Modern Optical Spectroscopy: From Fundamentals to Applications in Chemistry, Biochemistry and Biophysics [3 ed.] 3031172213, 9783031172212

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William W. Parson Clemens Burda

Modern Optical Spectroscopy From Fundamentals to Applications in Chemistry, Biochemistry and Biophysics Third Edition

Modern Optical Spectroscopy

William W. Parson • Clemens Burda

Modern Optical Spectroscopy From Fundamentals to Applications in Chemistry, Biochemistry and Biophysics Third Edition

William W. Parson Department of Biochemistry University of Washington Seattle, WA, USA

Clemens Burda Department of Chemistry Case Western Reserve University Cleveland, OH, USA

ISBN 978-3-031-17221-2 ISBN 978-3-031-17222-9 https://doi.org/10.1007/978-3-031-17222-9

(eBook)

# The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2007, 2015, 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. Cover Illustration: The reaction center and surrounding LH1 antenna complex of the photosynthetic bacterium Rhodobacter sphaeroides (Tani, K., et al., Nat. Commun. 12, 6300 (2021)). This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

1

2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Beer-Lambert Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Regions of the Electromagnetic Spectrum . . . . . . . . . . . . . . . . 1.4 Absorption Spectra of Proteins and Nucleic Acids . . . . . . . . . . 1.5 Absorption Spectra of Mixtures . . . . . . . . . . . . . . . . . . . . . . . 1.6 The Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Techniques for Measuring Absorbance . . . . . . . . . . . . . . . . . . 1.8 Pump-Probe and Photon-Echo Experiments . . . . . . . . . . . . . . 1.9 Linear and Circular Dichroism . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Distortions of Absorption Spectra by Light Scattering or Nonuniform Distributions of the Absorbing Molecules . . . . . . 1.11 Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 IR and Raman Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 1.13 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 4 6 8 10 11 15 16

Basic Concepts of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . 2.1 Wavefunctions, Operators and Expectation Values . . . . . . . . . 2.1.1 Wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Operators and Expectation Values . . . . . . . . . . . . . . . . 2.2 The Time-Dependent and Time-Independent Schrödinger Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Superposition States . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Spatial Wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 A Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 A Particle in a Box . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Atomic Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Molecular Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Wavefunctions for Large Systems . . . . . . . . . . . . . . . .

35 35 35 36

18 20 25 28 30 31 32

42 47 49 49 50 53 57 60 66 v

vi

Contents

2.4 2.5

3

4

Spin Wavefunctions and Singlet and Triplet States . . . . . . . . . Transitions Between States: Time-Dependent Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Lifetimes of States and the Uncertainty Principle . . . . . . . . . . . 2.7 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Electrostatic Forces and Fields . . . . . . . . . . . . . . . . . . 3.1.2 Electrostatic Potentials . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Electromagnetic Radiation . . . . . . . . . . . . . . . . . . . . . 3.1.4 Energy Density and Irradiance . . . . . . . . . . . . . . . . . . 3.1.5 Electromagnetic Momentum . . . . . . . . . . . . . . . . . . . . 3.2 The Black-Body Radiation Law . . . . . . . . . . . . . . . . . . . . . . . 3.3 Linear and Circular Polarization . . . . . . . . . . . . . . . . . . . . . . . 3.4 Quantum Theory of Electromagnetic Radiation . . . . . . . . . . . . 3.5 Superposition States and Interference Effects in Quantum Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Refraction, Evanescent Radiation, and Surface Plasmons . . . . . 3.7 The Classical Theory of Dielectric Dispersion . . . . . . . . . . . . . 3.8 Nonlinear Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Birefringence and Electro-Optic Effects . . . . . . . . . . . . . . . . . 3.10 Optical Wavepackets and Mode-Locked Lasers . . . . . . . . . . . . 3.11 Local-Field Correction Factors . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 87 87 88 91 98 101 103 105 107

Electronic Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Interactions of Electrons with Oscillating Electric Fields . . . . . 4.2 The Rates of Absorption and Stimulated Emission . . . . . . . . . . 4.3 Transition Dipoles and Dipole Strengths . . . . . . . . . . . . . . . . . 4.4 Calculating Transition Dipoles for π Molecular Orbitals . . . . . . 4.5 The Role of Molecular Symmetry in Electronic Transitions . . . 4.6 Using Group Theory to Determine Whether a Transition Is Allowed by Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Linear Dichroism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Configuration Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Calculating Electric Transition Dipoles with the Gradient Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transition Dipoles for Excitations to Singlet and Triplet 4.10 States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 The Born-Oppenheimer Approximation, Franck-Condon Factors, and the Shapes of Electronic Absorption Bands . . . . . 4.12 Spectroscopic Hole Burning . . . . . . . . . . . . . . . . . . . . . . . . . .

77 80 83 85

110 115 118 122 124 127 130 132 134 137 137 142 146 156 158 164 176 180 184 193 195 204

Contents

Effects of the Surroundings on Molecular Transition Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.14 The Electronic Stark Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 4.15 Spectroscopy of Transition-Metal Complexes . . . . . . . . . . . . . 4.16 Thermodynamics of Photoexcitation . . . . . . . . . . . . . . . . . . . . 4.17 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

4.13

5

6

Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Einstein Coefficients for Absorption and Emission . . . . . . 5.2 The Stokes Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Mirror-image Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The Strickler-Berg Equation and Other Relationships Between Absorption and Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Quantum Theory of Absorption and Emission . . . . . . . . . . . . . 5.6 Fluorescence Yields and Lifetimes . . . . . . . . . . . . . . . . . . . . . 5.7 Fluorescent Probes and Tags . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Quantum Dot Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Photobleaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Fluorescence Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Single-molecule Fluorescence and High-resolution Fluorescence Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Fluorescence Correlation Spectroscopy . . . . . . . . . . . . . . . . . . 5.13 Intersystem Crossing, Phosphorescence, and Delayed Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14 Electron Transfer from Excited Molecules . . . . . . . . . . . . . . . 5.15 Solar Cells and Light-emitting Diodes . . . . . . . . . . . . . . . . . . 5.16 Aggregation-induced Emission . . . . . . . . . . . . . . . . . . . . . . . . 5.17 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vibrational Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Vibrational Normal Modes and Wavefunctions . . . . . . . . . . . . 6.2 Vibrational Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Selection Rules and Effects of Anharmonicity . . . . . . . . . . . . . 6.4 Comparisons of IR and Raman Spectroscopy . . . . . . . . . . . . . 6.5 Effects of Molecular Symmetry in IR and Raman Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Rotational Absorption and Fine Structure . . . . . . . . . . . . . . . . 6.7 Infrared Spectroscopy of Proteins and Nucleic Acids . . . . . . . . 6.8 Vibrational Stark Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . IR Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 6.10 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

207 216 223 228 234 236 245 245 248 250 253 259 264 271 274 276 277 284 289 295 298 306 309 311 312 331 331 340 342 345 348 353 357 362 364 365 367

viii

7

8

Contents

Resonance Energy Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Förster Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Using Energy Transfer to Study Fast Processes in Single Protein Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Exchange Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Energy Transfer in Photosynthetic Antennas . . . . . . . . . . . . . . 7.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exciton Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Stationary States of Systems with Interacting Molecules . . . . . 8.2 Effects of Exciton Interactions on the Absorption Spectra of Oligomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Transition-Monopole Treatments of Interaction Matrix Elements and Mixing with Charge-Transfer Transitions . . . . . . 8.4 Exciton Absorption Band Shapes and Dynamic Localization of Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Exciton States in Photosynthetic Antenna Complexes . . . . . . . 8.6 Excimers and Exciplexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

377 377 379 394 396 398 402 404 409 409 418 424 428 431 433 436 439

9

Circular Dichroism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Magnetic Transition Dipoles and n - π  Transitions . . . . . . . . 9.2 The Origin of Circular Dichroism . . . . . . . . . . . . . . . . . . . . . . 9.3 Circular Dichroism of Dimers and Higher Oligomers . . . . . . . . 9.4 UV Circular Dichroism of Proteins and Nucleic Acids . . . . . . . 9.5 Vibrational Circular Dichroism . . . . . . . . . . . . . . . . . . . . . . . 9.6 Magnetic Circular Dichroism . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

445 445 456 462 467 470 473 476 478

10

Coherence and Dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Oscillations Between Quantum States of an Isolated System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 The Stochastic Liouville Equation . . . . . . . . . . . . . . . . . . . . . 10.4 Effects of Stochastic Relaxations on the Dynamics of Quantum Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 A Density-Matrix Treatment of Absorption of Weak, Continuous Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 The Relaxation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 More General Relaxation Functions and Spectral Lineshapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 Anomalous Fluorescence Anisotropy . . . . . . . . . . . . . . . . . . .

483 483 487 493 495 501 504 513 519

Contents

ix

10.9 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 11

Pump-Probe Spectroscopy, Photon Echoes, Two-Dimensional Spectroscopy and Vibrational Wavepackets . . . . . . . . . . . . . . . . . . 11.1 First-Order Optical Polarization . . . . . . . . . . . . . . . . . . . . . . . 11.2 Third-Order Optical Polarization and Non-linear Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Pump-Probe Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Photon Echoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Two-Dimensional Electronic and Vibrational Spectroscopy . . . 11.6 Transient Gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Vibrational Wavepackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 Wavepacket Pictures of Spectroscopic Transitions . . . . . . . . . . 11.9 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

529 529 538 543 547 553 558 561 568 572 573

Raman Scattering and Other Multi-photon Processes . . . . . . . . . . . 12.1 Types of Light Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The Kramers-Heisenberg-Dirac Theory . . . . . . . . . . . . . . . . . . 12.3 The Wavepacket Picture of resonance Raman Scattering . . . . . 12.4 Selection Rules for Raman Scattering . . . . . . . . . . . . . . . . . . . 12.5 Surface-enhanced Raman Scattering . . . . . . . . . . . . . . . . . . . . 12.6 Applications of Raman Spectroscopy . . . . . . . . . . . . . . . . . . . 12.7 Coherent (Stimulated) Raman Scattering . . . . . . . . . . . . . . . . . 12.8 Multi-photon Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9 Quasielastic (Dynamic) Light Scattering (Photon Correlation Spectroscopy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.10 Mie scattering by Larger Particles . . . . . . . . . . . . . . . . . . . . . 12.11 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

583 583 587 596 598 601 601 603 605

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase Shift and Modulation Amplitude in Frequency-Domain Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CGS and SI Units and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . Harmonic-Oscillator Wavefunction Integrals . . . . . . . . . . . . . . . . . . . .

623 623 625 627

12

608 612 614 615

631 633 634

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639

List of Boxes

Box 2.1 Box 2.2 Box 2.3 Box 2.4 Box 2.5 Box 2.6 Box 3.1 Box 4.1 Box 4.2 Box 4.3 Box 4.4 Box 4.5 Box 4.6 Box 4.7 Box 4.8 Box 4.9 Box 4.10 Box 4.11 Box 4.12 Box 4.13 Box 4.14 Box 5.1 Box 5.2 Box 5.3 Box 6.1 Box 7.1

Operators for Observable Properties Must Be Hermitian . . . . . .. . . . Commutators and Formulations of the Position, Momentum and Hamiltonian Operators . . .. .. . .. .. . .. .. . .. . .. .. . .. .. . .. .. . .. .. . .. . The Origin of the Time-Dependent Schrödinger Equation . . . . . . . . Linear Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hermite Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boltzmann, Fermi-Dirac and Bose-Einstein Statistics . . . . . . . . . . . . . Maxwell’s Equations and the Vector Potential . . . . . . . . . . . . . . . . . . . . . Energy of a Dipole in an External Electric Field . . . . . . . . . . . . . . . . . . . Multipole Expansion of the Energy of a Set of Charges in a Variable External Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Behavior of the Function [exp(iy)-1]/y as y goes to 0 . . . . . . . . The Function sin2x/x2 and Its Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Oscillating Electric Dipole of a Superposition State . . . . . . . . . . The Mean-Squared Energy of Interaction of an External Field with Dipoles in an Isotropic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Constants and Conversion Factors for Absorption of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evaluating Configuration-Interaction Coefficients . . . . . . . . . . . . . . . . . The Relationship between Matrix Elements of the Electric Dipole and Gradient Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matrix Elements of the Gradient Operator for Atomic 2p Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection Rules for Electric-Dipole Excitations of Linear Polyenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recursion Formulas for Vibrational Overlap Integrals . . . . . . . . . . . . Thermally Weighted Franck-Condon Factors . . . . . . . . . . . . . . . . . . . . . . Electronic Stark Spectroscopy of Immobilized Molecules . . . . . . . . The ν3 Factor in the Strickler-Berg Equation . . . . . . . . . . . . . . . . . . . . . . . Creation and Annihilation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Binomial, Poisson and Gaussian Distributions . . . . . . . . . . . . . . . . . . . . . Normal Coordinates and Molecular-dynamics Simulations . . . . . . . Dipole-dipole Interactions . . . . .. . . . . .. . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . .

38 39 43 51 55 74 93 138 140 143 145 147 151 153 181 185 188 192 199 201 220 256 263 291 332 383 xi

xii

Box 8.1 Box 8.2 Box 8.3 Box 8.4 Box 9.1 Box 9.2 Box 10.1 Box 10.2 Box 10.3 Box 10.4 Box 10.5 Box 12.1

List of Boxes

Why Must the Secular Determinant Be Zero? . . . .. . . . .. . . . .. . . . .. . . Avoided Crossings and Conical Intersections of Energy Surfaces . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . Exciton States Are Stationary in the Absence of Further Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Sum Rule for Exciton Dipole Strengths . . . . . . . . . . . . . . . . . . . . . . . Quantum Theory of Magnetic-Dipole and Electric-Quadrupole Transitions . . . .. . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . Ellipticity and Optical Rotation . . . .. . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . Time Dependence of the Density Matrix for an Isolated Three-State System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The “Watched-Pot” or “Quantum Zeno” Paradox . . . . . . . . . . . . . . . . . The Relaxation Matrix for a Two-State System . . . . . . . . . . . . . . . . . . . . Dephasing by Static Inhomogeneity . . .. .. . .. .. . .. .. . .. . .. .. . .. .. . .. . Orientational Averages of Vector Dot Products . . . . . . . . . . . . . . . . . . . . Quantum Theory of Electronic Polarizability . . . . . . . . . . . . . . . . . . . . . .

412 415 417 421 453 458 491 499 508 512 522 593

1

Introduction

1.1

Overview

Their extraordinary sensitivity and speed make optical spectroscopic techniques well suited for addressing a broad range of questions in molecular and cellular biophysics. Photomultipliers sensitive enough to detect a single photon make it possible to measure the fluorescence from individual molecules, and lasers providing light pulses with widths of less than 10–14 s can be used to probe molecular behavior on the time scale of nuclear motions. Spectroscopic properties such as absorbance, fluorescence, scattering and linear and circular dichroism can report on the identities, concentrations, interactions, conformations, or dynamics of molecules and can be sensitive to small changes in molecular structure or surroundings. Resonance energy transfer provides a way to probe intermolecular distances. Because they usually are not destructive, spectrophotometric techniques can be used with samples that must be recovered after an experiment. They also can provide analytical methods that avoid the need for radioisotopes or hazardous reagents. When combined with genetic engineering and microscopy, they provide insight into the locations, dynamics, and turnover of particular molecules in living cells. While this book describes many types of optical spectroscopies and their applications to chemistry, biochemistry, and biophysics, the spirit of the book is broader. It is about light, how light interacts with matter, and what we can learn from these interactions. These are topics that have puzzled and astonished people for thousands of years, and continue to do so today. To understand how molecules respond to light we first must inquire into why molecules exist in well-defined states and how they change from one state to another. Thinking about these questions underwent a series of revolutions with the development of quantum mechanics, and today quantum mechanics forms the scaffold for almost any investigation of molecular properties. Although most of the molecules that interest biophysicists are far too large and complex to be treated exactly by quantum mechanical techniques, their properties often can be rationalized by quantum mechanical principles that have been refined on simpler systems. We’ll discuss these principles in Chap. 2. For now, # The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. W. Parson, C. Burda, Modern Optical Spectroscopy, https://doi.org/10.1007/978-3-031-17222-9_1

1

2

1 Introduction

the most salient points are just that a molecule can exist in a variety of states depending on how its electrons are distributed among a set of molecular orbitals, and that each of these quantum states is associated with a definite energy. For a molecule with 2n electrons, the electronic state with the lowest total energy usually is obtained when there are two electrons with antiparallel spins in each of the n lowest orbitals and all the higher orbitals are empty. This is the ground state. In the absence of external perturbations, a molecule placed in the ground state will remain there indefinitely. Chapter 3 will discuss the nature of light, beginning with a classical description of an oscillating electromagnetic field. Exposing a molecule to such a field causes the potential energies of the electrons to fluctuate with time so that the original molecular orbitals no longer limit the possibilities. The result of this can be that an electron moves from one of the occupied molecular orbitals to an unoccupied orbital with higher energy. Two main requirements must be met for such a transition to occur. First, the electromagnetic field must oscillate at the right frequency. The required frequency (ν) is ν = ΔE=h,

ð1:1Þ

where ΔE is the difference between the energies of the ground and excited states and h is Planck’s constant (6.63 × 10–34 joule  s, 4.12 × 10–15 eV  s, or 3.34 × 10–11 cm–1  s). This expression is in accord with our experience that a given type of molecule, or a molecule in a particular environment, absorbs light of some colors and not of others. In Chap. 4 we will see that the frequency rule emerges straightforwardly from the classical electromagnetic theory of light, as long as we treat the absorbing molecule quantum mechanically. It is not necessary at this point to use a quantum mechanical picture of light. The second requirement is perhaps less familiar than the first and has to do with the shapes of the two molecular orbitals and the disposition of the orbitals in space relative to the polarization of the oscillating electrical field. The two orbitals must have different geometrical symmetries and must be oriented appropriately with respect to the field. This requirement rationalizes the observation that absorption bands of various molecules vary widely in strength. It also explains why the absorbance of an anisotropic sample depends on the polarization of the light beam. The molecular property that determines both the strength of an absorption band and the optimal polarization of the light is a vector called the transition dipole, which can be calculated from the molecular orbitals of the ground and excited state. The square of the magnitude of the transition dipole is termed the dipole strength and is proportional to the strength of absorption. Chapter 4 develops these notions more fully and examines how they arise from the principles of quantum mechanics. This provides the theoretical groundwork for discussing how measurements of the wavelength, strength, or polarization of electronic absorption bands can provide information on molecular structure and dynamics. In Chaps. 10 and 11 we extend the quantum mechanical treatment of absorption to large ensembles of molecules that

1.2 The Beer-Lambert Law

3

interact with their surroundings in a variety of ways. Various types of vibrational spectroscopy are discussed in Chaps. 6, 11, and 12. An atom or molecule that has been excited by light can decay back to the ground state by several possible paths. One possibility is to reemit energy as fluorescence. Although spontaneous fluorescence is not simply the reverse of absorption, it shares the same requirements for energy matching and appropriate orbital symmetry. Again, the frequency of the emitted radiation is proportional to the energy difference between the excited and ground states and the polarization of the radiation depends on the orientation of the excited molecule, although both the orientation and the energy of the excited molecule usually change in the interval between absorption and emission. As we will see in Chap. 7, the same requirements underlie another mechanism by which an excited molecule can decay, the transfer of energy to a neighboring molecule. The relationship between fluorescence and absorption is developed in Chap. 5, where the need for a quantum theory of light finally comes to the front.

1.2

The Beer-Lambert Law

A beam of light passing through a solution of absorbing molecules transfers energy to the molecules as it proceeds, and thus decreases progressively in intensity. The decrease in the intensity or irradiance (I) over the course of a small volume element is proportional to the irradiance of the light entering the element, the concentration of absorbers (C), and the length of the path through the element (dx): dI = - ε0 I C: dx

ð1:2Þ

The proportionality constant (ε′) depends on the wavelength of the light and the absorber’s structure, orientation, and environment. Integrating Eq. (1.2) shows that if light with irradiance Io is incident on a cell of thickness l, the irradiance of the transmitted light will be: I = I o expð- ε0 C lÞ = I o 10 - ε C l  I o 10 - A :

ð1:3Þ

Here A is the absorbance or optical density of the sample (A = ε C l) and ε is called the molar extinction coefficient or molar absorption coefficient (ε = ε′/ln(10) = ε′/ 2.303). The absorbance is a dimensionless quantity, so if C is given in units of molarity (1 M = 1 mol/l) and c in cm, ε must have dimensions of M–1  cm–1. Equations (1.2) and (1.3) are statements of Beer’s law, or more accurately, the Beer-Lambert law. Johann Lambert, a physicist, mathematician, and astronomer born in 1728, observed that the fraction of the light that is transmitted (I/Io) is independent of Io. Wilhelm Beer, a banker and astronomer who lived from 1797 to 1850, noted the exponential dependence on C. The Beer-Lambert law assumes that the concentration of the chromophore is low enough so that intermolecular interactions are negligible.

4

1 Introduction

In the classical electromagnetic theory of light, the oscillation frequency ν is related to the wavelength (λ), the velocity of light in a vacuum (c), and the refractive index of the medium (n) by the expression ν = c=n λ:

ð1:4Þ

Light with a single wavelength, or more realistically, with a narrow band of wavelengths is called monochromatic. The light intensity, or irradiance (I) in Eqs. (1.2) and (1.3) represents the flux of radiant energy per unit cross-sectional area of the beam (joules per second per square cm or watts per square cm). We usually are concerned with the radiation in a particular frequency interval (Δν), so I has units of joule per frequency interval per second per square cm. For a light beam with a cross-sectional area of 1 cm2, the amplitude of the signal that might be recorded by a photomultiplier or other detector is proportional to I(ν)Δν. In the quantum theory of light that we’ll discuss briefly in Sect. 1.6 and at greater depth in Chap. 3, intensities often are expressed in terms of the flux of photons rather than energy [photons per frequency interval per second per square cm]. A beam with an irradiance of 1 watt cm–2 has a photon flux of 5.05x(λ/ nm) × 1015 photons  cm–2. The dependence of the absorbance on the frequency of light can be displayed by plotting A or ε as a function of the frequency (ν), the wavelength (λ), or the wavenumber (ν). The wavenumber is simply the reciprocal of the wavelength in a vacuum: ν = 1/λ = ν/c, and has units of cm–1. Sometimes the percent of the incident light that is absorbed or transmitted is plotted. The percent absorbed is 100 × (Io – I)/ Io = 100 × (1 – 10–A), which is proportional to A if A l, where n is any integer. The energies associated with these wavefunctions are E n = n2 h2 =8ml2 :

ð2:25Þ

Each eigenfunction (ψ n) thus is determined by a particular value of an integer quantum number (n), and the energies (En) increase quadratically with this number. Figure 2.2 shows the first five eigenfunctions, their energies, and the corresponding probability density functions (jψ nj2). Outside the box, the wavefunctions must be zero because there is no chance of finding the particle in a region where its potential energy would be infinite. To avoid discontinuities, the wavefunctions must go to zero at both ends of the box, and it is this boundary condition that forces n to be an integer. There are several important points to note here. First, the energies are quantized. Second, excluding the trivial solution n = 0, which corresponds to an empty box, the lowest possible energy is not zero as it would be for a classical particle at rest in a box, but rather h2/8ml2. The smaller the box, the higher the energy. Finally, the number of nodes (points where the wavefunction crosses zero, or in a multidimensional system, surfaces where this occurs) increases linearly with n, so that

2.3 Spatial Wavefunctions

51

A

B 25

25 5

5

4

/l

2 -1

20

En/(h /8ml ); |

n|

4

15

2

15

10

2

5

3

2

3

2

2

En/(h /8ml );

-1/2 n/l

20

10

2

5

1

0

0

1

0.5 x/l

1.0

0

0

0.5 x/l

1.0

Fig. 2.2 (A) Eigenvalues (dotted horizontal green lines) and eigenfunctions (solid blue curves) for the first five eigenstates (n = 1, 2, . . . 5) of a particle in a one-dimensional rectangular box with length l and infinitely high walls. The eigenfunctions are displaced vertically to align them with the corresponding energies. (B) probability densities (solid blue curves) for the same five eigenstates, displaced vertically for alignment with the energies (dotted green lines) as in (A)

the probability distribution becomes more uniform with increasing n (Fig. 2.2). The momentum of the particle is discussed in Box 2.4. Box 2.4 Linear Momentum Because we took the potential energy inside the box in Fig. 2.2 to be zero, the energies given by Eq. (2.25) are entirely kinetic energy. From classical physics, a particle with kinetic energy En and mass m should have a linear momentum p with magnitude p = jpj =

pffiffiffiffiffiffiffiffiffiffiffi 2mE n = nh=2l:

ðB2:4:1Þ

Referring to Fig. 2.2A, we see that wavefunction ψ n is equivalent to a standing wave with a wavelength λn of 2 l/n. We therefore also could write the momentum (nh/2 l ) as h/λn or hν , where ν is the wavenumber (1/λ). This is consistent with deBroglie’s expression linking the momentum of a free particle to the wavenumber of an associated wave (Eq. (B2.3.1)). (continued)

52

2

Basic Concepts of Quantum Mechanics

Box 2.4 (continued) But the expectation value of the momentum of the particle in a box with infinitely high walls is not hν ; it’s zero. We can see this by using Eqs. (2.4) and (2.5): e n i = hψ n jðħ=iÞ∂ψ n =∂xi hψ n jpjψ D E = ðħ=iÞ ð2=lÞ1=2 sinðnπx=lÞjð2=lÞ1=2 ∂sinðnπx=lÞ=∂x Zl sinðnπx=lÞ cosðnπx=lÞdx = 0:

= ðħ=iÞð2=lÞðnπ=lÞ

ðB2:4:2Þ

0

This result makes sense if we view each of the standing waves described by Eq. (2.22) as a superposition of two waves moving in opposite directions. Individual measurements of the momentum then might give either positive or negative values but would average to zero. From a more formal perspective, we cannot predict the outcome of an individual measurement of the momentum because the wavefunctions given by Eq. (2.22) are not eigenfunctions of the momentum operator. This is apparent because the action of the momentum operator on wavefunction ψ n gives   e pψ n = ðħ=iÞ∂ψ n =∂x = ðħ=iÞ 21=2 nπ=l3=2 cosðnπx=lÞ, ðB2:4:3Þ which is not equal to a constant times ψ n. However, the expectation value obtained by Eq. (B2.4.2) still gives the correct result for the momentum of a standing wave (zero). The Schrödinger equation does have solutions that are eigenfunctions of the momentum operator for an electron moving through an unbounded region of constant potential. These can be written ψ ± = A expf2π i ðνx ∓ νt Þg,

ðB2:4:4Þ

where A is a constant and the plus and minus signs are for particles moving in the two directions. Applying the momentum operator to these wavefunctions gives e pψ ± = ðħ=iÞ∂ψ ± =∂x = ðħ=iÞAð2π iνÞ expf2π iðνx ∓ νt Þg = hνψ ± , ðB2:4:5Þ which leads to the correct expectation value of the momentum of a free particle (hν).

2.3 Spatial Wavefunctions

53

Although the probability of finding the particle at a given position varies with the position, the eigenfunctions given by Eq. (2.24) extend over the full length of the box. Wavefunctions for a particle that is more localized in space can be constructed from linear combinations of these eigenfunctions. A superposition state formed in this way from vibrational wavefunctions is called a wavepacket. The combination ψ 1 - ψ 3 + ψ 5 - ψ 7, for example, gives a wavepacket whose amplitude peaks strongly at the center of the box (x = l/2), where the individual spatial wavefunctions interfere constructively. At positions far from the center, the wavefunctions interfere destructively and the summed amplitude is small. Because the time-dependent factors in the complete wavefunctions [exp(-iEnt/ħ)] oscillate at different frequencies, the wavepacket will not remain fixed in position, but will move and change shape with time. Wavepackets provide a way of representing an atom or macroscopic particle whose energy eigenvalues are uncertain relative to the separation between the eigenvalues. The spacing between the energies for a particle in a one-dimensional box is inversely proportional to both the mass of the particle and l2 (Eq. (2.25)), and will be very small for any macroscopic particle with a substantial mass or size. If the potential energy outside the square box described in Sect. 2.3.2 is not infinite, there is some probability of finding the particle here even if the potential energy is greater than the total energy of the system. For a one-dimensional square box with infinitely thick walls of height V, the wavefunctions in the region x > l are given by ψ n = An expf- ðx - lÞζ g,

ð2:26Þ

where ζ = [2 m(V – En)]1/2ħ-1 and An is a constant set by the boundary conditions. Inside the box, ψ n has a sinusoidal form similar to that of a wavefunction for a box with infinitely high walls (Eq. (2.24a)), but its amplitude goes to An rather than zero at the boundary (x = l). Both the amplitudes and the slopes of the wavefunctions are continuous across the boundary. A quantum mechanical particle thus can tunnel into a potential-energy wall, although the probability of finding it here decreases exponentially with the distance into the wall. A classical particle could not penetrate the wall because the condition V > En requires the kinetic energy to be negative, which is not possible in classical physics.

2.3.3

The Harmonic Oscillator

The potential well that restrains a chemical bond near its mean length is described reasonably well by the quadratic expression V ð xÞ =

1 2 kx , 2

ð2:27Þ

54

2

Basic Concepts of Quantum Mechanics

where x is the difference between the length of the bond and the mean length, and k is a force constant. If the bond is stretched or compressed, a restoring force proportional to the distortion (F = -kx) acts to return the bond to its mean length, provided that the distortion is not too large. Such a parabolic potential well is described as harmonic. At larger departures from the equilibrium length the potential well becomes increasingly anharmonic, rising more steeply for compression than for stretching, as illustrated in Fig. 2.1. A classical particle of mass m in a harmonic potential well oscillates about its equilibrium position with a frequency given by  1=2 1 k : ð2:28Þ υ= 2π m Such a system is called a harmonic oscillator. Equation (2.28) also applies to the frequency of the classical vibrations of a pair of bonded atoms if we replace m by the reduced mass of the pair (mr = m1m2/(m1 + m2), where m1 and m2 are the masses of the individual atoms. The angular frequency ω is 2πυ = (k/m)1/2. The sum of the kinetic and potential energies of a classical harmonic oscillator is E classical =

1 1 jpj2 þ kx2 , 2mr 2

ð2:29Þ

which can have any non-negative value including zero. The quantum mechanical picture again is significantly different. The eigenvalues of the Hamiltonian operator for a one-dimensional harmonic oscillator are an evenly spaced ladder of energies, starting not at zero but at (1/2)hυ: En = ðn þ 1=2Þhυ,

ð2:30Þ

with n = 0, 1, 2, . . . . The corresponding wavefunctions are   χ n ðxÞ = N n H n ðuÞ exp - u2 =2 :

ð2:31Þ

Here Nn is a normalization factor, Hn(u) is a Hermite polynomial, and u is a dimensionless positional coordinate obtained by dividing the Cartesian coordinate x by (ħ/2πmrυ)1/2: u = x=ðħ=2π mr υÞ1=2 :

ð2:32Þ

The Hermite polynomials and their normalization factors Nn are given in Box 2.5 and the first six wavefunctions are shown in Fig. 2.3.

2.3 Spatial Wavefunctions

55

5

5 Probability Density x 5; E/h

B 6

Wavefunction Amplitude; E/h

A 6

4

3

2

3

2

1

1

0 -4

4

-2

0 u

2

4

0 -4

-2

0 u

2

4

Fig. 2.3 (A) The first 5 wavefunctions (blue curves) and eigenvalues of the total energy (En, dotted green horizontal lines) of a harmonic oscillator. The dimensionless quantity plotted on the abscissa (u) is the distance of the particle from its equilibrium position divided by (ħ/2πmrυ)1/2, where mr is the reduced mass and υ is the classical oscillation frequency. The dashed black curve represents the potential energy. Energies are expressed in units of hυ. The wavefunctions are displaced vertically to align them with the corresponding eigenvalues. (B) probability densities for the same five eigenstates, displaced vertically for alignment with the eigenvalues as in (A)

Box 2.5 Hermite Polynomials The Hermite polynomials are defined by the expression   dn  2 H n ðuÞ = ð- 1Þn exp u2 n exp - u : du

ðB2:5:1Þ

They can be generated by starting with Eq. (B2.5.1) to find H0 = 1 and H1 = 2u, and then using the recursion formula H nþ1 ðuÞ = 2uH n ðuÞ - 2nH n - 1 ðuÞ:

ðB2:5:2Þ

The first six Hermite polynomials are: (continued)

56

2

Basic Concepts of Quantum Mechanics

Box 2.5 (continued) H 0 = 1,

H 3 = 8u3 - 12u,

H 1 = 2u, H 2 = 4u2 - 2,

H 4 = 16u4 - 48u2 þ 12,

ðB2:5:3Þ

H 5 = 32u5 - 160u3 þ 120u:

The normalization factor Nn in (Eq. 2.31) is h i1=2 : N n = ð2πυ=ħÞ1=2 =ð2n n!Þ

ðB2:5:4Þ

Note that changing the sign of the positional coordinate u has no effect on the even-numbered Hermite polynomials (H2n(-u) = H2n(u)) but inverts the odd-numbered polynomials ((H2n + 1(-u) = -H2n + 1(u)). The even-numbered Hermite polynomials therefore are said to have gerade (German “even”) symmetry while the odd-numbered polynomials have ungerade (“odd”) symmetry. Multiplying a Hermite polynomial by u changes its symmetry from gerade to ungerade or vice versa. These symmetries are important when products of vibrational wavefunctions are integrated over the positional coordinate. Integrals of the forms hχ mjχ ni and hχ mjujχ ni play important roles in the selection rules for electronic and vibrational absorption, fluorescence and Raman scattering, and will come up frequently in Chapters 4–6 and 12. If χ m and χ n are vibrational eigenfunctions of the same electronic state (or, more generally, of states that have congruent harmonic potential energy curves with minima at the same value of u), hχ mjχ ni is non-zero only if m = n because the wavefuntions are normalized and orthogonal. The integral hχ mjujχ ni, on the other hand, can be non-zero only if m = n ± 1. You can see this by rewriting the recursion relationship for Hermite polynomials (Eq. B2.5.2) as uH n =

1 þ nH n - 1 , H 2 nþ1

ðB2:5:5Þ

which implies that hχ m jujχ n i =

 1

χ jujχ nþ1 þ nhχ m jujχ n - 1 i: 2 m

ðB2:5:6Þ

Both of the integrals on the right-hand side of Eq. (B2.5.6) are zero unless m = n ± 1. Representative plots of χ mχ n, χ muχ n and their integrals are shown in Appendix A6. The eigenvalues of the harmonic oscillator Hamiltonian usually are described in terms of the wavenumber (ω = υ/c) in units of cm-1. The minimum energy, (1/2) hυ or (1/2)ħω, is called the zero-point energy.

2.3 Spatial Wavefunctions

57

Although the eigenvalues of a harmonic oscillator increase linearly instead of quadratically with n, and the shapes of the wavefunctions are more complex than those of a particle in a square well, the solutions of the Schrödinger equation for these two potentials have several features in common. Each eigenvalue corresponds to a particular integer value of the quantum number n, which determines the number of nodes in the wavefunction, and the spatial distribution of the wavefunction becomes more uniform as n increases. As in the case of a box with finite walls, a quantum mechanical harmonic oscillator has a definite probability of being outside the region bounded by the potential energy curve (Fig. 2.3B). Finally, as mentioned above and discussed in more detail in Chap. 11, a wavepacket for a particle at a particular position can be constructed from a linear combination of harmonicoscillator wavefunctions. The position of such a wavepacket oscillates in the potential well at the classical oscillation frequency υ.

2.3.4

Atomic Orbitals

Spatial wavefunctions for the electron in a hydrogen atom, or more generally, for a single electron with charge –e bound to a nucleus of charge +Ze, can be written as products of two functions, Rn,l(r) and Yl,m(θ,ϕ), where the variables r, θ, and ϕ specify positions in polar coordinates relative to the nucleus and an arbitrary z axis: ψ nlm = Rn,l ðr ÞY l,m ðθ, ϕÞ:

ð2:33Þ

(See Fig. 4.4 for an explanation of polar coordinates.) The subscripts n, l and m in these functions denote integer quantum numbers with the following possible values: principal quantum number: angular momentum (azimuthal) quantum number: magnetic quantum number:

n = 1, 2, 3, . . ., l = 0, 1, 2, . . ., n-1, m = -l, . . ., 0, . . ., l.

The energy of the orbital depends mainly on the principal quantum number (n) and is given by E n = -16π 2 Z 2 mr e4 =n2 h2 ,

ð2:34Þ

where mr is the reduced mass of the electron and the nucleus. (This expression uses the cgs system of units, which is discussed in Sect. 3.1.1.) The angular momentum or azimuthal quantum number (l ) determines the electron’s angular momentum, while the magnetic quantum number (m) determines the component of the angular momentum along a specified axis and relates to a splitting of the energy levels in a magnetic field (Sect. 9.5). The wavefunctions for the first few hydrogen-atom orbitals are given in Table 2.1, and Figs. 2.4, 2.5, and 2.6 show their shapes. Atomic orbitals with l = 0, 1, 2 and 3 are conventionally labeled s, p, d and f. The s wavefunctions peak at the nucleus and are spherically symmetrical (Figs. 2.4 and 2.5). The 1 s

58

2

Basic Concepts of Quantum Mechanics

Table 2.1 Hydrogen-atom wavefunctions Yl,m(θ,ϕ)b pffiffiffi - 1 ð2 π Þ pffiffiffi - 1 ð2 π Þ

n 1

l 0

2s

2

0

0

ðZ=ao Þ3=2 pffiffi 2 2

2pz

2

1

0

ðZao Þ3=2 pffiffi 2 6

ρ expð- ρ=2Þ

(3/4π)1/2 cos θ

2p-

2

1

-1

3=2 ðZap o Þffiffi 2 6

ρ expð- ρ=2Þ

ð3=8π Þ1=2 sin θf cos ϕ - i sin ϕÞ

1

3=2

ρ expð- ρ=2Þ

2p+

2

1

m

Rn,l(r)a (Z/ao)3/22 exp(-ρ)

orbital 1s

0

ðZap o Þffiffi 2 6

ð2 - ρÞ expð- ρ=2Þ

= ð3=8π Þ1=2 sin θ expð- iϕÞ ð3=8π Þ1=2 sin θf cos ϕ þ i sin ϕÞ = ð3=8π Þ1=2 sin θ expðiϕÞ

ρ = Zr/ao, where Z is the nuclear charge, r is the distance from the nucleus and ao is the Bohr radius (0.529 Å) b θ and ϕ are the angles with respect to the z and x axes in polar coordinates (Fig. 4.4) a

wavefunction has the same sign everywhere, whereas 2 s changes sign at r = 2ħ2/ mre2Z, or 2ao/Z, where ao (the Bohr radius) is ħ2/mre2 (0.5292 Å). Because the volume element (dσ) between spherical shells with radii r and r + dr is (for small values of dr) 4πr2dr, the probability of finding an s electron at distance r from the nucleus (the radial distribution function) depends on 4πr2ψ(r)2dr. The radial distribution function for the 1 s wavefunction peaks at r = ao (Fig. 2.4E, F). The p orbitals have nodal planes that pass through the nucleus (Figs. 2.5 and 2.6). The choice of a coordinate system for describing the orientations of the orbitals is arbitrary unless the atom is in a magnetic field, in which case the z axis is taken to be the direction of the field. The 2p orbital with m = 0 is oriented along this axis and is called 2pz. The 2p orbitals with m = ±1 (2p+ and 2p-) are complex functions that are maximal in the plane normal to the z axis and rotate in opposite directions around the z axis with time. However, they can be combined to give two real functions that are oriented along definite x- and y-axes and have no net rotational motion (2px and 2py). The 2px and 2py wavefunctions then are identical to 2pz except for their orientations in space. This is essentially the same as constructing standing waves from wavefunctions for electrons moving in opposite directions, as described in Box 2.4. If we use the scaled coordinates z = ρcosθ, x = ρsinθcosϕ and y = ρsinθsinϕ, with ρ = Zr/ao, the three 2p wavefunctions can be written: 2pz = ðZ=ao Þ5=2 ð32π Þ - 1=2 z expð- ρ=2Þ,

ð2:35aÞ

1 2px = pffiffiffi 2p - þ 2pþ = ðZ=ao Þ5=2 ð32π Þ - 1=2 x expð- ρ=2Þ, 2

ð2:35bÞ

i 2py = pffiffiffi 2p - þ 2pþ = ðZ=ao Þ5=2 ð32π Þ - 1=2 y expð- ρ=2Þ: 2

ð2:35cÞ

and

2.3 Spatial Wavefunctions

59

8

A

1s

B

2s

C

1s

D

2s

y / a0

4 0 -4

Wavefunction Amplitude

-8 2

1

0 -8

-4

0 x / a0

8 -8

4

-4

0 x / a0

4

8

8

E

F

1s

2s

2 2

4πr ψ a0

6 4 2 0

0

4 r / a0

8

0

4 r / a0

8

Fig. 2.4 (A) Contour plots (lines of constant amplitude in the plane of the nucleus) of the hydrogen-atom 1 s wavefunctions. The Cartesian coordinates x and y are expressed as dimensionless multiples of the Bohr radius (ao = 0.529 Å). The contour intervals for the amplitude are 0.2ao3/2. (B) Same as A, but for the 2 s wavefunctions with contour intervals of 0.05ao3/2. Positive amplitudes are shown as blue lines that get lighter as the amplitude increases, negative amplitudes as red lines, and zero as a dotted purple line. (C), (D) Amplitudes of the 1 s and 2 s wavefunctions as functions of the x coordinate with y fixed at 0. (E), (F) Radial distribution functions of the 1 s and 2 s wavefunctions

60

2

Basic Concepts of Quantum Mechanics

Fig. 2.5 Perspective drawings of the “boundary surfaces” (regions where an electron is most likely to be found) of the 1 s, 2p, and 3d wavefunctions of the hydrogen atom. The wavefunctions have a constant sign in the regions with light blue shading and the opposite sign in the regions with purple shading. These drawings are only approximately to scale. Fig. 4.30 shows similar drawings of the 4d wavefunctions

The 3d orbitals with m = ±1 or ± 2 also are complex functions that can be combined to give a set of real wavefunctions. The boundary surfaces of the real wavefunctions are shown in Fig. 2.5.

2.3.5

Molecular Orbitals

The Schrödinger equation has not been solved exactly for electrons in molecules larger than the H2+ ion; the interactions of multiple electrons become too complex to handle. However, the eigenfunctions of the Hamiltonian operator provide a complete set of functions, and as mentioned in Sect. 2.2.1, a linear combination of such functions can be used to construct any well-behaved function of the same coordinates. This suggests the possibility of representing a molecular electronic

2.3 Spatial Wavefunctions

61

Fig. 2.6 (A) Contour plot of the amplitude of the hydrogen-atom 2py wavefunction in the xy plane, with blue lines for positive amplitudes, red lines for negative amplitudes, and a purple dot-dash line for an amplitude of zero. Lighter colors represent larger absolute amplitudes. Distances are given as dimensionless multiples of the Bohr radius. The contour intervals for the amplitude are 0.01ao3/2. (B) The amplitude of the 2py wavefunction in the xy plane as a function of position along the y-axis

10

A

x / ao

5

0

-5

Wavefunction Amplitude / ao

2/3

( z = 0)

-10 0.1

B

0

(x = 0, z = 0)

-0.1 -10

-5

0 y / ao

5

10

wavefunction by a linear combination of hydrogen atomic orbitals centered at the nuclear positions. In principle, we should include the entire set of atomic orbitals for each nucleus, but smaller sets often provide useful approximations. For example, the π orbitals of a molecule with N conjugated atoms can be written as ψk ≈

N X n=1

Ckn ψ 2zðnÞ ,

ð2:36Þ

where ψ 2z(n) is an atomic 2pz orbital centered on atom n, and coefficient Ckn indicates the contribution that ψ 2z(n) makes to molecular orbital k. If the atomic wavefunctions are orthogonal and normalized (hψ 2z(n)jψ 2z(n)i = 1 for all n, and hψ 2z(n)jψ 2z(m)i = 0 for m ≠ n) the molecular wavefunctions can be normalized by scaling the coefficients so that N X n=1

jCkn j2 = 1:

ð2:37Þ

62

2

Basic Concepts of Quantum Mechanics

The highest occupied molecular orbital (HOMO) of ethylene, a bonding π orbital, can be approximated reasonably well by a symmetric combination of carbon 2pz orbitals centered on carbon atoms 1 and 2 and oriented so that their z-axes are parallel:     → → → ψ π r ≈ 2 - 1=2 ψ 2pzð1Þ þ 2 - 1=2 ψ 2pzð2Þ = 2 - 1=2 ψ 2pz r - r 1   → → þ 2 - 1=2 ψ 2pz r - r 2 , ð2:38Þ →

→

where r 1 and r 2 denote the positions of the two carbon atoms (Fig. 2.7). The contributions from ψ 2z(1) and ψ 2z(2) combine constructively in the region between the two atoms, leading to a build-up of electron density in this region. The molecular orbital resembles the 2pz atomic orbitals in having a node in the xy plane. The lowest unoccupied molecular orbital (LUMO) of ethylene, an antibonding (π*) orbital, can be represented as a similar linear combination but with opposite sign:     → → → ψ π r ≈ 2 - 1=2 ψ 2pzð1Þ - 2 - 1=2 ψ 2pzð2Þ = 2 - 1=2 ψ 2pz r - r 1   → → - 2 - 1=2 ψ 2pz r - r 2 : ð2:39Þ In this case, ψ 2z(1) and ψ 2z(2) interfere destructively in the region between the two carbons, and the molecular wavefunction has a node in the xz plane as well as the xy plane (Fig. 2.7C). The combinations described by Eqs. (2.38) and (2.39) are called symmetric and antisymmetric, respectively. This treatment of the π molecular orbitals of ethylene is known as linear combination of atomic orbitals (LCAO). A general mathematical procedure for finding the orbital energies and coefficients C ki for molecules with delocalized π systems was introduced by the German physical chemist Eric Hückel in 1930 [26, 27]. Although it gives only approximate energies, the Hückel molecular orbital method is still widely used to obtain initial guesses of the orbitals for more accurate treatments. The method involves constructing a Hamiltonian matrix in which each of the diagonal elements is the energy of an individual carbon 2p orbital relative to the energy of an unbound electron in a vacuum. This energy (α) is on the order of -11.4 eV. The offdiagonal matrix element for the interaction of neighboring carbons that form a covalent bond represents the further decrease in energy resulting from the delocalization of an electron over adjacent π atoms. This resonance energy (β) depends on the distance between the two atoms but typically is on the order of -1 eV. Off-diagonal elements for interactions of non-bonded carbons are taken to be zero. Diagonalizing the Hückel matrix gives the energies and atomic coefficients of the π orbitals. We’ll explain the diagonalization procedure in more detail in the following section. C. A. Coulson and G. S. Rushbrook pointed out that particularly simple relationships hold among the orbital energies and coefficients that the Hückel method gives for a large family of aromatic hydrocarbons called alternant

2.3 Spatial Wavefunctions

63

4

A

z / ao

2 0 -2 x=0

-4 4

C

B

z / ao

2 0 -2 x=0

x=0

-4

x / ao

2

D

E

0

-2 -4

z = ao

-2

0 y / ao

2

z = ao

4 -4

-2

0 y / ao

2

4

Fig. 2.7 Combining atomic 2p wavefunctions to form π and π* molecular orbitals. (A) Contour plot of a 2pz wavefunction of an individual carbon atom. Coordinates are given as multiples of the Bohr radius (ao), and the carbon 2pz wavefunction is represented as a Slater-type 2pz orbital (Eq. (2.40)) with ζ = 3.071/Å (1.625/ao). The plot shows lines of constant amplitude in the yz plane, with blue lines for positive amplitudes, red lines for negative amplitudes, and a dot-dashed purple line for zero. (B) Contour plot of the amplitude in the yz plane for the bonding (π) molecular orbital formed by a symmetric combination of 2pz wavefunctions of two carbon atoms separated by 2.51ao (1.33 Å) in the y-direction. The line colors and styles are the same as in A. (C) Same as (B), but for the antibonding (π*) molecular orbital created by an antisymmetric combination of the atomic wavefunctions. (D), (E) Same as (B) and (C), respectively, but showing the amplitudes of the wavefunctions in the plane parallel to the xy plane and ao above xy (horizontal dashed lines at z = ao in B, C). A map of the amplitudes ao below the plane of the ring would be the same except for an interchange of positive and negative signs. The contour intervals for the amplitude are 0.1ao3/2 in all five panels

64

2

Basic Concepts of Quantum Mechanics

Fig. 2.8 Alternant and non-alternant aromatic molecules. Benzene (A), naphthalene (B) and pyrene (C) are alternant; azurin (D) and indole (E) are non-alternant. The carbon atoms in each molecule have been sorted into the two groups indicated by black or blue-filled spheres. In alternant molecules, this can be done so that each atom forms covalent bonds only with atoms of the other group. In non-alternant molecules, some atoms inevitably form bonds to one or more other atoms of the same group. The red ovals in D and E indicate such bonds for the groups shown here; different groupings for azurin or indole would give bonds between other pairs of atoms within a group

hydrocarbons [28, 29]. In an alternant hydrocarbon, the π atoms can be sorted into two equal groups such that there are covalent bonds only between atoms of different groups. The atoms of ethylene, benzene, naphthalene, and pyrene can be sorted in this way (Fig. 2.8). The same cannot be done with azulene, which has 5- and 7-membered rings of conjugated atoms; it is, therefore, a nonalternant hydrocarbon. The aromatic heterocyclic molecule indole, which has a 5-membered ring, also is nonalternant. More generally, any conjugated hydrocarbon that does not have an odd-numbered ring of conjugated atoms is an alternant hydrocarbon. The Coulson-Rushbrook theorem [28, 29] makes several observations about alternant hydrocarbons. First, the energies of the Hückel π orbitals for these molecules lie at regular intervals above and below a mean value that turns out to be the diagonal energy of the Hückel matrix (α). Orbitals with energies below α are occupied with two electrons in the molecule’s ground state, while those with energies above α are unoccupied. Further, the neighboring occupied (or unoccupied) orbitals are separated by the resonance stabilization energy β. The energies thus can be written Ej = α ± kβ, where k = 1, 2, . . . . For each occupied orbital with energy α -kjβj, there is a corresponding unoccupied orbital with energy α + kjβj (Fig. 2.9). Second, the atomic expansion coefficients C ki for a given unoccupied orbital are identical to those for the corresponding occupied orbital except that the signs of the coefficients are inverted for all the atoms in one of the two groups (Fig. 2.9). And finally, the squares of the coefficients for all the atoms sum to unity in each orbital in accord with Eq. (2.37). The electrons in each π orbital therefore contribute a total electric charge of -1 to the molecule. You can check that

2.3 Spatial Wavefunctions

65

Fig. 2.9 Hückel coefficients and energies for benzene π orbitals [28, 30]. The dashed green line indicates the mean energy (α ≈ -11.4 eV). The three orbitals with lower energies are occupied in the ground state, and those with higher energies are unoccupied. Approximate energies were calculated using β = -0.8 eV. Coefficients that have different signs in the unoccupied orbitals are indicated in blue

these statements hold for ethylene by examining Eqs. (2.38) and (2.39) above. Figure 2.9 gives the Hückel coefficients and approximate orbital energies for benzene so you can verify that Coulson and Rushbrook’s relations hold there also. The Coulson-Rushbrook relations provide considerable insight into the spectroscopic properties of alternant hydrocarbons and are useful for tracking orbital energies along photochemical reaction paths [31]. Protonation of azulene on its 5-membered ring removes one of the carbons from the π system, converting the molecule to an alternant hydrocarbon and modifying its absorption and fluorescence properties substantially [32]. Further work by D. R. Hartree, J. C. Slater, and V. A. Fock in the period around 1930 led to improved treatments that incorporated electronic spin and made each wavefunction antisymmetric overall. We’ll come back to the need for antisymmetric wavefunctions in Sect. 2.4. R. Hoffman later extended the Hückel treatment to include σ bonds, and density-functional methods that considered electron exchange and correlation interactions were introduced to quantum chemistry in the 1990s. These methods have been refined extensively by comparing calculated molecular energies, dipole moments, vibrational frequencies, and other properties with experimental measurements. Numerous software packages for such calculations now are available [5, 6, 10, 33–44]. The basis wavefunctions employed in these descriptions usually are not the atomic orbitals obtained by solving the Schrödinger equation for the hydrogen atom, but rather idealized wavefunctions with mathematical forms that are easier to manipulate. They include parameters that are adjusted semiempirically

66

2

Basic Concepts of Quantum Mechanics

to model a particular type of atom and to adjust for overlap with neighboring atoms. A standard form is the Slater-type orbital, 

ψ nlm = Nr n

-1

expf- ζr gY l,m ,

ð2:40Þ

where N is a normalization factor, n* and ζ are parameters related to the principal quantum number (n) and the effective nuclear charge, and Yl,m is a spherical harmonic function of the polar coordinates θ and ϕ. The Slater 2pz orbitals of carbon, nitrogen, and oxygen, for example, take the form (ζ 5/π)1/2rcos(θ)exp.(-ζr) with ζ = 3.071, 3.685, and 4.299 Å-1, respectively. Gaussian functions also are commonly used because, although they differ substantially in shape from the hydrogen-atom wavefunctions, they have particularly convenient mathematical properties. For exam→ → ple, the product of two Gaussians centered at positions r 1 and r 2 is another Gaussian centered midway between these points. Current programs use linear combinations of three or more Gaussians to replace each Slater-type orbital. See [37] for further information on these and other semiempirical orbitals. Figure 2.10 shows contour plots of calculated amplitudes of the wavefunctions for the HOMO and LUMO of 3-methylindole, which is a good model of the sidechain of tryptophan. Note that the HOMO has two nodal curves in the plane of the drawing, while the LUMO has three; the LUMO thus has less bonding character. The wavefunction for a system comprised of several independent components often can be approximated as a product of the wavefunctions of the components. Thus a wavefunction for a molecule can be written, to a first approximation, as a product of electronic and nuclear wavefunctions: Ψðr, RÞ ≈ ψ ðrÞχ ðRÞ,

ð2:41Þ

where r and R are, respectively, the electronic and nuclear coordinates. In this approximation, the total energy of the system is simply the sum of the energies of the electronic and nuclear wavefunctions. Similarly, a wavefunction for a molecule with N electrons can be approximated as a product of N one-electron wavefunctions.

2.3.6

Wavefunctions for Large Systems

Extending the ideas described in the previous section, linear combinations of molecular wavefunctions can be used to generate approximate wavefunctions for systems containing more than one molecule. For example, a wavefunction representing an excited state of an oligonucleotide can be described as a linear combination of wavefunctions for the excited states of the individual nucleotides. The basis wavefunctions used in such constructions are said to be diabatic, which means that they are not eigenfunctions of the complete Hamiltonian; they do not consider all the intermolecular interactions that contribute to the energy of the actual system. A measurement of the energy must give one of the eigenvalues associated

2.3 Spatial Wavefunctions

67

6 HOMO

4 y / ao

2

LUMO

N

N

0 -2 -4 -6

-6 -4 -2 0 2 x / ao

4

6

-6 -4 -2

0 2 x / ao

4

6

Fig. 2.10 Contour plots of the wavefunction amplitudes for the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) of 3-methylindole. Positive amplitudes are indicated by blue lines, negative amplitudes by red lines, and zero by dot-dashed purple lines. The plane of the map is parallel to the plane of the indole ring and is above the ring by ao as in Fig. 2.7, panels D and E. The contour intervals for the amplitude are 0.05ao3/2. Small contributions from the carbon and hydrogen atoms of the methyl group are neglected. The dark green lines represent the carbon and nitrogen skeleton of the molecule. The atomic coefficients for the molecular orbitals were obtained as described by Callis [45–47]. Slater-type atomic orbitals (Eq. (2.40)) with ζ = 3.071/Å (1.625/ao) and 3.685/Å (1.949/ao) were used to represent C and N, respectively

with the adiabatic wavefunctions of the full Hamiltonian, which usually will not be an eigenvalue of the Hamiltonian for any one of the basis functions. The energies of the full system are, however, given approximately by Ek ≈

N X n=1

jCkn j2 En ,

ð2:42Þ

where En is the energy of basis function ψ n representing molecule n, Ckn is the coefficient for ψ n in state k of the larger system (Eq. (2.36)), and the sum runs over all N molecules. The accuracy of this approximation depends on the choice of the basis functions, the number of terms that are included in the sum, and the reliability of the coefficients. Although the individual basis wavefunctions are not eigenfunctions of the full Hamiltonian, it is possible in principle to find linear combinations of these wavefunctions that do give such eigenfunctions, at least to the extent that the basis functions are a complete, orthonormal set. Equation (2.42) then becomes exact. The coefficients and eigenvalues are obtained by solving the simultaneous linear equations

68

2

Basic Concepts of Quantum Mechanics

Ck1 H 11 þ C k2 H 12 þ C k3 H 13 þ ⋯ þ C kn H 1n = C k1 E k Ck1 H 21 þ C k2 H 22 þ C k3 H 23 þ ⋯ þ C kn H 2n = C k2 E k Ck1 H 31 þ C k2 H 32 þ C k3 H 33 þ ⋯ þ C kn H 3n = C k3 E k ⋮ Ck1 H n1

þ

C k2 H n2

þ

C k3 H n3

þ ⋯ þ C kn H nn = C kn E k

ð2:43Þ

D E e k , and H e is the complete Hamiltonian of the system, including Here H jk = ψ j jHjψ e kk) and terms that terms that act on the basis functions for the individual molecules (H e “couple” or “mix” two basis functions (Hjk with j ≠ k). We’ll go through the derivation of Eqs. (2.43) for a system of two molecules in Chap. 8. Equations (2.43) can be written compactly as a matrix equation by using the rules for matrix multiplication (Appendix A2): H  Ck = E k Ck ,

ð2:44Þ

where H denotes a matrix of the Hamiltonian integrals (Hjk), 2

H 11

6H 6 21 H=6 4 H n1

⋯ H 1n

H 12

3

H 22 ⋮

⋯ H 2n 7 7 7, 5

H n2

⋯ H nn

and Ck is a column vector of coefficients: 2

C k1

ð2:45aÞ

3

6 Ck 7 6 7 C k = 6 2 7: 4⋮5

ð2:45bÞ

C kn In general, there will be n eigenvalues of the energy (Ek) that satisfy Eq. (2.44), each with its own eigenvector Ck. The eigenvectors can be arranged in a square matrix C, in which each column corresponds to a particular eigenvalue: 2 1 3 2 n C1

1 C = 4 C2

C 1n

C1 C 22 ⋮ C 2n

⋯ ⋯

C1 C n2

⋯

C nn

5:

ð2:46aÞ

If we then find C-1 (the inverse of C), the product C-1HC turns out to be a diagonal matrix with the eigenvalues on the diagonal (see Appendix A.2):

2.4 Spin Wavefunctions and Singlet and Triplet States

2

E1 6 0 6 C-1  H  C = 6 4 0

69

3 0 0 7 7 7: 5

0 E2

⋯ ⋯

⋮ 0

⋯ En

ð2:46bÞ

The problem of finding the eigenvalues and eigenfunctions therefore is to diagonalize the Hamiltonian matrix H by finding another matrix C and its inverse such that the product C-1HC is diagonal. The diagonal elements of C-1HC are the eigenvalues of the adiabatic states, and column k of C is the set of coefficients corresponding to the eigenvalue Ek. Computer programs for diagonalizing even large matrices are available [48] and are parts of all contemporary quantum mechanical packages. Before the advent of digital computers, the problem often was solved by evaluating a determinant formed from the Hamiltonian as we describe in Chap. 8. The Hamiltonian matrix is always Hermitian, and for all the cases that will concern us is symmetric (Appendix A.2). Its eigenvectors (Ck) are, therefore, always real. In addition, there is always an orthonormal set of eigenvectors (CiCj = 0 for i ≠ j, and CiCi = 1) [48]. Density functional theory provides an alternative way of dealing with multielectron systems. In this approach, the problem of n interacting electrons moving in a static potential field from many nuclei is replaced by n expressions similar to the one-electron Schrödinger equation for an effective potential that incorporates Coulombic, exchange, and correlation interactions among the electrons in addition to the external potential. An iterative procedure for finding a self-consistent effective potential was developed by Walter Kohn and Leu Ju Sham in 1965 [49] and has become impressively accurate and efficient [35, 41, 42].

2.4

Spin Wavefunctions and Singlet and Triplet States

Electrons, protons, and other nuclei have an intrinsic angular momentum or “spin” that is characterized by two spin quantum numbers, s and ms. The magnitude of the angular momentum is [s (s + 1)]1/2ħ. For an individual electron, s = 1/2, so the angular momentum has a magnitude of (31/2/2)ħ. The component of the angular momentum parallel to a single prescribed axis (z), also is quantized and is given by msħ, where ms = s, s - 1, . . ., -s. For s = 1/2, ms is limited to ±1/2, making the momentum along the z-axis ±ħ/2. The magnitude of the spin and the component of the spin in this direction do not commute with the components of the momentum in orthogonal directions, so these components are indeterminate. The angular momentum vector for an electron therefore can lie anywhere on a cone whose half-angle (θ) with respect to the z-axis is given by cos(θ) = ms/(s(s + 1))1/2 (θ ≈ 54.7°; Fig. 2.11). The two possible values of ms for an electron can be described formally by two spin wavefunctions, α and β, which we can view as “spin up” (ms = +1/2) and “spin down” (ms = -1/2), respectively. These functions are orthogonal and normalized so

70

2

Basic Concepts of Quantum Mechanics

that hαjαi = hβjβi = 1 and hαjβi = hβjαi = 0. The variable of integration here is not a spatial coordinate, but rather a spin variable that represents the orientation of the electron. The notion of electronic spin was first proposed by Uhlenbeck and Goudsmit in 1925 to account for the splitting of some of the lines seen in atomic spectra. They and others showed that the electronic spin also accounted for the anomalous effects of magnetic fields (Zeeman effects) on the spectra of many atoms. However, it was necessary to postulate that the magnetic moment associated with electronic spin is not simply the product of the angular momentum and e/2mc, as is true of orbital magnetic moments, but rather twice this value. The extra factor of 2 is called the Landé g factor. When Dirac [50] reformulated quantum mechanics to be consistent with special relativity, both the intrinsic angular momentum and the anomalous

Fig. 2.11 A vectorial representation of the electronic spins in the four possible spin states of a system with two coupled electrons. Each red arrow represents a spin with angular momentum 31/2ħ/2 constrained to the surface of a cone with half-angle θ = cos-1(±3–1/2) relative to the z-axis (54.7° for spin α and 125.3° for β). In the singlet state (left), the two vectors are antiparallel, giving a total spin quantum number (S) of zero. In the triplet states (right), S = 1 and Ms = 1, 0 or - 1. This requires the two individual spin vectors to be arranged so that their resultant angular momentum (not shown) has magnitude 21/2ħ and lies on a cone with half-angle 45°, 90°, or 135° from z for Ms = 1, 0 or - 1, respectively

2.4 Spin Wavefunctions and Singlet and Triplet States

71

factor of 2 emerged automatically without any ad hoc postulates. Dirac shared the Physics Nobel Prize with Schrödinger in 1933. In a system of two interacting electrons, the total spin (S) is quantized and can be either 1 or 0 depending on whether the individual spins are the same (e.g., both β) or different (one α, the other β). The component of the total spin along a prescribed axis is Msħ with Ms = S, S –1, . . ., -S, i.e., 0 if S = 0 and either 1, 0 or -1 if S = 1. For most organic molecules in their ground state, the HOMO contains two electrons with different antiparallel spins, making both S and Ms zero. But because we cannot tell which electron has spin α and which has β, the electronic wavefunction for the ground state must be written as a combination of expressions representing the two possible assignments: h i ð2:47Þ Ψa = ½ψ h ð1Þψ h ð2Þ 2 - 1=2 αð1Þβð2Þ - 2 - 1=2 αð2Þβð1Þ : Here ψ h denotes a spatial wavefunction that is independent of spin, and the numbers in parentheses are labels for the two electrons; α( j)β(k) means that electron j has spin wavefunction α and electron k has β. Note that the complete wavefunction Ψ a as written in Eq. (2.47) changes sign if the labels of the electrons (1 and 2) are interchanged. W. Pauli pointed out that the wavefunctions of all multielectronic systems have this property. The overall wavefunction invariably is antisymmetric for an interchange of the coordinates (both positional and spin) of any two electrons. This assertion rests on experimental measurements of atomic and molecular absorption spectra: absorption bands predicted on the basis of antisymmetric electronic wavefunctions are seen experimentally, whereas bands predicted on the basis of symmetric electronic wavefunctions are not observed. Its most important implication is the Pauli exclusion principle, which says that a given spatial wavefunction can hold no more than two electrons. This follows if an electron can be described completely by specifying its spatial and spin wavefunction and electrons have only two possible spin wavefunctions (α and β). Now consider an excited state in which either electron 1 or electron 2 is promoted from the HOMO to the LUMO (ψ l). The complete wavefunction for the excited state must incorporate the various possible assignments of spin α or β to the electrons in addition to the two possible ways of assigning the electrons to the two orbitals, and again the wavefunction must change sign if we interchange the coordinates of the electrons. If there is no change in the net spin of the system during the excitation, as is usually the case, the spin part of the wavefunction remains the same as for the ground state so that both S and Ms remain zero. The spatial part is more complicated: h ih i 1 Ψb = 2 - 1=2 ψ h ð1Þψ l ð2Þ þ 2 - 1=2 ψ h ð2Þψ l ð1Þ 2 - 1=2 αð1Þβð2Þ - 2 - 1=2 αð2Þβð1Þ : ð2:48Þ The choice of a + sign for the combination of the spatial wavefunctions in the first braces satisfies the requirement that the overall wavefunction be antisymmetric for

72

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Basic Concepts of Quantum Mechanics

an exchange of the two electrons. If antisymmetric combinations were used in both brackets, the overall product of the spatial and spin wavefunctions would be symmetric, in conflict with experiment. The state described by such a wavefunction is referred to as a singlet state because when the spatial wavefunction is symmetric there is only one possible combination of spin wavefunctions (the one written in Eq. (2.48)). The ground state (Eq. (2.47)) also is a singlet state. Singlet states often are indicated by a superscript “1” as in Eq. (2.48). If, instead, we choose the antisymmetric combination for the spatial part of the wavefunction in the excited state, there are three possible symmetric combinations of spin wavefunctions that make the overall wavefunction antisymmetric: h i 3 þ1 ð2:49aÞ Ψb = 2 - 1=2 ψ h ð1Þψ l ð2Þ - 2 - 1=2 ψ h ð2Þψ l ð1Þ ½αð1Þαð2Þ 3

h ih i Ψ0b = 2 - 1=2 ψ h ð1Þψ l ð2Þ - 2 - 1=2 ψ h ð2Þψ l ð1Þ 2 - 1=2 αð1Þβð2Þ þ 2 - 1=2 αð2Þβð1Þ ð2:49bÞ 3

i h Ψb- 1 = 2 - 1=2 ψ h ð1Þψ l ð2Þ - 2 - 1=2 ψ h ð2Þψ l ð1Þ ½βð1Þβð2Þ:

ð2:49cÞ

These are the three excited triplet states corresponding to S = 1 and Ms = 1, 0, and - 1, respectively. Figure 2.11 shows a vectorial representation of the angular momenta of the two electrons in the singlet and triplet states when the vertical (z) axis is defined by a magnetic field. The vectors are drawn to satisfy the quantization of the spin and the zcomponent of the spin simultaneously for both the individual electrons and the combined system. Although the x and y components of the individual spins are still indeterminate, the angle between the two spins is fixed. In the triplet state with Ms = 0, the resultant vector representing the total spin is in the xy plane; in the states with Ms = ±1, it lies on a cone with a half-angle of 45° or 135° with respect to the z-axis. The triplet state 3 Ψ0b almost invariably has lower energy than the corresponding singlet state, 1Ψ b. This statement, which is often called Hund’s rule, is a consequence of the different spatial wavefunctions, not the spin wavefunctions. The motions of the two electrons are correlated in a way that tends to keep the electrons farther apart in the triplet wavefunction, decreasing their repulsive interactions. The difference between the two energies, called the singlet-triplet splitting, is given by 2Khl, where Khl is the exchange integral:  2   e    K hl = ψ l ð1Þψ h ð2Þ ψ h ð1Þψ l ð2Þ : ð2:50Þ r 12 Khl is always positive [51]. A special situation arises when the two orbitals are equivalent in the sense that they have the same energy and can be transformed into each other by a symmetry operation such as rotation. The triplet state then can be lower in energy than a singlet

2.4 Spin Wavefunctions and Singlet and Triplet States

73

state in which both electrons reside in one of the orbitals. This is the case for O2, for which the ground state is a triplet and the lowest excited state is a singlet. The z-axis that defines the orientation of an electronic spin often is determined uniquely by an external magnetic field. Because of their different magnetic moments, the three triplet states split apart in energy in the presence of a magnetic 3 -1 moving down; 3 Ψ0b is not affected. field, with 3 Ψþ1 b moving up in energy and Ψb Although the triplet states are degenerate in the absence of a magnetic field, orbital motions of the electrons can create local magnetic fields that lift this degeneracy. This zero-field splitting can be measured by imposing an oscillating microwave field to induce transitions between the triplet states. The magnetic dipoles of the three zero-field triplet states generally can be associated with x, y and z structural axes of the molecule. Wavefunctions for more than two electrons can be approximated as linear combinations of all the allowable products of electronic and spin wavefunctions for the individual electrons. Combinations that satisfy the Pauli exclusion principle can be written conveniently as determinants:   ψ a ð1Þαð1Þ   ψ ð1Þβð1Þ  a   1=2  ψ b ð1Þαð1Þ Ψ = ð1=N!Þ   ψ b ð1Þβð1Þ   ⋮   ψ ð1Þβð1Þ N

ψ a ð2Þαð2Þ

ψ a ð3Þαð3Þ

⋯

ψ a ð2Þβð2Þ ψ b ð2Þαð2Þ

ψ a ð3Þβð3Þ ψ b ð3Þαð3Þ

⋯ ⋯

ψ b ð2Þβð2Þ ⋮

ψ b ð3Þβð3Þ ⋮

⋯

ψ N ð2Þβð2Þ

ψ N ð3Þβð3Þ ⋯

 ψ a ðN ÞαðN Þ   ψ a ðN ÞβðN Þ   ψ b ðN ÞαðN Þ  , ð2:51Þ ψ b ðN ÞβðN Þ    ⋮  ψ ðN ÞβðN Þ  N

where the ψ i are the individual one-electron spatial wavefunctions and N is the total number of electrons. Equation (2.47), for two electrons and a single spatial wavefunction, consists of the 2 × 2 block at the top left corner of Eq. (2.51). This formulation guarantees that the overall wavefunction will be antisymmetric for the interchange of any two electrons because the value of a determinant always changes sign if any two columns or rows are interchanged. Such determinants are called Slater determinants after J. C. Slater, who developed the procedure. They often are denoted compactly by omitting the normalization factor (1/N!)1/2, listing only the diagonal terms of the determinant, representing the combinations ψ j(k)α(k) and ψ j(k)β(k) by ψ j(k) and ψ j(k), respectively, and omitting the indices for the electrons: Ψ = jψ a ψ a ψ b ψ b ⋯ψ N j:

ð2:52Þ

In the nomenclature of Eq. (2.48), the Slater determinant for the HOMO and LUMO “spin-orbital” wavefunctions of the ground, excited singlet, and excited triplet states (Eqs. (2.47–2.49)) would be written: Ψa = jψ h ψ h j,

ð2:53aÞ

74

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Basic Concepts of Quantum Mechanics

Ψb = 2 - 1=2 ðjψ h ψ l j þ jψ l ψ h jÞ,

ð2:53bÞ

- 1=2 ðjψ h ψ l j - jψ l ψ h jÞ, Ψþ1 b =2

ð2:53cÞ

Ψ0b = 2 - 1=2 ðjψ h ψ l j - jψ l ψ h jÞ,

ð2:53dÞ

Ψb- 1 = 2 - 1=2 ðjψ h ψ l j - jψ l ψ h jÞ:

ð2:53eÞ

1 3

3

and 3

Transitions between singlet and triplet states will be discussed in Sect. 4.9. Protons resemble electrons in having spin quantum numbers s = 1/2 and ms = ±1/2. Wavefunctions of systems containing multiple protons, or any other particles with half-integer spin (fermions), also resemble electronic wavefunctions in being antisymmetric for interchange of any two identical particles. Wavefunctions for systems containing multiple particles with integer or zero spins (bosons), by contrast, must be symmetric for the interchange of two identical particles. Deuterons, which have spin quantum numbers s = 1 and ms = -1, 0 of 1, fall in this second group. Because interchanging bosons leaves their combined wavefunction unchanged, any number of bosons can occupy the same wavefunction. Fermions and bosons obey different statistics that become increasingly distinct at low temperatures (Box 2.6). Box 2.6 Boltzmann, Fermi-Dirac and Bose-Einstein Statistics As Eqs. (2.47–2.52) illustrate, wavefunctions of a system comprised of multiple non-interacting components can be written as linear combinations of products of the form Ψ = Ψa ð1ÞΨb ð2ÞΨc ð3Þ⋯:

ðB2:6:1Þ

The Boltzmann distribution law says that, if the individual components are distinguishable and the number of components is very large, the probability of finding a given component in a state with wavefunction Ψ m at temperature T is Pm = Z - 1 gm exp½ - E m =k B T :

ðB2:6:2Þ

Here Em is the energy of state m, gm is the degeneracy, or multiplicity, of the state (the number of substates with the same energy), kB is the Boltzmann constant (1.3806 × 10-23 m2kgs-2K-1 = 1.3806 × 10-16 ergK-1 = 8.617 × 10-5 eVK-1), and Z (the partition function) is a temperaturedependent factor that normalizes the sum of the probabilities over all the states: (continued)

2.4 Spin Wavefunctions and Singlet and Triplet States

75

Box 2.6 (continued) Z=

X gm exp½ - E m =kB T :

ðB2:6:3Þ

m

Electronic states with the same spatial wavefunction but different spins (α or β) can be described as having a multiplicity (gm) of 2 if the energy difference between the two spin sublevels is negligible. Alternatively, the spin states can be enumerated separately, each with gm = 1. Writing combined wavefunctions of the form of Eq. (B2.6.1) becomes problematic if the individual components of the system are indistinguishable. As we have discussed, wavefunctions that are symmetric for the interchange of two identical particles are not available to fermions, while wavefunctions that are antisymmetric for such interchanges are not available to bosons. One consequence of this is that systems of fermions or bosons follow different distribution laws. Fermions obey the Fermi-Dirac distribution, Pm = fAF gm exp½E m =k B T  þ N g - 1 ,

ðB2:6:4Þ

where N is the number of particles and AF is defined to make the total probability 1. Bosons follow the Bose-Einstein distribution, Pm = fAB gm exp½Em =kB T  - N g - 1 ,

ðB2:6:5Þ

with AB again defined to give a total probability of 1. The Boltzmann, FermiDirac and Bose-Einstein distribution laws can be derived by defining the entropy of a distribution as S = kBlnΩ, where Ω is the number of different ways that the particles could be assigned to various states, and then maximizing the entropy while keeping the total energy of the system constant. The formulas for Ω depend on whether the particles are distinguishable and on whether or not more than one particle can occupy the same state (Sect. 4.15 and Fig. 4.31). Figure 2.12 illustrates Boltzmann, Fermi-Dirac, and Bose-Einstein distributions of four particles among five states with equally spaced energies. Each state is assumed to have a multiplicity of 2. The populations of the five states are plotted as functions of kBT/E, where E is the energy difference between adjacent states. The populations all converge on 0.2 at high temperatures, where a given particle has nearly equal probabilities of being in any of the five states. However, the three distributions differ notably at low temperatures. Here the Fermi-Dirac distribution puts two particles in each of the states with the two lowest energies, making the probability of finding a given particle in a given one of these states 0.5 (Fig. 2.12B). This (continued)

76

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Basic Concepts of Quantum Mechanics

Box 2.6 (continued) corresponds to the Pauli exclusion principle that a spatial wavefunction can hold no more than two electrons, which must have different spins. The Boltzmann and Bose-Einstein distributions, in contrast, both put all four particles in the state with the lowest energy (Fig. 2.12A, C). The BoseEinstein distribution differs from the Boltzmann distribution in changing more gradually with temperature. The Bose-Einstein distribution law was derived by S. N. Bose in 1924 to describe a photon gas. Einstein extended it to material gasses. Fermi developed the Fermi-Dirac distribution law in 1926 by exploring the Pauli exclusion principle, and Dirac obtained it independently in the same year by considering antisymmetric wavefunctions.

