volume, considered as a representative of a portion of the inside of the human body to be imaged. This region contains protons. When they are placed in a high magnetic field (B0 ), they will − → generate an equilibrium magnetization (say, M 0 ) along the direction of the magnetic field (say, z). Along with this field is applied a small field gradient Gz (= ∂∂Bz0 ). So at a particular value of z, the net field is (B0 + zGz ). Now let us apply a 90° pulse of a radio frequency magnetic field (Brf ) along the x-axis of frequency v = (B0 + zG z )γ = v0 + zγ G z

(9.46)

ν 0 = γ B0 being the resonance (Larmor) frequency at the position (x, y, z = 0, 0, 0), called isocentre of the applied magnetic field. Since it is a pulse, it will have a frequency distribution and the amplitude is largest for which the frequency matches − → with that of the applied field (9.46). This 90° pulse will rotate the magnetization M 0 about x-axis by 90°. So the protons lying within a small slice around a particular xy-plane whose position in the z-axis is given by Eq. (9.46) will precess about B0 in the xy-plane with the same Larmor frequency (9.46) after the pulse is turned off. Protons in the other planes will remain more or less unperturbed. That means we have selected a particular thin slice of xy-plane. This is shown in Fig. 9.16. Once this slice is selected, slice selection gradient is turned off. Phase encoding Phase encoding means to give specific phase angle to the transverse magnetization vector. This is achieved by the application of a phase encoding field gradient in the direction perpendicular to the applied field, i.e. say in the x-direction (Gx = ∂B0 /∂x). So the protons will precess about the field direction with Larmor frequencies depending on their positions on the x-axis. vx = (B0 + x GG x )γ = v0 + xγ G x

(9.47)

So when the pulse of the phase encoding gradient is turned off, although all the transverse magnetizations precess at the same Larmor frequency, their phase angles

374

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

Selected slice GZ

Y B0

X Fig. 9.16 Slice selection

Fig. 9.17 Phases of the transverse magnetization vectors after the Gx pulse is turned off

X ……………

..

..

..

..

………….…

Y

are different. The phase angles are ϕ x = 2π ν x t, t being the duration of the Gx pulse. This phase angle can be taken as the angle between the y-axis and the position of the transverse vector at the time when the pulse has been turned off. This is shown in the following Fig. 9.17. Frequency encoding After the phase encoding gradient is turned off, another gradient is applied along y-direction. The transverse magnetization vectors will now precess at frequency depending on their positions on the y-axis ) ( v y = B0 + yGG y γ = v0 + yγ G y

(9.48)

The signal (called raw data or k-space data) emitted by these precessing vectors in the form of free induction decay (fid) will be specified by the frequency ν y . This is called frequency encoding. Note that phases of the radiation vary in the x-direction and their magnitude is ν x t (t, being the duration of the phase encoding pulse). Thus we see that now each magnetization vector is characterized by its frequency and phase. So if we can measure the frequency and phase of the signal emitted by the magnetization vector, we can find its position in the slice. This is done by two dimensional Fourier

9.1 Nuclear Magnetic Resonance (NMR)

375

transform. That is the time-dependent signal is Fourier transformed to frequency and phase which makes the spatial encoding and form the image. After selection of the slice (turning off the slice selection gradient), the entire process is repeated 128 or 256 times to collect all the data needed for recording the image with different values of the phase encoding gradient Gx (varying Gx in equal steps). Time between the repetitions of the process is called repetition time (TR). Each time the process is repeated, the phase encoding gradient is changed and this change is made in equal steps between the maximum amplitude of the gradient and the minimum value. This technique of acquisition of the image is called FT tomographic imaging. Moreover, the position of the slice along the z-direction can be can be selected by the application of the slice selection gradient and the 90°—radio frequency pulse of appropriate frequency simultaneously. Another important thing is the image contrast to know which part of the tissue is anatomically normal and which part is abnormal is determined by the contrast characteristics. Voxel (which is the representation of a point in three dimension) intensity of a given tissue type (i.e. white matter versus grey matter) depends on the proton density of the tissue. Actually the proton density (PD) is measured as the ratio of the concentration of protons in the given tissue to that of water of same volume and at the same temperature. PD for CSF (cerebrospinal fluid, water) is 1 and that of white matter (high fat) is 0.6. Besides this, the contrast also depends on the longitudinal (T 1 ) and transverse (T 2 ) relaxation times. During longitudinal relaxation, the signal is generated in the detector coil as an induced electric current as the magnetization rotate past the coil. So if the protons in a tissue return to the equilibrium magnetization faster than other tissues, then that tissue appears brightest on the T1 weighted imaging (or scan). On the other hand while precessing, the protons of a tissue in the transverse plane become out of phase with one another and the measured signal decreases with the transverse relaxation constant (T 2 ). Thus if the protons in a tissue remain in phase with one another longer than other tissues, then that tissue appears brightest on the T2 weighted imaging (or scan). By minimizing the effects of both T 1 and T 2 contrasts, proton density (PD) imaging (or scan) is obtained. In this imaging, voxel or tissue intensity depends on the concentration of protons which means more and more a tissue is proton rich brighter is the image. There are two scanning parameters, namely repetition time (TR which is the time between the two successive radiofrequency pulses) and echo time (TE which is the time between the application of the radio frequency pulse and signal or echo received). By controlling the relative values of these two parameters, image contrast may be adjusted in different cases. For long repetition time (TR), protons in all the tissues relax back to the full equilibrium magnetization. For short TR, protons in some tissues fail to relax to the initial alignment along the magnetic field resulting in the decrease of the signal. For long echo time (TE), the dephasing of the transverse magnetization is more which results in the decrease of the signal. Reverse is the case for short echo time (TE). Tissues in white and grey matters may appear less or more bright in the above cases depending on their nature. But for fluids, protons remain in phase for a longer

376

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

time because they are not constrained by axons or neurons. Contrast of the image is controlled by the choice of the TR and TE values in the following manner. (i)

When anatomic structures are differentiated based on the proton density, proton density (PD) weighted imaging is used. This is achieved with long TR and short TE to minimize T1 and T2 relaxation effects. (ii) T1 weighted imaging is used when anatomic structures are differentiated on the basis of T1 relaxation. This is achieved using short TR and short TE to minimize T2 relaxation effect. Tissues with high fat (e.g. white matter) appear bright, and regions filled with water (e.g. CSF) appear dark. This is good for demonstrating anatomy. (iii) T2 weighted imaging is used when anatomic structures are differentiated on the basis of T2 relaxation. This is achieved using long TR and long TE to minimize T1 relaxation effect. Tissues with high fat (e.g. white matter) appear dark and regions filled with water (e.g. CSF) appear bright. This is good for demonstrating pathology since damage and injuries of an organ are associated with the increase of water. (iv) When TR is short and TE is long, the contrast is poor.

9.2 Electron Spin or Paramagnetic Resonance (ESR/EPR) 9.2.1 Basic Principle Electron spin resonance (ESR) or electron paramagnetic resonance (RPR) is the electronic analogue of NMR. It arises in atoms or molecules in the electronic states of nonzero spins. Let us start our discussion with the hydrogen atom in the ground state (2 S1/2 ). This atom when placed in a magnetic field B→ possesses a potential energy E mag = −μ → mag · B→ = ge

e → → S·B 2m e c

eh m s B = ge μ B m s B 2m e c = 2.00239μ B m s B

= ge

(9.49)

where Bohr magneton, μB = eè/2me c = 9.2732 × 10−21 erg/gauss, −e being the electronic charge and ms = ± 1/2 = magnetic quantum number associated with electronic spin (Fig. 9.18). Thus in a magnetic field B = 3000 gauss, the amount of splitting is ΔE mag = ge μB B = 5.56392 × 10−17 ergs. Hence if the atom is irradiated with a radiation of frequency 5.56392 × 10–17 / 6.626 × 10–27 ≈ 8.4 kMz, it absorbs this radiation lying in the microwave region and its spin flips from ms = −1/2 to ms = +1/2.

9.2 Electron Spin or Paramagnetic Resonance (ESR/EPR) Fig. 9.18 Splitting of the ground state energy level of hydrogen atom in a magnetic field. E 0 is its energy in the absence of the magnetic field

377

ms = 1/2 ΔEmag= geμBB

E0

ms = -1/2

9.2.2 ESR Spectrometer A block diagram of an ESR spectrometer is shown below in Fig. 9.19. Microwave is generated by the Klystron source (a single-frequency microwave radiation generator), and its power is adjusted in the attenuator. Then the wave enters a region called ‘circulator’ which behaves like a traffic circle. After entering this region, the wave is routed towards the cavity. The cavity is a rectangular or cylindrical hollow box whose dimension is matched with the microwave frequency, and the sample is mounted in this cavity. Microwave is reflected back from the cavity and is routed towards the diode detector. The diode is mounted along the electric vector of the plane polarized microwave radiation. So the current produced in the microammeter is proportional to the intensity of the wave reflected from the cavity. If the cavity absorbs any radiation, a decrease in the current in the microammeter is observed. Any radiation reflected from the diode is completely absorbed by the load. L o a d

Klystron

Diode detector

Attenuator

Circulator

μ - ammeter M a g n e t

Fig. 9.19 Block diagram of the ESR spectrometer

M a g n e Cavity

378

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

Fig. 9.20 a ESR absorption spectra. b First derivative spectra observed with field modulation

Actually the dc measurements are very noisy. To reduce this noise, a small amplitude magnetic field modulation is introduced on the dc magnetic field by means of small coils embedded in the walls of the cavity. When the field is in the vicinity of resonance, it is swept back and forth in this region and generates a ac component of the diode current. The ac output from the detector is amplified by using a frequency selective amplifier. The modulation amplitude is normally less than the line width. Thus, the detected ac signal is proportional to the change in the sample absorption and appears as the first derivative absorption spectra, shown in Fig. 9.20.

9.2.3 Effect of Nuclear Spin (Hyperfine and Super Hyperfine Interactions) The hyperfine interaction may arise due to two distinct ways. Firstly, it arises due to dipole–dipole interaction. This interaction is directional because it depends on the angle between the magnetic field and the line joining the two dipoles. So it is anisotropic. Moreover, the magnitude of this interaction is inversely proportional to the cube of the dipolar distance (i.e. 1/r 3 , r being the dipolar distance). For a system (say, organic free radical) in solution, due to rapid change of orientation of the radical with respect to the magnetic field, this effect averages out to zero. So hyperfine structure in solution must arise from some other mechanism. The second process arises due to Fermi contact interaction which becomes important when the electron comes to the nucleus. This is satisfied when the unpaired electron belongs to the s-orbital, because only the s-orbital has a nonzero electron density at the nucleus. All other orbitals (p-, d-, f -, etc.) have nodes at the nucleus. This interaction does not depend on the orientation with respect to the magnetic induction and so isotropic. This interaction is given by Hamiltonian − → H H F = a S · I→

(9.50)

9.2 Electron Spin or Paramagnetic Resonance (ESR/EPR)

379

where ‘a’ is the hyperfine interaction constant, given by a=

8π ge g N μ B μ N |ψ(0)|2 3

(9.51)

Ψ(0) being the wave function of the electron at the nucleus. Thus for a isotropic system having an unpaired electron (free radical), the Hamiltonian in the presence of a magnetic field B→ is − → − → H = ge μ B B→ · S→ − g N μ N B→ · I + a S · I→

(9.52)

If the system contains several nuclei with which the unpaired electron can interact effectively, then the Hamiltonian becomes H = ge μ B B→ · S→ −

E i

− → g N μ N B→ · I i +

E i

− → − → ai S · I i

(9.52a)

| ) The corresponding eigenkets are |Sm s I m I ) and | Sm s . . . Ii m Ii . . . . The first two terms on the right-hand side of Eqs. (9.52) and (9.52a) are the electronic and nuclear Zeeman terms and the remaining one is the hyperfine interaction. Since the second term is about 2000 times smaller than the first term, so for the consideration of the ESR spectra the second term, that is nuclear Zeeman term (in 9.52 and 9.52a) may be neglected. Remember that when the interaction occurs between the spins of the unpaired electron and the nucleus to which the electron is attached, it is called hyperfine interaction and when it is with neighbouring nucleus it is called super hyperfine interaction. So the strength of hyperfine interaction is greater than super hyperfine interaction. The hyperfine interaction constant with the ith nucleus is proportional to the electron density (ρ i ) at that nucleus, i.e. ai ∼ Q ρi ,

(9.53)

Q being the proportionality constant, which varies from nucleus to nucleus.. Let us now discuss some specific examples. (a) One unpaired electron interacting with two nuclei. These two nuclei may be equivalent and also may not be equivalent. First consider the case of two non-equivalent nuclei each having spin ½. Then the energy of the system, neglecting nuclear Zeeman term, is (Fig. 9.21) E = ge μ B Bm S +

E i

ai m S m Ii

= ge μ B Bm S + a1 m S m I1 + a2 m S m I2

(9.54)

where ms = ± 1/2 and mI1 = ± 1/2 = mI2 . So mI = mI1 + mI2 = 0, ± 1. For ESR spectra, the selection rules are Δms = ± 1 and ΔmIi = 0. Let a1 > a2 . Thus the energy levels, allowed transitions and the nature of the ESR spectrum are shown in

380

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

,

a1/4 and

a2/4

αn1

αe

βn1 βn1

βe

αn1

(a) a1/2

a1/2 a2/2

Relative

a2/2

a2/2

a2/2 geμB

(b) Fig. 9.21 a Energy levels and the allowed transitions, and b ESR spectrum for an unpaired electron coupled to two non-equivalent nuclei each with spin ½. (a1 > a2 )

the Fig. 9.18. We get four lines of frequencies v1 = [ge μ B B + 1/2(a1 + a2 )]/h, [m I 1 = 1/2 = m I 2 ]

(9.55a)

v2 = [ge μ B B + 1/2(a1 − a2 )]/h, [m I 1 = 1/2 = −m I 2 ]

(9.55b)

v3 = [ge μ B B − 1/2(a1 − a2 )]/h, [−m I1 = 1/2 = m I2 ]

(9.55c)

v4 = [ge μ B B − 1/2(a1 + a2 )]/h, [m 11 = −1/2 = m I 2 ]

(9.55d)

and their intensities are equal. (b) One unpaired electron coupled to two equivalent nuclei each with spin ½. Here a1 = a2 = a (say). So the energy of the system without the nuclear Zeeman term may be written as

E = ge μ B Bm S + a1 m S m I1 + a2 m S m I2 = ge μ B Bm S + am I m S

(9.56)

9.2 Electron Spin or Paramagnetic Resonance (ESR/EPR)

(a)

381

αn1 αe βn1 βn1 βe αn1

(b)

a

geμBB

a

Fig. 9.22 a Energy levels and the allowed transitions and b ESR spectrum for an unpaired electron coupled to two equivalent nuclei each with spin ½. (a1 = a2 = a)

where mI can take up values 0, + 1 and − 1. So according to the selection rules Δms = ± 1 and ΔmI = 0, we see that there are three ESR lines as shown in Fig. 9.22). The intensities are not equal. The lines appearing from the transitions, | βe βn1 αn2 ) → | αe βn1 αn2 ) and | βe αn1 βn2 ) → | αe αn1 βn2 ) , have same frequency ge μe B/h. So this line has twice the intensity of the two side lines of frequencies (ge μe B ± a)/h. This is shown in Fig. 9.22. So from the successive separation of the lines, the hyperfine interaction constant (a) can be determined. Similar kind of spectra is observed in the case of an unpaired electron coupled to a nucleus of spin 1, but in that case the relative intensities are in the ratio 1:1:1, because the statistical weight of all the three levels of MI (+1, 0, −1) is one. (c) One unpaired electron coupled with a single set of equivalent nuclei, each having spin ½. Let the number of equivalent nuclei in the set be n. For different values of n, there are different values of total nuclear magnetic quantum number (M I ).∑A particular M I may be obtained from various combinations of mIi such that i m i = M I . For example, in the case of three equivalent protons, different M I values are obtained from different combinations of mIi ’s as shown in the following Table 9.3. From the above table, it is seen that M I = + 3/2 or − 3/2 arises from one combination of mIi values, whereas M I = + 1/2 or − 1/2 arises from three combinations of mIi values. So the statistical weight for the levels M I = ± 3/2 is one, whereas for the level M I = ± 1/2, it is three. Thus, the ESR spectra of this system will exhibit four equidistant lines of relative intensity 1:3:3:1. Thus, the number of lines observed and their relative intensities in the ESR spectra of a single unpaired electron coupled to different numbers (n) of equivalent protons are given in Table 9.4.

382

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

Table 9.3 MI values for different combination of mIi ’s for three equivalent protons (n = 3) Set

mI1

mI2

mI3

MI

I

1/2

1/2

1/2

3/2

II

1/2

1/2

−1/2

1/2

III

1/2

−1/2

1/2

1/2

IV

−1/2

1/2

1/2

1/2

V

1/2

−1/2

−1/2

−1/2

VI

−1/2

1/2

−1/2

−1/2

VII

−1/2

−1/2

1/2

−1/2

VIII

−1/2

−1/2

−1/2

−3/2

Table 9.4 Number of lines and their relative intensities in the ESR spectra for different number of equivalent protons coupled to a single unpaired electron

n

Number of lines

Relative intensities

0

1

1

1

2

1:1

2

3

1:2:1

3

4

1:3:3:1

4

5

1:4:6:4:1

5

6

1:5:10:10:5:1

6

7

1:6:15:20:15:6:1

Fig. 9.23 ESR spectra of benzene radical anion

As an example, the ESR spectra of benzene anion radical are shown in Fig. 9.23. Following Eq. (9.54), it is seen that the resonance magnetic field (for the fixed resonating frequency ν) is found to be

9.2 Electron Spin or Paramagnetic Resonance (ESR/EPR)

B= =

383

E a hν − mI ge μ B ge μ B i i hν a − M I , for equivalen nuclei. ge μ B ge μ B

(9.57)

For benzene, since there are six equivalent nuclei, M I can take up seven values 3, 2, 1, 0, −1, −2, −3. So there are seven lines with relative intensities 1:6:15:20:15:6:1, since the statistical weights for the levels M I = ± 3, ± 2, ± 1 and 0 are 1, 6, 15 and 20 respectively. The hyperfine coupling constant (which is the separation of the successive lines), geaμ B is, found to be 3.75 G. Generally, the unpaired electron is distributed over the ring structure, but some amount of spin density of the unpaired electron is leaked out onto the protons which give rise to this hyperfine interaction constant. (d) One unpaired electron coupled to multiple sets of equivalent nuclei. In such case, the splitting arising from the strongest interaction is to be considered first. Each of these split components undergoes further splitting by interacting with the next strongest interaction and so on. To demonstrate this, consider the anion radical of pyrazine. Here the two nearest nitrogen nuclei (equivalent in nature) in the ring will provide the strongest interaction. Since nitrogen (14 N) has spin 1, the magnetic quantum numbers (MN ) of these two equivalent nitrogens are ± 2, ± 1 and 0 having statistical weights 1, 2 and 3 respectively. So the hyperfine interaction with these two nitrogen nuclei gives rise to a quintet with relative intensity 1:2:3:2:1. Each of these five split components will again undergo further splitting into a quintet due to the hyperfine interaction with the four equivalent protons, having relative intensities 1:4:6; 4:1. Thus, these two hyperfine interactions will give rise to 25 lines and their relative intensities are then determined accordingly. The ESR spectra of pyrazine anion radical are shown in Fig. 9.24. Another example is naphthalene radical anion. In naphthalene, there are two sets of four equivalent protons at the α and β positions. Each set is expected to produce a quintate of relative intensity 1:4:6:4:1 due to hyperfine interaction. Since the two sets are different, their hyperfine interaction constants ai ’s are also different. So in the overall spectra, a quintet of quintets, i.e. 25 lines, will be observed. The ESR spectrum of naphthalene anion radical is shown in Fig. 9.25. Below the spectrum are shown the relative intensities of the lines of each quintet arising from α and β protons. The hyperfine interaction constants are found to be, aα = 5.01 G and aβ = 1.79 G. The ratio aα : aβ = 5.01: 1.79 ≈ 2.80 which is very nearly equal to the corresponding ratio (2.62) of the electron densities at the α and β positions, predicted for the lowest unoccupied MO (LUMO) from simple Huckel theory (9.53). Note that analysis of the ESR spectrum provides information about the number and type of nuclei coupled to the unpaired electron. The magnitude of the hyperfine interaction constant (a) determines the extent to which the unpaired electrons are delocalized. In the case of metallic complex, the magnitude of g-value determines whether the unpaired electrons are mainly confined to the metallic ion or to the

384

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic … N − N

Fig. 9.24 ESR spectra of pyrazine anion radical anion

Fig. 9.25 ESR spectrum of naphthalene radical anion

adjacent ligand also. Generally, the g-value of the ligand is very near to that for the free electron (2.0023), but for the metal ion it generally varies from 0.2 to 8.0. For example, for V4+ (d1) compound, g ~ 1.9; for Cu2+ (d9) g > 2.00 (typically 2.2) and for Mn2+ (d5, L = 0) g = 2.00. Ionic crystals show g-value very close to that of the free electron when the unpaired electron is in the s-state (L = 0, so no spin orbit interaction) or when the crystal field is very strong for which the spin orbit interaction breaks down.