Population

1.0

B

A

0.8 Fermi-Dirac

Boltzmann

0.6 0.4 0.2 0

0

Population

1.0

C

2

4 6 k BT / E

8

10

0.8 Bose-Einstein

0.6 0.4 0.2 0

0

2

6 4 kBT / E

8

10

Fig. 2.12 Temperature dependence of the Boltzmann (A), Fermi-Dirac (B), and Bose-Einstein (C) distributions of four particles among five states with energies of 0, E, 2E, 3E, and 4E in arbitrary energy units. Each state is assumed to have a multiplicity of 2. The populations of the five states are plotted as functions of kBT/E. In each panel, the upper-most (heavy red) curve is the population in the state with the lowest energy, and the bottom (dark blue) curve is that in the state with the highest energy

2.5 Transitions Between States: Time-Dependent Perturbation Theory

2.5

77

Transitions Between States: Time-Dependent Perturbation Theory

The wavefunctions obtained by solving the time-independent Schrödinger equation (Eq. (2.14)) describe stationary states. A system that is placed in one of these states e=H e o, because the energy of the system evidently will stay there forever as long as H e is independent of time. But suppose H changes with time. We could, for example, switch on an electric field or bring two molecules together so that they interact. This will perturb the system so that the original solutions to the Schrödinger equation are no longer entirely valid. e is relatively small, we can write the total Hamiltonian of the If the change in H eo and a smaller, time-dependent perturbed system as a sum of the time-independent H e term, H′(t): e0 ðt Þ: e =H eo þ H H

ð2:54Þ

To find the wavefunction of the perturbed system, let’s express it as a linear combination of the eigenfunctions of the unperturbed system: Ψ = C a Ψa þ C b Ψ b þ ⋯ :

ð2:55Þ

where the coefficients Ck are functions of time. The value of jCkj2 at a given time represents the extent to which Ψ resembles the wavefunction of basis state k (Ψ k). This approach makes use of the fact that the original eigenfunctions form a complete set of functions, as discussed in Sect. 2.2.1. Suppose we know that the molecule is in state Ψ a before we introduce the e 0 . How rapidly does the wavefunction begin to resemble that of perturbation H some other basis state, say Ψ b? The answer should lie in the time-dependent Schrödinger equation (Eq. (2.9)). Using Eqs. (2.54) and (2.55), we can expand the left-hand side of the Schrödinger equation to: h i e 0 ðt Þ ½Ca ðt ÞΨa þ C b ðt ÞΨb þ ⋯ eo þ H H 0

e ½C a Ψa þ Cb Ψb þ ⋯ e o ½ C a Ψa þ C b Ψb þ ⋯  þ H ¼H e o Ψa þ C b H e o Ψb þ ⋯ þ C a H e 0 Ψa þ C b H e 0 Ψb þ ⋯: ¼ Ca H

ð2:56Þ

The right-hand side of the Schrödinger equation can be expanded similarly to iħ½Ψa ∂Ca =∂t þ Ψb ∂Cb =∂t þ ⋯ þ C a ∂Ψa =∂t þ C b ∂Ψb =∂t þ ⋯:

ð2:57Þ

e CkΨ = CkH e Ψ , which means that the operator H e and We have assumed that H multiplication by Ck commute; the result of performing the two operations is independent of the order in which they are performed. As discussed in Box 2.2, this assumption must be valid if the energy of the system and the values of the coefficients can be known simultaneously.

78

2

Basic Concepts of Quantum Mechanics

eoΨ a = iħ∂Ψ a/∂t and H eoΨ b = iħ∂Ψ b/ For the unperturbed system, we know that H ∂t, because each eigenfunction satisfied the Schrödinger equation before we e Canceling the corresponding terms on opposite sides of the Schrödinger changed H. eoΨ a from Eq. (2.56) and Caiħ∂Ψ a/∂t from Eq. (2.57)) equation (e.g., subtracting CaH leaves: e 0 Ψa þ Cb H0 Ψb þ ⋯ = iħ½Ψa ∂C a =∂t þ Ψb ∂C b =∂t þ ⋯: Ca H

ð2:58Þ

We can simplify this equation by multiplying each term by Ψ b* and integrating over all space because this allows us to use the orthogonality relationships (Eqs. (2.19–2.20)) to set many of the integrals to 0 or 1: D E D E e 0 jΨa þ Cb Ψb jH e 0 jΨb þ ⋯ Ca Ψb jH = iħ½hΨb jΨa i∂C a =∂t þ hΨb jΨb i∂C b =∂t þ ⋯ = iħ∂C b =∂t:

ð2:59Þ

If we know that the system is in the state a at a particular time, then Ca must be 1, and Cb and all the other coefficients must be zero. So all but one of the terms on the lefte ai = iħ∂Cb/∂t, or hand side of Eq. (2.59) drop out. This leaves us with hΨ bjH′jΨ D E D E e 0 jΨa = ð- i=ħÞ Ψb jH e 0 jΨa : ð2:60Þ ∂C b =∂t = ð1=iħÞ Ψb jH Equation (2.60) tells us how coefficient Cb increases with time at early times when there is still a high probability that the system is still in state a. But the wavefunctions Ψ a and Ψ b* in the equation are themselves functions of time. From the general solution to the time-dependent Schrödinger equation (Eq. (2.16)) we can separate the spatial and time-dependent parts of these wavefunctions as follows:   → ð2:61aÞ Ψa = ψ a r expð- iE a t=ħÞ and   → Ψb  = ψ b  r expðiE b t=ħÞ:

ð2:61bÞ

Inserting these relationships into Eq. (2.60) yields the following result for the growth of Cb with time: D E e 0 jψ a ∂Cb =∂t = -ði=ħÞ expðiE b t=ħÞ expð- iE a t=ħÞ ψ b jH D E e 0 jψ a = - ði=ħÞ exp½iðE b - Ea Þt=ħ ψ b jH = - ði=ħÞ exp½iðE b - Ea Þt=ħH 0ba :

ð2:62Þ

2.5 Transitions Between States: Time-Dependent Perturbation Theory

79

Equation (2.62) factors ∂Cb/∂t into an oscillatory component that hinges on the difference between the energies of states a and b (exp[i(Eb – Ea)t/ħ]) and an integral e and the spatial wavefunctions that depends on the time-dependent perturbation (H′) 0 e e The ψ a and ψ b. The integral hψ bjH′jψ ai, or H ba , is called a matrix element of H′. terminology is the same as that used in connection with Eq. (2.45a), except that here e ′ is just the perturbation term in the Hamiltonian rather than the complete H D E e 0 jψ j , could be written to Hamiltonian A similar matrix element, H 0kj = ψ k jH describe the build-up of the coefficient Ck for any other state of the system. To obtain the value of Cb after a short interval of time, τ, we need to integrate Eq. (2.62) from 0 to τ: Zτ C b ðτ Þ =

ð∂C b =∂t Þdt:

ð2:63Þ

0

During the time that the perturbation is applied, the system cannot be said to be in either state a or state b because these are no longer eigenstates of the Hamiltonian. So what physical interpretation should we place on the coefficients Ca and Cb for these states in Eq. (2.56)? Suppose that at time τ we perform a measurement that has different expectation values for states a and b in the unperturbed system. If the e , the expectation value for operator corresponding to the measurement is A observations on the perturbed system will be * + D E D E XX X X e j Ck ψ k = e jψ k e jΨ = C j ψ j jA C j C k ψ j jA A = ΨjA j

k

j

k

XX

 XX

 X = C j C k ψ j jAk ψ k = C j C k Ak ψ j jψ k = jCk j2 Ak , j

k

j

k

ð2:64Þ

k

where Ak is the expectation value for observations on an unperturbed system in state k. Equation (2.64) is a generalization of Eq. (2.42), which gives the expectation value of the energy. The magnitude of jCb(τ)j2 thus tells us the extent to which an arbitrary measurement on the perturbed system at time τ will resemble Ab, the result of the same measurement on a system known to be in state b. Now suppose we turn off the perturbation abruptly after we make a measurement on the perturbed system. With the perturbation removed, the basis states again become eigenstates of the system, and the evolution of the coefficients Ca and Cb comes to a halt. It thus seems reasonable to view jCb(τ)j2 as the statistical probability that the system has evolved into state b at time τ. If this probability increases linearly with time for small values of τ so that jCb(τ)j2 = κτ where κ is a constant, we can identify κ as the rate constant for the transition. To evaluate ∂Cb/∂t for any particular case, we need to specify the timee 0 more explicitly. We’ll do this in Chap. 4 for an oscillating dependence of H electromagnetic field. In Chaps. 5–8 we’ll consider several types of perturbations

80

2

Basic Concepts of Quantum Mechanics

introduced by bringing two molecules together so that they interact, and in Chaps. 10 and 11 we’ll consider the effects of randomly fluctuating interactions between a e 0 we can see system and its surroundings. But even without going into the nature of H that if the energies of the reactant and product states (Ea and Eb) are very different, the factor exp[i(Eb - Ea)t/ħ] in Eq. (2.62) will oscillate rapidly with time and will average to zero. On the other hand, if the two energies are the same so that the argument of the exponent is zero, and if H′ba is constant, Cb will increase linearly with time. Transitions from state a to state b therefore occur at a significant rate only if Eb is the same as or close to Ea. This is the condition for resonance between the two states and is the quantum mechanical expression of the classical principle that the transition must conserve energy. For transitions that involve absorption or emission of light, the energy of one of the states includes the energy of the photon that is absorbed or emitted. The general DconclusionE that transitions from state a to state b depend on the e 0 jψ a merits a few additional comments. Note that H′ba is an matrix element ψ b jH off-diagonal matrix element of the time-dependent perturbation to the Hamiltonian. In Sect. 2.3.6, we discussed how linear combinations of basis wavefunctions can be used to construct wavefunctions for more complex systems. Finding the sets of coefficients that give eigenfunctions of the complete Hamiltonian, and so give stationary states of the system, requires diagonalizing the Hamiltonian. A timedependent perturbation thus can drive transitions between diabatic states a and b, which are non-stationary in the presence of the perturbation, but it cannot drive transitions between the linear combinations of these states that make the Hamiltonian diagonal.

2.6

Lifetimes of States and the Uncertainty Principle

As discussed in Sect. 2.1.2, the position and momentum operators do not commute; the combined action of the two operators gives different results, depending on which operator is used first: p rψ = iħψ: ½er, e pψ =ee r pψ - ee

ð2:65Þ

With some algebra, it follows from Eq. (2.65) that the product of the uncertainties (root-mean-square deviations) in the expectation values for position and momentum must be ≥ħ/2 [4]. This is a statement of Heisenberg’s uncertainty principle. The potential energy of a particle can be specified precisely as a function of e also includes a term for kinetic position. However, the Hamiltonian operator H e does not commute with er. energy. Because kinetic energy depends on momentum, H We therefore cannot specify both the energy and the position of a particle simultaneously with arbitrary precision. Similarly, because the dipole moment of a molecule depends on the positions of all the electrons and nuclei, we cannot specify the dipole moment together with the energy to arbitrary precision. If we know that a system is

2.6 Lifetimes of States and the Uncertainty Principle

81

in a state with a particular energy, a measurement of the dipole moment will give a real result, but we cannot be sure exactly what result will be obtained on any given measurement. However, the average result of many such measurements is given by the expectation value, which is an integral over all possible positions. The expectation values of both the energy and the dipole moment can, therefore, be stated precisely, at least in principle (Box 2.2). There is no uncertainty principle comparable to the one for momentum and position that links the energy of a state with the state’s lifetime. Indeed, there is no quantum mechanical operator for the lifetime of a state. There is, nevertheless, a relationship between the lifetime and our ability to assign the state a definite energy. One way to view this relationship is to recall that the full wavefunction for a system with energy Ea is an oscillating function of time and that the oscillation frequency is proportional to the energy (Eq. (2.16)): Ψðr, t Þ = ψðrÞ expð- iE a t=ħÞ:

ð2:66Þ

According to this expression, if Ea is constant the probability of finding the system in the state is independent of time (P = Ψ *Ψ = ψ*ψ). Conversely, if a system remains in one state indefinitely we can specify its oscillation frequency (Ea/h), and thus its energy, with arbitrarily high precision. But if the particle can make a transition to another state the probability density for the initial state clearly must decrease with time. Suppose the probability of finding the system in the initial state decays to zero by first-order kinetics with a time constant T: Pðt Þ = hΨðr, t ÞjΨðr, t Þi = hΨðr, 0ÞjΨðr, 0Þi expð- t=T Þ,

ð2:67Þ

where Ψ (r,0) is the amplitude of the wavefunction at zero time. Equations (2.66) and (2.67) then require the wavefunction to be an oscillatory function that decreases in amplitude with a time constant of 2 T: Ψðr, t Þ = ψ ðrÞ expð- iE a t=ħÞ expð- t=2T Þ = ψ ðrÞ expf- ½ðiE a =ħÞ þ ð1=2T Þt g:

ð2:68Þ

The time dependence of such a function is illustrated in Fig. 2.13. We can equate the time-dependent function in Eq. (2.68) to a superposition of many oscillating functions, all of the form exp.(-Et/ħ) but with a range of energies: Z1 expf- ½ðiEa =ħÞ þ ð1=2T Þ t g =

GðEÞ expð- iE t=ħÞ dE:

ð2:69Þ

-1

Inspection of Eq. (2.69) shows that the distribution function G(E) is the Fourier transform of the time-dependent part of Ψ (see Appendix A3). In general, for such an equality to hold, G(E) must be a complex quantity. The real part of G(E)dE, Re [G(E)]dE, can be interpreted as the probability that the energy of the system is in the interval between E - dE/2 and E + dE/2, which can be normalized so that

82

2

Basic Concepts of Quantum Mechanics

1.0

Re[ψ ]

0.5 0.0 -0.5 -1.0

0

1

2

3

4

5

Time / h/Ea Fig. 2.13 Wavefunctions for particles with different lifetimes. The green dotted curve is the real part of an undamped wavefunction ψ oexp(-iEat/ħ) with ψ o = 1; this represents a particle with an infinite lifetime. The energy (Ea) is defined precisely. The cyan solid curve is the real part of the wavefunction ψ oexp(-iEat/ħ)exp.(-t/2 T) with a damping time constant 2 T that here is set equal to 2 h/Ea; this represents a particle with an energy of Ea but a finite lifetime of T = h/Ea

Z1 Re ½GðE Þ dE = 1:

ð2:70Þ

-1

The imaginary part of the Fourier transform, Im[G(E)], relates to the phases of the different oscillation frequencies. The phases must be such that the oscillations interfere constructively at t = 0, where jΨ j is maximal. As time increases, the interference must become predominantly destructive so that jΨ j decays to zero. The solution to Eq. (2.69) is that Re[G(E)] is a Lorentzian function:   ħ=2T 1 : ð2:71Þ Re ½GðEÞ = π ðE - E a Þ2 þ ðħ=2T Þ2 This function peaks at E = Ea, but has broad wings stretching out on either side (Fig. 2.14A). It falls to half its maximum amplitude when E = Ea ± ħ/2T, and its full width at the half-maximal points (FWHM) is ħ/T. (ħ is 5.308 × 10-12 cm-1 s, where 1 cm-1 = 1.240 × 10-4 eV = 2.844 cal/mol.) A Lorentzian has wider wings than a Gaussian with the same integrated area and FWHM (Fig. 2.14B). If we interpret the FWHM of the Lorentzian as representing an uncertainty δE in the energy caused by the finite lifetime of the state, we see that δE ≈ ħ=T:

ð2:72Þ

This uncertainty, or lifetime broadening, puts a lower limit on the width of an absorption line associated with exciting a molecule into a transient state.

2.7 Questions

83

1.0 Re[G] X 100

Relative Amplitude

A

1.0

T = 100 fs

0.5 50 fs 20 fs

0 -400

-200

0

200

(E - Ea) / cm

-1

400

B

0.5

0 -10

-5

0 x/a

5

10

Fig. 2.14 (A) Distribution of energies around the mean energy (Ea) of a system that decays exponentially with a time constant (T ) of 100, 50, or 20 fs (blue, cyan, and red curves, respectively). The Lorentzian distribution function (Eq. (2.71)) is normalized to keep the area under the curve constant. (B) Comparison of Lorentzian (cyan solid curve) and Gaussian (green dotted curve) functions with the same integrated area and the same full width at half-maximum amplitude (FWHM). The Gaussian a-1(2π)-1/2exp(-x2/2a2) has a peak height of a-1(2π)-1/2 and a FWHM of (8ln2)1/2a. The Lorentzian π -1a/(x2 + a2) has a peak height of a-1π -1 and a FWHM of 2a

Equation (2.71) is used to describe the shapes of magnetic resonance absorption lines in terms of the “transverse” relaxation time constant (T2). We’ll use it similarly in Chaps. 4 and 10 to describe the shapes of optical absorption bands

2.7

Questions

1. (a) What did the double-slit experiment by Thomas Young in 1802 show? (b) How does Young’s experiment relate to Davisson and Germer‘s experiments in the years 1923–27? (c) What do these experiments have in common and what was surprising at first? 2. Let’s look at a baseball with the mindset of Louis de Broglie and assign it some wave properties. Given a baseball’s weight of 150 g and a pitch speed of 80 mph, what is the wavelength of its particle-wave? 3. A plot of the radial charge density distribution, 4πr2(Rn,l)2, against the distance from the atomic nucleus describes the probability of finding an electron in a thin spherical shell of radius r and thickness dr, which has volume 4πr2dr. Using the information in Table 2.1, calculate the peak of the probability distribution for the 1 s atomic orbital of a Be3+ ion. Also, determine the eigenvalue of this orbital. 4. (a) Let ψ1 be the complete, normalized wavefunction of an enzyme, and ψ2 the complete, normalized wavefunction of the substrate. What wavefunction would you use as a first approximation for the enzyme-substrate complex? Your wavefunction should be consistent with the fact that the enzyme and the

84

2

Basic Concepts of Quantum Mechanics

substrate both exist simultaneously as well as individually, and should be normalized. (b) Why is your wavefunction for the combined system only an approximation? 5. Given a complete set of orthonormalized eigenfunctions of the Hamiltonian describe any arbitrary wavefunction as a operator, ψ1, ψ2, . . ., ψn, you could P linear combination of the form Ψ = Ci ψ i . (a) Show that Ψ is normalized if i P  C i C i = 1. (b) When might it be appropriate to describe a system by such a i

linear combination? (c) Why would this be a poor choice for the enzymesubstrate complex considered in problem 1? 6. Using the treatment described in question 5, you find that only two eigenfunctions (ψ1 and ψ2) make significant contributions to Ψ. Suppose the system is twice as likely to be in state ψ1 as to be in ψ2. (a) Find the values of C1 and C2, assuming that both values are real. (b) If the energies of ψ1 and ψ2 are E1 and E2, what is the energy of Ψ? 7. (a) Consider two spatial wavefunctions for a free, one-dimensional particle of  pffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffi  mass m, ψ 1 = A exp i 2mE 1 x=ħ and ψ 2 = A exp i 2mE 2 x=ħ . Show that the momentum operator (e p = ‐iħ∂=∂x) conforms to the relationship pjψh1 i = i hiψ 2 je h  e = - A, e B e A e hψ 1 je pjψ 2 i for these wavefunctions. (b) The relationship B, e e for any two hoperators i A and B follows simply from the definition of the e B e (Box 2.2). Show that it holds in particular for the commutator A,

8.

9.

10.

11.

one-dimensional position and momentum operators by evaluating ½e x, e pψ and ½e p, e xψ explicitly for an arbitrary wavefunction ψ. Assuming that the energies of the atomic orbitals of carbon increase in the same order as those of hydrogen and that Hund’s rule holds, what are the electron configurations of (a) the ground state and (b) the first excited state of atomic carbon? Find the expectation values of the position of an electron in the first two non-trivial eigenstates (n = 1 and 2) of a particle in a one-dimensional rectangular box with infinitely high walls. (a) Write a Slater determinant for a singlet-state wavefunction of a system of four electrons. (b) Expand the determinant to write out the combination of spatial and spin wavefunctions that it represents. (c) Using whichever of the two representations you prefer, show that the wavefunction is antisymmetric for the interchange of two electrons. Consider two eigenstates of a one-dimensional system, with singlet wavefunctions ψ a(x,t) and ψ b(x,t). Show that, according to first-order perturbae 0 to the Hamiltonian can cause transitions between tion theory, a perturbation H

e 0 is a function of position (x). the two states only if H 12. The absorption spectra of naphthalene’s radical anion and radical cation are almost identical. Why is this? Would the same situation hold for pentacene?

References

85

References 1. Dirac, P.M.: The Principles of Quantum Mechanics. Oxford University Press, Oxford (1930) 2. Pauling, L., Wilson, E.B.: Introduction to quantum mechanics. McGraw-Hill, New York (1935) 3. van der Waerden, B.L. (ed.): Sources of Quantum Mechanics. Dover, New York (1968) 4. Atkins, P.W.: Molecular Quantum Mechanics, 2nd edn. Oxford Univ. Press, Oxford (1983) 5. Levine, I.N.: Quantum Chemistry. Prentice-Hall, Englewood Cliffs, NJ (2000) 6. Szabo, A., Ostlund, N.S.: Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory. Macmillan, New York (1982) 7. Jensen, F.: Introduction to Computational Chemistry. Wiley, New York (1999) 8. Simons, J., Nichols, J.: Quantum Mechanics in Chemistry. Oxford University Press, New York (1997) 9. Engel, T.: Quantum Chemistry and Spectroscopy. Benjamin Cummings, San Francisco (2006) 10. McQuarrie, D.A.: Quantum Chemistry. University Science Books, Sausalito, CA (2008) 11. Atkins, P.W.: Quanta: a Handbook of Concepts, p. 434. Oxford University Press, Oxford (1991) 12. Born, M.: The quantum mechanics of the impact process. Z. Phys. 37, 863–867 (1926) 13. Reichenbach, H.: Philosophic foundations of quantum mechanics, p. 182. University of California Press, Berkeley & Los Angeles (1944) 14. Pais, A.: Max Born’s statistical interpretation of quantum mechanics. Science. 218, 1193–1198 (1982) 15. Jammer, M.: The Philosophy of Quantum Mechanics: the Interpretation of Quantum Mechanics in Historical Perspective. Wiley, New York (1974) 16. Einstein, A.: Uber einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Ann. der Phys. 17, 132–146 (1905) 17. de Broglie, L.: Radiations - ondes et quanta. Comptes rendus. 177, 507–510 (1923) 18. Schrödinger, E.: Quantisierung als eigenwertproblem. Ann. der Phys. 79, 489–527 (1926) 19. Schrödinger, E.: Collected Papers on Wave Mechanics. Blackie & Son, London (1928) 20. Jammer, M.: The Conceptual Development of Quantum Mechanics. McGraw-Hill, New York (1966) 21. Morse, P.M.: Diatomic molecules according to the wave mechanics. II. Vibrational levels. Phys. Rev. 34, 57–64 (1929) 22. ter Haar, D.: The vibrational levels of an anharmonic oscillator. Phys. Rev. 70, 222–223 (1946) 23. Marton, L., Simpson, J.A., Suddeth, J.A.: Electron beam interferometer. Phys. Rev. 90, 490–491 (1953) 24. Carnal, O., Mlynek, J.: Young’s double-slit experiment with atoms: a simple atom interferometer. Phys. Rev. Lett. 66, 2689–2692 (1991) 25. Monroe, C., Meekhof, D.M., King, B.E., Wineland, D.J.: A “Schrödinger cat” superposition state of an atom. Science. 272, 1131–1136 (1996) 26. Hückel, E.: Quantentheoretische Beiträge zum Benzolproblem. Z. Phys. 70, 204–286 (1931) 27. Coulson, C.A., O’Leary, B., Mallion, R.B.: Hückel Theory for Organic Chemists. Academic Press, New York (1978) 28. Coulson, C.A., Rushbrook, G.S.: Note on the method of molecular orbitals. Proc. Cambridge Phil. Soc. 36, 193–200 (1940) 29. Mallion, R.B., Rouvray, D.H.: The golden jubilee of the Coulson-Rushbrooke pairing theorem. J. Math. Chem. 5, 1–21 (1990) 30. Coulson, C.A.: The electronic structure of some polyenes and aromatic molecules. VII. Bonds of fractional order by the molecular orbital method. Proc. Roy. Soc. Lond. Ser. A. 169, 413–428 (1939) 31. Dougherty, R.C.: A perturbation molecular orbital treatment of photochemical reactivity. The nonconservation of orbital symmetry in photochemical pericyclic reactions. J. Am. Chem. Soc. 93, 7187–7201 (1971)

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32. Murfin, L.C., Lewis, S.E.: Azulene - a bright core for sensing and imaging. Molecules. 26, 253/ 1-19 (2021) 33. Pople, J.A., Beveridge, D.L.: Approximate Molecular Orbital Theory. McGraw-Hill, New York (1970) 34. Pople, J.A.: Nobel lecture: quantum chemical models. Rev. Mod. Phys. 71, 1267–1274 (1999) 35. Parr, R.G., Yang, W.: Density-functional Theory of Atoms and Molecules. Oxford Univ. Press, New York (1989) 36. Ayscough, P.B.: Library of physical chemistry software, vol. 2. Oxford University Press & W. H. Freeman, New York (1990) 37. McGlynn, S.P., Vanquickenborne, L.C., Kinoshita, M., Carroll, D.G.: Introduction to Applied Quantum Chemistry. Holt, Reinhardt & Winston, New York (1972) 38. Kong, J., White, C.A., Krylov, A., Sherrill, D., et al.: Q-chem 2.0: A high-performance ab initio electronic structure program package. J. Comput. Chem. 21, 1532–1548 (2000) 39. Angeli, C.: DALTON, a molecular electronic structure program, Release 2.0 (2005). See http:// www.kjemi.uio.no/software/dalton/dalton.html. (2005) 40. Barca, G.M.J., Bertoni, C., Carrington, L., Datta, D., et al.: Recent developments in the general atomic and molecular electronic structure system. J. Chem. Phys. 152, 154102 (2020) 41. Becke, A.D.: Perspective: fifty years of density-functional theory in chemical physics. J. Chem. Phys. 140, 18A301–18A318 (2014) 42. Burke, K., Werschnik, J., Gross, E.K.U.: Time-dependent density functional theory: past, present, and future. J. Chem. Phys. 123, 062206 (2005) 43. Kühne, T.D., Iannuzzi, M., Del Ben, M., Rybkin, V.V., et al.: CP2K: an electronic structure and molecular dynamics software package - Quickstep: efficient and accurate electronic structure calculations. J. Chem. Phys. 152, 194,103/1–47 (2020) 44. Frisch, M.J., Trucks, G.W., Schlegel, H.B., Scuseria, G.E., et al.: Gaussian 16. 2016. Gaussian, Inc., Wallingford CT (2016) 45. Callis, P.R.: Molecular orbital theory of the 1Lb and 1La states of indole. J. Chem. Phys. 95, 4230–4240 (1991) 46. Slater, L.S., Callis, P.R.: Molecular orbital theory of the 1La and 1Lb states of indole. 2. An ab initio study. J. Phys. Chem. 99, 4230–4240 (1995) 47. Callis, P.R.:1La and 1Lb transitions of tryptophan: applications of theory and experimental observations to fluorescence of proteins. Meth. Enzymol. 278, 113–150 (1997) 48. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes: the Art of Scientific Computing, 3rd edn. Cambridge University Press, Cambridge (2007) 49. Kohn, W., Sham, L.J.: Self-consistent equations including exchange and correlation effects. Phys Rev. A. 140, A1133–A1138 (1965) 50. Dirac, P.M.: The quantum theory of the electron. Part II. Proc. Roy. Soc. A118, 351–361 (1928) 51. Roothaan, C.C.J.: New developments in molecular orbital theory. Rev. Mod. Phys. 23, 69–89 (1951)

3

Light

3.1

Electromagnetic Fields

In this chapter, we consider classical and quantum mechanical descriptions of electromagnetic radiation. We develop expressions for the energy density and irradiance of light passing through a homogeneous medium, and we discuss the Planck black-body radiation law and linear and circular polarization. Readers anxious to get on to the interactions of light with matter can skip ahead to Chap. 4 and return to the present chapter as the need arises.

3.1.1

Electrostatic Forces and Fields

The classical picture of light as an oscillating electromagnetic field provides a reasonably satisfactory basis for discussing the spectroscopic properties of molecules, provided that we take the quantum mechanical nature of matter into account. To develop this picture, let’s start by reviewing some of the principles of classical electrostatics. Charged particles exert forces that conventionally are described in terms of electric and magnetic fields. Consider two particles with charges q1 and q2 located at positions r1 and r2 in a vacuum. According to Coulomb’s law, the electrostatic force acting on particle 1 is F=

q 1 q2 ^r 12 , jr12 j2

ð3:1Þ →

→

where r12 = r1 - r2 and ^r 12 is a unit vector parallel to r 12. F is directed along r12 if the two charges have the same sign, and in the opposite direction if the signs are different. The electric field E at any given position is defined as the electrostatic force on an infinitesimally small, positive “test” charge at this position. For two particles in a vacuum, the field at r1 is simply the derivative of F with respect to q1: # The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. W. Parson, C. Burda, Modern Optical Spectroscopy, https://doi.org/10.1007/978-3-031-17222-9_3

87

88

3

Eðr1 Þ = lim

q1 → 0

∂Fðr1 Þ q = 2 2 ^r 12 : ∂q1 jr12 j

Light

ð3:2Þ

Fields are additive: if the system contains additional charged particles the field at r1 is the sum of the fields from all the other particles. The magnetic field (B) at position r1 is defined similarly as the magnetic force on an infinitesimally small magnetic pole m1. Magnetic fields are generated by moving electrical charges, and conversely, a changing magnetic field generates an electrical field that can cause an electrical charge to move. Equations (3.1) and (3.2) are written in the electrostatic or cgs system of units, in which charge is given in electrostatic units (esu), distance in cm, and force in dynes. The electron charge e is -4.803 × 10–10 esu. The electrostatic unit of charge is also called the statcoulomb or franklin. In the MKS units adopted by the System International, distance is expressed in meters, charge in coulombs (1 coulomb = 3 × 109esu; e = -1.602 × 10–19C), and force in newtons (1 newton = 105 dynes). In MKS units, the force between two charged particles is F=

1 q 1 q2 ^r , 4πεo jr12 j2 12

ð3:3Þ

where ℰo is a constant called the permittivity of free space (8.854 × 10–12C2N–1m–2). In the cgs system ℰo is equal to 1/4π so the proportionality constants in Eqs. (3.1 and 3.2) are unity. Because this simplifies the equations of electromagnetism, the cgs system continues to be widely used. Appendix A4 gives a table of equivalent units in the two systems.

3.1.2

Electrostatic Potentials

What is the energy of electrostatic interaction of two charges in a vacuum? Suppose we put particle 2 at the origin of the coordinate system and hold it there while we bring particle 1 in from infinity. To move particle 1 at a constant velocity (i.e., without using any extra force to accelerate it), we must apply a force Fapp that is always equal and opposite to the electrostatic force on the particle. The electrostatic energy (Eelec) is obtained by integrating the dot product Fapp(r) × dr, where r represents the variable position of particle 1 and dr is an incremental change in position during the approach. Because the final energy must be independent of the path, we can assume for simplicity that particle 1 moves in a straight line directly toward 2, so that dr and Fapp are always parallel to r. We then have (using cgs units again):

3.1 Electromagnetic Fields

89

Zr12 E elec =

Zr12 Fapp  dr = -

1

Zr12 FðrÞ  dr = -

1

1

q q q1 q2 ^r  dr = 1 2 : 2 jr12 j jrj

ð3:4Þ

The scalar electrostatic potential Velec at r1 is defined as the electrostatic energy of a positive test charge at this position. This is just the derivative of Eelec with respect to the charge at r1. In a vacuum, the potential at r1 created by a charge at r2 is V elec ðr1 Þ =

∂Eelec ðr1 Þ q = 2 : jr12 j ∂q1

ð3:5Þ

The electrostatic energy of a pair of charges in a vacuum is simply the product of charge q1 and the potential at r1: Eelec = q1 V elec ðr1 Þ:

ð3:6Þ

In a system with more than two charges, the total electrostatic energy is given similarly by Eelec =

1X 1 X X qj qi V elec ðri Þ = q , 2 i 2 i i j ≠ i jrij j

ð3:7Þ

where Velec(ri) is the electrostatic potential at ri resulting from the fields from all the other charges; the factor of 1/2 prevents counting the pairwise interactions twice. Figure 3.1 shows a contour plot of the electrostatic potential resulting from a pair of positive and negative charges. Note that Eq. (3.7) still refers to a set of stationary charges. Introducing a charge at ri will change the potential at this point if the field from the new charge causes other charged particles to move. In the cgs system, potentials have units of statvolts (ergs per esu of charge). In the MKS system, the potential difference between two points is one volt if one joule of work is required to move one coulomb of charge between the points. 1 volt = 1 J/ C = (107ergs)/(3 × 109esu) = 3 × 10-2erg/esu = 3 × 10-2(dyne - cm)/esu. The electric field at a given point, which we defined above in terms of forces (Eq. (3.2)), also can be defined as e elec ðrÞ, EðrÞ = - ∇V

ð3:8Þ

e elec is the gradient of the electrostatic potential at that point. The gradient where ∇V of a scalar function V is a vector whose components are the derivatives of V with respect to the coordinates: e V = ð∂V=∂x, ∂V=∂y, ∂V=∂zÞ = ^x∂V=∂x þ ^y∂V=∂y þ ^z∂V=∂z — →

ð3:9Þ →

(Eq. (2.5)). Thus, the electric field at r 1 generated by a charged particle at r 2 , is

90

3

B

z / r12

A

Light

y / r12 Fig. 3.1 Contour plots of the electric potential (Velec) generated by an electric dipole oriented along the z-axis, as functions of the position in the yz plane. The dipole consists of a unit positive charge at (y, z) = (0, r12/2) and a unit negative charge at (0, -r12/2). Blue lines represent positive potentials, becoming lighter as the potential increases; red lines represent negative potentials. The purple dot-dashed line represents zero potential. The contour intervals are 0.2e/r12 in A and 0.001 e/d12 in B; lines for jVelecj > 4e/r12 are omitted for clarity. The electric field vectors (not shown) are oriented normal to the contour lines of the potential, pointing in the direction of more positive potential. Their magnitudes are inversely proportional to the distances between the contour lines

 q2 = jr12 j     1 0  ∂ jr12 j - 1 ∂ jr12 j - 1 ∂ jr12 j - 1 A - q2 @ , , ∂x1 ∂y1 ∂z1

e elec ðrÞ = - ∇ e EðrÞ = - ∇V



ð x1 - x2 , y1 - y2 , z 1 - z 2 Þ q2 ^r 12 , = q2 h i3=2 = jr12 j2 ðx1 - x2 Þ2 þ ðy1 - y2 Þ2 þ ðz1 - z2 Þ2

ð3:10Þ

where |r12| = [(x1 - x2)2 + (y1 - y2)2 + (z1 - z2)2]1/2 and ^r 12 = ðx1 - x2 , y1 - y2 , → z1 - z2 Þ=j r 12j: This is the same as Eq. (3.2). Equation (3.8) implies that the line integral of the field over any path between two points r1 and r2 is just the difference between the potentials at the two points: Zr2 E  dr = - ½V ðr2 Þ - V ðr1 Þ = V ðr1 Þ - V ðr2 Þ:

ð3:11Þ

r1

This expression is similar to Eq. (3.4), in which we integrated the electrostatic force acting on a charged particle as another particle came in from a large distance. Here we integrate the component of the field that is parallel to the path element dr at each

3.1 Electromagnetic Fields

91

point along the path. Again, the result is independent of the path. Taking Eq. (3.11) one step further, we see that the line integral of the field over any closed path must be zero: I E  dr = 0: ð3:12Þ We’ll use this result later in this chapter to see what happens to the electric field when light enters a refracting medium.

3.1.3

Electromagnetic Radiation

The electric and magnetic fields (E and B) generated by a pair of positive and negative charges (an electric dipole) are simply the sum of the fields from the individual charges. If the orientation of the dipole oscillates with time, the fields in the vicinity will oscillate at the same frequency. It is found experimentally, however, that the fields at various positions do not all change in phase: the oscillations at larger distances from the dipole lag behind those at shorter distances, with the result that the oscillating fields spread out in waves. The oscillating components of E and B at a given position are perpendicular to each other, and at large distances from the dipole, they also are perpendicular to the position vector (r) relative to the center of the dipole (Fig. 3.1). They fall off in magnitude with 1/r, and with the sine of the angle (θ) between r and the dipole axis. Such a coupled set of oscillating electric and magnetic fields together constitute an electromagnetic radiation field. The strength of electromagnetic radiation often is expressed as the irradiance, which is a measure of the amount of energy flowing across a specified plane per unit area and time. The irradiance at a given position is proportional to the square of the magnitude of the electric field strength, |E|2 (Sect. 3.1.4). At large distances, the irradiance from an oscillating dipole therefore decreases with the square of the distance from the source and is proportional to sin2(θ), as shown in Fig. 3.2. The irradiance is symmetrical around the axis of the oscillations. A spreading radiation field like that illustrated in Figs. 3.1 and 3.2 can be collimated by a lens or mirror to generate a plane wave that propagates in a single direction with constant irradiance. The electric and magnetic fields in such a wave oscillate sinusoidally along the propagation axis as illustrated in Fig. 3.3, but are independent of the position normal to this axis. Polarizing devices can be used to restrict the orientation of the electric and magnetic fields to a particular axis in the plane. The plane wave illustrated in Fig. 3.3 is said to be linearly polarized because the electric field vector is always parallel to a fixed axis. Because E is confined to the plane normal to the axis of propagation, the wave also can be described as planepolarized. An unpolarized light beam propagating in the y-direction consists of electric and magnetic fields oscillating in the xz plane at all angles with respect to the z-axis.

92

3

Fig. 3.2 Irradiance of electromagnetic radiation from an electric charge that oscillates in position along a vertical axis. In this polar plot, the angular coordinate is the radiation angle (θ) in degrees relative to the oscillation axis. The radial distance of the green curve from the origin gives the relative irradiance of the wave propagating in the corresponding direction, which is proportional to sin2(θ). For example, the irradiance of the wave propagating at 60o (black arrow) is 75% of that of the wave propagating at 90o

Light

θ

Electric Field

1

0 Magnetic Field

-1 1

0

2

y/

Fig. 3.3 Electric and magnetic fields at a given time in a linearly polarized plane wave propagating in the y-direction, as functions of position along the propagation axis. The solid green curve is the component of the electric field parallel to the polarization axis, relative to the maximum amplitude (2|Eo|); λ is the wavelength. The dashed blue curve with shading to the propagation axis is a perspective view of the magnetic field, which is perpendicular to the electric field. The green and blue arrows indicate the directions and relative amplitudes of the fields at y/λ = 1

The properties of electromagnetic fields are described empirically by four coupled equations that were set forth by J.C. Maxwell in 1865 (Box 3.1). These very general equations apply to both static and oscillating fields, and they encapsulate the salient features of electromagnetic radiation. In words, they state that:

3.1 Electromagnetic Fields

93

1. Both E and B are always perpendicular to the direction of propagation of the radiation (i.e., the waves are transverse). 2. E and B are perpendicular to each other. 3. E and B oscillate in phase. 4. If we look in the direction of propagation, a rotation from the direction of E to the direction of B is clockwise. Box 3.1 Maxwell’s Equations and the Vector Potential Maxwell’s equations describe experimentally observed relationships between the electric and magnetic fields (E and B) and the densities of charge and current in the medium. The charge density (ρq) at a given point is defined so that the total charge in a small volume element dσ including the point is q = ρqdσ. If the charge moves with a velocity v, the current density (J) at the point is J = qv. In cgs units, Maxwell’s equations read divE =

4πρq , ε

divB = 0, curlE = -

1 ∂B , c ∂t

ðB3:1:1Þ ðB3:1:2Þ ðB3:1:3Þ

and curlB =

4π ε ∂E Jþ , c c ∂t

ðB3:1:4Þ

where c and ε are constants, and the vector operators div and curl are defined as follows: e  A = ∂Ax þ ∂Ay þ ∂Az divA = ∇ ∂x ∂y ∂z and     ∂Az ∂Ay ∂Ax ∂Az ^x þ ^y ∂y ∂z ∂z ∂x   ∂Ay ∂Ax ^z þ ∂x ∂y

e ×A= curlA = ∇

ðB3:1:5Þ

(continued)

94

3

Light

Box 3.1 (continued)   ^x   =  ∂=∂x   Ax

    ∂=∂z :  Az 

^y

^z

∂=∂y Ay

ðB3:1:6Þ

(Appendix A1.) The constant ε in Eqs. (B3.1.1 and B3.1.4) is the dielectric constant of the medium, which is defined as the ratio of the energy density (energy per unit volume) associated with an electric field in a medium to that for the same field in a vacuum. As we’ll discuss later in this chapter, the difference between the energy densities in a condensed medium and a vacuum reflects polarization of the medium by the field. In free space, or more generally, in a uniform, isotropic, nonconducting medium with no free charges, ρq and J are zero and ε is independent of position and orientation, and Eqs. (B3.1.1 and B3.1.4) simplify to divE = 0 and curlB = (ε/c)∂E/∂t. E and B then can be eliminated from two of Maxwell’s equations to give: 2

e 2E = ε ∂ E ∇ c2 ∂t 2

ðB3:1:7Þ

and 2

e 2B = ε ∂ B , ∇ c2 ∂t 2

ðB3:1:8Þ

e on a vector A is defined as where the action of the Laplacian operator ∇ !  2 2 2 2 2 2  2 ∂ A A A ∂ ∂ A A A ∂ ∂ ∂ y y y x x x e A= ^y ^x þ þ þ þ þ ∇ ∂x2 ∂y2 ∂z2 ∂x2 ∂y2 ∂z2  2 2 2  ∂ Az ∂ Az ∂ Az ^z: ðB3:1:9Þ þ þ þ ∂x2 ∂y2 ∂z2 2

Equations (B3.1.7 and B3.1.8) are classical, three-dimensional wave equations for waves that move through space with the velocity pffiffiffi u = c= ε: ðB3:1:10Þ Since ε = 1 in a vacuum, the constant c that appears in Eqs. (B3.1.7–B3.1.8) must be the speed at which electromagnetic waves travel in a vacuum. This was an unanticipated result when Maxwell discovered it. He had obtained the value of the constant from experimental data on the magnetic field generated (continued)

3.1 Electromagnetic Fields

95

Box 3.1 (continued) by a steady current, and there had been no reason to think that it had anything to do with light. The realization that c was, within a very small experimental error, the same as the measured speed of light led Maxwell [1] to suggest that light consists of electromagnetic waves. For a plane wave propagating in the y-direction with E polarized parallel to the z-axis, Eq. (B3.1.7) reduces to 2

2

∂ Ez ε ∂ Ez = 2 : c ∂2 ∂y2

ðB3:1:11Þ

Solutions to Maxwell’s equations also can be obtained in terms of a vector potential V and a scalar potential ϕ, which are related to the electric and magnetic fields by the expressions E= -

1 ∂V e - ∇ϕ c ∂t

ðB3:1:12Þ

and B = curlV :

ðB3:1:13Þ

This description has the advantage that only four parameters are needed to specify the electromagnetic fields (the magnitude of ϕ and the three components of V), instead of the six components of E and B. The description is not unique because adding any arbitrary function of time to ϕ does not affect the values of the physical observables E and B, which makes it possible to simplify the description further. If ϕ is chosen so that divV þ

1 ∂ϕ = 0, c ∂t

ðB3:1:14Þ

the scalar potential drops out and an electromagnetic radiation field can be represented in terms of the vector potential alone [2]. This choice of ϕ is called the Lorenz gauge. An alternative choice called the Coulomb gauge is often used for static systems. Using the Lorenz gauge, V for a uniform, isotropic, nonconducting medium with no free charges is determined by the equations 2

e 2V = ε ∂ V ∇ c2 ∂t 2

ðB3:1:15Þ

and (continued)

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Box 3.1 (continued) divV = 0,

ðB3:1:16Þ

while E is parallel to V and is given by E= -

1 ∂V : c ∂t

ðB3:1:17Þ

Equation (B3.1.13) still holds for B. See [1, 3] for Maxwell’s description of electromagnetism, and [2, 4–6] for additional discussion. For our purposes, we will not need to use Maxwell’s equations themselves; we can focus on a solution to these equations for a particular situation such as the plane wave of monochromatic, polarized light illustrated in Fig.3.3. In a uniform, homogeneous, nonconducting medium with no free charges, Maxwell’s equations for E in a one-dimensional plane wave reduce to 2

2

ε ∂ E ∂ E = 2 , c ∂t 2 ∂y2

ð3:13Þ

where c is the velocity of light in a vacuum and ε is the dielectric constant of the medium (Box 3.1). An identical expression holds for the magnetic field. Eq. (3.13) is a classical wave equation for a wave that moves with a velocity u through the medium, with u given by pffiffiffi u = c= ε: ð3:14Þ Solutions to Eq. (3.13) can be written in exponential notation as E = Eo f exp½2πiðνt - y=λ þ δÞ þ exp½- 2πiðνt - y=λ þ δÞg,

ð3:15aÞ

or, by using the identity exp(iθ) = cos(θ) + i sin(θ) and the relationships cos(-θ) = cos(θ) and sin(-θ) = - sin(θ), as E = 2Eo cos ½2π ðνt - y=λ þ δÞ:

ð3:15bÞ

In Eqs. (3.15a, b), Eo is a constant vector that expresses the magnitude and polarization of the field (parallel to the z-axis for the wave shown in Fig. 3.3), ν is the frequency of the oscillations, λ is the wavelength, and δ is a phase shift that depends on an arbitrary choice of zero time. The frequency and wavelength are linked by the expression λ = u=v

ð3:16Þ

3.1 Electromagnetic Fields

97

More generally, we can describe the electric field at point r in a plane wave of monochromatic, linearly polarized light propagating in an arbitrary direction (^k) by Eðr, t Þ = Eo f exp½2πiðνt - k  r þ δÞ þ exp½- 2πiðνt - k  r þ δÞg,

ð3:17Þ

where k, the wavevector, is a vector with magnitude 1/λ pointing in direction ^k. Note that each of the exponential terms in Eq. (3.17) could be written as a product of a factor that depends only on time, another factor that depends only on position, and a third factor that depends only on the phase shift. We’ll return to this point in Sect. 3.4. As discussed in Box 3.1, convenient solutions to Maxwell’s equations also can be obtained in terms of a vector potential V instead of electric and magnetic fields. Using the same formalism as Eq. (3.17) but omitting the phase shift for simplicity, the vector potential for a plane wave of monochromatic, linearly polarized light can be written V ðr, t Þ = V o f exp½2πiðνt - k  rÞ þ exp½- 2πiðνt - k  rÞg:

ð3:18Þ

We will use this expression in Sect. 3.4 when we consider the quantum mechanical theory of electromagnetic radiation. The velocity of light in a vacuum, 2.9979 × 1010cm/s, has been denoted almost universally by c since the early 1900s, probably for celeritas, the Latin word for speed. The first accurate measurements of the velocity of light in air were made by A. Fizeau in 1849 and L. Foucault in 1850. Fizeau passed a beam of light through a gap between teeth at the edge of a spinning disk, reflected the light back to the disk with a distant mirror, and increased the speed of the disk until the returning light passed through the next gap. Foucault used a system of rotating mirrors. Today, c is taken to be an exactly defined number rather than a measured quantity and is used to define the length of the meter. If monochromatic light moves from a vacuum into a non-absorbing medium with a refractive index n, the frequency ν remains the same but the velocity and wavelength decrease to c/n and λ/n, respectively. Equation (3.14) indicates that the refractive index, defined as c/u, can be equated with ε1/2: pffiffiffi n  c=u = ε: ð3:19Þ Most solvents have values of n between 1.2 and 1.6 for visible light. The refractive index of most materials increases with ν, and such media are said to have positive dispersion. As we’ll discuss in Sects. 3.1.4 and 3.5, Eqs. (3.14 and 3.19) do not necessarily hold in regions of the spectrum where n varies significantly with the wavelength. At frequencies where the medium absorbs light, the refractive index can vary strongly with ν and the velocity at which energy moves through the medium is not necessarily given simply by c/n, particularly if the light includes a broad band of frequencies. We will be interested mainly in the time-dependent oscillations of the electrical and magnetic fields in small, fixed regions of space. Because molecular dimensions

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typically are much smaller than the wavelength of visible light, the amplitude of the electrical field at a particular time will be nearly the same everywhere in the molecule. We also will restrict ourselves initially to phenomena that relate to averages of the field over many cycles of the oscillation and do not depend on coherent superposition of light beams with fixed phase relationships. With these restrictions, we can neglect the dependence of E on position and the phase shift, and write Eðt Þ = Eo ½ expð2πiνt Þ þ expð- 2πiνt Þ = 2Eo cosð2πνt Þ:

ð3:20Þ

We will have to use a more complete expression that includes changes of the fields with position in a molecule when we discuss circular dichroism. We’ll need to include phase shifts when we consider the light emitted by ensembles of many molecules.

3.1.4

Energy Density and Irradiance

Because electromagnetic radiation fields cause charged particles to move, they clearly can transmit energy. To evaluate the rate at which absorbing molecules take up energy from a beam of light, we will need to know how much energy the radiation field contains and how rapidly this energy flows from one place to another. We usually will be interested in the energy of the fields in a specified spectral region with frequencies between ν and ν + dν. The amount of energy per unit volume in such a spectral interval can be expressed as ρ(v)dv, where ρ(v) is the energy density of the field. The irradiance, I(ν)dν, is the amount of energy in a specified spectral interval that crosses a given plane per unit area and time. In a homogeneous, nonabsorbing medium, the irradiance is I ðvÞdv = uðvÞρðvÞdv,

ð3:21Þ

where u, as before, is the velocity of light in the medium. Several different measures are used to describe the strengths of light sources. The radiant intensity is the energy per unit time that a source radiates into a unit solid angle in a given direction. It usually is expressed in units of watts per steradian. Luminance, a measure of the amount of visible light leaving or passing through a surface of unit area, requires correcting the irradiance for the spectrum of sensitivity of the human eye, which peaks near 555 nm. Luminance is given in units of candela (cd) per square meter, or nits. For arcane historical reasons having to do with the apparent brightness of a hot bar of platinum, one candela is defined as the luminous intensity of a source that emits 540-nm monochromatic light with a radiant intensity of 1/683 (0.001464) watt per steradian. The total luminous flux from a source into a given solid angle, the product of the luminance and the solid angle, is expressed in lumens. One lux is one lumen per square meter. Bright sunlight has an illuminance on the order of 5 × 104to 1 × 105 lux, and luminance of 3 × 103 to 6 × 103cd  m-2.

3.1 Electromagnetic Fields

99

From Maxwell’s equations, one can show that the energy density of electromagnetic radiation depends on the square of the electric and magnetic field strengths [4– 6]. For radiation in a vacuum, the relationship is h i ð3:22Þ ρðνÞ = jE ðνÞj2 þ jBðνÞj2 ρν ðνÞ=8π, where the bar above the quantity in brackets means an average over the spatial region of interest and ρv(v)dv is the number of modes (frequencies) of oscillation in the small interval between ν and ν + dν. An oscillation mode for electromagnetic radiation is analogous to a standing wave for an electron in a box (Eq. (2.23a)). But the drawing in Fig. 3.3, which represents the electric field for such an individual mode (monochromatic light), is an idealization. In practice, electromagnetic radiation is never strictly monochromatic: it always includes fields oscillating over a range of frequencies. We’ll discuss the nature of this distribution in Sect. 3.6. Equation (3.22) is written in cgs units, which are particularly convenient here because the electric and magnetic fields in a vacuum have the same magnitude: jBj = jEj:

ð3:23Þ

(In MKS units, jB j = j E j /c.) The energy density of a radiation field in a vacuum is, therefore, ρðνÞ = jE ðνÞj2 ρν ðνÞ=4π:

ð3:24Þ

If we now use Eq. (3.15b) to express the dependence of E on time and position, we have jEj2 = ½2Eo cosð2πνt - 2πy=λ þ δÞ2 = 4jEo j2 cos 2 ð2πνt - 2πy=λ þ δÞ:

ð3:25Þ

Since the fields in a plane wave are by definition independent of position perpendicular to the propagation axis ( y), the average denoted by the bar in Eq. (3.25) requires only averaging over a distance in the y-direction. If this distance is much longer than λ (or is an integer multiple of λ), the average of cos2(2πvt - 2πy/λ + δ) equals 1/2, and Eq. (3.25) simplifies to jEj2 = 2jEo j2 :

ð3:26Þ

So, for a plane wave of light in a vacuum: ρðνÞ = jEo j2 ρν ðνÞ=2π

ð3:27Þ

I ðνÞ = cjEo j2 ρν ðνÞ=2π:

ð3:28Þ

and

If a beam of light in a vacuum strikes the surface of a refractive medium, part of the beam is reflected while another part enters the medium. We can use Eq. (3.28) to

100

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Light

relate the irradiances of the incident and reflected light to the amplitudes of the corresponding fields because the fields on this side of the interface are in a vacuum. But we need a comparable expression that relates the transmitted irradiance to the amplitude of the field in the medium, and for this, we must consider the effect of the field on the medium. As light passes through the medium, the electric field causes electrons in the material to move, setting up electric dipoles that generate an oscillating polarization field (P). In an isotropic, nonabsorbing, and nonconducting medium P is proportional to E and can be written P = χ e E:

ð3:29Þ

The proportionality constant χ e is called the electric susceptibility of the medium, and materials in which P and E are related linearly in this way are called linear optical materials. The field in the medium at any given time and position (E) can be viewed as the resultant of the polarization field and the electric displacement (D), which is the field that hypothetically would be present in the absence of the polarization. In a vacuum, P is zero and E = D. In cgs units, the field in a linear medium is given by E = D - 4π P = D - 4πχ e E = D=ð1 þ 4πχ e Þ = D=ε

ð3:30Þ

where, as in Maxwell’s equations (Box 3.1), ε is the dielectric constant of the medium. Rearranging these relationships gives ε = 1 + 4πχ e. We will be interested mainly in electromagnetic fields that oscillate with frequencies on the order of 1015 Hz, which is too rapid for nuclear motions to follow. The polarization described by P therefore reflects only the rapidly oscillating induced dipoles created by electronic motions, and the corresponding dielectric constant in Eq. (3.30) is called the high-frequency or optical dielectric constant. For a nonabsorbing medium, the electric susceptibility is independent of the oscillation frequency, and the high-frequency dielectric constant is equal to the square of the refractive index. The magnetic field of light passing through a refractive medium is affected analogously by induced magnetic dipoles, but in nonconducting materials, this is a much smaller effect and usually is negligible. In a linear, nonabsorbing medium, the relationship between the amplitudes of the magnetic and electric fields in cgs units becomes [4–6]. pffiffiffi jBj = εjEj, ð3:31Þ and the energy density of electromagnetic radiation is h i h i pffiffiffi ρðνÞ = EðνÞ  DðνÞ þ jBðνÞj2 ρν ðνÞ=8π = EðνÞ  εEðνÞ þ j εEðνÞj2 ρν ðνÞ=8π

3.1 Electromagnetic Fields

101

= εjEðνÞj2 ρν ðνÞ=4π = εjEo j2 ρν ðνÞ=2π

ð3:32Þ

We have assumed again that we are interested in the average energy over a region that is large relative to the wavelength of the radiation. Equation (3.32) indicates that, for equal field strengths, the energy density in a refracting medium is ε times that in a vacuum. The additional energy resides in the polarization of the medium. But we are not quite through. To find the irradiance in the medium, we also need to know the velocity at which energy moves through the medium. This energy velocity or group velocity (u) is not necessarily simply c/n because waves with different frequencies will travel at different rates if n varies with ν. The group velocity describes the speed at which a packet of waves with similar frequencies travels as a whole (Sect. 3.5). It is related generally to c, n and ε by [7, 8]. u = cn=ε,

ð3:33Þ

which reduces to u = c/n (Eq. (3.19)) if ε = n2, as it does if n is independent of ν. Combining Eq. (3.33) with (3.21) and (3.32) gives the irradiance in the medium: I ðνÞ =

c ρðνÞ = cnjEo j2 ρν ðνÞ=2π: n

ð3:34Þ

This is an important result for our purposes because it relates the irradiance of a light beam in a condensed medium to the refractive index and the amplitude of the electric field. We’ll need this relationship in Chap. 4 to connect the strength of an electronic absorption band to the electronic structure of a molecule. As discussed in Sect. 3.6, Eq. (3.34) also can be used to find the fractions of a light beam that are reflected and transmitted at a surface.

3.1.5

Electromagnetic Momentum

The flux of electromagnetic energy through a medium is described by a vector that is called the Poynting vector after J. H. Poynting, who derived it in 1884. The Poynting vector, S, is given by the cross product of the electric and magnetic fields vectors: S = E × B:

ð3:35Þ

This result is in accord with Maxwell’s equations stating that the electric and magnetic fields are perpendicular to each other and to the direction in which the fields move. It also fits with Einstein’s conclusion that photons convey energy and momentum (Box 2.3). Poynting’s derivation was based on the conservation of energy, which requires the electromagnetic energy entering a region of space to equal the sum of the electromagnetic energy leaving the region and the energy that is turned into heat or other forms there. The momentum of a beam of light changes direction when light undergoes refraction at an interface between media with different refractive indices. Because the total momentum of the system must be conserved, changes in the momentum of

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Fig. 3.4 Focused laser beams can act as “optical tweezers” to manipulate small objects. In this illustration, a collimated laser beam propagating along the laboratory z-axis encounters a transparent, spherical object whose refractive index (n2) exceeds that of the surrounding solution (n1). The center of the sphere is in the xz plane at distance x0 from the center of the beam. Light rays (red arrows) reaching the sphere along two axes parallel to z at equal distances on either side of x0 are labeled ia and ib, and the refracted rays leaving the sphere are labeled ra and rb. Light propagating on x0 (dashed red arrow) passes straight through the sphere. Light propagating on ia is diffracted in the +x direction, gaining momentum in this direction, and must impart an equal momentum in the -x direction to the sphere. Light on ib gains momentum in the -x direction and must impart momentum in the +x direction to the sphere. Because the irradiance (purple curve) is greater along ia and ra than along ib and ib, the net effect is to push the sphere in the -x direction, toward the center of the beam. A similar analysis shows that a converging beam of light passing through a refracting sphere pushes the sphere toward the focal point

another part of the system must balance the changes in the momentum of the photons. Light thus must be able to exert forces even if it is not absorbed. In the first clear demonstration of this remarkable effect, Arthur Ashkin [9] showed that a laser beam could impart motion to a transparent, micron-sized sphere of latex suspended in water (Fig. 3.4). Ashkin showed further that two laser beams focused through such a sphere in opposite directions could hold the sphere at a fixed position, overcoming both Brownian motion and the force of gravity [9]. Instruments that use laser beams in these ways (optical tweezers or optical traps) have found numerous applications in studies of single molecules, viruses, and cells [10, 11]. Ashkin received the Nobel Prize in Physics in 2018 for the development of optical tweezers and their application to biological systems. Donna Strickland and Gérard Mourou shared the prize for methods of generating ultra-short optical pulses. Woodside et al.

3.2 The Black-Body Radiation Law

103

have used optical tweezers to study the unfolding of individual protein molecules and transitions between single- and double-stranded DNA [12–14].

3.2

The Black-Body Radiation Law

It has long been known that the radiation emitted by a heated object shifts to higher frequencies as the temperature is increased. When we discuss fluorescence in Chap. 5, we will need to consider the electromagnetic radiation fields inside a closed box whose walls are at a given temperature. J. W. Strutt, who had become the third Baron Rayleigh (Lord Rayleigh), derived an expression for the energy distribution of this black-body radiation in 1900 by considering the number of possible modes of oscillation (standing waves) with frequencies between ν and ν + dν in a cube of volume V. Considering the two possible polarizations of the radiation (Sect. 3.3), taking the refractive index (n) of the medium inside the cube into account, and including a correction pointed out by Jeans, the number of oscillation modes in the frequency interval dν is  ð3:36Þ ρv ðvÞVdv = 8πn3 v2 V=c3 dv [15]. Following classical statistical mechanics, Rayleigh assumed that each mode would have an average energy of kBT, independent of the frequency. Since ρv increases quadratically with ν (Eq. (3.36)), this analysis led to the alarming conclusion that the energy density (the product of ρv and the average energy per mode) goes to infinity at high frequencies. Experimentally, the energy density was found to increase with frequency in accord with Rayleigh’s prediction at low frequencies, but then to pass through a maximum and decrease to zero. Max Planck saw that the observed dependence of the energy density on frequency could be reproduced by introducing the ad hoc hypothesis that the material in the walls of the box emits or absorbs energy only in integral multiples of hν, where h is a constant. He assumed further that, if the material is at thermal equilibrium at temperature T, the probability of emitting an amount of energy, Ej, is proportional to exp(‐Ej/kBT) where kB is the Boltzmann constant (Box 2.6). With these assumptions, the average energy of an oscillation mode with frequency ν becomes !, ! X X   E= E j exp - E j =k B T exp - Ej =kB T ð3:37aÞ j

=

X

j

!, jhv expð- jhv=k B T Þ

j

X

! expð- jhv=kB T Þ :

ð3:37bÞ

j

The sums can be evaluated by letting x = exp(-hv/kBT ), using the!expansion P P j P j jx = xd x =dx. This ð1 - xÞ - 1 = 1 þ x þ x2 þ . . . = x j , and noting that j

yields

j

j

104

A

Light

4

5

B 1.0

1.0 6000 K

6000 K

0.8 Relative I(ν)

0.8 Relative I(λ)

3

0.6 5000

0.4 0.2

5000

0.6 0.4

4000

0.2

4000

3000

3000

0

0

0.5

1.0 1.5 l / microns

2.0

0

0

1

2

3

hν / eV

Fig. 3.5 The intensity (I ) of the black-body radiation emitted by an object at a temperature of 3000 (red curves), 4000 (orange), 5000 (cyan), or 6000 K (blue), plotted as a function of wavelength (λ, A) or photon energy (hν, B). The surface temperature of the sun is about 6600 K, and a tungstenhalogen lamp filament typically has an effective temperature of about 3000 K

E=

hv : expðhv=kB T Þ - 1

ð3:37cÞ

(Note that E denotes a thermal average of the energy, a scalar quantity, not the electric field vector E.) Finally, multiplying Planck’s expression for E (Eq. (3.37c)) by ρv (Eq. (3.36)) gives  ρðvÞ = E ðvÞρv ðvÞ = 8πhn3 v3 =c3 =½ expðhv=kB T Þ - 1, ð3:38aÞ or expressed as irradiance,  I ðvÞ = ρðvÞc=n = 8πhn2 v3 =c2 =½ expðhv=k B T Þ - 1:

ð3:38bÞ

Figure 3.5 shows plots of ρ(v) as functions of the wavelength and wavenumber of the radiation. The predictions are in accord with the measured energy density of black-body radiation at all accessible frequencies and temperatures. In particular, Eq. (3.38a) accounts for the observations that the energy density of blackbody radiation increases with the fourth power of T (the Stefan-Boltzmann law) and that the wavelength of peak energy density is inversely proportional to T (the Wien displacement law). Although the derivation outlined above invokes a Boltzmann distribution, the same result can be obtained by using the Bose-Einstein distribution for a photon gas at thermal equilibrium [16]. Planck’s theory did not require the radiation field itself to be quantized, and Planck did not conclude that it is [17]. Because the radiation inside a black-body box

3.3 Linear and Circular Polarization

105

is emitted and absorbed by the walls of the box, only the energy levels of the material comprising the walls must be quantized. The theory is consistent with quantization of the radiation field as well but does not demand it. In addition to its pivotal contribution to the development of quantum theory, the black-body radiation law has practical applications in spectroscopy. Because it describes the spectrum of the light emitted by an incandescent lamp with a filament at a known temperature, Eq. (3.38b) can be used to calibrate the frequencydependence of a photodetector or monochromator. However, the actual emission spectrum of a lamp can depart somewhat from Eq. (3.38b), depending on the material used for the filament [18].

3.3

Linear and Circular Polarization

In the quantum theory described in the following section, the eigenfunctions of the Schrödinger equation for a radiation field have angular momentum quantum numbers s = 1 and ms = ± 1. The two possible values of ms correspond to left (ms = + 1) and right (ms = - 1) circularly polarized light. Figure 3.6 illustrates this property. In this depiction, the electric field vector E has a constant magnitude but its orientation rotates with time at frequency ν. The component of E parallel to any given axis

Fig. 3.6 Electric and magnetic fields in right (top) and left (bottom) circularly polarized light. In this illustration, both beams propagate diagonally upward from left to right. The solid green arrows in the disks indicate the orientations of the electric field (E) at a given time as a function of position along the propagation axis; the dotted blue arrows show the orientations of the magnetic field (B). The field vectors both rotate so that their tips describe right- or left-handed corkscrews, making one full turn in a distance corresponding to the wavelength of the light (λ)

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Fig. 3.7 Representation of linearly polarized light as a superposition of right and left circularly polarized light. The dotted blue arrows indicate the electric field vectors of right and left circularly polarized beams propagating to the right. Approximately half an oscillation period is shown. The vector sum of the two circularly polarized fields (solid green arrows) oscillates in amplitude parallel to a fixed axis

normal to the propagation axis oscillates at the same frequency and oscillates along the propagation axis with wavelength λ. For radiation propagating in the y-direction, the time dependence of the rotating field can be written (neglecting an arbitrary phase shift): E ± = 2Eo ½ cosð2πνt Þ^z ± sinð2πνt Þ^x½ cosð2πy=λÞ^z ± sinð2πy=λÞ^x,

ð3:39aÞ

B ± = 2E o ½ cosð2πνt Þ^x ∓ sinð2πνt Þ^z½ cosð2πy=λÞ^x ∓ sinð2πy=λÞ^z,

ð3:39bÞ

where Eo is a scalar amplitude and the + and - subscripts refer to left and right circular polarization, respectively. These expressions are solutions to the general wave equation that satisfies Maxwell’s equations for a non-conducting medium with no free charges (Eqs. (B3.1.7a, b) in Box 3.1). The different algebraic combinations of the z and x components in Eqs. (3.39a, b) keep the magnetic field perpendicular to the electric field for both polarizations. A linearly polarized beam of light can be treated as a coherent superposition of left and right circularly polarized light, as shown in Fig. 3.7. Changing the phase of one of the circularly polarized components relative to the other rotates the plane of the linear polarization. Unpolarized light consists of a mixture of photons with left and right circular polarization and with electric fields rotating at all possible phase angles, or equivalently, a mixture of linearly polarized light with all possible orientation angles. If linearly polarized light passes through a sample that preferentially absorbs one of the circularly polarized components, the transmitted beam will emerge with elliptical polarization. The ellipticity is defined as the arctangent of the ratio Iminor/ Imajor, where Iminor and Imajor are the light intensities measured through polarizers parallel to the minor and major axes of the ellipse. Ellipticity can be measured by using a quarter-wave plate to convert the elliptically polarized light back to a linearly polarized beam that is rotated in alignment relative to the original beam and determining the angle of rotation. This is one way to measure CD. (We will discuss quarter- and half-wave plates later in this chapter.) However, CD usually is measured by modulating the circular polarization electronically as described in Sects. 1.9 and 3.6, because this technique is less subject to artifacts.

3.4 Quantum Theory of Electromagnetic Radiation

3.4

107

Quantum Theory of Electromagnetic Radiation

The Schrödinger equation was first applied to electromagnetic radiation in 1927 by Paul Dirac [19]. The notion of a quantized radiation field that emerged from this work reconciled some of the apparent contradictions between earlier wave and particle theories of light, and as we will see in Chap. 5, led to a consistent explanation of the “spontaneous” fluorescence of excited molecules. To develop the quantum theory of electromagnetic radiation, it is convenient to describe a radiation field in terms of the vector potential V that is introduced in Box 3.1. Consider the vector potential associated with a plane wave of light propagating in the y-direction and polarized in the z-direction, and suppose the radiation is confined within a cube with edge L. As discussed in Sect. 3.2, the radiation with frequencies in a specified interval is restricted to a finite number of oscillation modes, each with a discrete wavelength λj = L/2πnj where nj is a positive integer. The total energy of the radiation is the sum of the energies of these individual modes, and the total vector potential evidently is a similar sum of the individual vector potentials: X Vj: ð3:40Þ V= j

According to Eq. (3.18), the contribution to V from mode j can be written  2 1=2  

   4πc ^z exp - iωj t exp iy=λj þ exp iωj t exp - iy=λj ð3:41aÞ Vj = 3 L = qj ðt ÞAj ðyÞ þ qj  ðt ÞAj  ðyÞ,

ð3:41bÞ

 qj = exp - iωj t ,

ð3:42aÞ

where ωj = 2πvj = c/λj,

and  Aj =

4πc2 L3

1=2  ^z exp iy=λj :

ð3:42bÞ

The vector fields Vj, which are real quantities, are written here in terms of products of two complex functions and their complex conjugates. The first function (qj) is a scalar that depends only on time; the second (Aj) is a vector function of position. The position-dependent factors are normalized so that hAj j Aji = 4πc2, while the factors for different modes are orthogonal: hAi j Aji = 0 for i ≠ j. Equations (3.41a–3.42b) hold for progressive waves as well as for standing waves, although the restriction on the number of possible modes applies only to standing waves. To put the energy of the radiation field in the Hamiltonian form, we now define two real variables,

108

3

Qj ðt Þ = qj ðt Þ þ qj  ðt Þ,

Light

ð3:43Þ

and P j ðt Þ =

n o ∂Qj = - iωj qj ðt Þ - qj ðt Þ : ∂t

ð3:44Þ

From Eqs. (B3.1.17, 3.24, 3.41a, b, and 3.42a, b), the contribution of mode j to the energy of the field, integrated over the volume of the cube, then takes the form [2, 4]. Ej =

1 2 Pj þ ωj 2 Qj 2 : 2

ð3:45Þ

A little algebra will show that Qj obeys a classical wave equation homologous to Eqs. (3.13) and (3.17), and that Qj and Pj have the formal properties of a timedependent position (Qj) and its conjugate momentum (Pj) in Hamiltonian’s classical equations of motion: ∂Ej ∂Qj = , ∂Pj ∂t

ð3:46aÞ

∂Pj ∂Ej : =∂Qj ∂t

ð3:46bÞ

and

In addition, Dirac noted that Eq. (3.45) is identical to the classical expression for the energy of a harmonic oscillator with unit mass (Eq. (2.29)). The first term in the braces corresponds formally to the kinetic energy of the oscillator; the second, to the potential energy. It follows that if we replace Pj and Qj with momentum and position e j , respectively, the eigenstates of the Schrödinger equation for ej and Q operators P electromagnetic radiation will be the same as those for harmonic oscillators. In particular, each oscillation mode will have a ladder of states with wavefunctions χ jðnj Þ and energies  E jðnj Þ = nj þ 1=2 hvj ,

ð3:47Þ

where nj = 0, 1, 2. . . The transformation of the time-dependent function Pj into a momentum operator is consistent with Einstein’s description of light in terms of particles (photons), each of which has momentum hv (Sect. 1.6 and Box 2.3). We can interpret the quantum number nj in Eq. (3.47) either as the particular excited state occupied by oscillator j or as the number of photons with frequency Vj. The oscillating electric and magnetic fields associated with a photon can still be described by Eqs. (3.41a, b) and (3.42a, b) if the amplitude factor Eo is scaled appropriately. However, we will be less concerned with the spatial properties of photon wavefunctions themselves than

3.4 Quantum Theory of Electromagnetic Radiation

109

e These matrix elements play a with the matrix elements of the position operator Q. central role in the quantum theory of absorption and emission, as we’ll discuss in Chap. 5. The total energy of a radiation field is the sum of the energies of its individual modes, and according to Eq. (3.47), the energy of each mode increases with the number of photons in the mode. But, like the harmonic oscillator, an electromagnetic radiation field has a zero-point or vacuum energy when every oscillator is at its lowest level (nj = 0). Because a radiation wave in free space could have an infinite number of different oscillation frequencies, the total zero-point energy of the universe appears to be infinite, which may seem a nonsensical result. One way to escape this dilemma would be to argue that the universe is bounded so that there is no such thing as completely free space. In this picture, the zero-point energy of the universe becomes an unknown, but finite constant. However, we can arbitrarily set the zeropoint energy of each oscillation mode to zero by simply subtracting a constant (hvj/2) from the Hamiltonian of Eq. (3.45), making the energy associated with each mode E jðnj Þ = nj hvj :

ð3:48Þ

This is common practice for other types of energy, which can be expressed with respect to any convenient reference. The relativistic rest-energy mc2, for example, usually is omitted in discussions of the nonrelativistic energy of a particle. See [2, 4, 20] for further discussion of this point. The zero-point eigenstate is a critical feature of the quantum theory. It suggests, surprisingly, that a radiation field might interact with a molecule even if the number of photons in the field is zero! In Chap. 5, we will discuss how this interaction gives rise to fluorescence. We also will see there that most transitions between different states of a radiation field change the energy of the field by ±hvj, and result in the creation or disappearance of a single photon. The identification of the quantum states of the radiation field with those of harmonic oscillators makes it possible to evaluate the matrix elements for such transitions. Experimental evidence for the existence of the vacuum radiation field has come from observations of the Casimir effect. First predicted by the Dutch physicist H. Casimir in 1948, this is an attractive force between reflective objects in a vacuum. Consider two polished square plates with parallel faces separated by distance L. Standing waves in the gap between the plates must have wavelengths of L/2n where n is a positive integer. Because the number of such possible radiation modes is proportional to L, the total energy of the vacuum field decreases when the plates move closer together. The attractive force has been measured for objects of a variety of shapes and found to agree well with predictions [21–23]. The fact that they have integer spin (ms = ± 1) implies that photons obey BoseEinstein statistics (Box 2.6). This means that any number of photons can have the same energy (hν) and spatial properties, and conversely, that an individual radiation mode can have any number of photons. Accumulation of many photons in a single radiation mode makes possible the coherent radiation emitted by lasers.

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In addition to accounting for the quantization of radiation, Dirac introduced a relativistic theory of electrons. But even with these advances, the quantum theory at this stage left fundamental questions unresolved, including the mechanism by which charged particles interact at a distance in a vacuum. What does it mean to say that an electron gives rise to an electromagnetic field? An answer to this question came from work by R. Feynman, J. Schwinger, and S.-I.Tomonaga in the 1950s. In the theory of quantum electrodynamics that emerged from these studies, charged particles interact by exchanging photons, and the charge of a particle is a measure of its tendency to absorb or emit photons. However, photons that move from one particle to another are termed “virtual” photons because they cannot be intercepted and measured directly. Contrary to the principles of classical optics, the theory of quantum electrodynamics asserts that photons do not necessarily travel in straight lines. To find the probability that a photon will move from point A to point B, we must sum the amplitudes of wavefunctions for all possible paths between the two points, including even round-about routes via distant galaxies and paths in which the photon splits transiently into an electron-positron pair. Although there are an infinite number of paths between any two points, destructive interferences cancel most of the contributions from all the indirect paths, leaving only small (but sometimes significant) corrections to the laws of classical optics and electrostatics. This is because small differences between the indirect routes have large effects on the overall lengths of the paths, causing phase shifts of the oscillations that photons traveling by these routes contribute at the destination. Feynman [24] has provided a readable introduction to the theory of quantum electrodynamics. For more complete treatments see [25–27].