9.2 Electron Spin or Paramagnetic Resonance (ESR/EPR)

385

9.2.4 Anisotropic Systems So far we have confined our discussion on the ESR spectra of molecular radicals in symmetrical environments, i.e. in liquids or solutions. But the case becomes somewhat different when the system, the free radical, is trapped in glasses or in crystals and polycrystalline powders. In that case the medium becomes anisotropic. So the g-values and the hyperfine interaction constant, which arises from dipole–dipole coupling between the magnetic moments of the electron and nucleus, no longer remain scalar but become second rank tensors. So the spectral characteristics change with the orientation of the system in the fixed environment with respect to the applied field. We shall consider the two anisotropies one by one. g-anisotropy To simplify our discussion, let us first neglect the hyperfine interaction. For an isotropic environment, the g-value is found from the relation (9.49) as g=

hν E+ − E− = μB B μB B

(9.58)

ν being the transition frequency. For free radical, this g-value is around 2.0039 which is very nearly equal to the free electron value 2.0023. But in an anisotropic environment (e.g. crystal, frozen solution or powder crystal), the case is not so simple. − → Actually, not only the spin ( S ) of the unshared electron but the orbital angular − → momentum ( L ) of the atom to which it is attached also contributes to the magnetic dipole moment of the electron, i.e. → μ → e = −μ → B (ge S→ + L)

(9.59)

In general, the orbital angular momentum is zero for an electron in the ground state (S = 0). But mixing of the ground state with excited states through spin–orbit coupling adds small amount of orbital angular momentum to the ground state and thus the Eq. (9.59) becomes ( ) μ → e = −μ → B ge S→ + spin-orbit contribution

(9.59a)

But it is a general practice to express μ → e as an entity proportional to the spin angular momentum S→ i.e. μ → e = −μ B g S→

(9.59b)

Here ge of the free electron is replaced by an entity g, different from ge . This difference is very small for organic free radicals. Therefore for organic free radicals with only H, C, N and O atoms, spin–orbit contribution is very small and g is very near to ge . However for ions of heavy element, like metal ions, this contribution no longer

386

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

remains small and g significantly differs from ge . Thus, g can be taken as a fingerprint of the molecule. We know that all the atomic orbitals (excepting l = 0, i.e. s-states, which has a spherically symmetric charge distribution) have directional properties. So g no longer remains a scalar quantity, it becomes a tensor (g) of rank − → two and the resonance condition depends on the direction of the applied field, B . Magnetic dipole moment of the electron in that case can be written as, μ → = −μ B g · S→

(9.60)

Here g is given in a matrix form ⎛

⎞ gx x gx y gx z g = ⎝ g yx g yy g yz ⎠ gzx gzy gzz

(9.61)

So μ → and S→ no longer remain parallel to each other as in the case of isotropic − → medium. The interaction energy of the magnetic dipole with the magnetic field B can be written as, ( ) E = μ B B→ · g · S→ = μ B B→ · g · S→ = μ B ge B→e f f · S→ (9.62) Here B→e f f is the effective magnetic field felt by an observer on the electron. Then the transition energy is hν = E + − E − = μ B ge Be f f = μ B ge (Be + δ B)

(9.63)

where δB is the additional magnetic field arising due to apparent nuclear motion around the concerned unshared electron and so it carries many information of the surrounding environment. But in practice what we do in the experiment is, measure the magnetic field at the resonance condition. Equation (9.63) can also be written in the form hν = E + − E − = ΔE = μ B (ge + δge )B = μ B g B = μ B g B

(9.64)

The last term corresponds to the anisotropy of the medium. So finding the reso→ it is possible to find the g-matrix, which contains information of the nance field B, molecular environment. Now we shall discuss how the meaningful elements of the tensor g are determined. If lx , ly and lz are the direction cosines of the magnetic field, then ) ( ˆ y + kl ˆz ˆ x + jl B→ = B il Thus from Eq. (9.64), we get

(9.65)

9.2 Electron Spin or Paramagnetic Resonance (ESR/EPR)

⎛ ⎞⎛ ⎞⎛ ⎞ gx x g yx gzx gx x gx y gx z lx ) (ΔE)2 = μ2B B l x l y l z ⎝ gx y g yy gzy ⎠⎝ g yx g yy g yz ⎠⎝ l y ⎠ gx z g yz gzz gzx gzy gzz lz ⎛ ⎞ ( )( ) l x = μ2B B 2 l x l y l z g 2 ⎝ l y ⎠ lz ⎛ ⎞⎛ ⎞ gx2x gx2y gx2z lx ( )⎜ ⎟ = μ2B B 2 l x l y l z ⎝ g 2yx g 2yy g 2yz ⎠⎝ l y ⎠ 2 2 2 lz gzx gzy gzz

387

( 2

(9.66)

This g2 matrix is symmetric, i.e. g2 ij = g2 ji . In general, the set of axes (x, y, z) chosen in the crystal is not the principal axes (X, Y, Z) of the crystal. So if we diagonalize the (g2 )-matrix, (x, y, z) axes coincide with the principal axes (X, Y, Z) of the crystal and in the diagonalized form, the (g2 )-matrix is ⎛ 2 ⎞ gX 0 0 ( 2) g = ⎝ 0 gY2 0 ⎠ 0 0 g 2Z

(9.66a)

→ ϕ), If the orientation of the magnetic field with respect to the principal axes is B(θ, then g = g(θ, ϕ) ( )1/2 = l 2X g 2X + lY2 gY2 + l 2Z g 2Z

(9.67)

where the direction cosines (lX , lY and lZ ) are ⎫ l X = sinθ · cosϕ ⎬ lY = sinθ · sinϕ ⎭ l Z = cosθ

(9.68)

For a crystal of cubic (cubal, octahedral or tetrahedral) symmetry, gX = gY = gZ = g (say). In that case, a single resonance peak is observed at B=

ΔE μB g

(9.69)

For a crystal of axial symmetry, let Z be the symmetry axis of the crystal. In that case gZ = g and gX,Y = g . Then ( )1 2 g = g(θ ) = g||2 cos2 θ + g⊥ sin2 θ 2

388

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

Fig. 9.26 Angular (θ) dependence of the resonance field (B) and g in an axially symmetric single crystal for rotation about X- (or Y-) axis, θ being the angle between Z-axis and the field direction. (g > g )

g║

g(θ)

B(θ)

00

900

1800

g┴

θ

) [ 2 ( 2 ]1/2 = g⊥ − g⊥ − g||2 cos2 θ

(9.70)

So peaks are observed at two different resonance magnetic fields for orientation along Z- and along any direction in the X, Y-plane corresponding to g and g . So if the crystal is rotated by an angle θ from the Z-axis about X- (or Y-) direction, variation of the ESR signal is found and is shown in Fig. 9.26 for (g > g ). For rhombic crystal, gX /= gY /= gZ . By rotating the crystal about X-, Y- and Z-directions, gX , gY and gZ can be determined in a similar way. But in general, it is − → not easy to apply the magnetic field B along the the directions of principal axes. So what we do is this. Let us consider that the magnetic field is applied along any direction in the xz-plane which makes an angle θ with the z-direction of the axes (x, y, z) which are not the principal axes. In that case, the direction cosines are lx = sin θ, l y = 0 and lz = cos θ

(9.71)

( )1/2 2 g = gx2x sin2 θ + 2gx2z sin θ cos θ + gzz cos2 θ

(9.71a)

Then

So rotating the crystal along y-direction, we get g = gzz when θ = 0°, g = gxx when θ = 90°. Knowing these values, gxz can be determined when θ is equal to say, 135°. For a similar rotation about the x-axis )1/2 ( 2 cos2 θ g = g 2yy sin2 θ + 2g 2yz sin θ cos θ + gzz

(9.71b)

and for rotation about z-axis (for θ being the angle between the field direction and the x-axis) )1/2 ( g = gx2x cos2 θ + 2gx2y sin θ cos θ + g 2yy sin2 θ

(9.71c)

9.2 Electron Spin or Paramagnetic Resonance (ESR/EPR)

389

Thus from the last two equations, we can determine the other elements gyy , gyz and gxy by measuring g at the appropriate orientations of the field. Thus, we can determine the g2 matrix since it is symmetrical. Then diagonalization of this matrix gives the three values of the principal elements gX , gY and gZ . But it is not easy to get single crystal. What is much easily done is by doing the measurements in the frozen solution or in the powder form of the paramagnetic sample. In these samples, the magnetic fields are oriented arbitrarily with respect to the sample. Then the probability of finding the field in the region of the angular direction between θ and θ + dθ with the z-axis (symmetry axis) of the sample is P(θ )dθ =

2πr 2 sinθ dθ sinθ dθ = 4πr 2 2

(9.72)

Corresponding probability of the magnetic field lying in the range between B and B + dB is proportional to this directional probability, i.e. P(B)d B = P(θ )dθ = C ·

sinθ dθ 2

(9.73)

Therefore, sinθ P(B) = C · ( d B ) 2 dθ

(9.74)

where C is the proportionality constant. Thus, we see that the magnetic field in all the directions is not equally probable, and it depends on the gradient (dB/dθ ). For example, in axially symmetric sample (having more than twofold symmetry along the Z-axis), B is given by Eqs. (9.69 and 9.70). So ( μ )2 ( ) dB B 2 g⊥ − g||2 sin θ cos θ = −B 3 dθ hν

(9.75)

Therefore P(B) = −

C 2

(

hν μB

)2 B3

(

2 g⊥

1 ) − g||2 cos θ

(9.76)

since B = hν/gμB , B(g ) < B(g ). Moreover, the probability of finding the system oriented along the symmetry axis is less than that in the plane perpendicular to it. So the intensity of the resonance peak is higher in intensity at the perpendicular position than that along the axis. This is shown in Fig. 9.27. For rhombic system, same procedure is applied, but the things are more complicated. So we shall not discuss it here.

390

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

Fig. 9.27 ESR spectra of an axially symmetric paramagnetic sample in the powder form. (g > g )

Anisotropy in hyperfine coupling In many oriented systems, hyperfine interaction is found to be anisotropic, and this anisotropy is quite large. So if we rotate the crystal about an axis, ESR spectra will change. For example, in axially symmetric paramagnetic crystal containing Cu(II) ion with I = 3/2, aZ (axial) may be as high as 200 gauss, whereas the transverse hyperfine interaction constant is around 50 gauss. Thus, the ESR spectra of the paramagnetic crystal split into four components for g absorption each separated from its nearby one by 200 gauss and the g absorption also splits into four components but with successive separation of 50 gauss. This anisotropic behavior of the hyperfine interaction can also be studied as above (Eq. 9.71a, 9.71b, 9.71c). The hyperfine interaction constants are measured by rotating the crystal about x, y and z axes (not the principal axes). They can be expressed as before for rotation by angle θ (9.71a, 9.71b, 9.71c), (about x-axis), a 2 = a 2yy sin2 θ + 2a 2yz sin θ cos θ 2 cos2 θ + azz

(9.77a)

(about y-axis), a 2 = ax2x sin2 θ + 2ax2z sin θ cos θ 2 + azz cos2 θ

(about z-axis), a 2 = ax2x sin2 θ + 2ax2y sin θ cos θ

(9.77b)

9.2 Electron Spin or Paramagnetic Resonance (ESR/EPR)

391

+ a 2yy cos2 θ

(9.77c)

Here also the tensor is symmetric, and it has six independent components. After determining these components, by diagonalization of the tensor, the principal values of the hyperfine constants (aX , aY and aZ ) are found.

9.2.5 Zero Field Splitting If a system contains two or more electrons, there arises some electron–electron interactions which split the energy levels even in the absence of the external magnetic field. This is called zero field splitting (zfs). The Hamiltonian associated with this interaction (spin Hamiltonian) can be written in the form ∨

Hz f s ∼ S→ · D · S→

(9.78)

∨

Here D is a symmetric tensor representing this interaction. This interaction arises due to two causes: (a) electron–electron dipolar interaction, and (b) mixing of ground and excited electronic states by spin–orbit coupling. The first one is important for organic molecules in the triplet state, and the second one is dominant in transition metal ions. We shall now consider zfs appearing from the first cause. The electron–electron dipolar interaction Hamiltonian between a pair of electrons (which appear in the triplet state of a molecule, for example) is of the form [ ] →1 · μ →2 → 2 · r→) μ0 μ → 1 · r→)(μ (μ (9.79) Hdi p = Hz f s = − 3 4π r3 r5 μ0 being permeability of free space. Since μ → = μ B g→s , this equation becomes, ] [ μ0 μ2B g1 g2 s→1 · s→2 (→s1 · r→)(→s2 · r→) − 3 4π r3 r5 ] [ s→1 · s→2 (→s1 · r→)(→s2 · r→) =C −3 3 r r5

Hz f s =

μ μ2 g g

μ μ2 g 2

(9.80)

B 1 2 Here C = 0 4π = 0 4πB is a constant [where g1 = g2 = g, slightly different from ge of electron, due to spin–orbit coupling (Eq. 9.59)]. If we form a total spin − → → → s 1+− vector, S = − s 2 , then after doing some lengthy algebra using the properties of the spin angular momentum, it can be shown that

392

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

⎛ Hz f s

)⎜ ( 1 = C · Sx S y Sz ⎝ 2

r 2 −3x 2 − 3xr 5y − 3xr 5z r5 2 2 − 3xr 5y r −3y − 3yz r5 r5 3x z 3x z r 2 −3z 2 − r5 − r5 r5

⎞⎛

⎞ Sx ⎟⎝ ⎠ ⎠ Sy Sz

∨

= C S→ · D · S→

(9.81)

∨

D being a symmetric tensor with zero trace. This matrix can be diagonalized in the principal axes, i.e. in the molecular axes (X, Y, Z) system. In this system, this Hamiltonian (apart from the constant term, C/2) is ⎞⎛ ⎞ SX DX 0 0 = S→ · D · S→ = (S X SY S Z )⎝ 0 DY 0 ⎠⎝ SY ⎠ SZ 0 0 DZ ⎛

Hz f s

∨

= D X S X2 + DY SY2 + D Z S Z2 (D X + DY ) + (D X − DY ) 2 (D X + DY ) − (D X − DY ) 2 SX + SY = 2 2 + D Z S Z2 ) DZ 2 3 (D X − DY ) ( 2 S + Dz S Z2 + S X − SY2 =− 2) 2 (2 ) ( 2 1 2 2 (9.82) = D S Z − S + E S X − SY2 3 where we have used the relation, D X + DY + D Z = 0

(9.83a)

∨

(since the trace of the D –matrix is zero) and the parameters ) ( D = (3/2)D Z and E = Dx − D y /2

(9.83b)

Putting S(S + 1) as the eigenvalue of S 2 and using the relations S + = (S X + iS Y ) and S − = (S X − iS Y ), we get ] [ ) 1 ( 1 Hz f s = D S Z2 − S(S + 1) + E S+2 + S−2 3 2

(9.84)



(See Appendix 2 for the construction of the matrix D ). The eigen functions and eigenvalues of this Hamiltonian are: | X) =

| αα) − | ββ) | 11) − | 1 − 1) 1 = , EX = D − E √ √ 3 2 2

(9.85a)

9.2 Electron Spin or Paramagnetic Resonance (ESR/EPR)

|Y) =

393

| αα) + | ββ) | 11) + | 1 − 1) 1 = , EY = D + E √ √ 3 2 2 | Z) =

| αβ) + | βα) = |10 , √ 2

ΙX>

2 Ez = − D 3

(9.85b) (9.85c)

E

ΙY> 0 ΙZ> (for E being negative)

Now we include the field term, so that the total Hamiltonian is H = Hfield + Hz f s = μ B g B→ · S→ ] [ ) 1 ( 1 + D S Z2 − S(S + 1) + E S+2 + S−2 3 2

(9.86)

Note that the eigenkets of the triplet state of the unperturbed Hamiltonian (i.e. the field part, H field ) are | αα) , | ββ) and

| αβ) + | βα) √ 2

(9.87)

Introduction of zfs in the total Hamiltonian breaks down this degeneracy of the triplet state (9.87) and makes the wave functions linear combinations of these as shown in Eq. (9.85). With the basis functions (9.85), the matrix elements of this total Hamiltonian (9.86) are |X )

|Y )

|Z )

( X|

1 3D

(Y|

gμ B Bl Z

1 3D

( Z|

igμ B BlY

gμ B Bl X

−E

gμ B Bl Z +E

−igμ B BlY gμ B Bl X − 23 D

(see Appendix 3)

− → lX , lY and lZ are the direction cosines of the applied magnetic field, B . If the magnetic field is applied along Z-direction, then lX = 0, lY = 0 and lZ = 1. In that case, the above matrix elements become

394

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic … |X )

|Y )

( X|

1 3D

(Y|

gμ B B

1 3D

( Z|

0

0

−E

|Z )

gμ B B +E

0 0 − 23 D

The solutions of the wave equation with the Hamiltonian (9.86) in the states given by the Eq. (9.85) for the energies are 2 E Z = − D and 3 ]1/2 [ 1 E ± = D ± E 2 + (gμ B B)2 3

(9.88)

When the magnetic fields are applied along the other directions X and Y, other similar solutions are found. The breaking of the degeneracy, for various directions of the applied field, is shown in Fig. 9.28. When the field is applied along Z-direction two lines are observed in the ESR spectra (following the selection rules ΔM S = ± 1). Similarly for the orientations of the magnetic field along the other two directions, a pair of lines is observed at different frequencies. So this effect is anisotropic. From these measurements, D tensor and hence the parameters D and E can be determined. Another very interesting thing is observed in the ESR spectra. Transition ΔM s = ± 2 is in general forbidden. But such lines are observed in the ESR spectra. Generally, this transition arises between two extreme levels as shown in Fig. 9.28 (by the dotted line). The reason for the violation of the selection rule, ΔM s = ± 1, is that at low value of the applied field or high zfs, the H field and H zfs are of the same order. Under such condition, it is not possible for the two spins to be quantized independently and hence not possible to assign M s values to the energy states. So the rule ΔM s = 2 is

Fig. 9.28 Triplet state zero field splitting and Zeeman splitting for the magnetic field applied along the three cartesian axes of the molecule. The full arrows correspond to the selection rule ΔM s = ± 1 and the dashed arrows to ΔM s = ± 2

9.2 Electron Spin or Paramagnetic Resonance (ESR/EPR)

395

somewhat meaningless, but it is commonly used to refer to the transition between the two extreme energy levels. The forbidden line is found enhanced if the rf field is parallel (and not perpendicular as is normal) to the applied filed. In a single crystal, this line has about 2% intensity of the line with ΔM s = 1. Since the transition ΔM s = ± 1 is anisotropic, it is not observed in glass. On the other hand, the forbidden transition ΔM s = ± 2 is relatively isotropic and so observed in glass. It is possible to determine the values of D and E from the forbidden transition.

9.2.6 ESR Spectra of Transition Metal Ion In a system with a number (say, n) of unpaired electrons, there arise several magnetic interactions between the spins of the electrons. This interaction, called ‘zero field splitting (zfs)’ interaction, as said earlier, is present even in the absence of external magnetic field. The corresponding Hamiltonian can be shown to be of the same forms as of Eqs. (9.82) and (9.84), ⎞⎛ ⎞ SX DX 0 0 = S→ · D · S→ = (S X SY S Z )⎝ 0 DY 0 ⎠⎝ SY ⎠ SZ 0 0 DZ ⎛

Hz f s



= D X S X2 + DY SY2 + D Z S Z2 ) [ ] ( = D S Z2 + S(S + 1) + E S+2 + S−2

(9.89)

 − → where the matrix D having zero trace, S being the total spin vector of the system and the parameters D (axial) and E (rhombic) are given by Eq. (9.83). Transition metals, rare earth and actinide ions belong to this system. We shall discuss here the way the atomic orbitals split in the crystal field and the spectral characteristics of the ESR spectra of the first transition series metal ions having 3d n unpaired electrons. We know that the d-states of an atom have fivefold degeneracy, and their orbital wave functional forms are pictorially represented in Fig. 9.29. In the transition metal ions, the d-orbitals are in the crystal field of the ligands (i.e. ligand field). Generally, this field belongs to either octahedral (Oh ) or tetrahedral (T d ) symmetry or some distorted forms of these symmetries. Let us take the case of octahedral symmetry. Here the ligands are considered as point negative charges or point dipoles whose negative ends are pointing towards the positive metal ion, placed at the centre of a cube. The six ligands, equidistant from the metal ion, are at the midpoints of the three pairs of opposite surfaces of the cube through which the three Cartesian axes pass. In this octahedral field, the electronic charges in the d x2 − y2 - and d z2 -orbitals of the metal ion see the ligands directly and hence undergo a stronger electrostatic repulsion with respect to those of the other three orbitals d xy , d yz and d zx . Because electronic

396

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

Fig. 9.29 A schematic representation of the angular part of the d-orbitals

charges in the latter three orbitals do not see the ligands directly and rather, because of their similar form, they experience a coulombic attraction of equal magnitude with the ligand ions. Hence, the crystal field of the ligands split the fivefold degeneracy of the d-orbitals into two groups, one triplet (d xy , d yz and d zx ) which become more stabilized and one doublet (d x2−y2 and d z2 ) which become more destabilized. Just the reverse effect occurs in the case of the tetrahedral crystal field. This is demonstrated in Fig. 9.30. In the latter case, the four ligands are at the alternate corners of a cube. The d xy , d yz and d zx orbitals are more directed to the metal ion than the d x2−y2 and d z2 ones. So the net effect is an electrostatic repulsion in the former set of orbitals and an electrostatic attraction in the case of the latter set of orbitals. Because of the greater amount of directional effect of the d-orbitals towards the ligands, the amount of splitting (Δo ) in octahedral complex is higher than that (Δt ) in the tetrahedral one. This is also shown in Fig. 9.30. From group theoretical consideration, it can be shown that the triplet and doublet are of the representations t 2g and eg under octahedral symmetry and e and t2 under tetrahedral symmetry. This can be done by determining the characters of the reducible representations, generated from the five d-orbitals (schematic forms of which are shown in Fig. 9.29) in the octahedral (Oh ) and tetrahedral (T d ) groups and reducing them to different irreducible representations in the way described in Chap. 8. There are two very important theorems, which are very relevant to the analyses of the ESR spectra of the transition metal (here 3d series) ions. They are Jahn–Teller theorem and Kramers theorem. Jahn–Teller theorem states that in any orbitally degenerate ground state, there will be a distortion to remove the degeneracy. The exception to this theorem is linear molecules and systems having Kramers doublets. Kramers theorem states that the system having odd number of electrons, the zero field ground state will be at least twofold degenerate. This degeneracy can only be removed by magnetic field.

9.2 Electron Spin or Paramagnetic Resonance (ESR/EPR)

397

z

L− z

o

−

L

o

M+

o

L−

y oL−

M+ y

x L−

L−

o

L−

L−

x

L−

o

L−

(b)

(a) eg

dx2-y2

dz2

3Δ0/5

2Δ0/5

t2g

Free ion d - orbitals dx2-y2 dz2 dxy dyz dzzx

dxy

dyz dzx

2Δt/5 3Δt/5 dx2-y2

dxy

dz2

t2 e

dyz dzzx

Octahedral field (Oh)

(c)

Tetrahedral field (Td)

(d)

Fig. 9.30 Splitting of d-orbitals in octahedral (Oh ) and tetrahedral (T d ) crystal fields. a, c belong to octahedral and b, d to tetrahedral symmetries

In analysing the ESR spectra, the first thing is to determine the environment of the metal ion, i.e. the crystal lattice or the complex in which it is situated. Then we determine the relative strength of different interactions. The strongest interaction is considered first, and the other interactions are considered in order of their decreasing strength. Any way the effect of the externally applied magnetic field is considered last. Three cases arise in this respect. The first is the weak field case which is found in the rare earths. Here the 4f electrons are well shielded by the 5 s and 5p electrons and so the spin–orbit interaction is considered before the crystal field effect. Next is the moderate field case which is found in the first transition series. Here the magnitude of the crystal field exceeds the spin–orbit interaction. Last is the strong field case, which is found in the 4d and 5d transition series. In the strong field case, the d-orbitals are split up into a triplet and a doublet as discussed above (Fig. 9.30). Then the electrons are inserted in these orbitals following Hund’s rules and Pauli’s exclusion principle. In large number of cases, the metal ion is surrounded by octahedral field, which is the case of many 3d group of ions. Ground states and degeneracies of the first transition series are shown in Table 9.5. We shall now interpret the ESR spectra of d n transition metal ion complexes. d1 , S = 1/2 [ex.: Sc(2), Ti(3)]: There is one unpaired electron in these ions in 1 3d configuration. Its free ion ground state is 2 D (Table 9.5). In the octahedral field, it splits up into two states: a lowered triply degenerate state, 2 T2 and an elevated doubly degenerate state, 2 E. For tetragonal distortion (elongation along z-axis and

398

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

Table 9.5 Ground states and degeneracies of the first transition series in octahedral fields Value of n in d n –configuration

Ground state of the free ion

Total spin S (free ion)

Orbital degeneracy of the free ion

Ground term in the octahedral field

Total spin S (in the octahedral field)

Examples

1

2D

1/2

5

2T

2

1/2

Sc(2), Ti(3)

2

3F

1

7

3T

1

1

Ti(2),V(3), Cr(4)

3

4F

3/2

7

4A 2

3/2

V(2), Cr(3), Mn(4)

4

5D

2

5

5E

1

Cr(2), Mn(3)

5

6S

5/2

1

6A 1

1/2

Mn(2), Fe(3), Co(4)

6

5D

2

5

5T

2

0

Fe(2)

7

4F

3/2

7

4T

1

1/2

Fe(1), Co(2), Ni(3)

8

3F

1

7

1

Ni(2), Cu(3)

9

2D

1/2

5

3A 2 2E

1/2

Cu(2)

contraction along x-/y-axes) the 2 T2 state splits into the Kramer’s doublet, the ground state 2 B2 and excited state 2 E and the excited state 2 E into 2 A1 and 2 B1 . Spin–orbit coupling does not further split the non-degenerate ground state (2 B2 ) but lowers its energy. In the external magnetic field, this lowest state 2 B2 splits into two states with ms = + 1/2 and ms = −1/2. The ESR spectrum arises due to transition between them. If the tetragonal distortion is small, the Kramers doublets lie close to each other which lead to a short spin–lattice relaxation time resulting in the broad spectral line. So in order to observe a good quality of the spectrum, temperature is to be lowered. The splitting of the levels and ESR transitions are shown in Fig. 9.31a. d2 , S = 1 (ex.:Ti(2): The ground state of the free ion is 3 F. In the octahedral field, this level splits into two triply degenerate and a non-degenerate states, 3 T1 , 3 T2 and 3 A1 , in order of increasing energy. In 3d2 ions, generally the distortion is trigonal and this distortion splits the lowest state 3 T1 into the ground state 3 A2 and excited state 3 E. Spin orbit coupling further spits the ground state into two, the lowest state with Ms = 0 and the doubly degenerate upper state with Ms = ± 1. In the presence of the magnetic field, two ESR transitions are possible as shown in Fig. 9.31b. If the zero field splitting is large, the spectrum will not be observed at the usual magnetic induction, but a transition Δ Ms = 2 may be observable due to mixing of states. d3 , S = 3/2 (ex.:Cr(3): In a regular octahedral field, the free ion ground state (4 F) is split into three states 4 A2 , 4 T2 and 4 T1 in order of increasing energy as shown in Fig. 9.31c. Jahn–Teller theorem does not apply here, since the lowest state is non-degenerate. The excited states are well removed from the ground state. So the spin–lattice relaxation time is high and the observed ESR spectrum is sharp and is readily observable.

9.2 Electron Spin or Paramagnetic Resonance (ESR/EPR)

399 3

A1

2

E

3

T2

+1 3

D

3

F

2

2

E

E

3

T1

2

T2

Free ion

Free ion

±1

3

A2

2

B2

Octahedral field

0

+1/2

±1/2 Tetragonal spin-orbit distortion coupling

Octahedral field

-1/2

0 spin-orbit Trigonal distortion coupling

-1

B

B

(a)

(b)

4

+3/2

T1

+2

5

T2

4

T2 5

D

4

F

5

A1

Free ion 4

A2

+1/2

±3/2

4

B1

5

E

±1/2

+1

±2 5

OctaheTetra- spin-orbit dral field gonal coupling distortion

Octahedral field

-1/2 -3/2

B1 ±1 Tetra0 gonal distortion spin-orbit coupling

0 −1

B

B

(c)

(d) +5/2

(e) +3/2

±5/2 6

S

6

Free ion

Octahedral Field

A1

6

A1

Tetragonal distortion

±3/2

+1/2

±1/2

−1/2

Spin-0rbit coupling

−3/2 −5/2

B

Fig. 9.31 Splitting of the states of a d1 ion, b d2 ion, c d3 ion, d d4 ion and e d5 ion

−2

400

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

d4 , S = 2 (ex.:Cr(2): The ground state 5 D, in an octahedral crystal field, splits into two states, 5 E and 5 T2 , of which the former one is the ground state. Jahn–Teller distortion is applicable and in a tetragonal distortion the 5 E state is split into a lower 5 B1 state and an upper state 5 A1 . The lower state is split into three components due to spin–orbit coupling. So four ESR transitions are possible. But the zero field splitting is often large in d4 metal ions and since M s = 0 state is the lowest one, the spectra are not readily obtainable. d5 , S = 5/2 (ex.:Fe(3): The free ion ground state is 6 S. It does not split in crystal field. The sixfold degeneracy is lifted by spin–orbit interaction giving three components. The splitting and the possible ESR transitions are shown in Fig. 9.31e. Low temperatures are often required to detect the absorption. Splitting of the energy levels and possible ESR spectral transitions of other metal ions having d n -electronic configuration (with n = 6 to 9) can be explained likewise.

9.3 Nuclear Quadrupole Resonance (NQR) Nuclear quadrupole resonance (NQR) is a branch of radio frequency spectroscopy just like nuclear magnetic resonance. But the origins of the two processes are different. In the former case, nuclear quadrupole moment of a nucleus interacts with the electric field gradient of the surrounding environment at that nucleus which split the nuclear level and transitions among the levels yield the NQR spectra. On the other hand in NMR, the magnetic dipole moment of a nuclei interacts with the externally applied magnetic field which breaks the nuclear level degeneracy and the transition among the split levels yield NMR spectra. We shall present below the basic principle of NQR spectroscopy.

9.3.1 Basic Principle of NQR Nucleus with spin I ≥ 1 is not spherical, and it possesses an electric quadrupole moment. Interaction between this aspherical nucleus with the asymmetric electric field gives rise to a set of quantized energy levels. NQR spectra deal with the transitions among these levels. Consider such a nucleus is surrounded by an asymmetric electric field. Then the − → − → Hamiltonian arising from the interaction between the external field ( E = − ∇ V ) with the nuclear charge of density ρ n is ( Hint ==

ρn V dτn =

E α

eV (xα )

(9.90)

9.3 Nuclear Quadrupole Resonance (NQR)

401

where α is the αth proton of the nucleus and the sum is(over all the)protons of the −→ nucleus, e being the protonic charge. Assuming the field E→ = −∇V varies slowly over the nucleus, the potential function can be expanded about the mass center, chosen as the origin, in a Taylor’s series, V = V0 +

E [E α

i

( xαi

∂V ∂ xαi

)

1E + i, j 2 0

(

∂2V ∂ xαi ∂ xα j

)

] xαi xα j + H O T

0

(9.91) HOT are the higher-order terms. With this form of potential function, | the) interaction Hamiltonian matrix between the two nuclear states | I m) and | I m ' becomes \ E ( ) / eV (xα )|I m ' I m|Hint |I m ' = I m| α \ E ( ∂V ) E / ( ) ' ' I m| xαi |I m = eV0 I m|I m + e i α ∂ x αi 0 / \ E ( ∂2V ) 1 E I m| + e xαi xα j |I m ' + H O T α ∂ x αi ∂ x α j 2 i, j 0 E E (∂V ) ( ( ) ) = eV0 I m|I m ' + e I m|xαi |I m ' α i ∂ xi 0 ( 2 ) E E ( ) ∂ V 1 I m|xαi xα j |I m ' + H O T (9.92) + e α i, j 2 ∂ xi ∂ x j 0 The first term is only a constant term (which is nonzero only when m = m' ) added to the energy, so can be neglected. The second term corresponds to the dipole interaction term. This term can also be neglected since there is no evidence of a nucleus having dipole moment. The third term corresponds to nuclear quadrupole interaction term. For a nucleus with spin I, maximum order of nuclear electrical multipole moment possible is 22I . So the lowest possible value of I for a nucleus having a quadrupole moment is 1. When the quadrupole moment is nonzero, the contribution of the HOT is generally much smaller than the quadrupole interaction term and so can be neglected. Thus, the interaction matrix becomes (

1 E I m|Hint |I m = e i, j 2 '

)

(

∂2V ∂ xi ∂ x j

) E ( 0

α

I m|xαi xα j |I m '

)

(9.93)

( 2 ) If the matrix ∂ ∂xi ∂Vx j is diagonalized, i.e. the axes are transformed to the principal 0 axes system X, Y, Z, where all the off-diagonal elements vanish, then (

( ) E ( ) 1 E ∂2V 2 I m|Hint |I m = e I m|X αi |I m ' 2 i ∂ X i2 0 α '

)

(9.94)

402

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

The charges producing the field at the nucleus are absolutely external to the nucleus, so according to Laplace equation ∂2V ∂2V ∂2V + + =0 ∂2 X2 ∂ 2Y 2 ∂2 Z2

(9.95)

Let us choose the principal axes such that | 2 | | 2 | | 2 | | ∂ V | |∂ V | | ∂ V | | | | | | | | ∂ 2 Z 2 | ≥ | ∂ 2Y 2 | ≥ | ∂ 2 X 2 |

(9.96)

Since the sum of the diagonal elements of the second derivatives of V (9.95) is zero, only two of the diagonal elements of it are independent. Let us express them by two parameters ∂2V and ∂(2 Z 2 ) ) ( 2 2 2 ∂2V ∂ V − ∂∂2 XV2 − ∂∂2 YV2 ∂2Y 2 ∂2 X2 )= ( 2 ) η= ( 2 ∂ V ∂2V ∂ V + 2 2 2 2 ∂ Y ∂ X ∂2 Z 2

eq =

(9.97)

where η is called the asymmetric parameter. It measures the deviation of the electric field gradient from the axial symmetry (assumed to lie along the Z-axix). Its magnitude lies between 0 and 1. When it is zero, the field gradient is symmetrical about Z-axis. Thus using the Eqs. (9.95) and (9.97), the Eq. (9.94) becomes (

) 1 ( ) ) ( I m|Hint |I m ' = eq I m|3Z 2 − R 2 + η X 2 − Y 2 |I m ' 4

(9.98)

(This expression is actually a summation over all the protons (α), but this summation symbol has been dropped after the Eq. (9.94) for convenience). This matrix element is determined with the help of Wigner-Eckart theorem. This theorem is related to the determination of the matrix element of a spherical tensor operator. So we shall first know what the spherical tensors are. Spherical tensors Tq(k) of rank (k) having 2k + 1components with q-values equal to (−k, −k + 1, −k + 2, …, k−2, k−1, k) are defined by those tensors which follow the following − → commutation relations with the components of the angular momentum operator j [

and

[

] Jz , Tq(k) = hqTq(k)

√ ] (k) J± , Tq(k) = h (k ∓ q)(k ± q + 1) Tq±1

(9.99a) (9.99b)

Another way of recognizing is that the spherical tensors Tq(k) transform in the m when k → l and q → m under rotation. same way as the spherical harmonics Y(l)

9.3 Nuclear Quadrupole Resonance (NQR)

403

As an example, Cartesian tensor of rank (2) can be formed from the dyadic of two − → − → vectors U and V . There are nine components, U i V j which can be written as Ui V j − U j Vi U→ · V→ Ui V j = δi j + + 3 2

(

Ui V j + U j Vi U→ V→ − δi j 2 3

) (9.100)

The first term on the right-hand side of Eq. (9.100) is a scalar product, invariant under rotation and so a tensor of rank (0). The second term, an ( antisymmetric tensor, ) → → can be expressed in the form of the component of a vector ei jk U × V . There are k three such independent components and is a tensor of rank (1). The last one within the bracket is a component of a symmetric traceless tensor of rank (2) having five independent components (5 = 6 − 1, 1 comes from traceless condition). These three tensors transform in the same way as the spherical harmonicsY00 , Y10,±1 and Y20,±1,±2 respectively under rotation. Another way of constructing the spherical tensors from spherical harmonics of various ranks is given below. /

/ / 3 3 z 3 (1) = cos θ = → T0 = Uz 4π 4π r 4π / / 3 3 x ± iy (1) ±1 ±iϕ Y1 = ∓ =∓ → T±1 sin θ e √ 8π 4π 2r / ( ) Ux ± iU y 3 = ∓ √ 4π 2 / / 15 15 (x ± i y)2 (2) Y2±2 = → T±2 sin2 θe±2iϕ = 32π 32π r2 / )2 15 ( = Ux ± iU y 32π / / 15 15 (x ± i y)z (1) Y2±1 = ∓ → T±2 sin θ cos θ e±iϕ = ∓ 8π 8π r2 / ) 15 ( =∓ Ux ± iU y Uz 8π / / ) ) 5 ( 5 1( 2 2 Y20 = 3z → T0(2) − r 3cos2 θ − 1 = 16π 16π r 2 / ) 5 ( 2 = 3Uz − U 2 16π Y10

(9.101a)

(9.101b)

(9.101c)

(9.101d)

(9.101e)

With this much knowledge of spherical tensor, we can go to the statement of Wigner-Eckart theorem. This theorem states that the matrix element of a

404

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

| ) tensor operator Tq(k) between two states | γ jm) and | γ ' j ' m ' is proportional to the ( ' ' ) Clebsch-Gordan ) coefficient j m | jkqm (which is the transformation coefficient ( ' ' of j m | jkqm generated from the normalized linear combination of | j m) and |k q))) and the proportionality constant is independent of any magnetic quantum proportionality constant is expressed by the standard numbers ( m, m||' or ||q. This ) notation γ ' j ' ||T (k) ||γ j , which looks like the original matrix without the magnetic quantum numbers and the double bar. The mathematical form of this theorem is | | || || ) ) ( )( γ ' j ' m ' |Tq(k) |γ jm = j ' m ' | jkqm γ ' j ' ||T (k) ||γ j

(

(9.102)

Here γ and γ ' are the other quantum numbers needed to specify completely the respective states. Now let us come back to our original problem associated with the determination of the matrix element on the right-hand side of Eq. (9.98). In accordance with the of quadrupole moment of an electric charge distribution, Q i j = (general definition ∑ ρ(3ri r j − r 2 δi j )dτ ≡ e α (3rαi rα j − rα2 δi j ), it is customary to define the nuclear quadrupole moment (Q) as eQ = eQ 33 =

E( α

|( )| ) I I | 3z α2 − rα2 | I I

(9.103)

e being the charge of the proton. According to Wigner-Eckart theorem, matrix elements of any two tensors of same rank are proportional to each other. Therefore, following the Eq. (101e), nuclear quadrupole moment can be written as eQ =

E( α

|( )| ) ( |( )| ) I I | 3z α2 − rα2 | I I = c I I | 3Iz2 − I 2 | I I

] = c 3I 2 − I (I + 1) = cI (2I − 1) [

(9.104)

The proportionality constant c is thus given by c=

eQ I (2I − 1)

(9.105)

2 2 Again X 2 − Y 2 in Eq. (9.98) can be written as (R+ + R− )/2 (where (R± = X ± iY )) which is also a tensor of rank (2) (Eq. 9.101c). So the matrix element I m|Hint |I m ' in Eq. (9.98) becomes (after substituting c from Eq. (9.105))

(

) I m|Hint |I m ' =

) ) ( e2 q Q ( I m|3Iz2 − I 2 + η I X2 − IY2 |I m ' 4I (2I − 1)

(9.106)

9.3 Nuclear Quadrupole Resonance (NQR)

405

9.3.2 Axially Symmetric System For axially symmetric system η = 0 (Eq. (9.97)), so the perturbed energy is given by (Hint )mm = (I m|Hint |m) =

] e2 q Q [ 2 3m − I (I + 1) 4I (2I − 1)

= E ±m

(9.107)

The magnetic quantum number (m) can take up 2I + 1 values, −I, −I + 1, −I + 2, …, I − 2, I − 1, I. So the energy levels are split into different m-values. The selection rule for transitions among these levels is Δm = ±1

(9.108)

So the frequency of all the transitions, |m| → |m| + 1 (for all values of m) are given by ν=

3e2 q Q (2|m| + 1) 4I (2I − 1)h

(9.109)

There will be (I − ½) frequencies for half integral spins and I for integral spins. e2 qQ/h is called the nuclear quadrupole constant (NQR) and has the unit of frequency. The transition frequencies lie in the range 100 kHz to 1000 MHz. For nuclei with half integral spin, say, I = 3/2 (35 Cl, 79 Br), only a single frequency of transition is possible which according to Eq. (9.109), is ν=

1 2 e qQ 2h

(9.110)

For nuclei with spin 5/2 (121 Sb, 127 I), there exists three levels and so two transitions are possible with frequencies, ν1 =

3 2 6 2 e q Q and ν2 = e qQ 20h 20h

(9.111)

These are illustrated in the diagram (9.28) (Fig. 9.32). For nuclei with integral spin, say I = 1 (14 N), the level splitting and transition frequency are shown below in Fig. 9.33. There will be two levels, one non-degenerate with m = 0 and one doubly degenerate with m = ± 1.

406

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

Fig. 9.32 Energy levels and transitions in nuclei with spin, a I = 3/2 and b I = 5/2

Fig. 9.33 Energy levels and transitions in nuclei with spin (I = 1)

m ±1

0

9.3.3 Non-axially Symmetric System For non-axially symmetric system, the energy levels are determined from Eq. (9.106) with η /= 0. In order to determine the matrix element in this equation, the operator I X2 − IY2 is expressed in the convenient form (I+2 + I−2 )/2, where I± = (I X ± i IY − ). Thus, the interaction Hamiltonian becomes (

) I m|Hint |I m ' =

\ )|| e2 q Q / || 2 η( I m |3Iz − I 2 + I+2 + I−2 |I m ' 4I (2I − 1) 2

(9.112)

Unlike the axially symmetric system, the off-diagonal elements for non-axially symmetric system also exist for m' = m ± 2 and the interaction matrix elements are given by (

) } e2 q Q [{ 2 3m − I (I + 1) δmm ' I m|Hint |I m ' = (Hint )mm ' = 4I (2I − 1) η {√ + (I − m ' ))(I + m ' + 1)(I − m ' − 1)(I + m ' + 2) δm,m ' +2 2 }] } √ + (I + m ' ))(I − m ' + 1)(I + m ' − 1)(I − m ' + 2) δm,m ' −2 (9.113)

9.3 Nuclear Quadrupole Resonance (NQR)

407

Here the energy solutions are obtained from the secular equation || || ||(Hint )mm ' − Eδmm ' = 0||

(9.114)

The solutions of these systems are to some extent complicated. For half integral spin, we shall present only the case with nuclear spin I = 3/2, because for higher spins, the solutions are obtained numerically. For I = 3/2 nuclei, m and m' values (for which the matrix elements are non-vanishing) are −3/2, −1/2, +1/2 and +3/2. It can be shown that (Hint )mm = (Hint )± 21 ,± 21 = − (Hint )mm = (Hint )± 23 ,± 23 =

e2 q Q 3e2 q Q =− , 4I (2I − 1) 4

(9.115a)

e2 q Q 3e2 q Q = , 4I (2I − 1) 4

(9.115b)

(Hint ) 23 ,− 21 = (Hint ) 21 ,− 23 = (Hint )− 23 , 21 = (Hint )− 21 , 23 √ √ η e2 q Q 12 η e2 q Q 12 . = = 2 4I (2I − 1) 2 12

(9.115c)

So the above secular Eq. (9.114) becomes || 2 || e q Q || − 4 √− E || η e2 q Q 12 || 2

12

√ η e2 q Q 12 2 12 e2 q Q −E 4

|| || || || = 0 ||

(9.116)

which gives [ ]1 η2 2 e2 q Q 3 1+ E± = Em = + for m = ± 4 3 2 1 [ 2 2 ]2 η 1 e qQ 1+ for m = ± =− 4 3 2

(9.117)

So according to the selection rule Δm = ±1, only one transition frequency is possible and it is ν=

[ ]1/2 η2 e2 q Q 1+ 2h 3

(9.118)

This is shown in Fig. 9.34. From this single frequency, nuclear quadrupole coupling constant, e2 qQ/h (in Hz) and the asymmetry parameter (η) cannot be determined simultaneously. However for small η (in the case of singly coordinated atoms, e.g. chlorine), this asymmetry parameter is taken as zero and the nuclear quadrupole constant becomes nearly equal to twice the transition frequency. In order to determine

408

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

Fig. 9.34 Level splitting and transitions in nucleus with spin 3/2 for η /= 0 compared to the case, η = 0 compare to

nuclear quadrupole constant more accurately along with the asymmetry parameter η, the sample is placed in a weak external magnetic field. This weak field interacts with the nuclear spin and exhibits Zeeman splitting of the energy levels, and hence, the transitions among these split levels give rise to a number of lines instead of only one. From the frequencies of these split components, the above two parameters are determined. For integral spin I = 1, one of the most important nuclei is 14 N. In this case (from Eq. 9.112), we get e2 q Q e2 q Q , (Hint )−1,−1 = (Hint )1,1 = + , 2 4 e2 q Q η, = (Hint )−1,1 = 4 = (Hint )−1,0 = (Hint )0,1 = (Hint )0,−1 = 0

(Hint )0,0 = − (Hint )1,−1 (Hint )1,0

(9.119)

Thus the secular Eq. (9.114) in this case is | | |A−E 0 Aη || | | 0 −2 A − E 0 || = 0 | | Aη 0 A−E|

(9.120)

where A = e2 qQ/4 and the solution is E 0 = −2 A and E ±1 = A(1 ± η)

(9.121)

The energy levels and transition frequencies are shown in Fig. 9.35. The frequencies of the three transitions are ν0 =

η) 1 e2 q Q 3 e2 q Q ( η and ν± = 1± 2 h 4 h 3

(9.122)

Appendix 1

409

Fig. 9.35 Level splitting and transitions in nucleus with spin 1 for η /= 0 compared to the case, η =0

The frequency ν 0 appears due to mixing of states. The quadrupole constant of 14 N is of the order of 4 MHz. So the line ν 0 can only be observed when the asymmetry parameter η is large to bring it into the possible range of detection. However from the transition frequencies, the nuclear quadrupole constant e2 qQ/h and the asymmetry parameter η can be determined excepting in nucleus with spin 3/2. Some of the uses of NQR spectroscopy are just mentioned below. If the readers are interested to know further details of these topics, they may concern the relevant literatures. (1) To determine the electronic structure of molecules. (2) Ionic character and information about hybridization of a bond can be determined by comparing the nuclear quadrupole constant in the atom with that in the molecule of that nucleus. (3) Information about chemical bonding in the solid state. (4) Structural studies of charge transfer complexes. (5) Crystal imperfections can be investigated because small amount of imperfection destroy the symmetry of the internal electric field causing splitting or broadening of NQR lines. (6) NQR is nowadays used to detect explosive and landmines.

Appendix 1 ) )( ) ( ( Proof of the relation 1−3cos2 θ = − 21 1−3cos2 β 1−3cos2 α in Eq. (9.45).

410

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

Ba(z) S

α θ

β

Let the unit vectors in the rotating frame of the vector r→ be iˆ , jˆ and k where the ' ' unit vector k is along the axis of rotation S→ (along z ' -direction) and jˆ is perpendicular ' to the zz plane (i.e. the plane containing the applied field Bz and the axis of rotation → The polar and the azimuthal angles of unit vector rˆ in the rotating frame are β S). and ψ (say). Then '

So

'

'

' ' ' rˆ = iˆ sin β cos ψ + jˆ sin β sin ψ + kˆ cos β

(9.123)

) ( ) ( ( ) zˆ · rˆ = sin β cos ψ zˆ · iˆ' + sin β sin ψ zˆ · jˆ' + cos β zˆ · k '

(9.124)

So cos2 θ = sin2 β cos2 ψ sin2 α + cos2 β cos2 α + 2sinβ cosψ sin α cosβ cos α

(9.125)

Therefore cos2 θ = sin2 βcos2 ψsin2 α + cos2 β cos2 α + 2sinβcosψ sin α cosβ cos α 1 = sin2 β sin2 α + cos2 β cos2 α 2 )( ) 1( = 1 − cos2 β 1 − cos2 α + cos2 β cos2 α 2 ) 1 3 1( = + cos2 β cos2 α − cos2 β + cos2 α 2 2 2

(9.126)

Thus ) 3 9 2 3( − cos β cos2 α + cos2 β + cos2 α 2 2 2 )( ) 1( = − 1 − 3cos2 β 1 − 3cos2 α 2

1 − 3cos2 θ = 1 −

(9.127)

Appendix 2

411

Appendix 2 ∨

Formation of ( D -Matrix) of Eq. (9.81). From Eq. 9.80, we get ] s→1 · s→2 (→s1 · r→)(→s2 · r→) =C −3 r3 r5 [ ( ) ( ) 1 = C 5 r 2 s1x s2x + s1y s2y + s1z s2z − 3 xs1x + ys1y + zs1z (r ) ( )( )] r 2 s1x s2x + s1y s2y + s1z s2z − 3 xs1x + ys1y + zs1z xs2x + ys2y + zs2z 1 [ = C 5 (r 2 − 3x 2 )s1x s2x + (r 2 − 3y 2 )s1y s2y + (r 2 − 3z 2 )s1z s2z r( ) ( ) ] −3x y s1x s2y + s1y s2x − 3yz s1y s2z + s1z s2y − 3zx( s1z s2x + s1x s2z ) (9.128) [

Hz f s

(This is the given Eq. (9.81) in matrix form). Now let us form a vector, ( ) − → S = s→1 + s→2 = i (s1x + s2x ) + j s1y + s2y + k (s1z + s2z ) Ʌ

Ʌ

Ʌ

(9.129)

So ) ( 2 ) ( 2 2 2 S 2 = s1x + s2x + 2s1x .s2x + s1y + s2y + 2s1y .s2y ( 2 ) 2 + s1z + s2z + 2s1z .s2z = Sx2 + S y2 + Sz2

(9.130)

Let us take s+ = sx + is y and s− = sx − is y Therefore sx =

s+ + s− s+ − s− and s y = 2 2i

(9.131a) (9.131b)

Again s+ + s− 1 | α) = | β) and 2 2 1 s+ + s− | β) = | α) sx | β) = 2 2 sx | α) =

s y | α) =

i s+ − s− | α) = | β) and 2i 2

(9.132a)

412

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

sx | β) =

s+ − s− i | β) = − | α) 2 2

1 | α) and 2 1 sz | β) = − | β) 2 | ) | ) (in unit of è) and where | α) = | 21 21 and | β) = | 21 − 21 . So

(9.132b)

sz | α) =

1 | α) , sx2 | β) 4 1 = | β) and so also for the y and z componenets. 4 ( 2 ) 2 Sx2 | α/β) = s1x + s2x + 2s1x · s2x | α/β) ( ) 1 1 + + 2s1x · s2x | α/β) = 4 4