3.5

Superposition States and Interference Effects in Quantum Optics

Sections 2.2.1 and 2.3.2 introduced the idea of a superposition state whose wavefunction is a linear combination of two or more eigenfunctions with fixed phases. We asserted there that a system must be described by such a linear combination if it cannot be assigned uniquely to an individual eigenstate. The interference terms in expectation values for superposition states lead to some of the most intriguing aspects of quantum optics, including the fringes in Young’s classical double-slit experiments. Figure 3.8A shows an experiment that illustrates the point well [28–34]. The apparatus is called a Mach-Zender interferometer. Photons enter the interferometer at the upper left and are detected by a pair of photon counters, D1 and D2. The light intensity is low enough so that no more than one photon is in the apparatus at any given time. BS1 is a beamsplitter that, on average, transmits 50% of the photons and reflects the other 50%. Mirrors M1 and M2 reflect all the photons reaching them to a second beamsplitter (BS2), which again transmits 50% and reflects 50%. If beam-splitter BS2 is removed, a photon is detected by detector D1 half the time and by D2 the other half, just as we would expect. If either of the two paths between BS1 and BS2 is blocked, half the photons get through, and again half

3.5 Superposition States and Interference Effects in Quantum Optics

111

Fig. 3.8 (A) Single-photon interference in a Mach-Zender interferometer with equal arms. Light beam entering the interferometer (green line) encounters two identical beam-splitters (BS1 and BS2), which have coefficients CT and CR for transmission and reflection, respectively (jCTj2 = jCRj2 = 1/2); M1 and M2 are mirrors (100% reflecting); and D1 and D2 are photoncounting detectors. The dependence of photon wavefunction Ψ on time and the distance along the optical path is not indicated explicitly. If BS2 is removed, or if either path is blocked before BS2, photons are detected at D1 and D2 with equal probability; but when BS2 is present and both paths are open, photons are detected only at D1. (B) Two-photon quantum interference. Short pulses of light with frequency ν (dark blue arrow) are focused into a crystal with nonlinear optical properties (XTL) to generate pairs of photons with frequency ν/2 (wavefunctions Ψ a and Ψ b, orange arrows), which are reflected by mirrors (M1, M2) to a 50:50 beam-splitter (BS) and two photon-counting detectors (D1, D2). The relative lengths of the paths from the crystal to D1 and D2 can be controlled by a translation stage like that sketched in Fig. 1.9 (not shown here). If the beam-splitter either reflects both photons or transmits both photons of a given pair, one would expect D1 and D2 each to get one of the photons; however, quantum interference prevents this unless the photons are distinguishable by their arrival times or polarizations. (C), (D) After an unpolarized light beam passes through a vertical polarizer (V ), it will not pass through a horizontal polarizer (H ). But if a polarizer with an intermediate orientation (ϕ) is placed between V and H, some of the light passes through both this polarizer and H

of these are detected at D1 and half at D2. That also seems expected. But if both paths are open, all the photons are detected at D1 and none at D2! To account for this surprising result, note that when there are two possible paths of equal length between B1 and B2, we have no way of knowing which path a given photon follows. According to the prescriptions of quantum electrodynamics (Sect. 3.4), we therefore must write the wavefunction for a photon reaching D1 or D2 as a sum of the two possibilities. Suppose that each of the beamsplitters transmits

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photons with probability |CT|2 and reflects photons with probability |CR|2. The wavefunction for photons reaching D1 then is the weighted sum: ΨD1 = CR CT Ψ þ C T CR Ψ = ðC R CT þ C T CR ÞΨ,

ð3:50aÞ

where Ψ(r, t) is the wavefunction for an individual photon (Fig. 3.8a). The wavefunction for photons reaching D2 is, similarly, ΨD2 = C T CT Ψ þ C R C R Ψ = ðC T CT þ C R C R ÞΨ:

ð3:50bÞ

We have omitted the coefficient for reflection at mirrors M1 and M2 since this has a magnitude of 1.0 and there is one mirror in each pathway. From Eqs. (3.50a, b), the probability that a given photon will be detected at D1 is PD1 = hðC R C T þ CT C R ÞΨjðCR C T þ CT C R ÞΨi = 4jC R j2 jCT j2 ,

ð3:51aÞ

whereas the probability that the photon is detected at D2 is PD2 = hðC T C T þ CR CR ÞΨjðC T CT þ C R C R ÞΨi = jC T j2 jCT j2 þ jCR j2 jC R j2 þ ðC T  CR Þ2 þ ðC R  C T Þ2 :

ð3:51bÞ

Now consider the coefficients CT and CR. Because the probabilities of reflection and transmission at a 50:50 beamsplitter are the same, we know that jCTj2 = jCRj2 = 1/ 2. This, however, leaves open the possibility that one of the coefficients is imaginary, which in fact proves to be necessary. If both coefficients were real, we would have CT = ± CR = ± 2-1/2, which on substitution in Eqs. (3.51a, b) gives PD1 = PD2 = 1. That cannot be correct because it means that a single photon would be detected with 100% certainty at both D1 and D2, violating the conservation of energy. Trying CT = ± CR = ± i2-1/2 (i.e., making both coefficients imaginary) gives the same unacceptable result. Either CT or CR, but not both, therefore must be imaginary. If we choose CR to be imaginary, we have CT = ± 2-1/2 and CR = ± i2-1/2. Inserting these values in Eqs. (3.51a, b) gives PD1 = 1 and PD2 = 0 in agreement with experiment. The experiment just discussed illustrates destructive quantum mechanical interference in the wavefunction of an individual photon. Can similar interference occur between different photons? To investigate this question, Hong et al. [35] generated pairs of photons by focusing short pulses of light from a laser into a crystal with non-linear optical properties. When certain conditions are met for matching the phases of the incident and transmitted light, such a crystal can “split” photons with incident frequency ν into two photons with frequency ν/2. (We will discuss such frequency doubling further in Sect. 3.8.) Hong et al. sent the two photons along separate paths to a 50:50 beamsplitter and on to a pair of detectors (D1 and D2) as shown in Fig. 3.8B. At the beamsplitter, there are four possibilities. Both photons might be transmitted; both might be reflected; photon a might be transmitted and photon b reflected; or photon a might be reflected and b transmitted. In the first two cases, detectors D1 and D2 each would receive a photon; in the third and fourth cases, one detector would get two photons and the other none. If the paths from the crystal to D1 and D2 are of different lengths, the four possible outcomes have equal

3.5 Superposition States and Interference Effects in Quantum Optics

113

probabilities. But when the paths are adjusted to be the same, a surprising thing happens: the two photons of each pair always go to the same detector. To see how this occurs, note that in cases 3 and 4 (one photon transmitted and the other reflected), the overall wavefunction acting on the detectors can be written as a product of the individual waveforms (Ψaand Ψb) with the applicable coefficients for transmission and reflection (CRΨa  CTΨb or CTΨa  CRΨb). Inserting the values of CT and CR gives a non-zero amplitude. On the other hand, if the detectors can not distinguish between the two photons by their arrival times, frequencies or polarizations, we have no way to distinguish between cases 1 and 2 (both photons transmitted or both photons reflected). We therefore must write the overall waveform for these cases as a sum of CTΨa  CTΨb and CRΨa  CRΨb. The amplitude of this sum evaluates to zero: Ψ = 2 - 1=2 ðC T Ψa C T Ψb ± C R Ψa C R Ψb Þ   = 2 - 1=2 2 - 1=2 2 - 1=2 ± i2 - 1=2 i2 - 1=2 Ψa Ψb :

ð3:52Þ

Two-photon quantum interference sometimes is ascribed to local interference of the photons at the beamsplitter. However, Kwiat et al. [36] and Pittman et al. [37] demonstrated experimentally that the critical question is whether the wavefunctions are distinguishable at the detectors. The photons do not need to be at the beamsplitter simultaneously. Linearly polarized light provides another instructive illustration of superposition states in quantum optics. As discussed in Sect. 3.3, unpolarized light can be viewed as a mixture of photons with all possible linear polarizations. Light that is polarized at an angle θ with respect to an arbitrary “vertical” axis can be viewed as a coherent superposition of vertically and horizontally polarized light with coefficients CV = cos (θ) and CH = sin (θ): Ψθ = C V ΨV þ C H ΨH = cosðθÞΨV þ sinðθÞΨH :

ð3:53Þ

The probability that a photon with this polarization will pass through a vertical polarizer is cos2(θ). After passage through such a polarizer, the light is completely polarized in the vertical direction, and the probability that it will pass through a horizontal polarizer is zero (Fig. 3.8C). But if a polarizer with a different orientation, ϕ, is placed between the vertical and horizontal polarizers, photons will go through this second polarizer with probability cos2(ϕ), and (assuming that 0 < cos2(ϕ) < 1) some of those photons now will pass through the horizontal polarizer (Fig. 3.8D). The interpretation is that the vertically polarized light consists of a coherent superposition of light polarized parallel and perpendicular to the second polarizer, both of which have a non-zero projection on the horizontal axis. The intensity of the light passing the horizontal polarizer peaks at ϕ = 45∘, where it is 1/8 relative to the intensity reaching the second polarizer. Photons and other particles can remain in quantum mechanical superposition states even if they move apart on widely diverging paths. This happens when a Ca atom decays from a highly excited state by emitting two photons with parallel

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polarization. Although the angle of the polarization is entirely random if it is measured for either photon alone, a measurement of the polarization for one of the photons makes that for the other photon predictable. Pairs of photons with similarly “entangled” polarization can be created by frequency down-conversion in non-linear optical crystals [38, 39]. The polarizations of the two photons are correlated even if independent observers measure them at places so far apart that information about one observer’s result could not possibly reach the other observer in time to influence the result there. They persist even if the observers change the axes for their measurements while the photons are in transit. Einstein famously described such entanglement as “spooky action at a distance”. In a paper with B. Podolsky and N. Rosen [40], he illustrated the paradox that entanglement poses for properties determined by non-commuting operators such as position and momentum. According to the uncertainty principle, a measurement of either observable precludes exact knowledge of the other. But for two particles in a quantum mechanical superposition state, measuring one property for particle 1 evidently also determines its value for particle 2. By then measuring the other property for particle 2, we apparently can learn the values of both observables for that particle, in conflict with the uncertainty principle. Einstein, Podolsky, and Rosen (EPR) concluded that properties of particles in an entangled state must be predetermined by “hidden variables” that quantum mechanics fails to explain. This suggestion stimulated spirited debates with Niels Bohr on whether quantum mechanics has such hidden variables, whether a physical property becomes “real” only when it is measured, and whether physical reality is purely “local” or can apply to properties of widely separated objects as well. The Northern-Irish physicist John S. Bell provided an answer to the EPR paradox in 1964 when he discovered that theories of local reality are fundamentally inconsistent with hidden variables [41, 42]. Bell showed that all theories of strictly local reality with a hidden variable put limits on statistical correlations between observations for a pair of particles, but that particles in a quantum mechanical superposition state can exceed these limits. His proof involves the probability that, if one photon passes through a polarizer oriented at angle ϕ, the other will pass through a polarizer at angle (ϕ+θ). For a quantum-mechanical superposition, this combined probability depends on cos2θ. Hidden-variable theories predict a linear dependence on θ, which gives different results for some values of θ. Clauser et al. [43, 44] recast Bell’s theorem in terms of correlation functions that were amenable to experimental tests, and John Clauser, Alain Aspect, and Anton Zeilinger received the 2022 Nobel prize in physics for demonstrating that photons in superposition states do indeed exceed limits described by Bell’s theorem [45–47]. In one experiment [47], frequency down-conversion of light from an argon-ion laser provided pairs of photons that were sent via optical fibers to observers 400 m apart. Photons reaching the observation sites were switched randomly between two polarizers in different orientations, giving four correlation functions for the analysis. Although views of its philosophical implications still vary, quantum entanglement has become a basis for information transfer, cryptography, and quantum computing. It also has been invoked in attempts to develop a quantum mechanical theory of time [48].

3.6 Refraction, Evanescent Radiation, and Surface Plasmons

3.6

115

Refraction, Evanescent Radiation, and Surface Plasmons

We now consider in more detail what happens to the electric field and irradiance when a beam of light moves from a vacuum into a refractive, but non-absorbing medium. This will return us to the topic of polarization of the medium, which we introduced in Sect. 3.1.4. It will lead from there to birefringence and non-linear optical effects that are important for generating laser radiation at new frequencies. To start, suppose a light beam with irradiance Iinc is incident on a dielectric medium. At the interface, some of the radiation is transmitted, giving an irradiance Itrans that continues forward in the medium, while a portion with irradiance Irefl is reflected. Itrans must equal Iinc ‐ Irefl to balance the flux of energy across the interface (Fig. 3.9A): I trans = I inc - I refl :

ð3:53Þ

The fraction of the incident irradiance that is transmitted depends on the angle of incidence and the refractive index of the medium (n). Suppose the incident beam is normal to the surface so that the electric and magnetic fields are in the plane of the surface. For fields in the plane of the interface, the instantaneous electric field on the medium side of the interface (Etrans) must be equal to the field on the vacuum side, which is the sum of the fields of the incident and reflected beams (Einc and Erefl): Etrans = Einc þ Erefl

ð3:54Þ

This follows from the fact that the path integral of the field around any closed path is zero (Eq. (3.12) and Fig. 3.9a). By using Eq. (3.32), we can replace Eq. (3.53) with a second relationship between the fields: njEtrans j2 = jEinc j2 - jErefl j2 :

ð3:54Þ

Eliminating Erefl from Eqs. (3.53) and (3.54) then gives Etrans =

2 E : n þ 1 inc

ð3:55Þ

And finally, using Eq. (3.32) again, I trans = cnjEtrans j2 ρν ðνÞ=2π =

4n I inc : ð n þ 1Þ 2

ð3:56Þ

For a typical refractive index of 1.5 for glass, Eqs. (3.55–3.56) give jEtransj = 0.8jEvacj and Itrans = 0.96Ivac. The same approach can be used for other angles of incidence to generate Snell’s law, which relates the angle of the refracted beam to the refractive index [6]. In general, when light passes from a nonabsorbing medium with refractive index n1 to a second nonabsorbing medium with refractive index n2,

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Fig. 3.9 (A) When a light beam propagating in a vacuum enters a refractive medium (cyan shading), the incident irradiance (Iin) must equal the sum of the transmitted and reflected irradiances (Itrans and Irefl). In addition, the electric field of the transmitted (refracted) beam in the plane of the interface (Etrans) must equal the sum of the electric fields of the incident and reflected beams (Einc + Erefl) because the path integral of the field over any closed loop must be zero (Eq. (3.12)). Here, the angle of incidence is normal to the surface and the light is linearly polarized so that the electric field is parallel to edges ab and cd of rectangle abcd (vectors b - a and d - c), and perpendicular to edges bc and ad. The path integral of the field over the loop a → b → c → d → a is (Einc + Erefl)  (b ‐ a) + Etrans  (d ‐ c) = (Einc + Erefl‐Etrans)  (b - a)  (B) - (D) If a beam propagating in a medium with refractive index n1 (cyan shading) encounters an interface with a medium with refractive index n2 (no shading), the angle of the refracted beam relative to the normal (θ2) is related to the angle of incidence (θ1) by n2 sin (θ2) = n1 sin (θ1). The refracted beam is bent toward the normal if n2 > n1 (B) and away from the normal if n2 < n1 (C). Total internal reflection occurs if n2 < n1 and θ1 ≥ θc, where θc = arcsin (n2/n1) (D). In this situation, an evanescent wave (dotted red arrow) propagates along the interface and penetrates a short distance into the second medium. (E) If the surface of the medium with the higher refractive index is coated with a semitransparent layer of silver (darker blue shading) and the angle of incidence at the interface with the second medium matches a resonance angle qr of about 60o, total internal reflection creates surface plasmons in the metal coating, greatly enhancing the evanescent field (heavier dotted red arrow)

3.6 Refraction, Evanescent Radiation, and Surface Plasmons

n2 sin θ2 = n1 sin θ1 ,

117

ð3:57Þ

where θ1 and θ2 are the angles of the incident and refracted beams relative to an axis normal to the surface. If n2 > n1, the refracted beam is bent toward the normal (Fig. 3.5B); if n2 < n1, it is bent away (Fig. 3.5C). For n2 < n1, the refracted beam becomes parallel to the interface (θ2 = 90∘) when the angle of incidence reaches the “critical angle” θc defined by sinθc = n2/n1. Values of θ1 > θc give total internal reflection: all the incident radiation is reflected at the interface and no beam continues forward through the second medium (Fig. 3.5D). The critical angle is about 61.1∘ at a glass/water interface (n1 = 1.52 and n2 = 1.33) and 41.1∘ at a glass/ air interface. Because the electric and magnetic fields must be continuous across the interface, the radiation must penetrate a finite distance into the second medium even in the case of total internal reflection. Constructive interference of the incident and reflected fields in this situation creates a wave of evanescent (“vanishing”) radiation that propagates parallel to the interface but drops off quickly in amplitude beyond the interface. The fall-off of the field with distance (z) in the second medium is given by  - 1=2 λ1 =4π , which typically gives a E = E0 exp(-z/d ) with d = n21 sin 2 θ1 - n22 penetration depth on the order of 500 to 1000 nm [49, 50]. For angles of incidence slightly greater than θc, Eo ≈ Einc + Erefl ≈ 2Einc.. The intensity of the evanescent radiation at z = 0 is therefore about four times that of the incident radiation. The intensity decreases gradually as the angle of incidence is raised above θc. The existence of evanescent radiation can be demonstrated from the effects of objects in the second medium close to the interface. For example, if a third medium with a higher refractive index is placed near the interface between the first and second materials, radiation can tunnel through the barrier imposed by the second medium. This process, called attenuated total internal reflection, is essentially the same as the tunneling of an electron between two potential wells separated by a region of higher potential (Sect. 2.3.2). Isaac Newton is said to have discovered the phenomenon when he placed a convex lens against the internally-reflecting face of a prism: the spot of light entering the lens was larger than the point where the two glass surfaces touched. As we’ll discuss in Chap. 5, evanescent radiation also can excite fluorescence from molecules situated close to the interface. However, an absorbing medium does not obey Snell’s law because, for plane waves entering the medium at other than normal incidence, the amplitude of the surviving light is not constant across a wavefront of constant phase. A striking phenomenon called surface plasmon resonance can occur at an interface between glass and a medium such as air or water if the glass is coated with a partially transmitting layer of gold or silver (Fig. 3.5E). As the angle of incidence is increased above θc, total internal reflection of the light back into the glass first occurs just as at an uncoated surface. But at an angle of about 56∘ (the exact value depends on the wavelength, the metal coating, and the refractive indices of the two media), the intensity of the reflected beam drops almost to zero and evanescent radiation with intensity as high as 50 times the intensity of the incident

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radiation can be detected in the water or air on the far side of the interface [51–54]. The strong evanescent radiation reflects movements of electron clouds (“plasmon surface polaritons” or “surface plasmons” for short) in the conduction band of the metal, and the loss of the reflected beam at the resonance angle results from destructive interference of the reflected and surface waves [55]. In analogy with the concerted vibrational motions that comprise a phonon, the term “plasmon“refers to a coherent oscillation of the electron cloud at a particular frequency. Plasmons resemble photons and phonons in having oscillation modes with quantized frequencies, and also in being bosons in the sense that a metallic surface or particle can support multiple plasmons simultaneously. Radiation with a particular frequency can excite surface plasmon modes in resonance with that frequency. The crosssections for such excitations are especially large in metallic nanoparticles with diameters less than the wavelength of the radiation [56, 57]. Interactions of surface plasmons with molecules close to a metallic surface can be detected by their fluorescence, their strongly enhanced Raman scattering, or their effects on the resonance angle. If an antigen or other protein is tethered to the coated surface, changes in the amplitude or angle of surface plasmon resonance can be analyzed quantitatively for rapid assays of the concentration or binding properties of antibodies or other macromolecules in a test solution flowing over the surface [58]. Especially intense fields from surface plasmons can be generated in colloidal gold or silver, metal-coated nanoparticles that are small relative to the wavelength of the radiation and also in microscopically patterned surfaces made by lithography [57, 59–61]. Plasmons in metallic nanoparticles much smaller than the excitation wavelength are distributed over the whole particle and, because they are not propagating, are called “localized surface plasmons“(LSPs). The resonance condition of an LSP depends on the size of the nanoparticle and the dielectric constant of the surrounding medium. Spherical gold nanoparticles have a strong LSP resonance band near 520 nm that redshifts as either the size of the particle or the dielectric constant of the medium increases [61, 62]. The improved sensitivity of assays on nanostructured metallic surfaces derives from the local-field enhancement at edges and peaks on the surface. Enhancement of the field by factors up to 103 have been observed between nanoparticles separated by approximately 1 nm [63]. The plasmonic field enhancement falls off with the cube of the distance from the surface, making assays that depend on the field highly selective towards molecules that bind directly to the nanoparticle or are within the first few solvation layers above the surface [64].

3.7

The Classical Theory of Dielectric Dispersion

In classical physics, dielectric media were modeled by considering an electron bound to a mean position by a force (k) that increased linearly with the displacement (x) from this position. The potential energy of the electron then is proportional to x2 (a parabolic function of x) and the oscillations of the electron around the energy minimum are said to be harmonic. The classical equation of motion for such an electron is

3.7 The Classical Theory of Dielectric Dispersion

me

d2 x dx þ g þ kx = 0, 2 dt dt

119

ð3:58Þ

where me is the electron mass and g is a damping constant. The damping factor is included to account for the loss of energy as heat, which in the classical theory is assumed to depend on the electron’s velocity. The solution to Eq. (3.58) is x = exp (iω0t - γt/2) with γ = g/me and ωo = (k/me - γ 2/4)1/2, or ωo ≈ (k/me)1/2 if γ is small. The displacement thus executes damped oscillations at frequency ωo, decaying to zero at long times. An external electromagnetic field E(t) oscillating at angular frequency ω perturbs the positions of the electrons, setting up an oscillating polarization field P(t) that contributes to the total field. The equation of motion for an electron under these conditions becomes me

d2 x dx 4π þ g þ kx = eEðt Þ þ ePðt Þ, 2 dt 3 dt

ð3:59Þ

where e is the electron charge. The total polarization for a unit volume of homogeneous material is proportional to the number of electrons in that volume (N ), the electron charge, and the time-dependent displacement: Pðt Þ = xðt ÞeN

ð3:60Þ

Combining Eqs. (3.59) and (3.60) gives a differential equation for P(t) with the following solution when γ is small [4]:  2 Eðt Þ Ne Pðt Þ = : ð3:61Þ  3me ðωo Þ2 - ω2 - 4πNe2 =3me þ iγω According to this expression, P oscillates at the same frequency as E, but with an amplitude that is a complex function of the frequency. The complex susceptibility (χ e) is defined as the ratio of the polarization to the field:  2 1 Ne χ e ðωÞ = Pðt Þ=Eðt Þ = : ð3:62Þ  3me ðωo Þ2 - ω2 - 4πNe2 =3me þ iγω The oscillating electron described above is said to have an oscillator strength of unity, which means that it responds to light’s electric field exactly as predicted by classical physics. From a contemporary perspective, the oscillator strength is the ratio of an electron’s actual response, as explained by the quantum mechanical treatment we develop in Chap. 4, to the classical prediction (see Eq. 4.18). The classical theory continues by assuming that each molecule in a dielectric medium could have a set of S oscillation frequencies (ωs), damping factors (γ s), and dimensionless oscillator strengths (Os), with the possible values of Os ranging from 0 to 1 [4]. To allow for various values of Os, a characteristic response frequency Ωs for each oscillation frequency can be defined by writing

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  4πN m e2 Os , ð Ω s Þ = ð ωs Þ 3me 2

2

Light

ð3:63Þ

where Nm is the number of molecules per unit volume. The complex susceptibility then becomes a weighted sum of the S oscillator strengths:   S Οs N m e2 X : χe = me s = 1 ðΩs Þ2 - ω2 þ iγ s ω

ð3:64Þ

Equation (3.64) is the fundamental equation of classical dispersion theory [4]. Because χ e is related to the high-frequency dielectric constant by Eq. (3.30), and the high-frequency dielectric constant is the square of the refractive index, it appears that the dielectric constant and refractive index also should be treated as complex quantities. To indicate this, we’ll rewrite Eqs. (3.19) and (3.30) using εc and nc to distinguish the complex dielectric constant and refractive index from, ε and n, the more familiar, real quantities that apply to non-absorbing media: ðnc Þ2 = εc = 1 þ 4πχ e :

ð3:65Þ

The meaning of the complex refractive index will become clearer if we consider what happens when a plane wave of monochromatic light passes from a vacuum into an absorbing, but non-scattering dielectric medium. If the propagation axis ( y) is normal to the surface, the electric field at the interface can be written E = Eo exp½iðωt - κyÞ þ Eo exp½- iðωt - κyÞ,

ð3:66Þ

where ω is the angular frequency, κ = nω/c, and n is the ordinary refractive index. According to Lambert’s law (Eq. (1.3)), the intensity of the light will decrease exponentially with position as the ray moves through the medium. If the absorbance is A, the amplitude of the electric field falls off as exp.(-ay), where a = Aln(10)/2. The field in the medium thus is E = Eo exp½iðωt - κyÞ - ay þ Eo exp½- iðωt - κyÞ - ay n h   io n h   io n a n a = Eo exp iω t 1 - i y þ Eo exp - iω t 1-i y c κ c κ     n h io n n = Eo exp iω t - c y þ Eo exp - iω t - c y , ð3:67Þ c c where nc = n - iðn=κ Þa = n - iðc=ωÞa:

ð3:68Þ

Equation (3.68) shows that the imaginary part of nc is proportional to a, and thus to the absorbance, whereas the real part of nc pertains to refraction. The real part of nc cannot, however, be used in Snell’s law to calculate the angle of refraction for light

3.7 The Classical Theory of Dielectric Dispersion

121

entering an absorbing medium, because Snell’s law does not hold in the presence of absorption. Let’s examine the behavior of the complex refractive index in the region of an absorption band representing an individual oscillator with natural frequency ωs and damping constant γ s. If ωs is well removed from the frequencies of the other oscillators in the medium, we can rewrite Eqs. (3.64) and (3.65) as  2 Οs Ne ðnc Þ2 = ðno Þ2 þ 4π , ð3:69Þ me ðΩs Þ2 - ω2 þ iγ s ω where no is the contribution of all the other oscillators to the real part of nc. Absorption due to these other oscillators is assumed to be negligible over the frequency range of interest. Making the approximations ω ≈ Ωs and Ω2s - ω2 = ðΩs þ ωÞðΩs - ωÞ ≈ 2Ωs ðΩs - ωÞ, we obtain   1 4πNe2 Οs ðnc Þ2 = ðno Þ2 þ me Ωs 2ðΩs - ωÞ þ iγ s ω #  " 2 ð Ω s - ωÞ γs 4πNe2 Οs 2 = ð no Þ þ -i : ð3:70Þ me Ωs 4 ð Ω s - ωÞ 2 þ ð γ s Þ 2 4 ð Ω s - ωÞ 2 þ ð γ s Þ 2 From Eq. (3.68), we also have ðnc Þ2 = n2 - ðca=ωÞ2 - 2iðca=ωÞ: Equating the real and imaginary parts of Eqs. (3.70) and (3.71) gives #  " Ωs - ω 8πNe2 Οs 2 2 2 n - ðca=ωÞ - ðno Þ = me Ω s 4ðΩs - ωÞ2 þ ðγ s Þ2

ð3:71Þ

ð3:72Þ

and 

2πNe2 Οs ca=ω = me Ω s

"

# γs : 4ðΩs - ωÞ2 þ ðγ s Þ2

ð3:73Þ

If (n - no) and ca/ω are small relative to no, the left-hand side of Eq. (3.72) is approximately equal to 2no(n - no), and #  " Ωs - ω 4πNe2 Οs n ≈ no þ : ð3:74Þ no me Ωs 4 ð Ω s - ωÞ 2 þ ð γ s Þ 2 Figure 3.10A shows plots of ca/ω and n - no as given by Eqs. (3.73) and (3.74). In agreement with experiment, the theory predicts that the contribution of the oscillator to the refractive index will increase with frequency below the absorption maximum at Ωs, but then will change sign. The inversion of the slope near the absorption peak is known as anomalous dispersion. The classical dispersion theory also reproduces the Lorentzian shape of a homogenous absorption line.

A

B

Refractive Index

3

(n - no) / (F/no), (ca/ω ) / F

122

2.4 2.3 2.2

no ne

2.1

0.5

(ω − Ωs) / γs

Light

1.0 Wavelength / μ m

1.5

Fig. 3.10 (A) Plots of the absorption (ca/ω, solid green curve) and refractive index (n – no, dashed blue curve) as functions of angular frequency (ω) in the region of an absorption band centered at frequency Ωs, as predicted by the classical theory of dielectric dispersion (Eqs. (3.73, 3.74)). Frequencies are plotted relative to the damping constant γ s; ca/ω, relative to the factor F = 2πNe2fs/meΩs; and (n - no), relative to the factor F/no. (B) Refractive indices of crystalline lithium niobate as functions of wavelength for light propagating along the ordinary (no, cyan curve) or extraordinary (ne, red) axes. The curves were generated by a three-term Sellmeier equation with coefficients that reproduce the measure refractive indices [82]

In spectral regions that are far from any absorption bands, the term iγ sωs in Eq. (3.64) drops out and the refractive index becomes purely real. The predicted frequency dependence of the refractive index then becomes  n2 - 1 =

 S Οs 4πN m e2 X : 2 2 me ð Ω Þ s -ω s=1

ð3:75Þ

Refractive indices measured experimentally (again, for frequencies remote from absorption bands) can be fit well by including two or three terms in the sum and adjusting the values ofPOs and Ωs phenomenologically (Fig. 3.10B). The empirical relationship n2 = 1 þ Ss = 1 As =ðBs - ω2 Þ with adjustable coefficients AS and BS is known as the Sellmeier equation after Wilhelm Sellmeier, who proposed it in 1872. The quantum theory of electric susceptibility is discussed in Box 12.1.

3.8

Nonlinear Optics

The classical theory described in the previous sections considers a linear dielectric, in which the polarization of the medium (P) is directly proportional to the electric field (E) (Eq. 3.29). This linearity can break down at high field strengths, revealing components of P that depend on the square or higher powers of E. The polarization then becomes a sum of terms,

3.8 Nonlinear Optics

123

P = χ ð1Þ E þ χ ð2Þ E2 þ χ ð3Þ E3 ⋯,

ð3:76Þ

with coefficients χ (1), χ (2) and χ (3) called the first-, second-, and third-order susceptibilities. If the field strength oscillates at frequency ω, the second-order component of P will oscillate as cos2ωt, and because cos2ω = [1 + cos(2ω)]/2, this term results in oscillations of the polarization at twice the fundamental frequency. Because cos3ω = [3cosω + cos(3ω)]/4, the third-order term leads to oscillations at 3ω. Further, if the radiation field includes two frequencies, ω1 and ω2b, the secondorder term can result in oscillations of P at ω1 ± ω2. Studies of nonlinear spectroscopic phenomena blossomed with the development of pulsed lasers, which can provide electromagnetic fields strong enough to bring even terms as high as χ (5 into play [65–71]. Quantum mechanical treatments of how radiation fields interact with matter will be presented in later chapters. Although these treatments have largely supplanted the classical theory, the classical treatment leads to several conclusions that are pertinent to the laser applications of nonlinear spectroscopy that we discuss in Sections 3.9 and 3.10 of the present chapter and in Chap. 4. The most important point is that the oscillating induced dipoles created by polarization of a dielectric medium can themselves serve as a source of radiation. Induced electric dipoles oscillating at frequencies other than the radiation’s fundamental frequency therefore can emit radiation with those frequencies. Nonlinear optical effects such as frequency doubling and three-wave mixing depend on this emission. In Chap. 4, we discuss how the dielectric material and its orientation with respect to a light beam can be chosen to optimize the transfer of radiation energy to a particular frequency. A second point to note is that the interaction of an oscillating electromagnetic field with a material depends on the positions of all the atoms. Consider a molecule such as nitrobenzene (C6H5 ‐ NO2). An oscillating field can cause electron density to move back and forth between the phenyl ring and the NO2 group, creating an induced dipole that oscillates along the CN axis. Depending on its orientation, the field also could shift electron density back and forth in a direction approximately perpendicular to this axis within the NO2 group. The magnitudes of the induced dipoles along these different axes are not independent, because the transfer of electron density from the phenyl ring raises the effective number of electrons for induced dipoles within the NO2 group, and the transfer of electron density from the NO2 group to the phenyl ring increases the effective number of electrons for induced dipoles in the ring. This coupling of polarization along different axes creates a quadratic dependence of the total polarization on the field, making χ (2)E2 non-zero. Now consider a molecule such as p-dinitrobenzene, which is centrosymmetric in the sense that each atom with coordinates ri = (x, y, z) relative to the center of electron density has a counterpart at rj = ‐ ri = (‐x, ‐y, ‐z). Again, an oscillating electric field E(t) polarized along the N⋯C6H4⋯N axis creates an induced dipole oscillating along that axis, and a field with perpendicular polarization creates induced dipoles with this orientation in the NO2 groups. But in this case, oscillations of the induced dipoles along the two axes do not reinforce each other because the

124

3

Light

transfer of electron density from the phenyl ring to one of the NO2 groups is balanced by a simultaneous transfer to the ring from the other NO2 group. The quadratic term χ (2)E2 in the polarizability thus is zero for a centrosymmetric molecule. Considerations along these lines explain why nonlinear spectroscopic effects vary widely among materials with different symmetry and can be used to design materials for non-linear optics. Radhakrishnan [72] gives examples for a variety of organic materials. The higher-order terms in Eq. (3.76) also can be related to classical potentialenergy surfaces for the displacement of electrons from their mean positions. If the restoring force for return to the mean position is directly proportional to the displacement, the potential energy will be a parabolic (harmonic) function of the displacement, and the magnitude of the induced dipole will be directly proportional to E(t) as described in Eq. (3.60). But if the field is large enough for the departure of the potential-energy surface from a parabola (anharmonicity) to be significant, higher-order terms of the polarizability become important. Odd powers of the displacement usually dominate over even powers in this respect, because they represent asymmetry of the potential-energy function: displacing an electron in one direction typically increases the energy somewhat differently than displacing it by the same amount in the opposite direction. Anharmonicities thus tend to make χ (3)E3 and other odd-numbered terms in the polarizability more important than the even-numbered terms.

3.9

Birefringence and Electro-Optic Effects

The classical theory of dielectric dispersion relates a material’s refractive index to electronic motions induced by the oscillatory electric field of light. In anisotropic materials, the amplitudes and frequencies of these motions generally depend on the orientation of the field vector (E) relative to one or more molecular axis. Materials that have different refractive indices for light propagating in different directions or light beams with different polarizations are said to be birefringent and to exhibit birefringence. Rasmus Bartholin, a Danish physician, is credited with discovering the birefringence of calcite (CaCO3) crystals in 1669, although Viking sailors used the angle of the sun viewed through “Iceland spar“for navigation at much earlier times. Christiaan Huygens, Isaac Newton, and others struggled to account for the phenomenon, but it was not until 1822 that the great French physicist Augustin Fresnel provided a satisfactory explanation. Calcite is a uniaxial crystal, which means that it has a single optical axis (Fig. 3.11A). The refractive index for a beam of light traveling through calcite along this axis is independent of the polarization of the light, whereas a beam propagating normal to the optical axis experiences a refractive index of 1.602 for 590 nm light polarized perpendicular to this axis and 1.486 for parallel polarization. As a result, unpolarized light entering a calcite crystal normal to the surface but at an angle to the optical axis is split into two rays with orthogonal polarizations. One ray, called the ordinary ray, passes straight through the crystal, while the extraordinary ray diverges from that path. If these rays

3.9 Birefringence and Electro-Optic Effects

125

Fig. 3.11 (A, B) Calcite crystals comprise alternating layers of Ca2+ (yellow) and CO23 - (carbonate, gray and red) ions. The crystallographic unit cell has edge lengths a = b = 4.990 Å and c=17.062Å and interaxial angles α = β = 90.0∘ and γ = 120.0∘. A shows a 2 × 2 × 2 lattice of unit cells viewed along the c axis, and B shows the same lattice viewed along the axis normal to b and c. The crystallographic axes in the plane of each image are labeled and an individual unit cell is outlined in black. Calcite‘s optic axis is parallel to c. (C, D) Crystal structure of lithium niobate (LiNbO3) with lithium ions represented in purple, niobium in cyan, and oxygen in red. The unit cell has edge lengths a = b = 5:148Å and c = 13:863Å and angles α = β = 90.0∘ and γ = 120.0∘. The lattice, perspectives of the images, labels, and outlines of unit cells in C and D are similar to those in A and B.

are reflected back through the crystal, an observer sees a double image of the reflecting surface. Ordinary ice also is a uniaxial crystal, having refractive indices of 1.3090 and 1.3104 for the two polarizations [73]. Figure 3.10B shows the wavelength dependence of the refractive indices for light propagating along the ordinary and extraordinary axes of crystalline lithium niobate, another uniaxial material that is used extensively in nonlinear spectroscopy. Mica and lead oxide are examples of biaxial crystals, which have two optical axes and three refractive indices. The polarization of a light beam can be rotated by passage through a wave plate made by cutting a crystal of calcite, quartz, or another uniaxial material so that its optical axis is on the front surface of the plate. The ordinary and extraordinary axes then also lie in the plane of the surface, with the extraordinary axis parallel to the optic axis and the ordinary axis at 90o (see Fig. 3.12). Waveplates commonly are designed to act on a beam of light that is incident normal to the surface and polarized at ±45o with respect to both o and e and. The beam’s electric vector can be viewed as the sum of vectors oscillating parallel to o and e that are in phase where the beam enters the plate but get progressively out of phase as the beam proceeds. Passage

126

3

Light

Fig. 3.12 A half-wave plate is made from a uniaxial birefringent crystal cut so that its optic axis (dashed vertical black line) is in the plane of the plate’s front surface (white square). Axis e is parallel to the optic axis, and o is an orthogonal axis in the same plane. A beam of light (dotted red arrow) with its electric field vector (dark blue arrow) polarized at ±45∘ relative to e and o is incident on the crystal normal to the surface. The light blue and green arrows represent the components of the light’s electric field vector parallel and perpendicular to the optic axis, respectively. As the beam passes through the crystal (yellow block), the e and o components get out of phase by an amount Γ. If the thickness of the plate (L ) is chosen to give Γ = π, the resultant of the e and o components in the beam emerging from the plate (dark blue arrow in the gray square) is rotated 90∘ relative to the incident light.

through a plate of thickness L leads to a phase difference of Γ = 2πΔnL/λ0, where Δn is the difference between the refractive indices for light rays parallel and perpendicular to the optic axis and λ0 is the wavelength of the light in free space. A half-wave plate is cut to give a phase difference of π between the o and e components in the light exiting the plate, which has the effect of rotating their resultant by 90o relative to the polarization of the incident light (Fig. 3.12). Quarter-wave plates cut to give Γ = π/2 act to interconvert linear and circular polarization. Some materials that ordinarily are isotropic become birefringent if they are placed in an electric field. John Kerr discovered in 1875 that liquids exhibit a birefringence that increases quadratically with the strength of the field, a phenomenon now called the Kerr effect. The Kerr effect occurs to some extent in all liquids but is strongest in polar liquids such as nitrobenzene, which become increasingly ordered in an electric field. Birefringence that increases linearly with the strength of the field is known as the Pockels effect after Friedrich Pockels, who discovered it in 1893. The Pockels effect is seen in some glasses and crystals that lack inversion symmetry, such as lithium niobate (Fig. 3.11C, D), where it dominates strongly over the Kerr effect. Laser Q-switches can be made by placing a Pockels or Kerr cell between crossed polarizers that spoil passage of light through the laser cavity in the absence of an electric field. Switching a high-voltage field rapidly on and off creates a transient birefringence that rotates the polarization of light between the polarizers, allowing

3.10

Optical Wavepackets and Mode-Locked Lasers

127

lasing to occur for a limited period. Electro-optical devices working on the same principle can be used to modulate the amplitude, polarization, or phase of light beams at high frequencies. In the optical Kerr effect, the electric field that creates birefringence comes from the light itself, with the result that the change in the index of refraction is proportional to the irradiance of the light beam. Light traveling through a material that responds to the irradiance in this way can create a lens through which the beam focuses itself. The birefringence of uniaxial materials such as lithium niobate plays an important role in the use of nonlinear optical techniques to generate second-harmonics and other frequencies. For these techniques to work efficiently, radiation with two frequencies should remain in phase as light travels through an optical component where second- or higher-order polarization occurs. This may not be possible in a non-birefringent material, because the refractive index generally varies with the frequency (Fig. 3.10). It can be achieved in birefringent materials by adjusting the orientation of the nonlinear device so that the difference between the refractive indices for the ordinary and extraordinary rays cancels out the difference due to dispersion. Such adjustments are referred to as phase matching. In some materials, phase matching can be obtained by controlling the temperature of the optical element. Birefringence can be seen in anisotropic biological macromolecules such as collagen and in materials that become anisotropic when they are stretched or compressed mechanically. It has important applications in microscopy, where birefringent objects can stand out in fields illuminated with polarized light and viewed through a crossed polarizer. Different dependencies of birefringence on the wavelength of light can be used to distinguish between some superficially similar materials. For example, crystals of monosodium urate and calcium pyrophosphate, which must be identified to distinguish between the disorders gout and pseudogout, take on characteristic colors under a polarizing microscope.

3.10

Optical Wavepackets and Mode-Locked Lasers

The idealized wave described by Eq. (3.15a) or (3.15b) continues indefinitely for all time and all values of y. Any real beam of light must start and stop at some point, and so cannot be described completely by this expression. It can, however, be described by a linear combination of idealized waves with a distribution of frequencies. Such a combination of waves is called a wave group or wavepacket. The details of the distribution function determine the width and shape of the pulse. This description is essentially the same as using a linear combination of wavefunctions for a localized particle in a box (Sect. 2.3.2) or a harmonic potential well (Sect. 2.3.3 and Chap. 11). The pulses of laser light produced by Q-switching as described in Sects. 1.13 and 3.6 include a band of frequencies that depend partly on the spectrum of stimulated emission from the active medium, and partly on the length of the path between the mirrors that reflect light back and forth through the medium. Two flat mirrors

128

3

Light

separated by a distance L form what is called a Fabry-Pérot cavity for light with wavelengths λ = 2L/n where n is any positive integer. Oscillations with these wavelengths create standing waves in the cavity, while destructive interference quenches oscillations with other wavelengths. The wavelengths of light that satisfy the condition λ = 2L/n, and for which stimulated emission outweighs losses from light scattering and misdirected reflection or refraction, constitute the oscillation modes of the laser. The 300-nm spectrum of stimulated emission from a Ti:sapphire crystal is broad enough to support approximately 250,000 modes. In a Q-switched laser, oscillations in numerous modes occur independently with random phases, giving an approximately Gaussian distribution of frequencies around the peak for stimulated emission. The resulting wavepackets typically have widths on the order of 20 ns. To obtain shorter pulses, it is necessary to bring oscillations of multiple modes into fixed phase relationships, so that they interfere constructively at regular intervals and destructively at other times. This is called mode-locking and lasers with such fixed phase relationships are said to be modelocked or phase-locked. Methods of mode-locking are classified as either “passive” or “active”. Mode locking can occur passively if the laser cavity includes an optical element that bleaches transiently when it is excited (a saturable absorber), or that undergoes self-focusing by the optical Kerr effect described in Sect. 3.6. These elements preferentially transmit the strongest pulses passing through them, allowing their wavepackets to undergo further amplification while weaker light is absorbed or dispersed. Active mode-locking can be achieved by including an electro-optic modulator to favor pulses that pass through it at intervals corresponding to the cavity’s round-trip time, 2L/c. Other possibilities are to pump the laser synchronously with light from another pulsed laser or to use an acousto-optic device in which a piezo-electric driver modulates diffraction of light by a crystal in the cavity. Acousto-optic modulators also can be used to “dump” radiation from the laser cavity at intervals, allowing a larger buildup of wavepackets in the cavity between dumps. Further details on techniques for generating and working with short laser pulses are available in the books by Fleming [74] and Demtröder [75, 76]. The wavepackets produced by mode-locked Ti:sapphire or dye lasers typically have Gaussian or sech2(t) shapes. If all the oscillation modes are in phase at t = 0, when the electric field of the light peaks, the dependence of the field strength on time in a Gaussian pulse can be written  - 1=2  exp - t 2 =2τ2 cosðωo t Þ, Eðt Þ = 2π τ2

ð3:77Þ

where τ is a time constant and ωo is the center angular frequency of the light (radians per second, or 2π times the center frequency in herz). The Gaussian function exp(-t2/2τ2) has a full width at half-maximal amplitude (FWHM) of (8  ln 2)1/2τ, or 2.355 τ. The factor (2πτ2)‐1/2 normalizes the area under the curve to 1. The measured intensity of the pulse, being proportional to |E(t)|2 or exp(‐t2/τ2), would be narrower by a factor of 21/2. Its FWHM thus is 2(ln2)1/2τ, or 1.665τ.

3.10

Optical Wavepackets and Mode-Locked Lasers

129

We can equate the time-dependent function on the right side of Eq. (3.76) to a frequency-dependent function 1 jEðt Þj = pffiffiffiffiffi 2π

Z1 GðωÞ expðiωt Þdω,

ð3:78Þ

-1

where G(ω) is the distribution of angular frequencies in the pulse. This means that the temporal shape of the field pulse, jE(t)j, is the Fourier transform of the frequency distribution G(ω), and conversely, that G(ω) is the Fourier transform of jE(t)j (Appendix A3). For the Gaussian shape function exp(‐t2/2τ2), the solution to G(ω) is   n o τ p ffiffiffiffiffi GðωÞ = ð3:79Þ exp - ðω - ωo Þ2 τ2 =2 : 2π The field thus includes a Gaussian distribution of angular frequencies around ωo, with a FWHM of (8  ln 2)1/2τ, or 2.35/τ radians/s. In frequency units, the FWHM is (8  ln 2)1/2/2πτ, or 0.375/τHz. The sharper the pulse, the broader the spread of frequencies. The measured spectrum of the intensity again is narrower by a factor of 21/2 and has a FWHM of 0.265/τHz. A Gaussian pulse with τ = 6 fs (a FWHM of 10 fs, the order of magnitude of the shortest pulses that can be generated by current mode-locked Ti:sapphire lasers) includes a span of about 6.25 × 1013 Hz, which corresponds to an energy (hν) band of 2.0 × 103 cm-1. If the spectrum is centered at 800 nm (12, 500 cm-1), its FWHM is 274 nm. Figure 3.13 shows the energy distribution for such a pulse and for pulses with FWHM’s of 20 and 50 fs. For a Gaussian pulse, the product of the measured temporal width (1.665/τ) and the frequency width (0.265/τ) is fixed at 0.441. Fleming [74] gives corresponding expressions for pulses with other shapes. For a square pulse, the product of the measured temporal and frequency widths is 0.886; for a pulse in which the intensity is proportional to sech2(t), the product is 0.315. These expressions assume that the frequency distribution arises solely from the finite length of the pulse. Such a pulse is said to be transform-limited. Light from an incoherent source such as a xenon flash lamp contains a distribution of frequencies that are unrelated to the length of the pulse because atoms or ions with many different energies contribute to the emission. Figure 3.14 shows the electric field at a fixed time as a function of position in wave groups with three different distributions of wavelengths. If the distribution is very narrow relative to the mean wavelength (λ0), the wave group resembles a pure sine wave over many periods of the oscillation. With broader distributions, the envelope of the oscillations is more bunched, and the envelope moves through space with a group velocity dispersion (GVD) that can be written [4].   c λ dn , ð3:80Þ GVDðλÞ = 1þ n n dλ

130

3

0.4

-1

G / (1000 cm )

Fig. 3.13 Distributions of energy (G(hν) with hν in units of cm‐1) in the electric fields of radiation for homogeneously broadened Gaussian pulses with widths (FWHM) of 10, 20, and 50 fs (Gaussian width parameters τ = 6, 12, and 30 fs; cyan, green and dark blue curves, respectively)

0.3

0.2

0.1

0.0

1000 2000 -2000 -1000 0 -1 hν / cm

1.0 0.5 E / Eo

Fig. 3.14 Amplitude of the electric field as a function of position along the propagation axis ( y) in wave groups with three different Gaussian distributions of wavelengths. The widths (FWHM) of the distributions are 0.1% (dotted blue curve), 2% (dashed cyan curve), or 5% (solid green curve) of the mean wavelength (λ0). The amplitudes are normalized at y=0

Light

0 -0.5 -1.0 -5.0

-2.5

0 y / λo

2.5

5.0

where n is the usual refractive index. Group velocity dispersion smears out the envelope with time as the individual oscillations get increasingly out of phase. For most liquids and the wavelengths of visible light, the GVD is within 5% of c/n.

3.11

Local-Field Correction Factors

To conclude this chapter, we consider an absorbing molecule dissolved in a medium with linear polarizability. If the molecular polarizability is different from the polarizability of the medium, the local electric field “inside” the molecule (Eloc) will differ from the field in the medium (Emed). The ratio of the two fields (| Eloc| /| Emed| ), or localfield correction factor ( f ), depends on the shape and polarizability of the molecule and the refractive index of the medium. One model for this effect is an empty spherical

3.11

Local-Field Correction Factors

131

Fig. 3.15 The effective electric field acting on a molecule (empty or stippled circle) in a polarizable medium (light blue box) is Eloc = fEmed, where Emed is the field in the medium and f is the local-field correction factor. In the cavity-field model (A) Eloc is the field that would be present if the molecule were replaced by an empty cavity (Ecav); in the Lorentz model (B) Eloc is the sum of Ecav and the reaction field (Ereact) resulting from the polarization of the medium by induced dipoles within the molecule

cavity embedded in a homogeneous medium with dielectric constant ε. For highfrequency fields (ε = n2), the electric field in such a cavity is given by   3n2 Ecav = ð3:81Þ E = f cav Emed 2n2 þ 1 med [77]. Although a spherical cavity is a very simplistic model for a molecule, models of this type are useful in quantum mechanical theories that include explicit treatments of a molecule’s electronic structure. The macroscopic dielectric constant ε or n2 can be used to describe the electronic polarization of the surrounding medium, while the intramolecular electrons are treated microscopically. Equation (3.81) neglects the reaction field due to polarization of the medium by the molecule itself. The reaction field results partly from interactions of the medium with oscillating dipoles that are induced in the molecule by the electromagnetic radiation (Fig. 3.15). (Again, we are concerned only with electronic induced dipoles that can follow the high-frequency oscillation of electromagnetic radiation. If the medium contains molecules that can rotate or bend so as to realign their permanent dipole moments, the reaction field also includes a static component.) The highfrequency field acting on a spherical molecule can be taken to be the sum of the reaction field and the cavity field described by Eq. (3.36). An approximate expression for this total field, due to H. A. Lorentz [78], is  2  n þ2 Emed = f L Emed : ð3:82Þ EL = 3

132

3

2.0 Local Field Correction Factor

Fig. 3.16 The dimensionless cavity-field ( fcav, dashed blue curve) and Lorentz ( fL, solid green curve) correction factors for the local electric field acting on a spherical molecule in a homogeneous medium, as a function of the refractive index of the medium

Light

1.5

1.0

0.5

0 1.0

1.2

1.6 1.4 Refractive Index

1.8

The factor (n2 + 2)/3 is called the Lorentz correction. Liptay [79] gives expressions that include the molecular radius, dipole moment, and polarizability explicitly. More elaborate expressions for f also have been derived for cylindrical or ellipsoidal cavities that are closer to actual molecular shapes [80, 81]. Figure 3.16 shows the local-field correction factors given by Eqs. (3.81) and (3.82). The Lorentz correction is somewhat larger and may tend to overestimate the contribution of the reaction field because the cavity-field expression agrees better with experiment in some cases (Fig. 4.5). With the local-field correction factor, the relationships between the energy density and irradiance in the medium (ρ(ν) and I(ν)) and the amplitude of the local field (| Eloc(o)| ) become: ρðνÞ = n2 jElocðoÞ j2 ρν ðνÞ=2πf 2

ð3:83Þ

I ðνÞ = cnjElocðoÞ j2 ρν ðνÞ=2πf 2 :

ð3:84Þ

and

Local-field corrections should be used with caution, bearing in mind that they depend on simplified theoretical models and cannot be measured directly.

3.12

Questions

1. Consider three particles with the following charges (esu) and coordinates (cm): q1 = 0.5, r1 = (1.0, 0.0, 0.5); q2 = - 0.6, r2 = (-0.5, 0.5, 0.0); q3 = - 0.4, r3 = (1.5, 1.0, -0.5). Calculate the following electrostatic quantities and give the

3.12

2.

3.

4.

5. 6.

7.

8. 9.

Questions

133

results in both CGS and MKS units: (a) the electrostatic field at particle 1 from particles 2 and 3; (b) the potential at particle 1; (c) the energy of electrostatic interaction of particle 1 with with field; (d ) the electrostatic force acting on particle 1; and (e) the total electrostatic energy of the system. (a) How long does it take a photon to travel from the sun to the earth, assuming that the photon takes the most direct path (mean distance, 1.496 × 1011m) and neglecting the effects of the earth’s atmosphere? (b) Suppose a 500-nm photon travels simultaneously on two paths that differ in length. What difference in length would make the radiation arrive 180∘ out of phase so that the resultant field strength is zero? Suppose a laser emits light at 800 nm with a power of 1 W. (a) What is the photon flux (photons/s) of the laser beam. (b) If the area of the beam profile is 0.1 cm2, what is the energy flux (W/cm2)? (c) what would the amplitude of the electric field be? Consider a plane wave of light passing through a medium with a refractive index of 1.2. (a) Neglecting local-field corrections, what is the magnitude (|E0|) of the oscillating electric field at frequency ν if the irradiance I(ν)dν is 1 watt cm-2? (b) What is the total energy density (ρ(v)dv) in frequency interval dν? (c) Including the cavity-field correction, calculate the magnitude of the field acting on a molecule in a spherical cavity embedded in the medium. (d) Calculate the magnitude of the field acting on a molecule using the Lorentz correction. (e) What component of the field is included (approximately) in the Lorentz correction but not in the cavity-field correction. What is the energy density of radiation at 500 nm emitted by a black-body source at the surface temperature of the sun (6600 K)? (a) What are the frequencies (s-1) of the first six modes of a radiation field in a rectangular box with dimensions 1000 × 1500 × 2000Å3? (“First” here means the modes with lowest energies.) (b) What is the total energy of these six modes if there are no photons in any of the modes. (Indicate the convention you use for zero energy.) (c) Using the same convention for zero energy, what is the total energy of the six modes if there is one photon in the first mode, two in the second, and none in the higher modes? (a) A light beam propagating along the x-axis of a Cartesian coordinate system is passed through a polarizer oriented in either the y- or the z-direction. The intensity of the light measured with the y orientation of the polarizer is 1.3 times that measured with z. What is the ellipticity of the light? (b) What ellipticity corresponds to complete circular polarization? What quantities determine the energy flux (energy per cm2 per second) of quantized monochromatic electromagnetic radiation? What are the spectral bandwidths (FWHM of the frequency distributions) of pulses of 800-nm light that have a Gaussian temporal shape with widths of (a) 1 fs and (b) 1 ps? Show the steps to get to your answer.

134

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References 1. Maxwell, J.C.: A dynamical theory of the electromagnetic field. Philos. Trans. Roy. Soc. 155, 459–512 (1865) 2. Hameka, H.: Advanced Quantum Chemistry. Addison-Wesley, Reading, MA (1965) 3. Maxwell, J.C.: A Treatise on Electricity and Magnetism. Clarendon Press, Oxford (1873) 4. Ditchburn, R.W.: Light, 3rd edn. Academic, New York (1976) 5. Schatz, G.C., Ratner, M.A.: Quantum mechanics in chemistry. Prentice-Hall, Englewood Cliffs, NJ. 325 (1993) 6. Griffiths, D.J.: Introduction to Electrodynamics, 3rd edn. Prentice-Hall, Upper Saddle River, NJ (1999) 7. Brillouin, L.: Wave Propagation and Group Velocity. Academic, New York (1960) 8. Knox, R.S.: Refractive index dependence of the Förster resonance excitation transfer rate. J. Phys. Chem. B. 106, 5289–5293 (2002) 9. Ashkin, A.: Acceleration and trapping of particles by radiation pressure. Phys. Rev. Lett. 24, 156–159 (1970) 10. Ashkin, A., Dziedzic, J.M., Bjorkholm, J.E., Chu, S.: Observation of a single-beam gradient force optical trap for dielectric particles. Opt. Lett. 1, 288–290 (1986) 11. Ashkin, A., Dziedzic, J.M.: Optical trapping and manipulation of viruses and bacteria. Science. 235, 1517–1520 (1987) 12. Woodside, M.T., Block, S.M.: Reconstructing folding energy landscapes by single-molecule force spectroscopy. Annu. Rev. Biophys. 43, 19–39 (2014) 13. Neupane, K., Foster, D.A.N., Dee, D.R., Yu, H.Z., Wang, F., Woodside, M.T.: Direct observation of transition paths during the folding of proteins and nucleic acids. Science. 352, 239–242 (2016) 14. Hoffer, N.Q., Woodside, M.T.: Probing microscopic conformational dynamics in folding reactions by measuring transition paths. Curr. Opin. Chem. Biol. 53, 68–74 (2019) 15. Atkins, P.W.: Molecular Quantum Mechanics, 2nd edn. Oxford Univ. Press, Oxford (1983) 16. Landau, L.D., Lifshitz, E.M.: Statistical Physics. Addison-Wesley, Reading, MA (1958) 17. Planck, M.: The Theory of Heat Radiation (2nd Ed, Engl transl by M. Masius). Dover, New York (1959) 18. Touloukian, Y.S., DeWitt, D.P.: Thermophysical Properties of Matter, vol. 7. Metallic Elements and Alloys. IFI/Plenum, New York, Thermal Radiative Properties (1970) 19. Dirac, P.M.: The Principles of Quantum Mechanics. Oxford University Press, Oxford (1930) 20. Heitler, W.: Quantum Theory of Radiation. Oxford University Press, Oxford (1954) 21. Bordag, M., Mohideen, U., Mostepanenko, V.M.: New developments in the Casimir effect. Phys. Rep. 353, 1–205 (2001) 22. Bressi, G., Carugno, G., Onofrio, R., Ruoso, G.: Measurement of the Casimir force between parallel metallic surfaces. Phys. Rev. Lett. 88, , Art. No. 041804 (2002) 23. Lamoreaux, S.K.: Demonstration of the Casimir force in the 0.6 to 6 mm range. Phys. Rev. Lett. 78, 5–8 (1997) 24. Feynman, R.P.: QED: the Strange Theory of Light and Matter. Princeton University Press, Princeton, NJ (1985) 25. Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965) 26. Craig, D.P., Thirunamachandran, T.: Molecular Quantum Electrodynamics: an Introduction to Radiation-Molecule Interactions. Academic, London (1984) 27. Peskin, M.E., Schroeder, D.V.: An Introduction to Quantum Field Theory. Perseus Press, Reading, MA (1995) 28. Rioux, F.: Illustrating the superposition principle with single-photon interference. The Chem. Educator. 10, 424–426 (2005) 29. Scarani, V., Suarez, A.: Introducing quantum mechanics: one-particle interferences. Am. J. Phys. 66, 718–721 (1998)

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30. Zeilinger, A.: General properties of lossless beam splitters in interferometry. A. J. Phys. 49, 882–883 (1981) 31. Glauber, R.J.: Dirac's famous dictum on interference: one photon or two? Am. J. Phys. 63, 12 (1995) 32. Monroe, C., Meekhof, D.M., King, B.E., Wineland, D.J.: A “Schrödinger cat” superposition state of an atom. Science. 272, 1131–1136 (1996) 33. Marton, L., Simpson, J.A., Suddeth, J.A.: Electron beam interferometer. Phys. Rev. 90, 490–491 (1953) 34. Carnal, O., Mlynek, J.: Young's double-slit experiment with atoms: a simple atom interferometer. Phys. Rev. Lett. 66, 2689–2692 (1991) 35. Hong, C.K., Ou, Z.Y., Mandel, L.: Measurement of subpicosecond time intervals between two photons by interference. Phys. Rev. Lett. 59, 2044–2046 (1987) 36. Kwiat, P.G., Steinberg, A.M., Chiao, R.Y.: Observation of a “quantum eraser”: a revival of coherence in a two-photon interference experiment. Phys. Rev. A. 45, 7729–7739 (1992) 37. Pittman, T.B., Strekalov, D.V., Migdall, A., Rubin, M.H., Sergienko, A.V., Shih, Y.H.: Can two-photon interference be considered the interference of two photons? Phys. Rev. Lett. 77, 1917–1920 (1996) 38. Kwiat, P.G., Mattle, K., Weinfurter, H., Zeilinger, A., Sergienko, A.V., Shih, Y.: New highintensity source of polarization-entangled photon pairs. Phys. Rev. Lett. 75, 4337–4341 (1995) 39. Kwiat, P.G., White, A.G., Appelbaum, I., Eberhard, P.H.: Ultrabright source of polarizationentangled photons. Phys. Rev. A. 60, R773–R776 (1999) 40. Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935) 41. Bell, J.S.: On the Einstein Podolsky Rosen paradox. Physics. 1, 195–200 (1964) 42. Bell, J.S.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447– 452 (1966) 43. Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Proposed experiment to test local hiddenvariable theories. Phys. Rev. Lett. 23, 880–884 (1969) 44. Clauser, J.F., Shimony, A.: Bell’s theorem: experimental tests and implications. Rep. Prog. Phys. 41, 1881–1927 (1978) 45. Freedman, S.J., Clauser, J.F.: Experimental test of local hidden-variable theories. Phys. Rev. Lett. 28, 938–941 (1972) 46. Aspect, A., Dalibard, J., Roger, G.: Experimental test of Bell inequalities using time-varying analyzers. Phys. Rev. Lett. 49, 1804–1807 (1982) 47. Weihs, G., Jennewein, T., Simon, C., Weinfurter, H., Zeilinger, A.: Violation of Bell’s inequality under strict Einstein locality conditions. Phys Rev. Lett. 23, 5039–5043 (1998) 48. Moreva, E., Brida, G., Gramegna, M., Giovannetti, V., Maccone, L.: Time from quantum entanglement: an experimental illustration. Phys. Rev. A. 89, 052122 (2014) 49. de Fornel, F.: Evanescent Waves: from Newtonian Optics to Atomic Optics. Springer, Berlin (2001) 50. Bekefi, G., Barrett, A.H.: Electromagnetic Vibrations, Waves, and Radiation. MIT Press, Cambridge, MA (1987) 51. Liebermann, T., Knoll, W.: Surface-plasmon field-enhanced fluorescence spectroscopy. Colloids Surfaces A: Physiochem. Eng. Aspects. 10, 115–130 (2000) 52. Moscovits, M.: Surface-enhanced spectroscopy. Rev. Mod. Phys. 57, 783–826 (1985) 53. Knoll, W.: Interfaces and thin films as seen by bound electromagnetic waves. Ann. Rev. Phys. Chem. 49, 565–634 (1998) 54. Aslan, K., Lakowicz, J.R., Geddes, C.D.: Plasmon light scattering in biology and medicine: new sensing approaches, visions and perspectives. Curr. Opin. Chem. Biol. 9, 538–544 (2005) 55. Maier, S.A.: Plasmonics: Fundamentals and Applications. Springer (2007) 56. Kelly, K.L., Coronado, E., Zhao, L.L., Schatz, G.C.: The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment. J. Phys. Chem. B. 107, 668–677 (2002)

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57. Amendola, V., Pilot, R., Fasconi, M., Marago, O.M., Lati, M.A.: Surface plasmon resonance in gold nanoparticles: a review. J. Phys - Condensed Matter. 29, 203002/1-48 (2017) 58. Rich, R.L., Myszka, D.G.: Higher-throughput, label-free, real-time molecular interaction analysis. Anal. Biochem. 361, 1–6 (2007) 59. Haynes, C.L., Van Duyne, R.P.: Plasmon-sampled surface-enhanced Raman excitation spectroscopy. J. Phys. Chem. B. 107, 7426–7433 (2003) 60. Wang, Z.J., Pan, S.L., Krauss, T.D., Du, H., Rothberg, L.J.: The structural basis for giant enhancement enabling single-molecule Raman scattering. Proc. Natl. Acad. Sci. U. S. A. 100, 8638–8643 (2003) 61. Kreibig, U., Vollmer, M.: Optical Properties of Metal Clusters, vol. 535. Springer, Berlin (1995) 62. Liao, J., Ji, L., Zhang, J., Gao, N., Li, P., Huang, K., Yu, E.T., Kang, J.: Influence of the substrate to the LSP coupling wavelength and strength. Nanoscale Res. Lett. 13, 280/1-11 (2018) 63. Toscano, G., Raza, S., Xiao, S., Wubs, M., Jauho, A., Bozhevolnyi, S.I., Mortensen, N.A.: Surface-enhanced Raman spectroscopy: nonlocal limitations. Opt. Lett. 37(13), 2538–2540 (2012) 64. Lati, M.A., Lidorikis, E., Saija, R.: Modeling of enhanced electromagnetic fields in plasmonic nanostructures. In: de la Chapelle, M.L., Lidgi-Guigui, N., Gucciardi, P.G. (eds.) Handbook of Enhanced Spectroscopy, pp. 101–140. Pan Stanford Publ, Singapore (2016) 65. Armstrong, J.A., Bloembergen, N., Ducuing, J., Pershan, P.S.: Interactions between light waves in a nonlinear dielectric. Phys. Rev. 127, 1918–1939 (1962) 66. Bloembergen, N.: Nonlinear Optics. Benjamin, New York (1965) 67. Shen, Y.R.: The Principles of Nonlinear Optics. Wiley, New York (1984) 68. Butcher, P.N., Cotter, D.: The Elements of Nonlinear Optics. Cambridge Univ. Press, Cambridge (1990) 69. Mukamel, S.: Principles of Nonlinear Optical Spectroscopy. Oxford Univ. Press, Oxford (1995) 70. Byer, R.L.: Nonlinear optical phenomena and materials. Ann. Rev. Mater. Sci. 4, 147–190 (1974) 71. Boyd, R.: Nonlinear Optics, 4th edn. Academic, London (2020) 72. Radhakrishnan, T.P.: Molecular structure, symmetry, and shape as design elements in the fabrication of molecular crystals for second harmonic generation and the role of moleculesin-materials. Acc. Chem. Res. 41, 367–376 (2008) 73. Hobbs, P.V.: Ice Physics., pp. 202. Oxford Univ. Press, New York (2010) 74. Fleming, G.R.: Chemical Applications of Ultrafast Spectroscopy. Oxford University Press, New York (1986) 75. Demtröder, W.: Laser Spectroscopy 1: Basic Principles, 5th edn. Springer, New York (2014) 76. Demtröder, W.: Laser Spectroscopy 2: Experimental Techniques, 5th edn. Springer, New York (2015) 77. Böttcher, C.J.F.: Theory of Electric Polarization, 2nd edn. Elsevier, Amsterdam (1973) 78. Lorentz, H.A.: The Theory of Electrons. Dover, New York (1952) 79. Liptay, W.: Dipole moments of molecules in excited states and the effect of external electric fields on the optical absorption of molecules in solution. In: Sinanoglu, O. (ed.) Modern Quantum Chemistry Part III: Action of Light and Organic Crystals. Academic, New York (1965) 80. Chen, F.P., Hansom, D.M., Fox, D.: Origin of stark shifts and splittings in molecular crystal spectra. 1. Effective molecular polarizability and local electric field. Durene and naphthalene. J. Chem. Phys. 63, 3878–3885 (1975) 81. Myers, A.B., Birge, R.R.: The effect of solvent environment on molecular electronic oscillator strengths. J. Chem. Phys. 73, 5314–5321 (1980) 82. Zelmon, D.E., Small, D.L., Jundt, D.: Infrared corrected Sellmeier coefficients for congruently grown lithium niobate and 5 Mol. % magnesium oxide-doped lithium niobate. J. Opt. Soc. Am. B. 14, 3319–3322 (1997)

4

Electronic Absorption

4.1

Interactions of Electrons with Oscillating Electric Fields

This chapter begins with a discussion of how the oscillating electric field of light can raise a molecule to an excited electronic state. We then explore the factors that determine the wavelength, strength, linear dichroism, and shapes of molecular absorption bands. Our approach is to treat the molecule quantum mechanically with time-dependent perturbation theory (Chap. 2) but to consider light, the perturbation, as a purely classical oscillating electric field. Because many of the phenomena associated with the absorption of light can be explained well by this semiclassical approach, we defer considering the quantum nature of light until Chap. 5. Interactions with the magnetic field of light will be discussed in Chap. 9. Let’s start by considering the interaction of an electron with the oscillating electrical field (E) of linearly polarized light as in Eq. (3.15a): Eðt Þ = Eo ðt Þ½ expð2πiνt Þ þ expð- 2πiνt Þ:

ð4:1Þ

The oscillating field adds a time-dependent term to the Hamiltonian operator for the electron. To an approximation that often proves acceptable, we can write the perturbation as the dot product of E with the dipole operator, e μ: e0 ðt Þ = - Eðt Þ  e μ: H

ð4:2Þ

The dipole operator for an electron is simply e μ = ere= er,

ð4:3Þ

where e is the electron charge (-4.803 × 10-10 esu in the cgs system or -1.602 × 10-19 C in MKS units), er is the position operator, and r is the position of the electron. Thus Eq. (4.2) also can be written

# The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. W. Parson, C. Burda, Modern Optical Spectroscopy, https://doi.org/10.1007/978-3-031-17222-9_4

137

138

4

Electronic Absorption

e0 ðt Þ = - eEðt Þ  r = - ejEo j½ expð2πiνt Þ þ expð- 2πiνt Þjrj cos θ, H

ð4:4Þ

where θ is the angle between Eo and r. Moving a classical particle with charge e by a small distance dr in an electric field changes the potential energy of the particle by dV = -eE(r)dr (Box 4.1). So if the field is independent of position, moving the particle from the origin of the coordinate system to position r changes the classical energy by -eEr. Quantum mechanically, if an electron is described by wavefunction Ψ , interaction with a uniform field changes the potential energy of the electron by -ehΨ| E  r| Ψi, assuming that the field does not alter the wavefunction itself. Box 4.1 Energy of a Dipole in an External Electric Field In Sect. 3.1 we discussed the energy of a charged particle in the electric field from another charge. The same considerations apply to a set of charged particles in an external electric field, such as the field between the plates of a capacitor. Consider a pair of parallel, oppositely charged plates separated by a small gap. The field in the region between the plates (E) is normal to the plates, points from the positive plate to the negative, and (if we are sufficiently far from the edges of the plates) is independent of position. The electrostatic potential (Velec) therefore increases linearly from the negative plate to the positive. Let’s put the origin of our coordinate system at the center of the negative plate and express Velec(r) relative to the potential here. The field from the plates then changes the electrostatic energy of particle i by Zri E q, field = qi V elec ðri Þ = - qi

E  dr = - qi E  ri ,

ðB4:1:1Þ

0

where qi and ri are the charge and position of the particle. Summing over all the charged particles between the plates gives the total energy of interaction of the particles with the external field: ( ) X X o qi E  ri = - E  RQ þ qi r i = - E E Q, field = i

fRQ þ μg:

i

ðB4:1:2Þ

Here Q is the net charge of the system (Σqi); R is the center of charge defined as (continued)

4.1 Interactions of Electrons with Oscillating Electric Fields

139

Box 4.1 (continued) R=

1X Rq; Q i i i

ðB4:1:3Þ

rio is the position of charge i with respect to the center of charge (roi = ri - R), and μ is the electric dipole, or electric dipole moment of the system of charges: X μ= qi r i o : ðB4:1:4Þ i

In the convention used here, the dipole of a pair of charges with opposite signs points from the negative to the positive charge. We’ll show later in this chapter that the term RQ in Eq. (B4.1.2) drops out of the interactions of a molecule with the oscillating field of light so that the choice of the coordinate system is immaterial for these interactions. RQ also drops out for static fields if we use the center of charge as the origin of the coordinate system, or if the net charge (Q) is zero. In these situations, we can write μ simply as X qi ri : ðB4:1:5Þ μ= i

Equation (4.4) makes several approximations. In addition to treating light classically, we have neglected the magnetic component of the radiation field. This is often an acceptable approximation because the effects of the electric field usually are much greater than those of the magnetic field. We will return to this point in Chap. 5 when we discuss the quantum theory of absorption and emission, and again in Chap. 9 when we take up circular dichroism. We also have assumed that Eo is independent of the position of the electron within the molecular orbital. This also is a reasonable approximation for most molecular chromophores, which typically are small relative to the wavelength of visible light (~5000Å). We could, however, consider the variation in the field with position by expanding the field strength in a Taylor series about the origin of the coordinate system. For light polarized in the z-direction, this gives e0 ðx, y, z, t Þ = H (

"  # )    ∂jEðt Þj ∂jEðt Þj - ez jEðt Þjx,y = 0 þ x þy þ ⋯ - ⋯: ð4:5Þ ∂x ∂y x,y = 0 x,y = 0

140

4

Electronic Absorption

Fig. 4.1 Systems of charges with no net charge but with either a dipole moment (A) or a quadrupole moment but no dipole moment (B). A blue “+” represents a unit positive charge; a red “-“, a unit negative charge. The energy of interaction of the system with a constant external electric field depends on the dipole moment, and so is zero in B. The quadrupole moment becomes important if the magnitude of the external electric field (|E|) varies with position. The shaded yellow backgrounds here represent a field that increases in strength with the position in the x-direction

The leading term in the expansion represents the dipole interaction; the subsequent terms represent quadrupolar, octupolar, and higher-order interactions that usually are much smaller in magnitude (Box 4.2 and Fig. 4.1). Box 4.2 Multipole Expansion of the Energy of a Set of Charges in a Variable External Field Figures 4.1A and B show two sets of charges in a field (E) that points in the ydirection and increases in strength with the position in the x-direction. For simplicity, suppose that all the charges are in the xy plane and the field strength is independent of the y-coordinate. The energy of the interactions of the particles with the field then can be written yi X Z qi Eðxi Þ  dy, E Q, field = i

ðB4:2:1Þ

0

where qi and (xi, yi) are the charge and position of particle i. One way to evaluate this sum is to choose the center of charge as the origin of the coordinate system and expand E as a Taylor series around this point. For the systems shown in Fig. 4.1, this gives: (continued)

4.1 Interactions of Electrons with Oscillating Electric Fields

141

Box 4.2 (continued) ( )   X  ∂jEj X ^ x i y i qi þ ⋯ , E Q, field = - E  ^y jEj yi qi þ ∂x i i 

ðB4:2:2Þ

^ and ^y are unit vectors parallel to the field and the y axis, and E and its where E derivatives are evaluated at the center of charge. Factor E  ^y is just +1 in this illustration. The first term in the brackets in Eq. (B4.2.2) is non-zero for the set of charges shown in Fig. 4.1A, which have an electric dipole with a component in the y-direction. This term vanishes for the set of charges in Fig. 4.1B, where the contribution from one pair of positive and negative charges cancels the contribution from the other. The second term in the brackets vanishes if the field is constant, but not if the field strength changes with x. Equation (B4.2.2) can be written in a more general way by using matrices and matrix operators: h i X e Θ þ⋯ qi Eðri Þ  ri = - E  RQ - E  μ - Tr ∇E EQ, field = i

h i e  Θ þ ⋯: = V elec ðRÞQ - E  μ - Tr ∇E

ðB4:2:3Þ

Here R again is the center of charge, and E and its derivatives are evaluated at this point; Q is the total charge of the system; Velec(R) is the electrostatic potential at R; μ is the electric dipole calculated with respect to R as in Eq. (B4.1.4) (or with respect to any coordinate system if Q is zero); Θ is a matrix called the electric quadrupole moment h of the i system of charges (see e e below); ∇E is the gradient of E; and Tr ∇E  Θ means the trace of the e  Θ. (See Appendix A2 for definitions of the gradient of a matrix product ∇E vector, the product of two matrices, and the trace of a matrix.) Equation (B4.2.3) is a multipole expansion of the interaction of a set of charges with an external field. The first term on the right side (Velec(R)Q) is the interaction of the net charge of the system with the potential at the center of charge. The second term (-E  μ) describesthe interaction h iof the dipole e moment with the external field, and the third - Tr ∇E  Θ describes the interaction of the quadrupole moment with the gradient of the field. The ellipsis represents terms for the electric octupole and higher-order moments interacting with progressively higher derivatives of E. In most of the situations that arise in optical spectroscopy, these higher-order terms are very small relative to the terms given in Eq. (B4.2.3), and even the quadrupole term usually is negligible compared to the dipole term. (continued)

142

4

Electronic Absorption

Box 4.2 (continued) The matrix elements of the quadrupole moment of a system of charges are defined as X qi r αðiÞ r βðiÞ , ðB4:2:4Þ Θα,β = i

where rα(i) and rβ(i) denote the x, y, or z coordinate of charge i with respect to the center of charge. For example, letting α and β be 1, 2 or 3 for x, y or z, respectively, Θ1, 2 = ∑ qixiyi and Θ3, 1 = ∑ qizixi. Note that Θα, β = Θβ, α, so Θ is symmetric. With the recipes for matrix operations given in Appendix A2, the quadrupole term in Eq. (B4.2.3) becomes h i XXX e Θ = r αðiÞ r βðiÞ ∂E α =∂rβ Tr ∇E i

=

X i

xi xi

α

β

∂Ey ∂E y ∂E x ∂E z ∂E ∂Ez þ þ yi xi þ zi xi þ xi y i x þ yi y i þ z i yi ∂x ∂x ∂x ∂y ∂y ∂y þ xi zi

∂E y ∂Ex ∂E z : þ yi z i þ zi zi ∂z ∂z ∂z

ðB4:2:5Þ

If the field is oriented along the y axis and its magnitude depends only on x, as in Fig. 4.1, this expression reduces to h i X e Θ = qi yi xi ∂jEj=∂x, ðB4:2:6Þ Tr ∇E i

which is the same as the second term in the braces on the right side of Eq. (B4.2.2).

4.2

The Rates of Absorption and Stimulated Emission

Suppose that before we turn on the light our electron is in a state described by wavefunction Ψ a. In the presence of the oscillating radiation field, this and the other solutions to the Schrödinger equation for the unperturbed system become unsatisfactory; they no longer represent stationary states. However, we can represent the wavefunction of the electron in the presence of the field by a linear combination of the original wavefunctions, CaΨ a + CbΨ b + . . ., where the coefficients Ck are functions of time (Eq. (2.55)). As long as the system is still in Ψ a, Ca = 1 and all the other coefficients zero; but if the perturbation is sufficiently strong Ca will decrease with time while Cb or one or more of the other coefficients increases. We

4.2 The Rates of Absorption and Stimulated Emission

143

can find the expected rate of growth of Cb by incorporating Eqs. (4.1 and 4.2) into Eq. (2.62): ∂C b =∂t = ði=ħÞ exp½iðE b - Ea Þt=ħ½ expð2πiνt Þ þ expð- 2πiνt ÞEo μjψ a i  hψ b je

ð4:6aÞ

= ði=ħÞf exp½iðEb - E a þ hνÞt=ħ þ exp½iðE b - Ea - hνÞt=ħgEo μjψ a i,  hψ b je

ð4:6bÞ

where Ea and Eb are the energies of states a and b. The probability that the electron has made a transition from Ψ a to Ψ b by time τ is obtained by integrating Eq. (4.6b) from time t = 0 to τ (Eq. (2.63)) and then evaluating |Cb(τ)|2. Integrating Eq. (4.6b) is straightforward, and gives the following result:   exp½iðE b - E a þ hνÞτ=ħ - 1 exp½iðE b - Ea - hνÞτ=ħ - 1 E0 C b ðτ Þ = þ E b - Ea þ hν E b - Ea - hν μjψ a i: hψ b je

ð4:7Þ

Note that the two fractions in the large parentheses in Eq. (4.7) differ only in the sign of the term hν. Suppose that Eb > Ea, which means that Ψ b lies above Ψ a in energy. The denominator in the second term in the braces then becomes zero when Eb Ea = hv. The numerator of this term is a complex number, but its magnitude also goes to zero when Eb - Ea = hv, and the ratio of the numerator to the denominator becomes iτ/ħ (Box 4.3). On the other hand if Eb < Ea (i.e., if Ψ b lies below Ψ a), then the ratio of the numerator to the denominator in the first term in the braces becomes iτ/ħ when Eb - Ea = hv. If |Eb - Ea| is very different from hν, both terms will be very small. (The amplitudes of the numerators cannot exceed 2 for any values of Ea, Eb, or hν, whereas the denominators usually are large.) So something special happens if hν is close to the energy difference between the two states. We’ll see shortly that the second term in the braces in Eq. (4.7) accounts for absorption of light when Eb Ea = hv, and that the first term accounts for induced or stimulated emission of light when Ea - Eb = hv. Stimulated emission, a downward electronic transition in which light is given off, is just the reverse of absorption. Box 4.3 The Behavior of the Function [exp(iy)-1]/y as y goes to 0 To examine the behavior of Eq. (4.7) when hv ≈ |Eb - Ea|, let y = (Eb - Ea hv)τ/ħ. The term for absorption then is (continued)

144

4

Electronic Absorption

Box 4.3 (continued)   expðiyÞ - 1 τ exp½iðEb - E a - hvÞτ=ħ - 1 = y Eb - E a - hv ħ   2 1 þ iy - y =2! þ . . . - 1 τ = , y ħ

ðB4:3:1Þ

which goes to iτ/ħ as y approaches zero. Alternatively, we could write     expðiyÞ - 1 τ cosðyÞ þ i sinðyÞ - 1 τ = ðB4:3:2Þ y y ħ ħ which also goes to iτ/ħ as y goes to zero. The fact that iτ/ħ is imaginary has no particular significance here because we are interested in |Cb(τ)|2. Equation (4.7) describes the effects of light with a single frequency (ν). As we discussed in Chap. 3, light always includes multiple oscillation modes spanning a range of frequencies. The rates of excitation caused by these individual modes are additive. To obtain the total rate of excitation, we therefore must integrate |Cb(τ)|2 over all the frequencies included in the radiation. The integral will be very small unless the region of integration includes a frequency for which hv = |Eb - Ea|. This means that we can safely take the integral from ν = 0 to 1, which is convenient because the result then appears in standard tables (Box 4.4). Also, as explained above, we only need to consider either the term for absorption or that for stimulated emission, depending on whether Eb is greater than or less than Ea. Integrating the term for absorption gives: Z1

Cb  ðτ, vÞC b ðτ, vÞρv dv

0

Z1 " 2

ðEo  hψ b je μjψ a iÞ

= 0

# ! f exp½-iðE b -E a -hνÞτ=ħ-1gf exp½iðE b -E a -hνÞτ=ħ -1g ρν ðνÞ dν ðEb -E a -hνÞ2

ð4:8aÞ μjψ a iÞ2 ρν ðνo Þ τ=ħ2 = ðEo  hψ b je

ð4:8bÞ

= ðEo  μba Þ2 ρν ðνo Þ τ=ħ2 ,

ð4:8cÞ

where ρv(v)dv is the number of modes of oscillation in the frequency interval μjψ a i. We have factored between ν and v + dv, vo = (Eb - Ea)/h, and μba  hψ b je (Eo  μba)2ρv(v) out of the integral in Eq. (4.8a) on the assumption that the field is essentially independent of ν over the small frequency interval where hν is close to Eb - Ea. The factor ρv(vo) in the final expression therefore pertains to this interval.

4.2 The Rates of Absorption and Stimulated Emission Fig. 4.2 The function sin2(x)/x2. In Eq. (B4.4.1), x = s/2 = (Eb - Ea - hv)τ/2ħ

145

1.0

0.6

2

sin (x)/x

2

0.8

0.4 0.2 0 -4

-2

0 x/π

2

4

Additional details of the derivation are in Box 4.4. Equation (4.8c) is a special case of a general expression that is often called the golden rule of quantum mechanics, which we met in Chap. 2 and will encounter again in a variety of contexts. Box 4.4 The Function sin2x/x2 and Its Integral To evaluate Eq. (4.8a), first pull the term (Eo  μba)2ρ(v) out of the integral and let s = (Eb - Ea - hv)τħ as in Box 4.3. With this substitution, ds = - 2πτdv and the limits of integration are from s = 1 to s = - 1. The integral then can be evaluated as follows: )   Z- 1  Z- 1 ( ½ expð- isÞ - 1½ expðisÞ - 1 1 - cosðsÞ -τ ds = ds s2 πħ2 s2 ðħ=τÞ2 ð- 2πτÞ 1

1



 Z1



 Z1 

τ = πħ2 =

τ πħ2

-1

-1

2 sin 2 ðs=2Þ ds s2 2 sin ðs=2Þ d ðs=2Þ ðs=2Þ

ðB4:4:1Þ

  = τ=πħ2 π = τ=ħ2 :

ðB4:4:2Þ

Figure 4.2 shows the function sin2x/x2 that appears in Eq. (B4.4.1). The function, sometimes called sinc2x, has a value of 1 at x = 0 and drops off rapidly on either side of zero. But could the spectrum of the light be so sharp that it would cover only a small fraction of the region where sin2(s/2)/(s/2)2 is significantly different from zero? Note that s includes a product of an energy (continued)

146

4

Electronic Absorption

Box 4.4 (continued) difference (Eb - Ea - hv) and time (τ), and remember from Chaps. 2 and 3 that such products must cover a minimum range of approximately h. The spread of s/2 therefore must be at least on the order of h/2ħ, or π, which would include a substantial part of the integral. This analysis has led us to the resonance condition for absorption of light: hv = Eb - Ea. We also have obtained a very general expression for the rate at which a molecule will be excited from state a to state b when the resonance condition is met (Eq. (4.8c)). Integrating the term for stimulated emission in Eq. (4.7) gives the same result except that ρv(vo) refers to the frequency where hv = Ea - Eb. You may be surprised to see that we did not have to introduce the notion of photons with quantized energy (hν) to obtain the resonance condition. Our description of light was completely classical. Although the requirement for hν to match jEb - Eaj is a quantum mechanical result, it emerges in this treatment as a consequence of the quantization of the states of the absorber, not the quantization of light. However, we will see in Chap. 5 that the same result is obtained by a fully quantum mechanical treatment that includes a quantized radiation field. Equation (4.7) has the curious feature that, when |Eb - Ea| ≈ hv, Cb(τ) is proportional to -iτ/ħ (Box 4.3). This means that the probability density |Cb(τ)|2 is proportional to τ2, at least for short times when the system still is most likely to be in state Ψ a. In other words, the probability that the system has made a transition to state Ψ b increases quadratically with time! By contrast, Eqs. (4.8b and c) say that the probability that the system has made the transition grows linearly with time, which seems more in keeping with everyday observations. The quadratic time dependence predicted by Eq. (4.7) results from considering light with a single frequency, or equivalently, from considering a system with a single, sharply defined value of Eb Ea. Integrating over a distribution of frequencies gives Eq. (4.8). We could have obtained the same linear dependence on τ by considering a very large number of molecules with a distribution of energy gaps clustered around hν, or by considering a single molecule for which Eb - Ea fluctuates rapidly with time. In Chap. 11 we will see that the dynamics of absorption are expected to be nonlinear on very short time scales, although the extent of this nonlinearity depends on the fluctuating interactions of the system with the surroundings.

4.3

Transition Dipoles and Dipole Strengths →

μ j ψ a that we denote as μba in Eq. (4.8c) is called a The matrix element ψ b je transition dipole. Transition dipoles are vectors whose magnitudes have units of μjψ a i, or μaa, which is the charge times distance. Note that μba differs from hψ a je contribution that an electron in orbital ψ a makes to the permanent dipole of the molecule. The total electric dipole of the molecule is given by a sum of terms

4.3 Transition Dipoles and Dipole Strengths

147

corresponding to μaa for all the wavefunctions of all the charged particles in the molecule, including both electrons and nuclei. As we’ll discuss in Sect. 4.10, the change in the permanent dipole when the system is excited from ψ a to ψ b (μbb μaa) bears on how interactions with the surroundings affect the energy difference between the excited state and the ground state. The transition dipole, on the other hand, determines the strength of the absorption band associated with the excitation. The transition dipole can be related to the oscillatory component of the dipole in a superposition of the ground and excited states (Box 4.5). Box 4.5 The Oscillating Electric Dipole of a Superposition State Classically, energy can be transferred between a molecule and an oscillating electromagnetic field only if the molecule has an electric or magnetic dipole that oscillates in time at a frequency close to the oscillation frequency of the field; otherwise, the interactions of the molecule with the field will average to zero. A molecule in a state described by a superposition of its ground and excited states (ψ a and ψ b) can have such an oscillating dipole even though the individual states do not. Figure 4.3 illustrates this point for the first two eigenfunctions of an electron in a one-dimensional box. The dotted and dashed curves in panel A show the amplitudes of ψ a and ψ b at time t = 0 (Eq. (2.23) and Fig. 2.2). The solid curve shows the sum (Ψ a + Ψ b) multiplied by the normalization factor 2-1/2. Because the time-dependent parts of ψ a and ψ b [exp(-iEat/ħ) and exp(-iEbt/ħ), where Ea and Eb are the energies of the pure states] are both unity at zero time, the superposition state at this time is simply the sum of the spatial parts of the wavefunctions (ψ a + ψ b). The corresponding probability functions are shown in Fig. 4.3C. The pure states have no dipole moment because the electron density (e|ψ(x)|2) at any point where x > 0 is balanced by the R electron density at a corresponding point with x < 0, making the integrals e|ψ(x)|2xdx zero. This symmetry is broken in the superposition state, which has a higher electron density for negative values of x (Fig. 4.3C, solid curve). To see that the dipole moment of the superposition state oscillates with time, consider the same states at time t = (1/2)h/(Eb - Ea). For the particle in a box, the energy of the second eigenstate is four times that of the first, Eb = 4Ea (Eq. (2.24)), so (1/2)h/(Eb - Ea) = (1/6)h/Ea. The individual wavefunctions at this time have both real and imaginary parts. For the lower-energy state, we find by using the relationship exp(-iθ) = cos(θ) - i sin(θ) that Ψa ðx, t Þ = ψ a ðxÞ expð- iE a t=ħÞ = ψ a ðxÞ expð- iE a h=6E a ħÞ = ψ a ðxÞ expð- iπ=3Þ = ψ a ðxÞ½ cosðπ=3Þ - i sinðπ=3Þ:

ðB4:5:1Þ (continued) (continued)

148

4

Electronic Absorption

Box 4.5 (continued) For the higher-energy state, which oscillates four times more rapidly, Ψb ðx, t Þ = ψ b ðxÞ expð- iE b t=ħÞ = ψ b ðxÞ expð- i4π=3Þ = ψ b ðxÞf cosð4π=3Þ - i sinð4π=3Þg = - ψ b ðxÞ  f cosðπ=3Þ - i sinðπ=3Þg:

ðB4:5:2Þ

Because cos(4π/3) = - cos(π/3). and sin(4π/3) = - sin(π/3), ψ b has changed sign relative to ψ a. The wavefunction of the superposition state at t = (1/2)h/ (Eb - Ea) is, therefore, 2 - 1=2 fΨa ðx, t Þ þ Ψb ðx, tÞg = 2 - 1=2 fψ a ðxÞ - ψ b ðxÞg  f cosðπ=3Þ - i sinðπ=3Þg:

ðB4:5:3Þ

This differs from the superposition state at zero time in that it depends on the difference between ψ a(x) and ψ b(x) instead of the sum (Fig. 4.3B). Inspection of the electron density function in Fig. 4.3D shows that the electric dipole of the superposition state has reversed direction relative to the orientation at t = 0. The wavefunction of the superposition state returns to its initial shape at t = h/(Eb - Ea), when the phases of the time-dependent parts of Ψ a and Ψ b are 2π/3 and 8π/3, respectively. The spatial part of Ψ a + Ψ b thus oscillates between symmetric and antisymmetric combinations of ψ a and ψ b(ψ a + ψ b and ψ a ψ b) with a period of h/(Eb - Ea), and the electric dipole oscillates in concert. We can relate the amplitude of the oscillating dipole of the superposition state to the transition dipole (μba) as follows. For a superposition state CaΨ a + CbΨ b with Ψ k = ψ k exp(-iEkt/ħ), the expectation value of the dipole is μ j C a Ψa þ C b Ψb C a Ψ a þ C b Ψb j e = jC a j2 Ψa je μ j Ψa þ jC b j2 Ψb je μ j Ψb þ C a  Cb Ψa je μ j Ψb þ C b  Ca Ψb je μ j Ψa = jCa j2 μaa þ jC b j2 μbb þ Ca  C b μab exp½iðEa - Eb Þt=ħ þ Cb  C a μba exp½iðE b - Ea Þt=ħ

ðB4:5:4aÞ

= jC a j2 μaa þ jC b j2 μbb þ 2 Re fC b C a μba exp½iðE b - Ea Þt=ħg:

ðB4:5:4bÞ

We have used the equality θ + θ = 2 Re (θ) where Re(θ) is the real part of the complex number θ. Equation (B4.5.4b) shows that the dipole moment of the superposition state includes a component that oscillates sinusoidally with a period of h/|Eb - Ea|, and that the amplitude of this component is proportional to μba. (continued)

4.3 Transition Dipoles and Dipole Strengths

149

Box 4.5 (continued) Now suppose that a molecule in the superposition state is exposed to the oscillating electric field of light (E). If the frequency of the light is very different from that of the oscillating molecular dipole (|Eb - Ea|/h), the interactions of the molecule with the radiation field will average to zero. On the other hand, if the two frequencies match and are in phase, the interaction energy will be proportional to μba  E, and in general, will be non-zero. The oscillating dipole of the superposition state thus appears to rationalize the dependence of the absorption of light on both the resonance condition and μba. However, this argument has the problem that, in addition to being proportional to μba, the oscillating dipole of the superposition state depends on the product of the coefficients Ca and Cb (Eq. (B4.5.4b)). If we know the system is in the ground state, then Cb = 0, and the amplitude of the oscillating dipole is zero. The perturbation treatment presented in Sect. 4.2 does not encounter this dilemma, and indeed predicts that the rate of absorption will be maximal when Ca = 1 and Cb = 0. We’ll resolve this apparent contradiction in Chap. 10, when we discuss polarization of a medium by an electromagnetic radiation field. The magnitudes of both permanent and transition dipole moments commonly are expressed in units of debyes (D) after Peter Debye, who received the Chemistry Nobel Prize in 1936 for showing how dipole moments can be measured and related to molecular structure. One debye is 10-18 esucm in the cgs system and 3.336 × 1030 Cm in MKS units). Because the charge of an electron is -4.803 × 10-10 esu, and 1Å = 10-8 cm, the dipole moment associated with a pair of positive and negative elementary charges separated by 1 Å is 4.803 debyes. According to Eq. (4.8c), the absorption strength is proportional to the square of the magnitude of μba, which is called the dipole strength: (Dba): μjψ a ij2 : Dba = jμba j2 = jhψ b je

ð4:9Þ

Dipole strength is a scalar with units of debye2. Suppose that a sample is illuminated with a light beam of irradiance I, with I defined as in Chap. 3 so that IΔv is the flux of energy (e.g., in joules s-1) in the frequency interval Δv crossing a plane with an area of 1 cm2. According to Eqs. (1.1 and 1.2), the light will decrease in intensity by ICε l ln(10), where C is the concentration of the absorbing molecules (M), l is the sample’s thickness (cm), and ε is the molar extinction coefficient in the frequency interval represented by Δv(M-1cm-1). The rate at which a sample with a 1 cm2 area absorbs energy in frequency interval Δv is, therefore,

150

4

ψa, ψb, ψa+ ψb

1

A

Electronic Absorption

B Ψa

Ψa

0 Ψb

Ψb

t = (1/2)h /(Eb - Ea)

t=0

-1

Relative Electron Density

1.5

D

C

1.0

0.5

0.0 -1

0 x/l

1 -1

0 x/l

1

Fig. 4.3 Wavefunction amplitudes (A, B) and probability densities (C, D) for two pure states and a superposition state. The blue dotted and green dashed curves are for the first two eigenstates of an electron in a one-dimensional box of unit length, as given by Eq. (2.23) with n = 1 (Ψ a) or 2 (Ψ b). The cyan solid curves are for the superposition 2-1/2(Ψ a+Ψ b). A, C The signed amplitudes of the wavefunctions and the probability densities at time t = 0, when all the wavefunctions are real. B, D The corresponding functions at time t = (1/2)h/(Eb - Ea), where Ea and Eb are the energies of the pure states

dE=dt = IΔvCεl ln ð10Þ:

ð4:10Þ

Assume for now that ε has a constant value across the frequency interval Δv and is zero everywhere else. Then Eq. (4.10) must account for all the absorption of light by the sample. Equation (4.8c), on the other hand, indicates that the rate at which molecules are excited is (Eo  μba)2Ngρu(v)/ħ2 molecules per second, where Ng is the number of molecules in the ground state in the illuminated region. By combining these two expressions we can relate the molar extinction coefficient, an experimentally measurable quantity, to the transition dipole (μba) and the dipole strength (|μba|2).

4.3 Transition Dipoles and Dipole Strengths

151

If the light beam has a cross-sectional area of 1 cm2, the volume of the illuminated region of interest is l cm3 and the total number of molecules in this volume is N = 10-3lCNA where NA is Avogadro’s number. We usually can use N in place of the number of molecules in the ground state (Ng) because, in most measurements with continuous light sources, the light intensity is low enough and the decay of the excited state is fast enough so that depletion of the ground-state population is negligible. Equation (4.10) thus can be rewritten as dE/dt = IΔv103 ln (10)εN/NA. The dot product Eo  μba in Eq. (4.8c) depends on the cosine of the angle between the electric field vector of the light (Eo) and the molecular transition dipole vector (μba), and this angle usually varies from molecule to molecule in the sample. To find the average value of (Eo  μba)2, imagine a Cartesian coordinate system in which the z-axis is parallel to Eo. The x and y axes can be chosen arbitrarily as long as they are perpendicular to z and to each other. In this coordinate system, the vector μba for an individual molecule can be written as (μx, μy, μz), where μz = Eo  μba/|Eo| and μ2x þ μ2y þ μ2z = jμba j2 . If the sample is isotropic (i.e., if the absorbing molecules have no preferred orientation), then the average values of μ2x ,μ2y and μ2z must all be the same, and so must be |μba|2/3. The average value of (Eo  μba)2 for an isotropic sample is, therefore, ðE o  μba Þ2 = ð1=3ÞjE o j2 jμba j2 :

ð4:11Þ

Box 4.6 describes a more general approach that leads to the same result. Box 4.6 The Mean-Squared Energy of Interaction of an External Field with Dipoles in an Isotropic System The average value of (Eo  μba)2 is jEo j2 jμba j2 cos 2 θ , where θ is the angle between Eo and μba for an individual absorbing molecule and cos 2 θ means the average of cos2θ over all the molecules in the system. The average value of cos2θ in an isotropic system can be obtained by representing θ as the angle of a vector r with respect to the z-axis in polar coordinates and integrating cos2θ over the surface of a sphere (Fig. 4.4). If we let r be the length of r, evaluating the integral gives: 0 cos 2 θ = @r 2

Z2π

Zπ dϕ

0

1, 0 cos 2 θ sin θdθA

0

@r 2

Z2π

Zπ dϕ

0

1 sin θdθA

0

= ð4π=3Þ=4π = 1=3: ðB4:6:1Þ The denominator in this expression is just the integral over the same surface without the weighting of the integrand by cos2θ. Though more cumbersome (continued)

152

4

Electronic Absorption

Box 4.6 (continued) than the justification of Eq. (4.11) given in the text, this analysis illustrates a more general way of dealing with related problems that arise in connection with fluorescence polarization (Chaps. 5 and 10). If we now use Eq. (3.34) to relate |Eo|2 to the irradiance (I ), still considering an isotropic sample, Eq. (4.11) becomes:     ðE o  μba Þ2 ρν ðνÞ = 2πf 2 =3cn jμba j2 I = 2πf 2 =3cn Dba I,

ð4:12Þ

where c is the speed of light in a vacuum, n is the refractive index of the solution, and f is the local-field correction factor. Thus the excitation rate is:   - dN g =dt = 10 - 3 lCN A 2πf 2 =3cnħ2 Dba I molecules s - 1 cm - 2 :

ð4:13Þ

Because each excitation increases the energy of a molecule by Eb ‐ Ea, or hν, the rate at which energy is transferred from the radiation field to the sample must be   ð4:14Þ dE=dt = 10 - 3 hνlCN A 2πf 2 =3cnħ2 Dba I: Finally, by equating the two expressions for dE/dt (Eqs. (4.10 and 4.14)) we obtain   3000 ln ð10Þnhc ε Dba = Δν: ð4:15Þ ν 8π 3 f 2 N A

Fig. 4.4 The average value of cos2θ can be obtained by integrating over the surface of a sphere. The arrow represents a vector with length r parallel to the transition dipole of a particular molecule; the z-axis is the polarization axis of the light. The area of a small element on the surface of the sphere is r2 sin θdϕdθ. The polar coordinates used here can be converted to Cartesian coordinates by the transformation z = r cos(θ), x = r sin(θ) cos(ϕ), y = r sin(θ) sin(ϕ)

4.3 Transition Dipoles and Dipole Strengths

153

In deriving Eq. (4.15) we have assumed that all the transitions expected based on dipole strength Dba occur within a small frequency interval Δν over which ε is constant. This is fine for atomic transitions, but not for molecules. As we will discuss in Sect. 4.10, molecular absorption bands are broadened because a variety of nuclear transitions can accompany the electronic excitation. To include all of these transitions, we must relate Dba to an integral over the absorption band:  Z 3000 ln ð10Þhc nε Dba = dν ≈ 9:186 8π 3 N A f 2ν  Z ε D2 -3 n , ð4:16aÞ × 10 dν ν M - 1 cm - 1 f2 or Z

 2   2  f 4π 2 N A M - 1 cm - 1 ε f , ð4:16bÞ dν ≈ jμba j2 jμba j2 = 108:86 n n ν 3000 ln ð10Þħc D2

where ε is the density of molar extinction coefficients (M-1cm-1 per unit dν) at frequency ν. The values of the physical constants and conversion factors in Eqs. (4.16a, 4.16b) are given in Box 4.7. Box 4.7 Physical Constants and Conversion Factors for Absorption of Light The values of the physical constants in Eqs. (4.16a, 4.16b) are: NA = 6.0222 × 1023 moleculesmol-1 ħ = 1:0546 × 10 - 27 erg  s = 6:5821 × 10 - 27 eV  s c = 2:9979 × 1010 cm  s - 1 ln10 = 2.30259 1D = 10 - 18 esu  cm 1dyn = 1esu2  cm - 2 4π 2 = 39:4784 1erg = 1dyn  cm = 1esu2  cm - 1 : If μba is given in debyes, then (continued)

154

4

Electronic Absorption

Box 4.7 (continued) 

 4π 2 N A jμba j2 3000 ln ð10Þħc

      2 2 D2 39:4784 × 6:0222 × 1023 molecules × 10 - 36 esuDcm × jμba j2 molecule 2 mol =       esu2  3 3 × 103 cml × 2:30259 × 1:0546 × 10 - 27 erg s × 2:9979 × 1010 cm s × 1 erg cm

= 108:86 mol - 1  cm - 1  1 = 108:86 M - 1  cm - 1 : If we use the cavity-field expression for f (Eq. (3.35)) and n = 1.33 (the refractive index of water), then the factor ( f2/n) is 9n3/(2n2 + 1)2 = 1.028. We have assumed that the refractive index (n) and the local-field correction factor ( f ) are essentially constant over the spectral region of the absorption band so that the ratio n/f 2 can be extracted from the integral in Eq. (4.16a). As discussed in Chap. 3, f depends on the shape and polarizability of the molecule, and usually cannot be measured independently. If the Lorentz expression (Eq. (3.82)) is used for f, as some authors recommend [1, 2], Eq. (4.16a) becomes Z 9n ε -3 ð4:17aÞ dν D2 : Dba = 9:186 × 10 2 2 ν ð n þ 2Þ Using the cavity-field expression (Eq. (3.81)) gives Dba = 9:186 × 10 - 3

ð2n2 þ 1Þ 9n3

2

Z

ε dv D2 : v

ð4:17bÞ

Myers and Birge [3] give additional expressions for n/f 2 that depend on the shape of the chromophore. Figure 4.5 illustrates how the treatment of the local-field correction factor in Eq. (4.16a) affects the dipole strength calculated for bacteriochlorophyll-a from the measured absorption spectrum [4]. For this molecule, using the Lorentz correction or just setting f = 1 leads to values for Dba that change systematically with n, whereas the cavity-field expression gives values that are nearly independent of n. Excluding specific solvent-solute interactions such as hydrogen bonding, which could affect the molecular orbitals, Dba should be an intrinsic molecular property that is relatively insensitive to the solvent. The cavity-field expression thus works reasonably well for bacteriochlorophyll, although the actual shape of the molecule is hardly spherical. The dependence of Dba on n for chlorophyll-a is essentially the same as that for bacteriochlorophyll-a [5]. The strength of an absorption band sometimes is expressed in terms of the oscillator strength, a dimensionless quantity defined as

Dipole Strength / debye

2

4.3 Transition Dipoles and Dipole Strengths

155

60

40

20

0

1.3

1.5 1.4 Index of Refraction

Fig. 4.5 Dipole strength of the long-wavelength absorption band of bacteriochlorophyll-a, calculated by Eq. (4.16a) from absorption spectra measured in solvents with various refractive indices. Three treatments of the local-field correction factor ( f ) were used: green triangles, f = 1.0 (no correction); cyan circles, f = Lorentz factor (Eq. 4.17a); blue squares, f = cavity-field factor (Eq. 4.17b). The dashed lines are least-squares fits to the data. Spectra measured by Connolly et al. [182] were converted to dipole strengths as described by Alden et al. [4] and Knox and Spring [5]

8π 2 me ν 2:303 × 103 me c Dba = 2 3e h πe2 N A Z   n ε dν, × 10 - 19 f2

Οba =

Z   n ε dν ≈ 1:44 f2 ð4:18Þ

where me is the electron mass (Box 3.3). Here the units of ν do matter; the numerical factors given in Eq. (4.18) are for ν in s-1. The oscillator strength relates the rate of absorption of energy to the rate predicted for a classical electric dipole oscillating at the same frequency (ν). It is on the order of 1 for the strongest possible electronic absorption band of a single chromophore. According to the Thomas-Reiche-Kuhn or Kuhn-Thomas sum rule [6–8], the sum of the oscillator strengths for all the absorption bands of a molecule in the ground state is equal to the total number of electrons in the molecule; however, this rule usually is of little practical value because many of the absorption bands at high energies are not measurable. We will see other sum rules for spectroscopic transitions in Chaps. 8 and 9. The strength of an absorption band also can be expressed as the absorption crosssection (σ), which (for ε in units of M-1cm-1) is given by 10-3 ln (10)ε/Na, or 3.82 × 10-21ε in units of cm2. If the incident light intensity is I photons cm-2s-1 and the excited molecules return to the ground state rapidly relative to the rate of excitation, a molecule with absorption cross-section σ will be excited Iσ times per second. This result is independent of the concentration of absorbing molecules in the

156

4

Electronic Absorption

sample, although I drops off more rapidly with depth in the sample if the concentration is increased.

4.4

Calculating Transition Dipoles for p Molecular Orbitals

Theoretical dipole strengths for molecular transitions can be calculated by using linear combinations of atomic orbitals to represent the molecular wavefunctions of the excited and unexcited system. Discussing an example of such a calculation will help to bring out the vectorial nature of transition dipoles. Consider a molecule in which the highest normally occupied molecular orbital (HOMO) and the lowest normally unoccupied orbital (LUMO) are both π orbitals. We can describe these orbitals by writing, as in Eq. (2.35), X X C ht pt and ψ l ≈ C lt pt , ð4:19Þ ψh ≈ t

t

where superscripts h and l stand for HOMO and LUMO, the sums run over the conjugated atoms of the π system, and pt represents an atomic 2pz orbital on atom t. In the ground state, ψ h usually contains two electrons. If we use the notation ψ k( j) to indicate that electron j is in orbital ψ k, we can express the wavefunction for the ground state as a product: Ψa = ψ h ð1Þψ h ð2Þ:

ð4:20Þ

We have factored out all the filled orbitals below the HOMO and have omitted them in Eq. (4.20) on the simplifying assumption that the electrons in these orbitals are not affected by the movement of one of the outer electrons from the HOMO to the LUMO. This represents only a first approximation to the actual rearrangement of the electrons that accompanies the excitation. In the excited state, either electron 1 or electron 2 could be promoted from the HOMO to the LUMO. Since we cannot distinguish the individual electrons, the wavefunction for the excited state must combine the various possible ways that the electrons could be assigned to the two orbitals: Ψb = 2 - 1=2 ψ h ð1Þψ l ð2Þ þ 2 - 1=2 ψ h ð2Þψ l ð1Þ:

ð4:21Þ

Again, we have factored out all the orbitals below the HOMO. For now, we also neglect the spins of the two electrons and assume that no change of spin occurs during the excitation. The wavefunctions written for both the ground and the excited state pertain to singlet states, in which the spins of electrons 1 and 2 are antiparallel (Sect. 2.4). We will return to this point in Sect. 4.9. By using Eqs. (4.20 and 4.21) for Ψ a and Ψ b, the transition dipole can be reduced to a sum of terms involving the atomic coordinates and the molecular orbital coefficients Cht and C lt for the HOMO and LUMO:

4.4 Calculating Transition Dipoles for p Molecular Orbitals →

μjΨa i = hΨb je μ ð 1Þ þ e μð2ÞjΨa i μ ba  hΨb je D E = 2 - 1=2 ðψ h ð1Þψ l ð2Þ þ ψ h ð2Þψ l ð1Þje μ ð 1Þ þ e μð2Þjψ h ð1Þψ h ð2Þ

157

ð4:22aÞ ð4:22bÞ

=2 -1=2 hψ l ð1Þje μð1Þjψ h ð1Þihψ h ð2Þjψ h ð2Þiþ2 -1=2 hψ l ð2Þje μð2Þjψ h ð2Þi ð4:22cÞ  hψ h ð1Þjψ h ð1Þi * + X pffiffiffi X l pffiffiffi h μ j C t pt μðkÞjψ h ðkÞi ≈ 2 Cs ps je ð4:22dÞ = 2hψ l ðk Þje s

t

pffiffiffi X pffiffiffi XX l h C s C t hps jerjpt i ≈ 2 e Cls Cht rt , = 2 e s

t

ð4:22eÞ

t

where ri is the position of atom i. In this derivation, we have separated the dipole operator e μ into two parts that make identical contributions to the overall transition dipole. One part, e μð1Þ, acts only on electron 1 while e μð2Þ acts only on electron 2; μð2Þjψ h ð1Þi are zero. The final step of the derivation μð1Þjψ h ð2Þi and hψ l ð1Þje hψπ ð2Þje uses the fact that pt jer j pt = rt and the approximation jps jerjpt j ≈ 0 for s ≠ t. As an example consider ethylene, for which the HOMO and LUMO can be described approximately as symmetric and antisymmetric combinations of carbon 2p orbitals: Ψ a = 2-1/2( p1 + p2) and Ψ b = 2-1/2( p1 - p2), respectively (Fig. 2.7). The corresponding absorption band occurs at 175 nm. Equation (4.22e) gives a transition dipole of (21/2/2)e(r1 - r2) = (21/2/2)er12, where r12 is the vector from carbon 2 to carbon 1. The transition dipole vector thus is aligned along the C=C bond. The calculated dipole strength is Dba = |μba|2 = (e2/2)|r12|2, or 11.53|r12|2 debye2 if r12 is given in Å. From this result, it might appear that the dipole strength would increase indefinitely with the square of the C=C bond length. But if the bond is stretched much beyond the length of a typical C=C double bond, the description of the HOMO and LUMO as symmetric and antisymmetric combinations of the two atomic pz orbitals begins to break down. In the limit of a large interatomic distance, the orbitals are no longer shared by the two carbons but instead are localized entirely at one site or the other. Equation (4.22e) then gives a dipole strength of zero because either Chi or Cli is zero in each of the products that enters into the sum. Note that although the transition dipole calculated by Eq. (4.22e) has a definite direction, flipping it over by 180∘ would not affect the absorption spectrum because the extinction coefficient depends on the dipole strength (|μba|2) rather than on μba itself. This makes sense, considering that the electric field of light oscillates rapidly in sign. However, we later will consider transitions that are best described by linear combinations of excitations in which an electron moves between any of several different pairs of molecular orbitals, rather than simply from the HOMO to the LUMO (Sect. 4.7 and Chap. 8). Because the overall transition dipole in this situation is a vector combination of the weighted transition dipoles for the individual excitations, the relative signs of the individual contributions are important.

158

4.5

4

Electronic Absorption

The Role of Molecular Symmetry in Electronic Transitions

We have seen that the transition dipole of ethylene is aligned along the bond between the two carbon atoms. Because the strength of absorption depends on the dot product of μba and the electrical field of the light (Eq. (4.8c)), absorption of light polarized parallel to the C=C bond is said to be allowed, whereas absorption of light polarized perpendicular to the bond is forbidden. It often is easy to determine whether or not an absorption band is allowed, and if so for what polarization of light, by examining the symmetries of the molecular orbitals involved in the transition. Note first that the transition dipole depends on the integral over all space of a product of the position vector and the amplitudes of the two wavefunctions: Z μjΨa i = ehΨb jrjΨa i  e Ψb  rΨa dσ: ð4:23Þ μba = hΨb je At a given point r, each of these three quantities could have either a positive or a negative sign depending on our choice of the origin for the coordinate system. However, the magnitude of a molecular transition dipole does not depend on any particular choice of coordinate system. This should be clear in the case of ethylene, where Eq. (4.22) shows that |μba| depends only on the length of the carbon-carbon bond (|r12|) and is oriented along this bond. More generally, if we shift the origin by adding any constant vector R to r, Eq. (4.23) becomes μba = ehΨb jr þ RjΨa i = ehΨb jrjΨa i þ ehΨb jRjΨa i

ð4:24aÞ

= ehΨb jrjΨa i þ eRhΨb jΨa i = ehΨb jrjΨa i,

ð4:24bÞ

which is the same as before. The term eRhΨ b j Ψ ai is zero as long as Ψ a and Ψ b are orthogonal. Similarly, rotating the coordinate system modifies the x, y, and z components of μba but does not change the magnitude or orientation of the vector. Returning to ethylene, let’s put the origin of the coordinate system midway between the two carbons and align the C=C bond on the y-axis, making the z axes of the individual atoms perpendicular to the bond as shown in Fig. 4.6 (see also Fig. 2.7). Like the atomic pz orbitals (Fig. 2.6B), the HOMO (Ψ a) then has equal magnitudes but opposite signs on the two sides of the xy plane; it is said to be an odd or antisymmetric function of the z coordinate (Fig. 4.6A). In other words, Ψ a(x, y, z) = - Ψ a(x, y, -z) for any fixed values of x and y. The same is true of the LUMO: Ψ b(x, y, z) = - Ψ b(x, y, -z) (Fig. 4.6B). The product of the HOMO and the LUMO, on the other hand, has the same sign on the two sides of the xy plane, and thus is an even or symmetric function of z: Ψ b(x, y, z)Ψ a(x, y, z) = Ψ b(x, y, -z)Ψ a(x, y, -z) (Fig. 4.6C). The product Ψ bΨ a also is an even function of x, but an odd function of y (Fig. 4.6D). To evaluate the x, y, or z component of the transition dipole, we have to multiply Ψ bΨ a by, respectively, x, y, or z and integrate the result over all space. Since z has opposite signs on the two sides of the xy plane, whereas Ψ bΨ a has the same sign, the

4.5 The Role of Molecular Symmetry in Electronic Transitions

159

4

B

A

z/a

2 0 -2 ψa

ψb

x=0

x=0

-4

C

2 x/a

z/a

2 0 -2

ψbψa -2

x=0 0 y/a

2

D

0 -2

ψbψa -2

z = ao 0 y/a

2

Fig. 4.6 Orbital symmetry in the first π → π  electronic transition of ethylene. Panels A, B Contour plots of the amplitudes of the HOMO (π, A) and LUMO (π , B) wavefunctions. C, D Contour plots of the products of the two wavefunctions. The C=C bond is aligned with the y-axis, and the atomic z axes are parallel to the molecular z-axis. In A–C, the plane of the drawing coincides with the yz plane; in D, the plane of the drawing is parallel to the molecular xy plane and is above this plane by   the Bohr radius a0 = 0:529Å . The wavefunctions are constructed as in Fig. 2.7. Blue curves represent positive amplitudes; red curves, negative. Distances are plotted as dimensionless 3=2 3=2 multiples of ao, and the contour intervals are 0:05a0 in A and B and 0:02a0 in C and D. The arrows in C and D show the transition dipole in units of eÅ=ao as calculated by Eq. (4.22e)

quantity zΨ bΨ a is an odd function of z and will vanish if we integrate it along any line parallel to the z-axis: Z1 zΨb ðx, y, zÞΨa ðx, y, zÞdz = 0

ð4:25Þ

-1

The z component of μba therefore is zero. The same is true for the x component. By contrast, the quantity y Ψ bΨ a is an even function of y and will give either a positive result or zero if it is integrated along any line parallel to y (Fig. 4.6C, D). y Ψ bΨ a also is an even function of x and an even function of z, so its integral over all space must μjΨ a i thus has a nonzero y component. be nonzero. The transition dipole hΨ b je A particle in a one-dimensional box provides another simple illustration of these principles. Inspection of Fig. 2.2A shows that the wavefunctions for n = 1, 3, 5 . . .

160

4

Electronic Absorption

are all symmetric functions of the distance from the center of the box (Δx) whereas those for even n = 2, 4, . . . are all antisymmetric. The product of the wavefunctions for n = 1(ψ 1) and n = 2(ψ 2) thus has the same symmetry as Δx. The quantity Δx ψ 1ψ 2 therefore will give a non-zero result if we integrate it over all values of Δx, which means that excitation from ψ 1 to ψ 2 has a non-zero transition dipole oriented along x. The same is true for excitation from ψ 1 to any of the higher states with even values of n, but not for transitions to states with odd values of n. In this system, the selection rule for absorption is simply that n must change from odd to even or vice versa. To generalize the foregoing results, we can say that the j component ( j = x, y, or μjΨ a i will be zero if the product j Ψ aΨ b has odd z) of a transition dipole hΨ b je reflection symmetry with respect to any plane (xy, xz, or yz). Excitation from Ψ a to Ψ b by light polarized in the j direction then is said to be forbidden by symmetry. Simple considerations of molecular symmetry thus often can determine whether such a transition is forbidden or allowed. The selection rules imposed by the symmetry of the initial and final molecular orbitals can be expressed in still more general terms in the language of group theory → μjΨ a i to be non-zero, the product Ψ b r Ψ a must have a (Sect. 4.6). For hΨ b je component that is totally symmetric with respect to all the symmetry operations that apply to the molecule. The applicable symmetry operations depend on the molecular geometry, but can include reflection in a plane, rotation around an axis, inversion through a point, and a combination of rotation and reflection called an “improper rotation”. By saying that a quantity is totally symmetric with respect to a symmetry operation such as rotation by 180∘ about a given axis, we mean that the quantity does not change when the molecule is rotated in this way. If the operation causes the quantity to change sign but leaves the absolute magnitude the same, then the integral of the quantity over all space will be zero. For the π - π transition of ethylene, for example, the product Ψ bz Ψ a changes sign upon reflection in the xy plane, which is one of the symmetry operations that apply to the molecule when the x, y, and z axes are defined as in Fig. 2.7. The z component of the transition dipole therefore is zero. Ψ bx Ψ a does the same upon reflection in the yz plane, so the x component of the transition dipole is also zero. Ψ by Ψ a, however, is unchanged by such a reflection or any of the other applicable symmetry operations (reflection in the xz or yz plane, rotation by 180o around the x, y, or z-axis, or inversion of the structure through the origin), so the y component of the transition dipole is non-zero. But that is all we can say based simply on the symmetry of the molecule. To find the actual magnitude of the y component of μba, we have to evaluate the integral. To illustrate the vectorial nature of transition dipoles for a larger molecule, Fig. 4.7 shows the two highest occupied and two lowest unoccupied molecular orbitals of bacteriochlorophyll-a. These four wavefunctions are labeled ψ 1 - ψ 4 in order of increasing energy. The products of the wavefunctions for the four possible excitations (ψ 1 → ψ 3, ψ 1 → ψ 4, ψ 2 → ψ 3 and ψ 2 → ψ 4) are shown in Fig. 4.8. The conjugated atoms of bacteriochlorophyll-a form an approximately planar π system, and the wavefunctions and their products all have a plane of reflection

4.5 The Role of Molecular Symmetry in Electronic Transitions

10

ψ3

ψ4

ψ1

ψ2

161

y/ao

5 0 -5 -10

10

y/ao

5 0 -5 -10 -10

-5

0 x /ao

5

10 -10

-5

0 x /ao

5

10

Fig. 4.7 Contour plots of the two highest occupied orbitals (ψ 1 and ψ 2) and the two lowest unoccupied molecular orbitals (ψ 3 and ψ 4) of bacteriochlorophyll-a. The plane of each drawing is parallel to the plane of the macrocyclic ring and is above the ring by the Bohr radius, ao (Fig. 2.7, panels D, E). Blue curves represent positive amplitudes; red curves, negative amplitudes. The contours for zero amplitude are omitted for clarity. Distances are given as multiples of ao, and the 3=2 contour intervals are 0:02a0 . The skeleton of the π system is shown with green lines. The coefficients for the atomic pzorbitals were obtained with the program QCFF/PI [30, 183]

symmetry that coincides with the plane of the macrocyclic ring. Because the wavefunctions have opposite signs on the two sides of this plane (z), their products are even functions of z. The z component of the transition dipole for excitation from ψ 1 or ψ 2 to ψ 3 or ψ 4b therefore will be zero: all the transition dipoles lie in the plane of the π-system. Inspection of Fig. 4.8 shows further that the products ψ 1ψ 4 and ψ 2ψ 3 both are approximately odd functions of the y (vertical) coordinate in the figure, and approximately even functions of the x (horizontal) coordinate. The transition dipoles for ψ 1 → ψ 4 and ψ 2 → ψ 3 are therefore oriented approximately

162

4

10

ψ2ψ3

ψ1ψ4

ψ1ψ3

ψ2ψ4

Electronic Absorption

y/ao

5 0 -5 -10

10

y/ao

5 0 -5 -10 -10

-5

0 x /ao

5

10 -10

-5

0 x /ao

5

10

Fig. 4.8 Contour plots of products of the four molecular wavefunctions of bacteriochlorophyll-a shown in Fig. 4.7. The planes of the drawings and the line colors are as in Fig. 4.7. The contour intervals are 0:002a30 . The arrows show the transition dipoles calculated by Eq. (4.22e), with the length in units of eÅ=5a0

parallel to the y-axis. The other two products, ψ 1ψ 3 and ψ 2ψ 4, are approximately odd functions of x and approximately even functions of y, so the transition dipoles for ψ 1 → ψ 3 and ψ 2 → ψ 4 must be approximately parallel to the x-axis. The calculated transition dipoles confirm these qualitative predictions (Fig. 4.8). The fact that the transition dipoles for the ψ 1 → ψ 3 and ψ 2 → ψ 4 excitations point in opposite directions reflects arbitrary choices of the signs of the wavefunctions and has no particular significance. Figure 4.9 shows similar calculations of the transition dipoles for the two highest occupied and lowest empty molecular orbitals of 3-methylindole, a model of the side chain of tryptophan. Again, the transition dipoles for excitation from one of these orbitals to another must lie in the plane of the π system. The transition dipole

4.5 The Role of Molecular Symmetry in Electronic Transitions

4

A

163

B

y/ao

2 0

-2

ψ3

-4 4

C

ψ4

D

y/ao

2 0

-2

ψ1

-4 4

E

ψ2

F

y/ao

2 0

-2

ψ2ψ3

-4 4

G

ψ1ψ4

H

y/ao

2 0

-2

ψ1ψ3

-4 -6 -4 -2

0 2 x/ao

4

6

ψ2ψ4 -6 -4 -2

0 2 x /ao

4

6

Fig. 4.9 Contour plots of the two lowest unoccupied molecular orbitals (ψ 3 and ψ 4, A, B) and the two highest occupied orbitals (ψ 1 and ψ 2, C, D) of 3-methylindole, and products of these wavefunctions (ψ 2ψ 3, ψ 1ψ 4, ψ 1ψ 3 and ψ 2ψ 4, E–H). Positive amplitudes are indicated with blue lines; negative amplitudes with red. Small contributions from the methyl group are neglected. The 3=2 contour intervals are 0:02a0 in A–D and 0.005a03in E–H. The arrows in E–H show the transition dipoles calculated by Eq. (4.22e), with the length in units of eÅ=2:5a0 . The atomic coefficients for the orbitals were obtained as described by Callis [42, 43, 184]

164

4

Electronic Absorption

calculated for the ψ 2 → ψ 3 and ψ 1 → ψ 4 excitations are oriented approximately 30∘ from the x-axis in the figure (Fig. 4.9E, F). That calculated for the ψ 1 → ψ 3 and ψ 2 → ψ 4 excitations is approximately 120∘ from the x-axis (Fig. 4.9G, H). Transitions that are formally forbidden by symmetry with the dipole operator sometimes are weakly allowed by the quadrupole or octupole terms in Eq. (4.5). Forbidden transitions also can be promoted by vibrational motions that perturb the symmetry or change the mixture of electronic configurations contributing to the excited state. This is called vibronic coupling. Finally, some transitions with small electric transition dipoles can be driven by the magnetic field of light. We’ll return to this point in Chap. 9. Figure 9.4 of that chapter also illustrates how the electric transition dipoles for the first four excitations of trans-butadiene depend on the symmetries of the molecular orbitals. The symmetry of the atomic orbitals allows excitation from an s orbital to a p orbital but forbids excitation from s to f. More generally, if we neglect the intrinsic spin of electrons for now, a selection rule for atomic absorption is that the azimuthal quantum number l that determines the electron's orbital angular momentum must change by ±1. For absorption of circularly polarized light, the magnetic quantum number m that determines the projection of the electron's angular momentum on a particular axis also must change by either -1 or +1, depending on whether the photon has left (ms = + 1) or right (ms = - 1) circular polarization, respectively (Sect. 3.3). These changes in l and m by ±1 keep both the total angular momentum and the projection of the angular momentum on any arbitrary axis constant. No change in m is needed for absorption of linearly polarized light because a linearly polarized photon is in a superposition of states with left and right circular polarization. The magnetic selection rule for atomic absorption of linearly polarized light is, therefore, Δm = 0. Selection rules that depend on the electronic spins of the groundand excited-state orbitals will be discussed in Sect. 4.9.

4.6

Using Group Theory to Determine Whether a Transition Is Allowed by Symmetry

Molecular structures and orbitals can be classified into various point groups according to the symmetry elements they contain. Symmetry elements are lines, planes, or points with respect to which various symmetry operations such as rotation can be performed without changing the structure. Point groups and their symmetry operations are commonly described with a notation that is called the Schönflies notation after the German mathematician Arthur Moritz Schönflies (1853–1926). The symmetry operations that concern us here are as follows: e n). If rotation by 2π/n radians (1/n of a full rotation) about a particular Rotation (C axis generates a structure that (barring isotopic labeling) is indistinguishable from the original structure, the molecule is said to have a Cn axis of rotational symmetry. σ). The xy plane is said to be a plane of reflection symmetry or a mirror Reflection (e plane (σ) if moving each atom from its original position (x,y,z) to (x,y,-z) gives an

4.6 Using Group Theory to Determine Whether a Transition Is Allowed by Symmetry

165

identical structure. A mirror plane often is designated as σ h if it is normal to a principal rotation axis, or σ v if it contains this axis. Inversion (ei). A molecule has a point of inversion symmetry (i) if moving each atom in a straight line through the point to the opposite side of the molecule gives the same structure. If we use the point of inversion symmetry as the origin of the coordinate system, the inversion operation moves each atom from its original position (x,y,z) to (-x,-y,-z). Improper rotation (e Sn ). Improper rotation is a rotation by 2π/n followed by reflection through a plane perpendicular to the rotation axis. An axis of improper rotation (Sn) is an axis about which this operation leaves the structure of a molecule unchanged. e The identity operator leaves all the particles in a molecule where it Identity (E). finds them, which is to say that it does nothing. It is, nevertheless, essential to group theory. All molecules have the identity symmetry element (E). Translation, though an important symmetry operation in crystallography, is not included here because we are concerned with the symmetry of an individual molecule. A molecule whose center of mass has been shifted is, in principle, distinguishable from the original molecule. Point groups are sets of symmetry elements corresponding to symmetry operators that obey four general rules of group theory: e 1. Each group must include the identity operator E. e in the group, the group must include an inverse operator A e -1 2. For each operator A e -1  A e =A e A e - 1 = E. e with the properties that A e e B e e e and B eA 3. If operators A and B are members of a group, then the products A also must be in the group. (As in Chap. 2, the operator written on the right in such a product acts first, followed by the one on the left. The order may or may not matter, depending on the operators and the point group. A symmetry operator e but not necessarily with the necessarily commutes with its inverse and with E, other operators.) 4. Multiplication   of the symmetry operators must be associative. This means that e B e = A e B e for any three operators in the group. eC e C A Let’s look at a few examples. Ethylene has three perpendicular axes of two-fold rotational symmetry [C2(x), C2( y), and C2(z)], three planes of reflection symmetry [σ(xz), σ(yz) and σ(xy)], and a center of inversion symmetry (i) (Fig. 4.10A). When combined with the identity element E, these symmetry elements obey the general rules for a group. They are called the D2h point group. The conjugated atoms of porphin have a C4 axis (z), four C2 axes in the xy plane, four planes of reflection symmetry containing the z-axis (σ v), a center of inversion symmetry, and identity (Fig. 4.10B). These elements form the D4h point group.

166

4

Electronic Absorption

Fig. 4.10 Symmetry elements in ethylene (A), porphyrin (B), water (C), and a peptide (D). In A, ethylene is drawn in the xy plane. The x, y, and z axes here are all axes of two-fold rotational symmetry (C2), and the xy, xz, and yz axes are planes of mirror symmetry. If we take z to be the “principal” axis of rotational symmetry, the mirror planes that contain this axis (xz and yz) are called “vertical” mirror planes (σ v) and the mirror plane normal to z (xy) is called a “horizontal” plane of mirror symmetry (σ h). In B, porphyrin is viewed along an axis (z, filled circle) normal to the plane of the macrocycle. The z-axis is an axis of four-fold rotational symmetry (C4) and is the principal symmetry axis. There are four C2 axes in the xy plane (dotted lines), four vertical planes of mirror symmetry (σ v), one horizontal plane of mirror symmetry (xy), and a point of inversion symmetry at the center. Water, drawn in the yz plane in C, has one C2 axis (z) and two vertical planes of mirror symmetry (xz and yz). The peptide bond (D) has a plane of mirror symmetry (the plane of the drawing), but no other symmetry elements

Water has a C2 axis that passes through the oxygen atom and bisects the H-O-H angle, and two perpendicular reflection planes that contain the C2 axis, and again, identity (Fig. 4.10C). It is in point group C2v. The backbone of a peptide has only one symmetry element other than identity, a mirror plane that includes the central N, C, O and Cα atoms (Fig. 4.10D). This puts it in point group Cs. For general procedures for the algebraic treatment of molecular point groups and finding the point group for a molecule see [9–11]. As shown in Table 4.1 for the D2h point group, the products of the operations in a point group can be collected in a multiplication table. The entries in such a table are the results of performing the symmetry operation for the symmetry element given in row 1, followed by the operation for the element given in column 1. In the D2h point group, the operators all commute with each other and each of the operators is its own inverse. For example, rotating around a C2 axis by 2π/2 and then rotating by an additional 2π/2 around the same axis returns all the atoms to their original positions,

4.6 Using Group Theory to Determine Whether a Transition Is Allowed by Symmetry

167

Table 4.1 Products of symmetry operations for the D2h point group E C2(z) C2( y) C2(x) i σ h(xy) σ v(xz) σ v(yz)

E E C2(z) C2( y) C2(x) i σ h(xy) σ v(xz) σ v(yz)

C2(z) C2(z) E C2(x) C2( y) σ h(xy) i σ v(yz) σ v(xz)

C2( y) C2( y) C2(x) E C2(z) σ v(xz) σ v(yz) i σ h(xy)

C2(x) C2(x) C2( y) C2(z) E σ v(yz) σ v(xz) σ h(xy) i

i i σ h(xy) σ v(xz) σ v(yz) E C2(z) C2( y) C2(x)

σ h(xy) σ h(xy) i σ v(yz) σ v(yz) C2(z) E C2(x) C2( y)

σ v(xz) σ v(xz) σ v(yz) i σ h(xy) C2( y) C2(x) E C2(z)

σ v(yz) σ v(yz) σ v(xz) σ h(xy) i C2(x) C2( y) C2(z) E

Fig. 4.11 Projection drawings of the effects of symmetry operations in point group D2h. The large red circles represent regions of space containing the atoms in the group; horizontal and vertical lines indicate the x and y axes; the z-axis is normal to the plane of the paper. The small dark blue circles represent points above the xy plane; the small cyan circles, points below the plane. The symmetry operations in point group D2h move an atom from its initial position (x,y,z) to the indicated positions

 n - 1 e2  C e 2 = E. e n in any point group is C en e More generally, the inverse of C so C . To work out the other products, it is helpful to make projection drawings of the type shown in Fig. 4.11, where the small filled circles represent points above the xy-plane, and the empty circles represent points below this plane.

168

4

Electronic Absorption

Symmetry operators and their products can be represented conveniently by matrices. Suppose an atom of a molecule in the D2h point group has coordinates r = (x,y,z) relative to the molecule’s center of mass. If we represent the coordinates by a column vector and use the expression for multiplication of a vector by a matrix (Eq. (A.20) in Appendix A2), the actions of the symmetry operators on the atom’s location can be written: 2 30 1 0 1 x 1 0 0 x 7B C B C e  r=6 ð4:26aÞ = E 0 1 0 y 4 5@ A @ y A z z 1 2 30 1 0 -x 0 x -1 0 C 7B C B eI  r = 6 ð4:26bÞ -1 0 5@ y A = @ - y A 4 0 -z 0 0 -1 z 3 2 30 1 2 -1 0 0 cosð2π=2Þ sinð2π=2Þ 0 x 7 7B C 6 e 2 ðzÞ  r = 6 C -1 05 4 - sinð2π=2Þ cosð2π=2Þ 0 5@ y A = 4 0 0 0 1 0 0 1 z 1 0 1 0 -x x C B C B ð4:26cÞ @yA=@ -yA 0 0

1

z

z 2

1 6 e C2 ðxÞ  r = 4 cosð2π=2Þ - sinð2π=2Þ 1 0 1 0 x x C B C B @yA=@ -yA

0 0

30 1 2 1 0 0 x 7B C 6 sinð2π=2Þ 5@ y A = 4 - 1 0

0

cosð2π=2Þ

z

0

0

3 0 7 0 5 -1 ð4:26dÞ

-z

z

30 1 2 cosð2π=2Þ 0 sinð2π=2Þ x 6 7B C 6 e C2 ðyÞ  r = 4 0 1 0 5@ y A = 4 - sinð2π=2Þ 0 cosð2π=2Þ z 1 0 1 0 -x x C B C B @yA=@ y A -z z 2 30 1 0 1 0 0 x 6 7B C B e σðxyÞ  r = 4 0 1 0 5@ y A = @ 2

0

0

-1

z

-1 0

0 1

0

0

3 0 7 0 5 -1 ð4:26eÞ

x

1

C y A -z

ð4:26fÞ

4.6 Using Group Theory to Determine Whether a Transition Is Allowed by Symmetry

2

1 6 e σðxzÞ  r = 4 0

0 -1

0

0

169

30 1 0 1 0 x x 7B C B C 0 5@ y A = @ - y A 1 z z

ð4:26gÞ

30 1 0 1 0 x -x 7B C B C 0 5 @ y A = @ y A: 1 z z

ð4:26hÞ

and 2

-1 6 e σðyzÞ  r = 4 0 0

0 1 0

By using the expression for the product of two matrices (Eq. (A.2.6)) and the fact that matrix multiplication is associative, it is straightforward to show that these e 2 ðzÞ  matrices have the same multiplication table as the operators. The products C e e e C2 ðzÞ and i  C2 ðzÞ, for example, are 2 32 30 1 -1 0 0 -1 0 0 x 6 76 7B C e e C 2 ðzÞ  C 2 ðzÞ  r = 4 0 - 1 0 54 0 - 1 0 5@ y A 2

1 6 =40 0

0 0 1 30 1 0 1 x 0 0 x 7B C B C 1 0 5@ y A = @ y A 0 1

z

0

0

1

z ð4:27aÞ

z

and 2

32 -1 0 0 -1 6 7 6 e 2 ðzÞ  r = 4 0 eI  C -1 0 54 0 0 0 -1 0 1 2 30 1 0 x 1 0 0 x C 6 7B C B =40 1 0 5@ y A = @ y A: -z 0 0 -1 z

0 -1 0

30 1 0 x 7B C 0 5@ y A 1

z ð4:27bÞ

You can verify these results by referring to Fig. 4.11. Multiplying any of the matrices e (the identity matrix) leaves the first matrix unchanged, by the matrix representing E as it should: 3 30 1 2 32 2 -1 0 0 x -1 0 0 1 0 0 7 7B C 6 76 e 2 ðzÞ  r = 6 eC -1 05 E - 1 0 5@ y A = 4 0 4 0 1 0 54 0 0 0 0 1 1 0 1 0 -x x C B C B × @ y A = @ - y A: z z

0

1

z

0

0

1 ð4:28Þ

170

4

Electronic Absorption

Since (a) the products of the matrices have the same multiplication table as the operators in the point group, (b) matrix multiplications are associative, (c) each of the products is the same as one of the original matrices, and (d) every matrix with a non-zero determinant has an inverse (Appendix A2), the set of matrices representing the symmetry operators in a point group (Eqs. 4.26a–4.26h) meets the criteria stated above for a group. The matrix group thus provides a representation of the point group, just as the individual matrices provide representations of the individual symmetry operators. The vector (x,y,z) that we used in this example is said to form a basis for a representation of the D2h point group. There are an infinite number of possible choices for such a basis, and the matrices representing the symmetry operators depend on our choice. We could, for example, use the coordinates of the six atoms of ethylene, in which case we would need an 18×18 matrix to represent each of the operators. Other possibilities would be a set of bond lengths and angles, the molecular orbitals of ethylene, or other functions of the coordinates. But the choices can be reduced systematically to a small set of representations that are mathematically orthogonal to each other. The number of these irreducible representations depends on how many different classes of symmetry operations make up the point e group. We will not go into this in detail here other than to say that two operators X e and Y are in the same class, and are said to be conjugate operators, if and only if eZ e where Z e is some other operator. The procedure of transforming X e to Y e e =Z e‐1 X Y ‐1 e e e by forming the product Z X Z is called a similarity transformation. In the D2h point group, each of the eight symmetry operators is of a different class, so there are eight different irreducible representations. Any more complicated (reducible) representation can be written as a linear combination of these eight irreducible representations, just as a vector can be constructed from Cartesian x, y, and z components. Information on the irreducible representations of the various point groups is presented in tables that are called character tables because they give the characters of the symmetry operations in the point group. The character of a symmetry operation is the trace (the sum of the diagonal elements) of the matrix that represents that operation. Referring to Eqs. (4.26e–4.26h), we see that the matrices representing the e 2 ðyÞ, C e 2 ðzÞ, e e 2 ðxÞ, C σ ðxyÞ, e σ ðxzÞ , and e σ ðyzÞ in the D2h point group all operations C have characters of either ±1. These operations are one-dimensional in the sense that we can represent them as multiplication by a one-dimensional matrix, which amounts to multiplication by a single number. Multiplication by +1 means that the operation does not change the property being represented. It could, for example, mean that the sign of a wavefunction or the direction of a vibrational motion remains the same. A character of -1 would mean that the symmetry operation changes the sign of the wavefunction or reverses the direction of motion. A character of ±2 indicates an operation of multiplying by a 2-dimensional matrix. The point group C3v, for example, includes the operation of rotation by 2π/3 around the z-axis, which can be represented as  multiplication of (x,y) by the 2×2 matrix  cos ð2π=3Þ - sin ð2π=3Þ . A character of ±3 similarly signals multiplication sin ð2π=3Þ cos ð2π=3Þ by a 3-dimensional matrix.

4.6 Using Group Theory to Determine Whether a Transition Is Allowed by Symmetry

171

Character tables for the D2h, C2v, C3v and C4v point groups are presented in Table 4.2. Various classes of the symmetry elements of the point group are displayed in the top row of each table, and the conventional names, called Mulliken symbols, of the irreducible representations that comprise the point groups are given in the first column. Letters A and B are the Mulliken symbols for representations comprised of operations with a character of ±1 or zero. The operations in the D2h and C2v point groups are all of this type. Mulliken’s E, which stands for the German term entartet (degenerate), denotes representations that require a two-dimensional matrix to handle pairs of bases that have identical energies (two-fold degeneracies). T (sometimes replaced by F) denotes representations that use three-dimensional matrices for threefold degeneracies. We’ll have more to say about this in Chap. 6 when we consider representations of degenerate vibrational modes. Representations of types A and B differ in being, respectively, symmetric and antisymmetric with respect to a Cn axis of rotation. Subscripts g and u stand for the German terms gerade and ungerade and indicate whether the representation is even (g) or odd (u) with respect to a center of inversion. The numbers 1, 2, and 3 in the subscripts distinguish different Table 4.2 Character tables for the D2h, C2v, C3v and C4v point groups D2h

E

C2(z)

C2( y)

C2(x)

i

σ h(xy)

σ v(xz)

σ v(yz)

Ag B1g B2g B3g Au B1u B2u B3u

1 1 1 1 1 1 1 1

1 1 -1 -1 1 1 -1 -1

1 -1 1 -1 1 -1 1 -1

1 -1 -1 1 1 -1 -1 1

1 1 1 1 -1 -1 -1 -1

1 1 -1 -1 -1 -1 1 1

1 -1 1 -1 -1 1 -1 1

1 -1 -1 1 -1 1 1 -1

Functions x2, y2, z2 xy xz yz z y x

C2v

E

C2

σ v(xz)

σ 0v ðyzÞ

Functions

A1 A2 B1 B2

1 1 1 1

1 1 -1 -1

1 -1 1 -1

1 -1 -1 1

z

C3v

E

2C3

3σ v

Functions

A1 A2 E

1 1 2

1 1 -1

1 -1 0

z Rz [x, y][Rx, Ry]

x2, y2, z2 xy xz yz

x y x2 + y2, z2

[x2 - y2, xy][xz, yz]

C4v

E

2C4

C2

2σ v

2σ d

Functions

A1 A2 B1 B2 E

1 1 1 1 2

1 1 -1 -1 0

1 1 1 1 -2

1 -1 1 -1 0

1 -1 -1 1 0

z

x2 + y2, z2

[x, y][Rx, Ry]

x2 - y2 xy xz, yz

172

4

Electronic Absorption

representations of the same general type. The three columns on the right side of the character table give some linear and quadratic functions of the coordinates that have the same symmetry as the various irreducible representations. These are referred to as basis functions for the irreducible representations. Looking at the character table for the D2h point group, we see again that each symmetry operation either leaves the representation unchanged, in which case the character is +1, or changes the sign of the representation, making the character –1. Ag is the totally symmetric representation. Like the quadratic functions x2, y2 and z2, it is unaffected by any of the symmetry operations of the D2h point group. Au also is unaffected by rotation about the x, y, or z-axis but changes sign on inversion or reflection across the xy, xz, or yz plane. It thus has the same symmetry properties as the product xyz in this point group. B1u, B2u, and B3u have the same symmetry as the coordinates z, y, and x, each of which changes sign on rotation around two of the three C2 axes, inversion, or reflection across one plane. The behaviors of the basis functions on the right side of the character table for the D2h point group are described by saying that x2, y2, and z2 transform as Ag in this point group, xy transforms as B1g, z transforms as B1u, and so forth. Turning to the C2v and C4v point groups, note that here z transforms as the totally symmetric irreducible representation, A1. Rotation around the C2 axis does not change the z coordinate in this point group, because the single C2 axis coincides with the zaxis (Fig. 4.7C). Reflection across the xz or yz plane also leaves the z coordinate unchanged. There is no center of inversion in the C2v or C4v point group, so none of the Mulliken symbols has a g or u subscript. The entries 2C4, 2σ v and 2σ d in the top row of the character table for the C4v point group mean that this point group has two independent symmetry operations in each of the C4, σ v and σ d classes. The C4 class includes both the C4 operation itself and the inverse of this operation, C4-1, which is the same as C43. The basis functions for the two-dimensional irreducible representation (E) in the last row are pairs of coordinate values (x, y) or pairs of products of these values. Character 2 here means that the identity symmetry preserves both values, as it should, and character –2 indicates that the C2 operation changes the sign of both values. Character tables for virtually all the point groups that are encountered in chemistry are available [9, 10]. The full tables also include information on how the symmetry operations affect the direction of molecular rotation around the x, y, or z-axis, which is pertinent to the rotational spectroscopy of small molecules. Because molecular orbitals must recognize the symmetry of a molecule, a symmetry operation must either preserve the value of the wavefunction or simply change the sign of the wavefunction. The wavefunction therefore provides a basis for a representation of the molecule’s point group. Inspection of the HOMO and LUMO of ethylene (Fig. 4.6) shows that the HOMO transforms as the irreducible representation B1u and z in the D2h point group (Table 4.2), whereas the LUMO transforms as B3g and yz. Perhaps the most important feature of character tables for our present purposes is that we can determine the symmetry of a product of two irreducible representations

4.6 Using Group Theory to Determine Whether a Transition Is Allowed by Symmetry

173

simply by looking at the products of the characters of these representations in the same point group. For example, the product of the HOMO and LUMO of ethylene transforms as B2u in D2h, as one can see by comparing the products of the characters of B3g and B1u with the corresponding characters of B2u. Alternatively, we can just note that the product of B3g and B1u transforms as the product of z and yz, or yz2, which is equivalent to y. Further, the product of these two wavefunctions and y transforms as the square of B2u, or y2,which is the same as the totally symmetric representation, Ag. The electronic transition from the HOMO to the LUMO is, therefore, allowed upon excitation with radiation polarized along the y-axis. In general, excitation of a centrosymmetric system is forbidden if the initial and final orbitals both have the same symmetry for inversion (either both g or both u). This principle is known as Laporte’s rule. Like molecules and wavefunctions, the repeating structural units (unit cells) of a crystal can be characterized by symmetry operations. Only certain symmetry operations are consistent with a packing of the cells into a 3-dimensional lattice. This restriction leads to 32 possible symmetry operations, each of which is consistent with only one of seven different types of lattice. In order of increasing overall symmetry, the crystallographic lattice systems are referred to as triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. Table 4.3 lists the symmetry operations that are possible in each type of lattice. Also given in the table are the angles and edge lengths that must be specified to describe the size and shape of a unit cell. This requirement decreases as the overall lattice becomes more symmetric. The unit cell of a triclinic crystal structure, which has no restrictions imposed by symmetry, requires specifying six independent parameters: three unit-cell edge lengths (a, b, and c) and three angles between the axes (α, β and γ). The unit cell of a cubic structure, with the highest symmetry, can be described by a single length (a = b = c) and one angle (α = β = γ = 90∘). Unlike descriptions of individual molecules or wavefunctions, a complete description of a crystal lattice requires information on translational relationships between the unit cells in addition to the point-group symmetry of the individual cells. The full space groups usually are described with the Hermann-Mauguin notation, which is also known as the international notation. We will not elaborate on translational aspects of crystal structures because they are less pertinent to spectroscopy. Table 4.3 gives examples of the many crystalline materials that have practical applications in spectroscopy. The highly symmetric crystals of the cubic crystal system are isotropic with respect to their electro-optical properties and, thus, are not birefringent. Minerals belonging to this group include rock salt and diamond, which has the highest hardness and thermal conductivity of any natural material. Diamond is used in prisms and optical windows for UV and infrared spectroscopy and attenuated total reflectance spectroscopy because of its transparency over a broad band of wavelengths. Because it has the highest hardness and thermal conductivity of any natural material, diamond is the preferred material for optical anvils, which are used to view the optical properties of a small piece of material under extreme pressure.

1 6-fold axis of rotation: C6, C3 h, C6 h, C6v D6, D3 h, D6 h

4 3-fold axes of rotation: T, Th, Td, O, Oh

Cubic

Symmetry operations none: C1, Ci 1 2-fold axis of rotation or a mirror plane: C2, C1 h, C2 h 3 2-fold axes of rotation or 1 2-fold axis of rotation & 2 mirror planes: C2v, D2, D2 h C 4 ,C3v ,C4 h ,D4 1 4-fold axis of rotation: D2 h ,D4 h ,S4 1 3-fold axis of rotation: C3, C3i, C3v, D3, D3 d

Hexagonal

Trigonal

Tetragonal

Crystal system Triclinic Monoclinic Orthorhombic

Table 4.3 Crystal systems

2 : a = b ≠ c α = β = 90∘ γ = 120∘ 1 : a = b = c α = β = γ = 90∘

2 : a = b = c;α = β = γ ≠ 90∘

2 : a = b ≠ c; α = β = γ = 90∘

Cell parameters 6 : a ≠ b ≠ cα ≠ β ≠ γ 4 : a ≠ b ≠ c;α = γ, β ≥ 90∘ 3 : a ≠ b ≠ c;α = β = γ = 90∘

Diamond, rock salt, Y3Al5O12

Calcite, quartz, LiNbO3 proustite Ice, beryl

Examples Ulexite Mica, borax Olivine, perovskite, topaz Rutile, MgF2, KH2PO4

None

Uniaxial

Uniaxial

Uniaxial

Birefringence Biaxial Biaxial Biaxial

174 4 Electronic Absorption

4.6 Using Group Theory to Determine Whether a Transition Is Allowed by Symmetry

175

Yttrium aluminum garnet (YAG, Y3Al5O2) crystallizes in a complex cubic structure containing Y3+ ions in dodecahedral sites, Al3+ ions in both octahedral and tetrahedral sites, and three different oxygen polyhedra. Since the invention of the Nd:YAG laser in 1964, Nd:YAG has become the most widely used medium for solid-state lasers, including both low-power continuous-wave lasers and high-power Q-switched (pulsed) lasers [12]. Nd:YAG lasers produce IR light with a wavelength of 1064 nm. Frequency doubling provides green light at 532 nm, and tripling gives UV light at 355 nm. Of the crystal forms with low symmetry, the uniaxially birefringent minerals calcite and lithium niobate have trigonal crystal structures, as do quartz and ruby. Lithium niobate is a synthetic, ferromagnetic material with wide applications in Pockels cells and other electro-optical modulators. Chapters 1 and 3 discuss some of these applications, including the use of uniaxially birefringent crystals to form quarter- and half-wave plates. Quartz is used to make acousto-optic modulators, in which the application of pressure distorts the crystallographic unit cell and thus modulates the angle at which the crystal diffracts light. Its strength and transparency extending into the UV also make quartz a good material for optical windows, lenses, and prisms. Fused quartz, which is not birefringent, often becomes the material of choice for optical windows when birefringence is undesirable. The first working laser made by Theodore Maiman in 1960 was a ruby laser. Q-switched ruby lasers produce pulses at 694.3 nm with widths on the order of 10 ns. Proustite (Ag3AsS3)another member of the trigonal group, is transparent at IR wavelengths beyond 13 μm, which makes it an ideal material for IR lasers. Its exceptionally large coefficient for nonlinear susceptibility, about 300 times that of potassium dihydrogen phosphate, is useful for nonlinear optical effects such as the generation of second harmonics and sum frequencies. Minerals belonging to the hexagonal and tetragonal systems also can have uniaxial birefringence, whereas biaxial birefringence requires lower symmetry. Rutile, which is the most common natural form of TiO2, has a tetragonal unit cell. It has one of the highest refractive indices of any known crystal and exhibits strong birefringence (n = 2.86 and 2.58 at 632.8 nm for extraordinary and ordinary polarizations, respectively). These properties make rutile useful for coupling light into optical fibers and polarization optics at wavelengths between about 600 nm and 4.5 μm. Magnesium fluoride (MgF2), which also has tetragonal unit cells, is transparent over an exceptionally broad range extending from the vacuum UV to 8 μm and is used widely for optical windows, lenses, and prisms. Its refractive index of 1.37 is lower than those of flint glasses (1.45 to 2.00), making MgF2 suitable for inexpensive anti-reflective coatings. Potassium dihydrogen phosphate (KH2PO4, or KDP as it often is called) was among the first crystals used for nonlinear optical effects such as doubling the frequency of the output of Nd-YAG lasers to produce 632-nm light for microscopy or spectroscopy. KDP crystals that are transparent from 200 to 1500 nm are easily grown. Their high threshold for optical damage makes these crystals well suited for laser applications.

176

4

Electronic Absorption

Mica and ulexite, which are biaxially birefringent, belong to the monoclinic and triclinic systems, respectively. Micas include a broad group of minerals that can be split into thin, elastic plates. These have been used for optical fibers for transmitting light, and for wave plates and optoelectric applications. Ulexite (NaCaB5O6(OH)6  5H2O) is a structurally complex material that is used extensively in optical fibers. Its fiber-optic effect involves internal reflection that results partly from its biaxial birefringence. Perovskites are materials that crystalize with the composition ABX3. The original perovskite mineral, discovered in 1839 and named after the Russian mineralogist Lev Perovski, consisted of CaTiO3. CaTiO3 is orthorhombic and is biaxially birefringent. However, perovskites with larger cations in the A site, such as SrTiO3 and the synthetic metal-halide perovskites used in solar cells become cubic and are not birefringent. Applications of perovskites in solar cells are discussed in Chap. 5.

4.7

Linear Dichroism

As discussed above, the quantity (E0  μba)2 in Eq. (4.8c) is equal to |Eo|2|μba|2cos2θ, where θ is the angle between the molecular transition dipole (μba) and the polarization axis of the light (Eo). Because the absorption strength depends on cos2θ, rotating a molecule by 180o does not affect the absorption. However, if the molecules in a sample have a fixed orientation, the strength of the absorption can depend strongly on the angle between the orientation axis and the axis of polarization of the incident light. Such a dependence of the absorbance on the polarization axis is called linear dichroism. Linear dichroism is not seen if a sample is isotropic, i.e., made up of randomly oriented molecules so that θ takes on all possible values. In this case, the absorbance is simply proportional to (1/3) |Eo|2|μba|2 (Box 4.6). But anisotropic materials are common in biology, and purified macromolecules often can be oriented experimentally by taking advantage of their molecular asymmetry. Nucleic acids can be oriented by flowing them through a narrow capillary. Proteins frequently can be oriented by embedding them in a polymer such as polyvinyl alcohol or polyacrylamide and then stretching or squeezing the specimen to align the highly asymmetric polymer molecules. Phospholipid membranes can be aligned by magnetic fields or by layering on flat surfaces. One application of measurements of linear dichroism is to explore the structures of complexes containing multiple chromophores. An example is the “reaction center” of purple photosynthetic bacteria, which contains four molecules of bacteriochlorophyll, two molecules of bacteriopheophytin, and several additional pigments bound to a protein. (Bacteriopheophytin is the same as bacteriochlorophyll except that it has two hydrogen atoms instead of Mg at the center of the macrocyclic ring system.) If reaction centers are oriented in a stretched film or a squeezed polyacrylamide gel, the absorption bands of the various pigments exhibit linear dichroism relative to the orientation axis [13–18]. Figure 4.12B shows the linear dichroism spectrum of such a sample, expressed as A⊥ - Ak, where A⊥ and Ak are the absorbances measured with light polarized, respectively, perpendicular and parallel

4.7 Linear Dichroism

177

Absorbance

1.0

A

0.5

0

Linear Dichroism

1.0

B

0.5 0 -0.5 -1.0 800 900 1000 Wavelength / nm

Fig. 4.12 Absorption (A) and linear dichroism (B) spectra of photosynthetic reaction centers of Blastochloris (formerly called Rhodopseudomonas) viridis at 10 K [17, 18]. Aggregates of the pigment-protein complex were embedded in a polyacrylamide gel and oriented by uniaxial squeezing. The spectra are normalized relative to the positive peak near 830 nm. Linear dichroism is expressed as A⊥ - Ak, where A⊥ and Ak are the absorbances measured with light polarized perpendicular and parallel to the compression axis. Transition dipoles that lie within about 35° of the plane of the aggregate’s largest cross-section give positive linear dichroism, whereas transition dipoles that are closer to normal to this plane give negative linear dichroism. The absorption bands in the spectra shown here represent mixed Qy transitions of the reaction center’s four molecules of bacteriochlorophyll-b and two molecules of bacteriopheophytin-b (Sect. 4.7, Chap. 8). See [17, 18] for spectra extending to shorter wavelengths

to the orientation axis. The absorption spectrum measured with unpolarized light is shown in Fig. 4.12A. The bands between 830 and 1000 nm represent transitions of the bacteriochlorophylls, while the bands at 790 and 805 nm are assigned to the bacteriopheophytins. Note that the linear dichroism of the bacteriopheophytin bands is negative, whereas that of the bacteriochlorophyll band around 1000 nm is positive, indicating that the bacteriochlorophylls that contribute the long-wavelength band are oriented with their transition dipoles approximately perpendicular to the transition dipoles of the bacteriopheophytins. We’ll return to the absorption spectrum of reaction centers in Sect. 4.8 and in several later chapters. The biphasic linear

178

4

Electronic Absorption

dichroism in the 830-nm region in Fig. 4.12B is complicated by the effects of exciton interactions that we discuss in Chap. 8. A classic application of linear dichroism to study molecular orientations and motions in a complex biological system was R. Cone’s study of induced dichroism in retinal rod outer segments [19]. Rhodopsin, the light-sensitive pigment-protein complex of the retina, contains 11-cis-retinal attached covalently to a protein (opsin) by a Schiff base linkage (Fig. 4.13A, B). Its transition dipole is oriented

Fig. 4.13 (A) Crystal structure of bovine rhodopsin viewed from a perspective approximately normal to the membrane. The polypeptide backbone is represented by a cyan ribbon and the retinylidene chromophore by a red licorice model. The coordinates are from Protein Data Bank file 1f88.pdb [86]. Some parts of the protein that protrude from the phospholipid bilayer of the membrane are omitted for clarity. (B) The 11-cis-retinylidene chromophore is attached to a lysine residue by a protonated Schiff base linkage. Excitation results in isomerization around the C11-C12 bond to give an all-trans structure. (C and D) Schematic depictions of a field of rhodopsin molecules in a rod cell disk membrane, viewed normal to the membrane. Individual rhodopsin molecules are represented by cyan ovals with red arrows for the chromophore's transition dipole. (Each disk in a human retina contains approximately 1000 rhodopsins.) The transition dipoles lie approximately in the plane of the membrane but have no preferred orientation in this plane. A polarized excitation flash (horizontal double-headed arrow in C) selectively excites molecules that are oriented with their transition dipoles parallel to the polarization axis. This causes some of these molecules to isomerize, changing their absorption spectrum (yellow arrows in the ovals of D). (E) Absorbance changes at 580 nm measured with probe light polarized either parallel (ΔAk, blue curve) or perpendicular (ΔA⊥, green) to the excitation polarization. The abscissa is the time after the excitation flash [19]. The absorbance change initially depends on the polarization of the probe, but this dependence disappears as rhodopsin molecules rotate in the membrane. The smooth curves shown here are the functions ΔAk = 2 + fi exp (-t/τ) and ΔA⊥ = 2 - fi exp (-t/τ) with parameters fi and τ adjusted to fit the measured traces ( fi = 0.7, τ = 27.5 μs [19]). (F) Time dependence of the ratio of the absorbance changes measured with probe polarizations parallel and perpendicular to the excitation (ΔAk/ΔA⊥)

4.7 Linear Dichroism

179

approximately along the long axis of the retinylidene chromophore. Rhodopsin is an integral membrane protein and resides in flattened membrane vesicles (“disks”) that are stacked in the outer segments of rod cells. Previous investigators had shown that when rod cells were illuminated from the side, light polarized parallel to the planes of the disk membranes was absorbed much more strongly than light polarized normal to the membranes. However, if the cells were illuminated end-on, there was no preference for any particular polarization in the plane of the membrane. These measurements indicated that the rhodopsin molecules are aligned so that the transition dipole of the chromophore in each molecule is approximately parallel to the plane of the membrane, but that the transition dipoles point in random directions within this plane (Fig. 4.13C). When rhodopsin is excited with light the 11-cis-retinyl chromophore is isomerized to all-trans, initiating conformational changes in the protein that ultimately result in vision (Fig. 4.13B). If rod cells are illuminated end-on with a weak flash of polarized light, the light is absorbed selectively by rhodopsin molecules that happen to be oriented with their transition dipoles parallel to the polarization axis (Fig. 4.13C, D). Molecules whose transition dipoles are oriented at an angle θ with respect to the polarization axis are excited with a probability that falls off with cos2θ. On the order of 2/3 of the molecules that are excited undergo isomerization to an all-trans structure, and then evolve through a series of metastable states that can be distinguished by changes in their optical absorption spectrum. Cone [19] measured the absorbance changes associated with these transformations, again using polarized light that passed through the rod cells end-on. In the absence of the excitation flash, the absorbance measured with the probe beam was, as stated above, independent of the polarization of the probe. However, the absorbance changes resulting from the excitation flash were very different, depending on whether the measuring light was polarized parallel or perpendicular to the excitation light (Fig. 4.13E). The polarized excitation flash, by preferentially exciting molecules with a certain orientation, thus created linear dichroism that could be probed at a later time. The difference between the signals measured with parallel and perpendicular polarizations of the probe beam decayed with a time constant of about 20 μs. Cone interpreted the decay of the induced dichroism as reflecting rotations of rhodopsin in the plane of the membrane. These experiments provided the first quantitative measurements of the fluidity of a biological membrane. The rotation dynamics are still of current interest, however, because measurements by atomic force microscopy indicate that rhodopsin may exist as dimers and paracrystalline arrays under some conditions [20]. In another application, Junge et al. [21, 22] measured the rate of rotation of the γ subunit of the chloroplast ATP synthase when the enzyme hydrolyzes ATP. Rotation of the γ subunit relative to the α and β subunits appears to couple transmembrane movement of protons to the synthesis or breakdown of ATP. Another example is the use of linear dichroism to examine the orientation of CO bound to myoglobin. Compared to free heme, myoglobin discriminates strongly against the binding of CO relative to O2. This discrimination initially was ascribed to steric factors that prevent the diatomic molecules from sitting along an axis normal to the heme; it was suggested that the molecular orbitals of O2 were more suitable for

180

4

Electronic Absorption

an orientation off the normal than those of CO. Lim et al. [23] examined the orientation of bound CO by exciting carboxymyoglobin with polarized laser pulses at a wavelength absorbed by the heme. Excitation of the heme causes photodissociation of the bound CO, which remains associated with the protein in a pocket close to the heme. This results in an absorption decrease in an infrared band that reflects CO attached to the Fe, and in the appearance of a new band reflecting CO in the looser pocket. Polarized infrared probe pulses following the excitation pulse thus can be used to determine the orientations of CO molecules in the two sites relative to the transition dipole of the heme. Lim et al. found that the C-O bond of CO attached to the Fe is approximately normal to the heme plane, indicating that other factors must be responsible for the preference of myoglobin for O2.