(9.132c)

sx2 | α) =

(9.133)

(9.134)

etc. and hence the operators s1x .s2x =

1 2 1 S − , 2 x 4

(9.135)

so also for the other components. Again ) ( ) ( Sx S y + S y Sx = (s1x + s2x ) s1y + s2y + s1y + s2y (s1x + s2x ) ) ( )} {( = s1x s1y + s1y S1x + s2x s2y + s2y s2x ( ) + 2 s1x s2y + s1y s2x

(9.136)

But ( ) 1[ s1x s1y + s1y s1x |α/ β) = (s1+ + s1− )(s1+ − s1− ) 4i ] +(s1+ − s1− )(s1+ + s1− ) |α/ β) ) 1( 2 2 |α/ β) = 0 s1+ − s1− = 2i

(9.137)

Hence from Eq. (9.136), we get ( ) 1( ) s1x s2y + s1y s12x = Sx S y + S y Sx , 2

(9.138)

Appendix 3

413

(similarly for the other components). Substituting (9.138) and (9.141) in (9.128), we get, ) ) ) ( ( 1 1 [( 2 C 5 r − 3x 2 Sx2 + r 2 − 3y 2 s y2 + r 2 − 3z 2 Sz2 2 r ] −3x y Sx S y − 3yzS y Sz − 3zx Sz Sx ⎞⎛ ⎞ ⎛ 2 2 3x y r −3x 3x z − − Sx 5 5 5 r 2 r ( )⎜ r 1 2 3yz ⎟⎝ = C · Sx S y Sz ⎝ − 3xr 5y r −3y Sy ⎠ − ⎠ r5 r5 2 2 2 Sz − 3xr 5z − 3xr 5z r −3z r5

Hz f s =

∨

= C S→ · D · S→

(9.139)

Appendix 3 Construction of the matrix element Dij ( ) |α α) − |β β) H | X ) = Hfield + Hz f s √ 2 [ )] ( ) ( |α α) − |β β) 1 = μ B g B l x Sx + l y S y + l z Sz + D−E √ 3 2

(9.140)

Now, Sx

Sy

|α α) − |β β) |α α) − |β β) = (s1x + s2x ) √ √ 2 2 1 = √ (|β α) + |α β) ) − (| αβ) + | βα) ) 2 2 =0 ) |α α) − |β β) ( |α α) − |β β) = s1y + s2y √ √ 2 2 i = √ (| βα) + | αβ) ) + (| αβ) + | βα) ) 2 2 | αβ) + | βα) =i √ 2 Sz

|α α) − |β β) |α α) − |β β) = (s1z + s2z ) √ √ 2 2 1 = √ (|α α) − |β β) ) 2

(9.141)

(9.142)

(9.143)

414

9 Nuclear Magnetic Resonance (NMR), Electron Spin or Paramagnetic …

Therefore, |[ )] ( ) ( (α α| − (β β| || 1 μ D − E + g B l S + l S + l S √ B x x y y z z | 3 2 |α α) − |β β) (α α| − (β β| = √ √ 2 2 [ )] ( |α β) + |β α) |α α) + |β β) μ B g B il y + lz √ √ 2 2 ( ) ( ) 1 1 + D−E = D−E (9.144a) 3 3 |[ )] ( ) ( (α α| + (β β| || 1 (Y |H |X ) = √ | μ B g B l x Sx + l y S y + l z Sz + 3 D − E 2 | (α α| + (β β| || |α α) − |β β) = √ √ | 2 2 [ )] ( |α β) + |β α) |α α) + |β β) μ B g B il y + lz √ √ 2 2 (9.144b) = μ B g Bl z = (X |H |Y )

(X |H |X ) =

|[ )] ( ( ) (α β| + (β α| || 1 μ D − E g B l S + l S + l S + √ B x x y y z z | 3 2 | | (α β| + (β α| | |α α) − |β β) = √ √ | 2 2 [ )] ( |α β) + |β α) |α α) + |β β) μ B g B il y + lz √ √ 2 2 (9.144c) = iμ B g Bl y = −(X |H |Z )

(Z |H |X ) =

since the matrix Hij is symmetric. In a similar way we get, ( ) |α α) + |β β) H | Y ) = H f ield + Hz f s √ 2 )] ( [ ) ( |α α) + |β β) 1 D+E = μ B g B l x Sx + l y S y + l z Sz + √ 3 2 ) ( |β α) + |α β) |α α) + |β β) + lz = μ B g B lx √ √ 2 2 ( ) |α α) + |β β) 1 + D+E (9.145) √ 3 2

References and Suggested Reading

415

Therefore, ( (Y |H |Y ) =

1 D+E 3

) (9.146a)

| ) ( |β α) + |α β) |α α) + |β β) α(β | + β(α | || (Z |H |Y ) = + lz √ √ √ |μ B g B l x 2 2 2 ( ) |α α) + |β β) 1 + D+E (9.146b) = μ B g Bl x = (Y |H |Z ) √ 3 2 Hence, the H ij matrix (where i, j ≡ X, Y, Z) is. |X ) |X )

1 3D

−E

|Y )

|Z )

gμ B Bl Z

−igμ B BlY

|Y )

gμ B Bl Z

1 3D

|Z )

igμ B BlY

gμ B Bl X

+E

gμ B Bl X − 23 D

References and Suggested Reading 1. G. Aruldhas, Molecular Structure and Spectroscopy (Prentice Hall of India, 2001) 2. C.N. Banwell, Fundamentals of Molecular Spectroscopy (Tata McGraw Hill Publishing Company Limited, New Delhi, India, 1983) 3. B.P. Sraughan, S.Walker, Spectroscopy, vol. 1 (Chapman and Hall, 1976) 4. T. Soderberg, Organic Chemistry with a Biological Emphasis by University of Minnesota, Morris 5. J.P. Hornak, The basics of MRI—copy right reserved to J.P. Hornak 1996–2010 6. P.S.M. Brady, Basics of MRI (Department of Engineering Science, Oxford University, Michaelmas, 2004) 7. E. Duin, Electron Paramagnetic Resonance Theory. https://www.icmmo.u-psud.fr›media›epr slides3 8. (G tensor).pdf https://www2.chemistry.msu.edu/courses/cem987/CMepr2.pdf 9. Dr. P. Rieger, Electron Spin Resonance (Copyright-author, Brown University. https://fen.nsu. ru›physmethods›lit›NMR›add) 10. G. Baym, Lectures on Quantum Mechanics (W.A. Benzamine, 1969) 11. D. Bohm, Quantum Theory (Dover publications, 1989) 12. J.J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994) 13. A.R. Edmonds, Angular Momentum in Quantum Mechanics (Pinceton University Press, 1957) 14. M.E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957) 15. P.M. Mathews, K. Venkatesan, A Text Book of Quantum Mechanics (Tata McGraw Hill publishing company, New Delhi, 1991) 16. Dr. A. Mishra, Nuclear Quadrupole Resonance. Lecture notes. http://www.anilmishra.name/ notes/nqr1.pdf 17. M. Kaplansky, Application of Nuclear Quadrupole Resonance Spectroscopy (Thesis submitted to the Faculty of graduate studies and research of McGil University, Montreal, Canada, in the partial fulfillment of the requirements of the degree of Doctor of Philosoplhy, 1967)

Chapter 10

Mossbauer Spectroscopy

Abstract Nuclear recoil, Doppler Effect and resonance absorption and emission processes have been discussed, and on these bases, the principle of Mossbauer spectroscopy (along with the technique of the experiment for recording the spectra) has been described. The isomer shift and the effect of nuclear quadrupole interaction and magnetic fields are also investigated.

When an atom emits a radiation, it is highly possible for this radiation to be absorbed by a second similar atom present in the closed surrounding of the former one. This phenomenon is called resonance absorption. Unlike radiation in the optical or further lower energy region, resonance absorption is not common in the case of nuclear radiation. Although in 1920s, possibility of γ-ray resonance absorption was predicted, and experimental scientists had to wait several years to make it a success. The main reason for their failure was nuclear recoil and thermal broadening. Mossbauer in 1958 utilized some special technique to make it possible for which he was awarded Noble prize in 1961.

10.1 Nuclear Recoil and Doppler Effect If a nucleus emits a γ-ray of energy E γ , the nucleus experiences a recoil velocity v. According to the conservation of linear / momentum, the recoil momentum of the nucleus is |M v→ | = | p→recoil | = pγ = E γ c. Thus, the recoil energy is E recoil ≈

pγ2 E γ2 p2 p2 = recoil = = 2M 2M 2M 2MC 2

(10.1)

As an example, consider a γ-radiation of energy 14.4 keV from Fe57 . For this transition, the recoil energy is

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 P. K. Mallick, Fundamentals of Molecular Spectroscopy, https://doi.org/10.1007/978-981-99-0791-5_10

417

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10 Mossbauer Spectroscopy

E recoil

)2 ( 14.4 × 103 = ≈ 1.94 × 10−3 ev 2 × 57 × 938 × 106

(10.2)

Although this loss of energy is much smaller than the γ-ray energy itself but is much larger than the line widths of γ-rays (10–7 to 10–8 eV). So the emission line will not overlap with the absorption line, i.e. the recoil energy shifts the absorption out of resonance with the emission. Mossbauer was able to minimize the recoil energy by fixing the nucleus in a lattice. Thus instead of recoil of a single nucleus, recoil is to be considered of the entire lattice. Thus by increasing the mass of the recoil substance enormously, the recoil energy is brought down to 10−9 eV. Besides this, there will be a Doppler broadening of the γ-radiation due to thermal motion of the nuclei. So the energy of the emitted γ-photon is ( ) v E γ ' = E γ 1 + cos θ c

(10.3)

where v is the velocity of the emitting nucleus and θ is the angle between the direction of emitted γ-ray and the direction of motion of the nucleus. For the atoms in thermal ( ) 21 equilibrium at temperature T, v can be approximated to 2kT , and thus, the spread M / Eγ 2kT of energy is about ± c cos θ . Thus, both the absorption and emission lines M will be broadened. Since absorption will only occur when there is sufficient overlap between the emission and absorption lines, the amount of resonance absorption will be greatly reduced (see Fig. 10.1). The overlapping of emission and absorption is small in the case of nuclear radiation, but it is sufficiently high in the case of atomic radiation. Thus, resonance absorption is very common in atomic spectra but not so in nuclear spectra. Since 2E recoil is the difference between the peaks of the absorption and emission lines, the condition of resonance absorption is Fig. 10.1 Emission and absorption overlaps in atoms (a) and nuclei (b)

Eγ' – ERecoil

Eγ' + ERecoil

Emission

Absorption Overlapping region

(a) Atomic radiation Eγ' – ERecoil Emission

Eγ' +ERecoil Absorption

Overlapping region

(b) Nuclear radiation

E

10.2 Earlier Experiments on Resonance Absorption

2E recoil =

E γ2 MC 2

419

< [(natural half width of the line)

(10.4)

When this condition is fulfilled, the emission and absorption curve will overlap to a good extent.

10.2 Earlier Experiments on Resonance Absorption When γ-rays from a source impinge on a system containing the resonant nuclei, nuclear resonance is observed by determining the degree of absorption or scattering intensity. First successful resonance scattering experiment was performed by Moon in 1951. He used a mechanical technique to compensate for the loss of the recoil energy. An Au198 source was mounted on a rotor tip which had velocities of the order of 800 m/sec. Hg198* is formed in the excited state by β-decay of Au198 . Excited mercury (Hg198* ) decays to its ground state by emitting a γ-ray of energy 411 keV. The γrays were scattered by liquid mercury scatterer containing 10% of Hg198 (natural mercury). The intensity of the scattered γ-rays was measured at a certain angle as a function of velocity, and a maximum intensity was observed when the velocity was about 700 m/sec which can be understood from the following calculation. From Eqs. 10.3 and 10.4, we see Eγ ·

E γ2 v p = Eγ · = c Mc Mc2 (10.5)

i.e. v=

Eγ · c 411 × 10 × 3 × 10 = ∼ 664 m/sec 2 Mc 198 × 938 × 106 3

8

In another method, emitting nuclei of the atoms are in thermal motion of a gas. There is a Maxwellian distribution, and hence, both the absorption and emission lines are broadened. Resonance absorption is obtained when the broadened lines begin to overlap. None of these techniques was sensitive enough to the detect small changes in γ-ray energy. Mossbauer showed that when both the emitting and the absorbing nuclei are confined in a solid, recoil energy of the lattice is reduced enough to be compensated by imparting a relative velocity between the emitting and absorbing nuclei. Before going to describe the experimental technique of Mossbauer spectroscopy, we shall discuss the principal of the process.

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10 Mossbauer Spectroscopy

10.3 Principal of Mossbauer Spectroscopy The most commonly used nucleus in Mossbauer spectroscopy is Fe57 which is produced by the radioactive decay of Co57 having a relatively longer life time (270 days). The decay scheme is shown below in Fig. 10.2. In the decay scheme, Co57 isotope decays to the excited Fe57 * ( we see that the ) −9 to another of energy 136.4 keV τ1/ 2 = 8.6 × 10 (second , most of which decays ) excited state of Fe57 * of energy 14.4 keV τ1/ 2 = 1.5 × 10−7 second from which it decays to its ground state by emitting a γ-ray of frequency ν = 3.5 × 1018 Hz. If a second Fe57 nucleus is exposed in this radiation, the latter is expected to absorb this. But as discussed above, due to sufficient recoil of the former one, the latter one is unable to absorb it because of the sharpness (low band width) of the line. However if a proper relative velocity is introduced between the emitting and absorbing nuclei, the absorption (resonance) can take place, and this technique was utilized by Mossbauer. Let us clarify this idea. The life time of the emitting state is t1/ 2 = 1.5 × 10−7 / sec. So according to uncertainty principal, the bandwidth of the γ-line is Δν1/ 2 = 1 t1/ 2 ∼ /106 Hz. This band frequency is much smaller than the frequency of the γ-line, Δν1/ 2 ν ∼ 10−12 . The recoil velocity (10.5) is v=

Eγ · c 14.4 × 103 × 3 × 108 = ∼ 81 m/sec Mc2 57 × 938 × 106

(10.6)

This recoil velocity will produce a Doppler shift of the emitting frequency Fig. 10.2 Decay scheme leading Mossbauer transition in 57 Fe

I

Co57 (T1/2 = 270 days)

7/2 K- capture 5/2

Γ2 (10.68%)

Fe57*: 136.4 kev (8.6x10-9sec)

γ1 (85.6%)

3/2

Fe57 : 14.4 kev (1.5x10-7sec) ν = 3.5 x 1018 Hz

1/2

Fe57 : 0 kev

10.4 Experimental Set-up of Mossbauer Spectroscopy

421

3.5 × 1018 × 81 νv = ∼ 1012 Hz (10.7) c 3 × 108 ( ) Although this shift is much smaller than the frequency 3.5 × 1018 Hz of the γ-radiation, but it is much larger than the half width of the line (106 Hz). This shift corresponds to millions of the line width. So resonance absorption will not take place. In order to get rid of this mismatch, Mossbauer introduced two remedial steps. Firstly, he fixed the emitting nucleus in a crystal lattice in which, due to its heavy mass, the recoil energy can be dissipated. Secondly, he, by some experimental trick, introduced a relative velocity between the source and the absorber which made the absorption possible for a proper relative velocity. Furthermore, he made arrangement to cool both the source and the sample to reduce the thermal motion of the lattice atoms. Δv =

10.4 Experimental Set-up of Mossbauer Spectroscopy We have seen that a relative velocity of about 81 m/sec produces a huge Doppler shift ~ 1012 Hz. So a relative velocity of about 1 cm/sec is expected to produce a Doppler shift ~ 108 Hz which is equivalent to 100 line widths. So in Mossbauer spectroscopy, the following technique is utilized to make the absorption experimentally possible, and it is shown in Fig. 10.3. A piece of the radioactive source (57 Co) is mounted on a disc to which is attached a screw thread. Through this screw, the source is given steady back and forth velocity. A Geiger counter is mounted behind the sample to collect the γ-rays transmitted through it. When the velocity of the disc is proper to compensate the recoil effect, there occurs a sudden fall in the counter rate which corresponds to absorption. Since

a

Source

Sample

Geiger Counter

b

~

To Detector, Amplifier, Multi channel analyser and Computer

Fig. 10.3 Experimental arrangements of Mossbauer spectroscopy: a screw thread arrangement and b loudspeaker coil arrangement

422

10 Mossbauer Spectroscopy

the Doppler shift is constant for a particular velocity, the spectrum is obtained by creating several values of relative velocities in a range, say from −1 to + 1 cm/sec. In a more convenient arrangement, instead of the screw thread, the source is mounted on a plate attached to one end of the coil of a loud speaker. An alternating current of a few cycle/sec is passed through the coil to generate a back and forth movement of the source creating a relative velocity in the range between two extremes at which the relative velocity is zero and maximum at their midpoint. The output of the Geiger counter is fed to a multichannel analyzer. This analyzer collects the signal as a function of relative velocity and sums it over for each cycle. The final Mossbauer spectrum displays the counts/sec as a function of relative velocity between the source and the absorber. A time of several minutes to few hours is sufficient to record a good spectrum. So we see that the principal of the experimental arrangement is simple. The source is expected to have comparatively long life time (not less than few weeks) so that during the experiment, the emitting γ-ray intensity remains constant. Another point is noteworthy in this connection. The relative velocity should be controlled very precisely, since an error in the measurement of the relative velocity of about 0.01 cm/sec shifts the frequency more than one line width which could render the absorption undetectable. The source and the sample are maintained at liquid helium temperature to reduce thermal broadening of the spectrum. Now, we shall discuss about some application of Mossbauer spectroscopy.

10.5 Isomer Shift (Chemical Shift) Instead of considering the nucleus as a point charge, let us consider that the electric of the )nucleus is uniformly distributed in a sphere of radius R ( charge (Ze) ∼ 1.2 × 10−13 A1/ 3 cm . So in the presence of a surrounding molecular environment, the probability density of the electronic charge inside the nucleus affects the zeroth-order potential energy. This gives rise to a change in the absorption frequency with respect to the free nucleus. From this shift, some chemical information about the nuclear environment can be obtained. So this shift is called chemical shift and sometimes also isomer shift. We shall use perturbation theory to calculate the change in the nuclear energy level due to the finite volume (of radius R) of the nucleus, however, small it may be, as stated above. The electrostatic potential of the nuclear charge Ze at a distance r from centre of the nucleus is given by V (r ) =

[ ( r )2 ] Ze , for r < R 3− 2R R

(10.8a)

ze , for r > R r

(10.8b)

V (r ) =

10.5 Isomer Shift (Chemical Shift)

423

The difference of these two potentials within the nucleus is the perturbation potential, and it vanishes outside the nucleus. So the change in the potential energy of the system due to the finite size of the nucleus is the shift ΔE of the nuclear energy level,

[ ( r )2 2R ] Ze R 3− ρ(r )dτ ΔE = − 2R 0 R r

R[ ( r )2 2R ] Ze 3− ρ(0)4π r 2 dr ≈ − 2R R r 0 ) ( R3 Ze = (−e)|ψ(0)|2 4π − 2R 5 2π 2 Z e |ψ(0)|2 R 2 = 5

(10.9)

In the above calculation, the electronic charge density ρ(r ) inside the nucleus is assumed to be constant and is approximated to ρ(0) = −e|ψ(0)|2 . Since all the atomic orbitals p-, d-, f-, etc., excepting the s-orbitals have nodes at the nucleus, so they will not contribute to |ψ(0)|2 , only the s-orbitals will contribute to it. However, p-, d-, f-, etc., orbitals indirectly affect |ψ(0)|2 through the screening effect produced on the s-orbitals. If we apply this change in energy to both the ground and the excited states of the nucleus, for which we are interested, then we get ( ) 2π 2 2 2 Z e |ψ(0)|2 Rex − Rgr 5 ) ( 2π 2 Z e |ψ(0)|2 2R Rex − Rgr = 5 dR 4π 2 Z e |ψ(0)|2 R 2 = 5 R

∂(ΔE) =

(10.10)

where dR = Rex −Rgr and R are the average radius of the nucleus. In Mossbauer spectroscopy, the isomer shift or the chemical shift is determined by the shift of this energy difference of the absorber with respect to the source δ = ∂(ΔE)a − ∂(ΔE)s ] 4π 2 2 d R [ |ψa (0)|2 − |ψs (0)|2 Ze R = 5 R

(10.11)

This formula (10.11) is extremely helpful to compare the electron density inside the nucleus in different compounds. Besides this, the sign of the isomeric shift depends on the sign of dR. dR is not necessarily positive. In fact, dR for Fe57 is negative, i.e. dimension of the excited state nucleus of Fe57 is less than that of the ground state. Thus, the isomer shift yields information about the s-electron density at the nucleus which in turn determines the valence state of an atom in a compound. Let us illustrate this with the case of iron. It is found that the isomer shift of Fe2+ and Fe3+ ions is different. The valence electronic configurations of the two ions are

424

10 Mossbauer Spectroscopy

Table 10.1 Chemical shift of some Sn119 compounds

Valence state

Electron configuration

Chemical shift (mm/sec)

Sn4+

5s0 5p0

0

Sn (4-covalent)

5(sp3 )

2.1

Sn2+

5s2 5p0

3.7

3d6 4s0 and 3d5 4s0 , respectively. The extra 3d electron in the Fe2+ ion shields the core 1, 2 and 3 s electrons from the positive nuclear in Fe2+ ion than in Fe3+ ∑3 charge more 2 |ψ ion. This results in the slight decrease in n=1 ns (0)| in the case of Fe2+ . Since R 2 dRR is negative in 57 Fe, so the chemical shift is greater in Fe2+ ion. This means that the Mossbauer peak of Fe2 + compound is shifted to more positive velocity than the Fe3+ compound. Another interesting example is related with the two nuclei of iodine, 127 I and 129 I. The two respective nuclei which are found in the excited states with energies 57.6 keV (for 127 I) and 27.7 keV (for 129 I) above the ground state are generated from the β–decay of the radioactive nuclei 127 Te and 129 Te in the compound zinc telluride. It is found that the absorption peaks of the two nuclei are on the opposite sides of zero velocity. dR is negative for the nuclei 127 I and positive for 129 I. Thus, the excited state of 129 I is larger in size than its ground state, whereas the reverse is the case of the nucleus 127 I. Moreover since |ψs (0)| > |ψa (0)|, resonance peak is observed on the positive side in 127 I and on the negative side in 129 I of zero velocity in the Mossbauer spectra. Measurement of the relative s-electron density is also helpful in determining the bond characteristic of the atoms attached to the Mossbauer nucleus. For example, the outer electronic configuration of tin (Sn119 ) is 5s2 5p2 . But in three different chemical environments, the chemical shifts of this Mossbauer nucleus are different as shown in Table 10.1. All the shifts are compared with respect to Sn4+ configuration. In the tetrahedral compound, the four electrons in the four hybridized orbitals 5(sp3 ) essentially correspond to one s-electron. Thus, we see from the Table 10.1 that the isomer shift increases with the increase in the number of s-electrons (i.e. s-electron density).