4.8

Configuration Interactions

Although many molecular absorption bands can be ascribed predominantly to transitions between the HOMO and LUMO, it often is necessary to take other transitions into account. One reason for this is that the HOMO and LUMO wavefunctions pertain to an unexcited molecule that has two electrons in each of the lower-lying orbitals, including the HOMO itself. The interactions among the electrons are somewhat different if there is an unpaired electron in each of the HOMO and LUMO. In addition, the wavefunctions themselves are approximations of varying reliability. Better descriptions of the excited state often can be obtained by considering the excitation as a linear combination of transitions from several of the occupied orbitals to several unoccupied orbitals. Each such orbital transition is termed a configuration, and the mixing of several configurations in an excitation is called configuration interaction. In general, two transitions will mix most strongly if they have similar energies and involve similar changes in the symmetry of the molecular orbitals. Equation (4.22e) can be expanded straightforwardly to include a sum over the various configurations that contribute to an excitation: X pffiffiffi X Cjt C kt rt , ð4:29Þ μba ≈ 2 e Aa,b j,k j,k

t

where Aa,b j,k is the coefficient for the configuration ψ j → ψ k in the overall excitation from state a to state b. Box 4.8 describes a procedure for finding these coefficients. Box 4.8 Evaluating Configuration-Interaction Coefficients The procedure for finding the configuration-interaction (CI) coefficients Aa,b j,k involves constructing a matrix in which the diagonal elements are the energies of the individual transitions after corrections for the altered orbital occupancies (continued)

4.8 Configuration Interactions

181

Box 4.8 (continued) in the excited states [24–30]. For excited singlet states of π molecular orbitals, the off-diagonal matrix elements that couple two configurations, ψ j1 → ψ k1 and ψ j2 → ψ k2 with j1 ≠ j2 and k1 ≠ k2, take the form   * + XX  D E   2 e e =r s,t ψ j2 → k2 ψ j1 → k1 jHjψ j2 → k2 = ψ j1 → k1   s t  ≈

XX s

 k1 j2 k2 j1 j2 k1 k2 2C j1 s C s C t C t - C s C s C t C t γ s,t ::

ðB4:8:1Þ

t

Here Cjt represents the contribution of atom t to wavefunction ψ j, as in Eqs. (4.19–4.22 and 2.42), γ s, t is the distance between atoms s and t, and γ s, t is a semiempirical function of this distance. A typical expression for γ s, t, obtained by maximizing the agreement between the calculated and observed spectroscopic properties of a large number of molecules, is γ s,t = A expð- Brs,t Þ þ C=ðD þ rs,t Þ,

ðB4:8:2Þ

with A = 3:77 × 104 , B = 0:232=Å, C = 1:17 × 105 Å, and D = 2.82 [30]. The CI coefficients are obtained by diagonalizing the matrix as we describe in Sect. 2.3.6. The restrictions that orbital symmetry imposes on configuration interactions can be analyzed in the same manner as the selection rules for the individual transition dipoles. As Eq. (B4.8.1) indicates, the coupling of two transitions, ψ j1 → ψ k1 and ψ j2 → ψ k2, depends on the product of the four wavefunctions (ψ j1, ψ k1, ψ j2 and ψ k2). In the language of group theory (Sect. 4.6) we can say that, if the product of ψ j1 and ψ k1 and the product of ψ j2 and ψ k2 both transform as x, the product of all four wavefunctions will transform as x2 and will be totally symmetric. Mixing of the two transitions then will be allowed by symmetry. On the other hand, if the product of ψ j1 and ψ k1 transforms as x, say, while that of ψ j2 and ψ k2 transforms as y, the overall product will transform as xy and will integrate to zero. Inspection of Figs. 4.10 and 4.11 shows that if we denote the top two filled molecular orbitals and the first two empty orbitals of bacteriochlorophyll or 3-methylindole as ψ 1 - ψ 4 in order of increasing energy, the product ψ 1ψ 3 has the same symmetry as ψ 2ψ 4, whereas ψ 2ψ 3 has the same symmetry as ψ 1 ψ 4. The ψ 1 → ψ 3 transition thus should mix with ψ 2 → ψ 4 and the ψ 2 → ψ 3 transition should mix with ψ 1 → ψ 4. Porphyrins and their chlorin and bacteriochlorin derivatives provide numerous illustrations of the importance of configuration interactions [31, 32]. In porphin (Fig. 4.14, left), the highly symmetrical parent compound, the two highest filled molecular orbitals (ψ 1 and ψ 2 in order of increasing energy) are nearly isoenergetic,

182

4

Electronic Absorption

Fig. 4.14 Structures, energy diagrams, and excitations of porphin, chlorin, and bacteriochlorin. The horizontal green bars represent the energies of the highest two occupied molecular orbitals (ψ 1 and ψ 2) and the first two unoccupied orbitals (ψ 3 and ψ 4), relative to the energy of ψ 1 in each molecule. Arrows indicate excitations from one of the occupied orbitals to an empty orbital. x and y correspond to the molecular axes shown as dotted lines with the porphin structure and convey the symmetry of the product of the initial and final wavefunctions for each configuration. Configurations with the same symmetry (e.g., ψ 1 → ψ 4 and ψ 2 → ψ 3) mix in the excited states, and the transition dipole for the mixed excitation is oriented approximately along the x or y molecular axis. Reduction of one or both of the tetrapyrrole rings in chlorin and bacteriochlorin moves the lowest-energy excitation to progressively lower energy and increases its dipole strength

as are the two lowest unoccupied orbitals (ψ 3 and ψ 4). The ψ 1 → ψ 4 and ψ 2 → ψ 3 transitions have the same symmetry and energy and thus mix strongly, as explained in Box 4.8; the ψ 1 → ψ 3 transition mixes similarly with ψ 2 → ψ 4. Because of the near degeneracy of the transition energies the two CI coefficients (Aa,b j,k ) are simply -1/2 ±2 in each case, but the signs of the coefficients can be either the same or opposite. The four orbitals thus give rise to four different excited states, which are commonly referred to as the By, Bx, Qx, and Qy states. By and Qy consist of the configurations 2-1/2(ψ 1 → ψ 4) ± 2-1/2(ψ 2 → ψ 3); Bx and Qx, of 2-1/ 2 (ψ 1 → ψ 3) ± 2-1/2(ψ 2 → ψ 4). However, the transition dipoles cancel each other almost exactly in two of the combinations and reinforce each other in the other two. The result is that the two lowest-energy absorption bands (Qx and Qy) occur at

4.8 Configuration Interactions

183

100

A

B cytochrome c

bacteriochlorophyll

-1

ε / mM cm

-1

essentially the same energy and are very weak relative to the two higher-energy bands (Bx and By). In chlorin (Fig. 4.14, center), one of the four pyrrole rings is partially reduced, removing two carbons from the π system. This perturbs the symmetry of the molecule and moves ψ 2 and ψ 4 up in energy relative to ψ 1 and ψ 3. As a result, the lowest-energy absorption band moves to a lower energy and gains dipole strength, whereas the highest-energy band moves to higher energy and loses dipole strength. The trend continues in the bacteriochlorins, in which two of the pyrrole rings are reduced (Fig. 4.14, right). Hemes, which are symmetrical iron porphyrins, thus absorb blue light strongly and absorb yellow or red light only weakly (Fig. 4.15A), while bacteriochlorophylls absorb intensely in the red or near-IR (Fig. 4.15B). This four-orbital model rationalizes a large body of experimental observations on the spectroscopic properties of metalloporphyrins, chlorophylls, bacteriochlorophylls, and related molecules [31, 33, 34], and has been used to analyze the spectroscopic properties of photosynthetic bacterial reaction centers and antenna complexes [4, 35, 36]. In the spectrum of Bl. viridis reaction centers shown in Fig. 4.12A, the absorption bands between 750 and 1050 nm reflect Qy transitions of the bacteriopheophytins and bacteriochlorophylls, which are mixed by exciton interactions as discussed in Chap. 8. The corresponding Qx bands (not shown in the figure) are in the regions of 530–545 and 600 nm. The indole side chain of tryptophan provides another example. Its absorption spectrum again involves significant contributions from the orbitals lying just below the HOMO and just above the LUMO (ψ 1 and ψ 4), in addition to the HOMO and LUMO themselves (ψ 2 and ψ 3). Transitions among the four orbitals give rise to two overlapping absorption bands in the region of 280 nm that are commonly called the 1 La and 1Lb bands, and two higher-energy bands (1Ba and 1Bb) near 195 and 221 nm [37–43]. The 1La excitation has somewhat higher energy and greater dipole strength than 1Lb, and, as we will see in Sect. 4.12, results in a larger change in the permanent dipole moment. Neglecting small contributions from higher-energy configurations, the excited singlet states associated with the 1La and 1Lb excitations can be described

50

0

400 500 600 Wavelength / nm

400 600 800 Wavelength / nm

Fig. 4.15 (A) Absorption spectra of cytochrome c with the bound heme in its reduced (blue curve) and oxidized (red curve) forms. (B) Absorption spectrum of bacteriochlorophyll a in methanol

184

4

4

Electronic Absorption

B

A

y/ao

2 0

-2

1

1

Lb

La

-4 -6 -4 -2

0 2 x /ao

4

6

-6 -4 -2

0 2 x /ao

4

6

Fig. 4.16 Transition dipoles for the 1 La (A) and 1 Lb (B) excitations of 3-methylindole calculated from linear combinations of products of the molecular orbitals shown in Fig. 4.11. Contour plots of the functions 0.917ψ 2ψ 3 - 0.340ψ 1ψ 4 (A) and 0.732ψ 1ψ 3 + 0.634ψ 2ψ 4 (B) are shown with red lines for positive amplitudes, blue lines for negative amplitudes, and contour intervals of 0.005a03. The arrows show the transition dipoles calculated by Eq. (4.29), with the lengths in units of eÅ=5a∘

reasonably well by the combinations 1La ≈ 0.917(ψ 2 → ψ 3) - 0.340(ψ 1 → ψ 4) and 1 Lb ≈ 0.732(ψ 1 → ψ 3) + 0.634(ψ 2 → ψ 4). Figure 4.16 shows the two transition dipoles calculated by Eq. (4.29). If an improved description of an excited electronic state is wanted for other purposes such as calculations of exciton interactions (Chap. 8), the CI coefficients can be adjusted empirically to maximize the agreement between the calculated and observed transition energy or dipole strength. In one application of this idea [36], the CI coefficients for the Qx and Qy absorption bands of bacteriochlorophyll and bacteriopheophytin were adjusted so that the dipole strengths calculated using the transition gradient operator (Sect. 4.9) matched the measured dipole strengths. The discrepancy between these dipole strengths and the values calculated with the dipole operator then was used to correct the calculated energies of dipole-dipole interactions among the bacteriochlorophylls and bacteriopheophytins in photosynthetic bacterial reaction centers.

4.9

Calculating Electric Transition Dipoles with the Gradient Operator

When contributions from transitions involving the top two or three filled orbitals and the first few unoccupied orbitals are considered, dipole strengths calculated by using Eqs. (4.22a–e) typically agree with experimentally measured dipole strengths to within a factor of 2 or 3, which means that the magnitude of the transition dipole is correct to within about ±50%. Better agreement sometimes can be obtained by using e = ð∂=∂x, ∂=∂y, ∂=∂zÞ, instead of e the gradient operator, ∇ μ . Matrix elements of the gradient and dipole operators are related by the expression

4.9 Calculating Electric Transition Dipoles with the Gradient Operator

D

E e a = - ðEb - E a Þme hΨb je μjΨa i, Ψb j∇jΨ ħ2 e

185

ð4:30Þ

where me is the electron mass (Box 4.9). D Thus, if theEenergy difference Eb - Eais e a and vice versa. Transition known, we can obtain hΨ b je μjΨ a i from Ψ b j∇jΨ e should be identical if the molecular orbitals are dipoles calculated with e μ and ∇ exact, but with approximate orbitals, the two methods usually give somewhat different results. The dipole strengths calculated with the dipole operator often are too large, while those obtained with the gradient operator agree better with experiment [34, 36, 44]. Box 4.9 The Relationship between Matrix Elements of the Electric Dipole and Gradient Operators e to the Equation (4.30) can be derived by relating the gradient ∇ h operator i e e commutator of the Hamiltonian and dipole operators H, μ . (See Box 2.2 e includes for an introduction to commutators.) The Hamiltonian operator H e e terms for both potential energy (V) and kinetic energy (T); however, h wei only e e e e need to consider T because V commutes with the position operator ( V, x = 0) e is just ∂/∂x, the and e μ is simply ee x. For a one-dimensional system, in which ∇ e commutator of H and e μ is i h i h i  h 2 e e e ee ðB4:9:1aÞ H, μ = T, x = - ħ2 e=2m ∂ =∂x2 , x       = - ħ2 e=2m d2 =dx2 x - x d2 =dx2       = - ħ2 e=2m 2ðd=dxÞ þ x d2 =dx2 - x d 2 =dx2     e = - ħ2 e=m d=dx = - ħ2 e=m ∇

ðB4:9:1bÞ ðB4:9:1cÞ ðB4:9:1dÞ

Generalizing to three dimensions, and treating the commutator as an operator gives i E D h E  D  e  e a : e Ψb  H, μ Ψa = - ħ2 e=m Ψb j∇jΨ ðB4:9:2Þ We can relate the matrix element on the left side of Eq. (B4.9.2) to the → transition dipole ( μ ba ) by expanding ψ b and ψ a formally in the basis of all e ðΨ k Þ and using the procedure for matrix multiplication the eigenfunctions of H (Appendix A2). This gives (continued)

186

4

Electronic Absorption

Box 4.9 (continued) i E D D h E    e  e μjΨa - Ψb je e a e μ Ψa  Ψb jHe μHjΨ Ψb  H, =

XD k

E D E X e j Ψk hΨk je e j Ψa Ψb j H μ j Ψa i μ j Ψk i Ψk j H hΨb je

ðB4:9:3aÞ ðB4:9:3bÞ

k →

= Eb hΨb je μ j Ψa i - hΨb je μ j Ψa iE a = ðE b - Ea Þ μ ba ,

ðB4:9:3cÞ

where Eb and Ea are the energies of states a and b. Eq. (B4.9.3b) reduces to B4.9.3c because, in the absence of additional perturbations, the only non-zero Hamiltonian matrix elements involving Ψ b or Ψ a are hΨ b| H| Ψ bi and hΨ a| H| Ψ ai. Equating the right-hand sides of Eqs. (B4.9.2 and B4.9.3c) gives Eq. (4.30). In the quantum theory ofD absorption Ethat we discuss in Chap. 5, the transition e a arises directly, rather than just as an gradient matrix element Ψ b j∇jΨ e also play a μjΨ a i . Matrix elements of ∇ alternative way of calculating hΨ b je fundamental role in the theory of circular dichroism (Chap. 9). Figure 4.17 illustrates the functions that enter into the transition gradient matrix element for the HOMO → LUMO excitation of ethylene. Contour plots of the HOMO and LUMO (ψ a and ψ b) are reproduced from Fig. 4.6B in panels A and B for reference. As before, the C=C bond is aligned along the y axis and ψ a and ψ b are constructed from atomic 2pz orbitals. Panel C shows a contour plot of the derivative ∂ψ a/∂y, and D shows the result of multiplying this derivative by ψDb. Integrating E the e product ψ b∂ψ a/∂y over all space gives the y component of ψ b j∇jψ a . By inspecting Fig. 4.17D you can see that ψ b∂ψ a/∂y is an even function of y, so except for points in the xy plane, where ψ a and ψ b both go through zero, integration along any line parallel to the y axis will give a non-zero result. (Although the contour plots in the figure show only the amplitudes in the yz plane, the corresponding plots for any other plane parallel to yz would be similar because ψ a and ψ b are both even, monotonic functions of x.) By contrast, the function ψ b∂ψ a/∂z (Fig. 4.17F) is an odd function of both y and z, which means that integrating this product overDall space will E e give zero. This is true also of ψ b∂ψ a/∂x (not shown). The vector ψ b j∇jψ a is μjψ a i. oriented along the C=C bond, whichDis just what E we found above for hψ b je e Calculating the matrix element ψ b j∇jψ a for a transition between two π molecular orbitals is somewhat more cumbersome than calculating hψ b je μjψ a i , but is still relatively straightforward if the molecular orbitals are constructed of linear combinations of Slater-type atomic orbitals. The transition matrix element for excitation to an excited singlet state then takes the same form as Eq. (4.22e):

4.9 Calculating Electric Transition Dipoles with the Gradient Operator

187

Fig. 4.17 Components of the transition matrix element of the gradient operator for excitation of ethylene. (A, B) Contour plots of the amplitudes of the HOMO (ψ a) and LUMO (ψ b) molecular orbitals in the yz plane. The C=C double bond lies on the y-axis. (C, E) The derivatives of ψ a with respect to y and Dz, respectively. (D, F) The products of these derivatives with ψ b. The y and E e z components of ψ j∇jψ are obtained by integrating ψ b∂ψ a/∂y and ψ b∂ψ a/∂z, respectively, b

a

over all space. The symmetry of the wavefunctions is such that this integral is zero for ψ b∂ψ a/∂z and (not shown) ψ b∂ψ a/∂x but non-zero for ψ b∂ψ a/∂y. Positive amplitudes are shown with blue lines; negative with red lines; and zero with purple dot-dash lines

D E pffiffiffiXX D E e a = 2 e Ψb j∇jΨ Cbs Cat ps j∇jp t , s

ð4:31Þ

t

on where C bs and C at are the expansion coefficients for atomic 2pz orbitals centered E D e atoms s and t ( ps and pt) in molecular orbitals Ψ b and Ψ , respectively, and ps j∇jpt e for the two atomic orbitals. Box 4.10 outlines a general is the matrix element of ∇

188

4

D procedure for evaluating

e ps j∇jp t

Electronic Absorption

E that allows the atomic orbitals to have any

orientation with D respect E to each other. D Equation E (4.31) D can be E simplified slightly by e e e noting that pt j∇jpt is zero and pt j∇jps = - ps j∇jpt , as you can verify by studying Fig. 4.17. With these substitutions, the sum over atoms s and t becomes D E pffiffiffiXX  E D e a = 2 e Ψb j∇jΨ 2 C as C bt - Cbs Cat ps j∇jp ð4:32Þ t , st

t

which sometimes can be approximated well by a sum over just the pairs of bonded atoms [34, 36, 44–46]. Box 4.10 Matrix Elements of the Gradient Operator for Atomic 2p Orbitals D E e The integral ps j∇jp in Eqs. (4.31 and 4.32) is the matrix element of the t e for atomic 2p orbitals on atoms s and t. To allow the gradient operator (∇) local z-axis of atom s to have any orientation relative to that of atom t, the atomic matrix elements can be written D E   e ^ ps j∇jp t ≈ ηx0 ,s ηy0 ,t þ ηy0 ,s ηx0 ,t ∇xy i   þ ηy0 ,s ηy0 ,t ∇σ þ ηx0 ,s ηx0 ,t þ ηz0 ,s ηz0 ,t ∇π ^j:   þ ηz0 ,t , ηy0 ,t þ ηy0 ,s , ηz0 ,t ∇zy ^k:

ðB4:10:1Þ

Here ηx0 ,t ,ηy0 ,t , and ηz0 ,t are direction cosines of the atomic z-axis of orbital t with respect to a Cartesian coordinate system (x′, y′, z′) defined so that atom t is at the origin and the y′ axis points along the line from atom t to atom s (for example, ηy0 ,t is the cosine of the angle between y′ and the local z-axis of orbital t); ^i,^j and ^k are unit vectors parallel to the x′, y′, and z′ axes; and — σ , — π and e for pairs of Slater 2p orbitals in three — zy are the matrix elements of ∇ canonical orientations. — σ is for the end-on orientation shown in Fig. 4.18A;— π, for side-by-side orientation with parallel z-axes along z′ or x′ (Fig. 4.18B); and — zy, for one orbital displaced along y′ and rotated by 90∘ around an axis parallel to x′ (Fig. 4.18C). — xz is 0, and — xy is the same as — zy. Inspection of Eq. (B4.10.1) and Fig. 4.18 shows that if the z axes of atoms t and s are parallel to each other and perpendicular to y′, as is approximately e j p points from atom t to atom s and has the case for π-electron systems, ps j ∇ t magnitude — π. The Slater 2pz orbital, in polar coordinates centered on atom s (rs, θs, ϕs), is (continued)

4.9 Calculating Electric Transition Dipoles with the Gradient Operator

189

Box 4.10 (continued)  1=2 ps = ζ 5s =π r s cosθs expð- ζ s r s =2Þ,

ðB4:10:2Þ

where θs is the angle with respect to the z-axis and ζs = 3.071, 3.685, and -1 4:299 Å for C, N, and O, respectively. The dependences of — σ , — π and — zy on the interatomic distance (R) and the Slater orbital parameters for the two atoms (ζ s and ζ t) can be evaluated as follows, using expressions described originally by Mulliken et al. [47] and Král [48] (see [4, 49]): First, define the two functions Z1 Ak =

Z1 w expð- PwÞdw and Bk =

wk expð- QwÞdw,

k

ðB4:10:3Þ

-1

1

where P = (ζs + ζ t)R/2 and Q = (ζ s - ζ t)R/2. Then,   ∇σ = ðζ s ζ t Þ5=2 R4 =8 fA0 B2 - A2 B0 þ A1 B3 - A3 B1 þðζ t R=2Þ½A1 ðB0 - B2 Þ þ B1 ðA0 - A2 Þ þ A3 ðB4 - B2 Þ þ B3 ðA4 - A2 Þg, ðB4:10:4Þ   ∇π = ðζ s ζ t Þ5=2 ξt R5 =32 × ½ðB1 -B3 ÞðA0 -2A2 þA4 Þþ ðA1 -A3 ÞðB0 -2B2 þB4 Þ,

ðB4:10:5Þ

and   ∇zy = ðζ s ζ t Þ5=2 R4 =8 fA0 B2 - A2 B0 þ A1 B3 - A3 B1 þðζ t R=4Þ½ðA3 - A1 ÞðB0 - B4 Þ þ ðB3 - B1 ÞðA0 - A4 Þg:

ðB4:10:6Þ

Ak and Bk can be calculated with the formulas [50]: Ak = ½ expð- sÞ þ kAk - 1 =P, Bk = 2

3 X

Q2i =ð2iÞ!ðk þ 2i þ 1Þ for k even,

ðB4:10:7Þ ðB4:10:8Þ

i=0

and Bk = - 2

3 X

Q2iþ1 =ð2i þ 1Þ!ðk þ 2i þ 2Þ for k odd: i=0

ðB4:10:9Þ

D E e See [34, 36, 45, 46] for other semiempirical expressions for ps j∇jp t .

190

4

Electronic Absorption

Fig. 4.18 Canonical orientations of 2pz orbitals of two carbon atoms. The blue and purple regions represent boundary surfaces of the wavefunctions as in Fig. 2.5. The x′y′z′ Cartesian coordinate system is centered on atom 1 with the y′ axis aligned along the interatomic vector. The transition gradient matrix elements — σ , — π and — zy are for the orientations shown in (A, B, and C), respectively. Matrix elements for an arbitrary orientation can be expressed as linear combinations of these canonical matrix elements

Figure 4.19 illustrates the use of this approach to calculate transition matrix elements for trans-butadiene, which provides a useful model for carotenoids and retinals. Panels A - D show the two highest filled molecular orbitals and the two lowest unoccupied orbitals (ψ 1 to ψ 4 in order of increasing energy). The vector diagram in panel E shows the direction magnitude of the matrix element D and relative E e e for each pair of bonded atoms, p j∇jp , weighted by the coefficients for that of ∇ s t  b a pair of atoms in wavefunctions ψ 2 and ψ 3 C s C t . Combining the vectors for each pair of atoms gives an overall transition gradient matrix element D E PP e ( s t Cbs Cat ps j∇jp ) oriented on the long axis of the molecule. (Contributions t from pairs of non-bonded atoms do not affect the overall matrix element significantly.) Excitation of an electron from ψ 2 to ψ 3, which is the lowest-energy configuration in the excited singlet state, therefore has a non-zero transition dipole with this orientation. Figure 4.19H is a similar vector diagram for the highest-energy configuration in an excited singlet state of the four wavefunctions (ψ 1 → ψ 4). The transition-gradient dipole for this excitation has a smaller magnitude and a different orientation than that D E b a e for ψ 2 → ψ 3 because the C s C t ps j∇jpt vector for the central two atoms cancels more of the contributions from the outer pairs of atoms.

D

2

H

2

y/ao

0

y/ao

4.9 Calculating Electric Transition Dipoles with the Gradient Operator

0

-2

C

ψ4

-2

G

0

-2

B

ψ3

F

-2

A

ψ2

E

-2

ψ1 -2

2

ψ4

ψ2

ψ3

-2

0 x/ao

0

-2 0 x/ao

ψ2

2

y/ao

y/ao

0

ψ3

0

-2

2

ψ1

2

y/ao

y/ao

0

ψ4

0

-2

2

ψ1

2

y/ao

y/ao

2

191

2

Fig. 4.19 (A–D) Contour plots of the two highest occupied and the two lowest unoccupied molecular orbitals in the ground state of trans-butadiene (ψ 1 - ψ 4 in order of increasing energy). The green filled circles connected by green lines indicate the positions of the atoms. The contour lines give the amplitudes of the wavefunctions in a plane parallel to the plane of the π system, and above that plane by the Bohr radius (ao). Blue lines indicate positive amplitudes; red lines, negative; purple dot-dashed lines, zero. The wavefunctions were calculated with QCFF-PID[30, 183E ]. (E–H) e Vector diagrams of the directions and relative magnitudes of the products C b C a p j∇jp (green s

t

s

t

arrows) for pairs of bonded atoms in the first four excitations of trans-butadiene. The initial and

192

4

Electronic Absorption

Fig. 4.19 (continued) final molecular orbitals are indicated in each panel. C bs and C at are the coefficients 2pz orbitals of atoms s and t in the final and initial wavefunction, respecD for atomic E e tively; ps j∇jpt is the matrix element of the gradient operator for the two atomic orbitals. The empty green circles indicate the positions of the atoms. The transition gradient matrix element for each excitation is given approximately by the vector sum of the arrows. (The contributions from the pairs of nonbonded atoms do not change the overall matrix element significantly)

The electric transition dipoles for excitations ψ 2 →D ψ 4 andEψ 1 → ψ 3 of transe butadiene are zero (Fig. 4.19F, G). In both cases, Cbs C at ps j∇jp t for the central pair of atoms is zero because the initial and final wavefunctions have the same symmetry with respect to a center of inversion (gerade for ψ 1 → ψ 3 and ungerade for ψ 2 → ψ 4). The contributions from the other pairs of atoms are not zero individually, but give antiparallel vectors that cancel in the sum. These two excitations are therefore forbidden if we consider only interactions with the electric field of light. As we will see in Chap. 9, they are weakly allowed through interactions with the magnetic field, but the associated dipole strength is much lower than that for ψ 2 → ψ 3 and ψ 1 → ψ 4. Box 4.11 restates the selection rules for excitation transbutadiene and related compounds in the language of group theory and describes the nomenclature that is used for these excitations. Box 4.11 Selection Rules for Electric-Dipole Excitations of Linear Polyenes The basic selection rules for electric-dipole excitations of trans-butadiene and other centrosymmetric polyenes can be described simply in terms of the symmetry of the molecular orbitals. In order of increasing energy, the four wavefunctions shown in Fig. 4.19A–D have Ag, Bu, Ag, and Bu symmetry. As explained in Sect. 4.6, the Mulliken symbols A and B refer to functions that are, respectively, symmetric and antisymmetric with respect to rotation about the C2 axis, which passes through the center of trans-butadiene and is normal to the plane of the π system; subscripts g and u denote even and odd symmetry with respect to inversion through the molecular center. The excitations ψ 2 → ψ 3, ψ 2 → ψ 4, ψ 1 → ψ 3 and ψ 1 → ψ 4 are characterized by the symmetries of the direct products of the initial and final wavefunctions: Bu × Ag = Bu for ψ 2 → ψ 3, Bu × Bu = Agfor ψ 2 → ψ 4, Ag × Ag = Ag for → ψ 1 → ψ 3, and Ag × Bu = Bu for ψ 1 → ψ 4. Since the position vector ( r ) has Bu → → → → symmetry, the products r ψ 2 ψ 3 , r ψ 2 ψ 4 , r ψ 1 ψ 3 and r ψ 1 ψ 4 transform asAg, Bu,Bu, and Ag, respectively. The electric dipole transition matrix elements are, therefore, nonzero for the first and fourth excitations (the two excitations with Bu symmetry), and zero for the second and third (the two with Agsymmetry). Similar considerations apply to longer polyenes including carotenoids, (continued)

4.10

Transition Dipoles for Excitations to Singlet and Triplet States

193

Box 4.11 (continued) although twisting and bending distortions leave these molecules only approximately centrosymmetric. Note that the vectors in Fig. 4.19E–H pertain to the pure configurations ψ 2 → ψ 3, ψ 2 → ψ 4, ψ 1 → ψ 3 and ψ 1 → ψ 4. An accurate description of the actual excited states of trans-butadiene and other such polyenes requires extensive configuration interactions [51]. The ψ 2 → ψ 4 excitation mixes strongly with ψ 1 → ψ 3 and other higher-energy configurations, with the result that an excited state with this symmetry moves below the first Bu state in energy [51–53]. The lowest-energy absorption band therefore reflects excitation to the second excited state rather than the first, which has significant consequences for the functions of carotenoids as energy donors and acceptors in photosynthesis (Sect. 7.4). The “forbidden” first excited state of several polyenes has been detected by Raman spectroscopy [52], two-photon spectroscopy (Chap. 12 and [54]), and time-resolved measurements of absorption and fluorescence [53, 55–57]. In the nomenclature that is commonly used, the ground state of a centrosymmetric polyene is labeled 11Ag-; the first excited singlet state with Ag symmetry is 21Ag-, and the first with Busymmetry (the second excited singlet state in order of energy) is 11 Bþ u . The first number indicates the position of the state in order of increasing energy among states with the same symmetry. The first superscript identifies the spin multiplicity (singlet or triplet), and the superscript “+” or “-” indicates whether the state is predominantly covalent (-) or ionic (+). Because the same selection rules also hold as a first approximation in larger polyenes that are not strictly centrosymmetric, this nomenclature also is used for retinals and complex carotenoids.

4.10

Transition Dipoles for Excitations to Singlet and Triplet States

As we discussed in Chap. 2 (Sect. 2.4), electrons have an intrinsic angular momentum or “spin” that is characterized by spin quantum numbers s = 1/2 and ms = ± 1/2. The different values of ms can be described by two spin wavefunctions, α (“spin up”) for ms = + 1/2and β (“spin down”) forms = - 1/2. For systems with more than one electron, wavefunctions that include the electronic spins must be written in a way so that the complete wavefunction changes sign if we interchange any two electrons. We skipped over this point quickly when we derived expressions for the transition dipole for forming an excited singlet state (Eqs. (4.22a–4.22e)), so let’s check whether we get the same expressions if we write the spin wavefunctions explicitly. We’ll also examine the transition dipoles for excitation to triplet states.

194

4

Electronic Absorption

Using the notation of Eqs. (2.47), the transition dipole for excitation from a singlet ground state to an excited singlet state of a system with two electrons takes the form →

 μð1Þ þ e μð2ÞjΨa Ψb je o Dn ¼ 2 - 1=2 ½ψ h ð1Þψ l ð2Þ þ ψ h ð2Þψ l ð1Þ2 - 1=2 ½αð1Þβð2Þ - αð2Þβð1Þ ð4:33Þ n oE × je μð1Þ þ e μð2Þj ψ h ð1Þψ h ð2Þ2 - 1=2 ½αð1Þβð2Þ - αð2Þβð1Þ :

μ ba ¼

1

Because the electric dipole operator does not act on the spin wavefunctions, integrals such as hα(1) j α(1)i and hα(1) j β(1)i can be factored out of the overall integral. By doing this and making the same approximations we made above in Eqs. (4.22a– 4.22c), we obtain →

μð1Þjψ h ð1Þihψ h ð2Þjψ h ð2Þi μ ba ¼ 2 - 3=2 ½hψ l ð1Þje

ð4:34Þ

þ hψ l ð2Þje μð2Þjψ h ð2Þihψ h ð1Þjψ h ð1Þi × ½hαð1Þjαð1Þihβð2Þjβð2Þi - hαð1Þjβð1Þihβð2Þjαð2Þi - hαð2Þjβð2Þihβð1Þjαð1Þi þ hαð2Þjαð2Þihβð1Þjβð1Þi: The factors hψ h(1) j ψ h(1)i and hψ h(2) j ψ h(2)i in Eq. (4.34) are both unity. The spin integrals also can be evaluated immediately because the spin wavefunctions are orthogonal and normalized: hα(1) j α(1)i = hβ(1) j β(1)i = 1, and hα(1) j β(1)i = hβ(1) j α(1)i = 0. The terms in the second square brackets thus are 1 × 1 - 0 × 0 0 × 0 + 1 × 1 = 2, so pffiffiffi pffiffiffi X μðkÞjψ h ðkÞi ≈ 2 e C li Chi ri : μba = 2 hψ l ðk Þje

ð4:35Þ

i

This is the same result as Eq. (4.22e). The transition dipoles for transitions from the singlet ground state to the excited triplet states are very different from the transition dipole for forming the excited singlet state. The transition dipoles for forming the triplet states all evaluate to zero because terms with opposite signs cancel or because each term includes an integral of the formhα(1) j β(1)i. For 3 Ψ0b , we have: 3 0    μð1Þ þ e μð2ÞjΨa = 2 - 1 ½ψ h ð1Þψ l ð2Þ - ψ h ð2Þψ l ð1Þ½αð1Þβð2Þ þ αð2Þβð1Þ Ψb je n oE × je μð1Þ þ e μð2Þj ψ h ð1Þψ h ð2Þ2 - 1=2 ½αð1Þβð2Þ - αð2Þβð1Þ ð4:36aÞ

The Born-Oppenheimer Approximation, Franck-Condon Factors, and the. . .

4.11

195

= 2 - 3=2 ½hψ l ð2Þje μð2Þjψ h ð2Þψ h ð1Þ j ψ h ð1Þi - hψ l ð1Þje μð1Þjψ h ð1Þψ h ð2Þ j ψ h ð2Þi × ½hαð1Þjαð1Þihβð2Þjβð2Þi - hαð1Þjαð2Þihβð2Þjβð1Þi þ hαð2Þjαð1Þihβð1Þjβð2Þi - hαð2Þjαð2Þihβð1Þjβð1Þi: ð4:36bÞ μð2Þjψ h ð2Þi The products in the first line of Eq. (4.36b) reduce to ½hψ l ð2Þje μð1Þjψ h ð1Þi, which is zero because the transition dipole does not depend hψ l ð1Þje on how we label the electron. The sum of products of spin integrals in the second line evaluates to 1 × 1 – 0 × 0 + 0 × 0 – 1 × 1, which also is 0. Similarly, for 3 Ψþ1 b : 3

o  Dn - 1=2 μ ð 1 Þ þ e μ ð 2 ÞjΨ Ψþ1 ½ ψ ð 1 Þψ ð 2 Þ ψ ð 2 Þψ ð 1 Þ ½ α ð 1 Þα ð 2 Þ ×   = 2 je a h l h l b oE n μð1Þ þ e μð2Þj ψ h ð1Þψ h ð2Þ2 - 1=2 ½αð1Þβð2Þ - αð2Þβð1Þ je ð4:37aÞ

= 2 - 1 ½hψ l ð2Þje μð2Þjψ h ð2Þihψ h ð1Þjψ h ð1Þi - hψ l ð1Þje μð1Þjψ h ð1Þihψ h ð2Þjψ h ð2Þi × ½hαð1Þjαð1Þihαð2Þjβð2Þi - hαð1Þ j βð1Þihαð2Þ j αð2Þi:

:

ð4:37bÞ Here again, both the first and the second lines of the final expression are zero. Evaluating the transition dipole for 3 Ψb- 1 similarly gives the same result. Excitations from the ground state to an excited triplet state are, therefore, formally forbidden. In practice, weak optical transitions between singlet and triplet states sometimes are observable. Triplet states also can be created by intersystem crossing from excited singlet states, as we’ll discuss in Chap. 5. This process results mainly from the coupling of the magnetic dipoles associated with electronic spin and orbital electronic motion.

4.11

The Born-Oppenheimer Approximation, Franck-Condon Factors, and the Shapes of Electronic Absorption Bands

So far, we have focused on the effects of light on electrons. The complete wavefunction for a molecule must describe the nuclei also. But because nuclei have very large masses compared to electrons, it is reasonable for some purposes to view them as being more or less fixed in position. The Hamiltonian operator for the electrons then will include slowly changing fields from the nuclei, while the Hamiltonian for the nuclei includes the nuclear kinetic energies and averaged fields from the surrounding clouds of rapidly-moving electrons. Using the electronic Hamiltonian in the Schrödinger equation leads to a set of electronic wavefunctions ψ i(r, R) that depend on both the electron coordinates (r) and the coordinates of the nuclei (R). Using the nuclear wavefunction in the Schrödinger equation provides a

196

4

Electronic Absorption

set of vibrational-rotational nuclear wavefunctions χ n(i)(R) for each electronic state. The solutions to the Schrödinger equation for the full Hamiltonian, which includes the motions of both the nuclei and the electrons, can be written as a linear combination of products of these partial wavefunctions: XX Ψðr, RÞ = ai,n ψ i ðr, RÞχ nðiÞ ðRÞ: ð4:38Þ i

n

This description is most useful when the double sum on the right-hand side of Eq. (4.38) is dominated by a single term because we then can express the complete wavefunction as a simple product of an electronic wavefunction and a nuclear wavefunction: Ψðr, RÞ ≈ ψ i ðr, RÞχ nðiÞ ðRÞ:

ð4:39Þ

This is called the Born-Oppenheimer approximation. The Born-Oppenheimer approximation proves to be reasonably satisfactory under a wide range of conditions. This is of fundamental importance in molecular spectroscopy because it allows us to assign transitions as primarily either electronic, vibrational, or rotational in nature. In addition, it leads to tidy explanations of how electronic transitions depend on nuclear wavefunctions and temperature. A more complete discussion of the basis of the Born-Oppenheimer approximation and of the situations in which it breaks down can be found in [58, 59]. For a diatomic molecule, the potential energy term in the nuclear Hamiltonian is approximately a quadratic function of the distance between the two nuclei, with a minimum at the mean bond length. Such a Hamiltonian gives a set of vibrational wavefunctions with equally spaced energies (Eq. 2.29 and Fig. 2.3). The energy levels are En = (n + 1/2)hv, where n = 0, 1, 2, 3 . . ., and υ is the classical bond vibration frequency. A combination of vibrational and electronic wavefunctions χn and ψi is referred to as a vibronic state or level. Consider a transition from a particular vibronic level of the ground electronic state, Ψ a,n = ψ a(r, R)χ n(R), to a vibronic level of an excited state, Ψ b,m = ψ b(r, R)χ m(R). The transition could involve a change in vibrational wavefunction from χ n to χ m in addition to the change in electronic wavefunction from ψ a to ψ b. We can analyze the matrix element for this process by writing the dipole operator as a sum of separate operators for the electrons and the nuclei: X X e μnuc = eri þ zRj , ð4:40Þ μ=e μel þ e i

j

where ri is the position of electron i, and Rj and zj are the position and charge of nucleus j. For one electron and one nucleus, the transition dipole then is: μel jψ a ðr, RÞχ n ðRÞi μba,mn = hψ b ðr, RÞχ m ðRÞje μnuc jψ a ðr, RÞχ n ðRÞi þhψ b ðr, RÞχ m ðRÞje

4.11

The Born-Oppenheimer Approximation, Franck-Condon Factors, and the. . .

Z =e

χ m  ðRÞχ n ðRÞdR Z

þ z



Z

197

ψ b  ðr, RÞ ψ a ðr, RÞrdr

χ m ðRÞχ n ðRÞRdR

Z

ψ b  ðr, RÞ ψ a ðr, RÞdr:

ð4:41Þ

R → The integral ψ b ðr, RÞψ a ðr, RÞd r on the right side of Eq. (4.39) is just hψ b j ψ ai for a particular value of R, which is zero if the electronic wavefunctions are orthogonal for all R. Therefore, μel jψ a ðr, RÞχ n ðRÞi μba,mn = hψ b ðr, RÞχ m ðRÞje Z Z = e χ m  ðRÞχ n ðRÞdR ψ b  ðr, RÞ ψ a ðr, RÞrdr: ð4:42Þ The double integral in Eq. (4.40) cannot be factored rigorously into a product of the form ehχ m(R) j χ n(R)ihψ b(r)| r| ψ a(r)i because the electronic wavefunctions depend on R in addition to r. We can, however, write μel jψ a ðr, RÞχ n ðRÞi = hχ m ðRÞjχ n ðRÞihψ b ðr, RÞje μel jψ a ðr, RÞi hψ b ðr, RÞχ m ðRÞje = hχ m ðRÞjχ n ðRÞU ba ðRÞi,

ð4:43Þ

where Uba(R) is an electronic transition dipole that is a function of R. If Uba(R) does not vary greatly over the range of R where both χm and χn have substantial amplitudes, then μba,mn ≈ hχ m ðRÞjχ n ðRÞiU ba ,

ð4:44Þ

where U ba denotes an average of Uba(R) over the nuclear coordinates of the initial and final vibrational states. To a good approximation, the overall transition dipole μba, mn thus depends on the product of a nuclear overlap integral, hχm j χni, and an electronic transition dipole (μba) that is averaged over the nuclear coordinates. This is called the Condon approximation [60]. The contribution that a particular vibronic transition makes to the dipole strength depends on |μba, mn|2, and thus on the square of the nuclear overlap integral, |hχmjχni|2. The square of such a nuclear overlap integral is called a Franck-Condon factor. Franck-Condon factors provide quantitative quantum mechanical expressions of the classical notion that nuclei, being much heavier than electrons, do not move significantly on the short time scale of electronic transitions [60]. For a particular vibronic transition to occur, the Franck-Condon factor must be non-zero. The different vibrational wavefunctions for a given electronic state are orthogonal to each other. So if the same set of vibrational wavefunctions apply to the ground and excited electronic states, the nuclear overlap integral hχmjχni is 1 for m = n and zero for m ≠ n. If the vibrational potential is harmonic, the energies of the allowed vibronic transitions (Ψ a,n → Ψ b,n) will be the same for all n and the absorption

198

4

Electronic Absorption

Wavefunction Amplitude & Energy

spectrum will consist of a single line at the frequency set by the electronic energy difference, Ea - Eb, as shown in Fig. 4.20A. In most cases, the vibrational wavefunctions differ somewhat in the ground and excited electronic states because the electron distributions in the molecule are different. This makes hχmjχni non-zero for m ≠ n, and less than 1 for m = n, allowing transitions between different vibrational levels to occur in concert with the electronic transition. The absorption spectrum therefore includes lines at multiple frequencies corresponding to various vibronic transitions (Fig. 4.20B). At low temperatures, most of the molecules will be at the lowest vibrational level of the ground state (the zero-point level) and the absorption line with the lowest energy will be the (0-0) transition. At elevated temperatures, higher vibrational levels will be populated, giving rise to absorption lines at energies below the 0-0 transition energy. Approximate Franck-Condon factors for vibronic transitions can be obtained by using the wavefunctions of the harmonic oscillator (Eq. (2.30) and Fig. 2.3). Consider the vibronic transitions of ethylene. Because the HOMO is a bonding orbital and the LUMO is antibonding, the equilibrium length of the C=C bond is slightly longer when the molecule is in the excited state than it is in the ground state, but the change in the bond vibration frequency (υ) is relatively small. The vibrational potential energy wells for the two states have approximately the same shape but are displaced along the horizontal coordinate (bond length) as represented in

-4

2 0 -2 Vibrational Coordinate

4

-2

0 2 4 Vibrational Coordinate

Fig. 4.20 Harmonic vibrational potential energies (dashed black curves), eigenvalues (green dotted lines), and eigenfunctions (blue curves) for ground and excited electronic states. If the potential energy surfaces are the same in the ground and excited electronic states (A), the FranckCondon factors are nonzero only for vibronic transitions between corresponding vibrational levels, and these transitions (vertical green arrows) all have the same energy; the absorption spectrum consists of a single line at this energy. If the minimum of the potential energy surface is displaced along the nuclear coordinate in the excited state, as in B, Franck-Condon factors for multiple vibronic transitions are nonzero and the spectrum includes lines at multiple energies

4.11

The Born-Oppenheimer Approximation, Franck-Condon Factors, and the. . .

199

Fig. 4.20B. It is convenient to express the change in bond length (be - bg) in terms of the dimensionless quantity Δ defined by the expression pffiffiffiffiffiffiffiffiffiffiffiffi  Δ = 2π mr v=h be - bg , ð4:45Þ where mr is the reduced mass of the vibrating atoms. If we define the coupling strength S=

1 2 Δ , 2

ð4:46Þ

then the Franck-Condon factor for a transition from the lowest vibrational level of the ground state (χ0) to level m of the excited state (χm) can be written as: jhχ m j χ 0 ij2 =

Sm expð- SÞ : m!

ð4:47Þ

Corresponding expressions can be derived for the Franck-Condon factors for transitions that start in higher vibrational levels of the ground state (Box 4.12). The coupling strength S is sometimes called the Huang-Rhys factor. Box 4.12 Recursion Formulas for Vibrational Overlap Integrals Overlap integrals for harmonic-oscillator wavefunctions can be calculated using recursion formulas derived by Manneback [61]. Manneback treated the general case that the two vibrational states have different vibrational frequencies in addition to a displacement. Here we give only the results when the frequencies are the same. Let the dimensionless displacement of wavefunction χm with respect to χn be Δ, and define the Huang-Rhys factor or coupling strength, S, as Δ2/2. The overlap integral for the two zero-point wavefunctions then is: hχ 0 j χ 0 i = expð- S=2Þ:

ðB4:12:1Þ

Note that the vibrational wavefunctions in the bra and ket portions of this expression implicitly pertain to different electronic states. We have dropped indices a and b for the electronic states to simplify the notation. Overlap integrals for other combinations of vibrational levels can be built from hχ 0 j χ 0i by the recursion formulas: n o   χ mþ1 j χ n = ðm þ 1Þ - 1=2 n1=2 hχ m j χ n - 1 i - S1=2 hχ m j χ n i ðB4:12:2aÞ and 

n o  χ m jχ nþ1 = ðn þ 1Þ - 1=2 m1=2 hχ m - 1 jχ n i þ S1=2 hχ m jχ n i ,

ðB4:12:2bÞ

with hχmjχ-1i = hχ-1jχmi = 0. For the overlap of the lowest vibrational level of the ground state with level m of the excited state, these formulas give (continued)

200

4

Electronic Absorption

Box 4.12 (continued) hχ m j χ 0 i = expð- S=2Þð- 1Þm Sm=2 =ðm!Þ1=2 :

ðB4:12:3Þ

The Franck-Condon factors are the squares of the overlap integrals: jhχ m j χ 0 ij2 = expð- SÞSm =m!:

ðB4:12:4Þ

Borrelli and Peluso have described a more general treatment of Frank-Condon factors that considers changes in vibrational frequency and other effects [62– 64]. Figure 4.21 shows how the Franck-Condon factors for transitions from the lowest vibrational level of the ground state (χ0) to various levels of the excited state change as a function of Δ according to Eq. (4.47). If Δ = 0, only |hχ0 j χ0i|2 is nonzero. As jΔj increases, |hχ0 j χ0i|2 shrinks, and the Franck-Condon factors for higher-energy vibronic transitions grow, with the sum of all the Franck-Condon factors remaining constant at 1.0. If jΔ j > 1, the absorption spectrum peaks at an energy approximately Shυ above the 0-0 energy. As we’ll discuss in Chap. 6, a nonlinear molecule with N atoms has 3N 6 vibrational modes, each involving movements of at least two, and sometimes many atoms. The overall vibrational wavefunction can be written as a product of wavefunctions for these individual modes, and the overall Franck-Condon factor for a given vibronic transition is the product of the Franck-Condon factors for all the modes. When the molecule is raised to an excited electronic state some of its

1.0

|〈χk|χ0〉|

2

0.8 0.6

Δ=0

Δ = 0.5

Δ = 1.0

Δ = 1.5

Δ = 2.0

Δ = 2.5

Δ = 3.0

0.4 0.2 0

m

m

m

m

m

10

10 0

10 0

10 0

10 0

10 0

10 0

0

m

m

Fig. 4.21 Franck-Condon factors (green vertical bars) for vibronic transitions from the lowest vibrational level (n = 0) of a ground electronic state to various vibrational levels of an excited electronic state, as functions of the dimensionless displacement (Δ), for a system with a single, harmonic vibrational mode. The abscissa in each panel is the vibrational quantum number (m) of the excited state

4.11

The Born-Oppenheimer Approximation, Franck-Condon Factors, and the. . .

201

vibrational modes will be affected but others may not be. The coupling factor (S) provides a measure of these effects. Vibrational modes for which S is large are strongly coupled to the excitation, and ladders of lines corresponding to vibronic transitions of each of these modes will feature most prominently in the absorption spectrum. In the ground state, molecules are distributed among the different vibrational states depending on the energies of these states. At thermal equilibrium, the relative population of level nk of vibrational mode k is given by the Boltzmann expression (Eq. B2.6.3): Bk =

1 expð- nk hυk =kB T Þ, Zk

ð4:48Þ

where kB is the Boltzmann constant, T the temperature, and Zk the vibrational partition function for the mode: Zk =

1 X

expð- nk hυk =k B T Þ = ½1 - expð- hυk =kB T Þ - 1 :

ð4:49Þ

n=0

The absorption strength at a given frequency depends on the sum of the Boltzmannweighted Franck-Condon factors for all the vibronic transitions in which the change in the total energy (electronic plus vibrational) matches the photon energy hν (Box 4.13). Box 4.13 Thermally Weighted Franck-Condon Factors Vibronic spatial wavefunctions for the ground and excited states of a molecule with N harmonic vibrational modes can be written as products of wavefunctions of the individual modes: Ψa = ψ a

N Y k=1

χ ak,nk and Ψb = ψ b

N Y k=1

χ bk,mk ,

ðB4:13:1Þ

where ψ a and ψ b are electronic wavefunctions and χ ak,nk denotes the nth k vibrational wavefunction of mode k in electronic state a. If the vibrational frequencies (υk) do not change significantly when the molecule is excited, the strength of absorption at frequency ν and temperature T can be related to a sum of weighted Franck-Condon factors: W ðν, T Þ = (

  1X exp - E vib =k B T a,n Z n

) N D E X Y  b  a 2 × δðhν - E m,n Þ  χ k,mk χ k,nk  : m

ðB4:13:2Þ

k=1

(continued)

202

4

Electronic Absorption

Box 4.13 (continued) Here the bold-face subscripts m and n denote vectorial representations of the vibrational levels of all the modes in the two electronic states: m = (m1, m2, . . ., mN) and n = (n1, n2, . . ., nN). The other terms are defined as follows: E vib a,n =

N X

ðnk þ 1=2Þ hυk ,

ðB4:13:3aÞ

ðmk þ 1=2Þ hυk ,

ðB4:13:3bÞ

k=1

E vib b,m =

N X k=1

   elec  vib vib Em,n = E elec b þ E b,m - E a þ E a,n , X   exp - Evib Z= a,n =k B T

ðB4:13:3cÞ ðB4:13:4aÞ

n

=

Y Y Z k = ½1 - expð- hυk =kB T Þ - 1 , k

ðB4:13:4bÞ

k

elec and E elec a and E b are the electronic energies of the two states. The Kronecker delta function δ(hv - Emn) is 1 if the excitation energy hν is equal to the total energy difference given by Eq. (B4.13.3c), and zero otherwise. In the Condon approximation, the absorption strength at frequency ν depends on W(v, T )|μba|2, where μba is the electronic transition dipole averaged over the nuclear coordinates as in Eq. (4.44). The sum over n in Eq. (B4.13.2) runs over all possible vibrational levels of the ground state. A given level represents a particular distribution   of energy among the N vibrational modes, and its vibrational energy E vib a,n is the sum of the vibrational energies of all the modes (Eq. (B4.13.3a)). Each   vib level is weighted by the Boltzmann factor exp - E a,n =kB T =Z, where Z is the complete vibrational partition function for the ground state. The vibrational partition function for a system with multiple vibrational modes is the product of the partition functions of all the individual modes (Zk in Eq. (B4.14.4b)), as you can see by writing out the sum in (Eq. (B4.13.4a)) for a system with two or three modes. The sum over m in Eq. (B4.13.2) considers all possible vibrational levels of the excited state, but the delta function preserves only the levels for which Emn = h ν. If a level meets this resonance condition, the Franck-Condon factor for the corresponding vibronic transition is the product of the Franck-Condon factors for all the individual vibrational modes. The function W(ν,T ) defined by Eq. (B4.13.2) gives a set of lines at frequencies (νmn) where h ν = Emn. As we discuss below and in Chaps. 10

(continued)

4.11

The Born-Oppenheimer Approximation, Franck-Condon Factors, and the. . .

203

Box 4.13 (continued) and 11, each of these absorption lines typically has a Lorentzian or Gaussian shape with a width that depends on the lifetime of the excited vibronic state. To incorporate this effect, the delta function in Eq. (B4.13.2) can be replaced by a line-shape function wmn(ν - νmn) for each combination of m and n vectors that meets the resonance condition. At low temperatures, molecules in the electronic ground state are confined largely to the zero-point levels of their vibrational modes. The spectrum described by Eq. (B4.13.2) then simplifies to W ðν, T = 0Þ =

N X X k=1 m

δðhν - E m,0 Þ

N D E Y  b  a 2  χ k,mk χ k,0  :

ðB4:13:5Þ

k=1

By writing out the Franck-Condon factors and introducing the line-shape functions wm,0(v - vm0), we can recast this expression in the form W ðv, T = 0Þ = expð- St Þ

N X 1 X

ððSk Þm =m!Þwm,0 ,

ðB4:13:6Þ

k=1 m=0

where Sk is the Huang-Rhys factor (coupling strength) for mode k and St is the sum of the Huang-Rhys factors for all the modes that are coupled to the electronic transition. Molecules that are in higher vibrational levels in the electronic ground state contribute to the red (lower-energy) side of the absorption band. Most absorption bands therefore shift to the red with temperature as the populations of these levels grow. However, mixing with charge-transfer transitions can cause a band to shift in the opposite direction with temperature. We’ll discuss this in Chap. 8. The foregoing expressions consider only the homogeneous spectrum of a elec system in which all the molecules have identical values of Eelec a and E b . In an inhomogeneous system, the molecules will have a distribution of electronic energy differences, and the spectrum will be a convolution of this site-distribution function with Eqs. (B4.13.2) or (B4.13.6). For further discussion of the Franck-Condon factors, line-shape functions, and site-distribution functions for bacteriochlorophyll-a and related molecules see [65, 66]. Because the eigenfunctions of the harmonic oscillator form a complete set, the Franck-Condon factors for excitation from any given vibrational level of the ground electronic state to all the vibrational levels of an excited electronic state must sum to 1. |hχ0|χ0i|2 therefore gives the ratio of the strength of the 0-0 transition to the total

204

4

Electronic Absorption

dipole strength. This ratio is called the Debye-Waller factor. From Eq. (4.47), the Debye-Waller factor is exp(-S/2). The analysis of Franck-Condon factors described in Boxes 4.12 and 4.13 assumes that the chromophore’s vibrational modes are essentially the same in the excited and ground electronic states, differing only in the location of the energy minimum on the vibrational coordinate and a possible shift in the vibrational frequency. Breakdowns of this assumption are referred to as Duschinsky effects. They can be treated in some cases by representing the vibrational modes for one state as a linear combination of those for the other [67–69]. The width of an absorption band for an individual vibronic transition depends on how long the excited molecule remains in the state created by the excitation. According to Eq. (2.70), the spectrum for excitation to a state that decays exponentially with time should be a Lorentzian function of frequency. The shorter the lifetime of the excited state, the broader the Lorentzian (Fig. 2.12). A variety of processes can cause an excited molecule to evolve with time, and thus can broaden the absorption line. The molecule might, for example, decay to another vibrational state by redistributing energy among its internal vibrational modes or by releasing energy to the surroundings. Molecules in higher vibrational levels have a larger number of possible relaxation pathways that are thermodynamically favorable and thus relax more rapidly than molecules in lower levels. In addition, the energies of the individual molecules in a sample will fluctuate as a result of randomly changing interactions with the surroundings. These fluctuations cause the time-dependent parts of the wavefunctions of the molecules to get out of phase, a process termed pure dephasing. In Chap. 10 we will see that the composite time constant (T2) that determines the width of a Lorentzian vibronic absorption band depends on both the equilibration time constant for true decay processes (T1) and the time constant for pure dephasing (T2). The width of the Lorentzian at half-maximal amplitude is ħ/T2. If T2 is long, as it can be for molecules in the gas phase and for some molecules chilled to low temperatures in inert matrices, the absorption line can be very sharp. The width of such an absorption line for an individual molecule in a fixed environment, or an ensemble of identical molecules with the same solvation energies, is termed the homogeneous line width. An absorption band representing molecules that interact with their surroundings in a variety of ways is said to be inhomogeneously broadened and its width is termed the inhomogeneous line width. The spectrum generated by a family of Lorentzians with a Gaussian distribution of center energies is called a Voigt spectrum.

4.12

Spectroscopic Hole Burning

The homogeneous absorption lines that underlie an inhomogeneous spectrum can be probed experimentally by hole-burning spectroscopy at low temperatures [70– 72]. In photochemical hole-burning, the excited molecule evolves into a triplet state or is converted photochemically to another long-lived product, leaving a hole in the absorption spectrum at the frequency of the original excitation. In

4.12

Spectroscopic Hole Burning

205

nonphotochemical hole-burning, the excited molecule decays back to the original ground state with conversion of the excitation energy to heat. The thermal energy that is released causes rearrangements of the molecule’s immediate surroundings, shifting the absorption spectrum and again leaving a hole at the excitation frequency. At room temperature, non-photochemical spectral holes usually are filled in quickly by fluctuations of the surroundings on the picosecond time scale. This process, termed spectral diffusion, can be studied by picosecond pump-probe techniques. At temperatures below 4 K, non-photochemical spectral holes can persist almost indefinitely and can be measured with a conventional spectrophotometer. The shape of the hole depends on the lifetime of the excited state and the coupling of the electronic excitation to vibrational modes of the solvent, both of which depend in turn on the excitation wavelength. Excitation on the far-red edge of the absorption band populates mainly the lowest vibrational level of the excited state, which has a relatively long lifetime, and the resulting zero-phonon hole is correspondingly sharp (Fig. 4.22A). The zero-phonon hole typically is accompanied by one or more phonon sidebands that reflect vibrational excitation of the solvent in concert with electronic excitation of the chromophore. The sidebands are broader than the zerophonon hole because the excited solvent molecules relax rapidly by transferring the excess vibrational energy to the surroundings. In addition, the phonon sidebands sometimes represent a variety of vibrational modes or a quasi-continuum of closely lying vibrational states. Excitation on the blue side of the absorption band populates

1.0

A

B

Absorbance

0.8 0.6 0.4 0.2 0

-200

0 ν − νmax / cm

200 -1

-200

0 ν − νmax / cm

200 -1

Fig. 4.22 Non-photochemical hole-burning. An inhomogeneous absorption spectrum (cyan curve) usually is an envelope of spectra for chromophores in many different local environments. If an inhomogeneous sample is irradiated with light covering a narrow band of frequencies (vertical red arrow), the spectra of molecules that absorb here can be shifted to higher or lower frequencies, leaving a hole in the inhomogeneous spectrum (blue curve). If the sample is excited on the red (low-energy) edge of the absorption band, the hole often consists of a sharp zero-phonon hole with one or more broad phonon side-bands at somewhat higher energies (A). The width of the zerophonon hole provides information on the lifetime of the excited electronic state, while the phonon sidebands report on solvent vibrations that are coupled to the electronic excitation. Excitation on the blue side of the absorption band usually gives a broader, unstructured hole (B)

206

4

Electronic Absorption

higher-energy vibrational levels of both the chromophore and the solvent, which usually decay rapidly and give a broad, unstructured hole (Fig. 4.22B). Studies of antenna complexes from photosynthetic bacteria by Small and coworkers [73–75] provide a good illustration of nonphotochemical hole-burning. These complexes have extensive manifolds of excited electronic states that lie close together in energy. Excitation on the long-wavelength edge of the main absorption band populates mainly the lowest vibrational level of the lowest excited electronic state, which decays with a time constant on the order of 10 ps. Holes burnt in this region of the spectrum have correspondingly narrow widths of about 3 cm-1. Excitation at shorter wavelengths populates higher excited electronic states, which evidently decay to the lowest state in 0.01 to 0.1 ps. Holes burnt near the center of the absorption band therefore have widths on the order of 200 cm-1. In similar studies of photosynthetic reaction centers, the width of the zero-phonon hole for the reactive bacteriochlorophyll dimer was related to the time constant for electron transfer to a neighboring molecule [65, 76, 77]. Figure 4.23 shows a typical hole spectrum (the difference between absorption spectra measured with the excitation laser on and off) for a sample of reaction centers that was excited at 10,912 cm-1 at 5 K. The holes in this experiment resulted from the photochemical electrontransfer reaction followed by conversion of the bacteriochlorophyll dimer to an excited triplet state. They are broadened by strong vibronic coupling to motions of

Fig. 4.23 The spectrum of a photochemical hole burned in the long-wavelength absorption band of a sample of photosynthetic bacterial reaction centers at 5 K [77]. The gray curve is the difference between absorption spectra measured with the excitation laser on and off. The excitation frequency was 10, 912 cm-1. Note the sharp zero-phonon hole (ZPH, upward red arrow) at 10, 980 cm-1. The downward blue arrows indicate the centers of two discrete vibrational (phonon) bands that are linked to the zero-phonon transition. The solid black curve is a theoretical spectrum of the hole calculated as described in the text

4.13

Effects of the Surroundings on Molecular Transition Energies

207

the protein surrounding the bacteriochlorophylls. The holes generated by burning in the red edge of the spectrum reveal a zero-phonon line with a width of about 6 cm-1, which corresponds to an electron-transfer time constant of about 1 ps. They also have several discrete phonon sidebands that can be assigned to two characteristic vibrational modes. In the spectrum shown in Fig. 4.23, the zero-phonon hole is at 10,980 cm-1, and the prominent vibrational modes have frequencies of approximately 30 and 130 cm-1. Both the hole spectrum and the original absorption spectrum can be fit well by using Eq. (B4.13.6) with these two vibrational modes, a Gaussian distribution of zero-phonon transition energies, and relatively simple Lorentzian and Gaussian functions for the shapes of the zero-, one- and two-phonon absorption lines [65, 66, 77]. Each of the homogeneous lines in a vibronic absorption spectrum consists of a family of transitions between various rotational states of the molecule. The rotational fine structure in the spectrum can be seen for small molecules in the gas phase, but the rotational lines for large molecules are too close together to be resolved.

4.13

Effects of the Surroundings on Molecular Transition Energies

Interactions with the surroundings can shift the energy of an absorption band to either higher or lower energies, depending on the nature of the chromophore and the solvent. Such shifts are called solvatochromic effects. Consider, for example, an n-π* transition, in which an electron is excited from a nonbonding orbital of an oxygen atom to an antibonding molecular orbital distributed between O and C atoms (Sect. 9.1). In the ground state, electrons in the nonbonding orbital can be stabilized by hydrogen-bonding or dielectric effects of the solvent. In the excited state, these favorable interactions are disrupted. Although solvent molecules will tend to reorient in response to the new distribution of electrons in the chromophore, this reorientation is too slow to occur during the excitation itself. An n-π* transition therefore shifts to higher energy in more polar or H-bonding solvents relative to less polar solvents. A shift of an absorption band in this direction is called a hypsochromic or blue shift or a negative solvatochromic effect. The energies of π-π* transitions are less sensitive to the polarity of the solvent but still depend on the solvent’s high-frequency polarizability, which as we noted in Chap. 3, increases quadratically with the refractive index. Increasing the refractive index usually decreases the transition energy of a π-π* transition, causing a bathochromic or red shift of the absorption band. This is a positive solvatochromic effect. The terms “blue” and “red” often are used in this context without regard to the position of the absorption band relative to the spectrum of visible light. For example, a shift of an IR band to lower energies is generally called a redshift even though the band moves away from the region of the visible spectrum that we perceive as red. Solvatochromic effects have been used to establish scales for evaluating the effective polarities of solvents and micro-heterogeneous systems [78–84]. The first such scale, proposed by E. M. Kosower in 1958 [78], used

208

4

Electronic Absorption

Fig. 4.24 Three molecules with large solvatochromic effects

1-ethyl-4-(methoxycarbonyl)pyridinium iodide (Fig. 4.24A) as the chromophore. This molecule has a dipole moment of about 14 debyes in the ground state. Excitation results in the transfer of electron density from iodide ion to the pyridinium ring, decreasing the dipole moment by about 5 D. The absorption spectrum therefore shifts to the blue in polar solvents. The zwitterionic molecule 4-(2,4,6-triphenyl-1pyridinio)-2,6-diphenylphenolate (Fig. 4.24B), which is also called betaine-30 or Reichardt's dye after its discoverer [79–81], features the most dramatic negative solvatochromic effect known. Transfer of electron density from the phenolate to the pyridinium ring in the excited state decreases the dipole moment by about 7 D [85] and shifts the absorption maximum from 810 nm in diphenyl ether to 453 nm in water. The molecule with the strongest known positive solvatochromic effect is 5-(dimethylamino)-5'-nitro-2,2-bisthiophene (Fig. 4.24C), whose dipole moment increases substantially on excitation [82]. Its absorption maximum shifts from 466 nm in hexane to 577 nm in aqueous formamide. The visual pigments provide dramatic illustrations of how minor changes in protein structure can shift the absorption spectrum of a bound chromophore. As in many other vertebrates, the human retina contains three types of cone cells whose pigments (cone-opsins) absorb in different regions of the visible spectrum. Coneopsin from human “blue” cones absorbs maximally near 414 nm, while those from “green” and “red” cones have absorption maxima near 530 and 560 nm, respectively (Fig. 4.25). The visual pigments from other organisms have absorption maxima

Effects of the Surroundings on Molecular Transition Energies

Fig. 4.25 Normalized absorption spectra of rhodopsin from human retinal rod cells (cyan curve 2), and cone-opsins from human “blue” (curve 1), “green” (curve 3), and “red” (curve 4) cone cells. The spectra are drawn as described by Stavenga et al. [185]

Normalized Absorbance

4.13

1.0

1

209

2

3

4

0.5

0 400

500

600

700

Wavelength / nm

ranging from 355 to 575 nm. Raptors such as hawks use an ultraviolet-sensitive pigment to follow trails of urine left by rodents that they hunt. Yet all these pigments resemble rhodopsin, the pigment from retinal rod cells, in containing a protonated Schiff base of 11-cis-retinal. (The chromophores in some other organisms are based on retinal A2, which has one more conjugated double bond than retinal A1 and can push the absorption maximum as far to the red as 620 nm.) The proteins from vertebrate rods and cones have homologous amino acid sequences and, although a crystal structure currently is available only for bovine rhodopsin [86], probably have very similar three-dimensional structures [87, 88]. If the proteins are denatured by acid, the absorption bands all move to the region of 440 nm and resemble the spectrum of the protonated Schiff base of 11-cis-retinal in methanol. Resonance Raman measurements have shown that the vibrational structure of the chromophore does not vary greatly among the different proteins, indicating that the spectral shifts probably result mainly from electrostatic interactions with the surrounding protein rather than from changes in the conformation of the chromophore [89, 90]. Studies of proteins containing conformationally constrained analogs of 11-cis-retinal also have lent support to this view [91, 92]. Correlations of the absorption spectra with the amino acid sequences of the proteins from a variety of organisms, together with studies of the effects of sitedirected mutations, indicate that the shifts of the spectra of the visual pigments reflect changes in a small number of polar or polarizable amino acid residues near the chromophore [90, 93–96] (Fig. 4.26). The 30-nm shift of the human red cone pigment relative to the green pigment can be attributed entirely to changes of seven residues, including the replacement of several Ala residues by Ser and Thr. The more polarizable side chains of Ser and Thr probably facilitate delocalization of the positive charge from the N atom of the retinyl Schiff base toward the β-ionone ring in the excited state. The red and the green cone pigments also have a bound Cl‐ ion that contributes to the redshift of the spectrum relative to rhodopsin and the blue

210

4

Electronic Absorption

Fig. 4.26 Models of the region surrounding the retinyl chromophore in the visual pigments from human “blue” (A) and “red” (B) cone cells. The main absorption band is shifted to longer wavelengths by almost 150 nm in the red pigment. Carbon, oxygen, nitrogen, and some hydrogen atoms of the chromophore (RET) and the side chains of the lysine residue that forms a protonated Schiff base with the retinal (K296), the glutamate that acts as a counter ion (E113), and some of the other residues that probably contribute to regulation of the color of the human cone pigments are represented with licorice models in CPK colors. The residues are labeled with the rhodopsin numbering and the one-letter amino acid code. Wire models are used for other heavy atoms of the protein. The models were constructed by homology with bovine rhodopsin [87]. The Cl‐ ion that binds to the red opsin is not shown

cone pigment. Variations in the position of a Glu carboxylate group that serves as a counterion for the protonated Schiff base also may contribute to the spectral shifts, although the changes in the distance from the counterion to the proton probably are relatively small [89, 90, 97]. DNA photolyases, which use the energy of blue light to split pyrimidine dimers formed by UV irradiation of DNA, provide other examples of large and variable shifts in the absorption spectrum of a bound chromophore. These enzymes contain a bound pterin, (methylenetetrahydrofolate, MTHF) or deazaflavin, which serves to absorb light and transfer energy to a flavin radical in the active site [98]. The absorption maximum of MTHF occurs at 360 nm in solution but ranges from 377 to 415 nm in the enzymes from different organisms [99]. Solvatochromic effects on the transition energies of molecules in solution often can be related phenomenologically to the solvent’s dielectric constant and refractive index. The analysis is similar to that used for local-field correction factors (Sect. 3. 1.5). Polar solvent molecules around the chromophore will be ordered in response to the chromophore’s ground-state dipole moment (μaa), and the oriented solvent molecules provide a reaction field that acts back on the chromophore. A simple model of the system is a dipole at the center of a sphere of radius R embedded in a

4.13

Effects of the Surroundings on Molecular Transition Energies

211

homogeneous medium with dielectric constant εs. The reaction field felt by such a dipole is given approximately by [28]:   2μ ε -1 : ð4:50Þ E = 3aa s εs þ 2 R Now suppose that excitation of the chromophore changes its dipole moment to μbb. Although the solvent molecules cannot reorient instantaneously in response, the dielectric constant εs includes electronic polarization of the solvent in addition to orientational polarization, and changes in electronic polarization can occur essentially instantaneously in response to changing electric fields. The high-frequency component of the dielectric constant is the square of the refractive index (n) (Sects. 3. 1.4–3.1.5). If we subtract the part of the reaction field that is attributable to electronic polarization, the part due to the orientation of the solvent (Eor) can be written   2   2μaa n -1 εs - 1 - 2 Eor = 3 : ð4:51Þ εs þ 2 n þ2 R The solvation energies associated with interactions of the chromophore’s dipoles with the oriented solvent molecules are – (1/2) μaa  Eor in the ground state and –(1/2) μbb  Eor in the excited state. The factor of 1/2 here reflects the fact that, to orient the solvent, approximately half of the favorable interaction energy between the chromophore and Eor must be used to overcome unfavorable interactions of the solvent dipoles with each other. The change in excitation energy resulting from the orientation of the solvent in the ground state is, therefore, ðμ - μbb Þ  μaa ΔE or = ð1=2Þðμaa - μbb Þ  Eor = aa R3   2   εs - 1 n -1 - 2 : × εs þ 2 n þ2

ð4:52Þ

This expression shows that interactions with a polar solvent can either increase or decrease the transition energy, depending on the sign of (μaa - μbb)  μaa. The change in energy (ΔEor) is expected to be small if the excitation involves little change in dipole moment (μaa ≈ μbb), if the chromophore is nonpolar in the ground state (μaa ≈ 0), or if the solvent is nonpolar (εs ≈ n2). Equations (4.51 and 4.52) do not include the effects of electronic polarizability. The solvent can be polarized electronically by both the permanent and transition dipoles of the chromophore. Quantum mechanically, this inductive polarization can be viewed as the mixing of the excited and ground electronic states of the solvent under the perturbation caused by electric fields from the solute (Box 12.1). The chromophore experiences a similar inductive polarization by fields from the solvent. In nonpolar solvents, π-π* absorption bands of nonpolar molecules typically decrease in energy with increasing refractive index, and the decrease is approximately linear in the function (n2 – 1)/(n2 + 2). Some authors use the function (n2 – 1)/ (2n2 + 1), which gives very similar results (see [28] for a review of early work and

4

Electronic Absorption

Qy Transition Energy / cm

-1

212

13000

12800

12600 0.25

0.20

0.15 2

2

(n -1)/(2n +1) Fig. 4.27 Dependence of the transition energy of the long-wavelength (Qy) absorption band of bacteriochlorophyll-a on the refractive index (n) in nonpolar solvents. Experimental data from Limantara et al. [100] are replotted as a function of (n2 - 1)/(2n2 + 1). Extrapolating the cyan line to n = 1 (0 on the abscissa) gives 13, 810 cm-1 for the transition energy in a vacuum. A similar plot of the data vs (n2 - 1)/(n2 + 2) gives 13,600 cm-1

[100] for a more recent study). Figure 4.27 illustrates this shift for the longwavelength absorption band of bacteriochlorophyll-a. Extrapolation to n = 1 gives a “vacuum” transition energy on the order of 1000 cm-1 above the energy measured for bacteriochlorophyll in solution. The energy of interactions of a molecule with its surroundings can be treated more microscopically by using Eqs. (4.19–4.22) to describe the ground and excitedstate wavefunctions, ψ a and ψ b. The solvation energy of the molecule in the ground state is E solv a ≈ - 2e

N  X i=1

2 C ai V i þ E solv core ,

ð4:53Þ

where e is the electronic charge, Vi is the electric potential at the position of atom i, C ai is the atomic expansion coefficient for atom i in ψ a, N is the total number of atoms that participate in the wavefunction and Esolv core represents the effects of the surroundings on the nuclei and electrons in the orbitals other than ψ a. The contribution of ψ a  2 to the electronic charge on atom i is proportional to C ai and the factor of 2 before the sum reflects the assumption that there are two electrons in ψ a in the ground state. Similarly, the solvation energy in the excited state is: Esolv b ≈ -e

N h X i=1

C ai

2

 2 i þ C bi V i þ Esolv core ,

ð4:54Þ

4.13

Effects of the Surroundings on Molecular Transition Energies

213

where C bi is the coefficient for atom i in ψ b. The change in solvation energy upon excitation is the difference between the two solvation energies: solv E solv b - Ea ≈ - e

N h X i=1

Cbi

2

 2 i þ Cai V i:

ð4:55Þ

Note that these expressions consider only a single configuration, excitation from ψ a to ψ b. If several configurations contribute to the absorption band, the relative contribution of a given configuration to the changes in electronic charge is proportional to the square of the coefficient for this configuration in the overall excitation (Eq. (4.23)). If the structure of the surroundings is well defined, as it may be for a chromophore in a protein (see, for example, the visual pigments shown in Fig. 4.26), the electric potential at each point in the chromophore can be estimated by summing the contributions from the charges and dipoles of the surrounding atoms: ! X Q μk  rik k þ : ð4:56Þ Vi ≈ jrik j jrik j3 k ≠ i⋯ Here Qk is the charge on atom k of the surroundings, rik is the vector from atom k to atom i, and μk is the electric dipole induced on atom k by the electric fields from all the charges and other induced dipoles in the system. The sums in this expression exclude any atoms whose interactions with atom i must be treated quantum mechanically. In addition to atom i itself, other atoms that are part of the chromophore or are connected to atom i by three or fewer bonds typically require such special treatment. The induced dipoles that are included in Eq. (4.54) can be calculated from the atomic charges and polarizabilities by an iterative procedure [101]. Shifts in the absorption spectrum thus can be used to measure the binding of prosthetic groups to proteins or to probe protein conformational changes in the region of a bound chromophore. Equations (4.53 and 4.54) involve significant approximations, however, because interactions with the surroundings ideally should be taken into account when the molecular orbitals and eigenvalues are obtained in the first place. Figure 4.28 illustrates the calculated redistributions of charge that accompany excitation of the indole side chain of tryptophan. As discussed above, the 1La absorption band consists mainly of the configuration ψ 2 → ψ 3 and a smaller contribution from ψ 1 → ψ 4, where ψ 1 and ψ 2 are the second-highest and highest occupied molecular orbitals and ψ 3 and ψ 4 are the second-lowest and lowest unoccupied orbitals. Both these configurations result in a transfer of electron density from the pyrrole ring to the benzyl portion of the indole sidechain. The energy of the 1 La transition thus should be red-shifted by positively charged species near the benzene ring, and blue-shifted by negatively charged species in this region. An electron-withdrawing substituent such as a cyanyl, formyl, or nitro group attached to the benzyl ring also shifts the absorption to the red and increases its sensitivity to the polarity of the surroundings [102–105]. The 1Lb absorption band consists largely of

214

4

Electronic Absorption

6 4

B

ψ2 → ψ 4 N

N

2 y/ao

ψ 1→ ψ 4

A

0 -2 -4 -6 6 4

C

D

ψ1 → ψ 3 N

N

2 y/ao

ψ2 → ψ 3

0 -2 -4 -6 6 4

1

F

La

Lb

N

N

2 y/ao

1

E

0 -2 -4 -6

-6 -4 -2

0 2 x /ao

4

6

-6 -4 -2

0 2 x /ao

4

6

Fig. 4.28 Redistribution of charge upon excitation of 3-methylindole. (A–D) Contour plots show the changes in electron density (increases in negative charge) when an electron is excited from one of the two highest occupied molecular orbitals (ψ 1 or ψ 2) to one of the first two unoccupied orbitals (ψ 3 or ψ 4). Red lines denote increases in electron density; blue lines, decreases. The contour intervals are 0:01 ea30 . The planes of the drawings and the line types for positive and negative amplitudes are as in Fig. 4.11. (E, F) Similar plots for the combinations 0.841(ψ 2 → ψ 3) + 0.116 (ψ 1 → ψ 4) and 0.536(ψ 1 → ψ 3) + 0.402(ψ 2→ψ 4), which are approximately the contributions of these four orbitals in the 1 La and 1 Lb excitations, respectively. (Note that the coefficient for a given configuration here is the square of the corresponding coefficient for the transition dipole.) The contour intervals are 0.005 ea03. The black arrows indicate the changes in the permanent dipole (μbb - μaa, Eq. (4.58)) in units of eÅ=5a0

4.13

Effects of the Surroundings on Molecular Transition Energies

215

the configurations ψ 2 → ψ 4 and ψ 1 → ψ 3, the first of which results in a large shift of electron density to the benzyl ring. However, the ψ 1 → ψ 3 transition moves electron density in the opposite direction, making the net change of dipole moment associated with the 1Lb band smaller than that associated with 1La (Fig. 4.28E, F). For a more quantitative analysis of these effects, contributions from higher-energy configurations also need to be considered [42, 43, 106]. The gradient operator may also be preferable to the dipole operator here, but this has not been studied extensively. The situation takes on an additional dimension if the positions of the charged or polar groups near the chromophore fluctuate rapidly with time. One way to describe the effects of such fluctuations is to write the energies of the ground and excited electronic states (Ea and Eb) as harmonic functions of a generalized solvent coordinate (X): E a = E oa þ ðK=2ÞX 2 ,

ð4:57aÞ

Eb = E ob þ ðK=2ÞðX - ΔÞ2 ,

ð4:57bÞ

and

where Eoa and E ob are the minimum energies of the two electronic states, Δ is the displacement of the energy minima along the solvent coordinate, and K is a force constant. We use the term “solvent” here in a general sense to refer to a chromophore’s surroundings in either a protein or free solution. Because of the displacement Δ, a vertical transition starting from X = 0 in the ground state creates an excited state with excess solvation energy that must be dissipated as the system relaxes in the excited state (Fig. 4.29). The extra energy is termed the solvent reorganization energy and is given by Λs = KΔ2/2. The energy difference between the excited and ground states at any given value of the solvent coordinate thus is E b - Ea = E ob þ ðK=2ÞðX - ΔÞ2 - E oa - ðK=2ÞX 2 = Eo - KXΔ þ Λs ,

ð4:58Þ

where Eo = E ob - E oa : If we equate potential energies approximately with free energies, we also can say that the relative probability of finding a particular value of X when the chromophore is in the ground state is P(X) = exp(-KX2/2kBT ). Combining this expression with Eq. (4.58) and using the relationships Λs = KΔ2/2 and h v = Eb ‐ Ea gives an expression for the relative strength of absorption at energy hν: n o PðhvÞ = exp - K ðE0 þ Λs - hvÞ2 =2ðKΔÞ2 kB T n o = exp - ðEo þ Λs - hνÞ2 =4Λs kB T :

ð4:59Þ

216

4

Electronic Absorption

Fig. 4.29 Classical harmonic energy curves for a system with a chromophore in its ground and excited electronic states, as functions of a generalized, dimensionless solvent coordinate. The curve for the excited state is shifted by an amount Δ along this coordinate. If the chromophore is raised from the ground state to the excited electronic state with no change in nuclear coordinates, it is left with excess solvation energy relative to the energy minimum for the excited state. This is the reorganization energy (ΛS)

This Gaussian function of hν peaks at an energy Λs above Eo and has a width at halfmaximum amplitude (FWHM) of 2(2ΛskBT  ln 2)1/2/π, or 2(KkBT  ln 2)1/2Δ/π. Fluctuating interactions with the solvent thus broaden the vibronic absorption lines of the chromophore and shift them to higher energies relative to Eo. As discussed above, however, the mean energy of interaction can shift Eo either upward or downward depending on the chromophore and the solvent. We will discuss generalized solvent coordinates further in Chap. 5. Fluctuating electrostatic interactions can be treated microscopically by incorporating Eqs. (4.51–4.54) into molecular dynamics simulations (Box 6.1). The results can be used to construct potential energy surfaces similar to those of Eqs. (4.57a, 4.57b), or can be used in quantum calculations of the eigenvalues of the chromophore in the electric field from the solvent. Mercer et al. [107] were able to reproduce the width of the long-wavelength absorption band of bacteriochlorophyll in methanol well by this approach.