10.6 Nuclear Quadrupole Interaction It is known that nuclei with spin greater than ½ lack spherical symmetry. Such nuclei possess an electric quadrupole moment. In Chap. 9, we have seen that the quadrupole moment (eQ) of such nuclei interact with the electric field gradient at the nucleus and the interaction energy is Em =

]1 2 [ ] η2 / e2 q Q [ 2 3m − I (I + 1) 1 + 4I (2I − 1) 3

(10.12)

10.6 Nuclear Quadrupole Interaction

425

Here, eq = ∂ 2 V /∂Z 2 and η = (∂ 2 V /∂X 2 −∂ 2 V/∂Y 2 )/∂ 2 V /∂Z 2 are the asymmetry parameter of the field about | 2 gradient | | | | the |spin symmetry axis (Z). The field gradient | ∂ V | | ∂2V | | ∂2V | is so chosen that | ∂ 2 Z 2 | ≥ | ∂ 2 Y 2 | ≥ | ∂ 2 X 2 |. Let us consider the case of a Mossbauer nucleus in which the spins of the ground and the excited states are 1/2 and 3/2, respectively. Since the ground state has zero quadrupole moment, it will not split, only the excited state will split due to the above interaction. Thus, the level I = 3/2 will split into two levels with m = ± 3/2 and ± 1/2 with energies, E ±3/ 2 =

[ ]1 2 η2 / 3e2 q Q 1+ 4I (2I − 1) 3

[ ]1 2 η2 / 3e2 q Q 1+ E ±1/ 2 = − 4I (2I − 1) 3

(10.13a)

(10.13b)

So the selection rule Δm = 0, ± 1 gives two lines (Fig. 10.4) separated by a frequency Δν =

[ ]1 2 [ ]1 2 η2 / η2 / 6e2 q Q e2 q Q 1+ 1+ = 4I (2I − 1)h 3 2h 3

(10.14)

It is important to note that the spectrum is symmetrical only in the case of a transition 3/2 → 1/2. It is not possible to determine the sign of the quadrupole coupling constant (e2 qQ) and the asymmetry parameter η from such observation. If, at least, one of the nuclear states has spin greater than 3/2, there would be more than two lines and from there it is possible to determine the asymmetry parameter, η and in certain cases the sign of eq (i.e. electric field gradient ∂ 2 V /∂Z 2 ). Even for nucleus

Fig. 10.4 Quadrupole splitting in a system with I = 1/2 in the ground state and I = 3/2 in the excited state

426

10 Mossbauer Spectroscopy

with I = 3/2, it is possible to determine the direction and the sign of the electric field gradient from the studies of the spectra of single crystal. If no single crystal is available, the sign of field gradient can be determined if the spectrum is taken in a large external magnetic field of strength ~ 3–10 T. The electric field gradient V zz can originate in two ways. Each electron in the atom can contribute to the electric field gradient tensor V ij = ∂ 2 V/∂Ri ∂Rj, and if the orbital occupation is non-spherical, the total value of V ZZ is nonzero for this valence configuration. If there is an excess of electron density along Z-axis (electrons in pZ , d Z2 , d XZ , d YZ -orbitals), V ZZ is negative in sign, and if the excess is in the XY-plane (electrons in the pX , pY, d XY, d X2-Y2 -orbitals), V ZZ is positive in sign. If the immediate chemical bonding dominates V ZZ , it is possible to get an insight into the occupation of the orbitals. Distant charges (perhaps ionic charges) may also contribute to V ZZ . This is called lattice contribution. This is specially important for an s-state ion (for example d 5 electronic contribution in high spin Fe3+ ion) where the valence contribution is formally absent. However note that the lattice charge can also induce electric field gradient in the valence electrons by polarization. Thus, we see that the quadrupole interaction is a measure of asymmetry of the atomic environment which is related with the electron orbitals of the chemical bonds.

10.7 The Effect of Magnetic Field Mossbauer ( ) nucleus with nonzero spin (I /= 0) when interacts directly with a magnetic field B→ exhibits Zeeman splitting of the relevant levels, and the interaction energy is |   | E m = I m | Hmag | I m = −gμn Bm

(10.15)

where g is the nuclear g-factor and μn is nuclear magneton. Let us consider the case of a nucleus with its ground and excited states having spin ½ and 3/2, respectively. If the ground and the excited states are indicated by double and single primes, the Zeeman splitting of the two states is 1 '' 1 '' '' '' E +1 / 2 = − 2 g μn B, E −1/ 2 = 2 g μn B

(10.16a)

1 ' 1 ' ' ' E +1 / 2 = − 2 g μn B, E −1/ 2 = 2 g μn B, 3 ' 3 ' ' ' E +3 / 2 = − 2 g μn B and E −3/ 2 = 2 g μn B

(10.16b)

The selection rule Δm = 0, ± 1 gives six transitions which are shown in Fig. 10.5, and the transition frequencies are shown in Table 10.2. Although in this illustration,

10.7 The Effect of Magnetic Field ''

427

'

both gn and gn are taken as positive, but in real cases, the things may not be so. In 57 Fe '' ' gn and gn are 0.1804 and −0.1027, respectively, whereas in 119 Sn, the corresponding values are −2.0920 and 0.507. The six lines are expected to be equally separated if only magnetic effect is considered, but they do not remain equally separated if along with this, quadrupole interaction is also taken into consideration (Fig. 10.5). Another thing is to be noted in this connection. There are three pairs of transitions: (i) 1/2 ↔ 3/2, -1/2 ↔ -3/2, (ii) -1/2 ↔ -1/2, 1/2 ↔ 1/2 and (iii) 1/2 ↔ -1/2. Detailed calculations show that although the transition probabilities within each group are same but they are different in different groups. The relative values of the corresponding transition probabilities (i.e. intensities) are 3: 2:1. This is to be noted that the magnetic field may be applied externally or it may arise internally due to the interaction of the electrons with the angular momentum of the nucleus. The latter one is called the intrinsic magnetic field. If an atom has

m -3/2 -1/2 I′ = 3/2

1/2

3/2

hνo

-1/2 I′′ = 1/2 1/2

a

b

c

Fig. 10.5 Energy level splitting and transitions between the states with I ' = 1/2 and I ' = 3/2. a Free nucleus; b nucleus in a magnetic field only; c nucleus in a magnetic field with quadrupolar interaction. νo is transition frequency in the absence of magnetic and quadrupole interaction (In this ' illustration, both gn '' and gn are taken as positive)

Table 10.2 Transition frequencies when I ' = 3/2 and I '' = 1/2 (without quadrupole interaction)

m''

m'

Transition frequencies

−1/2

−3/2

ν0 + μn B (3gn −gn '' )/2h

'

'

−1/2

−1/2

ν0 + μn B (gn −gn '' )/2h

−1/2

1/2

ν0 −μn B (gn + gn '' )/2h

1/2

−1/2

ν0 + μn B (gn + gn '' )/2h

'

'

'

1/2

1/2

ν0 −μn B (gn −gn '' )/2h

1/2

3/2

ν0 −μn B (3gn −gn '' )/2h

'

428

10 Mossbauer Spectroscopy

any unpaired electron(s), this can also produce a large intrinsic magnetic field at the nucleus by creating a slight imbalance in the s–electron spin density at the nucleus which interacts differently with the parallel and antiparallel spins of other electrons. Calibrating with the external field, the strength of this field may be estimated. The intrinsic field may be as high as 100 T which is much greater than (5–10 T) obtained from superconducting magnets. But remember that this internal field is not extended throughout the region of the bulk sample but is confined to a very narrow region around the nucleus, i.e. it is a very localized field. Mossbauer spectroscopy is useful to determine the magnetic hyperfine fields in a variety of magnetic materials. Parameters such as crystalline environment, pressure, temperature and external fields affect the hyperfine fields. The internal fields in various ferrous (Fe2+ ) and ferric (Fe3+ ) compounds are found to be in the ratio of 4:5, i.e. proportional to the number of unpaired electrons. In a magnetically ordered solid, the direction of the unpaired spin and thus the internal field is effectively frozen, and this results in the magnetic hyperfine splitting. In a paramagnetic solid, the direction of the spin changes rapidly due to electronic spin relaxation and the time average of the field is usually zero within the life time of the Mossbauer nucleus in its excited state, and so no splitting is seen.

References and Suggested Reading 1. G. Aruldhas, Molecular structure and spectroscopy (Prentice Hall of India, New Delhi, India, 2001) 2. C.N. Banwell, Fundamentals of Molecular spectroscopy (Tata McGraw-Hill Publishing Company Limited, New Delhi, India, 1983) 3. B.P. Straughan, S. Walker, Spectroscopy, vol. 1 (Chapman and Hall, New York, 1976) 4. J.F. Duncan, Lectures on Chemical Applications of Mossbauer Effect (Tata Institute of fundamental Research, Bombay, 1968)

Chapter 11

Some Nonlinear Processes

Abstract Basic principle of nonlinear spectroscopy is discussed, and the relevant selection rules are derived. The following nonlinear phenomena have been extensively discussed: hyper Rayleigh and Raman Effects, coherent antistokes Raman scattering (CARS), stimulated Raman scattering, inverse Raman scattering, two photon and multiphoton absorption and ionization, multiphoton dissociation and laser isotope separation. Detailed theoretical backgrounds of the determination of susceptibilities of various orders, sum frequency and second harmonic generation, stimulated Raman scattering and coherent antistokes Raman scattering are given.

Whenever a molecule is exposed to an electric field, an electric dipole moment is induced in it. For weak field, the induced dipole moment has a linear dependence on the incident field intensity, and the proportionality factor is not an ordinary scalar constant but a tensor of rank 2. This factor is called the polarizability (or susceptibility) tensor of the molecule, and in this case, it is linear. Actually, the induced dipole moment does not only have a linear term but also has several nonlinear terms depending on various higher integral powers of the electric field. So there exists not only the linear polarizability, but also polarizabilities of higher and higher orders the magnitudes of which decrease with the increase of the order. For the ordinary strength of the electric field, the contributions of these nonlinear terms in the induced dipole moment are negligible, and they yield no new results. For example in the case of Raman effect, only the contributions of the linear polarizability are effective, and weak Raman lines are observed, both on the higher and the lower wavelength sides of the incident radiation (Rayleigh line). But for high intensity laser beams, viz. light from giant laser, the quadratic and other higher-order terms of the induced dipole may be significant, and from these terms, several interesting phenomena may arise. The spectroscopy arising from these phenomena is called nonlinear spectroscopy or laser spectroscopy, as they appear from the interaction of molecules with high intensity laser beams. In this chapter, we shall discuss some of these nonlinear phenomena which not only give some new insights into the molecules but also open some new areas of investigation.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 P. K. Mallick, Fundamentals of Molecular Spectroscopy, https://doi.org/10.1007/978-981-99-0791-5_11

429

430

11 Some Nonlinear Processes

11.1 Nonlinear Raman Effect In Chap. 4, we have seen that whenever a molecule is exposed to an incident radiation, it interacts with the electric field associated with the radiation giving rise to an induced dipole moment having a linear relation with various components of the field. Actually, - can be written in a more general way which includes the induced dipole moment (μ) the nonlinear terms along with the linear one. Considering the interaction of the radiation with a single isolated molecule, the induced dipole moment is − → − → − → − → − → − → μ - = α · E + β- · E · E + γ · E · E · E + · · · · · ·

(11.1)

where α, β, γ · · · · · · are the polarizability, hyperpolarizability, second hyperpolarizability, etc., which are, respectively, second, third, fourth, etc., rank tensors. Thus, the components of the induced dipole moment are given by μi = αij E j + βijk E j E k + γijk E j E k El + · · · · · ·

(11.2)

Here, α ij , β ijk and γ ijkl are different components of the polarizability, hyperpolarizability and second hyperpolarizability tensors where summation over the repetitive indexes, j, k, l (x, y, z), has been used.

11.1.1 Normal, Hyper and Second Hyper Rayleigh and Raman Scattering All the components of the polarizability and hyperpolarizability tensors are functions of the normal coordinates of the molecule concerned. So they will oscillate with the oscillation of the normal modes, Qk = Ak cos 2π ν k t. Let the electric field be represented by E- = E-0 cos 2π ν0 t, ν0 being the frequency of the monochromatic laser radiation. If we drop the subscripts of the vector and tensor components for the time being, the induced dipole moment component can be written (considering only one mode of vibration for simplicity) as μ = α(Q)E + β(Q)E 2 + γ (Q)E 3 + · · · · · · ) ( ] [ ∂a Ak cos 2π vk t E 0 cos 2π v0 t = α0 + ∂Q 0 ) ( ] [ ∂β Ak cos 2π vk t E 02 cos2 2π v0 t + β0 + ∂Q 0 ) [ ( ] ∂γ + γ0 + Ak cos 2π vk t E 03 cos3 2π v0 t + · · · · · · ∂Q 0

11.1 Nonlinear Raman Effect

431

) ( ) [( 1 1 ∂β 3 β0 E 02 + Ak E 02 cos 2π vk t + α0 E 0 + γ0 E 03 cos 2π v0 t 2 2 ∂Q 0 4 ( ( ) ) ) ( ∂α ∂γ 1 3 Ak {cos 2π(v0 + vk) t E0 + + E3 2 ∂Q 0 8 0 ∂Q 0 [ 1 β0 E 02 cos 2π (2v0 )t + cos 2π (v0 − vk )t}] + 2 ( ) ] 1 ∂β E 02 Ak {cos 2π (2v0 + vk )t + cos 2π (2v0 − vk )t} + 4 ∂Q 0 ( ) ( [ 3 ∂Y 1 γ0 E 03 cos 2π (3v0 ) + + 4 8 ∂Q 0 ] 3 E 0 Ak {cos 2π (3v0 + vk )t + cos 2π (3v0 − vk )t} + . . . . . . (11.3)

=

where the zeroes of the tensor elements correspond to the equilibrium nuclear configuration Q0 of the molecule. The first term on the right-hand side of the above equation is a constant term, and the second term corresponds to the radiation of a molecular frequency ν k . The terms, in the first square bracket, are responsible for the Rayleigh (ν 0 ) and Raman (ν 0 ± ν k ) scattering. The terms in the next square bracket are similarly responsible for the hyper Rayleigh (2ν 0 ) and hyper Raman (2ν 0 ± ν k ) scattering. These terms indicate that the induced dipole moment is proportional to the square of the field intensity for both the hyper Rayleigh and hyper Raman scattering. The hyper Rayleigh term yields some interesting results. It is known that the dipole moment changes sign under inversion operation. So for a centrosymmetric molecule, this is only possible when β0 vanishes. Therefore, no hyper Rayleigh scattering is possible in centrosymmetric molecules. Another interesting point is noteworthy in this context. Although in the centrosymmetric molecules, β 0 vanishes but not necessarily the gradient (∂β/∂Q)0 of the hyperpolarizability components. So along with other molecules, hyper Raman scattering is allowed in these molecules provided the transitions follow certain selection rules. As an extension of this result, we can say that second harmonic generation is not possible in any centrosymmetric crystal, i.e. from a coherent radiation of angular frequency ω, however, strong it may be, no coherent radiation of angular frequency 2ω can be generated in centrosymmetric molecule. Lastly, the terms in the third or the last square bracket of the last line of Eq. (11.3) give rise to second hyper Rayleigh (3ν 0 ) and second hyper Raman (3ν 0 ± ν k ) scattering.

11.1.1.1

Selection Rules

We know that the intensity of a spectral line arising from the molecular transition from the initial state |i) to a final state | f ) is proportional to the square of the transition dipole moment matrix. So any (kind | of | )Raman transition is allowed if the corresponding transition dipole moment i |μρ | f is non-vanishing for at least one component of the induced dipole moment μρ (ρ = x, y, z). So in order to have

432

11 Some Nonlinear Processes

a normal mode (Qk ) active( in Effect, at least one component of the | hyper| Raman ) hyperpolarizability matrix, i |∂βρσ δ | f f (Eq. 11.2) has to be non-vanishing. For the excitation of a Raman fundamental vibration, both the initial and the final states are vibronic states of the ground electronic state. So the initial state |i) is the ground vibrational state, and the final (| f )) state is the first excited vibrational state of the concerned normal mode (say kth, with, vk = 1) of the ground electronic state. Non-vanishing characteristics of the hyperpolarizability matrix can be determined from the viewpoint of group theory. Generally, the ground electronic and ground vibrational states of( most of) the molecules are totally symmetric. In order to have the matrix element i|βρσ δ | f non-vanishing, the direct product of the integrand ψ i • β ρσ δ • ψ f should belong to the totally symmetric representation of the point group to which the molecule belongs. This is only possible when the normal mode Qk belongs to the same representation as the hyperpolarizability component, i.e. [(Qk ) = [(β ρσ δ ). The reason for this is that the representation of the first excited vibration state vk = 1 of any molecule is same as that of the corresponding normal coordinate Qk, i.e. ([(Qk ) = [(ψ f (vk = 1)). So the hyper Raman selection rules are different from those of the Raman ones {[(Qk ) = [(α ρσ )}. So hyper Raman studies may provide some new information, and hence, it can be used to extend the gamut of vibrational investigation of molecules. Similarly, those vibrations are expected to be observed in the second hyper Raman spectra of molecules the symmetry of which are same as those of any second hyperpolarizability components γ ρσ δϕ .

11.1.2 Stimulated Raman Scattering (SRS) The stimulated Raman scattering was discovered by chance when Woodbury and Ng [1] were doing experiment by introducing nitrobenzene in a cell placed in the Ruby laser cavity where they found rather strong emission in the output of frequencies different from that of the ruby line. In normal Raman scattering, the intensity of Raman lines is weak. But when the excitation is made with a very high intensity (Q– switched) laser beam of frequency ν0 , the intensity of the stokes line of frequency ν 0 –ν M (ν M being the most intense Raman active molecular frequency) increases and is used to stimulate this radiation in the direction of the incident laser beam. So the stokes line becomes a strong coherent radiation of intensity about 50% of the incident beam in the forward direction observed along with the incident radiation. So this process is called a stimulated Raman gain spectroscopy (SRGS). This gain can be exponential and the transfer of energy from the original laser beam (ν 0 ) to the shifted line of frequency ν 0 –ν M may be sufficiently high leading to a substantial increase of the population of the molecular state v = v1 . This stokes line is then further used as an exciting line and produce another stokes line of frequency ν 0 –2ν M , this too also being a stimulated one. In this way, a few series of stimulated Raman lines of frequencies ν0 –nνM (n is about 1–4) are observed along with the original Q-switched laser frequency ν0 in the forward direction. This is illustrated in the Fig. 11.1. If a coloured photographic film is used to detect and analyze the outgoing radiation, a

433

ν0 – 3νM νM

V = V1

ν0 – 2νM ν0 – 2νM

ν0 – νM ν0 – νM

ν0

Laser beam

V = V0

(a)

Lens

Sample Cell (C)

V1 V2 V3

Detector (D)

11.1 Nonlinear Raman Effect

ν0 ( and ν0 – nνM) ν0 + νM ν0 + 2νM ν0 + 3νM ν0 + 4νM

(b)

Fig. 11.1 Stimulated Raman scattering. a Energy level diagram. Vi’s (associated with the dotted lines) are virtual states. b Experimental demonstration

few circular rings of various colours are also observed within 100 of the original laser beam direction. These coloured rings correspond to the frequencies ν 0 + nν M , with frequency increasing with the ring diameter as shown in the Fig. 11.1b. As in the case of stimulated Raman gain spectroscopy (SRGS), stimulated Raman loss spectroscopic (SRLS) phenomena also occur. Here, stimulated stokes Raman line of frequency ν0 -νM interacts with the molecule in the excited state v = v1 whose population is sufficiently increased by SRGS and generates a gain in the line of frequency ν0 at the cost of the loss of the stokes line ν 0 -ν M through the process of antistokes transition to the ground molecular state v = v0 = 0. Another point is noteworthy in this respect. Since the conversion efficiency of stokes Raman line is high in SRGS, this may be used as a shifting mechanism of a pulsed laser wavelength which is otherwise non-tunable.

11.1.3 Coherent Antistokes Raman Scattering (CARS) Coherent antistokes Raman scattering (CARS) is a kind of four wave mixing phenomena where three incident waves are mixed up and generate a new one. In a nonlinear medium (say, crystal), two laser waves of frequencies ν1 and ν 2 (say, ν 1 > ν 2 ) are made to overlap to generate a new one of frequency (2ν 1 −ν 2 ). Suppose, ν1 is held fixed, and ν2 is varied. In that case, a very effective mixing is observed when (ν 1 -ν 2 ) matches with one of the Raman active molecular frequency of the medium (i.e., ν M = ν 1 −ν 2 ). This radiation is coherent, and the phenomenon is called coherent antistokes Raman scattering (CARS). CARS was first observed experimentally by Maker and Terhune [6]. This phenomenon can be viewed in the following way. Under the condition mentioned above (i.e., when ν M = ν 1 −ν 2 ), the molecule

434

11 Some Nonlinear Processes

oscillates vigorously, and this is called Raman resonance. In the Raman resonance condition, the molecule is in the two vibrational states (one ground and one excited) at the same time, i.e. a coherent superposition of states. When the probe beam (ν 1 ) interacts with the molecule in this condition, antistokes Raman transition brings the molecule back from the upper vibrational state to the ground state where it no longer remains coherent. This gives rise to the CARS line (ν 1 + ν M = ν 1 + ν 1 −ν 2 = 2ν 1 −ν 2 ) detected in a direction different from those of the incident beams. Now if the frequency ν 1 is held fixed and the frequency ν 2 is varied over a range which covers the desired range of ν M , CARS spectra can be obtained. On CARS spectra, weak non-resonance scattering (2ν 1 –ν 2 ) is superposed as a background. CARS is highly directional and has a small divergence (Fig. 11.2). It is equally possible to observe the coherent stokes Raman spectra relative to the line of frequency ν2, and the phenomena are called coherent stokes Raman scattering (CSRS). The CSRS frequency is ν CSRS = ν 2 −ν M = ν 2 −(ν 1 −ν 2 ) = 2ν 2 −ν 1 which is also detected in a direction different from those of the exciting laser beams. Both CARS and CSRS are equally favourable and are shown in Fig. 1.2 with the display of an experimental demonstration. But there is a general tendency of the CSRS to overlap with the fluorescence spectra of the sample. For this reason, CARS technique is more convenient than CSRS for use to the study of the vibrational spectra. Moreover, the scattered intensity is further increased when either ν 1 or ν 2 coincides with an electronic transition, and in those cases, the phenomena are called resonance CARS or resonance CSRS. CARS ν1 νCSRS ν2 ν1 νCARS

ν1 ν2

(a)

ν2

ν1

νCARS = ν1 + νM νm

CSRS

ν2

ν1

ν2

νCSRS = ν2 - νM νm

(b) Fig. 11.2 a Diagrammatic description of experimental set-up of CARS and CSRS; b energy level diagram with transitions in CARS and CSRS

11.1 Nonlinear Raman Effect

435

11.1.4 Inverse Raman Scattering When a molecular system is simultaneously excited by a strong laser beam (of frequency ν L ) and a continuum of higher frequencies, two types of stimulated Raman scattering occur. Light of frequency ν L + ν M in the continuum, ν M being a molecular frequency, undergoes stokes Raman scattering through a molecular transition from its lower to higher vibrational state (ν M ) by emitting a radiation of frequency ν L which is stimulated by the already existing laser frequency ν L . The other possibility is that the laser radiation (ν L ) undergoes an antistokes Raman scattering, which brings the molecule from its higher (ν M ) to the ground state, by emitting a radiation of frequency ν L + ν M , which is stimulated by a similar radiation present in the continuum. In general, the former one is more efficient than the later not only because it is stokes scattering but also because of the stimulating radiation is more intense (laser) in the case of former one. In that case, the continuum shows an absorption line (a dip in intensity) at ν L + ν M . Thus if the continuum is broad enough, the entire vibrational spectra of the molecule can be collected as absorption lines or dip in the continuum corresponding to different vibrational frequencies (ν M ). This phenomenon is called inverse Raman effect. An experimental set-up for the inverse Raman Effect is shown in Fig. 11.3. A laser beam (ν) is split up into two parts by a plate P1. One part goes straight towards the sample cell, and the other part is allowed to enter into a second harmonic generator crystal (SHG). After passing through the SHG crystal, the second harmonic beam is allowed to enter into a dye cell as an excitation radiation which produces a strong fluorescence continuum in the region above (ν). This continuum and the initial laser beam meet on the second plate P2, and from there, both the laser beam and the fluorescence continuum are allowed to enter into the sample cell. After passing through the sample cell and the filter (which filters out the exciting radiation), the emergent radiation is detected as inverse Raman spectra of the sample. P1 LASER

P2

ν SAMPLE FILTER

ν

2ν SHG CRYSTAL

DYE CELL

Fig. 11.3 Demonstration of experimental set-up of Inverse Raman Scattering

SPECTROGRAPH/ DETECTOR

436

11 Some Nonlinear Processes

11.1.5 Two Photon and Multiphoton Absorption and Ionization In ordinary (linear) absorption phenomenon, a molecule absorbs a photon from the incident beam, and this takes the molecule from an initial (generally the ground) state to an excited state, both of which are real stationary states of it. In two photon absorptions, two photons are required to excite the molecule from the ground to the excited state. This is actually a two-step phenomenon because this process involves an intermediate state, used as a buffer state, which may be virtual or a real stationary state of the molecule to allow the two transitions to occur. This can be understood in the following way. Let us consider Raman scattering. It is a two-step process in which an incident photon takes the molecule from an initial state 1 to an intermediate state V where from it makes a downward transition to a final molecular state 2. This two processes cannot be considered as two independent processes, first an absorption and then emission, rather they occur more or less simultaneously. The intermediate state may be a real state (in the case of resonance Raman scattering) or a virtual state (in the case of normal Raman scattering). In two photon absorption, the molecule makes an upward transition in the second step, rather than a downward transition as in the case of Raman scattering, to a final molecular state 2. These are illustrated in Fig. 11.4. In Raman (stokes) scattering, the molecule in the initial state 1 goes to an intermediate virtual state V (normal Raman), and there from it makes a downward transition to a higher (vibrational, say) state 2 of the molecule. This is shown in Fig. 11.4a. In antistokes Raman scattering, only the energy of the initial state I is greater than that of the final one 2. In Fig. 11.4b, a two photon absorption occurs by absorbing two photons of equal energy (hν) via the virtual intermediate state V. Figure 11.4c is also a two photon absorption with different energy photons (hν1 and hν2 ), and the last Fig. 11.4d represents a three photon absorption occurring by absorbing three equal energy photons (hν) via two intermediate virtual states V 1 and V 2 . So if the final molecular state 2 is not accessible by single photon absorption, it may be accessed by a double photon absorption mechanism. Similar to two and

ν ν1

ν2

ν 1

(b)

V2

ν ν1

2 1

(a)

ν

V

ν2

2

2

2

V

V

V1

ν 1

1

(c)

(d)

Fig. 11.4 a Raman scattering; b two photon absorption with two photons of equal energy; c two photon absorption with two photons of unequal energies and d three photon absorption with three photons of equal energy. V, V1 and V2 are the intermediate states, here virtual

11.1 Nonlinear Raman Effect

437

three photon absorption, multiphoton absorption is also possible. In that case, many intermediate or virtual states are required to reach the final molecular state by successive transitions, first from the initial molecular state to the virtual state nearest to it, then from this virtual state to another and in this way through successive transitions, the molecular state 2 is reached in the last transition. In fact, such a process in a three photon absorption is shown in Fig. 11.4d. Remember also in these processes, all the transitions occur more or less simultaneously. All double and multiphoton absorptions are higher-order effects, so high intensity radiation is required to detect the absorption. For this reason high intensity, laser is used to observe these effects. Another point is to be noted in this regard. Selection rules depend on the order of the absorption process, so the spectral structure of a molecule is not same for single, double and multiphoton absorption. Also remember that the energy conservation rule is followed in these processes, so the energy difference between the final and initial molecular states is equal to the sum of the energies of the photons involved. Double and multiphoton transitions are generally monitored by two methods. This is illustrated in Fig. 11.5. In the first method, the molecule is excited with a tunable dye laser, and fluorescence emission is monitored. The fluorescence emission occurs if sum of the energies of two (in double photon absorption) or more photons (in multiphoton absorption) becomes equal to the energy of any rotationalvibrational level of an excited electronic (fluorescing) state of the molecule. The total and undispersed fluorescence intensity is monitored by varying the laser frequency. The fluorescence excitation spectra thus give the double or multiphoton absorption spectra of the molecule. In the second method, two photons from the incident tunable laser take the molecule to an excited electronic state, and the third photon ionizes the molecule into a positively charged molecular ion and an electron. This is called a 2 + 1 multiphoton ionization process. Similarly, other ionization precesses such as 1 + 1, 2 + 2 or 3 + 1 are also possible. The number of molecular ions collected by a plate with a negative potential are counted as a function of laser frequency to produce a double photon ionization spectra of the molecule. In a similar way, m + n multiphoton ionization spectra may also be recorded in which m photons are absorbed to take the molecule to an excited state and n photons ionize the molecule. Figure 11.5b is thus a 2 + 1 multiphoton ionization process. Multiphoton ionization is more advantageous when the quantum yield of the fluorescence emission created by two photon absorption is small.

11.1.6 Multiphoton Dissociation and Laser Isotope Separation It was found that some gases emit radiation (luminescence) when a high intensity pulsed CO2 laser beam is focused into it, and the emission was observed even after the pulse was stopped. The gas molecules which emit this type of radiation are found to have some normal modes whose frequencies are very nearly equal to any of the

438

Ionization continuum ν 2

2

Fluorescence

Fig. 11.5 Monitoring of double photon absorption spectra of a molecule. a Fluorescence excitation spectra. b Double photon ionization spectra where two absorbed photons (hν) take the molecule from the ground state 1 to the excited state 2, and the third photon ionizes it. V is the virtual intermediate state

11 Some Nonlinear Processes

ν

ν

ν V

V ν 1

1

(a)

(b)

frequencies of the three lines of the CO2 laser. For example, CF2 Cl2 , SiF4 and NH3 have vibrational–rotational band frequencies which are very nearly equal to those of the lines of CO2 laser. The species responsible for this kind of luminescence was found to be C2 , SiF and NH2, respectively, which are the products of the interaction of the high-power laser with the respective gas molecules. When the high intensity laser beam is focused into the gas, it gets dissociated into some components, one of which emits fluorescence. The dissociated products arise from the breaking of one of the bonds of the molecule. But the bond energy is much greater than the relevant normal frequency (or frequencies of the laser lines), so obvious question arises that how does this dissociation occur? In the presence of the high intensity infrared laser, not a single transition but a multiple transition is necessary to reach the continuum and break up the bond. About 30 or more transitions are required to provide appropriate energy to conduct the process. As discussed in the earlier chapters, a normal coordinate is actually a linear combination of small oscillations of several bonds and angles of a molecule. So each normal frequency is the sum of the fractional frequencies of oscillation of these bonds and angles. But there are some normal modes where the contribution of one bond is much greater than those of the other bonds and angles. In those cases, the corresponding mode is mostly associated with the oscillation of that particular bond. So when one such bond frequency exactly matches with any of the laser frequency, the molecule absorbs a photon and excites the molecule from the vibrational state v = 0 to the state v = 1 of that molecule. Due to anharmonicity of the vibrational levels, the laser frequency is not in exact resonance with the transition v = 1 to v = 2 states. But resonance condition is achieved if we consider the rotational motion of the molecule, and in that case, the laser frequency resonates with any of the rotational levels of that vibrational state. Such kind of resonance may occur up to a small number of vibrational states, say v = 3 leading to the successive vibrational

11.1 Nonlinear Raman Effect

439

transitions 0 → 1, 1 → 2, 2 → 3. In a polyatomic molecule, there is a high density of rotational-vibrational states arising from the excited states of the other normal modes of the molecule which will superpose with the levels v > 3 of the said mode. This high density rotational-vibrational state constitutes a quasicontinuous region of energy. This kind of quasicontinuity increases with the increase of the number of normal modes, i.e. with the increase of the number of atoms in the molecule. Due to this kind of quasicontinuity, superposing on the vibrational levels v > 3 of the mode under consideration and multiphoton transitions, 0 → 1, 1 → 2, 2 → 3, 3 → 4, 4 → 5, 5 → 6,….., vmax -1 → vmax , vmax → dissociation continuum, take place, and the molecule gets dissociated breaking the said bond. This is shown in Fig. 11.6. The presence of the dissociated products is confirmed from the fluorescence emission of a component of the product when excited with another probe laser. In some cases, chemi luminescence of the product is the evidence of this type of dissociation. Otherwise exciting the molecule with another laser (probe), luminescence from the product can be confirmed. In the latter experiment, the CO2 laser and the probe laser are focused into the molecule in two mutually perpendicular directions, and the total fluorescence from the product is observed in a direction perpendicular to both the two laser beams. In fact, production of NH2 from the dissociation of hydrazine (N2 H4 ) or methylamine (CH3 NH2 ) was confirmed by a probing laser tuning at a wavelength ´ corresponding to a vibronic band of an electronic transition of the NH . 5980 Å 2 Multiphoton dissociation is an effective technique used in isotope separation. Due to the high monochromaticity of the laser beam, it is possible to dissociate a selective species in the sample containing the desired isotope, and the dissociated product containing the desired isotope is allowed to combine with the scavenger molecule, used in this process along with the sample, to form a compound. This type of selective dissociation is useful for the enrichment of the selective isotope both in the sample and also in the product. Fig. 11.6 Multiphoton laser dissociation

Dissociation continuum

Quasi continuum

vmax

V

3 2 1 0

440

11 Some Nonlinear Processes

One application is the separation of the isotopes of boron, 10 B and 11 B. Their presence in natural abundance is 18.7 and 81.3, respectively. When BCl3 is excited by a selective wavelength from the high intensity CO2 laser, it will dissociate either of the species by tuning if the laser frequency matches with one of the normal frequencies of either of the species. When excited by the frequency 958 cm−1 , which is also the normal frequency of 11 BCl3 , the product will react with O2 and form 11 BO. This molecule exhibits chemiluminescence in the visible region since the product 11 BCl3 exists in an excited electronic state in the process. Similar chemiluminescence from 10 BCl3 is also possible if the laser is tuned to the corresponding mode of this species. Another example is SF6 . The natural abundances of the isotopes of 32 S, 33 S, 34 S, and 36 S are approximately 95.0, 0.74, 4.24 and 0.17%, respectively. When a mixture of SF6 /H2 or SF6 /NO is excited by the high power CO2 laser tuned at 945 cm−1 , which is also a strongly active infrared mode of 32 SF6 , this species will be dissociated by the infrared multiphoton dissociation (IRMPD) process and the additive H2 or NO which are used as radical scavenger of free radical energy generated from the dissociation of SF6 . The enrichment of the heavier isotope 34 S is then examined in the unreacted mixture of 32 SF6 / 34 SF6 using mass spectrometry, and it is found nearly as high as 1:1.

11.2 Theoretical Description We know that when a magnetic dipole or an electric dipole or multipole oscillates, it emits an electromagnetic radiation. But in general, the spectral lines or bands observed in atoms or molecules appear due to oscillation of electric dipoles, and so, they are called electric dipole radiation. In electric dipole radiation, the spectral line intensity is proportional to the square of the transition dipole (in the present case induced dipole) moment matrix for a transition from an initial molecular state |i) to - 2 . In the presence of an exciting radiation, the a final molecular state | f ), i.e. |( f |μ|i)| exciting electric field is a function of time, and so, the wave functions of the molecule no longer maintain their unperturbed form. So the matrix element is to be determined under time-dependent perturbation technique. Both the initial and the final states of the molecule at a particular instant of time (t) can be written as linear combinations of the unperturbed states. The total Hamiltonian of the molecular system in the presence of the exciting radiation is H = H0 + λV (t)

(11.4)

H 0 and V (t) being, respectively, the unperturbed and perturbed parts of the Hamiltonian, and λ is not any physical entity, its power denotes the order of the perturbation. The time-dependent wave equation is

11.2 Theoretical Description

441

ih

∂ψ = H ψ = (H0 + λV )ψ ∂t

(11.5)

where the unperturbed Hamiltonian follows the wave equation ih

∂ψn(0) = H0 ψn(0) = E n0 ψn0) = hωn0 ψn(0) ∂t

(11.6)

and the unperturbed wave function is ψn(0) = ψn(0) (r , t) = u n (r )e−i ( En /π )t = u n (r )e−iωn t 0

0

(11.7)

Expanding ψ in the power series of λ ψ = ψ(r , t) = ψ (0) (r , t) + λψ (1) (r , t) + λ(2) ψ (2) (r , t) + λ(3) ψ (3) (r , t) + . . . (11.8) and substituting Eq. (11.8) in (11.5) and equating the coefficients of the same order (say, Nth) on both the sides, we get ih

r , t) ∂ψ (M) (= H0 ψ (N ) (r , t) + V (t)ψ (N −1) (r , t) ∂t

(11.9)

Expanding ψ (N ) (r , t) in the field of the complete set of unperturbed wave functions, ψl0 (r , t), we write ψ (N ) (r , t) =

∑

a˙ l(N ) (t)ψl(0) (r , t) =

l

∑

al(N ) (t)u l(0) (r )e−iωl t

(11.10)

l

where al(N ) is the Nth order probability amplitude for the lth state. Substituting (11.10) in (11.9), we get ih

∑

a˙ l(N ) (t)u l(0) (r )e−iωi t =

l

∑

al(N −1) (t)V (t)u l(0) (r )e−i ωi t

(11.11)

l

Multiplying both sides by u (0)∗ r ) and integrating first in space and then with n (respect to time, we get an(N ) (t)

1 ∑ = ih l

(t

al(N −1) (t1 )Vnl (t1 )ei ωnl tl dt1

−∞

( =

1 ih

)2 ∑ ( t l,k −∞

(t1 Vnl (t1 )e

iωnl t1

dt1 −∞

ak(N −2) (t2 )Vlk (t2 )ei ωlk t2 dt2

442

11 Some Nonlinear Processes

( =

1 ih (t2

×

)3 ∑ ( t

(t1 Vnl (t1 )e

iωnl t1

Vlk (t2 )eiωlk t2 dt2

dt1

l,k, j−∞

−∞

−3) a (N (t3 )Vkj (t3 )eiωkj t3 dt3 j

(11.12)

−∞

= × ×

( 1 )N ih

(t2

(t

∑

l,k, j,...q,r,s −∞

Vnl (t1 )eiωnl t1 dt1

Vkj (t3 )eiωkj t2 dt3 · · ·

−∞ t N(−1 −∞

t N(−2 −∞

(t1 −∞

Vlk (t2 )eiωlk t2 dt2

Vqr (t N −1 )eiωqr t N −1 dt N −1

(11.12a)

as(0) Vrs (t N )eiωr s t N dt N

where Vnl (t) = (n|V (t)|l) etc.. Equation (11.12a), called Dyson equation, is the generalized form of the probability amplitude, and from this equation, the amplitude of any order can be determined. The electric field associated with the exciting radiation is ) ∑ 1( ( ) ∑ ' ' ε- p e−i ω p t + ε-∗p eiω p t ε- p cos ω p t ' = E- t ' = 2 n>0 n>0 ∑ ' = E- p e−iω p t

(11.13)

p

(Note the subscript p > 0 actually means ωp > 0). The incident radiation is assumed to be non-monochromatic and is supposed to have a number of discrete angular frequencies. In the last term of the above equation, the summation over the subscript ‘p’ includes both the positive and negative values of ωp . So, for example, in electric dipole approximation / | ∑ ( )|| \ ( ) ' | - · E- t ' |l = − Vnl t ' = − n |μ μ - nl · E- p e−iω p t

(11.14)

p

and hence, the probability amplitude for the nth state in Eq. (11.12) becomes an(N ) (t)

1 ∑ =− ih l, p

(t

) ( )( ' - nl · E- p e−i (ω p −ωnl )t dt ' al(N −1) t ' μ

(11.15)

−∞

We shall determine the probability amplitudes up to third order which will determine the susceptibilities up to third order, and this will serve our purpose for the discussion on the relevant spectroscopic phenomena.

11.2 Theoretical Description

443

11.2.1 First-Order Effect and Linear Susceptibility The first-order probability amplitude is an(1)

1 ∑ =− ih l, p

(t

) ( )( ' - nl · E- p e−i (ω p −ωnl )t dt ' al(0) t ' μ

(11.16)

−∞

( ) Let us assume that the system was initially in the state ‘g’. So al(0) t ' = δlg . So the first-order probability amplitude becomes an(1) (t)

1 = − ∑p ih

⎡

(t (

) μ - ng · E- p e−i (ω p −ωng )t1 dt1

−∞

) 1 ∑( μ - rg · E- p = lim⎣− :→0 ih p

(t

(11.17)

⎤ e−i (ω p −ωnp +it )t dt1 ⎦

(11.17a)

−∞

( ) −i (ω −ω )t p ng 1 ∑ e μ - ng · E- p −i = − ih ω p −ωng ) ( p ) i (ω −ω )t ∑( ng p μ - ng · E- p e ω −ω = h1 ( ng p) p

(11.17b)

To evaluate the above integral at the lower limit, a real and positive e(→ 0) is introduced by replacing ωng by ωng − ie in the intermediate step for our convenience to make the integral convergent which is supposed not to affect the final result. We may also keep this ie in the denominator of Eq. (11.17b), and in that case, 2e denotes the half width of the state |n). The absolute magnitude of the integral (11.17) is very large at ωp = ωng and not so at other values of ωp . How rapidly the magnitude of this integral diminishes on either sides of ωp = ωng depends on t. Anyway for large t, this decay is very rapid and for t(→ ∞, the) time integral in Eq. (11.17) takes the form of Dirac delta function, 2π δ( ω p − ωng . Thus, an(1) (∞) = an(1) (t → ∞) =

2πi ∑

)) (( (μ - ng · E- p )δ ω p − ωng

(11.18)

p

Note that for any finite value of t, an(1) (t)vsω curve(has a sharp) maximum around ω = ωp, and it behaves like a Dirac delta function δ( ω p − ωng for large t. So for large t, we can expect a resonance occurs when ωp = ωng, and this corresponds to a single photon absorption/induced emission at this frequency. → μ is (ψ|μ|ψ) The expectation value of − where ψ is expanded in terms (of the ) → μ is perturbation expansion (11.8) coefficients. So the first-order contribution to − given by

444

11 Some Nonlinear Processes

( (1) ) ( (0) ) ( ) - (1) + ψ (1) |μ|ψ - (0) μ = ψ |μ|ψ ∑[ ( ) ( )] - 0 + an(1)∗ n 0 |μ|g - 0 an(1) g 0 |μ|n = n

⎡ ⎤ ( ) ( )∗ -p -p μ μ μ μ · E · E ∑ ∑ gn ng ng ng 1 ⎢ ⎥ ) e−i ω p t + ( ) eiω p t ⎦ = ⎣ ( h p n ωng − ω p ωng − ω p ] ( ) ( )[ - gn μ - ng∗ - gn− μ - ng μ 1 ∑∑ μ ( )+( ) E- p e−iω p t = h p n ωng − ω p ωng + ω p

(11.19)

Since the subscript ‘p’ is summed over both positive and negative values, ωp is replaced by—ωp in the second term within the square bracket of the above equation. − → − → Using 11.13, we have used E p → E ∗p as ω p → −ω p . Comparing Eq. (11.19) with (11.1), the molecular polarizability (α) and linear susceptibility (χ ) are given by ] [ - ng - gn μ - gn μ μ - ng μ 1 ∑∑ ( )+( ) α = e0 χ = h p n ωng − ω p ωng + ω p

(11.20)

and their various elements are given by ] [ j j μign μng μgn μing 1 ∑∑ ( )+( ) αij = e0 χij = h p n ωng − ω p ωng + ω p

(11.20a)

The first term in the square bracket of the above equation corresponds to resonance and second one to non-resonant contribution to the polarizability or linear susceptibility matrix. The resonance condition occurs when ωp is positive and equals to ωng . For off resonant excitation, the contributions of both the terms are even. Note that the states |g) and |n) are the molecular states. This process is discussed in the next section in connection with two photon process (including the Raman one). Detailed studies on the intensities of Raman bands (which is proportional to the square of the polarizability matrix) for various region of excitation (ω) (i.e., examination of the excitation profiles of different Raman bands in the regions) may yield important information about the molecular properties in various excited electronic states.

11.2 Theoretical Description

445

11.2.2 Second-Order Effect and Second-Order Susceptibility The second-order probability amplitude as found from Eq. (11.15) is an(2) (t)

1 ∑ =− ih m,q

(t

) ( am(1) (t1 ) μ - nm · E-q e−i (ωq −ωnm )t1 dt1

−∞

) ( (t ( ) 1 2 ∑∑ μ - nm · E-q e−i (ωq −ωnm )t1 = − ih m p,q −∞

(t1 ( dt1

) μ - mg . E- p e−i (ω p −ωmg )t2 dt2

(11.21a)

−∞

∑ ∑ (μ- nm · E-q )(μ- mg · E- p ) (t −i (ω p +ωq −ωng )t1 e dt1 −i (ω p −ωmg ) m p,q −∞ ∑ (μ- nm · E-q )(μ- mg · E- p ) −i (ω p +ωq −ωng )t e ω +ωq −ωng )(ω p −ωmg ) m ( p

= − h12 =

1 h2

(11.21b)

Suppose the molecule is excited with a radiation containing two frequencies ωq = ω1 and ω p = ω2 . Then, an(2) (t)

) ( (t ( ) 1 2∑ μ - nm · E-1 e−i (ω1 −ωnm )t1 dt1 = − ih m −∞

×

(t1 (

) μ - mg · E-2 e−i (ω2 −ωmg )t2 dt2

−∞

( )( ) -1 μ -2 ( t μ · E · E ∑ nm mg 1 ) ( =− 2 e−i (ω1 +ω2 −ωng )t1 dt1 h m −i ω2 − ωmg −∞ ( )( ) μ - nm · E-1 μ - mg · E-2 1 ∑ ( )( ) e−i (ω1 +ω2 −ωng )t = 2 h m ω1 + ω2 − ωng ω2 − ωmg

(11.22)

In both the integrations, the contribution at the lower limit is made zero as before (see Eq. 11.17). Here also, the absolute value of an(2) (t) is large when ω1 + ω2 = ωn −ωg = ωng and small elsewhere. For large t → ∞, it becomes

446

11 Some Nonlinear Processes

( )( ) t→∞ - nm · E-1 μ - mg · E-2 ( ∑ μ 1 (2) ) ( an (t → ∞) = − 2 e−i (ω1 +ω2 −ωng )t1 dt1 h m −i ω2 − ωmg −∞ ( )( ) - nm · E-1 μ - mg · E-2 ( ) 2πi ∑ μ ∼ ( ) (11.23) δ ω1 + ω2 − ωng = 2 h m ωmg − ω2 Thus, we see that in the presence of the radiation field, the molecular system transits from the initial state |g) to the final state |n) through an intermediate state |m) . Moreover, the two transitions |g) → |m) and |m) → |g) occur more or less simultaneously and not independently. Another important thing is to be kept in mind. The intermediate state |m) is not necessarily a real state of the molecule; generally, it is a virtual state. In general, a virtual state is a linear combination of several actual states of the molecule. So for a virtual intermediate energy state (m), E m −E g = E mg need not be a real transition energy of the molecule. When the intermediate state |m) coincides with a real state, the coefficient of all the actual molecular states in the linear expansion of the virtual state is zero or negligibly small excepting the actual (resonating) molecular state |m). In that case, the process is a resonance phenomenon. Different phenomena arise due to different relative positions of the states |g), |m) and |n). These are shown in Fig. 11.7. When the state |m) lies between |g) and |n) and E g < E n , the process is a two photon absorption, and this is shown in Fig. 11.7a. When the states with energies E g and E n are two rotational/vibrational/rotationalvibrational states of the molecule, and these energies are less than E m , the process is stoke or antistoke Raman scattering process according to E g < E n or E g > E n . Here for stokes Raman ω2 > ω1 and for antistokes Raman scattering ω2 < ω1 (Fig. 11.7b,c). For E g = E n , the same process is called Rayleigh scattering. When the state |m) lies between |g) and |n) and E g > E n , the process is a two photon emission as shown in Fig. 11.7d. Each of these two photon processes can be expressed by second-order probability amplitude as given in Eq. (11.23). The number of terms (integrals) in this expression increases with the increase of the order of perturbation. (Note that in the energy level diagram, the upward transition corresponds to a positive frequency, and the downward transition corresponds to a negative frequency). For third and higher orders, there is another convenient way to determine the terms of the probability amplitude which may not be needed for the use in the second-order perturbation. However, it is a much simpler way to determine the terms of the probability amplitude, so we are describing it here. This is not that helpful for second-order perturbation, but for third- and higher-order perturbation, it reduces the complexity of determining the amplitude. This is a diagrammatic representation, called time ordered graph. This graph helps to determine the term(s) in the probability amplitude corresponding to any particular multiphoton process as given by Eq. (11.21a) for a second-order process. All these are shown below in Fig. 11.8 for the secondorder case. The rules used here to draw the time ordered graphs relevant to the Dyson equation corresponding to any multiphoton process are listed below.