4.14

The Electronic Stark Effect

If an external electric field is applied across an absorbing sample, the absorption bands can be shifted to either higher or lower energies depending on the orientation of the chromophores with respect to the field. This is the Stark or electrochromic

4.14

The Electronic Stark Effect

217

effect. The effect was discovered in 1913 by the physicist Johannes Stark, who found that electric fields on the order of 105 V/cm cause a splitting of the spectral lines of hydrogen into symmetrically placed components with different polarizations. The basic theoretical tools for extracting information on the dipole moment and polarizability of a molecule by Stark spectroscopy were worked out by Liptay [108– 110]. They have been extended and applied to a variety of systems by the groups of Boxer [111–114], Nagae [115], and others. In one application, Premvardhan et al. found that excitation of photoactive yellow protein or its chromophore (a thioester of p-coumaric acid) in solution causes remarkably large changes in the dipole moment and the polarizability of the chromophore [116–118]. The change in dipole moment (jΔμ j = 26 debyes) corresponds to moving an electric charge by 5.4 Å and seems likely to contribute importantly to the structural changes that follow the excitation. In the simplest situation for a molecular chromophore, the magnitude and direction of the shift depend on the dot product of the local electric field vector ( Eext = f Eapp, where Eapp is the applied field and f is the local-field correction factor) with the vector difference between the chromophore’s permanent dipole moments in the excited and ground states (Δμ): ΔE = - Eext  Δμ = - f Eapp  Δμ:

ð4:60Þ

The difference between the dipole moments in the two states can be related to the chromophore’s molecular orbitals by the expression Δμ = μbb - μaa ≈ e

N h 2   i X 2 ri C bi - Cai ,

ð4:61Þ

i

where ri is the position of atom i and C ai and C bi are the coefficients for this atom in the ground- and excited-state wavefunctions (Eqs. (4.22d–4.22e) and (4.53–4.55)). If the sample is isotropic, some molecules will be oriented so that the external field shifts their transition energies to higher values, while others will experience shifts to lower energies. The result will be a broadening of the overall absorption spectrum. If the system is anisotropic, on the other hand, the external field can shift the spectrum systematically to higher or lower frequencies. Equation (4.61) assumes that the electric field does not cause the molecules to reorient, but simply shifts the energy difference between the ground and excited states without changing μaa or μbb. The validity of this assumption depends on the molecule and the experimental apparatus. Although small polar molecules in solution can be oriented by external electric fields, this is less likely to occur for proteins, particularly if the direction of the field is modulated rapidly. It can be prevented by immobilizing the protein in a polyvinyl alcohol film. But to the extent that the chromophore is polarizable, the field will create an additional induced electric dipole that depends on the strength of the field, and this dipole can change when the molecule is excited if the polarizability in the excited state differs from that in the

218

4

Electronic Absorption

ground state. In general, the molecular polarizability should be treated as a matrix, or more formally a second-rank tensor, because it depends on the orientation of the molecule relative to the field and the induced dipole can have components that are not parallel to the field (Boxes 4.14 and 12.1); however, we’ll assume now that the polarizability can be described adequately by a scalar quantity with dimensions of ind cm-3. The induced dipole (μind aa or μbb for the ground or excited state, respectively) then will be simply the product of the polarizability (αaa or αbb) and the total field, including both the external field (Eext) introduced by the applied field and the “internal” field from the molecule’s surroundings (Eint). Interactions of Eext with the induced dipoles will change the transition energy by   ind ΔE ind = - Eext  μind ð4:62aÞ bb - μaa = - Eext  ðαbb - αaa ÞðEext þ Eint Þ   = - Δα jEext j2 þ Eext  Eint , ð4:62bÞ where Δα = αbb - αaa. The internal field Eint can have any orientation relative to the external field, and it usually has a considerably larger magnitude (typically on the order of 106 V/cm or more). But if the chromophore is bound to a highly structured system such as a protein, Eint will have approximately the same magnitude for all the molecules in a sample and its orientation will be relatively well-fixed with respect to the individual molecular axes. We then can consider the factor ΔαEint to be part of the dipole change Δμ in Eq. (4.61), rather than including it separately in Eq. (4.62a, b). The additional contribution to the transition energy from dipoles induced by the external field Eext then is just ΔE ind = - ΔαjEext j2 :

ð4:63Þ

According to Eq. (4.63), interactions of the external field with induced dipoles will shift the transition energies for all the molecules of a sample in the same direction, depending on whether Δα is positive or negative. The change in the transition energy increases quadratically with the strength of the external field. The contribution to ΔE from induced dipoles is often termed a “quadratic” Stark effect to distinguish it from the “linear” contribution from Δμ, which depends linearly on |Eext| as described by Eq. (4.60). As we will see shortly, however, the changes in the absorption spectrum attributable to Δα and Δμ both depend quadratically on |Eext|. Let’s assume that the internal field Eint has a fixed magnitude and orientation with respect to the molecular axes so that Eq. (4.63) is valid. If Δα and Δμ both are non-zero, the total change in the transition frequency for an individual molecule then will be   ð4:64Þ Δν = - Eext  Δμ þ jEext j2 Δα =h:

4.14

The Electronic Stark Effect

219

The effect on the overall absorption spectrum of an isotropic sample can be evaluated by expanding the absorption spectrum as a Taylor’s series in powers of Δν: 2

∂εðν, 0Þ 1 ∂ εðν, 0Þ Δν þ jΔνj2 þ ⋯ 2 ∂ν ∂ν2  ∂εðv, 0Þ  = εðv, 0Þ Eext  Δμ þ jEext j2 Δα h - 1 ∂v 2  2 1 ∂ εðv, 0Þ 2 þ E  Δμ þ Δα h - 2 þ ⋯, j jE ext ext 2 ∂v2

εðν, EÞ = εðν, 0Þ þ

ð4:65Þ

where ε(ν,0) represents the absorption spectrum in the absence of the external field. We next need to average this expression over all orientations of the molecules with respect to the field. If the solution is isotropic, terms that depend on the 1st, 3rd, or any odd power of |Eext| average to zero while terms that depend on even powers of |Eext| remain, and the average value of (Eext  Δμ)2 becomes (1/3) |Eext|2|Δμ|2 (Box 4.6). The total effect of the external field on the molar extinction coefficient at frequency ν then will be: !   2 ∂εðν, 0Þ ΔαjEext j2 1 ∂ εðν, 0Þ þ εðν, EÞ - εðν, 0Þ = h 2 ∂ν ∂ν2 ×

jEext j2 jΔμj2 þ⋯ 3h2

ð4:66Þ

This expression shows that the major contributions from Δα to the changes in the absorption spectrum depend on the first derivative of the spectrum with respect to ν, whereas the major contributions from Δμ depend on the second derivative. As stated above, both depend on the square of |Eext|. Stark spectra often include contributions from both induced and permanent dipoles, which can be separated experimentally by fitting the measured difference spectrum to a sum of first- and second-derivative terms (Fig. 4.30). If the measuring light is polarized, the spectrum also depends on the angle between the polarization axis and the applied field, and this dependence can be used to determine the orientation of Δμ relative to the transition dipole (Box 4.14).

220

4

A

B

C

D

Absorbance

1

Electronic Absorption

0

ΔA x 10

1

0

-1 -100

0

ν − νmax / cm

-1

100

-100

0

-1

100

ν − νmax / cm

Fig. 4.30 Idealized absorption spectra of an isotropic system in the absence (cyan curves) and presence (red curves) of an external electric field (A, B), and the changes in the spectra caused by the field (C, D, blue curves). In (A and C), the chromophore is assumed to have the same dipole moment but higher polarizability in the excited state than the ground state; the field shifts the spectrum to lower energies. In (B) and (D), the chromophore has the same polarizability but a higher dipole moment in the excited state than in the ground state; the field broadens the spectrum and decreases the peak absorbance. Stark spectra often show a combination of these two effects

Box 4.14 Electronic Stark Spectroscopy of Immobilized Molecules In Liptay’s treatment [108–110] as implemented by Boxer and coworkers [113], applying an external field E to a non-oriented but immobilized system changes the absorbance (A) at wavenumber ν ν by ( ) 2 Bχ ∂½AðνÞ=ν Cχ ∂ ½AðνÞ=ν 2 ν ΔAðνÞ = jEj Aχ AðνÞ þ : ν þ 15hc ∂ν ∂ν2 30h2 c2 ðB4:14:1Þ with (continued)

4.14

The Electronic Stark Effect

221

Box 4.14 (continued) Xn  2  h  2 i o 1 2 10 a þ 3 cos χ 1 3a a þ aij Aχ = ij ii jj 30jμba j2 i,j þ

io Xn  h 1 2 10μ b þ 3 cos χ 1 4μ b , ijj ijj ba ð i Þ ba ð i Þ 15jμba j2 i,j

i  h3 5 1 Bχ = TrðΔαÞ þ 3 cos 2 χ - 1 Δαμ - TrðΔαÞ 2 2 2 n 1 X 10μbaðiÞ aij Δμj þ jμba j2 i,j i  h þ 3 cos 2 χ - 1 3μbaðiÞ ajj Δμi þ μbaðiÞ aij Δμj ,

g

ðB4:14:2Þ

ðB4:14:3Þ

and    C χ = jμba j2 5 þ 3 cos 2 χ - 1 3 cos 2 ζ - 1 :

ðB4:14:4Þ

These expressions include averaging over all orientations of the chromophore with respect to the field and the polarization of the measuring light. The factors aij and bijj in Eqs. (B4.14.2 and B4.14.3) are elements of the transition polarizability tensor (a) and the transition hyperpolarizability (b), which describe the effects of the external field on the dipole strength of the absorption band. These effects probably are relatively minor in most cases and are neglected in Eq. (4.66). The transition polarizability tensor a is a 3 × 3 matrix, whose nine elements are defined as aij = (∂μba(i)/∂Ej), where μba(i) is the i component of the transition dipole (μba) and Ej is the j component of the field. The transition hyperpolarizability b is a 3 × 3 × 3 cubic array, or third-rank tensor. Including both polarizability and hyperpolarizability, the change in the → → → transition dipole (μba) caused by the field is a  E þ E  b  E : The change in the x component of μba, for example, is axxEx + axyEy + axzEz + bxxx|Ex|2 + byx| Ey|2 + bzxz|Ez|2. The hyperpolarizability terms are included in Eq. (B4.14.2) because, though small relative to μba, they can dominate over the polarizability terms for strongly allowed transitions. This is because, as we discussed in Sect. 4.5, an electronic transition can be strongly allowed only if the initial and final states have different symmetries. The transition polarizability, by contrast, tends to be small for states with different symmetries because it depends on the mixing of these states with other states ([113] and Box 12.1). If the initial and final states have different symmetries they generally cannot both mix well with a third state. (continued)

222

4

Electronic Absorption

Box 4.14 (continued) In Eqs. (B4.14.3 and B4.14.4), Δμ is the difference between the permanent dipole moments of the excited and ground states (μbb - μaa), and Δμx, Δμy, and Δμz are its components. Δα is the difference between the polarizabilities of the excited and ground state, with the polarizabilities again described as secondrank tensors. To first order, the field changes Δμ by Δα  Eext.Tr(Δα) is the sum of the three diagonal elements of Δα, and Δαμ is the component of Δα along the direction of μba(μba  Δα  μba/|μba|2). χ is the angle between Eext and the polarization of the measuring light and ζ is the angle between μba and Δμ. Stark effects usually are measured by modulating the external field at a frequency on the order of 1 kHz and using lock-in detection electronics to extract oscillations of a transmitted light beam at twice the modulation frequency. The three terms in the brackets on the right-hand side of Eq. (B4.14.1) can be separated experimentally by their dependence on, respectively, the 0th, 1st, and 2nd derivatives of A=ν with respect to ν . |Δμba|2 and |ζ| then are obtained by measuring the dependence of Cχ on the experimental angle χ. The experiment thus gives the magnitude of Δμ uniquely but restricts the orientation of Δμ only to a cone with half-angle ±ζ relative to Δμba. The factor Bχ depends on both Δα and cross-terms involving Δμ and the transition polarizability (Eq. (B4.14.3)). Although the transition polarizability (a) is expected to be small for strongly allowed transitions, its products with Δμ are not necessarily negligible relative to Δα. Conventional Stark measurements therefore do not yield unambiguous values for Δα. In some cases, it may be possible to obtain additional information by measuring the oscillations of the transmitted light beam at higher harmonics of the frequency at which the field is modulated and relating the signals to higher derivatives of the absorption spectrum [112, 113, 119, 120]. This technique is called higherorder Stark spectroscopy. Equation (4.66) assumes that the absorption band responds homogeneously to the external field. This assumption can break down if the band represents several different transitions, particularly if these have nonparallel transition dipoles, but information about the individual components sometimes can be obtained by combining Stark spectroscopy with hole-burning [121]. An illustration is a study by Gafert et al. [122] on mesoporphyrin-IX in horseradish peroxidase. The authors were able to evaluate the contribution of the internal field to Δμ and relate varying local fields to different conformational states of the protein. Pierce and Boxer [123] have described Stark effects on N-acetyl-Ltryptophanamide and the single tryptophan residue in the protein melittin. They separated the effects on the 1La and 1Lb bands by taking advantage of the different fluorescence anisotropy of the two bands (Sect. 5.6). In agreement with Fig. 4.28, the 1 La band exhibited a relatively large Δμ of approximately 6/f Debye, where f is the unknown local-field correction factor. The Δμ for the 1Lb band was much smaller.

4.15

Spectroscopy of Transition-Metal Complexes

223

Other interesting complexities can arise if the excited chromophore enters into a rapid photochemical reaction that is affected by the external field. This is the situation in photosynthetic bacterial reaction centers, where the excited bacteriochlorophyll dimer (P*) transfers an electron to a neighboring molecule (B) with a time constant on the order of 2 ps. The electron-transfer process generates an ion-pair state (P+B‐) whose dipole moment is much larger than that of either P* or the ground state. An external electric field thus can shift the ion-pair state to substantially higher or lower energy depending on the orientation of the field relative to P and B, and this shift can alter the rate of electron transfer. The resulting changes in the absorption spectrum are not described well by a simple sum of first- and second-derivative terms (Eq. (4.66)), but can be analyzed fruitfully by including higher-order terms [124–127]. Vibrational Stark spectroscopy is discussed in Sect. 6.4.

4.15

Spectroscopy of Transition-Metal Complexes

Ions of transition metals typically have between four and six ligands in a tetrahedral, square planar, or octahedral arrangement around the metal. The valence orbitals in these complexes can be viewed as linear combinations of s and p atomic orbitals of the ligand with one or another of the metal's five d orbitals (e.g., 3d z2 ,3dx2 - y2 ,3dxy ,3dxz or 3dyz for the first row of transition metals) and smaller contributions from the metal's s and p orbitals. Negative electric potentials stemming from the ligands split the d orbitals into subshells with different energies while conserving the overall energy [9, 128, 129], and simple considerations of symmetry can provide a useful qualitative depiction of the relative energies of the subshells (Figs. 4.31 and 4.32). Orbitals in the subshells with lowest energies usually are bonding, while those with the highest energies are antibonding and those with intermediate energies are nonbonding. By Hund's rule, each orbital in a subshell usually is occupied by a single electron before any orbitals are doubly occupied, and all the electrons in singly-occupied orbitals have the same spin. This distribution maximizes favorable electrostatic interactions of the electrons with the positively charged metal nucleus while minimizing unfavorable interactions between electrons. If an additional electron is added after all the d orbitals in a subshell are singly occupied, its disposition depends on the difference between the energies of the subshells. If the subshells are energetically relatively close together, the new electron tends to go to the next subshell, whereas a large gap between the energies favors pairing with an electron of opposite spin in one of the occupied orbitals. Ligands that cause large splitting of the d orbitals thus generally form complexes with low electron spin. These effects are referred to as ligand field effects. Most tetrahedral complexes of transition metals have a relatively small d-orbital splitting, resulting in a high electronic spin, but octahedral and square-planar complexes can be either "high spin" or "low spin" depending on the ligands and the ionization state of the metal. The Fe2+ ion, for

224

4

Electronic Absorption

Fig. 4.31 A depiction of the boundary surfaces of transition-metal d orbitals (orange) relative to the proximate atoms of the ligands (L ) in an octahedral complex. Parts of the wavefunctions with negative sign are shaded with darker color than those with positive sign. The coordinate system is shown at the top left

example, has six d electrons and forms octahedral complexes with all the d electrons in doubly-occupied orbitals (net electronic spin S = 0), and also both octahedral and tetrahedral complexes with four unpaired electrons (S = 2). Fe3+, with five d electrons, forms octahedral complexes with one unpaired electron (S = 1/2) and octahedral and tetrahedral complexes with five unpaired electrons (S = 5/2). Common ligands for transition metals can be ranked approximately as follows in order of increasing d-orbital splittings and increasing propensity for configurations with low electron spin: I‐ < Br‐ < S2‐ < SCN‐ < Cl‐ < NO‐3 < F‐ < OH‐ < H2 O < NCS‐ < CH3 CN < pyridine < NH3 < ethylenediamine < 2,2′-bipyridine < 1,10phenanthrene t t

ð9:8Þ

where m0ba is the result obtained before we change the coordinate system. We showed in Sect. 4.9 that the factor in square brackets is proportional to the electric transition dipole, μba. So shifting the coordinate system will not affect mba if |μba| is zero. This argument applies to either exact or inexact wavefunctions as long as we e rather than the dipole operator e μ. obtain μba with the gradient operator (∇) Values of mba calculated by Eq. (9.7) do depend on the choice of the coordinate system if the electric transition dipole is not zero. But this is no different from the formula for angular momentum in classical physics (r×p), which assumes that the motion is circular and that we know the center of the rotation: the result is physically meaningless if the motion is linear or if we use the wrong center. As we will see later in this chapter, we usually are less concerned with mba itself than with the dot product of mba and μba. This product is independent of the choice of the coordinate e in the same system as long as mba and μba are calculated consistently by using ∇ coordinate system. As an illustration of these points, consider trans-butadiene. The pertinent molecular orbitals are shown in Fig.D 4.19, along E with vector diagrams of the weighted e atomic matrix elements (C bs C at ps j∇jp t ) that combine to make the electric transition dipoles for the first four excitations. Figure 9.4 reproduces the vector diagrams for excitations from the HOMO (ψ 2) to the two lowest unoccupied molecular orbitals (ψ 3 and ψ 4). The figure also shows the position vectors Rmid(s, t) that are needed to DcalculateEmba by Eq. (9.7). For the excitation ψ 2 → ψ 3, combining the e weighted ps j∇jp vectors for atom pairs (s,t) = (2,1), (3,2), and (4,3) gives an t electric transition dipole that points along the principal axis of the molecule (Figs. 4.19E and 9.4A). The magnetic transition dipole calculated by Eq. (9.7) is zero for this excitation if we center the coordinate system at the molecule’s center of symmetry as shown in the figure. The contribution from atom pair (3,2)

452

9

Circular Dichroism

2

A

1

y / ao

3

0 2 4 2

3

-2 2

B

1

y / ao

3

0 2 4 2

-2 -2

4

0 x / ao

2

Fig. 9.4 (A) Vector diagram of contributions to the transition-gradient matrix elements for the lowest-energy excitation of trans-butadiene (Ψ2 → Ψ3). The initial and final molecular wavefunctions in the four-orbital model used for Fig. 4.19 are indicated. The atoms are labeled 1–4 by the empty circles indicating theEpositions. The solid blue arrows indicate the directions and D e for pairs of bonded atoms. (C b and Ca are the coefficients relative magnitudes of C b C a p j∇jp s

t

s

t

s

t

for D atomic E 2pz orbitals of atoms s and t in the final and initial wavefunction, respectively, and e ps j∇jp t is the matrix element of the gradient operator for the two atomic orbitals.) The vectors (Rmid(s, t)) from the origin of the coordinate system to points midway between the pairs of bonded atoms are indicated with dashed green arrows. The transition gradient matrix element for the excitation is given by the vector sum of the solid blue arrows. (Including contributions from the pairs of nonbonded atoms would not change the results.) The D magnetic E transition dipole (mba) is e . (B) Same as (A) but for obtained by summing the cross products Rmidðs,tÞ × Cb C a p j∇jp s

excitation from Ψ2 to Ψ4

t

s

t

9.1 Magnetic Transition Dipoles and n – p
Transitions



453

D E e Rmidð3,2Þ C b3 C a2 p3 j∇jp is zero because the position vector Rmid(3, 2) is zero in 2

this coordinateDsystem; the from E contributions D E atom pairs (2,1) and (4,3) sum to zero b a b a e e because C 2 C 1 p2 j∇jp1 = C 4 C 3 p4 j∇jp3 while Rmid(2, 1) = Rmid(4, 3). The lack of a magnetic transition dipole for this excitation is not surprising because the electron motions depicted by the vectors do not constitute a rotation around the molecular center of symmetry, or indeed around any point at a finite distance. Equation (9.7) would, however, give a non-zero result with no physical significance if we shifted the coordinate system off the center of symmetry. dipole The situation is E D reversedE for ψ 2 → ψ 4. Here, theDelectric transition E D is zero e e e because C b C a p j∇jp =0 and C b C a p j∇jp = -Cb Ca p j∇jp 3

2

3

2

2

1

2

1

4

3

4

3

(Figs. 4.19F and 9.4B), whereas the magnetic transition dipole is non-zero. Inspection of Fig. 9.4B shows that, although the electron motions represented by the vectors do not complete a circle, they occur in opposite directions on either D E D side of E b a b a e e the center of symmetry. The opposite signs of C 2 C 1 p2 j∇jp1 and C 4 C3 p4 j∇jp 3 D E e rectify the inversion of Rmid, making the sum of Rmidðs, tÞ × C bs C at ps j∇jp t over the atoms non-zero. As we would expect, shifting the origin of the coordinate system does not affect the calculated value of mba for this excitation. Transitions involving magnetic transition dipoles also can be treated by the quantum-mechanical approach we used in Chap. 5 to account for fluorescence. Remarkably, although the final results are the same and the problem of choosing the coordinate system remains, the quantum theory does not require distinguishing between the magnetic and electric fields of the radiation. Instead, as explained in Box 9.1, “magnetic dipole-allowed” transitions are related to the change in the amplitude of the vector potential with position across the chromophore. This approach provides a different perspective on the question of why magnetic-dipole transitions typically are much weaker than electric-dipole transitions. If the wavelength of the radiation is large compared to the size of the chromophore, as is usually the case in electronic spectroscopy, the variation in the vector potential across the chromophore will be relatively insignificant. Box 9.1 Quantum Theory of Magnetic-Dipole and Electric-Quadrupole Transitions To see how the quantum theory accounts for interactions of a chromophore with the magnetic field of light, we return to the expressions we used in Chap. 5 for the energy of interaction of an electron with a linearly polarized radiation field (Eqs. (5.32a, b)): (continued)

454

9

Box 9.1 (continued)

=-

Circular Dichroism

2 e0 = - ħe V  ∇ e þ e jVj2 H ime c 2me c2

ðB9:1:1aÞ

 

   eħπ 1=2 X e e ^ej  ∇ qj exp 2πikj  r þ e qj
exp -2πikj  r ime j

þ ⋯:

ðB9:1:1bÞ

Here V is the vector potential of the radiation field; ^ej and kj are, respectively, the polarization axis and wavevector of radiation mode j; and me is the electron mass. The ellipsis in Eq. (B9.1.1b) represents two-photon processes that we continue to defer to Chap. 12. In Sect. 5.5, our next step was to assume that the wavelength of the radiation was long enough so that the factors exp(±2πikj  r) could be set equal to 1 (Eq. (5.33)). Here we include the second-order term in a power series expansion of the exponential,  2 1 exp ± 2πikj  r = 1 ± 2πikj  r þ 2πikj  r þ ⋯: 2

ðB9:1:2Þ

For 300-nm light and a chromophore with dimensions in the range of 1–3 nm, the product kj  r in the second-order term is on the order of 0.01, which is small enough to neglect in most cases. But suppose that the electric transition dipole μba for a transition of an electron between wavefunctions ψ a and ψ b is zero by symmetry. The first term on the right in Eq. (B9.1.2) then does not contribute to the overall matrix element for the transition, so we must consider the second term (±2πikj  r). Using this term with Eq. (B9.1.1b) gives  E D E D   2πeħπ 1=2  e0 jψ χ e ψ ψ b  kj  r ^ej  ∇ ψ b χ jðmÞ jH a jðnÞ = a me D E e jðnÞ ,  χ jðmÞ jQjχ

ðB9:1:3Þ

e is the photon position operator and χ j(n) denotes the photon where Q wavefunction D Efor the nth excitation level of mode j. We analyzed the factor e χ jðmÞ jQj jχ jðnÞ in Sect. 5.5, and found that it includes separate terms for absorption (m = n -1)  and D  E emission (m = n + 1). Our interest now is the factor   e ψ . ψ b  kj  r ^ej  ∇ a Consider a radiation mode that propagates along the y-axis with wavelength λba and is polarized in the z-direction. The dot product kjr then reduces (continued)

9.1 Magnetic Transition Dipoles and n – p
Transitions

455

Box 9.1 (continued) e reduces to ∂/∂z, and the matrix element for to the y component of r; ^ej  ∇ excitation of ψ a to ψ b becomes -

 E D  3=2  1 2eħπ 3=2  e ψ = - 2eħπ ψ b  kj  r ^ej  ∇ a me me λba   E D   e ψ  ψ b ð^y  rÞ ^z  ∇ a =-

ðB9:1:4Þ

     ∂ 2eħπ 3=2 ψ b y ψ a me λba ∂z

2eħπ 3=2 1 =me λba 2                    ∂  ∂  ∂  ∂        þ ψ b y ψ a þ ψ b z ψ a :  ψ b y ψ a - ψ b z ψ a ∂z ∂y ∂z ∂y ðB9:1:5Þ The quantity in the first set of parentheses on the right side of this expression is, except for multiplicative constants, the x component of the magnetic transition dipole: -

2eħπ 3=2 1 me λba 2

=-



       D E  ∂  ∂ eħπ 3=2 e =ψ b jr × ∇jψ ψ b y ψ a - ψ b z ψ a a me λba ∂z ∂y

  eħπ 3=2 2me c 2π 3=2 c ðmba Þx = ðmba Þx = -i2π 3=2 νba ðmba Þx : ðB9:1:6Þ iλba me λba -iħe

Now, look at the quantity in the second set of square brackets on the right side of Eq. (B9.1.5). By using the relationship between the matrix elements of the gradient operator and the commutator of the Hamiltonian and dipole operators (Box 4.9), we can relate this term to the matrix element of the product yz [7, 8]. This gives -

2eħπ 3=2 1 me λba 2



          ∂  ∂ 2eħπ 3=2 1 2πme νba = ψ b y ψ a þ ψ b z ψ a me λba 2 ħ ∂z ∂y

 hψ b jyzjψ a i (continued)

456

9

Box 9.1 (continued) =

Circular Dichroism

2π 5=2 vba ehψ b jyzjψ a i: λba

ðB9:1:7Þ

Further, the factor ehψ b| yz| ψ ai in Eq. (B9.1.7) is recognizable as the yz element of the electric quadrupole interaction matrix (Eqs. (4.5) and (B4.2.4)). A similar analysis for radiation propagating along the x or z axis instead of y, or polarized along y or x instead of z, gives corresponding results with the y or z component of mba replacing the x component in Eq. (B9.1.6), and/or with different components of the quadrupole matrix replacing the yz element in Eq. (B9.1.7) [7, 8]. By including the second-order term in the dependence of the vector potential V on position we thus obtain a transition matrix element that commonly is ascribed to a magnetic transition dipole (Eq. (B9.1.6)), along with a matrix element representing the quadrupolar distribution of the wavefunction (Eq. (B9.1.7)). Including the third-order term in the distance dependence would add a matrix element for octupolar interactions. Since the magnetic-dipole and electric-quadrupole matrix elements both arise from the same term in the distance dependence, they should have comparable magnitudes and generally should be much smaller than electric-dipole matrix elements. As pointed out above, the quantum theory of absorption differs from the semiclassical theory in that it does not distinguish explicitly between the electric and magnetic fields of electromagnetic radiation. Although the vector potential was obtained originally as a solution to Maxwell’s equations, which generalize a large body of experimental observations on electric and magnetic effects, the distinction between electric and magnetic interactions no longer seems as clear as it did in classical physics.

9.2

The Origin of Circular Dichroism

The difference between the dipole strengths for left- and right circularly polarized light is characterized experimentally by the rotational strength ℜ of an absorption band: ℜ=

3000 ln ð10Þhc 32π 3 N A

Z ΔεðνÞ n dν ν f2

Z  ΔεðvÞ n dvðdebye Bohr magnetonÞ= M -1 cm-1 , ≈ 0:248 2 v f

ð9:9aÞ ð9:9bÞ

9.2 The Origin of Circular Dichroism

457

Fig. 9.5 A linearly-polarized beam of light passing through an optically active material becomes elliptically polarized. As in Figs. 3.8 and 3.9, the straight arrows in these diagrams represent the electric fields of light propagating away from the observer; the numbers indicate equal increasing intervals of time. (A) The electric field of linearly polarized light (double-headed solid cyan arrow) can be viewed as the resultant of equal fields from left- (dotted green arrows) and right- (dashed blue arrows) circularly polarized light. The circles indicate the bounds of the rotating fields and the curved dotted green and dashed blue arrows indicate the directions of rotation. (B) After passage through an optically active material, light with one of the two circular polarizations (here, the rightcircular polarization) is attenuated relative to that with the other. The resultant of the two fields now sweeps out an ellipse. (C) The ellipticity (θ) is defined as the arctangent of the ratio of the minor to the major half-axes of the ellipse. This figure greatly exaggerates the minor axis of the ellipse relative to the major. It neglects the rotation of the ellipse (optical rotation) caused by the difference between the refractive indices for left- and right circularly polarized light

where NA is Avogadro’s number, Δε is the difference between the molar extinction coefficients for left- and right-circularly polarized light (εl - εr ) in units of M-1 cm-1, ν is the frequency, n is the refractive index and f is the local-field correction. Rotational strengths commonly are expressed in units of debyeBohr magnetons (9.274 × 10-39esu2  cm2 or 9.274 × 10-39erg  cm3). Unlike the dipole strength, ℜ can be either positive or negative. For historical reasons, circular dichroism often is described in terms of the molar ellipticity, [θ]M, in the arcane units of 100×degreesM-1 cm-1. The angular units reflect the fact that plane-polarized light passing through an optically active sample emerges with elliptical polarization (Box 9.2 and Fig. 9.5). Ellipticity is defined as the arc tangent of the ratio dmin/dmax, where dmin and dmax are the short and long axes of the ellipse. The relationship between [θ]M and εl ‐ εr is ½θM =

100 lnð10Þ180∘ Δε = 3298Δε: 4π

ð9:10Þ

Early CD spectrometers actually measured ellipticity, but most modern instruments measure Δε more directly and sensitively by using an electro-optic modulator to switch a light beam rapidly between right- and left-circular polarization (Chaps. 1, 3, and 4). In studies of proteins or polynucleotides, the molar ellipticity usually is

458

9

Circular Dichroism

divided by the number of amino acid residues or nucleotide bases in the macromolecule to obtain the mean residue ellipticity. Circular dichroism (CD) is a very small effect, typically amounting to a difference of only about 1 part in 103 or 104 between the extinction coefficients for light with left- or right-circular polarization. But despite its small magnitude, CD proves to be a very sensitive probe of molecular structure. Box 9.2 Ellipticity and Optical Rotation As we discussed in Chap. 3, the electric and magnetic fields of linearly polarized light can be viewed as superpositions of fields from right- and leftcircularly polarized light (Figs. 3.6, 3.7, and 9.5A). Consider a beam of linearly polarized light after it has passed through 1 cm of a 1 M solution of an optically active material. If the molar absorption coefficient for left-circular polarization exceeds that for right-circular polarization by Δε, the ratio of the intensities of the transmitted light with left- and right-circular polarization will be Il/Ir = exp [-ln (10)Δε]. Since the electric field amplitude is proportional to the square root of the intensity, the ratio of the fields will be jEl j=jEr j = exp½- ln ð10ÞΔε=2:

ðB9:2:1Þ

The resultant field will oscillate in amplitude from |Emax| = |Er| + |El| when the two fields are aligned in parallel (the times labeled 0 and 4 in Fig. 9.5B) to | Emin| = |Er| ‐ |El| when they are antiparallel (time 2). The minor and major halfaxes of the ellipse swept out by the resultant field (Fig. 9.5B) are |Emin| and | Emax|, which have the ratio 1- exp½- ln ð10ÞΔε=2 ln ð10ÞΔε=2 jE min j = ≈ 2 jE max j 1 þ exp½- ln ð10ÞΔε=2 = ln ð10ÞΔε=4

ðB9:2:2Þ

when Δε > 1/ω). With this substitution, Eq. (10.39) gives ∂ρab =∂t = expðiω t Þ∂ρab =∂t þ iω  expðiω t Þρab = ði=ħÞðρbb - ρaa Þμab  E o exp ðiωt Þ þ ðiωba - 1=T 2 Þρab exp ðiωt Þ:

ð10:41Þ

Equation (10.41) can be solved immediately for ρab in a steady state, when ∂ρab =∂t = 0 :

10.5

A Density-Matrix Treatment of Absorption of Weak, Continuous Light

ρab =

ði=ħÞ ðρbb - ρaa Þ μab  Eo : iðω - ωba Þ þ 1=T 2

503

ð10:42aÞ

Similarly, setting ρba = ρba exp(‑iωt) gives ρba =

ði=ħÞ ðρbb - ρaa Þ μba  Eo  : iðω - ωba Þ - 1=T 2

ð10:42bÞ

Here we retained only the exp(-iωt) component of E, making Vba = - μba  Eo*exp (-iωt). This is consistent with the hermiticity of the Hamiltonian operator: if Vab = - μab  Eoexp(iωt)], then Vba = Vab = - μba  Eo*exp(-iωt). To find the steady-state rate of excitation of molecules from state a to state b in the absence of stochastic decay processes, we now can use Eq. (10.21c): ∂ρbb =∂t = ði=ħÞðρba V ab - ρab V ba Þ 

  ≈ ði=ħ

Þ expð- iω t Þρba ½ - μab  Eo expð iω t Þ - expð iω t Þρab ½ - μba  Eo expð- iω t Þg

jμ  E j2 1 1 = ðρbb - ρaa Þ ba 2 o iðωba - ωÞ þ 1=T 2 iðωba - ωÞ - 1=T 2 ħ ! jμba  Eo j2 2=T 2 = ðρaa - ρbb Þ : ð10:43Þ ħ2 ðωba - ωÞ2 - ð1=T 2 Þ2 Equation (10.43) is just the same as Eq. (10.35) for the particular case of absorption of light. It predicts again that the absorption spectrum will be a Lorentzian function of frequency. The width of the Lorentzian corresponds to the homogeneous distribution of transition energies when the effective lifetime of the excited state is T2/2, as we discussed in Sect. 10.4. However, this result depends on our assumption that the dephasing of the ensemble is described adequately by an exponential decay with a single time constant, T2. As we will discuss in Sect. 10.7, the absorption band shape depends on a Fourier transform of the dephasing dynamics, which usually is more complex than we have assumed here. Note also that Eq. (10.43) pertains only to times greater than T2. We’ll consider shorter times in the following chapter. The total steady-state rate of excitation also can be written as ∂ρbb =∂t = - ∂ρaa =∂t = ð2π=ħÞ ðρaa - ρbb Þ jμba  Eo j2 ρs ðEba Þ,

ð10:44Þ

where ρs(Eba)dE is the number of states for which the excitation energy is within dE of Eba. This again is equivalent to Fermi’s golden rule (Eqs. B5.3.3, 7.10, and 10.36). As in Eq. (10.36), the density of states ρs here has units of reciprocal energy (e.g., states per cm-1). Equations (4.8a, b) have an additional factor of h in the denominator because ρv, the corresponding density of oscillation modes in units of reciprocal frequency (modes per Hz‐1), is hρs.

504

10 Coherence and Dephasing

If the molecules in the ensemble have different orientations relative to the excitation, then Eqs. (10.43) and (10.44) require additional averaging over the orientations. For an isotropic sample, the average value of |μ21  Eo|2 is (1/3) |μ21|2| Eo|2 (Eq. 4.11 and Boxes 4.6 and 10.5). Equations (10.43) and (10.44) also describe the rate of stimulated emission. The corresponding rates for the situation that Hbb < Haa are obtained in the same manner by retaining the term exp(-iωt) instead of exp(iωt) in Vab and vice versa for Vba.

10.6

The Relaxation Matrix

Our discussion so far has focused on systems with only two quantum states, and our treatment of the time constants T1 and T2 has been entirely phenomenological. We now discuss the elements of the relaxation matrix R in a more general way and consider how they depend on the strengths and dynamics of interactions with the surroundings. Following work by Pound, Bloch, and others, Alfred Redfield [19] explored descriptions of relaxations of the density matrix by the stochastic Liouville equation (Eqs. 10.24 and 10.28–10.31). Redfield’s treatment showed clearly how the main elements of the relaxation matrix R depend on the frequencies and strengths of fluctuating interactions with the surroundings. His basic approach was to write the Hamiltonian matrix for the system as H(t) = H0 + V(t), where H0 is independent of time and V(t) represents interactions with fluctuating electric or magnetic fields from the surroundings. Redfield then used the von Neumann-Liouville equation (Eq. 10.24) to find the effects of V(t) on ρ for the system. In the following outline of the derivation, we use the Schrödinger representation of the density matrix for simplicity and consistency with Sects. 10.2 and 10.3. Redfield [19] and Slichter [20], who provided an excellent introduction to the theory, used the interaction representation and then returned to the Schrödinger picture for the final expressions. If we know the averaged reduced density matrix for the ensemble at zero time (ρ(0)), we can obtain an estimate of ρ for a later time (t) by integrating the von Neumann-Liouville equation (Eq. 10.24), using ρ(0) in the commutator: ρð t Þ = ρð 0 Þ þ

i ħ

Zt h

i ρð0Þ, Hðt 1 Þ dt 1 :

ð10:45Þ

0

For a better estimate, we could use Eq. (10.45) to find ρ at an intermediate time (t2) and then use ρðt 2 Þ instead of ρ(0) in the commutator: 3 20 1 Zt Zt2 i 6@ i 7 ½ρð0Þ, Hðt 1 Þ dt 1 A, Hðt 2 Þ5 dt 2 : ð10:46Þ ρð t Þ = 4 ρð 0 Þ þ ħ ħ 0

0

Differentiating this expression gives ∂ρ=∂t at time t:

10.6

The Relaxation Matrix

505

h i  2 ∂ρðt Þ i i = ρð0Þ, Hðt Þ þ ħ ħ ∂t

Zt h

i ½ρð0Þ, Hðt 1 Þ, Hðt Þ dt 1 ,

ð10:47Þ

0

where, as indicated above, H(t) = H0 + V(t). According to the ergodic hypothesis of statistical mechanics, the ensemble averages indicated by bars in Eq. (10.47) are equivalent to averages for a single system over a long period. If the fluctuating interactions with the surroundings (V(t)) vary randomly between positive and negative values, the matrix elements of V will average to zero: D E e ðt Þjψ m = 0: ð10:48Þ V nm ðt Þ = ψ n jV (Any constant interactions with the surroundings could be included in H0 and used simply to redefine the basis states.) [ρð0Þ,Vðt Þ] thus is zero and does not contribute to ∂ρðt Þ=∂t: The average value of (Vnm(t))2 or Vnm(t)Vjk(t), however, generally is not zero, and these factors contribute to ∂ρðt Þ=∂t through the integral on the right side of Eq. (10.47). The integrand in Eq. (10.47) includes a sum of terms of the form ρð0ÞV nm ðt 1 ÞV jk ðt 2 Þ , each of which involves the product of the density matrix at zero time with a correlation function or memory function, M nm,jk ðt 1 , t 2 Þ = V nm ðt 1 ÞV jk ðt 2 Þ. Mnm,jk(t1, t2) is the ensemble average of the product of Vnm at time t1 with Vjk at time t2. If Vnm and Vjk vary randomly, their average product should not depend on the particular times t1 and t2, but only on the difference, t = t2 - t1. The correlation function Mnm,jk therefore can be written as a function of this single variable: M nm,jk ðt Þ = V nm ðt 1 ÞV jk ðt 1 þ t Þ,

ð10:49Þ

where the bar implies an average over time t1 as well as an average over the ensemble. For jk = nm, Mnm,jk(t) is the same as the autocorrelation function of Vnm (Eq. 5.79). Panel B of Fig. 10.7 shows the autocorrelation function of the fluctuating quantity shown in panel A. The initial value of a correlation function is the mean product of the fluctuating quantities, M nm,jk ð0Þ = V nm V jk , which for an autocorrelation function is simply the mean-square amplitude of the fluctuations, M nm,nm ð0Þ = jV nm j2. At long times, Mnm, jk approaches the product of the two means, M nm,jk ð1Þ = V nm V jk , which is zero (Eq. 10.48). In the simplest model, Mnm,jk(t) decays exponentially to zero with a single time constant τc called the correlation time: Mnm, jk(t) = Mnm,jk(0) exp(-t/τc). The correlation time for more complex decays can be defined by the expression τc =

1

Z1 M nm,jk ðt Þ dt,

M nm,jk ð0Þ 0

ð10:50Þ

506

10 Coherence and Dephasing

A

V / cm

M / cm

-1

-2

B

C

J / ps cm

JSC / ps cm

-2

-2

D

-1

Frequency / ps

-1

Frequency / ps

Fig. 10.7 (A) A fluctuating interaction energy, V(t), with a mean value (V ) of zero and a mean square (M(0) = V 2 ) of 1.0 cm-2 (root mean square 1.0 cm-1). (B) The autocorrelation function, M(t), of V(t). (C) The spectral density function, J(ν), of V(t) as defined by Eq. (10.52). (Some authors use “spectral density function” to refer the product νJ(ν) or ωJ(ω).) (D) A semiclassical spectral density function JSC(v) = J(v)/[1 + exp(-hv/kBT )] for T = 295 K (kBT = 205 cm-1). Note the slight asymmetry of JSC(v) relative to J(ν)

which reduces to the exponential time constant if the decay is monoexponential. Redfield’s analysis does not make any assumptions about the detailed time course of the decay, but it does assume that the correlation functions decay rapidly relative to the phenomenological relaxation time constants T1 and T2 that we introduced in Eqs. (10.28) and (10.29) (i.e., that τc 1/ω21, but decreases the rate constant if τc < 1/ω21. At low frequencies (ω 0) causes the magnitude of the oscillations to decay to zero as exp(-σ 2t2/2ħ2), which gives a Gaussian dependence on time. jρjk ðt Þj falls to half its initial value in a time τ1/2 = (2ln2)1/2ħ/σ = 1.177ħ/σ. A standard deviation (σ) of 1 cm‐1 gives τ1/2 = 1.177 × 5.31 × 10-12(cm-1s)/(cm-1) = 6.2 ps. Figure 10.9 shows plots of ρjk ðt Þ=ρjk ð0Þ for Gaussian distributions of Ejk with several values of σ. Gaussian decays of coherence also can be seen in Fig. 10.2, but the models considered there include static heterogeneity in the interaction matrix element that couples the two quantum states (H21) in addition to the energy difference between the states.

10.7

More General Relaxation Functions and Spectral Lineshapes

We have seen that if the interactions of a quantum system with its surroundings fluctuate rapidly, the decay of coherence in an ensemble of such systems can be described by a relaxation matrix of microscopic first-order rate constants. But if the energies of the basis states vary statically from system to system, coherence decays with a Gaussian dependence on time (Box 10.4 and Figs. 10.2 and 10.9). We now seek a more general expression for pure dephasing to connect the domains in which the fluctuations of the surroundings are either very slow or very fast. To start, consider an off-diagonal density matrix element ρnm of an individual system when the energies of states n and m (Hnn and Hmm) vary with time (Eq. 10.15). Assume that n and m are stationary states so that the factors Cn and Cm are independent of time. The initial value of ρnm then is ρnm ð0Þ = Cn Cm . ρnm now begins to oscillate with an angular frequency ωnm(0) = (Hnn - Hmm)/ħ, which

514

10 Coherence and Dephasing

Fig. 10.9 The time dependence of ρjk (t) in ensembles in which the energy difference between states j and k (Ejk) has a Gaussian distribution with a mean of Eo and a standard deviation (σ) of 0.01Eo, 0.1Eo, or Eo. The oscillation period of ρjk is h/Eo

1.0

ρjk(t) / ρjk(0)

σ = 0.01Eo

σ = 0.1Eo

0.5 σ = Eo

0

1000 500 Time x Eo/h

0

1500

we take to be constant for a short interval of time Δt. By the end of this interval, ρnm has become ρnm ðt = Δt Þ = ρnm ð0Þ expf- i ½ωnm ð0ÞΔt g:

ð10:61Þ

Suppose that after time Δt, the oscillation frequency changes to some new value, ωnm(1), which then persists for the same increment of time. At the end of this second interval, ρnm is ρnm ðt = 2Δt Þ = ρnm ð0Þ expf- i ½ωnm ð0ÞΔt þ ωnm ð1ÞΔt g,

ð10:62Þ

and in general, 0 ρnm ðt Þ = ρnm ð0Þ exp@- i

Zt

1 ωnm ðτÞ dτA:

ð10:63Þ

0

In an ensemble of such systems, the detailed time dependence of ωnm will vary randomly from one system to the next. Let the average frequency for the ensemble be ωnm , and call the variable part for our particular system wnm, so that ωnm ðτÞ = ωnm þ wnm ðτÞ. Then, for an individual system, 0 1 Zt ð10:64aÞ ρnm ðt Þρnm  ð0Þ = ρnm ð0Þρnm  ð0Þ exp@- i ½ωnm þ wnm ðτÞ dτA 0

0 = jρnm ð0Þj exp@- i ωnm t - i

Zt

2

0

1 wnm ðτÞ dτA:

ð10:64bÞ

10.7

More General Relaxation Functions and Spectral Lineshapes

515

Assuming that |ρnm(0)|2 is not correlated with the fluctuations of ωnm, we can obtain a correlation function for the ensemble-averaged density matrix element by averaging the integral in Eq. (10.64b): ρnm ðt Þ ρnm  ð0Þ jρnm ð0Þj2

= exp½ - i ωnm t - gnm ðt Þ = expð- i ωnm t Þ ϕnm ðt Þ,

ð10:65Þ

where Zt gnm ðt Þ = i

wnm ðτÞ dτ:

ð10:66aÞ

ϕnm ðt Þ = exp½- gnm ðt Þ,

ð10:66bÞ

0

and

The dephasing of the ensemble thus is contained in the relaxation function ϕnm(t) = exp(-gnm(t)), with gnm(t) (sometimes called the lineshape or line-broadening function) defined in Eq. (10.66a). Ryogo Kubo [2, 3] showed that if wnm has a Gaussian distribution about zero, gnm(t) is given by Zt gnm ðt Þ =

Zτ1 wnm ðτ1 Þwnm ðτ2 Þ dτ2

dτ1 0

0

Zt = σ nm 2

M nm ðτ2 Þ dτ2 :

dτ1 0

ð10:67Þ

Zτ1 0

Here σ nm2 is the mean square of the frequency fluctuations (σ nm 2 = jw2nm j in units of radian2s-2) and Mnm(t) is a normalized autocorrelation function of the fluctuations: M nm ðt Þ =

1 w ðτ þ t Þwnm ðτÞ: σ nm 2 nm

ð10:68Þ

If Mnm(t) decays exponentially with time constant τc, evaluating the integrals in Eq. (10.67) is straightforward and yields  ϕnm ðt Þ = expð- gnm ðt ÞÞ = exp - σ 2nm τc 2 ½ðt=τc Þ - 1 þ expð- t=τc Þ : ð10:69Þ Figure 10.10 shows the behavior of this expression for several values of τc and σ nm. At short times (t > τc, Eq. (10.69) goes to ϕnm ðt Þ = exp - σ 2nm τc t . ρnm then

516

10 Coherence and Dephasing

A 1.0

B σ =

τc =

0.8

0.2

0.05

0.6 0.4

0.2

0.2

0.5

0.5 100

0

0.3

0.1

φ(t)

0

1

2

5

10

15

Time / arbitrary units

0

1

5

10

15

Time / arbitrary units

Fig. 10.10 The Kubo relaxation function ϕ(t) as given by Eq. (10.69) for an ensemble in which the correlation function decays exponentially with time constant τc. In (A) τc (indicated in arbitrary time units) is varied, while the RMS amplitude of the fluctuations (σ) is fixed at 1 reciprocal time unit. In (B) τc is fixed at 1 time unit and σ is varied

decays exponentially with a rate constant (1/T2) of σ nm2τc, in accord with the Redfield theory and Eqs. (10.29a, 10.29b). Kubo’s function thus captures both the exponential dephasing that results from fast fluctuations and the Gaussian dephasing associated with slow fluctuations. It shows that the operational terms “fast” and “slow” relate to the ratio of the observation time (t) to τc, and it reproduces the reciprocal relationship between τc and the rate of pure dephasing that we discussed in the previous section. If the off-diagonal matrix elements that describe the coherence between a ground state and an excited electronic state decay exponentially with time, the homogeneous absorption line should have a Lorentzian shape (Figs. 10.7 and 10.8 and Eqs. (2.70), (10.35)). More generally, as we discussed in Sect. 10.6, the spectral lineshape is the Fourier transform of the relaxation function: εðω - ω o Þ → / j μ ab j2 ω

Z1

eiðω - ωo Þt ϕðt Þdt,

ð10:70Þ

-1

where ωo is the angular frequency corresponding to the 0–0 transition and we have omitted the subscript nm to generalize the relationship. Figure 10.11 shows spectra calculated by using this expression with the relaxation function ϕ(t) from Eq. (10.69) and several values of the underlying correlation time (τc) and variance (σ) of the fluctuations. As we saw in Fig. 10.10, ϕ(t) can vary from Gaussian to exponential depending on t/τc. Because the Fourier transform of a Gaussian is another Gaussian, whereas the Fourier transform of an exponential is Lorentzian, the absorption lineshape can range from Gaussian to Lorentzian. A long correlation time gives a Gaussian absorption band, as we would expect for a spectrum that is

More General Relaxation Functions and Spectral Lineshapes

Fourier Transform (arbitrary units)

10.7

A

517

B

1.0

τc =

0.5

σ=

2 5

2 5

0

0.2 0.4 0 -0.4 -0.2 (ω − ωο ) / reciprocal time units

2 1 0 -1 -2 (ω − ωο ) / reciprocal time units

Fig. 10.11 Absorption spectral lineshapes calculated as the Fourier transform of the Kubo relaxation function ϕ(t). (A) τc (indicated in arbitrary time units) is varied, while τc is fixed at 1 reciprocal time unit; (B) τc is fixed at 1 time unit and σ is varied as indicated. To use the full Fourier transform (Eq. 10.70), ϕ(t) is treated as an even function of time (Fig. 10.12A)

inhomogeneously broadened; a short correlation time gives a Lorentzian band corresponding to the homogeneous lineshape. Equation (10.70) is a manifestation of a general principle called the FluctuationDissipation theorem, which describes how the response of a system to a small perturbation such as an electric field is related to the fluctuations of the system at thermal equilibrium. Derivations and additional discussion can be found in [25–31]. Although Kubo’s relaxation function can describe dephasing on time scales that are either shorter or longer than the energy correlation time, it rests on a particular model of the fluctuations and it still assumes that the correlation function (M(t)) decays exponentially with time. Sinusoidal components can be added to M(t) in Eq. (10.68) to represent coupling to particular vibrational modes of the molecule [29, 32, 33]. But more importantly, Eq. (10.68) also assumes that the mean energy difference between states n and m ðħ ωnm Þ is independent of time and is the same whether the system is in state n or state m. These assumptions are not very realistic because relaxations of the solvent around an excited molecule cause the emission to shift with time. This brings us back to the distinction between classical and semiclassical spectral density functions, which we discussed briefly in Sect. 10.6 in connection with the Redfield theory. We saw there that relaxations of a system to thermal equilibrium with the surroundings require a spectral density function that is not a purely even function of frequency (Eq. 10.54), or equivalently, a relaxation function that includes an imaginary component. Noting that the relaxation and lineshape functions should, in principle, be complex, Shaul Mukamel [23, 29, 34] obtained the following more general expression for g(t):

518

10 Coherence and Dephasing

B

Fourier Transform (arb. units)

A 1.0

0.5

φ(t) 0

-0.5

-0.2

0

0.2

time / arbitrary units

1.0

σ= 0.5 5 10

0

-4

-2

0

2

4

(ω − ωο ) / reciprocal time units

Fig. 10.12 (A) The real (solid line) and imaginary (dashed line) parts of the complex relaxation function ϕ(t) given by Eqs. (10.73) and (10.74) with τc = 1 arbitrary time unit and σ = 10 reciprocal time units. The dotted line shows the Kubo relaxation function (Eq. 10.69), treated as an even function of time (ϕ(-t) = ϕ(t)). (B) Absorption (solid lines) and emission (dashed lines) spectra calculated as the Fourier transforms of, respectively, the complex relaxation function ϕ(t) (Eqs. 10.72 and 10.73) and its complex conjugate ϕ*(t). The autocorrelation time constant τc was 1 arbitrary time unit, and σ was 2, 5, or 10 reciprocal time units, as indicated

Zt gð t Þ = σ

2

Zτ1 M ðτ2 Þdτ2 þ iΛs

dτ1 0

Zt

0

M ðτ1 Þdτ1 :

ð10:71Þ

0

Here as in Eq. (10.68), M(t) is the classical correlation function of the fluctuations, which is assumed to be the same in the ground and excited states. The scale factor Λs for the imaginary term is the solvent reorganization energy in units of angular frequency (Fig. 4.28). If M(t) decays exponentially with time constant τc, then evaluating the integrals in Eq. (10.71) and defining Γ = 1 - exp(-t/τc) yields gðt Þ = σ 2 τc 2 ½ðt=τc Þ - ð1 - expð- t=τc ÞÞ þ iΛs τc ð1 - expð- t=τc ÞÞ,

ð10:72Þ

and  ϕðt Þ = exp - σ 2 τc 2 ½ðt=τc Þ - Γ - iΛs τc Γ  = exp - σ 2 τc 2 ½ðt=τc Þ - Γ ½ cosðΛs τc ΓÞ - i sinðΛs τc ΓÞ:

ð10:73Þ

Fourier transforms of ϕ(t) and ϕ*(t) give the absorption and emission spectra, respectively [23, 29, 34, 35]. Figure 10.12A shows the real and imaginary parts of the relaxation function ϕ given by Eqs. (10.73) and (10.74a, b) when τcσ = 10. Kubo’s relaxation function (Eq. 10.69) is shown for comparison. Figure 10.12B shows the calculated absorption and emission spectra for τcσ = 2, 5, and 10. Including the term involving Λs in the relaxation function shifts the absorption to higher energies and the fluorescence to

10.8

Anomalous Fluorescence Anisotropy

519

lower energies. In agreement with the relationship we discussed in Chap. 5, the Stokes shift is 2Λs. Increasing σ increases the Stokes shift as it broadens the spectra. Relaxations of solvent-chromophore interactions can be studied experimentally by hole-burning spectroscopy, time-resolved pump-probe measurements, and photon-echo techniques that we discuss in the next chapter. If the temperature is low enough to freeze out pure dephasing, and a spectrally narrow laser is used to burn a hole in the absorption spectrum (Sect. 4.12), the zero-phonon hole should have the Lorentzian lineshape determined by the homogeneous lifetime of the excited state. The hole width increases with increasing temperature as the pure dephasing associated with σ 2 comes into play [36, 37]. The parameters σ 2 and τc also can be obtained by classical molecular-dynamics simulations in which one samples fluctuations of the energy difference between the ground and excited states during trajectories on the potential surfaces of ground and excited states. The variance and autocorrelation function of the fluctuations in the ground state provide σ 2 and τc for the absorption spectrum, and those in the excited state give the corresponding parameters for fluorescence. As we saw in Fig. 6.3, a Fourier transform of the autocorrelation function of the energy gap also can be used to identify the frequencies and dimensionless displacements of vibrational modes that are strongly coupled to an electronic transition [38, 39]. Further discussion of relaxation functions can be found in Chap. 11 and [27, 29, 30, 32–35, 40, 41].

10.8

Anomalous Fluorescence Anisotropy

In this section, we discuss a manifestation of electronic coherence in fluorescence spectroscopy: fluorescence anisotropy exceeding the classical maximum of 0.4. Such anisotropy has been seen at short times after excitation with femtosecond laser pulses that cover a sufficiently broad band of wavelengths to excite multiple optical transitions coherently. To examine the effect of coherence on fluorescence anisotropy, consider a system with a ground state (state 1) and two electronically excited states (2 and 3), and assume as before that the interaction matrix elements H12, H13, and H23 are zero in the absence of radiation. Suppose that an ensemble of systems in the ground state is excited with a weak pulse of light that is much shorter than the lifetime of the excited state (T1). In the impulsive limit, the population of excited state 2 generated by the pulse is μ12  ^ei j2 K 22 , ρ22 ð0Þ ≈ j^

ð10:74aÞ

where Z0 K 22 = Θ

Z1 dt

-1

0

ε21 ðνÞν - 1 I ðν, t Þdν:

ð10:74bÞ

520

10 Coherence and Dephasing

Here ^ei and μ ^21 are unit vectors defining the polarization of the excitation light and the orientation of the transition dipole μ12, respectively; Θ = (3000ln10)/hNavo; ε21(v) is the molar extinction coefficient for excitation to state 2; and I(ν,t) describes the dependence of the excitation flash intensity on frequency and time. The bars indicate averaging over the ensemble as usual, but we assume here that |μ12| is the same for all the molecules, so the averaging procedure needs to consider only the distribution of orientations relative to the polarization of the excitation light. To the same approximation, μ13  ^ei j2 K 33 , ρ33 ð0Þ ≈ j^

ð10:75aÞ

where Z0 K 33 = Θ

Z1 dt

-1

ε31 ðνÞν - 1 I ðν, t Þdν:

ð10:75bÞ

0

The initial value of ρ32 created by the excitation flash can be estimated similarly: μ21  ^ei j2 K 32 , μ31  ^ei j2 j^ ρ32 ð0Þ ≈ j^

ð10:76aÞ

with Z K 32 = Θ

Z dt

½ε31 ðνÞε21 ðνÞ1=2 ν - 1 I ðν, t Þ dν:

ð10:76bÞ

Here the square root can be taken arbitrarily to be positive. Equations (10.76a, 10.76b), which are a generalization of an expression suggested by Rahman et al. [42], arbitrarily assign ρ32 (0) a purely real value. In general, ρ23 (0) is complex and depends on the convolution of Eq. (10.76b) with the electric field in the excitation → pulse, E (ν,t). The main point is that a short excitation pulse can create coherence between states 2 and 3 if it overlaps the absorption bands for excitation to both states. Such overlap is a common feature of measurements made with femtosecond laser pulses, which inherently have large spectral widths. Following the excitation pulse, ρ32 (t) will execute damped oscillations with a period of h/|E32|. How does the transient coherence between states 2 and 3 affect the fluorescence intensity and anisotropy as a function of time after the excitation? Because fluorescence reflects radiative decay of ρ22 or ρ33 , let’s find steady-state expressions for ∂ρ22 =∂t and ∂ρ33 =∂t during a probe pulse of weak, broadband light. If the pump and probe pulses are well separated in time, these derivatives will tell us the rate of stimulated emission, which will be proportional to the intensity of spontaneous fluorescence. A more refined analysis that incorporates the detailed time courses of the pump and probe pulses requires evaluating the interaction of the probe field with the time-dependent third-order polarization as described in Chap. 11.

10.8

Anomalous Fluorescence Anisotropy

521

By proceeding as we did to obtain Eq. (10.39), but adding the term involving ρ32H13 from Eq. (B10.1.4), we obtain ∂ρ12 ðt Þ=∂t = ði=ħÞðρ22 - ρ11 Þμ12  Eo expð2π iν t Þ þ fði=ħÞðE21 Þ - 1=T 2 g ρ12 ðt Þ þ ði=ħÞμ13  Eo expð2π iν t Þ ρ32 ðt Þ: ð10:77Þ We have written this expression for the density matrix of an individual system rather than the averaged density matrix for an ensemble of systems. This is because, although the transition dipoles μ12 and μ13 for an individual system might have any angle with respect to the laboratory coordinates or the electric field vector of the pump pulse, we assume that their orientations do not change during the short time interval between the pump and probe pulses. We therefore need to average the product μ13  Eo exp(2πivt)ρ32 and the other similar products in Eq. (10.77) over all the orientations of the individual systems, rather than calculating the product of the ensemble averages of the separate components. We will do the required averaging after we find ∂ρ22/∂t and ∂ρ33/∂t for an individual system. Continuing as in Eqs. (10.39–10.42a, b) and making the steady-state approximation for ρ12 and ρ21 gives ρ12 ðνÞ ≈ ð - i=ħÞ ½μ12  Eo ðρ22 - ρ11 Þ þ μ13  Eo ρ32 =½ði=ħÞðE 21 - hνÞ - 1=T 2  ð10:78aÞ and ρ21 ðνÞ ≈ ði=ħÞ ½μ21  Eo ðρ22 - ρ11 Þ þ μ31  Eo ρ23 =½ði=ħÞðE21 - hνÞ - 1=T 2 : ð10:78bÞ And because H23 and H32 are assumed to be zero and we are considering only radiative transitions between states 1 and 2, the expression corresponding to Eq. (B10.1.2) becomes: ∂ρ22 =∂t = ði=ħÞðρ21 ðt ÞH 12 - ρ12 ðt ÞH 21 Þ = ði=ħÞð expð- 2πiνt Þρ21 ðνÞH 12 - expð2πiνt Þρ12 ðνÞH 21 Þ n = ðρ11 - ρ22 Þjμ21  Eo j2 - ð1=2Þ ½ðμ31  Eo Þðμ12  Eo Þρ23 þðμ13  Eo Þðμ21  Eo Þρ32 g W 12 ðνÞ o n = ðρ11 - ρ22 Þjμ21  Eo j2 - Re ðρ23 Þðμ21  Eo Þðμ31  Eo Þ W 12 ðνÞ, ð10:79Þ where W12(ν) is a Lorentzian lineshape function for fluorescence from state 2, h i W 12 ðνÞ = ð2=T 2 Þ= ðE 21 - hνÞ2 þ ðħ=T 2 Þ2 : ð10:80Þ Since Re(ρ23) oscillates and decays with time, Eq. (10.79) indicates that the amplitude of fluorescence from state 2 will execute similar damped oscillations,

522

10 Coherence and Dephasing

converging on a level that depends on the population difference ρ11 - ρ22. The oscillatory term in Eq. (10.79) also applies to fluorescence from state 3, because Re (ρ23) = Re (ρ32). To calculate the expected fluorescence anisotropy, we now evaluate the rate of fluorescence with frequency ν and polarization ^ef , following excitation with polarization ^ei . Because we are interested in the initial rate of radiative transitions from state 2 to state 1, we can set ρ11 to zero in Eq. (10.79). The density matrix elements ρ22 and ρ23 can be obtained from Eqs. (10.75a, b–10.76a, b). Averaging over all orientations then gives the initial rate of emission from state 2:   F 12 v, ^ei , ^ef h  2    i ^21 Þ2 ^ef  μ ^21 K 22 þ ð^el  μ ^21 Þð^el  μ ^31 Þ ^ef  μ ^21 ^ef  μ ^31 K 23 W 12 ðvÞ: = ð^el  μ ð10:81Þ In Chap. 5 we considered the fluorescence anisotropy of a molecule with only one excited state. In that case, or if the excitation pulse does not overlap with both the 1→2 and 1→3 transitions, K23 = 0. The second term on the right side of Eq. (10.81)  2 then drops out, leaving F 12 = ð^ei • μ ^21 Þ2 ^ef • μ ^21 K 22 W 12 ðνÞ: For an isotropic sample, the orientational average in this last expression evaluates to 1/5 if ^ef is parallel to ^ei , and to 1/15 if ^ef and ^ei are perpendicular (Box 10.5). The fluorescence polarized parallel to the excitation thus is three times the fluorescence with perpendicular polarization, which gives a fluorescence anisotropy of 0.4 in agreement with Eqs. (5.65). (Recall that the fluorescence anisotropy r is (Fk ‐ F⊥)/(Fk + 2F⊥)). Box 10.5 Orientational Averages of Vector Dot Products Orientational averages such as those in Eq. (10.81) can be evaluated by a general procedure involving the use of Euler angles [43]. Here are the results for several cases of interest. Let the excitation polarization ^ei be parallel to the laboratory x-axis, and the fluorescence detection polarization ^ef be parallel to x for measurement of Fk or → to y for measurement of F⊥; let ξ be the angle between transition dipoles μ 21 → and μ 31 . If the sample is isotropic, then ð^x  μ ^21 Þ2 = 1=3,

ðB10:5:1Þ

ð^x  μ ^21 Þ4 = 1=5,

ðB10:5:2Þ

^21 Þð^y  μ ð^x  μ ^21 Þ = 1=15,

ðB10:5:3Þ

ð^x  μ ^21 Þð^x  μ ^31 Þ = ð cos ξÞ=3,

ðB10:5:4Þ

(continued)

10.8

Anomalous Fluorescence Anisotropy

523

Box 10.5 (continued)   ^31 Þ2 = 1 þ 2 cos 2 ξ =15, ^21 Þ2 ð^x  μ ð^x  μ

ðB10:5:5Þ

  ð^x  μ ^31 Þ2 = 2 - cos 2 ξ =15, ^21 Þ2 ð^y  μ

ðB10:5:6Þ

and   ð^x  μ ^31 Þð^y  μ ^21 Þð^y  μ ^31 Þ = 3 cos 2 ξ - 1 =30: ^21 Þð^x  μ

ðB10:5:7Þ . The values 3/5 and 1/5 given in Eqs. (5.62) and (5.63) refer to ð^x  μ ^Þ4 ð^x  μ ^ Þ2 . and ð^x  μ ^Þ2 ð^y  μ ^Þ2 ð^x  μ ^Þ2 , respectively. Equations (B10.5.5) and (B10.5.6) account for the anisotropy of incoherent emission from a molecule with absorption and emission transition dipoles oriented at an angle ξ (Eq. 5.69). The text describes the use of the above expressions to calculate the fluorescence anisotropy of an ensemble of molecules in which states 2 and 3 are excited coherently. The same expressions also can be used to calculate the fluorescence anisotropy for an incoherent mixture of states 2 and 3. In the latter situation, h i

^21 Þ4 þ ρ33 ð^x  μ ^21 Þ2 ð^x  μ ^31 Þ2 K 22 Fk will be proportional to ρ22 ð^x21  μ h ^31 Þ4 þρ22 ð^x  μ ^31 Þ2 ð^x  μ ^21 Þ2 K 33 , where ρ22 and ρ33 are the þ ρ33 ð^x  μ populations of the two stateshand K22 and K33 are as defined in the text. F⊥iwill be

proportional

to

p2 ð^x  μ ^21 Þ2 ð^y  μ ^21 Þ2 þ p3 ð^x  μ ^21 Þ2 ð^y  μ ^31 Þ2 K 22

^31 Þ2 þ p3 ð^x  μ ^31 Þ2 ð^y  μ ^21 Þ2 K 33 : If K22 = K33 and ρ22 = þ½p2 ð^x  μ ^31 Þ2 ð^y  μ 2 ρ33 , the anisotropy will be (1 + 3cos ξ)/10, which is 0.4 for ξ = 0° and 0.1 for ξ = 90 ° . The second term on the right side of Eq. (10.81) represents the coherence between excited states 2 and 3. Because such coherences usually decay on a subpicosecond time scale, they are of little significance in conventional measurements of fluorescence amplitudes or anisotropy. We therefore neglected them in Chap. 5. However, coherence can have large effects on the fluorescence at short times [42–44]. Suppose for simplicity that K23 ≈ K22, and let the angle between the transition dipoles μ21 and μ31 be ξ. The initial anisotropy calculated from Eq. (10.81) and the orientational averages given in Box 10.5 then is

10 Coherence and Dephasing

Fig. 10.13 Anisotropy and isotropic fluorescence amplitude calculated by Eqs. (10.84) and (10.85) (with K23 = K22) for an ensemble with two coherently excited states, as functions of the angle (ξ) between the transition dipoles. The isotropic fluorescence is expressed relative to the amplitude expected for an incoherently excited ensemble of systems with the same value of ξ

Fluorescence Amplitude, Anisotropy

524

2.0 relative amplitude

1.5

1.0 anisotropy

0.5

0

0

45

90

135

180

ξ (degrees)

r= =

Fk - F⊥ F ðν, ^x, ^xÞ - F 12 ðν, ^x, ^yÞ = 12 F k þ 2F ⊥ F 12 ðν, ^x, ^xÞ þ 2F 12 ðν, ^x, ^yÞ

½1=5 þ ð1 þ 2 cos 2 ξÞ=15 - ½1=15 þ ð3 cos 2 ξ - 1Þ=30 7 þ cos 2 ξ = : 2 2 ½1=5 þ ð1 þ 2 cos ξÞ=15 þ 2 ½1=15 þ ð3 cos ξ - 1Þ=30 10 þ 10 cos 2 ξ ð10:82Þ

Figure 10.13 shows how the calculated anisotropy and the amplitude of the isotropic fluorescence (Fk + 2F⊥) depend on ξ. If ξ = 90 °, which as we discuss in Chap. 8 is the expected angle between the transition dipoles of the two exciton states of a dimer. Coherence between states 2 and 3 does not affect the amplitude of the isotropic fluorescence. The initial anisotropy, however, will be 0.7 instead of 0.4. The predicted anisotropy will drop to 0.4 as the off-diagonal density matrix elements ρ23 and ρ32 decay to zero. For a three-state system with three orthogonal transition dipoles, the initial anisotropy is predicted to be 1.0, which means that the fluorescence is completely polarized parallel to the excitation [43]! To generalize Eq. (10.81), consider a system with M excited states. If we can evaluate the product of the density matrix and the fluorescence operator for an ensemble of systems, we can use Eq. (10.14) to calculate the fluorescence from any given excited state. Equation (10.74) suggests that the operator e F for the dipole strength of fluorescence with polarization ^ef can be written symbolically as e μ  ^ef , ð10:83Þ F=e μ  ^ef jψ 1 ihψ 1 je    →  which has matrix elements F mn = ψ m je μ  ^ef jψ 1 ψ 1 je μ  ^ef jψ n , or μ m1  ^ef  →  μ n1  ^ef . Here ψ m, ψ n, and ψ 1 are the wavefunctions of states m, n, and the ground state, respectively. If we include the homogeneous emission spectrum W1n(v) as a

10.9

Questions

525

generalization of W12(v) (Eq. 10.80), it follows from Eq. (10.14) that the fluorescence from state n with polarization ^ef and frequency ν is " # M  X    F 1n ν, ^ei , ^ef = ρnn F nn þ ð1=2Þ ρnm F mn þ ρmn F nm W 1n ðνÞ, ð10:84Þ m≠n

where the ρnm F jk depend on the excitation polarization. If the density matrix immediately after the excitation pulse is given by Eqs. (10.75–10.76), the initial fluorescence from state n will be   X    F 1n ν, ^ei , ^ef = ð^ei  μ ^n1 Þð^ei  μ ^m1 Þ ^ef  μ ^n1 ^ef  μ ^m1 K nm W 1n ðνÞ: ð10:85Þ Note that Eq. (10.85) describes the fluorescence immediately after the excitation pulse. As the off-diagonal elements decay to zero, the fluorescence becomes simply  2 ð^el  μ ^n1 Þ2 ^ef  μ ^n1 K nn W 1n ðvÞ: Equation (10.85) also assumes that Wn1(v), Knn and Knm are the same for all the systems in the ensemble; if this is not the case, these factors must be averaged along with the geometric parameters. Initial fluorescence anisotropies greater than 0.4 have been measured in porphyrins, which have two degenerate excited states with orthogonal transition dipoles [44]. Anisotropies that appear to exceed 0.4 initially but decay on a time scale of about 30 fs have been seen in the stimulated emission from photosynthetic bacterial antenna complexes [45]. As we discussed in Chap. 8, these complexes have several allowed transitions with similar energies and approximately perpendicular transition dipoles. Excitation with a short pulse probably creates a coherent superposition of excited states that relaxes rapidly to an incoherent mixture of states with approximately the same populations, and then to a Boltzmann equilibrium.