11.2 Theoretical Description

447

Fig. 11.7 Energy level diagram for a two photon absorption, b stokes Raman scattering, c antistokes Raman scattering and d two photon emission

ω1 ω2

n t1 tn m

t2

tm

t1

t1

t1

t2

t2

(b)

(c)

g

(a)

t2

(d)

Fig. 11.8 Diagramatic representations of the four processes in the energy level diagram (11.7). a Two photon absorption, b stokes Raman scattering (ω2 > ω1 ), c antistokes Raman (ω2 < ω1 ) scattering and d two photon emission

(i). Time axis is shown by a vertical line and the time evolution on it is directed upward. During which time interval, the molecule is in which state is shown by different labels like g, m, n, etc., at different portions of the vertical line corresponding to the states of the molecule. For example, by the time (t m ), the molecule takes a transition to the state m by interacting with the perturbation radiation of frequency ω2 . Prior to this time, the molecule was in the state g. Similarly by the time t n , the molecule takes a transition to the state n from the state m by interacting with the perturbing radiation of frequency ω1 . (ii). The photon paths are represented by arrows. The arrows directed towards the time line correspond to absorption and those directed away from the time line corresponds to emission. The point of intersection of these arrows with the time line is called the interaction vertex. This point denotes the time at which the perturbation is applied to enable the molecule to take a transition at a time, from the state where it was at the interaction vertex (vertex state) to the state just above this.

448

11 Some Nonlinear Processes

Other rules related to the third and higher orders are specified later on during our discussions on those orders The four processes corresponding to the events in the Fig. 11.7 are shown in the time ordered graphs in Fig. 11.8. Note that the energy level diagram (11.7) is drawn following the energy conservation principle governed by the argument of the Dirac delta function in Eq. (11.23). Similar set of four diagrams (both energy level and time ordered graphs) may be drawn by interchanging the two frequencies ω1 and ω2 in Eq. (11.21a, b). The second-order dipole moment is (

) ( ) ( ) ( ) - (2) + ψ (1) |μ|ψ - (1) + ψ (2) |μ|ψ - (0) μ - (2) = ψ (0) |μ|ψ ∑[ ( ) ( ) ( )] - 0 + am(1)∗ an(1) m 0 |μ|n - 0 + am(2)∗ m 0 |μ|g - 0 an(2) g 0 |μ|n = m,n

)( ) ⎡ ( - mg · E- p μ - gn - nm · E-q μ 1 ∑∑⎣ μ ( )( ) e−i (ω p +ωq )t = 2 h m,n p,q ω p + ωq − ωng ω p − ωmg ( )∗ ( ) μ - mg · E- p μ - ng E-q - mn μ )( ) ei (ω p −ωq )t + ( ωmg − ω p ωng − ωq ⎤ ( )∗ ( )∗ μ - mg μ - mn · E-q μ - ng E- p ⎥ )( ) ei (ω p +ωq )t ⎦ + ( ω p + ωq − ωmg ω p − ωng

(11.24)

∑ using Eqs. (11.17 and 11.21). Since the p,q includes both positive and negative ωp and ωq , we can put ωp = − ωp in the second term and ωp , ωq = − ωp , − ωq in the third term in the square bracket of the above equation to get a simplified form from which the hyperpolarizability and second-order susceptibility can be determined. )( ) ⎡ ( - nm · E-q μ - mg · E- p μ - gn ( (2) ) 1 ∑∑⎣ μ ( )( ) μ = 2 h p,q m,n ω p + ωq − ωng ω p − ωmg ( ) ( ) μ - gm · E- p μ - mn μ - ng · E-q )( ) + ( ωmg + ω p ωng − ωq ( )( ) ⎤ - nm · E-q μ - gn · E- p μ - mg μ )( ) ⎦e−i (ω p +ωq )t + ( ω p + ωq + ωmg ω p + ωng ∑ ( ) = β ω p + ωq , ωq , ω p · E-q · E- p e−i (ω p +ωq )t p,q

=

∑ p,q

( ) e0 χ (2) ω p + ωq , ωq , ω p · E-q · E- p e−i (ω p +ωq )t

(11.25)

11.2 Theoretical Description

449

∗ since E −q = E q and E −∗ p = E p . Thus, the hyperpolarizability and second-order molecular susceptibility matrix elements (Eq. 11.1) are given by

( ) ( ) βi, j,k ω p + ωq , ωq , ω p = ε0 χ i,(2)j,k ω p + ωq , ωq , ω p ] j μign μnm μkmg 1 ∑ ( )( ) = 2 h m,n ω p + ωq − ωng ω p − ωmg j

μimn μng μkgm )( ) +( ωng − ωq ωmg + ω p j

μimg μkgn μnm )( ) + ( ω p + ωq + ωmg ω p + ωng

[ (11.26)

Here, p and q are used as interchangeable indexes. Cartesian indexes are to be permuted simultaneously. So actually, there are six terms, out of which only three terms are displayed in the above Eq. (11.26). Note that here also the states |g), |m) and |n) are molecular states.

11.2.3 Third-Order Effect and Third-Order Susceptibility The probability amplitude for the third-order perturbation as found from Dyson equation (11.15) is ) ( (t ( ) ( ) 1 3∑ (3) μ - np · E-1 e−i ω1 −ωnp t1 dt1 an (t) = − ih p,m −∞

×

(t1 (

−∞

) ( ) μ - pm · E-2 e−i ω2 −ωpm t2 dt2

(t2 ( ) ( ) μ - mg · E-3 e−i ω3 −ωmg t3 dt3 −∞

) ) ( )( ) ) t ( (t1 ( 1 3 ∑ ( μ - np · E-1 e−i ω1 −ωnp t1 dt1 μ - pm · E-2 μ - mg · E-3 = − ih (

p,m −∞ −∞ )3 ∑ (t ( )( )( ) ( −i (ω2 +ω3 −ωpg )t2 e 1 ) dt2 = − ( μ - np · E-1 μ - pm · E-2 μ - mg · E-3 × ih −i ω3 −ωmg p,m −∞ −1(ω1 +ω2 +ω3 −ωng )t2 e ( )( ) × dt2 (−i)2 ω2 +ω3 −ωpg ω3 −ωmg )3 ∑ ( )( )( ) ( 1 μ - np · E-1 μ - pm · E-2 μ - mg · E-3 = − ih p,m )3 ( e−1(ω1 +ω2)(+ω3 −ωng )t2 )( ( ) = −2π − 1 × 3 ω +ω +ω −ω iL ω ω (−i) +ω −ω −ω ng pg mg 1 2 3 2 3 3 ( − → )( − → )( − → ) ) ( ∑ μ- np · E 1 μ- pm · E 2 μ- mg · E 3 )( ) ( × δ ω1 + ω2 + ω3 − ωng ω ω +ω −ω −ω pg mg 2 3 3 p,m

(11.27a)

(11.27b)

450

11 Some Nonlinear Processes

Fig. 11.9 a Three photon absorption; b hyper Raman stokes (E g < E n ) and antistokes (E g > E n ) scattering; c sum frequency generation (also hyper Rayleigh scattering) and d three photon emission

Thus, we see that the molecular system transits from the initial state [ g) to the final state [ n) through two intermediate states [m) and [ p). This is a three photon process. Some of the three photon processes thus arise are shown in the following energy level diagram (11.9). In the first case, Fig. 11.9a, absorption of two photons of frequencies ω3 , ω2 takes the molecule first from the initial (ground) state [ g) to the state [ m) and then to the state [ p) and finally by absorbing another frequency ω1 , the molecule goes to the state [ n). All these transitions take place simultaneously. Here, [ g) and [ n) are the actual states of the molecule, whereas the other two, [ g) and [ n), are not necessarily so. In general, they are virtual states. This process is called three photon absorption which corresponds to the Dyson equation 11.27b. In this process, E n > E p > E m > E g . In the second case, Fig. 11.9b, E n > E g , but they are less in energy than the states [ m) and [ p). Here, the states [ g) and [ n) are the rotational-vibrational states of the molecule. This process corresponds to hyper stokes Raman scattering (ω3 + ω2 > ω1 ). When E n < E g , this process becomes hyper antistokes Raman scattering (ω3 + ω2 < ω1 ). When E n = E g , the process (Fig. 11.9c) becomes sum frequency generation (ω3 + ω2 = ω1 ). When ω3 = ω2 = ω (say), ω1 = 2ω, the process is called second harmonic generation (SHG)). The last case (Fig. 11.9d) is reverse of the three photon absorption process (11.9a) and is called three photon emission. The corresponding time ordered graphs of these are shown in Fig. 11.10a–d. One point is noteworthy in this context. In drawing the energy level diagram and also the time ordered graph from the Dyson equation, the energy conservation governed by the argument of Dirac delta function of the equation has to be followed. In our convention plus and minus signs appearing in the powers of the exponential in the Dyson equation correspond to emission and absorption of the corresponding angular frequencies. For example, ei ωi t corresponds to emission, and e−iωi t corresponds to absorption of radiation of photon energy èωi . Thus for another three photon process, the Dyson Eq. (11.28) corresponds to absorption of radiations of angular frequencies ω1 and ω2 and emission of radiation of angular frequency ω3 . This is illustrated in

11.2 Theoretical Description

451

n tn ω1 ω2

ω3

p

t 1 tp t2

m tm g

t3

(a)

(b)

(c)

(d)

Fig. 11.10 Time ordered graph. a Three photon absorption; b hyper Raman stokes (E g < E n ) and antistokes (E g > E n ) scattering; c sum frequency generation (also hyper Rayleigh scattering) and d three photon emission corresponding to the respective Fig. 11.9a–d

the time ordered graph (11.11a) and the energy level diagram (11.11b). an(3) (t)

( ) (t ( ) 1 3∑ μ - np · E-1 e−i (ω1 −ωnp )t1 dt1 = − ih p,m −∞

(t1 × −∞

(

) μ - pm · E-2 e−i (ω2 −ωpm )t2 dt2

(t2 ( )∗ μ - mg . E-3 e−i (−ω8 −ωmg )t3 dt3 −∞

( − → )( − → )( − → )∗ )3 ∑ μ ( μ μ · E · E · Eg np 1 pm 2 mg 1 ( )( ) = 2π − i ω2 − ωs − ωpg ωg + ωmg p,m ) ( δ ω1 + ω2 − ω3 − ωng

(11.28)

Another very interesting point is to be mentioned in this context. Instead of deriving the formula of the probability amplitude (for example 11.28) for any multiphoton process, we can easily write it down provided the corresponding time ordered graph is rightly drawn (as in Fig. 11.11) following the points mentioned previously (see the two points mentioned before the Fig. 11.8). In generating the formula for the probability amplitude, the following two other rules are also to be kept in mind (which is evident from Eq. 11.28): (i) The contribution of any interaction vertex, say at t3 on the time line in numerator of the probability amplitude should be a complex Fig. (11.11a), ( to the − → ) - mg · E 3 if it is a scattering and real if it is an absorption process, conjugate μ where |m) and |g) are the states lying above and below the transition time (indicated by small horizontal line crossing the time line) just above the vertex on the time line.

452

11 Some Nonlinear Processes

Fig. 11.11 a Time ordered graph corresponding to Eq. 11.28. b When E n /= E g , the process is stoke hyper Raman (for E n > E g ) and antistoke hyper Raman (for E n < E g ) scattering. When E n = E g , the process is a sum frequency generation

(a)

(b)

(ii) For the nth order perturbation, there will be (n–1) factors in the denominator of the formula. Rising from the bottom to top along the time line in the time order graph, the first (n–1) interaction vertices are to be considered. Each vertex will contribute a factor (ω−ωpg ) where | p is the state lying above that vertex state, |g is the initial state and ω is the total energy absorbed (+)/scattered(-) at all vertices starting from the lowest one, up to and including that vertex divided by è. These rules are very useful for writing down the formula from the time ordered graph of any multiphoton process. The third-order susceptibility, as per our discussions in the previous orders, can be determined from the fourth order probability amplitude which will be discussed in connection with CARS (Sect. 11.5).

11.3 Sum Frequency and Second Harmonics Generation In the nonlinear medium, the two incident waves of frequencies ω1 and ω2 are mixed up to generate a new wave of frequency ω3 = ω1 + ω2, and all these three waves will propagate in the medium. When ω1 = ω2 = ω, the new generated wave is the second harmonic of the original wave. Most of the optical lasers in the near UV region, used now a days, are the frequency doubled beam of dye lasers in the visible region. In order to describe the propagation of all the three waves in the medium, consider the Maxwell equations (in SI units) - =ρ ∇·D

(11.29a)

∇ · B- = 0

(11.29b)

11.3 Sum Frequency and Second Harmonics Generation

453

∂ B∇ × E- = − ∂t

(11.29c)

∂D ∇ × H- = + -j ∂t

(11.29d)

In a region where there is no free charge and current, taking curl of each side of (11.29c), we get ∇ × ∇ × E- = −∇ ×

∂2 D ∂ B= −μ0 2 ∂t ∂t

(11.30)

( ) Again ∇ × ∇ × E- = ∇ ∇ · E- − ∇ 2 E- = −∇ 2 E- (following the first Maxwell Eq. 11.29a). Thus, Eq. (11.30) becomes ∂2 D −∇ 2 E- + μ0 2 = 0 ∂t

(11.31)

- = e0 E- + PD

(11.32)

The displacement vector is

P- being the polarization and e0 the permittivity of free space. If the polarization is supposed to have a linear and a nonlinear term, i.e. P- = P-l + P-nl , then the Eq. (11.32) can be written as ) ( - 1 + P-nl - = e0 E- + P- = e0 E- + P-l + P-nl = D (11.33) D where − → D1 = e0 E- + P-l = e0 eˆ1 · E-

(11.33a)

Ʌ

ε1 being the dielectric constant of the medium. In general, it is a tensor, but in an isotropic medium, it is a scalar. Thus in such a medium, the wave Eq. (11.31) becomes −∇ 2 E- +

1 ∂ 2 P-nl ε1 ∂ 2 E=− 2 2 2 c ∂t ε0 c ∂t 2

(11.34)

In a dispersive medium, each frequency component of the vectors and dielectric constant is to be considered separately. Let the electric vectors and polarizations be the sum of the corresponding entities of different frequencies,

454

11 Some Nonlinear Processes

E- =

∑ n

( → ) ∑ (− ) r, t = r e−iωn t + cc E-n − E-n →

(11.35a)

n

and P-nl =

∑ n

(→ ) ∑ (→) −i ωn t r, t = r e + cc P-nl − P-nl,n −

(11.35b)

n

Then, each frequency component of the Eq. (11.34) becomes (ignoring the complex conjugate parts) ε1 (ωn )ωn2 ωn2 ∇ 2 E-n (r) + · E (r ) = − r) Pnl,n (n c2 ε0 c2

(11.36)

In a nonlinear medium, two intense (laser) input fields of frequencies ω1 and ω2 cause the polarization in the medium to generate a new field of frequency ω3 = ω1 + ω2 . We shall consider the medium to be non-dissipative, and the two input beams are collimated, monochromatic and continuous (Fig. 11.12) and incident normally (in the z-direction) on the medium. If we neglect the source term (nonlinear polarization, P-nl,n ) in Eq. (11.34), we can take the solution of the homogeneous equation (for the newly generated wave of frequency ω3 ) as a plane wave propagating in the zdirection as − → E-3 = E 3 (z, t) = A-3 e−i (ω3 t−k3 z) + CC

(11.37)

Since the electric field E-3 (z, t) is dependent only on z, replacing the ∇ 2 operator by d2 /dz2 and disregarding the vector notation, Eq. (11.34) becomes d2 e1 (ω3 ) ∂ 2 E 3 (z, t) 1 ∂ 2 Pnl,3 (z, t) E (z, t) + = − 3 dz 2 c2 ∂t 2 e0 c 2 ∂t 2

(11.38)

The source term can also be represented by Pnl,3 = Pnl,3 (z, t) = P3 (z)e−iω3 t + CC

(11.39)

Here, the amplitude of the nonlinear source term can be represented by P3 (z) = e0 χ E 1 E 2 ei(k1 +k2 )z Fig. 11.12 Generation of sum frequency

(11.40)

11.3 Sum Frequency and Second Harmonics Generation

455

(see Eq. 11.2). In the absence of the source term in Eq. (11.36), A3 in Eq. (11.37) is a constant. But in the presence of the source term Pnl,3 , this is not so. If we assume that the nonlinear source term is not too large, the solution of the inhomogeneous Eq. (11.38) can be taken more or less same (11.37) as that of the homogeneous equation (i.e. without the source term), but only the amplitude A3 varies slowly with z. Thus after substituting Eqs. (11.37) and (11.39), the Eq. (11.38) becomes (leaving the cc terms) [

d 2 A3 (z) + 2ik3 dAdz3 (z) − k32 A3 (z) dz 2 2 ω = − c23 χ A1 A2 e−{ω3 t−(k1 +k2 )z}

+

ε1 (ω3 )ω32 c2

] A3 (z) e−i(ω3 t−k3 z)

(11.41)

where A1 and A2 are the amplitudes of the respective input fields as in (11.37) Sincek3 = ω3√ [e1 (ω3 )]/c , the last two terms in the square bracket cancel out, and the Eq. (11.41) reduces to [

] d2 A3 (z) dA3 (z) ik3 z ω32 e + +2ik = − χ A1 A2 ei(k1 +k2 )z 3 dz 2 dz c2

(11.42)

Since A3 (z) is a slowly varying function of z, we can neglect its second derivative, and the equation becomes dA3 (z) i ω32 i ω32 χ A1 A2 ei(k1 +k2 −k3 )z = χ A1 A2 eiΔk.z = 2 dz 2k3 c 2k3 c2

(11.43)

where Δk = k 1 + k 2 −k 3 is the wave vector mismatch. Equation (11.43) is called the coupled amplitude equation, because it shows the variation of the amplitude of the wave of frequency ω3 due to its coupling with the input waves. Perfect phase matching is obtained when the generated wave has a fixed phase relation with the nonlinear polarization. In deriving this equation, we have assumed the medium is loss less, and so no loss term is introduced. Furthermore, it is to be pointed out that the spatial variation of the ω1 and ω2 waves must also be taken into consideration, and the variation of their respective amplitudes can be similarly derived as i ω12 d A1 (z) = χ A3 A∗2 e−iΔk.z dz 2k1 c2

(11.44)

i ω22 dA2 (z) = χ A3 A∗1 e−iΔk.z dz 2k2 c2

(11.45)

and

Another interesting point is to be mentioned in this context. Actually, χ is the nonlinear susceptibility matrix components (i.e. χijk etc.). It can be shown that in a loss less medium

456

11 Some Nonlinear Processes

χ ≡ χijk (ω3 = ω1 + ω2 ) = χjki (−ω1 = ω2 − ω3 ) = χjki (ω1 = −ω2 + ω3 ) = χkij (−ω2 = ω1 − ω3 ) = χkij (ω2 = ω3 − ω1 )

(11.46)

If Δk /= 0, the sum frequency generated amplitude (in Eq. 11.43) is an oscillatory one. Let us assume that the conversion of the input fields into sum frequency is not large so that the amplitudes of the input waves (A1 and A2 ) in (11.43) can be taken as constants. A special case arises when Δk = 0. This condition is called phase matching. In perfect phase matching, the amplitude increases linearly with the distance z (11.43), and hence, the intensity increases as the square of the distance covered. But in the absence of the perfect phase matching, the amplitude of the sum frequency generated wave is to be determined by integrating the Eq. (11.43) from 0 to L (the path length in the medium). Thus, L eiΔk.L − 1 ω32 i ω32 iΔk.z ∫ χ A A e dz = χ A A 1 2 1 2 2k3 c2 2k3 c2 Δk 0 iΔk.L/2 −iΔk.L/2 ω32 e −e = χ A1 A2 eiΔk.L/2 2k3 c2 Δk i SinΔk.L/2 ω32 χ A1 A2 LeiΔk.L/2 = 2k3 c2 Δk.L/2

A3 (L) =

(11.47)

Hence, the intensity of the wave emerging from the medium is | | 1 1 ω32 I1 I2 χ 2 L 2 || SinΔk.L/2 ||2 n 3 ε0 c|A3 |2 = 2 2 n 1 n 2 n 3 ε0 c3 | Δk.L/2 | |2 | | | max | SinΔk.L/2 | = I3 | Δk.L/2 |

I3 =

= I3max sin c2 (Δk · L/2)

(11.48)

The above expression shows that I 3 decreases drastically (with some oscillations occurring) when Δk.L/2 is large, i.e. the efficiency of mixing diminishes when Δk.L/2 is much away from the perfect phase matching condition, ΔK = 0 (Fig. 11.13). So when Δk /= 0, the generated wave becomes out of phase with its driving polarization after an approximate interference length L coh = 2/Δk, which is called the coherent length, and the output power can flow from the ω3 wave back into the input waves of frequencies ω1 and ω2 . Thus, the Eq. (11.48) can be written as | | 1 ω32 I1 I2 χ 2 L 2 || SinΔk.L/2 ||2 I3 = = I3max sin c2 (L/L coh ) 2 n 1 n 2 n 3 ε0 c3 | Δk.L/2 |

(11.49)

11.3 Sum Frequency and Second Harmonics Generation

457

Fig. 11.13 Variation of output intensity (I 3 ) with (Δk.L/2) curve

Phase matching condition is not easy to achieve in practical cases. For perfect phase matching, k 1 + k 2 = k 3 n 2 ω2 n 3 ω3 n 1 ω1 + = c c c

(11.50)

along with ω1 + ω2 = ω3 . Equation (11.50) gives n3 =

n 1 ω1 + n 2 ω2 ω3

(11.51)

and hence n3 − n2 =

n 1 ω1 + n 2 ω2 − n 2 ω3 n 1 ω1 − n 2 ω1 ω1 = = (n 1 − n 2 ) ω3 ω3 ω3

(11.52)

For normal dispersion in a loss less medium n1 < n2 < n3 (assuming ω1 < ω2 < ω3 ), this condition cannot be achieved, because the left side of equation is positive, whereas the right-hand side is negative. However, this condition can be achieved in the anomalous dispersion region, i.e. in the region of absorption. But in order to achieve the phase matching condition, generally, the birefringent medium (crystal) is used. In a uniaxial crystal (other than a cubic one, which is isotropic), any ray going through it is split up into an ordinary and an extraordinary ray. For the positive crystal, the refractive index of the ordinary ray (n0 ) is less than that of the extraordinary ray (ne ), and reverse is the case for the negative crystal. For ordinary ray, the refractive index is the usual refractive index no with polarization perpendicular to the plane - whereas for the containing the optic axis (c or z-axis) and the propagation vector k, extraordinary ray, the refractive index is ne with polarization in the said plane. Phase matching condition can be achieved in two ways in the birefringent uniaxial crystal.

458 Table 11.1 Types of phase matching in uniaxial crystal

11 Some Nonlinear Processes Type

Positive crystal (ne > no )

Negative crystal (ne < no )

Type I

n o3 ω3 n o3 ω3

n e3 ω3 = n o1 ω1 + n o2 ω2

Type II

= =

n e1 ω1 n o1 ω1

+ n e2 ω2 + n e2 ω2

n e3 ω3 = n e1 ω1 + n o2 ω2

The generated wave is chosen to have the same polarization as that of the wave of the lower of the two refractive indices, i.e. it is an ordinary ray for positive crystal and extraordinary ray for negative crystal. Since no < ne in positive crystal and no > ne in the negative crystal, phase matching can be obtained in two ways as shown in the following Table 11.1. The refractive index of the extraordinary ray is not a constant, and it is a function of θ, the angle between the propagation direction k- and the optic (c or z-) axis, given by the relation 1 sin2 (θ ) cos2 (θ ) = + 2 n˜ 2e n 2o n e (θ )

(11.53)

n˜ e is the principal value of the refractive index of the extraordinary ray. The refractive index ne (θ ) of the extraordinary ray is its principal value for θ = 90° and no for θ = 0° . The phase matching condition is achieved by varying the angle θ so that Δk = 0. By carefully controlling the refractive indices, the phase matching condition can be achieved. There are two methods of obtaining this, angle tuning and temperature tuning, described below. Angle Turning Method Let us demonstrate this with reference to the second harmonic generation in a negative uniaxial crystal. For this case ω1 = ω2 = ω and ω3 = 2ω. Here, the generated ray is chosen as an extraordinary ray and the fundamental one as an ordinary one. So perfect phase matching (11.50) for certain value of θ requires Fig. 11.14. n e (2ω, θ ) = n o (ω, θ )

(11.54)

It can be shown from Eq. (11.53) that this can be achieved if θ is such that sin θ = 2

Fig. 11.14 Phase matching of second harmonic generation in negative uniaxial crystal

1 1 − n 2 (2ω) n 2o (ω) o 1 1 − n 2 (2ω) n˜ 2e (2ω) o

(11.55)

11.4 Stimulated Raman Scattering

459

However, this relation may sometimes yield some unrealistic results, so care should be taken in this regard. For example, when the dispersion is large or birefringence is small, sin2 θ may become greater than unity. So for achieving perfect phase matching, not only the crystal has to be so oriented that the relation (11.55) is satisfied but it has to be such that the angle θ is real. Temperature Tuning This is to be noted that when the angle between the optic axis and the propagation vector is other than 0° and 90° , the direction of pointing vector S- and the propagation vector k- are not same. So the ordinary and the extraordinary rays diverge from each other as they propagates. This decreases the nonlinear mixing efficiency as they propagates in the crystal. It is found that in some crystal, for example, lithium niobate, birefringence depends strongly on the temperature. So keeping the angle θ fixed at 90° and by proper adjustment of the temperature of the crystal, perfect phase matching condition is attained.

11.4 Stimulated Raman Scattering Normal (or spontaneous) Raman scattering arises from electric dipole radiation spreading all around the sample in 4π solid angle, whereas the stimulated Raman scattering is confined within a very narrow cone in the forward and backward directions. So as the stimulated scattering moves in the said directions, it gains intensity. In order to see how this amplification occurs, let us first present a very simple model based on the photon occupation number in various field modes. Suppose at any instant of time, the photon occupation number of the stokes (Raman) mode is ms, and the number in the exciting laser mode is ml . Assuming the Raman intensity having a linear dependence on the exciting intensity, the rate of change of stokes photon occupation number is dm s = D · m l · (m s + 1) dt

(11.56)

The term unity that appears within the bracket on the right-hand side corresponds to the creation of a stokes mode due to a spontaneous Raman scattering. So the factor (ms + 1) is related to the number of stokes mode appearing from both stimulated and spontaneous scattering process. D is a constant which depends on the characteristics of the scattering material. Let us consider the stokes mode moving in the z-direction. In order to see the growth of the stokes radiation along the axis (z-direction) of the material medium, we can write down 1 dm s 1 dm s = = D · m l · (m s + 1) dz c/n dt c/n

(11.57)

460

11 Some Nonlinear Processes

Two extreme cases arise, ms > 1. In the first case, ms ω2 ) are applied to a material system. Two photons each of frequency (ω1 ) are absorbed, one photon of frequency ω2 is obtained by stimulated (stokes Raman) emission, and a third coherent radiation of frequency (2 ω1 −ω2 ) is also scattered. In CARS, the frequency (2ω1 −ω2 ) matches with an antistokes frequency such that ω1 −ω2 corresponds to a vibrational mode (= ωv ) of the system. So by tuning the lower frequency laser mode (ω2 ) keeping the other one (ω1 ) fixed, the vibrational spectra of the entire system can be scanned with intensities much greater than those of ordinary Raman spectra. Similarly in the case of coherent stokes Raman scattering (CSRS), the third coherent frequency generated is (2ω2 −ω1 ) which corresponds to a stokes Raman radiation ω2 −(ω1 −ω2 ) = ω2 –ωv . In practice, CARS is more frequently used than CSRS, so we shall discuss the former one only. Note that both CARS and CSRS are passive processes with no net change of energy of the material system. The evolution of the antistokes wave can be described by the coupled amplitude Eq. (11.93b)

11.5 Coherent Antistokes Raman Scattering (CARS)

467

dAa = −αa Aa + ka A∗2 eiΔk·z dz

(11.97)

where ωa χ R (ωa )|A1 |2 na c

(11.98)

ωa χ F (ωa )A21 2n a c

(11.99)

αa = −3i ka = 3i and

) ( Δk = Δk- · z = 2k-1 − k-s − k-a · zˆ Ʌ

(11.100)

The four wave mixing susceptibility for the antistokes wave for the current choice of antistokes frequency ωa = 2ω1 −ω2 is found from Eqs. (11.81 and 11.88) as

χ F (ωa ) =

N e0 6mωv

(

)2 ∂α ∂q 0

[ωa − (ω1 + ωv )] + iγ

=

Ne0 6mωv

(

)2 ∂α ∂q 0

[(ω1 − ω2 ) − ωv )] + iγ

(11.101)

Considering the contribution of the first term of Eq. (11.97), we see that at exact resonance, CARS signal is attenuated as it proceeds according to the discussion based on the Eq. (11.83). So the CARS signal is studied by the contribution of the second term of the Eq. 11.97. CARS signal is dominated when the gain of the stimulated Raman scattering is small. So for such condition and under perfect phase matching, the growth of the CARS signal is Aa = 3i

ωa χ F (ωa ) A21 A∗2 z 2n a c

(11.102)

Thus, we see that CARS signal is proportional to the square of the irradiance of the laser 1 (I 1 2 ) and the irradiance of the second laser 2 (I 2 ). This is also proportional to the square of the susceptibility χ F (ωa ). If the excitation beam I1 is held fixed and the other beam (I 2 ) is tuned to make the phase matching condition satisfied, strong antistokes signal is concentrated in a direction with very small divergence (~ 10–4 ) whose intensity exceeds that of the ordinary Raman scattering (which is spread in 4π solid angle) by several orders. CARS is advantageous specially for those samples where the scattered signal is accompanied by fluorescence. Due to the directional selectivity and intensity enhancement, CARS can be used to detect gases even at very low pressure (10–10 atm). Disadvantages of CARS are the sweeping of ω2 (whereas a single frequency is sufficient for recording the normal Raman spectra) to detect various vibrations and the sensitivity of the alignment for maintaining the momentum conservation for each mode of vibration in condensed samples. CARS transition mechanisms, that is, absorption of two photons of frequency ω1 and scattering of

468

11 Some Nonlinear Processes o

ω3 ω2

n

ω1

ω1

ω1

ω2

ω3

ω1 ω1

ω1

o

(a) ω3

ω1

ω3

p ω1

ω2

ω3

ω2

q

(b)

ω1

ω1

(c)

ω1 ω2 ω3

ω3 ω2

(d)

ω1

(e)

(f)

ω2 ω1 ω2

ω1 ω1

ω1

ω1 ω2

(g)

ω3

ω1

ω1

ω3

ω2

(h)

(i)

ω1

ω1

ω1

ω1 ω2

ω1

ω3

ω3

ω2

(j)

ω3

(k)

(l)

Fig. 11.16 Twelve time ordered graphs for CARS with two incident frequencies ω1 and two scattered photons of frequencies ω2 and ω3

two frequencies ω2 (by stimulated emission) and ω3 = 2ω1 −ω2 (which is coherent) are shown in the 12 time ordered graphs in Fig. 11.16. Using the diagrammatic technique, the corresponding fourth order probability amplitude for the CARS process, absorption of two photons, each of frequency ω1 and scattering of two others of frequencies ω2 and ω3 can be written, following the Fig. 11.16, as ( ) 1 4 2π − a4 (t → ∞) = aCARS (t → ∞) = (−i )3 ih ⎡( )∗ ( )∗ ( )( ) μ - qp · E-2 μ - pn · E-1 μ - no · E-1 - oq · E-3 ∑⎢ μ )( ) ⎣ ( −ω2 + 2ω1 − ωqo 2ω1 − ωpo (ω1 − ωno ) n, p,q ( )∗ ( )∗ ( )( ) μ - oq · E-2 μ - qp · E-3 μ - pn · E-1 μ - no · E-1 )( ) + ( −ω3 + 2ω1 − ωqo 2ω1 − ωpo (ω1 − ωno ) ( )∗ ( )( )∗ ( ) μ - oq · E-3 μ - qp · E-1 μ - pn · E-2 μ - no · E-1 )( ) +( −ω2 + 2ω1 − ωqo −ω2 + ω1 − ωpo (ω1 − ωno ) )∗ ( )( )∗ ( ) ( μ - qp · E-1 μ - pn · E-3 μ - no · E-1 μ - oq · E-2 )( ) +( −ω3 + 2ω1 − ωqo −ω3 + ω1 − ωpo (ω1 − ωno )

11.5 Coherent Antistokes Raman Scattering (CARS)

469

( )( )∗ ( )∗ ( ) μ - oq · E-1 μ - qp · E-3 μ - pn · E-2 μ - no · E-1 )( ) +( −ω3 − ω2 + ω1 − ωqo −ω2 + ω1 − ωpo (ω1 − ωno ) )( )∗ ( )∗ ( ) ( - qp · E-2 μ - pn · E-3 μ - no · E-1 μ - oq · E-1 μ )( ) +( −ω2 − ω3 + ω1 − ωqo −ω3 + ω1 − ωpo (ω1 − ωno ) ( )∗ ( )( )( )∗ μ - oq · E-3 μ - qp · E-1 μ - pn · E-1 μ - no · E-2 )( ) +( 2ω1 − ω2 − ωqo ω1 − ω2 − ωpo (−ω2 − ωno ) ( )∗ ( )( )( )∗ μ - oq · E-2 μ - qp · E-1 μ - pn · E-1 μ - no · E-3 )( ) +( 2ω1 − ω3 − ωqo ω1 − ω3 − ωpo (−ω3 − ωno ) )( )∗ ( )( )∗ - qp · E-3 μ - pn · E-1 μ - no · E-2 μ - oq · E-1 μ )( ) +( −ω3 + ω1 − ω2 − ωqo ω1 − ω2 − ωpo (−ω2 − ωno ) )( )∗ ( )( )∗ - qp · E-2 μ - pn · E-1 μ - no · E-3 μ - oq · E-1 μ )( ) +( −ω2 + ω1 − ω3 − ωqo ω1 − ω3 − ωpo (−ω3 − ωno ) )( )( )∗ ( )∗ μ - oq · E-1 μ - qp · E-1 μ - pn · E-3 μ - no · E-2 )( ) +( ω1 − ω3 − ω2 − ωqo −ω3 − ω2 − ωpo (−ω2 − ωno ) ⎤ )( )( )∗ ( )∗ − → - qp · E-1 μ - pn · E-2 μ - no · E-3 (μoq · E-1 μ ⎥ )( ) +( ⎦ ω1 − ω2 − ω3 − ωqo −ω2 − ω3 − ωpo (−ω3 − ωno ) δ(2ω1 − ω2 − ω3 )

(11.103)

The successive terms in the square bracket of Eq. 11.103 are corresponded with the respective time ordered diagrams in Fig. 11.16. It can be shown that (N + 1)th order probability amplitude is proportional to the Nth order susceptibility [which is a (N + 1) rank tensor] in the electric dipole approximation (where the interaction - So the third-order susceptibility (or the - · E). part of the Hamiltonian is V = −μ second-order nonlinear susceptibility) for CARS is proportional to the fourth order probability amplitude aCARS(∞), and its various components (apart from the constant) are ( ) ⎡ j j μioq μkqp μlpn μno + μpn μlno ∑ CARS ⎣ χikjl ≡ 2(ω3 − ωqo )(2ω1 − ωpo )(ω1 − ωno ) n, p,q ( ) j j μiqp μkoq μlpn μno + μpn μlno ) + ( 2 ω2 − ωqo (2ω1 − ωpo )(ω1 − ωno )

470

11 Some Nonlinear Processes j

j

μioq μkpn μno μlqp μipn μkoq μno μlqp )( ) ) ( +( + ω3 − ωqo Δ − ωpo (ω1 − ωno ) (ω2 − ωqo ) −Δ − ωpo (ω1 − ωno ) j

μiqp μkpn μno μloq )( ) +( −ω1 − ωqo Δ − ωpo (ω1 − ωno ) j

μipn μkqp μno μloq )( ) +( −ω1 − ωqo −Δ − ωpo (ω1 − ωno ) j

μioq μkno μpn μlqp )( ) +( ω3 − ωqo Δ − ωpo (−ω2 − ωno ) j

+

μino μkoq μpn μlqp ) ( (ω2 − ωqo ) −Δ − ωpo (−ω3 − ωno ) j

μiqp μkno μpn μloq )( ) +( −ω1 − ωqo Δ − ωpo (−ω2 − ωno ) j

μino μkqp μpn μloq )( ) +( −ω1 − ωqo −Δ − ωpo (−ω3 − ωno ) ( ) j j μipn μkno μqp μloq + μlqp μoq )( ) + ( 2 −ω1 − ωqo −2ω1 − ωpo (−ω2 − ωno ) ( ) ⎤ j j μino μkpn μqp μloq + μlqp μoq ⎦ ) + ( 2 −ω1 − ωqo (−2ω1 − ωpo )(−ω3 − ωno )

(11.104)

where we have used ω3 = 2ω1 −ω2 and Δ = ω1 −ω2 = ω3 −ω1 . The third, fifth, seventh and ninth terms in the square bracket of the above equation, corresponding to the time ordered graphs (c), (e), (g) and (i) in Fig. 11.16, become very large if èωpo = èΔ [= è(ω1 −ω2 ) = è(ω3 −ω1 )]. This is illustrated in Fig. 11.17. That is, when the difference of the two laser frequencies (ω1 −ω2 ) matches with a rotationalvibrational energy difference E p − E o of the molecule, the term [Δ − ωpo ] in the denominator of the corresponding entities becomes very small (not zero because of the width of the level |p > ). So keeping the laser frequency ω1 fixed and sweeping the other laser frequency ω2 , a sharp peak is observed at ω3 when it coincides with a antistokes Raman frequency [ω1 + (E p −E o )/è] of the molecular system. This condition corresponds to resonance CARs. The other terms in the CARS susceptibility χ i,k,j,l (11.104) only provide a non-resonant contribution to the scattering constituting a weak background intensity which varies very little with the sweeping of the laser frequency ω2 . It can be shown that the total scattering intensity of the CARS is I3 = K I12 I2 ω34

|2 ∑ || CARS | | E-i (ω3 ) E-k (ω2 ) E- j (ω1 ) E-l (ω1 )χi,k, j,l | i,k, j,l

(11.105)

11.5 Coherent Antistokes Raman Scattering (CARS)

471

│q> │n> ω1 ω2

ω1

ω3

ω1

ω2

ω1

ω3

│p> │o> (a) Nonresonant CARS

(b) Resonant CARS

Fig. 11.17 Energy level diagram for non-resonant and resonant CARS. For resonant, CARS è(ω1 −ω2 ) coincides with the energy difference E p –E o of the two rotational-vibrational levels |p > and |o > of the scattering system C ARS where K is a constant and χi,k, j,l is averaged over all molecular orientation. From Eq. 11.100, we see that CARS signal is obtained when the phase matching condition is satisfied

i.e. Δk = 0 or 2k1 = k2 + k3

(11.106)

Here, k-s = k-2 and k-a = k-3 . Since ki = n i ωi /c, in dilute gases, the refractive index ni varies little with the frequency ωi , so the CARS signal is obtained for the collinear geometry of the three waves of frequencies ω1 , ω2 and ω3 . But for liquid medium, ni increases with the increase of the frequency ωi . So perfect phase matching in such medium is obtained when the angle θ between the two incident laser beams is Fig. 11.18 4k12 + k22 − k33 4n 2 ω2 + n 22 ω22 − n 23 ω32 = 1 1 4k1 k2 4ω1 ω2 n 1 n 2 ( ) ( ) 4ω1 ω2 n 23 − 4ω12 n 23 − n 21 − ω22 n 23 − n 22 = 4ω1 ω2 n 1 n 2

cos θ =

(11.107)

In some kind of samples (specially biological), fluorescence background is a problem in the detection of the scattered radiation in the Raman spectra. So for Fig. 11.18 Phase matching in CARS

472

11 Some Nonlinear Processes

such samples, CARS is very advantageous to study the vibrational spectra. Besides its directional sensitivity, high peak intensity is very helpful to study gases at very low pressure. One of the main disadvantages, as mentioned earlier, is that unlike linear Raman spectroscopy where the entire vibrational spectra are acquired with a single frequency exciting radiation, in CARS a tunable laser is required to sweep the frequency ω2 for recording the spectra. Also the CARS sensitivity is limited by the background scattering contributed by the non-resonant terms of the CARS CARS susceptibility χi,k, j,l .

References and Suggested Reading 1. 2. 3. 4. 5. 6.

E.J. Woodbury, W.K. Ng, Ruby laser operation in the near IR. Proc. IRE. 50, 2347–2348 (1962) W. Struve, Fundamentals of Molecular Spectroscopy (Wiley, 1989) R.W. Boyd, Non Linear Optics (Academic Press, 2008) D.A. Long, Raman Spectroscopy (Wiley, 2002) J.M. Hollas, Modern Spectroscopy (Wiley, 2004) P.D. Maker, R.W. Terhune, Study of optical effects due to an induced polarization third order in the electric field strength. Phys. Rev. 137, A801 (1965) 7. M.D. Levenson, S.S. Kano, Introduction of Nonlinear Laser Spectroscopy (Academic Press, Boston, 1988) 8. N. Bloembergen, The stimulated Raman effect. Am. J. Phys. 35, 989 (1967) 9. M. Gorwitz, The South Korean Laser Isotope Separation Experience by (1996)

Index

A Acceptor, 245, 247–249, 252–254, 262–267, 270–272, 287 Anharmonic oscillator, 64 Asymmetric top molecules, 39 Auger Electron Spectroscopy (AES), 274

B Bloch equations, 355 Born – Oppenheimer approximation, 11 Breakdown of Born-Opppenheimer approximation, 11

C Character, 296, 305, 313–316, 324, 329, 332, 334, 341 Character table, 281, 297, 305–307, 310, 311, 316, 317, 321, 322, 331, 333, 335, 338, 340, 341 Charge transfer, 101, 103, 104 Chemical shifts, 284, 362, 371, 424 Coherent Anti stokes Raman Scattering (CARS), 433, 466 Condon parabola, 172, 173 Conjugated system, 219

D Depolarization ratio, 341 Determination of heat of dissociation, 184 Dexter, 266 Dissociation energy, 71, 184–186 Donor, 245, 247–249, 253, 254, 262–267, 270–272, 287

Doppler effect, 417 Double photon absorption, 436–438 Dyson equation, 442, 446, 449, 450

E Einstein coefficients, 285 Electronic energy, 208, 209, 305 Electronic spectra, 161, 215, 304 Electronic structure, 201 Electron Spin Resonance (ESR), 376 Electron transfer, 267, 269–273 Electron transfer phenomena, 267 Electron transfer process, 267, 269–272 ESR spectra of transition metal ion, 395 Estimation of different energies in molecules, 7 Excimers and Exciplexes, 255 Exciplex, 255–258

F Fluorescence, 226, 286, 287, 438 Förster, 266 Fourier Transform Spectroscopy, 147, 148 Free electron model, 220

G G-Matrix and Kinetic Energy, 135 Group, 289, 291, 346

H Huckel’s approximation, 222 Hund’s rules, 397

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 P. K. Mallick, Fundamentals of Molecular Spectroscopy, https://doi.org/10.1007/978-981-99-0791-5

473

474 Hybridization, 216 Hydrogen bonding, 252 Hydrogen molecule, 90, 189, 196, 277 Hydrogen molecule ion, 189 Hyper polarizability, 432, 448 Hyper Raman effect, 432

I Internal conversion, 225, 227, 241, 244 Internal Coordinate, 126, 137 Intersystem crossing, 225, 227, 241, 243, 244 Isomers, 217 Isomer shift, 422 Isotope separation, 437, 439

K Kekule structures, 224

L Lamda doubling (Ʌ), 182, 209 Linear molecules, 30, 31, 86 Longitudinal relaxation, 351, 356, 375

M Magic angle spinning, 371 Marcus inverted region, 272 Molar extinction coefficient and fluorescence quantum spectrum, 286 Molecular interactions, 353 Molecular orbital, 221, 253 Molecular orbital method, 189, 221 Molecular wave function, 89, 90, 200, 210, 230, 249 Morse potential, 64, 71, 184 Mossbauer spectroscopy, 417, 419–423, 428 Multiphoton absorption, 437 Multiphoton dissociation, 439 Multiphoton ionization, 437 Multiplication table, 291, 292

N Non linear spectroscopy, 429 Non-rigid rotator, 21 Normal coordinate, 105, 109 Normal modes, 114, 121 Nuclear magnetic resonance imaging, 372 Nuclear Magnetic Resonance (NMR), 347

Index Nuclear quadrupole, 424 Nuclear quadrupole coupling constant, 407 Nuclear Quadrupole Resonance (NQR), 400 Nuclear recoil, 417 Nuclear spin, 351

O Order, 3, 6–8, 11, 27, 34, 39, 41–45, 58, 64–68, 76, 78, 79, 96, 98, 100, 102, 106, 108, 110, 113, 114, 130, 132, 140, 142, 150, 185, 193, 199, 204, 206–208, 210–212, 219, 222, 225, 228, 237, 240, 241, 259, 265, 268, 291, 292, 297, 298, 302–304, 307–310, 312, 315, 322, 331, 336, 338, 339, 341, 351, 353, 354, 367, 370, 394, 397, 398, 401, 406, 407, 409, 419, 421, 422, 429, 431, 437, 440–443, 445, 446, 448, 449, 452, 457, 459, 462, 464, 468, 469 Orthogonality theorems, 296

P Permanents, 232 Phosphorescence, 226, 237, 340 Phosphorescence decay, 237 Photochemical process, 225 Photoelectron spectroscopy, 273, 275, 281, 283 Photophysical process., 225 Polarizability, 75, 76 Polarizability matrix, 84, 93, 96, 432, 444 Polarizability tensor, 80, 83, 114, 310, 342, 343, 346 Polarization characteristics, 241 Potential Energy Distribution (PED), 139 Potential Energy in terms of Internal Coordinates, 137 Predissociation, 187

Q Quadrupole hyperfine structure, 45 Quadrupole moment, 52 Quantum numbers of electronic states, 176 Quantum numbers of electronic states in diatomic molecules, 176 Quantum yield, 258

Index R Raman, 73–79, 81–87, 92, 93, 95–99, 102–104, 111, 113, 114, 117, 121, 310–312, 317, 321, 329, 333, 341–343, 345, 429–436, 444, 446, 447, 450–452, 459–467, 470, 471 Raman spectra, 79, 82, 83, 85–87, 91–93, 95, 97, 105, 111, 113, 114, 310, 341, 432, 434, 435, 466, 467, 471 Raman spectroscopy, 82, 93, 104, 472 Rayleigh’s scattering, 81 Reaction coordinate, 268 Relaxation, 225, 350 Representations, 293 Resonance absorption, 419 Rigid rotator, 14, 16, 18, 21–23, 34, 56, 59, 82, 89, 95 Rotational energy, 16 Rotational fine structure of vibronic bands in diatomic molecules, 167 Rotational Raman spectra, 82 Rotational spectra, 2, 13, 20, 24 Rotational vibrational spectra, 62

S Second harmonic generation, 431, 450, 458 Selection rules, 15, 57, 182–184, 303, 437 Solid state nmr, 369 Spherically symmetric molecules, 39 Spin orbit coupling, 336, 398 Stark effect, 42, 44 Sum frequency and second harmonics generation, 452 Surface Enhanced Raman Scattering (SERS), 97

475 Susceptibility, 358, 429, 443–445, 448, 449, 452, 455, 462–465, 467, 469, 470, 472 S – vectors, 152 Symmetric top molecules, 33, 86 Symmetry, 87, 292, 299, 308, 316–318, 329 Symmetry coordinate, 141, 318 Symmetry operations, 292, 293, 295, 298, 299, 319, 320, 322

T Transverse relaxation, 351, 356, 375

V Valence bond, 189 Validity of Born-Oppenheimer Approximation, 6 Vibrational coarse structure of electronic bands of a diatomic molecule, 162 Vibrational energy levels, 55 Vibrational spectra, 105, 308 Vibronic and spin-orbit interactions and n→π* transitions in organic Molecules, 227 Vibronic coupling, 228, 335

W Wigner-Eckart theorem, 402–404 Wilson, 140, 144, 147 Wilson G-F Matrix method, 140

Z Zero field splitting, 391, 395