10.9

Questions

1. Explain the meaning of the density matrix. 2. Suppose that molecule a of the dimer considered in Exercise 8.1 is excited at time t = 0, and that the excitation then oscillates between the two molecules. (a) Calculate the frequency of the oscillation if the excitation energy of the two molecules is the same. (b) What would the oscillation frequency be if the excitation energy of molecule a is 1000 cm-1 above that of molecule b? (c) Calculate the time-averaged probability of finding the excitation on molecule a for the second case (Eba = Ea - Eb = 1000 cm-1) in the absence of thermal relaxations, and compare your result with the probability of finding the excitation on this molecule at thermal equilibrium. 3. Consider the excited dimer as a system with two basis states, with ca(t) and cb(t) representing coefficients for the states in which the excitation is on, respectively, molecule a or molecule b. (a) Define the four elements of the density matrix (ρ) for this system. (b) Using the notation Haa, Hab, etc., for the Hamiltonian matrix

526

4. 5. 6.

7. 8.

10 Coherence and Dephasing

elements, write an expression for the time dependence of each element of ρ (e.g., ∂ρa/∂t) in the absence of stochastic relaxations. (c) What is the relationship between ρab(t) and ρba(t)? (d ) Suppose that interconversions of the two basis states are driven only by the quantum mechanical coupling element Hab, but that stochastic fluctuations of the energies cause pure dephasing with a time constant T2. What are the longitudinal (T1) and transverse (T2) relaxation times in this situation? (e) Write out the Stochastic Liouville expression for the timedependence of each element of ρ. ( f ) How would T1 and T2 be modified if the system also changes stochastically from state a to b with rate constant kab and from b to a with rate constant kab? (g) In what limit does the stochastic Liouville equation reduce to the golden-rule expression? How does the relaxation matrix introduced by the Redfield theory improve treatments of a system's density matrix. How do fluctuations in the energy difference between two coupled excited states affect the localization of the excitation? (a) Write an expression relating the stochastic rate constant for conversion of an ensemble of two-state quantum systems from state a to b (-Raa, aa, where R is the Redfield relaxation matrix) to the spectral density of fluctuating electric fields from the surroundings. Your expression should indicate that the rate constant depends on the fluctuations that occur at a particular frequency. (b) How does the important frequency depend on the energy difference between the two states (Eba)? (c) Relate the pertinent spectral density function to the autocorrelation function (memory function) of a quantum mechanical matrix element. (d ) If the autocorrelation function decays exponentially with time constant τc, how does the rate constant depend on the value of τc? (e) In what limit of time does the Redfield theory apply? ( f ) Outline the modifications or extensions that are needed to account for a directional relaxation such as the Stokes shift of fluorescence relative to absorption. What is meant by a rotating-wave approximation? Show that Fourier transformations of the relaxation functions ϕ(t) and ϕ(t)* give the absorption and emission spectra, respectively.

References 1. Atkins, P.W.: Molecular Quantum Mechanics, 2nd edn. Oxford University Press, Oxford (1983) 2. Kubo, R.: The fluctuation-dissipation theorem. Rept. Progr. Theor. Phys. 29, 255–284 (1966) 3. Kubo, R., Toda, M., Hashitsume, N.: Statistical Physics II: Nonequilibrium Statistical Mechanics. Springer, Berlin (1985) 4. Davidson, E.R.: Reduced Density Matrices in Quantum Chemistry. Academic Press, London (1976) 5. Lin, S.H., Alden, R.G., Islampour, R., Ma, H., Villaeys, A.A.: Density Matrix Method and Femtosecond Processes. World Scientific, Singapore (1991) 6. Blum, K.: Density Matrix Theory and Applications, 2nd edn. Plenum, New York (1996) 7. Haberkorn, R., Michel-Beyerle, M.E.: On the mechanism of magnetic field effects in bacterial photosynthesis. Biophys. J. 26, 489–498 (1979)

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8. Bray, A.J., Moore, M.A.: Influence of dissipation on quantum coherence. Phys. Rev. Lett. 49, 1545–1549 (1982) 9. Reimers, J.R., Hush, N.S.: Electron transfer and energy transfer through bridged systems. I. Formalism. Chem. Phys. 134, 323–354 (1989) 10. Kitano, M.: Quantum Zeno effect and adiabatic change. Phys. Rev. A. 56, 1138–1141 (1997) 11. Schulman, L.S.: Watching it boil: continuous observation for the quantum Zeno effect. Found. Phys. 27, 1623–1636 (1997) 12. Ashkenazi, G., Kosloff, R., Ratner, M.A.: Photoexcited electron transfer: short-time dynamics and turnover control by dephasing, relaxation, and mixing. J. Am. Chem. Soc. 121, 3386–3395 (1999) 13. Prezhdo, O.: Quantum anti-zeno acceleration of a chemical reaction. Phys. Rev. Lett. 85, 4413–4417 (2000) 14. Facchi, P., Nakazato, H., Pascazio, S.: From the quantum zeno to the inverse quantum zeno effect. Phys. Rev. Lett. 86, 2699–2703 (2001) 15. Kofman, A.G., Kurizki, G.: Frequent observations accelerate decay: the anti-Zeno effect. Zeit. Naturforsch. Sect. A. 56, 83–90 (2001) 16. Toschek, P.E., Wunderlich, C.: What does an observed quantum system reveal to its observer? Eur. Phys. J. D. 14, 387–396 (2001) 17. Parson, W.W., Warshel, A.: A density-matrix model of photosynthetic electron transfer with microscopically estimated vibrational relaxation times. Chem. Phys. 296, 201–206 (2004) 18. Chiu, C.B., Sudarshan, E.C.G., Misra, B.: Time evolution of unstable quantum states and a resolution of Zeno’s paradox. Phys. Rev. D. 16, 520–529 (1977) 19. Redfield, A.: The theory of relaxation processes. Adv. Magn. Res. 1, 1–32 (1965) 20. Slichter, C.P.: Principles of Magnetic Resonance with Examples from Solid State Physics. Harper & Row, New York (1963) 21. Silbey, R.J.: Relaxation processes. In: Funfschilling, J. (ed.) Molecular Excited States, pp. 243–276. Kluwer Academic Publishers, Dordrecht (1989) 22. Oxtoby, D.W.: Picosecond phase relaxation experiments. A microscopic theory and a new interpretation. J. Chem. Phys. 74, 5371–5376 (1981) 23. Yan, Y.J., Mukamel, S.: Photon echoes of polyatomic molecules in condensed phases. J. Chem. Phys. 94, 179–190 (1991) 24. Mercer, I.P., Gould, I.R., Klug, D.R.: A quantum mechanical/molecular mechanical approach to relaxation dynamics: calculation of the optical properties of solvated bacteriochlorophyll-a. J. Phys. Chem. B. 103, 7720–7727 (1999) 25. Callen, H.B., Greene, R.F.: On a theorem of irreversible thermodynamics. Phys. Rev. 86, 702–710 (1952) 26. Greene, R.F., Callen, H.B.: On a theorem of irreversible thermodynamics. II. Phys. Rev. 88, 1387–1391 (1952) 27. Berne, B.J., Harp, G.C.: On the calculation of time correlation functions. Adv. Chem. Phys. 17, 63–227 (1970) 28. de Groot, S.R., Mazur, P.: Non-equilibrium Thermodynamics. Dover, Mineola, NY (1984) 29. Mukamel, S.: Principles of Nonlinear Optical Spectroscopy. Oxford University Press, Oxford (1995) 30. McHale, J.L.: Molecular Spectroscopy. Prentice Hall, Upper Saddle River, NJ (1999) 31. May, V., Kühn, O.: Charge and Energy Transfer Dynamics in Molecular Systems. Wiley-VCH, Berlin (2000) 32. Joo, T., Jia, Y., Yu, J.-Y., Lang, M.J., Fleming, G.R.: Third-order nonlinear time domain probes of solvation dynamics. J. Chem. Phys. 104, 6089–6108 (1996) 33. de Boeij, W.P., Pshenichnikov, M.S., Wiersma, D.A.: System-bath correlation function probed by conventional and time-gated stimulated photon echo. J. Phys. Chem. 100, 11806–11823 (1996) 34. Mukamel, S.: Femtosecond optical spectroscopy: a direct look at elementary chemical events. Ann. Rev. Phys. Chem. 41, 647–681 (1990)

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35. Fleming, G.R., Cho, M.: Chromophore-solvent dynamics. Ann. Rev. Phys. Chem. 47, 109–134 (1996) 36. Volker, S.: Spectral hole-burning in crystalline and amorphous organic solids. Optical relaxation processes at low temperatures. In: Funfschilling, J. (ed.) Relaxation Processes in Molecular Excited States, pp. 113–242. Kluwer Academic Publ, Dordrecht (1989) 37. Reddy, N.R.S., Lyle, P.A., Small, G.J.: Applications of spectral hole burning spectroscopies to antenna and reaction center complexes. Photosynth. Res. 31, 167–194 (1992) 38. Warshel, A., Parson, W.W.: Computer simulations of electron-transfer reactions in solution and in photosynthetic reaction centers. Ann. Rev. Phys. Chem. 42, 279–309 (1991) 39. Warshel, A., Parson, W.W.: Dynamics of biochemical and biophysical reactions: insight from computer simulations. Q. Rev. Biophys. 34, 563–679 (2001) 40. de Boeij, W.P., Pshenichnikov, M.S., Wiersma, D.A.: Ultrafast solvation dynamics explored by femtosecond photon echo spectroscopies. Ann. Rev. Phys. Chem. 49, 99–123 (1998) 41. Myers, A.B.: Molecular electronic spectral broadening in liquids and glasses. Ann. Rev. Phys. Chem. 49, 267–295 (1998) 42. Rahman, T.S., Knox, R., Kenkre, V.M.: Theory of depolarization of fluorescence in molecular pairs. Chem. Phys. 44, 197–211 (1979) 43. van Amerongen, H., Struve, W.S.: Polarized optical spectroscopy of chromoproteins. Meth. Enzymol. 246, 259–283 (1995) 44. Wynne, K., Hochstrasser, R.M.: Coherence effects in the anisotropy of optical experiments. Chem. Phys. 171, 179–188 (1993) 45. Nagarajan, V., Johnson, E., Williams, J.C., Parson, W.W.: Femtosecond pump-probe spectroscopy of the B850 antenna complex of Rhodobacter sphaeroides at room temperature. J. Phys. Chem. B. 103, 2297–2309 (1999)

Pump-Probe Spectroscopy, Photon Echoes, Two-Dimensional Spectroscopy and Vibrational Wavepackets

11.1

11

First-Order Optical Polarization

In the last chapter, we used a steady-state treatment to relate the shape of an absorption band to the dynamics of relaxations in the excited state. Because establishing a steady state requires a period on the order of the electronic dephasing time, Eqs. (10.43) and (10.44) apply only on time scales longer than this. We need to escape this limitation if we hope to explore the relaxation dynamics themselves. Our first goal in this chapter is to develop a more general approach for analyzing spectroscopic experiments on femtosecond and picosecond time scales. This will provide a platform for discussing how pump-probe and photon-echo experiments can be used to probe the dynamics of structural fluctuations and the transfer of energy or electrons on these short time scales. To start, consider an ensemble of systems, each of which has two states (m and n) with energies Em and En, respectively. In the presence of a weakD radiationEfield e m can E(t) = Eo[exp(iwt) + exp (-iwt)], the Hamiltonian matrix element ψ n jHjψ be written H nm ðt Þ = H 0nm þ V nm ðt Þ, where H 0nm is the matrix element in the absence of the field, Vnm(t) = -μnm  Eo[exp(iwt) + exp (-iwt)], and μnm is the transition dipole. The commutator [ρ , H],  which  determines the time dependence of the density matrix, then is the sum ρ, H0 þ ½ρ, V. If the system is stationary in the 0 0 absence  of field (i.e., H nm = 0 for n ≠ m), the matrix elements of [ρ,H ] are 0 ρ, H nm = Emn ρnm = -E nm ρnm , where Enm = En – Em (Eqs. (10.21a, b, c–10.24)). We can simplify matters if we adjust all the elements of ρ by subtracting the Boltzmann-equilibrium values in the absence of the radiation field (0 for off-diagonal elements, and ρonn as prescribed by Eq. (10.26) for diagonal elements). The rate constant for stochastic relaxations of the adjusted density matrix element ρnm then can be written as γ nm, where γ nm = 1/T1 for n = m and 1/T2 for n ≠ m (Eq. 10.29). To simplify the notation further, let Enm /ħ = ωnm.

# The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. W. Parson, C. Burda, Modern Optical Spectroscopy, https://doi.org/10.1007/978-3-031-17222-9_11

529

530

11

Pump-Probe Spectroscopy, Photon Echoes, Two-Dimensional Spectroscopy. . .

With these definitions and adjustments, the stochastic Liouville equation (Eq. (10.30)) for the ensemble of two-state systems in the presence of the radiation field takes the form   ∂ρnm =∂t = ði=ħÞ ρ, H0 nm þ ði=ħÞ½ρ, Vnm - γ nm ρnm

ð11:1aÞ

= ði=ħÞ½ρ, Vnm - ðiωnm þ γ nm Þρnm :

ð11:1bÞ

The trick now is to expand ρnm (t) in a series of increasing orders of perturbation by the radiation field [1]: ð0Þ

ð1Þ

ð2Þ

ρnm = ρnm þ ρnm þ ρnm þ ⋯,

ð11:2Þ

ð0Þ

where ρnm is the density matrix at equilibrium in the absence of the radiation field, ð1Þ ρnm

ð0Þ

is the perturbation to ρnm in the limit of a very weak field (a linear, or first-order ð2Þ

perturbation), ρnm is a perturbation that is quadratic in the strength of the field, and so forth. If we view perturbations of progressively higher orders as developing sequentially in time, we can use Eq. (11.1b) to write their rates of change: ð0Þ

∂ρnm ð0Þ = -ðiωnm þ γ nm Þ ρnm , ∂t

ð11:3aÞ

ð1Þ h i ∂ρnm ð1Þ = ði=ħÞ ρð0Þ , V - ðiωnm þ γ nm Þ ρnm , ∂t nm

ð11:3bÞ

ð2Þ h i ∂ρnm ð2Þ = ði=ħÞ ρð1Þ , V - ðiωnm þ γ nm Þ ρnm , ∂t nm

ð11:3cÞ

and in general, h i ðk Þ ∂ρnm =∂t = ði=ħÞ ρðk-1Þ , V

ðk Þ

nm

- ðiωnm þ γ nm Þ ρnm :

ð11:4Þ

ðk Þ

The solution to Eq. (11.4) for ρnm at time τ is ðk Þ ρnm ðτÞ = ði=ħÞ

Zτ h i ρðk-1Þ , V

nm

exp½-ðiωnm þ γ nm Þ ðτ - t Þdt:

ð11:5Þ

0

The terms in Eq. (11.2) thus represent the results of sequential interactions of the ensemble with the radiation field, convolved with oscillations and decay of the states and coherences generated by these interactions. If an ensemble of two-state systems starts with all the systems in the ground state (a), a single interaction with the field creates one of the off-diagonal density matrix elements (ρab or ρba), which represent

11.1

First-Order Optical Polarization

531

Fig. 11.1 (A) Pathways in Liouville space. The white circles labeled a,a and b,b represent the diagonal elements of the density matrix (populations) for a two-state system; the yellow circles labeled a,b and b,a represent off-diagonal elements (coherences). Lines represent individual interactions with a radiation field, with vertical lines denoting interactions that change the lefthand (bra) index of the density matrix and horizontal lines those that change the right-hand (ket) index. In the convention used here, the zero-order density matrix (ρ(0)) is at the lower left, and time increases upwards and to the right; downward or leftward steps are not allowed. The coherences in the yellow circles are endpoints of the two one-step pathways [ρa,a → ρb,a( B) and ρa,a → ρa,b(C)]   that contribute to the first-order density matrix ρð1Þ and the first-order optical polarization (P(1)).

A second interaction with the radiation field (dotted line) can convert a coherence to the excited state (ρbb) or the ground (ρaa) state. The pathways in B and C are described as complex conjugates because one can be generated from the other by interchanging the two indices at each step

coherences of state a with the excited state (b). A second interaction can either create a population in state b (ρbb ) or regenerate ρaa . Such sequences of interactions are described as pathways in Liouville space, and can be represented schematically as shown in Fig. 11.1. A Liouville-space diagram consists of a square lattice, with each of the lattice points (circles in Fig. 11.1) labeled by the two indices of a density matrix element. A vertical line connecting two circles represents an interaction that changes the left index (bra) and a horizontal line represents an interaction that changes the right index (ket). The convention used here is that the density matrix of a resting ensemble begins at the lower left corner of the diagram and evolves upward or to the right on each interaction with an electromagnetic radiation field. The coherences or populations that are generated by a given number of interactions, and so make up that order of the density matrix, lie on an antidiagonal line. In Fig. 11.1, the coherences that contribute to ρð1Þ are highlighted. Figure 11.2 shows another useful representation called a double-sided Feynman diagram. The two vertical lines in this diagram represent the left and right indices of the density matrix, and each interaction with the field is represented by a wavy arrow pointing to or away from one of the lines. Time increases upwards. Arrows pointing toward a vertical line are associated with absorption of a photon; arrows pointing away, with emission. The diagram also can convey additional information such as the wavevector and frequency of the radiation [2–11]. The power-series expansion of ρ (t) can be used with Eq. (10.14) to find the expectation value of the macroscopic electric dipole for an ensemble of systems exposed to an electromagnetic field:

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11

Pump-Probe Spectroscopy, Photon Echoes, Two-Dimensional Spectroscopy. . .

Fig. 11.2 Double-sided Feynman diagrams. The pairs of purple vertical lines represent the evolution of the left (bra) and right (ket) indices of the density matrix. Time increases from the bottom to the top of the diagram. In the two-state system considered here, diagonal elements of ρ are labeled a,a and b,b for the ground and excited states, respectively, and a,b and b,a denote off-diagonal elements. Wavy red arrows labeled with a frequency (ω) indicate interactions with an electromagnetic radiation field. Incoming arrows are associated with absorption of a photon (an increase in one of the indices of ρ); outgoing arrows, with emission (a decrease in one of the indices). A single interaction with the field at time zero generates a coherence that contributes to the first-order optical polarization. A second interaction at time τ converts a coherence to either the excited state (A, C) or the ground state (B, D). The sequences depicted in (C) and (D) are the complex conjugates of those in (A) and (B), respectively. (A) and (B) correspond to the paths shown in Fig. 11.1B; (C and D), to those in Fig. 11.1C. Double-sided Feynman diagrams also can be used to convey additional information, such as the wavevectors of incoming and outgoing radiation in experiments involving multiple pulses [3, 4, 7]

hμ ðt Þi =

X PðkÞ ðt Þ,

ð11:6Þ

k

with   XX → ðk Þ ρnm ðt Þ μmn PðkÞ ðt Þ = Tr ρðkÞ ðt Þ μ = n

ð11:7Þ

m

μ j ψ n i. The terms P(k) are referred to as various orders of optical and μmn = hψ m j e polarization. The first-order optical polarization is the quantum mechanical analog of the classical linear polarization of a dielectric by an oscillating electromagnetic field (Box B3.3); the higher-order polarizations correspond to classical components that depend on higher powers of the field. But note that, according to Eqs. (11.4– 11.7), increasing orders of the optical polarization develop sequentially in time, whereas the classical polarizations with various dependencies on the field were assumed to form instantaneously and simultaneously.

11.1

First-Order Optical Polarization

533

The benefit of expanding the optical polarization in this way is that various optical phenomena can be assigned to the terms with particular values of k [2, 6, 12– 15]. The first-order, or linear optical polarization (P(1)) pertains to ordinary absorption and emission of light; the second-order optical polarization (P(2)), to the generation of sum and difference frequencies; and the third-order optical polarization (P(3)), to “four-wave mixing” experiments that include pump-probe spectroscopy, transient-grating spectroscopy, and photon echoes. More precisely, each class of optical phenomena can be related to interaction of the electric field with a particular order of the polarization. Classically, the optical polarization is viewed as a macroscopic oscillating dipole that can serve as either a source or an absorber of electromagnetic radiation. In a semiclassical picture in which we treat electromagnetic radiation classically, the energy of interaction of a radiation field E(t) with an optically polarized system is given by hH 0 ðt Þi = -hμðt Þi  Eðt Þ = -

X PðkÞ ðt Þ  Eðt Þ,

ð11:8Þ

k

or using Eq. (10.14),   hH 0 ðt Þi = Tr ρðt Þ Vðt Þ ,

ð11:9Þ

where V again is the interaction matrix (Vnm = -μnmP(t)). The rate of absorption of energy from the field is the derivative of this quantity with respect to time:     ∂ ðρ V Þ d hH 0 i d dV dρ þ Tr V : = Tr ðρ VÞ = Tr = Tr ρ dt dt dt dt ∂t

ð11:10Þ

The last term on the right side of Eq. (11.10) is zero. You can show this by using the von Neumann equation (Eq. (10.24)) and noting that, because cyclic permutation of three matrices does not change the trace of the product of the matrices (Appendix A2), TrðV ρ VÞ - TrðV V ρÞ = 0 : 

 dρ = ði=ħÞTrðV ½ρ, VÞ Tr V dt = ði=ħÞfTrðV ρ VÞ - TrðV V ρÞg = 0:

ð11:11Þ

If we drop this term, write the oscillating radiation field as in Eq. (10.37), and assume that the envelope of the field amplitude (Eo) changes only slowly relative to exp(iωt), Eq. (11.10) gives   d hH 0 i dV d = -Pðt Þ  ½Eo expðiωt Þ þ Eo  expð-iωt Þ = Tr ρ dt dt dt ≈ -iωPðt Þ  ½Eo expðiωt Þ þ Eo  expð-iωt Þ:

ð11:12Þ

534

11

Pump-Probe Spectroscopy, Photon Echoes, Two-Dimensional Spectroscopy. . .

The instantaneous rate of excitation thus is proportional to the dot product of the optical polarization and the field, P(t)E(t). We saw in Eqs. (10.40)–(10.42a, b) that the elements of the first-order density matrix contain factors that oscillate at the same frequency as the electromagnetic field that generates them. The same is true of the optical polarization. P therefore can be put in the form Pðt Þ = Po ðt Þ expðiωt Þ þ Po  ðt Þ expð-iωt Þ,

ð11:13Þ

where Po and Po, like Eo, Eo and the factors ρab and ρba in Eqs. (10.40–10.42), change comparatively slowly with time. With P written in this way, Eq. (11.12) becomes d hH 0 i = -iω½Po expðiωt Þ þ Po  expð-iωt Þ½Eo expðiωt Þ - Eo  expð-iωt Þ dt = iωðPo Eo  - Po  Eo Þ þ iω½Po Eo expð2iωt Þ - Po  Eo  expð-2iωt Þ:

ð11:14Þ

If we average this expression over a period of the oscillation, the factors containing exp(±2iωt) drop out, leaving d hH 0 i = iωðPo Eo  - Po  Eo Þ = 2ωImðPo Eo  Þ: dt

ð11:15Þ → ð1Þ

To illustrate the use of Eqs. (11.4–11.7) and (11.15), let’s evaluate ρð1Þ and P for an ensemble of two-state systems that are exposed to light. If the ensemble is at thermal equilibrium before the light is switched on, the initial density matrix is  ρð0Þ =

ρoaa 0

0 , ρobb

ð0Þ

ð0Þ

ð11:16Þ

with ρoaa and ρobb given by Eq. (10.13). ρab and ρba are both zero. Suppose that the amplitude of the oscillating electric field goes from zero to Eo abruptly at time zero and then remains constant at this level. Neglecting the initial rise, the perturbation matrix V for t ≥ 0 then is given by Vab = -μabE(t). We assume here that the transition dipoles are the same for all members of the ensemble (V ab = V ab), although Vab usually must be averaged together with ρ over molecules with different orientations relative to the incident radiation. Let’s also neglect any dependence of Eo and μab on ω, and assume further that the two basis states have no net charge or dipole moment, so that Vaa = Vbb = 0. h i With these assumptions, the commutator ρð0Þ , V is zero for t < 0, and becomes

11.1

First-Order Optical Polarization

" h i ρð0Þ , V =

535



0

 ρobb - ρoaa V ba

#  ρoaa - ρobb V ab 0

ð11:17Þ

for t ≥ 0. You can check these matrix elements by referring to Eqs. (10.21) and Box h i 10.1. For example, the entry for ρð0Þ , V h

ð1Þ

ab

, which pertains to the growth of ρab , is

i  X ð0Þ ð0Þ ρak V kb - V ak ρkb ρð0Þ , V = ab

k

 

 ð0Þ ð0Þ ð0Þ = ρaa - ρbb V ab þ ρba ðV bb - V aa Þ = ρoaa - ρobb V ab :

ð11:18Þ

h i ð0Þ ð0Þ Similarly, ρð0Þ , V = ρab V ba ‐ρba V ab = 0. h i aa from Eq. (11.18) with Vab from Eq. (10.37), Eq. (11.5) gives Using ρð0Þ , V ab

 ð1Þ ρab ðτÞ = ði=ħÞ ρobb - ρoaa

Zτ f exp½-ðiωab þ γ ab Þ ðτ - t Þ μab  Eo ½ expðiωt Þ 0

þ expð-iωt Þg dt

ð11:19Þ

The integrand in Eq. (11.19) includes factors of the forms exp[i(ωab + w)t] and exp [i(ωab ‐ ω)t], where ωab again is (Haa ‐ Hbb)/ħ. Because ωab = ‐ ωba, exp [i(ωab + ω)t] goes to 1 when ω ≈ ωba. The factor exp[i(ωab ‐ ω)t], on the other hand, becomes approximately exp(‐2iωt), which oscillates rapidly between positive and negative values and contributes little to the integral at times greater than 1/ω. Neglecting the term exp(‐2iωt) is essentially the same as the rotating-wave approximation we used in Sect. 10.6. Making this approximation, we have ð1Þ

ρab ðτÞ ≈

 i o ρbb - ρoaa μab ħ

Zτ

 Eo exp½-ði ωab þ γ ab Þτ

f exp ½ ðiωab þ iω þ γ ab Þt g dt 0

=

   exp½ ðiωab þ iω þ γ ab Þτ - 1 i o ρbb - ρoaa μab  Eo exp½-ðiωab þ γ ab Þτ ħ i ðωab þ ωÞ þ γ ab =

 expðiωτÞ - exp½-ðiωab þ γ ab Þτ i o : ρ - ρoaa μab  Eo ħ bb i ðωab þ ωÞ þ γ ab

ð11:20Þ

For comparison of Eq. (11.20) with the corresponding steady-state expression, ð1Þ

multiplying ρab by exp(‐iωt) to remove the rapid oscillation with time (Eq. (10.40)), and setting ω ≈ ωba = ‐ ωab, gives

536

Pump-Probe Spectroscopy, Photon Echoes, Two-Dimensional Spectroscopy. . .

11

 1 - exp½-ðiωab þ iω þ γ ab τÞ i o ρ - ρoaa μab  Eo ħ bb iðωab þ ωÞ þ γ ab

 1 - expð-γ ab τÞ i ≈ ρobb - ρoaa μab  Eo : ħ iðωab þ ωÞ þ γ ab

ð1Þ

ð1Þ

ρab = ρab expð-iωτÞ =

ð11:21Þ Except for the additional factor [1 - exp (-γ abτ)] in the numerator, this is the same as Eq. (10.42a). As τ increases, the factor [1 - exp (-γ abτ)] takes ρab expð-iωτÞ to its steady-state value with the time constant 1/γ ab, which is T2 in a two-state system. ð2Þ

Now that we have ρð1Þ , we can use Eq. (11.4) to find ∂ρbb =∂t, which will give us the time course of excitation to state b. The required element of the commutator [ρð1Þ ,V] is h

ρð1Þ ,

  i ð1Þ ð1Þ ð1Þ ð1Þ V = V ab ρba - V ba ρab = V ab ρab - ρab ,

ð11:22Þ

bb

h i and inserting ρð1Þ , V

bb

into Eq. (11.4) gives

ð2Þ ∂ρbb =∂t = ði=ħÞ V ab



ð1Þ ρab

ð1Þ - ρab

ð2Þ

- γ bb ρbb :

ð11:23Þ

If we’re interested in the rate of excitation at times that are short relative to the lifetime of the excited state (t 0) (Fig. 11.9). This path has two periods of coherence (delay periods t1 and t3) and the density matrix element in the second period (ρba) is the complex conjugate of that in the first (ρab). The two periods of coherence are separated by a “population” interval (t2) when the systems are in the excited state (ρbb). During period t1, both static and dynamic inhomogeneity in the energies of the molecules in the ensemble will cause the coherence to decay. But any truly static contribution to the energy of a given molecule will remain constant until period t3, when it will have the opposite effect that it had during period t1. This is because the Green function for ρba is the complex conjugate (more precisely, the Hermitian conjugate) of that for ρab (Eq. 11.37). Static inhomogeneity in the energy difference between states a and b creates a distribution of the factors exp(-iωbat) in the Green function for ρba, which disappears when it is multiplied by the same distribution of the factors exp(iωabt) in the Green function for ρab. Thus, the dephasing caused by static inhomogeneity in period t1 is reversed during period t3, regenerating a coherence that will appear as a pulse of emitted light when t3 ≈ t1. This is the photon echo. Dephasing due to dynamic fluctuations do not undergo such a reversal, because it enters the Green functions for both ρba and ρab in an identical manner, for example as exp(‐t/T2) for both Green functions if the coherence has a simple exponential decay. The amplitude of the photon echo therefore will be largest when t2 is short and will decrease as t2 is lengthened and fluctuations on a broader range of time scales invade the Hermitian relationship between the two Green functions. The dependence of the echo amplitude on t2 thus will report on the fluctuation dynamics with relatively little disturbance by the effects of static inhomogeneity. Now look at path R1, which can be obtained from R2 by simply delaying pulse 1 so that it follows pulse 2 (t1 < 0 in Figs. 11.8 and 11.9). R1 and its complex conjugate each have two periods with the same coherence (ρba) separated by a population interval. They do not result in echoes. The complex conjugate of R2 (not shown in Fig. 11.9) gives photon echoes that are the same as those of R2 but, as mentioned above, appear with wavevector ks = k1 - k2 - k3. R3 is similar to R2 in having two periods with conjugate coherences separated by a population period. However, because the system spends the population period in the ground state (ρaa), this path and its complex conjugate usually do not contribute an observable signal [49]. R4 and R4 have corresponding relationships to R1 and R1, and also give no echoes. In what is probably the most informative type of three-pulse photon-echo experiment, the time between pulses 1 and 2 (τ) is varied while that between pulses 2 and 3 (T) is held constant (Figs. 11.8 and 11.9). The echo signal in direction ks = k2 k1 + k3 is integrated over time t3, and the measurements are repeated with different values of T. The signal reflects the third-order polarization with wavevector ks, P(3)(ks), and can be detected either by collecting the emitted light directly (homodyne detection) or by mixing P(3)(ks) with a separate, stronger radiation field (heterodyne detection). With homodyne detection, the signal depends on the squared amplitude (modulus) of the polarization:

550

11

Pump-Probe Spectroscopy, Photon Echoes, Two-Dimensional Spectroscopy. . .

Z1 I ð ks Þ =

jPð3Þ ðks Þj2 dt 3 :

ð11:40Þ

0

If we consider only path R2, which dominates the signal if pulse 1 precedes pulse 2, the third-order polarization at time t is given by Zt Zt  3 Z t i dt 3 dt 2 dt 1 R2 ðt 3 , t 2 , t 1 Þ P ðks , t Þ = jμba  ^ej ħ ð3Þ

4

0

0

0

    × jEo3 ðt - t 3 Þj expðiω t 3 Þ × jEo2 ðt - t 3 - t 2 þ T Þj expð-iω t 2 Þ   × jEo1 ðt - t 3 - t 2 - t 1 þ τ þ T Þj expð-iω t 1 Þ , ð11:41Þ where j Eo1 ðt Þ j , j Eo2 ðt Þ j and j Eo3 ðt Þ j are the envelopes of the field amplitudes in the three pulses and the response function R2(t3, t2, t1) is related to the line-broadening function g(t) by Eq. (11.38b) [7, 49, 50]. We have assumed that the three pulses have the same frequency (ω) and that the field envelopes are all real functions of time. Now suppose that the light pulses are short relative to the time intervals t1, t2, and t3, so that t1 ≈ τ and t2 ≈ T. In this limit, Eqs. (11.40) and (11.41) reduce to   Z1 o 2 o 2 o 2 6 I ðks , τ > 0Þ = jμba  ^ej jE3 j jE2 j jE1 j =ħ jR2 j2 dt 3 , 8

ð11:42Þ

0

j,j j and j j are averages of the field strengths over the pulses. The where j notation “τ > 0” indicates that pulse 1 must precede pulse 2 to enforce the order of the first two interactions that make up path R2. Using Eq. (11.38b) for R2, we obtain Eo1

Eo2

Eo3

Z1  h  I ð ks , τ > 0 Þ /  exp -g ðt 3 Þ - g ðt 1 Þ þ gðt 2 Þ - gðt 2 þ t 3 Þ - g ðt 1 þ t 2 Þ 0

i þ g ðt 1 þ t 2 þ t 3 Þ 2 dt 3 : ð11:43Þ If pulse 1 is delayed so that it follows pulse 2 (τ < 0), R1(t3, t2, t1) replaces R2(t3, t2, t1), and using Eq. (11.38a) for R1 gives   Z1 o 2 o 2 o 2 6 I ðks , τ < 0Þ = jμba  ^ej jE3 j jE2 j jE1 j =ħ jR1 j2 dt 3 8

0

11.4

Photon Echoes

Fig. 11.10 Dependence of three-pulse photon-echo signals on delay time t3, as calculated in the impulsive limit with Eq. (11.43) for t1 = τ (the delay between pulses 1 and 2) = 5 (A) and with Eq. (11.44) for t1 = τ = - 5 (B). The delay between pulses 2 and 3 (t2 = T ) was 0, 10 or 100, as indicated. The units of time are arbitrary. All calculations used the Kubo relaxation function (Eq. (10.69)) with τc = 40 time units and σ = 0.1 reciprocal time units

551

A 1.0 t1 = 5

0.8

t2 = 0

0.6

10

2

|R2|

100

0.4 0.2 0

B 0.8 t1 = -5

0.6 2

|R1|

t2 = 100

0.4

10 0

0.2 0

0

10

20

30

t3 / arbitrary units Z1 /

j exp½-g ðt 3 Þ - gðt 1 Þ - g ðt 2 Þ þ g ðt 2 þ t 3 Þ þ gðt 1 þ t 2 Þ

0

- gðt 1 þ t 2 þ t 3 Þ j2 dt 3 :

ð11:44Þ

Figure 11.10 shows how the integrands in Eqs. (11.43) and (11.44) depend on time t3 for three values of t2 at a fixed value of t1. The Kubo relaxation function (Eq. (10.69)) was used for exp(g(t))]. If pulse 1 precedes pulse 2 so that Eq. (11.43) applies, and if t2 is close to zero, the signal peaks when t3 ≈ t1, demonstrating the expected rephasing by path R2 (Fig. 11.10A). The peak decreases in amplitude and moves toward t3 = 0 as t2 is lengthened and the coherence created by pulse 1 decays. When pulse 1 follows pulse 2 so that Eq. (11.44) applies, the peak occurs at t3 = 0 for all values of t2 because path R1 does not support rephasing (Fig. 11.10B). |R1|2 and |R2|2 become equivalent as t2 goes to 1 and the system loses all memory of the first pulse. Comparison of the curves shown with dotted blue lines in Figs. 11.10A and B show that |R1|2 and |R2|2 in the system considered there have become essentially identical when t2 is 100 time units.

11

Pump-Probe Spectroscopy, Photon Echoes, Two-Dimensional Spectroscopy. . .

Fig. 11.11 Dependence of integrated three-pulse photonecho signals on t1 (the delay between pulses 1 and 2), as calculated in the impulsive limit with Eq. (11.43) for t1 < 0 and with Eq. (11.44) for t1 > 0. The delay between pulses 2 and 3 (t2) was 0, 10, or 100, as indicated. The units of time are arbitrary. All calculations used the Kubo relaxation function (Eq. (10.69)) with τc = 40 time units and σ = 0.1 reciprocal time units

1.0 Integrated Signal

552

0.8

σ = 0.1, τc = 40

t2 = 0

0.6 10

0.4

100

0.2 0 -20

-10

0

10

20

30

t1 / arbitrary units

Figure 11.11 shows how the integrated signals depend on t1 for the same three values of t2. Eq. (11.44) was used when τ1 < 0 and Eq. (11.43) when τ1 > 0. Again, positive values of τ1 result in an echo peak that collapses toward the origin as t2 is increased. Fleming and coworkers [10, 49–54] have shown that the effect of t2 on the shift of this peak away from zero on the t1 axis (the three-pulse photon-echo peak shift) provides a particularly useful measure of the kinetics of dynamic dephasing. Plots of this effect are illustrated in Fig. 11.12 for several values of the correlation time constant τc in the Kubo relaxation function. As discussed in Sect. 10.7, the Kubo function includes both static (Gaussian) and dynamic (exponential) dephasing originating in energy fluctuations with a single correlation time constant. The three-pulse photon-echo peak shift recovers the dynamic component of this dephasing. Photon echoes also can be obtained by using only two pulses instead of three. These can be described with the same theoretical formalism by letting the second and third pulses coincide and setting k2 = k3 and j Eo2 ðt Þ j = j Eo3 ðt Þ j [55]. Since their first observation with ruby laser pulses lasting 10-8 s [56], photon echoes generated with a variety of pulsed lasers have been used to study solvation dynamics on time scales ranging from 10-14 to 10-1 s [10, 48–50, 52–54, 57– 64]. Applications to the motions of ligand-binding sites in proteins have included studies of myoglobin [65–67], Zn cytochrome c [68], bacteriorhodopsin [69], antibodies [70], and calmodulin [71]. Proteins generally show multiphasic relaxation dynamics consistent with hierarchical, richly textured potential surfaces. Relaxations with effective correlation times of 1.3 to 4 ns were found to occur in Zn cytochrome c even at 1.8 K [68]. IR photon echoes have been used to study the vibrational dynamics of peptides and small molecules [72, 73], and have been combined with isotopic labeling to probe the dynamics of the amide I vibration at a specific residue in a transmembrane peptide [74].

Two-Dimensional Electronic and Vibrational Spectroscopy

Fig. 11.12 Dependence of three-pulse photon-echo peak shift on delay time t2, calculated as in Fig. 11.11 using the Kubo relaxation function (Eq. 10.69) with τc = 10, 20, or 40 time units as indicated and σ = 0.05 (A) or 0.1 (B) reciprocal time units

A Peak Shift / arbitrary units

11.5

10

553

σ = 0.05

8

τc = 40

6 20

4 10

2 0

Peak Shift / arbitrary units

B σ = 0.1

6

τc =

4

40 20

2

0

10

0

20

40

60

80

100

t2 / arbitrary units

11.5

Two-Dimensional Electronic and Vibrational Spectroscopy

A powerful extension of pump-probe and photon-echo experiments is to vary the frequency of the light that is detected independently from the frequency of the excitation. A two-dimensional spectrum then can display the signal amplitude with the excitation frequency plotted on one axis and the detection frequency on the other. Such experiments now often employ heterodyne detection, in which third-order polarization of the sample following three short pulses of light is combined with the stronger field from a separate pulse called the “local oscillator,” and the combined radiation is dispersed in a spectrometer equipped with a diode-array detector (Fig. 11.13). The intensity of light with frequency ωs reaching the detector at time t is proportional to the modulus of the sum of the fields of the signal and the local oscillator (|Esig(t, ωs) + ELO(t, ωs)|2). This quantity has three components: |ELO(t, ωs)|2, which can be measured separately and subtracted, |Esig(t, ωs)|2, which

554

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Pump-Probe Spectroscopy, Photon Echoes, Two-Dimensional Spectroscopy. . .

Fig. 11.13 An apparatus for 2-dimensional spectroscopy. Short pulses of light from a laser enter at the top left and are directed into multiple paths by beam splitters (open rectangles) and mirrors (shaded rectangles). A parabolic mirror (PM) focuses pulses with three different wavevectors on the sample (S). Double arrows indicate movable mirrors that control the timing of the pulses. Radiation leaving the sample with a chosen wavevector (dashed line) is sent through a chopper (Ch) and combined with a beam of stronger pulses (LO, local oscillator). The combined radiation is dispersed in a spectrometer (Spec) and sent to a detector array (Det). Signals from the local oscillator alone are measured when the chopper blocks light coming from the sample and are subtracted. Polarizers and compensation plates are not shown. See [75, 91] for details and other optical schemes and information on data analysis

is negligible if the signal is much weaker than the local oscillator, and an interference term that depends on the product of the two fields. Mixing with the local oscillator thus can boost the signal substantially. If the local oscillator pulse is timed to be in phase with the third of the three pulses that impinge on the sample, cosine transforms of the interference term provide a one-dimensional spectrum of the signal amplitude versus ωs for a particular choice of the delay periods τ and T in Fig. 11.8 [75]. This technique depends on the use of sub-picosecond pulses, which inherently contain a broad spectrum of frequencies. To obtain the dependence of the signal on the excitation frequency (ωe), the measurements are repeated with different values of the delay between pulses 1 and 2 (τ in Fig. 11.8). A Fourier transform of the signal amplitude for a particular value of ωs has positive and/or negative peaks corresponding to frequencies where the pump light excites the sample. The whole experiment then is repeated with various values of the delay between pulses 2 and 3 (T ) to explore how the sample dephases or evolves in other ways after it is excited. The relative polarization of the pump and probe fields provides an additional independent variable. A two-dimensional plot displaying the excitation frequency on the ordinate and the signal frequency on the abscissa typically has one or more peaks on the diagonal representing ground-state bleaching, stimulated emission, or photon echoes at the excitation frequency. In addition, there can be off-diagonal signals that reflect excited states with different energies. These signals can reflect excited-state absorption, stimulated emission, coupling of excitations with different energies, conformational changes or energy transfer, and 2D spectroscopy can provide information on

11.5

Two-Dimensional Electronic and Vibrational Spectroscopy

555

Fig. 11.14 (A) Schematic depiction of a 2D spectrum of electronic transitions in a system of two weakly-interacting molecules. In the model considered here, molecule 1 has an absorption band centered at 12,500 cm-1 (800 nm) and molecule 2 has a band at 11,500 cm-1 (870 nm) with a similar dipole strength and band shape. The excitation pulses are assumed to cover a sufficiently broad band of frequencies so that they excite both molecules. Yellow to red colors indicate increases in the strength of the measured signal while green to blue colors indicate decreases. The positive signals along the diagonal represent ground-state bleaching and stimulated emission from the individual molecules immediately after the excitation (T = 0), and the negative off-diagonal signals represent excited-state absorption at the same time. (B) Spectrum for the same system measured with a delay of 5 ns between the excitation and probe pulses. Transfer of energy from molecule 1 to 2 increases the signals for molecule 2 and decreases those for 1. Structural fluctuations (spectral diffusion) make all the signals more symmetrical

the dynamics of these processes [76, 77]. Figure 11.14 illustrates 2D plots of electronic transitions in a system with two molecules that interact very weakly, which is the situation in Förster resonance energy transfer. The signals on the diagonal reflect ground-state bleaching of the individual molecules. These have positive signs because reduced absorbance increases |Esig(t, ωs)| at the detector. The off-diagonal signals with opposite sign represent excited-state absorption. In this illustration, both molecules have weak excited-state absorption bands on the lower-energy side of the ground-state absorption bands. The stretching of the signals along the diagonal at zero time (T = 0, Fig. 11.14A) results mainly from inhomogeneous broadening of the absorption spectra. The narrower width in the anti-diagonal direction reveals the homogeneous absorption bandshape of molecules that are excited selectively by light with a particular energy. Fluctuations of the structure and interactions with the surroundings (spectral diffusion) cause the signals to become more symmetrical with time (Fig. 11.14B). In addition, resonance energy transfer causes the signals representing the molecule that absorbs at lower energies to grow with time while those for the other molecule decay. Moca et al. [78] have described the effects of interactions with the solvent on 2D electronic spectra of chlorophyll a in solution. Figure 11.15 illustrates 2D plots of electronic transitions in a dimer with strong interactions between the two molecules. Here the positive signals on the diagonal

556

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Pump-Probe Spectroscopy, Photon Echoes, Two-Dimensional Spectroscopy. . .

Fig. 11.15 Spectra similar to those of Fig. 11.14 but for two molecules with strong electronic interactions. The dimer is assumed to have exciton absorption bands at 11,500 and 12,500 cm-1. Signals on the diagonal represent ground-state bleaching of these bands and stimulated emission from the exciton states. The positive (orange to red) off-diagonal signals at T = 0 (A) show that excitation in either exciton band causes bleaching of both bands. Excitation in either band also results in negative (blue) bands representing excitations to a doubly-excited state. At T = 50 ps after the excitation pulse, (B) both exciton bands remain bleached but the decrease in excited-state absorption by the higher-energy state indicates that the system has largely relaxed to the lowerenergy state

represent the bleaching of the two exciton bands. Excitation at the frequency of either band bleaches both bands immediately and results in the appearance of a negative signal representing excited-state absorption. Structural fluctuations make all the signals become more symmetrical with time but have little effect on their relative amplitudes. Excitation from level 0 to level 1 of a vibrational mode gives a diagonal peak from ground-state bleaching of this absorption, and an off-diagonal peak with opposite sign representing excitation of the same mode from level 1 to level 2, as shown schematically in Fig. 11.16A. The difference between the signal frequencies for these peaks reveals the anharmonicity of the vibrational mode. Such measurements have been described for NO and CO bound to myoglobin and other hemoproteins, where the ligand stretching mode has a fundamental frequency in the range of 1900 to 1930 cm-1 and an anharmonicity of about 30 cm-1 [79–82]. Two-dimensional IR (2D-IR) spectra of CO-myoglobin exhibit additional off-diagonal peaks that are attributable to switching between different conformational states [80]. These peaks are not seen at short times after excitation, but develop as the delay between pulses 2 and 3 (T) is increased, providing a direct measure of the conformational dynamics (Fig. 11.16B). In a system with multiple chromophores, energy transfer between pigments with differently oriented transition dipoles can give the off-diagonal signals a dependence on the relative polarization of the pump and probe fields. Measurements of such

11.5

Two-Dimensional Electronic and Vibrational Spectroscopy

557

Fig. 11.16 Schematic depictions of 2-dimensional IR spectra of stretching modes of CO and NO ligands bound to hemoproteins. The excitation frequency is displayed on the ordinate and the signal frequency on the abscissa. (A) The N–O stretching mode gives a diagonal peak in the region of 1900 cm-1 and an off-diagonal signal with an opposite sign near 1870 cm-1. The main peak represents excitation from level m = 0 to m = 1; the lower-energy peak, excitation from m = 1 to m = 2. The off-diagonal peak is seen with very short values of the population time (T ) as well as with longer delays. Structural disorder (inhomogeneous broadening) stretches the 0–1 peak along the diagonal at T = 0, and rapid structural fluctuations (spectral diffusion) make the peak more symmetrical with time. (B) Spectra of the C–O stretching mode in CO-myoglobin have two diagonal peaks that represent distinct conformational states. Off-diagonal peaks develop as T increases, indicating that interconversions between the two states occur on the time scale of 50 ps. Spectral diffusion again causes the peaks to become more symmetrical with time

two-dimensional spectra have provided insight into the pathways by which excitations move through photosynthetic antenna complexes and reaction centers [83–94]. One surprising result that has emerged from these studies is that oscillatory signals reflecting electronic coherences sometimes last longer than individual excitons, indicating that excitations move coherently through the structure, rather than hopping stochastically from pigment to pigment [87, 90, 91, 95–100]. One possible explanation of these long-lived coherences is that the dominant vibrational modes of the complex modulate the energies of the energy donor and acceptor in a

558

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Pump-Probe Spectroscopy, Photon Echoes, Two-Dimensional Spectroscopy. . .

correlated manner rather than incoherently. This could help to make energy capture robust to thermal fluctuations of the structure. Two-dimensional IR spectroscopy has been used to examine exciton interactions, internal Stark shifts, and relaxation mechanisms of amide I and II transitions in peptides and proteins [73, 75, 101–110]. Spreading the excitation energies along a second coordinate can separate components that are difficult to resolve in broad one-dimensional IR spectra. Two-dimensional UV spectroscopy has been applied to nucleic acid bases and nucleotides and holds promise for exploring the transfer of vibrational energy and the migration and localization of excitons in DNA [111–115]. These processes are pertinent to understanding how DNA avoids photodamage.

11.6

Transient Gratings

If two plane waves of light overlap in an absorbing medium, their fields create sinusoidal interference patterns like those shown in Fig. 11.17. In regions where the interference is constructive, the absorbance of the medium can change as a result of ground-state bleaching or excited-state absorption; where it is destructive, little is changed. The spacing of the excitation bands is λband = λex/2 sin (θex/2), where θex is the angle between the wavevectors of the two beams. The density of the medium also can change in the same pattern if the excitation causes local heating or volume changes, and this will affect the refractive index. Such bands of modified absorbance or refractive index can act as a diffraction grating to diffract a probe beam whose wavelength (λprobe) and angle of incidence relative to the mean of the two wavevectors (θprobe) satisfy the Bragg condition jλprobe = 2λband/cos (θprobe), where j is an integer. The grating, however, will disappear as the excited molecules return to their original state or diffuse away from their initial positions and as the heat generated by the excitation also diffuses away. The diffraction of the probe beam thus provides a way of measuring the decay dynamics of the excited state or volume changes caused by the excitation [116]. Figure 11.18 shows a typical pump-probe arrangement for studying transient gratings on picosecond time scales [116–119]. In this scheme, a train of pulses from a laser is split into three beams, two of which converge in the sample to create the grating. The third beam is used to generate probe pulses at the same or a different frequency. The probe pulses strike the sample at an adjustable time after the pump pulses, and the portion of the probe beam that diffracts off the grating passes through a slit to reach the detector. For measurements on time scales of nanoseconds or longer, a continuous beam from a separate laser is used as the probe, and the intensity of the diffracted beam is recorded in real-time with an oscilloscope or transient-digitizer [120, 121]. Contributions to a transient grating from absorbance changes can be distinguished from effects involving the real part of the refractive index by their different dependence on the probe frequency [118]. Gratings created by thermal expansion

11.6

Transient Gratings

559

1

1

D

y / λex

A 0

0

-1

-1

1

E

y / λex

B 0

-1

y / λex

1

1

F

C 0

0

-1

-1 -1

0 x / λex

1

-1

0

1

x / λex

Fig. 11.17 Formation of a transient grating by overlapping plane waves. (A, B) Image plots of the electric fields at zero time in two plane waves propagating to the right and downward (A) or upward (B) at an angle of 30° relative to the horizontal (x) axis (θex = 60°). The x and y coordinates are given relative to the wavelength of the radiation (λex). Black indicates positive fields; white, negative. (C) The sum of the fields in (A) and (B). (D–F) The same as (A–C), respectively, at time 1/0.25νex, where νex is the frequency of the radiation. The vertical nodes in (C) and (F) move from left to right with time (note, e.g., the vertical gray stripe at x/λex = 0 in (C) and at x/λex = 0.25 in (F), while the horizontal nodes (e.g., y/λex = ± 0.5) are stationary. The combined fields therefore will interact with the medium mainly in horizontal bands at y/λex = - 1, 0, + 1, etc. In general, plane waves intersecting at angle θex will give bands separated by y/λex = 1/2 sin (θex/2)

can be recognized by their rapid decay because thermal diffusion often is faster than other processes that follow the excitation [121, 122]. Transient-grating techniques thus offer a useful alternative to photoacoustic spectroscopy, in which thermal

560

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Pump-Probe Spectroscopy, Photon Echoes, Two-Dimensional Spectroscopy. . .

Fig. 11.18 (A) Apparatus for picosecond transient-grating experiments. Two parallel beams of picosecond pulses (heavy blue dotted lines) are focused by a lens so that they overlap in the sample. A probe beam of picosecond pulses (light cyan dotted lines) is focused by the same lens so that it strikes the overlap region at the Bragg angle. Probe light that diffracts off the transient grating in the sample passes through a slit to a detector. A variable delay path is used to control the time between the excitation and probe pulses as in ordinary pump-probe experiments. (Another delay stage, not shown, is used to adjust the path of one of the excitation beams so that the two excitation pulses reach the sample at the same time.) (B) Expanded view of the region where the excitation beams (light gray areas) intersect in the sample, showing the orientation of the transient grating (darker gray bands)

effects usually are identified by their dependence on the solvent’s coefficient of thermal expansion (see, e.g., [123, 124]). Applications have included studies of volume and enthalpy changes that follow the photodissociation of carbon monoxide from myoglobin and hemoglobin [125–130], excitation of photoactive yellow protein [131] and rhodopsin [132], and folding of cytochrome c [133, 134]. Transient gratings also have been used to study exciton migration in photosynthetic antenna complexes [135]. Transient gratings also can be examined on a femtosecond time scale as a function of the time between the two pulses that create the grating [136]. As our discussion in Sect. 11.3 suggests, the two radiation fields do not need to be present in the sample simultaneously; the second field can interfere constructively or destructively with coherence generated by the first. This makes femtosecond transientgrating experiments potentially useful for exploring relaxations that destroy such coherence. However, photon-echo experiments provide a more thoroughly developed path to this end.

11.7

11.7

Vibrational Wavepackets

561

Vibrational Wavepackets

Fluorescence from molecules that are excited with short flashes can exhibit oscillations that reflect coherent excitation of multiple vibrational levels. Suppose that an ensemble of molecules occupies the lowest nuclear wavefunction of the ground electronic state. The probability that the flash will populate vibrational level k of an excited electronic state then depends on the electronic transition dipole, the spectrum and intensity of the flash, and the overlap integral hχ k(e) j χ 0(g)i, where χ 0(g) and χ k(e) are the spatial parts of the ground and excited-state vibrational wavefunctions, respectively. If the flash includes a broad band of energies relative to the spacing of the vibrational eigenvalues, it will excite multiple vibronic levels ( j, k, l, ...) coherently, and this coherence will be expressed in the off-diagonal elements of the vibrational density matrix. Consider, for example, an individual molecule that has a single vibrational mode with energy hυ. If we neglect vibrational relaxations and decay of the excited electronic state following the flash, the off-diagonal density matrix elements will oscillate at frequencies that are various multiples of υ: 

  ρjk ðt Þ = ρjk ð0Þ exp -i E j - E k t=ħ = ρjk ð0Þ exp½-2πiðj - kÞυt :

ð11:45Þ

Constructive and destructive interference between these oscillations give rise to oscillatory features in the fluorescence. An excited ensemble with vibrational coherence can be described by a linear combination of vibrational wavefunctions: Χðu, t Þ =

X C k χ kðeÞ ðuÞ exp½-2πiðk þ 1=2Þυt :

ð11:46Þ

k

Here u represents a dimensionless nuclear coordinate and χ k(e)(u) again denotes the spatial part of basis function k. A system with such a combination of wavefunctions is called a wavepacket. The coefficients Ck represent averages over the ensemble. Making the Born-Oppenheimer approximation and neglecting relaxations of the excited state and nonlinear effects in the excitation, they are given by C k ≈ N -1

E X exp½-ðj þ 1=2Þhυ=kB T  D χ kðeÞ jχ jðgÞ I k,j , Z j

ð11:47Þ

where Ik, j is the spectral overlap of the excitation pulse with the homogeneous absorption band for the vibronic transition from level j of the ground state to level k of the excited state, kB and T are the Boltzmann constant and temperature, Z is the vibrational partition function, and N‐1 is a factor that depends on the electronic dipole strength and the intensity and width of the excitation flash. Figure 11.19B shows the probability function |X(u, t)|2 for such a wavepacket immediately after the excitation and at several later times. For this illustration, we used one-dimensional harmonic oscillator wavefunctions as the basis and assumed that all the molecules start in the

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Pump-Probe Spectroscopy, Photon Echoes, Two-Dimensional Spectroscopy. . .

B 0.10

A

1 2

3

4 5

|Χ(u,t)|

2

15

0.05 Energy

10

0 5

Vibrational Coordinate

0 Vibrational Coordinate Fig. 11.19 (A) Wavefunctions and relative energies for a one-dimensional harmonic oscillator with a displacement (Δ) of 2.0 between the ground and excited electronic states. The abscissa is the dimensionless coordinate u = (2πmrυ/h)-1/2x, where mr and υ are the reduced mass and classical vibration frequency (Eq. (2.28)); the origin is midway between the potential minima of the ground and excited states. The dashed lines show the potential energies. The 0–0 energy difference between the ground and excited states is arbitrary. (B) The probability function (|X(u, t)|2) of a wavepacket at t = 0, τ/8, τ/4, 3 τ/8, and τ/2 (curves 1–5, respectively), where τ = 1/υ. The vibrational coordinate is expressed relative to the minimum of the ground-state potential as in (A). Vibrational levels from k = 0 to 12 were included in the wavepacket with overlap integrals hχ k(e) j χ 0(g)i calculated as described in Box 4.13. The excitation flash was centered at the Franck-Condon absorption maximum, which for Δ = 2 is 1.5υ above the 0–0 energy. (The squares of the overlap integrals for k = 0 to 7 are 0.368, 0.520, 0.520, 0.425, 0.300, 0.190, 0.110 and 0.059.) The pulse included a broad band of energies (FWHM >> hυ), so that Ik, 0/N ≈ 1 in Eq. (11.47) for all the vibrational levels of the excited state that overlap significantly with χ 0(g). Vibrational relaxations and dephasing were neglected

lowest vibrational level of the ground state, which will be the case if T τ1. Show that the intersecting waves in a transient-grating experiment will create bands of excitation intensity separated by a distance λex/2 sin (θex/2). Explain why, in the semiclassical wavepacket picture, the width of an absorption band increases with the slope of the excited-state potential surface in the region of the Franck-Condon maximum, whereas the width of an emission band increases with the corresponding slope of the ground-state potential surface.

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Raman Scattering and Other Multi-photon Processes

12.1

12

Types of Light Scattering

The vibrational transitions discussed in Chap. 6 occur by absorption of a photon whose energy matches a vibrational energy spacing, hυ. Vibrational or rotational transitions also can occur when a molecule scatters light of higher frequencies; this is the phenomenon of Raman scattering. Raman scattering is one of a group of twophoton processes in which one photon is absorbed and another is emitted essentially simultaneously. Figure 12.1 illustrates the main possibilities. Rayleigh scattering and Mie scattering are elastic processes, in which there is no net transfer of energy between the molecule and the radiation field: the incident and emitted photons have the same energy (Fig. 12.1, transition A). Raman scattering is an inelastic process in which the incident and departing photons differ in energy and the molecule is either promoted to a higher vibrational or rotational level of the ground electronic state or demoted to a lower level. Raman transitions in which the molecule gains vibrational or rotational energy, called Stokes Raman scattering (Fig. 12.1, transition B), usually predominate over transitions in which energy is lost (anti-Stokes Raman scattering, Fig. 12.1, transition C) because resting molecules populate mainly the lowest levels of any vibrational modes with hυ > kBT. The strength of anti-Stokes scattering increases with temperature, and the ratio of anti-Stokes to Stokes scattering provides a way to measure the effective temperature of a molecule. Both Stokes and antiStokes Raman scattering increase greatly in strength if the incident light falls within a molecular absorption band (Fig. 12.1, transition D). The scattering then is termed resonance Raman scattering. Other types of light scattering involve a transfer of different forms of energy between the molecule and the radiation field. In Brillouin scattering, the energy difference between the absorbed and emitted photons creates acoustical waves in the sample; in quasielastic or dynamic light scattering, the energy goes into small changes in velocity or rotation. In two-photon absorption, the second photon is absorbed rather than emitted, leaving the molecule in an excited electronic state whose energy is the sum of the energies of the two photons. # The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. W. Parson, C. Burda, Modern Optical Spectroscopy, https://doi.org/10.1007/978-3-031-17222-9_12

583

584

12

Raman Scattering and Other Multi-photon Processes

Fig. 12.1 Types of light scattering. The solid green horizontal lines at the bottom represent vibrational levels of the ground electronic state. In Rayleigh and Mie scattering (A) a molecule or particle absorbs a photon and emits a photon with the same energy. Rayleigh scattering refers to scattering by molecules or particles that are small relative to the wavelength of the scattered light; Mie scattering, to scattering by larger particles. In Raman scattering (B–D), a photon with different energy is emitted, leaving the molecule at either a higher or lower vibrational level (Stokes (B) and anti-Stokes (C) Raman scattering, respectively). Resonance Raman scattering (D) occurs if the photon energies match transition energies to a higher electronic state (red line)

Raman scattering was discovered in 1928 by the Indian physicist C.V. Raman and his student K.S. Krishnan. In their early experiments, monochromatic light from a mercury arc lamp passed through various liquids and a spectrometer that Raman had invented was used to disperse the scattered radiation. Raman was awarded a Nobel Prize for this work in 1930. Today Raman scattering usually is measured by irradiating a sample with a narrow spectral line from a continuous laser, but timeresolved measurements also can be made by using a pulsed laser as the light source. Light scattered at 90° or another convenient angle from the axis of incidence is collected through a monochromator or with a diode array, and the intensity of the signal is plotted as a function of the difference in frequency or wavenumber between the excitation light and the scattered photons (νe - νs). The spectrum resembles an IR absorption spectrum (Fig. 12.2). However, the relative intensities of the Raman and IR lines corresponding to various vibrational modes generally differ, as we discussed in Chapter 6 and will elaborate below. Resonance Raman spectra of macromolecules also differ from IR spectra and off-resonance Raman spectra in that signals from bound chromophores can be much stronger than the background signals from the protein. In Fig. 12.2D, for example, the resonance Raman spectrum of the chromophore in GFP is readily observable whereas absorption due to the protein would completely overwhelm the IR absorption spectrum of the chromophore. (Also note

12.1

Types of Light Scattering

585

A

B

C

D

Fig. 12.2 Infrared absorption and Raman scattering spectra of the green fluorescent protein (GFP, green) and 4-hydroxybenzylidene-2,3-dimethyl-imidazolinone (HBDI, blue), a model of the GFP chromophore [1, 2]. (A) FTIR absorption spectrum of HBDI in a KBr pellet. (B) Off-resonance Raman emission spectrum of HBDI in ethanol with excitation at 532.0 nm (18,800 cm-1). (C) Resonance Raman emission spectrum of HBDI in ethanol with excitation at 368.9 nm (27,100 cm-1). (D) Resonance Raman emission spectrum of GFP in aqueous solution with excitation at 368.9 nm. The abscissa for the Raman spectra is the difference between the wavenumbers of the signal and the excitation light. The amplitudes of the Raman spectra are normalized to the peak near 1562 cm-1, which is assigned to an in-plane stretching mode of the C=N bond in the imidazolinone ring and the C=C bond between the phenolic and imidazolinone rings [1, 2]. The FTIR spectrum is normalized to the peak at 1605 cm-1, which represents a mode that is localized mainly to the phenolic ring. HBDI in neutral ethanol has an absorption maximum at 372 nm and GFP has a corresponding absorption band at 398 nm. See Fig. 5.9 for the structures of HBDI and the GFP chromophore

the higher signal/noise ratio in the resonance Raman spectrum of HDBI in Fig. 12.2C compared to the off-resonance Raman spectrum in Fig. 12.2B.) The Liouville-space diagrams in Fig. 12.3 help to clarify the main physical distinction between Raman scattering and ordinary fluorescence. Both processes require four interactions with a radiation field, and therefore four steps in Liouville space [3]. There are six possible pathways with four steps between the initial state whose population is indicated by a,a at the lower-left corner of Fig. 12.3A and the final state (b,b) at the upper right: the three paths shown in Fig. 12.3B–D and their

586

12

Raman Scattering and Other Multi-photon Processes

Fig. 12.3 Liouville-space diagrams for spontaneous fluorescence and Raman scattering. (A) Liouville-space pathways connecting an initial state (a), intermediate state (k) and a final state (b). (See Sect. 11.1, Figs. 11.1, 11.4 for an explanation of these diagrams.) Populations are shown as yellow filled circles; coherences with empty circles. (B–D) Three of the six possible paths from a to b with four steps (four interactions with a radiation field). The other three paths are the complex conjugates of the ones shown. All six paths contribute to spontaneous fluorescence; Raman scattering involves only path (D) (and its complex conjugate), in which the intermediate state is never populated. (E) A double-sided Feynman diagram for path (D)

complex conjugates. Ordinary fluorescence occurs by paths B and C, whereas Raman scattering and the other two-photon processes discussed in this chapter occur by path D. Inspection of the Liouville diagrams shows that paths B and C proceed through an intermediate state (k) that is transiently populated and thus is, in principle, measurable. Path D passes through two coherences with this state (a,k and k,b) and a coherence between the initial and final states (a,b), but never generates a population in state k. Raman scattering thus differs from fluorescence in involving only a “virtual” intermediate state that is not directly measurable. A double-sided Feynman diagram for path D is shown in Fig. 12.3E. Experimentally, Raman scattering differs from fluorescence in several ways. First, Raman emission lines are much narrower than fluorescence emission spectra. Raman lines for small molecules in solution typically have widths on the order of 10 cm-1 as compared to several hundred cm-1 for fluorescence. Second, the emission spectra have very different dependences on the frequency of the excitation light. Fluorescence emission spectra of many molecules are essentially independent of the excitation frequency, whereas Raman lines shift linearly with νe to maintain a

12.2

The Kramers-Heisenberg-Dirac Theory

587

constant value of jνs -νej. This reflects the requirement for the overall conservation of energy during the Raman process (νs + υ = νe for Stokes Raman scattering, νs υ = νe for anti-Stokes), in which the virtual intermediate has no opportunity to equilibrate thermally with the surroundings. Whereas spontaneous fluorescence typically has a lifetime of several ns, Raman scattering follows the time course of an excitation pulse with essentially no delay. Finally, the integrated strength of off-resonance Raman scattering usually is much lower than that of fluorescence. In the classical explanation of Raman scattering, the incident electromagnetic radiation field Eocosωe creates an oscillating induced dipole whose magnitude depends on the product of the field and the polarizability of the medium. The induced dipole constitutes the source of the radiation we detect as scattered light. If the polarizability (α) is modulated at a lower frequency by a molecular vibration (α = αo + α1cosωm), the induced dipole will be proportional to the product → E o cos ωe ðαo þ α1 cosωm Þ , which is the same as Eoαocos(ωe) + (Eoα1/2)[cos (ωe + ωm) + cos(ωe - ωm)]. The scattering thus will have components at frequencies ωe ± ωm in addition to ωe. The classical theory predicts correctly that the strength of Raman scattering depends on the extent to which a vibration changes the molecular polarizability as we discuss in Sect. 12.4, although it does not account readily for the difference between the strengths of Stokes and anti-Stokes scattering. Kramers and Heisenberg [4], who predicted the phenomenon of Raman scattering several years before Raman and Krishnan discovered it experimentally, advanced a semiclassical theory in which they treated the scattering molecule quantum mechanically and the radiation field classically. Dirac [5] soon extended the theory to include quantization of the radiation field, and Placzec, Albrecht, and others explored the selection rules for molecules with various symmetries [6, 7]. A theory of the resonance Raman effect based on vibrational wavepackets was developed by Heller, Mathies, Meyers, and their colleagues [8–13]. Mukamel [3, 14] presented a comprehensive theory that considered the nonlinear response functions for pathways in Liouville space. Having briefly described the pertinent pathways in Liouville space above, we will first develop the Kramers-Heisenberg-Dirac theory by a second-order perturbation approach and then turn to the wavepacket picture.

12.2

The Kramers-Heisenberg-Dirac Theory

Consider a molecule with ground-state wavefunction Ψ a and excited-state wavefunction Ψ k, and energies Ea and Ek. When a weak, continuous radiation field with frequency νe and amplitude Ee[exp(2πiνet) + exp (-2πiνet)] is introduced, the coefficient for state k (Ck) oscillates with time. We need an expression for Ck that incorporates uncertainty in the energies caused by electronic dephasing or decay of state k. We can find Ck at a short time (τ) by evaluating the density matrix element ρka ðτÞ for an ensemble of molecules exposed to steady-state illumination. Recall that, in the Schrödinger representation,

588

12

Raman Scattering and Other Multi-photon Processes

ρka ðτÞ = ck ðτÞca * ðτÞ = C k ðτÞC a * ðτÞ exp½ - iðE k - Ea Þτ=ħ,

ð12:1Þ

where the bars indicate averaging over the ensemble (Eqs. 10.8 and 10.12). If we replace Ca * ðτÞ by 1 on the assumption that virtually all the molecules are in the ground state, then Ck ðτÞ = ρka ðτÞ exp½iðEk - Ea Þτ=ħ:

ð12:2Þ

We found in Chap. 10 that the steady-state value of ρka can be written ρka ≈ ρka expðiωe t Þ,

ð12:3Þ

with ρka =

ði=ħÞðρkk - ρaa Þμka  Ee iðωe - ωak Þ þ 1=T 2 →

ð12:4aÞ

→

μ ka  E e ≈ , E k - Ea - hνe - iħ=T 2

ð12:4bÞ

where μka is the transition dipole for absorption and T2 is the time constant for decay of electronic coherence between states a and k (Eqs. 10.40 and 10.42a). We have dropped the term with +hνe in place of -hνe, which is negligible when Ek > Ea, and in Eq. (12.4b) we have again set ρaa ≈ 1 and ρkk ≈ 0. Combining Eqs. (12.2–12.4a, 12.4b), and omitting the bar over Ck to simplify the notation gives Ck ðτÞ =

μka  Ee exp½iðEk - E a - hνe Þτ=ħ: Ek - E a - hνe - iħ=T 2

ð12:5Þ

Now suppose that a second radiation field, Es[exp(2πiνst) + exp(-2πiνst)], couples state k to some other state, b. This could be either the radiation created by spontaneous fluorescence of state k or an incident field that might cause stimulated emission (Sect. 12.6) or excite the molecule to a state with higher energy (Sect. 12.7). However, we assume that neither the first nor the second radiation field can, by itself, convert state a directly to b. Neglecting any population of state b that is present at zero time, we can find how the coefficient for state b (Cb) grows with time by continuing the same perturbation treatment that we used to find Ck(τ). But here we’ll retain terms with either +hνs or -hνs in the denominator so that Eb can be either greater or less than Ek. By applying Eq. (4.6a, 4.6b) to the transition from k to b, and considering a time t that is short enough so that Cb is small and Ca is still close to 1, we obtain i C b ðt Þ ¼ ðμbk  Es Þ ħ

Zt n 0

exp½iðE b - Ek þ hνs Þτ=ħ

o þ exp½iðEb - E k - hνs Þτ=ħ C k ðτÞdτ

12.2

The Kramers-Heisenberg-Dirac Theory

¼ αba

8 < Zt :

589

exp½iðE b - Ea þ hνs - hνe Þτ=ħdτ

0

)

Zt þ

exp½iðEb - E a - hνs - hνe Þτ=ħdτ ,

ð12:6Þ

0

with αba =

ðμbk  Es Þðμka  Ee Þ : E k - E a - hνe - iħ=T 2

Evaluating the integrals in Eq. (12.6) gives  exp½iðEb - E a þ hνs - hνe Þt=ħ - 1 Cb ðt Þ = αba E b - E a þ hνs - hνe  exp½iðEb - E a - hνs - hνe Þt=ħ - 1 þ : E b - E a - hνs - hνe

ð12:7Þ

ð12:8Þ

The first term in the brackets on the right-hand side of Eq. (12.8) accounts for Rayleigh and Raman scattering; the second accounts for two-photon absorption, which we’ll discuss in Sect. 12.7. If Eb = Ea, as is the case for Rayleigh scattering, the first term goes to it/ħ when νs = νe (Box 4.3), while the second term is negligible for any positive values of νe and νs. If states a and b are different, as they are in Raman scattering, the first term goes to it/ħ when Eb - Ea = hνe - hνs. Finally, if Eb >> Ea, the first term in the braces usually is small but the second term goes to it/ħ when Eb - Ea ≈ hνe + hνs. We will see below that the quantity αba defined by Eq. (12.7) is a measure of the polarizability of the scattering material. The intensity of Rayleigh or Raman scattering of nearly monochromatic light should be proportional to the integral of Cb*Cb over a narrow band of the frequency difference νe - νs (Eq. 4.8a–4.8c). If we perform this integration for the first term in the braces in Eq. (12.8), the rate of scattering becomes Sba = jαba j2

ρν ðνÞ , ħ2

ð12:9Þ

where αba is given by Eq. (12.7), and ρv(ν) is the number of radiation modes that meet the condition hνe - hνs = Eb - Ea. Equation (12.9) can be viewed as another manifestation of an expression we have seen previously in connection with absorption, fluorescence, and resonance energy transfer, Fermi’s golden rule. To this point, we have considered only a single intermediate state (k) between states a and b. A molecule generally will have many excited electronic states, each with many vibrational levels, and any of these vibronic states could serve as a virtual state for Rayleigh or Raman scattering. If we assume for simplicity that the dephasing time constant T2 is approximately the same for all the important vibronic

590

12

Raman Scattering and Other Multi-photon Processes

levels (clearly a significant approximation), then summing the contributions to Cb(t) gives   exp½iðE b - E a þ hΔνÞt=ħ - 1 X ðμbk  Es Þðμka  Ee Þ C b ðt Þ ≈ , ð12:10Þ E b - E a þ hΔν Ek - E a - hνe - iħ=T 2 k where Δν = νs - νe and we have again retained only the first term from the braces in Eq. (12.8). The matrix element for light scattering thus becomes αba =

X ðμ  E s Þðμ  E e Þ bk ka , E E hν - iħ=T 2 k a e k

ð12:11Þ

or for the particular case of Rayleigh scattering (states a and b the same and μka = -μak), αaa =

X ðμ  Es Þðμ  Ee Þ ka ka : E k - Ea - hνe - iħ=T 2 k

ð12:12Þ

Resonance Raman scattering occurs when the incident light falls within an absorption band so that Ek(e) – Ea(g) ≈ hνe for a set of vibronic levels (k) of the excited electronic state. Because the vibronic levels of this state will dominate the sum in Eq. (12.11), we can use the Born-Oppenheimer and Condon approximations (Sect. 4.10) to factor the transition dipole μka into a vibrational overlap integral (hXk(e) j Xa(g)i) and an electronic transition dipole that is averaged over the nuclear coordinates (μeg). Pulling the electronic transition dipoles out of the sum then yields     X XbðgÞ jXkðeÞ XkðeÞ jXaðgÞ αba ≈ μge  Es μeg  Ee , ð12:13Þ EkðeÞ - E aðgÞ - hνe - iħ=T 2 k where hXi j Xji is the vibrational overlap integral for states i and j. Equations (12.9) and (12.12) indicate that the strength of Rayleigh scattering depends on jμka  Eej2jμka  Esj2, where Ee again is the excitation field and Es is the field of spontaneous emission from the virtual excited state. From Chap. 4, we know that jμka  Eej2ρv(ν) = (2πf2/3cn)DkaI, where Dka, I, n, and f are, respectively, the dipole strength for excitation to state k, intensity of the incident light, refractive index and local-field correction factor (Eq. 4.12). From Eq. (5.12), jμka  Esj2 = (8πhn3νs3/ c3)jμka  Eej2. Because νe = νs = ν for Rayleigh scattering, the strength of Rayleigh scattering in photons/s is proportional to Iν3. Converting to units of energy/s gives an additional factor of hν. The Kramers-Heisenberg-Dirac theory thus reproduces the observed dependence of Rayleigh scattering on the fourth power of the frequency. This dependence of Rayleigh scattering on ν4 accounts for the blue color of the sky because the short-wavelength blue light from the sun is scattered in all directions while red light passes through the atmosphere more directly. The theory also predicts the polarization of the scattered light correctly. (See Sect. 12.9 and refs. [13, 15] for further discussion of directional aspects of light scattering.)

12.2

The Kramers-Heisenberg-Dirac Theory

591

The matrix elements given by Eqs. (12.11–12.13) also depend on a sum of products of weighted overlap integrals of vibrational level k of the excited electronic state with the initial and final levels of the ground state. Each term in the sum is weighted inversely by (Ek(e) – Ea(g) - hνe – iħ/T2). But note that we have considered only a single vibrational level of the initial system. In a more complete description, hXk(e)jXa(g)i is replaced by a sum of thermally-weighted products of overlap integrals as described in Box 4.14. The dephasing factor iħ/T2 in Eqs. (12.11–12.13) makes the matrix elements for Rayleigh and Raman scattering complex quantities. The real and imaginary parts can be separated by multiplying each term in the sum by 1 in the form of (Ek(e) – Ea(g) hνe + iħ/T2)/(Ek(e) – Ea(g) - hνe + iħ/T2). Dissecting Eq. (12.13) in this way gives "   X E kðeÞ - E aðgÞ - hνe X bðgÞ jX kðeÞ X kðeÞ jX aðgÞ 2 αba = jμeg j  2 EkðeÞ - E aðgÞ - hνe þ ðħ=T 2 Þ2 k ð12:14Þ   # ðħ=T 2 Þ X bðgÞ jX kðeÞ X kðeÞ jX aðgÞ þi  : 2 E kðeÞ - EaðgÞ - hνe þ ðħ=T 2 Þ2 A comparison of Eq. (12.14) with Eq. (10.43) shows that the imaginary part of αaa (the matrix element for resonance Rayleigh scattering) is proportional to the matrix element for ordinary absorption. The real part of αaa can be related to the refractive index (Box 3.3 and [16]). Figure 12.4 shows spectra of jαbaj2 for a resonance Raman transition between vibrational levels 0 and 1 of the ground electronic state, as calculated by the Kramers-Heisenberg-Dirac theory (Eq. 12.13) for a molecule with a single harmonic vibrational mode. The spectra are plotted as functions of the excitation frequency (νe) for several values of T2 and the displacement of the vibrational coordinate in the excited state (Δ). Note that these are excitation spectra for resonance Raman scattering, not plots of the emission intensity as a function of νe – νs (cf. Figure 12.2), and note also that they do not consider inhomogeneous broadening. For comparison, the figure also shows the homogeneous absorption spectra calculated as Im (αaa) for the same systems. Spectra of jαbaj2 resemble a homogeneous absorption spectrum in having peaks at the 0–0 transition frequency and at integer multiples of υ above this, where the excitation energy matches the energy difference between the ground and excited vibronic states. However, the relative heights of the peaks differ. Note, for example, that a peak at (νe - νoo)/υ = 2 is seen in the jαbaj2 spectrum if Δ = 1 (Fig. 12.4A,B), but is too weak to be resolved if Δ = 2 (Fig. 12.4C,D). In addition, whereas changing Δ redistributes the ordinary absorption among the vibronic peaks without altering the integrated absorbance, it affects the integrated strength of Raman scattering. This point is Rillustrated in Fig. 12.5A, where the integrated Raman scattering crosssection, jαbaj2dν, is plotted as a function of Δ. Scattering into the first excited vibrational level peaks at Δ ≈ 0.9. The dephasing time constant T2 also has different effects on Raman and absorption spectra. An integrated absorption spectrum is independent of ħ/T2, whereas the

12

Raman Scattering and Other Multi-photon Processes

A

B 4

15 10

aa)

(arb. units)

592

Im(

2

|

2 ba| ,

5 0

C4

D 1.0

2

0.5

|

2 ba| ,

Im(

aa)

(arb. units)

0

0

0

2 (

e-

4 oo) /

0

0

2 (

e-

4 oo) /

Fig. 12.4 Calculated resonance Raman excitation spectra (|αba|2, solid cyan curves) and homogeneous absorption spectra (Im (αaa), dark blue dotted curves) for a molecule with a single harmonic vibrational mode of frequency υ. αaa is for scattering at the excitation frequency, and αba is for scattering from the zero-point vibrational level into the first excited level (hν1 - hν2 = hυ). The abscissa is the difference between the excitation frequency (νe) and the 0–0 transition frequency (νoo), in units of the vibrational frequency (υ). The absorption spectra are normalized to the Raman excitation spectra at the highest peaks in each panel. (A): ħ/T2 = 0.1hυ and Δ (dimensionless displacement of the vibrational coordinate in the excited electronic state) = 1.0. (B): ħ/T2 = 0.2hυ, Δ = 1.0. (C): ħ/T2 = 0.1hυ, Δ = 2.0. (D): ħ/T2 = 0.2hυ, Δ = 2.0. All vibrational levels of the excited state up to k = 25 were included in the sums. The overlap integrals were calculated as explained in Box 4.13

integrated strength of Raman scattering decreases as the dephasing becomes faster (Fig. 12.5B). Comparisons of the absolute cross sections for Raman scattering and ordinary absorption thus provide a way to measure the dynamics of dephasing [11, 12]. In the classical picture of light scattering, light passing through a polarizable medium generates oscillating induced electric dipoles that then can radiate light in various directions. The intensity of the scattering depends on the square of the induced dipole moment, and thus on the square of the polarizability. Box 12.1. describes a quantum mechanical treatment of electronic polarizability and shows that the matrix elements for light scattering are indeed proportional to the matrix elements for polarizability.

12.2

The Kramers-Heisenberg-Dirac Theory

(arb. units)

A

593

B

25

8

20

6

15 4 2

5

|

ba

2

|d

10

0

0 0

1

2

0

0.1 / T2

0.2

R Fig. 12.5 Integrated Raman scattering excitation cross section, |αba|2dν, for a molecule with a single harmonic vibrational mode with frequency υ. The model system is the same as in Fig. 12.4. (A) The integrated cross section as a function of Δ with ħ/T2 fixed at 0.1hυ. (B) The integrated crosssection as a function of ħ/T2 with Δ fixed at 1.0

Box 12.1 Quantum Theory of Electronic Polarizability Electronic polarizability generally is described by a second-rank tensor, α, which means that applying an electric field along a particular axis can generate an induced dipole with components perpendicular, as well as parallel, to this axis. However, it always is possible to choose a molecular coordinate system in which α is diagonal and polarizability can be described by a vector with magnitude α. Applying a field along the x, y, or z axis of this coordinate system generates an induced dipole only along the same axis. These are called the “principal axes” of polarizability. For an isotropic sample, the magnitude of the induced dipole is proportional to the scalar quantity α = (1/3)Tr(α), which does not depend on the choice of the coordinate system. In a quantum mechanical picture, polarizability reflects mixing of the ground state with higher-energy states when a molecule is perturbed by an external electric field. We’ll describe the theory first for a static field and then consider a time-dependent field. In the presence of a static field E, a molecule’s total electric dipole can be expressed as a Taylor’s series in powers of the field: μ = μ0 þ μind = μ0 þ α  E þ . . . ,

ðB12:1:1Þ

where μ0 is the permanent dipole in the absence of the field. μ0 and α also can be related to the first and second derivatives of the energy (E) with respect to the field: μ0 = -∂E=∂E,

ðB12:1:2Þ

and (continued)

594

12

Raman Scattering and Other Multi-photon Processes

Box 12.1 (continued) 2

α = -∂ E=∂E2 :

ðB12:1:3Þ

Equation (B12.1.2) is the same as Eq. (4.2). Equations (B12.1.2) and (B12.1.3) follow from a general theorem called the Hellman-Feynman theorem, which says that the derivative of the energy with respect to any parameter is equal to the expectation value of the derivative of the Hamiltonian with respect to that parameter. In the present case, the Hellman-Feynman theorem informs us that, for a system with wavefunction Ψ and eigenvalue E, ∂E=∂E = hΨ j∂H=∂EjΨ i = -hΨ j~ μjΨ i = -μ . To use this relationship, we first expand the energy as a Taylor’s series in powers of E:

 1 2 ðB12:1:4Þ E = E0 þ ð∂E=∂EÞ0 E þ ∂ E=∂E2 E2 þ . . . , 2 0 where all the derivatives are evaluated at E = (0,0,0). Differentiating Eq. (B12.1.4) with respect to E gives, according to the Hellman-Feynman theorem,

 2 ðB12:1:5Þ μ = ‐ð∂E=∂EÞ0 - ∂ E=∂E2 E - . . . , 0

Equating terms with the same powers of E in Eqs. (B12.1.1) and (B12.1.5) then yields Eqs. (B12.1.2) and (B12.1.3). Now consider a molecule with eigenfunctions Ψ k in the absence of external fields. The wavefunction in the presence of a static field can be written as a linear combination of these basis functions: X C k Ψk : ðB12:1:6Þ Ψ= k

To find αx by Eq. (B12.1.3), we must evaluate how the energy of this superposition state depends on the field. We can do this by following the procedure we have used to find how twostates with diabatic energies Ea and e 0 . Let state a be the ground state, and Ek are mixed by a weak perturbation H assume that the coefficient for any higher-energy state k is much smaller than that of state a(0 < j Ck j > 1, the terms of Eq. (12.27) for m1 ≠ m2 therefore sum to zero, leaving us with I s ðK Þ = jEsðoÞ j2 M

N X N X j=1 k=1

  

   exp 2πiK  rj - rk þ exp - 2πiK  rj - rk ,

ð12:28Þ

with both sums pertaining to the same molecule. Expanding the exponentials in this expression gives, to second order in jKj, I s ðK Þ = jEsðoÞ j2 M

N X N n X

o

 2 2 - 2πK  rj - rk - ⋯ :

ð12:29Þ

j=1 k=1

Inspection of Eq. (12.29) shows that the scattered irradiance at jKj = 0 is 2MN2jEs(o)j2. The intensity of the scattering at small angles thus provides a way of determining the number of segments in the molecule (N ), and from that, the molecular size [15, 134–136]. Since the segments are assumed to be identical, and so have the same mass, the sum of the quadratic terms in Eq. (12.29) is proportional to -(jKj2Rg2)/3, where Rg is the molecule’s radius of gyration [15]. The factor of (jKj2)/3 comes from summing the square of the dot product over random orientations of the vector (rj – rk) relative to K. The slope of a plot of Is versus jKj2 at small scattering angles therefore can be used to obtain Rg. Brownian diffusion and internal motions of a macromolecule cause the intensity of quasielastically scattered light to fluctuate with time, and the autocorrelation function of the scattering provides information on the dynamics of these motions as we discussed in Sect. 5.12 for fluorescence fluctuations. Berne and Pecora [15], Schurr [137, 138], Chu [136], and Brown [139] give expressions for autocorrelation functions that apply to various models for proteins and nucleic acids, along with further information on data collection and analysis. In addition to its applications in studies of macromolecules, dynamic light scattering has become a major analytical tool in nanotechnology [140–144]. It has, for example, been used with gold nanoparticles in an immunoassay for biomarkers of cancer [145]. The sizes of nanoparticles can be analyzed conveniently with

612

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desktop instruments that excite a sample with a laser in the visible or near-IR wavelength range and employ  a photomultiplier or an avalanche photodiode to detect the scattered light I S ðK Þ at several angles. For monodisperse particles, the normalized autocorrelation function of the signal often can be fit to an exponential decay function of the form

 ð12:30Þ Gðt Þ = exp - 2DjKj2 t , in which D is the particle’s translational diffusion coefficient; here jKj (the magnitude of scattering vector) again is (2/λ) sin (ϑ/2), where λ is the laser wavelength and ϑ is the scattering angle (Eq. 12.23). The translational diffusion coefficient of a spherical particle is related to the particle’s hydrodynamic radius (RH) by the StokesEinstein equation, RH = k B T=3πηD,

ð12:31Þ

where kB, T and η are, as usual, the Boltzmann constant, temperature, and viscosity of the medium. Smaller particles have a greater translational diffusion coefficient and produce a shorter autocorrelation time constant, which means more rapid fluctuations of the scattering. The hydrodynamic radius in Eq. (12.31) includes the solvent layers that move with the particle, along with any adsorbed ions or molecules. The autocorrelation function of dynamic light scattering by sample with a distribution of molecular sizes will have multiple components, and caution is necessary in interpreting data for such samples because the amplitude of the scattering generally is weighted in favor of the largest particles [141]. By measuring DLS under an oscillating electric field, it is possible to derive the surface potential (ζ ‐ potential) of nanoparticles, and measurements of ζ ‐ potentials have become an important component of nanoparticle research [146]. These measurements can provide guidelines for the rational design of nanoparticles for advanced uses in sensing, catalysis, and drug delivery [143].

12.10 Mie scattering by Larger Particles The theory described in the previous section pertains to scattering by particles that are small relative to the wavelength of the incident light. This restriction arises in our assumption that the amplitude of the electric field of the radiation (jjEej) is the same everywhere in the particle. Unlike Rayleigh scattering, which is emitted in all directions, scattering by larger particles becomes highly directional, proceeding mainly in the forward direction but with secondary peaks in other directions that depend on the size of the particles and the wavelength of the radiation. Its amplitude is much greater than that for Rayleigh scattering and varies in an oscillatory manner with the ratio of the particle diameter to the wavelength (Fig. 12.13). The peak scattering by suspensions of spherical gold nanoparticles, for example, changes from red to orange as their diameter increases from 200 to 800 Å. Such scattering is

12.10

Mie scattering by Larger Particles

613

Fig. 12.13 Relative amplitude of light scattering in the backward direction by spherical metallic particles as a function of the frequency of the radiation. The abscissa is the ratio of the particle’s circumference to the wavelength. In the Rayleigh regime of particles much smaller than the wavelength, the amplitude of the scattering increases as the fourth power of the wavelength (blue line). Particles whose sizes are comparable to the wavelength give Mie scattering with an amplitude that is an oscillatory function of the frequency. The amplitude of Mie scattering becomes constant (horizontal red line) when the particles are much larger than the wavelength. Adapted from [147]

generally called Mie scattering after Gustav Mie, who used Maxwell’s equations of electromagnetism to develop the theory for elastic scattering by colloidal suspensions of metallic spheres [148]. The complex angular dependence of the intensity and the oscillatory dependence on the size of the particles result from interference between radiation that is refracted in outer layers of the particles or surface plasmons and radiation that passes by on either side. When the size of the particle becomes much larger than the wavelength of the light, Mie scattering converges with classical geometric optics. Dust, pollen and the microscopic water droplets that form clouds are common causes of Mie scattering. Clouds appear white or grey because the water droplets scatter light of all wavelengths approximately identically. Mie scattering has been used to characterize the size and shapes of malarial parasites and red blood cells [149, 150]. Further information on Mie scattering is available in books by Kerker [135], van de Hulst [151], and Bohren and Huffman [152] and computer programs for calculating the amplitude of scattering as a function of wavelength and other parameters can be found on-line [153].

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12.11 Questions 1. The figure below shows an emission spectrum measured when a protein containing tryptophan was excited at 290 nm. (a) Suggest an assignment for the peak at 310 nm and point out the features of the spectrum that support the assignment. (c) How could you test your explanation?

2. Panels B, C and D of Fig. 12.3 show three of the six possible Liouville-space path diagrams for light-matter interactions leading from state a to b in four steps. (a) Draw the other three paths. (b) Indicate which of your newly drawn diagrams represents a Raman scattering process and explain this assignment. 3. The strength of resonance Raman scattering by a molecule typically decreases strongly with increasing temperature, while the absorbance and fluorescence change very little. Explain this observation using (a) the Kramers-HeisenbergDirac theory and/or (b) the semiclassical wavepacket theory. 4. (a) How many vibrational modes does carbontetrachloride (CCl4) have? (b) Assign the symmetries of the modes and identify which modes are active for IR absorption and which ones for Raman scattering. (c) Sketch the atomic movements in each mode. 5. (a) Explain why a symmetric vibrational mode that makes little or no contribution to the IR absorption spectrum can contribute strongly to the Raman spectrum. (b) Does the formal selection rule Δm = ±1, where m is the vibrational quantum number, apply to both resonance and off-resonance Raman scattering? 6. How does coherent anti-Stokes Raman scattering resemble, but differ from (a) ordinary anti-Stokes Raman scattering, and (b) ordinary stimulated emission? 7. (a) Why is two-photon absorption usually several orders of magnitudes weaker than one-photon excitation? (b) What are the potential advantages of two-photon excitation relative to one-photon excitation in fluorescence microscopy? (c) How do the selection rules for two-photon excitation differ from those for one-photon excitation?

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Appendix A

Vectors Vectors are used to represent properties that have both magnitude and direction. Scalars have a magnitude but no direction. Velocity, for example, is a vectorial property, while mass is a scalar. A vector in an N-dimensional coordinate space has N independent components (Ak), each parallel to one of the coordinate axes. In the text we denote a vector by a bold-face letter in italics or by enclosing a list of the individual components in parentheses: A = ðA1 , A2 , A3 , . . .Þ:

ðA1:1Þ

In a 3-dimensional coordinate system, for example, A = (Ax, Ay, Az) where Ax, Ay, and Az are the components parallel to the x, y, and z axes. The components can be arranged in either a row or a column. A vector with unit length parallel to the k axis is designated by a letter with a caret (^) on top (^k). The magnitude, modulus, or length of vector A is the square root of the sum of the squares of the individual components: !1=2 X 2 Ak : ðA1:2Þ jAj = k

The sum or difference of two vectors is obtained by simply adding or subtracting the corresponding components. In three dimensions, for example,   ðA1:3Þ A ± B = Ax ± Bx , Ay ± By , Az ± Bz : There are two types of vector products. The dot product or scalar product AB of vectors A and B is a scalar whose magnitude is the sum of the products of the corresponding components:

# The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. W. Parson, C. Burda, Modern Optical Spectroscopy, https://doi.org/10.1007/978-3-031-17222-9

623

624

Appendix A

A  B=

X Ak Bk :

ðA1:4Þ

k

The magnitude of A therefore can be written as |A| = (AA)1/2. In three dimensions, A  B = jAjjBj cos ðθÞ,

ðA1:5Þ

where θ is the angle between the two vectors. The cross product or vector product of two vectors, denoted A×B or A^B, is a vector that is perpendicular to both A and B and has magnitude |A||B|sin(θ). A×B is oriented in the direction in which a right-handed screw would advance if turning the screw rotates A onto B. Thus A×B and B×A have the same magnitude but point in opposite directions. In vector notation, A×B=

    Ay Bz - Az By , -½Ax Bz - Az Bx , Ax By - Ay Bx ,

which can be written in the form of a determinant:      ^x ^y ^z    A × B =  Ax Ay Az :    Bx By Bz 

ðA1:6Þ

ðA1:7Þ

The scalar triple product of three vectors, C  A × B = C  ðA × BÞ,

ðA1:8Þ

is a scalar whose sign changes if the order of any two of the vectors is interchanged: CA×B = -AC×B = BC×A = -BA×C. The scalar triple product is zero if any two of the three vectors are parallel. e is a vector whose The gradient of a scalar quantity A, which we will write ∇A, components are derivatives of A with respect to the corresponding coordinates. In three dimensions, the gradient operator is e = ð∂A=∂x, ∂A=∂y, ∂A=∂zÞ: ∇

ðA1:9Þ

Several other functions of the derivatives of vectors occur frequently in discussions of electromagnetic fields. The divergence of a vector A, written divA, is defined as e • A = ∂Ax þ ∂Ay þ ∂Az , div A = ∇ ∂x ∂y ∂z and the curl (curlA) is

ðA1:10Þ

Appendix A

625

Fig. A1.1 (A) The vector function -y^x + y^x has non-zero curl but zero divergence. (B) The function x^x + y^y has non-zero divergence but zero curl

   ^x ^y ^z     ∂Az ∂Ay   e curl A = ∇ × A =  ∂=∂x ∂=∂y ∂=∂z  = ^x   ∂y ∂z  Ax  Ay Az     ∂Ay ∂Ax ∂Ax ∂Az þ ^z : þ ^y ∂z ∂x ∂x ∂y

ðA1:11Þ

Figure A1.1a illustrates a vector function of x and y that has a non-zero curl but a divergence of zero. A vector with a non-zero divergence but a curl of zero is shown in Fig. A1.1b.

Matrices A matrix is an ordered, two-dimensional array of elements, Aij, with the first index (i) indicating the row in which the term is located in the array and the second ( j) indicating the column. Matrices are denoted in the text by letters in boldface or by enclosing the elements in square brackets. For example, if

41 73 A= , ðA2:1Þ 9 12 then A11 = 41, A12 = 73, A21 = 9 and A22 = 12. We are concerned mainly with square matrices, which are matrices in which the number of rows is the same as the number of columns. A diagonal matrix is a matrix in which non-zero elements occur only on the major diagonal, for example,

626

Appendix A



17 0 A= : 0 3

ðA2:2Þ

The trace, or character, of matrix A, denoted Tr[A], is the sum of the diagonal elements: X Akk : ðA2:3Þ Tr ½A = k

For example, the trace of the matrix in Eq. (A2.1) is Tr[A] = 41 + 12 = 53. The trace of a matrix obeys the distributive law of arithmetic: if C = A + B, Tr[C] = Tr[A] + Tr[B]. The sum or difference of two matrices A and B is obtained by adding or subtracting the corresponding elements. With 2×2 matrices, for example,





B11 B12 A11 ± B11 A12 ± B12 A11 A12 ± = : ðA2:4Þ A±B= A21 A22 B21 B22 A21 ± B21 A22 ± B22 The product of two matrices A and B (written AB or simply AB) is another matrix C whose elements are given by X C ij = Aik Bkj : ðA2:5Þ k

For example, the product of two 3×3 matrices is 3 2 2 B11 A11 A12 A13 7 6 6 A  B = 4 A21 A22 A23 5  4 B21 A31 A32 A33 B31

B12 B22 B32

B13

3

7 B23 5 B33

2

3 A11 B11 þ A12 B21 þ A13 B31 A11 B12 þ A12 B22 þ A13 B32 A11 B13 þ A12 B23 þ A13 B33 6 7 =4 A21 B11 þ A22 B21 þ A23 B31 A21 B12 þ A22 B22 þ A23 B32 A21 B13 þ A22 B23 þ A23 B33 5: A31 B11 þ A32 B21 þ A33 B31 A31 B12 þ A32 B22 þ A33 B32 A31 B13 þ A32 B23 þ A33 B33

ðA2:6Þ From Eqs. (A2.3 and A2.5), the trace of the product AB is XX XX Aik Bki = Aki Bik = Tr½BA: Tr½AB = i

k

k

ðA2:7Þ

i

And from this and the fact that ABC = A(BC) = (AB)C it follows that the trace of ABC is invariant to cyclic permutations: Tr½ABC = Tr½CAB = Tr½BCA: However, Tr[ABC] is not generally equal to Tr[CBA].

ðA2:8Þ

Appendix A

627

The product of a matrix A with a column vector B is a vector C with elements defined by X Ci = Aik Bk : ðA2:9Þ k

The transpose (AT) of matrix A is obtained by interchanging rows and columns, so that element Aij becomes Aji. The inverse (A-1) of A is a matrix that, when multiplied by A gives a diagonal matrix with all the diagonal terms equal to 1. So, for a 2×2 matrix, ‐1



A •A=

1 0 0 1

:

ðA2:10Þ

Such a diagonal matrix of 1’s is often denoted by a bold-face 1. Finding the inverse of a square matrix (inverting the matrix) is a common procedure that provides the solutions to sets of linear algebraic equations. Press et al. [1] give efficient algorithms for doing this. A matrix A is said to be symmetric if, for all its elements, Aij = Aji. It is Hermitian if, for all its elements, Aij = Aji*, where Aji* is the complex conjugate of Aji. All the matrices we discuss in the text are Hermitian. A matrix is called orthogonal if its transpose is the same as its inverse, so that AT • A = A‐1 • A = 1:

ðA2:11Þ

e is a matrix in which The gradient of a vector function A, which we write as ∇A, element Aij is the derivative of component i of the vector with respect to coordinate j. Thus if A = (Ax, Ay, Az), its gradient is 2 3 ∂Ax =∂x ∂Ay =∂x ∂Az =∂x 7 e =6 ðA2:12Þ ∇A 4 ∂Ax =∂y ∂Ay =∂y ∂Az =∂y 5: ∂Ax =∂z

∂Ay =∂z

∂Az =∂z

The solutions to many problems in quantum mechanics and spectroscopy require diagonalizing matrices. Given a non-diagonal matrix A, the task is to find another matrix C and its inverse C-1 such that the product C-1AC is diagonal. Computer algorithms are available for diagonalizing even large matrices rapidly [1].

Fourier Transforms The Fourier transform of a function f(t) of time is the integral Z1 f ðt Þ expð2π iν t Þ dt: F ðν Þ = -1

ðA3:1Þ

628

Appendix A

If f(t) is defined everywhere in the interval -1 < t < 1, and the integral of f(t)dt over this interval converges (i.e., is finite), the Fourier transform F(ν) also will converge. In addition, an inverse Fourier transform will regenerate the original function: Z1 F ðνÞ expð - 2π iν t Þ dν:

f ðt Þ =

ðA3:2Þ

-1

The pair of functions f(t) and F(ν) can be viewed as two different representations of the same physical quantity. For example, if f(t) expresses a quantity as a function of time (in seconds) F(ν) expresses the same quantity as a function of frequency (in cycles per second, or Hz). Sometimes it is convenient to use the angular frequency, ω = 2πν, in units of radians/sec; the transforms then must be scaled by a factor of (2π)-1/2: 1 F ðωÞ = pffiffiffiffiffi 2π

Z1 f ðt Þ expðiω t Þ dt

ðA3:3Þ

F ðωÞ expð - iω t Þ dω:

ðA3:4Þ

-1

and 1 f ðt Þ = pffiffiffiffiffi 2π

Z1 -1

The same expressions can be used with other pairs of variables. The Fourier transform of a function of position (in units of, say, Å) gives a function of inverse length (cycles per Å). The Fourier transform of an interferogram obtained in an FTIR spectrometer thus gives the intensity of radiation as a function of the wavenumber ν. In Chap. 2, we encountered a complex exponential function of the form f(t) = exp (-at-ibt) for t > 0 and f(t) = 0 for t < 0 (Eq. (2.69)). The Fourier transform of this function is  

1 1 i pffiffiffiffiffi F ð ωÞ = ðA3:5aÞ ħ 2π ω - b þ ia ! 

 a þ i ð ω - bÞ 1 i ω - b þ ia 1 = pffiffiffiffiffi = pffiffiffiffiffi , ðA3:5bÞ ħ 2π ω - b þ ia ω - b þ ia ħ 2π ðω - bÞ2 þ a2 where ω = E/ħ and dω = dE/ħ. Equation (2.71) is obtained by multiplying the real part of this expression by the normalization factor (2/π)1/2. The Fourier transform in Eqs. (3.5a, b) includes both real and imaginary parts. This is because the original function f(t) is not symmetrical around t = 0. For functions that are symmetrical around zero in the sense that jf(-t)j = jf(t)j, the nature of the Fourier transform depends on whether the function has the same or opposite signs on either side of zero. A function is said to be even if f(-t) = f(t), and odd if f(-t) = -f(t). The Fourier transform of any real, even function is also real and even,

Appendix A

629

whereas the transform of a real, odd function is purely imaginary and odd. The Fourier transform of exp(-|t/τ|) (a real and even function) thus has only a real part, which turns out to be a Lorentzian: f ðt Þ = expð-jt=τjÞ

ðA3:6Þ

and ( pffiffiffiffiffiffiffiffi F ðωÞ = 2=π

) 1=τ ð1=τÞ2 þ ω2

ðA3:7Þ

(Eq. (2.71) and Fig. 2.14). If a function does not have either even or odd symmetry, its Fourier transform is complex. The real and imaginary parts of the transform consist of the cosine and sine Fourier transforms discussed in Appendix 4. To illustrate these points, panels A and B of Fig. A3.1 show the even, two-sided decay function f2(t) = exp(-|t|/τ) and its Fourier transform (F2(ω)). Figure A3.1c shows the one-sided function f1(t) = exp(-t/τ) for t ≥ 0, f1(t) = 0 for t < 0, and Fig. A3.1d shows the Fourier transform of this function (F1(ω)) along with its real and imaginary parts. After scaling by a factor of 2 to compensate for the fact that it represents only positive values of t, the real part of F1(ω) is the same as F2(ω). The Fourier transform of a Gaussian function centered at t = zero (a real and even function of t) is another Gaussian:   f ðt Þ = exp -at 2 ðA3:8Þ and   F ðνÞ = ð2aÞ - 1=2 exp -ν2 =4a :

ðA3:9Þ

If the Gaussian is centered at some value m other than zero, the Fourier transform is multiplied by exp(imν), or cos(mν) + i sin(mν), and thus has an imaginary component. Fourier transforms provide a way of representing the Dirac delta function, δ(x), which is defined by the conditions δ(x) = 0 if x ≠ 0, and Za δðxÞdx = 1

ðA3:10Þ

-a

for a > 0. δ(x) is a function that peaks sharply at x = 0, in the limit that the width of the peak goes to zero while the height becomes infinite so that the area remains constant. It is useful for analyzing the dynamics of a process that occurs at a significant rate only when the energy difference between two states is close to zero. If a function f(x) is defined over the region x1 < x < x2 and X is a particular value of x in this region, then

630

Appendix A

1

1

B

f2(t)

F2( ) / (2/ )

1/2

A

0

1

1 2 F1( ) / (2/ )

1/2

0

f1(t)

C

D 2Re(F1)

0 2Im(F1)

0 -4

-2

2

0 t/

4

-4

-2

0 x

2

4

Fig. A3.1 (A) An even, two-sided decay function: f2(t) = exp(-|t|/τ). (B) F2(ω), the Fourier transform of f2(t), is a purely real Lorentzian. (C) A one-sided decay function: f1(t) = exp(-t/τ) for t ≥ 0, f1(t) = 0 for t < 0. (D) 2F1(ω), the Fourier transform of f1(t), (solid curve) and its real and imaginary parts (dotted curves). 2Re(F1(ω)) is identical to F2(ω). Note that the ordinate scales are different in B and D

Zx2 f ðxÞδðX - xÞdx = f ðX Þ:

ðA3:12Þ

x1

δ(x) can be expressed as (2π)-1/2 times the Fourier transform of the constant f(x) = 1: 1 δðxÞ = 2π

Z1 expði xyÞdy:

ðA3:13Þ

-1

One way to look at this relationship is to note that f(x) can be a constant only if its oscillation frequencies are distributed infinitely sharply around zero. The Fourier transform of cos(ωot) is δ(ω±ωo), a pair of delta functions located at ω=±ωo. The transform of sin(ωot) is a similar pair of delta functions, but with imaginary amplitudes. The transform of an arbitrary, fluctuating function can be viewed as a superposition of many such delta functions with amplitudes reflecting the contributions that oscillations at particular frequencies make to the overall function.

Appendix A

631

Tables of Fourier transforms of many other functions are available [2], and there are rapid computational methods for finding the Fourier transform of an arbitrary function [1]. For additional information on Fourier transforms, see [3].

Phase Shift and Modulation Amplitude in Frequency-Domain Spectroscopy To derive the expressions for the fluorescence phase shift (ϕ) and modulation amplitude (m) in Fig. 1.16, suppose the oscillatory part of the excitation light intensity is I(t) = sin(ωt), and that the fluorescence (F(t)) generated by an instantaneous pulse of exciting light decays exponentially with a time constant τ. The oscillatory part of the fluorescence signal observed at time t (S(t)) is obtained by integrating the fluorescence from the modulated excitation at all earlier times (t′): 0 t 1,0 1 1 Z Z Sðt Þ = @ I ðt 0 ÞF ðt - t 0 Þdt 0 A @ F ðt Þdt A 0

0

0

1, 0 1 1 Zt Z = @ sinðωt 0 Þ expð-ðt - t 0 Þ=τÞ dt 0 A @ expð-t=τÞ dt A: 0

ðA4:1Þ

0

The denominator in this expression is the total fluorescence generated by an instantaneous excitation pulse, which for a single-exponential decay is just τ. The convolution integral in the numerator can be evaluated straightforwardly: Zt

0

0

0

Zt

sinðωt Þ expð-ðt - t Þ=τÞ dt = expð-t=τÞ 0

sinðωt 0 Þ expðt 0 =τÞ dt 0

0

=

ð1=τÞsinðωt Þ - ω cosðωt Þ þ ω expð - t=τÞ : ð1=τÞ2 þ ω2

The term ωexp(-t/τ) goes to zero at long times (t >> τ), giving !, sinðωt Þ - ðωτÞcosðωt Þ ð1=τÞsinðωt Þ - ωcosðωt Þ : τ= Sð t Þ = 2 2 ð1=τÞ þ ω 1 þ ðωτÞ2

ðA4:2Þ

ðA4:3Þ

Equating S(t) to msin(ωt+ϕ) and using the relationship sin(ωt+ϕ) = sin(ωt)cos(ϕ) + cos(ωt)sin(ϕ) gives the desired expressions: m cosðϕÞ =

1 , 1 þ ðωτÞ2

ðA4:4Þ

632

Appendix A

m sinðϕÞ =

ωτ , 1 þ ðωτÞ2

ðA4:5Þ

tanðϕÞ = m sinðϕÞ=m cosðϕÞ = ωτ,

ðA4:6Þ



.

2 1 þ ðωτÞ2 , m2 = m2 cos 2 ðϕÞ þ m2 sin 2 ðϕÞ = 1 þ ðωτÞ2

ðA4:7Þ

and

-1=2 m = 1 þ ðωτÞ2 :

ðA4:8Þ

If the fluorescence response to an instantaneous excitation pulse is multiexponential, X F ðt Þ = Bk expð-t=τk Þ, ðA4:9Þ k

then Eq. (A4.3) becomes: Sð t Þ =

X k

!, ! X τk sinðωt Þ - ωτk 2 cosðωt Þ Bk Bk τ k : 1 þ ðωτk Þ2 k

ðA4:10Þ

The cosine and sine terms in this expression can be viewed as, respectively, normalized sine and cosine Fourier transforms of the fluorescence decay function. If we define the normalized sine and cosine Fourier transforms of F(t) as , Z1 !, ! Z1 X X ωτk 2 S sin ðωÞ = F ðt Þ sinðωt Þ dt F ðt Þ dt = Bk Bk τk 1 þ ðωτk Þ2 k k 0

0

ðA4:11Þ and , Z1

Z1 S cos ðωÞ =

F ðt Þ cosðωt Þ dt 0

F ðt Þ dt = 0

X k

τk Bk 1 þ ðωτk Þ2

!,

! X Bk τ k , k

ðA4:12Þ then Eqs. (A4.6 and A4.8) take the forms tanðϕÞ = S sin ðωÞ=S cos ðωÞ,

ðA4:13Þ



-1=2 : m = ðS sin ðωÞÞ2 þ ðS cos ðωÞÞ2

ðA4:14Þ

and

Appendix A

633

More generally, the sine Fourier transform Gsin(ω) of a function g(t) is defined as Z1

-1=2

G sin ðωÞ = ið2π Þ

gðt Þsinðωt Þ dt,

ðA4:15Þ

-1

which is zero if g is an even function of t. The cosine Fourier transform, G cos ðωÞ = ð2π Þ

-1=2

Z1 gðt Þcosðωt Þ dt,

ðA4:16Þ

-1

is zero for odd functions of t. The continuous Fourier transform defined in Eq. (A3.3) is the sum of the sine and cosine Fourier transforms in Eqs. (A4.15 and A4.16), as can be seen from the relationship exp(iθ) = cos(θ) + isin(θ). The factor i in Eq. (A4.15) is often omitted because only its product with the corresponding factor in the inverse transform (-i) is determined uniquely. Since the fluorescence decay function F(t) is zero for t < 0, taking the integrals in Eqs. (A4.11 and A4.12) from 0 to 1 rather than from -1 to 1 does not affect the results. See [4, 5] for further information on data analysis and extensions to fluorescence anisotropy.

CGS and SI Units and Abbreviations Physical quantity Electric current Energy Dipole moment Force Magnetic dipole moment

CGS unit abampere, biot (Bi) calorie (cal) debye (D) dyne (dyn) emu

Energy, work Electric charge

erg esu, statcoulomb or franklin (Fr) gauss (G)

Magnetic flux density (magnetic induction) Wavenumber Luminance Magnetic flux Magnetic field strength Illumination Dynamic viscosity Electric current Electric charge Potential Magnetic flux

kayser (cm-1) lambert (Lb) maxwell (Mx) oersted (Oe) phot poise (P) statampere statcoulomb statvolt unit pole

SI (MKS) equivalent 10 amperes (A) 4.1868 joule (J) 3.3356×10-30 coulombmeter (Cm) 10-5 newton (N) 10-3 amperemeter2 (Am2) 1.2566×10-3 tesla (T) 10-7 joule (J) 3.3356×10-10 coulomb 10-4 tesla (T) 100 per meter 3.1831×103 candelameter-2 (Cdm2) 10-8 weber (Wb) 79.577 amperemeter-1 104 lux 0.1 pascalsecond (Pas) 3.3356×10-10 ampere (A) 3.3356×10-10 coulomb (C) 299.79 volts (V) 1.2564×10-7 weber (Wb)

634

Appendix A

Harmonic-Oscillator Wavefunction Integrals The left-hand side of Figure A6.1 shows products of harmonic-oscillator eigenfunctions χ m and χ n as functions of the dimensionless positional coordinate u for n = 0 (the zero-point wavefunction) and m = 0 through 5. The plots on the right-hand side are the definite integrals R U of χ mχ n from u = -1 to U, the value of the coordinate given on the abscissa [ - 1 χ m ðuÞχ n ðuÞdu]. The eigenfunctions are orthonormalized so that the integral approaches 1 as the upper limit (U ) goes to +1 if m = n, while the integrals for m ≠ n all go to zero. See Fig. 2.3 for the individual eigenfunctions. Although the figure shows the functions only for n = 0, the results are the same for any value of n. Figure A6.2 shows similar plots of the products χ muχ n and their definite integrals, again forR n = 0 and m = 0 through 5. As the upper limit U goes to +1 , the definite U integral - 1 χ m ðuÞuχ n ðuÞdu goes to zero unless m = n ± 1. If χ muχ n is an ungerade function of u, contributions to the integral from u = -1 to 0 balance the contributions from u = 0 to +1; if χ muχ n is gerade, contributions from regions where χ muχ n < 0 balance those from regions where χ muχ n > 0 except in the special case that m = n ± 1. Again, other values of n give the same results.

Fig. A6.1 Products of harmonic-oscillator eigenfunctions χ m and χ n as functions of the dimensionless positional coordinate u (blue curves, left), and values of the definite integral of χ m(u)χ n(u) from u = -1 to the value of u given on the abscissa (red curves, right). The curves shown are for n = 0 and m from 0 to 5

Appendix A

635

Fig. A6.2 The functions χ muχ n for harmonic-oscillator eigenfunctions χ m and χ n as functions of the dimensionless positional coordinate u (blue curves, left), and values of the definite integral of χ m(u)uχ n(u) from u = -1 to the value of u given on the abscissa (red curves, right). The functions for n = 0 and m from 0 to 5 are shown

References

1. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes: the Art of Scientific Computing, 3rd edn. Cambridge University Press, Cambridge (2007) 2. Beyer, W.H.: CRC Standard Mathematical Tables. CRC Press, Boca Raton, FL (1973) 3. Butkov, E.: Mathematical Physics. Addison-Wesley, Reading, MA (1968) 4. Weber, G.: Theory of differential phase fluorometry: detection of anisotropic molecular rotations. J. Chem. Phys. 66, 4081–4091 (1977) 5. Lakowicz, J.R., Laczko, G., Cherek, H., Gratton, E., Limkeman, M.: Analysis of fluorescence decay kinetics from variable-frequency phase shift and modulation data. Biophys. J. 46, 463–477 (1984)

# The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 W. W. Parson, C. Burda, Modern Optical Spectroscopy, https://doi.org/10.1007/978-3-031-17222-9

637

Index

A Ab initio, 473 Absorbance, 1–4, 6–9, 11–13, 15–20, 24, 27, 31, 120, 176–179, 220, 472, 555, 558, 568, 591, 614 Absorption antenna complex, 183, 206, 601 bands, 2, 6, 7, 12, 13, 17–19, 26, 27, 71, 101, 121, 122, 445, 456, 458, 459, 463, 465, 466, 469, 470, 474, 475, 503, 516, 520, 529, 544, 546, 555, 561, 562, 573, 583, 585, 590, 598, 601band width, 204, 216, 231, 409, 429, 433, 573 coefficient, 122, 149, 182, 184, 213, 217, 447, 458, 483 cross section, 155, 177, 230, 231, 591, 592, 602, 607, 608 density-matrix treatment, 501–504 dimer, 465, 466 entropy, 228, 229, 231 excited-state, 2, 108, 147, 164, 180, 198, 200, 204, 205, 207–209, 217, 220, 226, 231, 503, 519, 529, 544, 546, 554–556, 561, 566, 570, 571, 573, 591, 592, 597, 605, 606 homogeneous, 204, 207, 516, 555, 561, 591, 592 infrared (IR), 13, 26, 27, 585, 598, 599, 601, 614 inhomogeneous, 204, 205 inhomogeneously broadened, 204, 392 MLCT, 226 oligomer, 7, 17, 20 π-π*, 211 plots, 13, 121, 122, 554, 555, 591 quantum theory, 109, 139, 186 rate, 6, 80, 461, 483, 501, 509, 533, 537, 571, 607

resonance, 83 selection rule, 160, 164, 614 spectral lineshape, 516, 517 spectrum, 4, 6–9, 12–15, 17–20, 26–28, 71, 84, 460, 466, 475, 476, 503, 516, 518, 519, 526, 555, 556, 569–571, 584, 585, 591, 592, 598, 601, 607, 608, 614 strength, 2, 5, 8, 10, 16, 17, 27, 101, 445, 465, 583, 591, 592, 601, 606 transition dipole, 2, 16, 146, 150, 157, 158, 160, 176–180, 182, 184, 202, 211, 219, 221, 222, 234, 342, 523, 588, 590, 607 two-photon, 27, 28, 193, 583, 605–608, 614 vibrational, 3, 27, 200, 203, 206, 207, 209, 470, 583, 584, 591, 614 wavepacket picture, 573 Absorption spectra cystine, 6 distortions, 18, 20 DNA, 7, 8, 469 flavins, 7 hemes, 15, 183 mixtures, 8, 9 NADH, 7, 235 phenylalanine, 6, 7 proteins, 6, 15 purines, 7 pyrimidines, 7 tryptophan, 6, 7 tyrosine, 6, 7 Adiabatic, 67, 69 Air mass, 307 Amide “amide A band”, 358 “amide I” band, 473 “amide II” band, 27, 358, 359 Anharmonicity, 48, 124, 556 Anharmonic vibrational mode, 343

# The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023 W. W. Parson, C. Burda, Modern Optical Spectroscopy, https://doi.org/10.1007/978-3-031-17222-9

639

640 Anisotropic, 2, 124, 127, 176, 217, 459 Antenna complex, 525, 557, 560, 602 Antisymmetric function, 158 Atkins, 35 ATP synthase, 179 Autocorrelation function, 292–294, 338, 339, 505, 506, 515, 519, 526, 566, 611, 612

B Bacteriochlorin, 181–183 Bacteriochlorophyll (BChl), 154, 176, 177, 181, 183, 184, 206, 212, 216, 223, 236, 426, 431, 546, 566, 601 Bacteriopheophytin, 176, 177, 183, 184 Bacteriorhodopsin, 15, 546, 552, 601 Basis functions, 67, 68, 172, 561, 562, 594 Basov, N., 29, 436 Beer, W., 3 Beer-Lambert law, 3, 4, 8 Betaine-30, 208 Betzig, E., 288 Biaxially birefringent, 176 Bioluminescence, 250, 394 Birefringence, 115, 124–127, 174–176 Birge, 607 Black-body radiation, 87, 103–105, 231, 232, 245, 247, 253, 254, 346 Bohr, N., 459 Bohr magneton, 446, 457 radius, 58, 59, 61, 63 Boltzmann constant, 6, 74, 103, 493, 507, 561, 612 distribution, 74–76, 104, 202, 230, 494, 612 entropy, 75, 228, 230 weighting, 201, 202, 427 Boltzmann factor, 202, 381, 427, 493 Born, M., 35, 38 Born-Oppenheimer approximation, 195–197, 199–204, 250, 251, 380, 470, 561 Bose-Einstein statistics, 74, 109, 232 Bosons, 74, 75, 118 Bra-ket notation, 36, 37 Brillouin scattering, 583 Bunsen, R. (1811–1899), 5 Butadiene, 164, 190–192, 234, 386, 387, 452

C Calcite, 124, 125 Candela, 98 Carbonyl group, 448, 450 Carotenoids, 190, 192, 193 CARS, 603, 604

Index Casimir effect, 109 Causality, 540 CdSe, 275, 276, 308 CD spectra, 17, 446, 460, 466–469, 477 CdTe, 275, 276 Celeritas, 97 Cgs units, 88, 93, 99, 100, 447 Character table, 170–172, 348, 365, 366 Charge-transfer (CT) band shapes, 431, 555 state, 236, 270, 273, 296, 424–428, 435 transition, 203, 226, 424–428, 430, 432, 601 Chlorin, 30, 182, 183 Chlorophyll, 183, 555 Chromophores, 17, 21, 22, 27, 139, 154, 155, 176, 178, 179, 204–211, 213, 215–218, 220, 221, 223, 445, 447, 448, 451, 453, 454, 556, 584, 585, 600–602, 606 Chung, H.S., 395 Circular dichroism (CD) alpha-helix, 17, 468, 469 beta-sheet, 18, 467–469 Circular polarization, 17, 87, 105, 106, 126, 133, 164 11-cis-retinal, 178, 209 Coherence, 16, 28, 30, 483–524, 530–532, 538, 540–542, 544, 545, 547, 549, 551, 557, 560, 561, 565–567, 572, 586, 588, 603, 605, 607 Commutator, 39–41, 84, 185, 455, 491, 495, 504, 529, 534, 536 Commute, 39, 69, 77, 80 Condon approximation, 197, 202, 234, 590 Configuration coefficient, 180, 190, 214, 234 interaction, 180, 181, 183, 193, 437 Confocal fluorescence microscopy, 25, 286, 608 Correlation functions, 281, 293, 505–508, 510–512, 515–518 time, 505–508, 510–512, 516–518, 552 Coupled-oscillator, 465 Coupling factor, 201, 497, 567 Coupling strength, 199, 203, 236, 248, 302, 303, 573 COVID-19, 602 Cramer’s rule, 412, 413 Crystallographic lattice systems, 173 Crystals biaxial, 125 uniaxial, 124–126 Cubic, 173–176, 221, 334 Cytochrome, 183, 552, 560, 601

Index Cytochrome-c oxidase, 15

D Davisson, E.R., 43, 83 Debye, P., 149 Debye-Stokes-Einstein, 281 Debye-Waller factor, 204, 236 de Broglie, L., 43, 83 Delayed fluorescence, 295–298 Density matrix, 487–493, 495, 504, 505, 507, 511, 513, 515, 521, 522, 524–526, 529–532, 534, 537, 538, 540, 549, 561, 566–569, 587, 595 Density of final states, 380 Dephasing, 483, 503, 511, 512, 515–517, 529, 545, 549, 552, 562, 565, 566, 568, 570, 587, 589, 591, 592, 596–598, 600, 602 Dexter, D.L., 396 Dextrorotatory, 472 Diabatic, 66, 80, 594 Dichroism circular, 1, 17, 18, 445–475 induced, 17 linear, 1, 16, 17, 27, 460, 608 Dielectric constant, 94, 96, 100, 118, 120, 131, 210, 211, 610 Dimers, 30, 462–467, 476, 524, 525, 546, 555, 556 Dipole electric, 91, 445, 447, 460, 461, 469, 472, 487, 531, 592 expectation value, 81, 148, 487, 531, 595 induced, 123, 124, 131, 587, 592, 593, 595 magnetic, 445, 446, 448, 456, 460 moment, 65, 80, 131, 132, 472, 534, 592, 595, 599 permanent, 131, 593, 595, 599 strength, 2, 91, 449, 456, 461, 463, 465–467, 469, 477, 555, 561, 587, 590, 595 transition, 146, 148, 149, 151–160, 162, 164, 178, 182–185, 190, 192–197, 203, 210, 211, 213, 214, 218, 221, 234, 235, 446, 450, 451, 454, 456, 459, 462–464, 469–472, 502, 520, 534, 555, 561, 599, 610 Dipole-dipole interactions, 184, 382, 383, 385, 386, 400 Dipole strength dimer, 467 Dirac, P., 35, 36, 38, 70, 107, 108, 587 Dispersion anomalous, 121 dielectric, 122, 124

641 Displacement, electric, 100 Distribution binomial, 289, 291, 292 Boltzmann, 74, 75, 202, 494, 612 Bose-Einstein, 75, 76, 104 donor-acceptor distances, 391, 392 energy, 50, 57, 75, 81, 83, 103, 129, 486, 499, 501, 514, 549, 597 Fermi-Dirac, 75, 76 frequencies in light pulses, 550, 573 Gaussian (normal), 83, 128, 129, 514, 515 Poisson, 291 site, 203 transition energy, 207, 503 Distribution function radial, 58, 59 for state with finite lifetime, 82 DNA photolyase, 210, 546 Duschinsky effect, 204, 344 Dynamic light scattering (DLS), 583, 611, 612

E Eigenfunction complete set, 47, 84, 203 normalized, 40, 47 orthogonal, 40 Eigenstate zero-point, 109 Eigenvalue, 38, 46–48, 51, 53–57, 66–69, 83, 198, 213, 216, 561, 594 Einstein, A., 10, 43 Einstein coefficients, 245–247, 571 Electric dipole moment, 139 Electric-magnetic coupling, 465 Electric quadrupole moment, 141 Electrochromic effect, 216 Electromagnetic irradiance, 87, 91, 92, 98 radiation, 6, 10, 25, 87, 91, 92, 94–101, 103, 107–110, 123, 131, 133, 147, 149, 445, 456, 501, 530, 532, 533, 538, 587 theory, 2, 4, 10, 97, 107–110, 587 Electron-transfer, 15, 19, 22, 206, 207, 223, 500, 545, 546, 566, 568, 569 Electro-optic effects, 124, 125, 127 Electrostatic/cgs system, 57, 88, 89, 137, 149 Electrostatic force, 87, 88, 90, 133, 309 Ellipticity, 17, 106, 133, 457, 458 El-Sayed, M., 296 El-Sayed’s rule, 296 Emission anisotropy, 226, 523, 525 stimulated, 29, 460, 603 strength, 263

642 Emission (cont.) vibrational, 566 Energy absorption of, 10, 27, 143, 146, 149, 155, 228, 230, 233, 483, 533, 583 avoided crossing, 415–416 conical intersection, 415–416 density, 10, 55, 98–101, 103, 499, 503, 568 derivatives, 42, 47, 89, 181, 495, 533, 593–595, 607 electronic interaction, 299 electrostatic, 88, 89, 138, 216, 223, 225, 569 excited-state, 28, 73, 198, 202, 211, 212, 215, 216, 228, 233, 549, 561, 563, 564, 567, 569, 571, 573 gap, 109, 495, 496, 501, 510, 511, 519, 566–569 kinetic, 10, 12, 48, 51, 53, 471, 547, 552 potential, 43, 44, 47–51, 53, 55–57, 89, 107, 118, 133, 519, 562–564, 567, 568 radiant, 4 vibrational, 28, 196, 198, 200, 201, 204, 205, 557, 558, 566, 570, 573, 584 zero-point, 109 Energy density field, 94, 98, 99, 101 Energy-gap law, 296 Enolization, 363 Enthalpy, 19, 560 Entropy vibrational, 233, 236 Equilibrium, 6, 20, 48, 54, 55, 103, 104, 198, 201, 230, 233, 486, 493, 494, 497, 507, 517, 525, 530, 534, 572 Ergodic hypothesis, 230, 505 Escherichia coli, 8 Ethylene, 62, 64, 65, 449 Euler angles, 522 Evanescent, 115–118, 284–286 Excimer, 30 Exciplex, 30 Excited state apparent temperature, 258 forbidden, 474, 607 singlet, 72, 447, 500 triplet, 72, 500 Exciton band, 464–466, 469, 476, 477, 556 band shapes, 428–431 CD, 467 excited-state, 450, 462, 464, 465, 471, 475, 556 Frenkel, 409

Index interactions, 178, 183, 184, 466, 469, 473, 555, 556, 558 localization, 428–431, 526, 558 states, 524, 556 Wannier, 410 Exclusion principle, 71, 73, 76, 350, 354 Expectation value, 35–42, 46, 49, 52, 79, 80, 84, 110, 487, 488, 594 External quantum efficiency, 307 Extinction coefficient, 3, 6–9, 17, 31, 149, 150, 157, 219, 225, 226, 231, 275, 344, 356, 447, 448, 457, 458, 471, 473, 520

F Facial, 351, 352 Fermi, E., 76 Fermions, 74, 75 Feynman double-sided diagram, 532, 540, 542, 544, 545, 547, 548, 572, 586, 605, 606 quantum electrodynamics, 110 Feynman, R., 110 Field cavity, 126, 130, 131 electric, 12, 16, 17, 77, 87, 89–92, 94, 97, 99, 100, 104–106, 115–117, 119, 120, 122–124, 126–133, 445, 446, 453, 456–458, 460, 501, 517, 520, 521, 533, 534, 538, 563, 593, 595, 610, 612 electromagnetic, 2, 16, 87, 94, 100, 119, 123, 445, 456, 532, 534, 624 energy density, 94, 98, 99, 101, 132, 133 local, 73, 130, 132, 457, 460, 553 magnetic, 57, 445, 446, 448, 453, 456, 458, 460, 471, 473–475 photon, 10–12, 109, 133, 531–533, 548, 563, 590, 606 radiation, 13, 91, 99, 104–107, 109, 115, 117, 118, 133, 445, 453, 502, 529–531, 533, 534, 538, 542, 544, 547, 549, 553, 560, 563, 572, 587, 588, 601, 608–610, 612 reaction, 131, 132 Fizeau, A, 97 Flavin, 7, 21 Fluctuation dissipation theorem, 517 Fluorescence anisotropy, 24–26, 222, 519–525, 607 antenna complex, 392, 525 bacteriochlorophyll, 280, 297 correlation spectroscopy, 289, 291–295 delayed, 24

Index decay kinetics, 24 dipole strength, 524 dyes, 21, 30 emission peak, 20 emission spectrum, 21, 22, 24, 27, 30, 518, 524, 565, 566, 571, 586, 606, 608 excimer and exciplex, 433–436 excitation spectrum, 21 flavins, 21, 271 fluctuations, 23, 24, 526, 611 lifetime, 23, 24, 32, 519, 587, 606 line narrowing, 284 loss in photobleaching (FLIP), 277 NADH, 21 near-field, 25 oscillations, 24, 103, 520, 521, 564–566 phenylalanine, 21 photobleaching, 608 photon counting histogram, 295 polarization, 3, 24, 26, 103, 152, 520, 522, 524, 525, 607, 610 proteins, 20, 21, 32, 611 quantum theory, 3 quenching, 23, 32 rate constant, 22, 231, 526 recovery after photobleaching (FRAP), 277 single molecule, 284–289 spectrum, 15, 565 tryptophan, 20, 21, 23, 32, 222 tyrosine, 21 upconversion, 24, 32, 250, 266, 566 wavepacket picture, 568–573, 587, 596–598 Y base, 21 yield, 21–24, 32 Fluorescence-depletion, 250, 285 Fluorescence loss in photobleaching (FLIP), 277 Fluorescence microscopy confocal, 25, 285, 286, 608 fluorescence depletion, 250 stimulated emission depletion (STED), 288 Fluorescence recovery after photobleaching (FRAP), 277 Fluorescence upconversion, 24, 32, 250, 266, 566 Fluorometer, 22, 27 Force, electrostatic, 87, 88, 90, 309 Förster radius, 387, 389, 390, 403 theories, 487, 500 Foucault, L., 97 Fourier-transform, 13, 14, 32, 81, 82, 129, 503, 506–508, 513, 516, 517, 519, 554, 566, 570, 571, 596, 627–631, 633

643 Fourier-transform IR (FTIR) spectrometer, 13 Four-wave mixing, 533, 544, 603 Franck-Condon factor Boltzmann (thermal) weighting, 201, 202, 381 Frauenhofer, J. von (1787–1826), 4 Free energy, 215, 228, 230 FTIR spectrometer, 14, 15, 27, 28, 628

G GAMESS, 348–350, 358 Gaussian function, 66, 128, 207, 216, 629 Gerade, 56, 606 Germer, 43, 83 Golden rule (Fermi’s golden rule), 145, 446, 499, 501, 503, 589 Göppert-Mayer, M., 606 Göppert-Mayer unit, 607 Gordon, J.P., 357 Gradient, 89, 450–452, 455, 459, 476 Gradient operator, 37, 184–188, 190, 192, 193, 215, 450–452, 455, 459, 476, 624 Grätzel, M., 306 Gramicidin A, 359 Green fluorescent protein (GFP) chromophore, 21, 273, 274, 312, 393, 585 IR absorption, 584, 585 Raman scattering, 585 red and yellow variants, 274, 393, 546 Green(’s) function, 539, 540, 549 Ground-state bleaching, 544, 554–556, 558, 572, 606 Group theory, 160, 164, 165, 168–170, 172, 173, 175, 176, 181, 192 Group velocity, 101, 129, 130 Gyromagnetic ratio, 446

H Hamiltonian diagonalization, 62 dimer, 377, 378 matrix, 62, 68, 69 matrix element, 62, 79, 80, 186, 525, 529 operator, 39–41, 46, 47, 50, 54, 60, 80, 84, 455, 491, 503 Harmonic oscillator, 53–57, 108, 109, 198, 199, 203, 249, 252, 261, 264, 332, 341, 342, 344, 540, 561, 562, 567, 598, 634, 635 Haugland, R.P., 390 Heisenberg commutation, 39 uncertainty principle, 80 Heisenberg, W., 38, 587

644 Heller, E.J., 570, 587 Hellman-Feynman theorem, 594 Hell, S.W., 286, 288 Heme, 7, 15, 28, 601 Hemoglobin, 560 Hermite polynomials, 54–56, 332, 342 Hessian matrix, 334, 336, 337 Heterodyne detection, 549, 553 Hexagonal, 173–175 Highest occupied molecular orbital (HOMO), 62, 66, 67, 71, 73, 234, 451 Hole-burning nonphotochemical, 205 photochemical, 205–207 Stark spectroscopy, 222 Homodyne detection, 549 Huang-Rhys factor (coupling strength), 199, 203, 236, 573 Hund’s rule, 72, 84 Hyperchromism, 421–423 Hyperpolarizability, 221 Hypochromism, 7, 17 I Iceland spar, 124 Indole, 64, 67, 469 Inelastic, 583 Infrared (IR) spectroscopy amide I band, 27, 358, 359, 558 amide II band, 27, 358, 359, 558 proteins, 331, 357–362, 558 Inhomogeneous broadening, 555, 557, 591 Integrating sphere, 18, 19 Interaction representation, 488, 504 Interference, 18, 28, 49, 82, 110–113, 117, 118, 128, 554, 558, 561, 570, 599, 608, 613 Interferometer, 14, 15, 110, 111 Internal conversion, 352, 546, 606 Internal quantum efficiency, 307 Intersystem crossing, 22, 195, 226, 265, 271, 295–297, 309, 352 Iron-proteins, 475 Irradiance, 3, 4, 91, 92, 98–102, 104, 115, 116, 127, 132, 133, 149, 152, 230, 231, 611 IR spectroscopy, see Infrared (IR) spectroscopy Isosbestic point, 9 Isotropic sample, 151, 152, 219, 504, 522, 593 J Jablonski diagram, 265, 295

K Kasha, M., 253, 266

Index Kasha’s rule, 606 Kennard-Stepanov expression, 258 Kerr, 126 Ketosteroid isomerase, 363 Kirchoff, G. (1824–1887), 5 Kramers-Heisenberg-Dirac theory, 587–596, 606, 614 Krishnan, K.S., 584 Kubo relaxation function, 516–518, 551–553 Kuhn-Thomas sum rule, 155

L Laguerre polynomial, 344 Lambert, J., 3 Landé g factor, 70 Laporte’s rule, 173, 226 Laser(s) diode, 30, 584 dye, 128, 546 excimer, 30, 436, 437 gas (Ar, He, Ne), 15, 28, 29, 435 IR, 15, 27, 29, 552, 584 mode-locked (mode locking), 128 pulsed, 128, 604, 606, 608 pulsed, 29, 552, 584, 607 Q-switched (Q switching), 128, 175 solid state (Nd-glass, ruby, Nd-YAG), 29, 30, 175, 552, 606 titanium-sapphire, 28–30, 128, 129, 364, 604 LASIK, 436 Levorotatory, 472 Lifetime broadening, 82 Ligand metal charge-transfer (LMCT), 227 Light absorption, 2–4, 12–20, 27, 29, 101, 109, 121, 122, 137, 139, 144, 146, 150, 155, 157, 158, 164, 176–179, 183, 205, 210, 219, 222, 230, 236, 476, 510, 554, 558, 570, 583, 584, 590, 601, 607, 608 emission, 3, 4, 15, 24–27, 105, 109, 123, 129, 510, 590, 603, 604 speed (velocity), 1, 4, 43, 94, 96–98, 446, 583, 608 Light-emitting diode (LED), 306–309, 352 Linear dichroism spectrum, 176–180 Linear momentum, 37, 51, 446 Linear polarization, 16, 106, 113, 532, 607 Line-broadening function, 515, 550 Lineshape, 513–519, 521, 543, 568 Line width homogeneous, 204, 436 inhomogeneous, 204

Index Liouville equation, 491 Liouville space diagram, 538, 585, 586 Lithium niobate, 122, 125–127, 175 Local-field correction factor, 130–132, 152, 154, 155, 217, 222, 245, 363, 590 Local-field corrections, 132, 133, 457, 460 Lorentz correction, 132, 133, 154 Lorentzian function, 82, 204, 501, 503, 512 Lowest unoccupied molecular orbital (LUMO), 62, 66, 67, 71, 73, 234 Luciferase, 250, 394 Lumens, 98 Luminance, 98 Lux, 98

M Mach-Zender, 110, 111 Magnetic circular dichroism (MCD), 473, 475, 476 Magnetic field, 17, 42, 57, 58, 70, 72, 73, 87, 88, 91–97, 99–101, 105, 106, 108, 115, 117, 137, 139, 164, 176, 192, 445, 446, 448, 453, 456, 458, 460, 471, 473–475, 504, 509 Magnetic transition dipole, 445–456, 459, 462, 465, 469, 471, 472 Manneback, C., 199 Marcus equation, 500 Marcus, R.A., 298 Matrix Hessian, 334, 336, 337 interaction, 62, 109, 147, 489, 493, 504, 505, 509, 510, 513, 519, 530, 531, 533, 538, 567 perturbation, 79, 80, 137, 186, 501, 530, 534, 540 Matrix element interaction, 62, 490, 495–498, 513, 540 magnetic transition, 446, 449, 450 off-diagonal, 62, 80, 495, 511, 513, 529, 538 Raman and Rayleigh scattering, 590, 591, 596 transition, 80, 495, 496, 501, 529, 599, 600 transition gradient, 186, 190, 192, 452 Maximum likelihood method, 394 Maxwell equations, 456 MCD, see Magnetic circular dichroism (MCD) Memory function, 505, 526 Meridional, 351, 352

645 Metal-to-ligand charge-transfer (MLCT), 226 3-Methylindole, 66, 67, 162, 163, 181, 184, 214, 607 Microwave absorption (emission), 355 Mie scattering, 583, 584, 612–613 Mirror-image law, 250–253, 255, 257 Mode normal, 26, 471, 599, 605 oscillation, 99, 103, 108, 109, 128, 561 radiation, 26, 99, 108, 109, 128, 454, 603, 604 vibrational, 26, 201, 233, 236, 471, 556, 568, 573, 592, 593, 599, 601, 603, 614 Moerner, W.E., 288 Molecular dynamics (MD) simulation, 298, 301, 332–338, 364, 519, 566, 568, 572 Momentum angular, 57, 69, 70, 105, 446, 447, 474 Monochromator, 12, 13, 15, 21, 26, 105, 584 Monoclinic, 173, 174, 176 Morse potential, 48 Mourou, G., 102 Mukamel relaxation function, 517 Mulliken symbols, 171, 172, 192 Multi-photon absorption, 605–608 Multipole expansion, 140, 141 Mutual exclusion principle, 347, 349 Myoglobin, 15, 468, 552, 556, 560

N NADH, 7 Nanocrystal, 274 Nanometer, 30 Nanoparticles, 118, 601, 602, 611, 612 Newton, I. (1642–1726), 4, 117, 124 N-H bond-stretching, 472 Niobium, 125 Nodes, 50, 57, 62, 559 Nonadiabatic reactions, 300, 500 Normal coordinate, 58, 138, 178, 472, 571, 597, 599 mode, 26, 471, 597, 599, 605 n-π* excitation, 448, 449 n-π* transition, 448, 450, 469, 470

O Operator(s) angular momentum, 446, 451 annihilation, 263–264 commutation, 39–41

646 Operator(s) (cont.) commutator, 455 creation, 263–264 dipole, 80, 455, 460, 595 gradient, 450–452, 455, 459, 476 Hamiltonian, 40, 41, 46, 47, 50, 54, 60, 79, 80, 137, 185, 186, 195, 196, 455, 491 Hermitian, 38, 491, 596 kinetic energy, 41, 42, 80, 108, 185, 195 Laplacian, 42, 94 Liouville space, 538 magnetic dipole, 445, 446, 460 momentum, 37–42, 52, 80, 81, 84, 108, 446 polarizability, 596 position, 36–42, 61, 80, 81, 84, 108, 109, 454, 539, 568 potential energy, 38, 41, 42, 50, 80, 108, 185, 196 time evolution, 539 Opsins, 178, 210 Optical density, 3 dielectric constant (high-frequency dielectric constant), 100 Kerr effect, 127, 128 rotation, 457–459 rotatory dispersion (ORD), 458, 460 sectioning, 286 tweezers, 102, 103 wavepacket, 127–130 Orbital antibonding, 6, 62, 63, 198, 207 antisymmetric, 234, 450 atomic, 61–64, 67, 70, 83, 447, 449, 450, 452, 463 bonding, 6, 62, 63 d, 475 molecular, 2, 6, 61–63, 447, 448, 451, 463 nonbonding, 207, 227, 448 p, 474 π, 61–65, 156–157, 437 π*, 6, 62, 63, 448, 450 Rydberg, 227 s, 450, 452 Slater-type, 63, 67 symmetric, 62 Order parameter, 283 O’Regan, B., 306 Organic light-emitting diode (OLED), 309 Orthorhombic, 173, 174, 176 Oscillator anharmonic, 342, 366 harmonic, 53–57, 108, 109, 199, 203, 249, 251, 252, 261, 262, 264, 332, 341, 342, 344, 540, 561, 562, 567, 598, 634–635

Index strengths, 119, 120, 154, 155, 465 Overlap integral nuclear (vibrational), 561, 590 spectral, 392 time-dependent, 570–572, 597 vibrational, 197, 199

P P700, 400 P870, 399 Partition function vibrational, 201, 202 Pauli, W., 71 Pauling, L., 35 Peptide, 6, 17, 27, 448, 467–469, 552, 558 Permittivity, 88, 227 Perovskite, 174, 176 Perturbation theory time-dependent, 137, 445, 483 Phase matching, 127 Phenylalanine, 6, 7, 21, 602 Phonon side bands, 205 Phosphorescence, 25 Photoacoustic spectroscopy, 19, 559 Photoactive yellow protein, 217, 546, 560 Photobleaching, 276–277, 284, 608 Photodiode, 12, 13, 16, 18, 604, 612 Photoelectric effect, 10, 11, 43 Photomultiplier, 1, 4, 11–13, 22, 24, 612 Photon absorption/emission, 228, 583 spin, 109, 235, 355 Photon echo three-pulse, 543, 573 two pulse, 15, 544, 552, 560 Photosynthesis, 15, 193, 500, 546 Physical constants (table), 153, 227 Phytochrome, 15 Photosystem I, 398–400 π-π* transition, 207, 448, 449, 469, 470 Planck black-body radiation, 87, 103–105, 247 constant, 2, 37 expression, 87 quantization, 105 Planck, M., 103 Plasmons, 118 Point-dipole approximation, 386, 396, 403, 419, 424, 437, 466, 467 Point groups operators, 166, 168, 170 representations, 170, 171 Polarity scale(s), 208

Index Polarizability quantum theory, 593–596 Polarization axis, 16, 92, 106, 113, 123–125, 454, 553, 605, 607, 610 circular, 445, 456–458, 460 electronic, 2, 100, 124, 131, 211, 221, 557 inductive, 211 linear, 100, 122, 126, 453, 457, 458, 533, 544 of the medium, 94, 100, 101, 115, 122, 131 optical, 100, 115, 123–125, 127, 531–534, 537–539 plane, 92, 106, 125, 457 Polyenes, 192–193 Porphin, 165, 181, 182 Porphyrins, 181, 183, 525 Potassium dihydrogen phosphate (KDP), 175 Potential electric, 12, 42, 95, 97, 612 electrostatic, 89, 132, 141 scalar, 89, 95 vector, 89, 90, 95, 97, 107, 453, 456 vibrational, 56, 519, 563 Power spectrum, 506 Probability density, 35, 46, 50, 51, 81, 83, 103 Prokhorov, A., 436 Protein Data Bank (PDB), 178 Proteins, 6, 7, 9, 16, 17, 21, 27, 28, 103, 118, 449, 457, 467, 469, 546, 552, 558, 569, 584, 585, 601, 602, 614 Proustite, 174, 175 Pseudomonas putida, 363 Pterins, 210, 377 Pump-probe, 15, 16, 24, 27, 205, 519, 529–572 Pure dephasing, 204, 494–498, 501, 507, 510, 511, 513, 516, 519, 526, 567, 572

Q Quadrupole moment, 476 Quantization, 72, 110, 146, 587 Quantum confinement, 274 Quantum dot, 607, 608 Quantum interference, 49, 111, 113 Quantum number electronics, 70, 164, 200 spin, 70 vibrational, 200 Quantum optics superposition states, 110–114 “Quantum Zeno” paradox, 499 Quencher, 22, 32

647 R Radial distribution function, 58, 59 Radiant intensity, 98 Radiation black-body, 104, 133 electromagnetic, 5, 6, 10, 25, 26, 87, 91, 92, 95, 98–100, 103, 107–109, 123, 131, 445, 456, 501, 531–533, 538, 587, 603 energy density, 87, 100, 104, 133 entropy, 232 evanescent, 117 quantum theory, 4, 105, 107, 109 reflection, 117, 284 solar, 232 spectrum, 5, 25, 105, 553 zero-point, 109, 259 Radiative lifetime, 264, 266, 295 Raman scattering anti-Stokes, 27, 583, 584, 587, 603, 614 classical explanation, 587 coherent (stimulated), 603, 604, 614 excitation spectrum, 597 Kramers-Heisenberg-Dirac theory, 587–596 matrix element, 591, 596, 597, 599, 600, 606 resonance, 28, 583–585, 587, 590, 591, 596–600, 602, 606, 614 selection rule, 56, 346, 587, 598–600, 614 Stokes, 27, 583, 584, 587, 604 surface-enhanced (SERS), 601, 602 wavepacket picture, 587 Raman spectroscopy, 25, 27, 28, 193, 331, 345–353, 355, 365, 572, 601–603 Raman, C.V., 584 Rate constant excitation, 22, 193, 230, 231, 525 fluorescence, 23, 193, 222, 230, 231, 526 protein conformation, 27 resonance energy transfer, 265, 380, 386–388, 390, 393, 395 Rayleigh black-body radiation, 103, 104 Reaction center photosynthetic, 601 photosynthetic bacterial, 183, 206, 223, 236, 338, 339, 363, 426, 546, 566 Redfield, A., 504, 506, 507, 511 Redfield theory, 516, 517, 526 Redistribution of energy, 345 Red shift, 207, 213, 270 Reduced density matrix, 493, 494, 504, 566 Refractive index, 4, 18, 20, 29, 30, 97, 100–103, 115–117, 120–122, 124–127, 130, 132, 133, 152, 154, 155, 175, 207, 210–212, 234, 457, 558, 590, 591

648 Reichardt’s dye, 208 Relaxation(s) function, 497, 499, 507, 512–519, 526, 540, 560, 562 matrix, 495, 497, 504, 505, 507, 508, 510, 511, 513, 516, 526 solvent, 517 stochastic, 493–495, 497, 499–501, 526, 529 vibrational, 204, 228, 517, 552, 566, 571, 600 Relaxation time longitudinal (T1), 494 spin-lattice, 494 transverse (T2), 83, 494, 526 Reorganization energy solvent, 518 vibrational, 233 Resonance, 1, 6, 25–28, 62, 64, 80, 83, 116–118, 483–485, 487, 494, 497, 499, 511, 555, 572, 583–585, 587, 589–592, 596–598, 600–602, 606, 608, 614 Resonance energy transfer, 379, 426 Coulomb interaction, 397, 400 Dexter mechanism, 396 distance dependence, 390, 398 exchange coupling, 396–398, 400, 403 exchange interaction, 397 Förster theory, 487 orientation factor, 386 single-molecule, 390, 394, 396 triplet-triplet, 396 Response function linear, 539 nonlinear, 541, 544 Resonance splitting, 415 Rhodamine, 21, 30 Rhodobacter sphaeroides, 360, 546, 569 Rhodopsin, 15, 178, 546, 560, 600, 601, 607 Rigid-rotor approximation, 355 Rocking, 349, 350 Rosenfeld equation, 460–462 Rosenfeld, L., 459 Ross, R.T., 253, 256, 257 Rotating-wave approximation, 511, 526, 535, 536, 544, 548 Rotational absorption (transition), 17, 207, 583 correlation time, 281–283 fine structure, 207, 353–357 strength, 456, 459, 462, 464–467, 469, 472, 475–477 Runge-Kutta, 495

Index Rydberg transition, 227 Rydberg, J., 227, 435

S Scattering dynamic, 1, 592, 608, 611 Mie, 583, 584, 612–613 quasielastic (dynamic), 583, 608–612 Raman, 15, 583–587, 589–593, 596, 597, 599–606, 614 Rayleigh, 583, 584, 589–591, 597, 612, 613 vector, 609–612 Schrödinger, E., 42 Schrödinger equation dimer, 411 time-dependent, 42, 44, 77, 78, 490 time-independent, 47, 49, 50, 77 Schrödinger representation, 488, 504, 587 Scissoring, 472 Second-order optical polarization, 533 Secular determinant, 336 Selection rule(s), 56, 160, 164, 181, 192, 193, 236, 587, 598–600, 606, 614 Sellmeier, 122 Shockley-Queisser, 307 Silicon, 12, 14 Single-molecule fluorescence, 284–286, 288, 289 Single-molecule SERS, 602 Single-photon counting, 24, 250 Singlet oxygen (Singlet O2), 276 Singlet-triplet splitting, 72 Slater determinant, 73, 84 Slichter, C.P., 504 Snell’s law, 115, 117, 120, 121 Solar cell, 176 Solvatochromic effect(s), 207, 208, 210 Solvent coordinate, 215, 216 Solvent polarity scale, 208 Spectral density, 509, 511, 512, 526 Spectral density function, 430, 506–508, 511, 517, 526 Spectral diffusion, 205, 555, 557 Spectrophotometer, 12–14, 17, 18, 32, 205 Spectrum fluorescence, 15, 21, 22, 27, 525, 561, 564, 565, 571, 606 homogeneous, 203, 503, 517, 519, 524, 591 inhomogeneously broadened, 516 Stark, 217, 219–223 Voigt spectrum, 204 Spin

Index integer, 109 Spin-orbit coupling, 296 Spin quantum number, 69, 70, 74, 193 Stark effect electronic, 216–223 higher order, 223 linear, 218 quadratic, 218 solvent, 216 spectrum, 219, 220 vibrational, 362–364 Stark tuning rate, 363 Statistics Boltzmann, 74–76 Bose-Einstein, 74–76 Fermi-Dirac, 74–76 Stationary state, 77, 80, 142, 493, 513 Stefan-Boltzmann law, 104 Stephens, P., 471, 474 Stern-Volmer equation, 267 modified plot, 268 quenching constant, 267, 311 Stimulated emission, 29, 30, 127, 128, 142–144, 146, 230, 246, 247, 250, 253, 259, 263, 266, 285, 288, 342, 357, 364, 365, 460, 504, 520, 525, 542, 544, 546, 554–556, 566, 572, 588, 606, 614 Stochastic Liouville equation, 493–495, 497, 499–501, 504, 511, 526, 530, 567 Stokes shift solvent, 248, 249 vibrational, 546 Strickland, D., 102 Strickler-Berg equation, 253–257 Stryer, L., 390 Sum rule exciton dipole strengths, 421–422 Kuhn-Thomas, 155 Superposition state dipole, 147–149, 487 dipole moment, 147, 148 wavefunction, 48, 49, 53, 110, 147, 148 Surface plasmons, 115–118, 601, 613 Surface potential (z-potential), 612 Susceptibility electric, 122 Symmetry configurations, 164, 180, 182, 193, 226, 234 elements, 164–166, 171, 348, 459 forbidden, 599, 606 operators/operations, 160, 164–168, 170, 174, 332, 338, 348, 350, 352

649 orbital, 72 Synchronization, 16

T Taylor series, 139, 140, 543 Tetragonal, 173–175 Thermal diffusion, 559 Third-order optical polarization, 533, 538–544 Thompson, P.G., 43 Three-state system, 491, 524, 544, 545 Total internal reflection, 116, 117, 277, 284–286 Totally symmetric, 160, 172, 173, 181 Townes, C.H., 29, 357, 436 Transform-limited pulse, 129 Transient grating, 533, 558–560, 573 Transition allowed, 465, 525, 599, 607 hyperpolarizability, 221 metals, 472, 475 n-π* excitation, 448 n-π* transition, 6, 207, 448, 450, 469, 470 π-π* transition, 6, 208, 448, 449, 469, 470 polarizability, 610 Transition dipole dimer, 465 electronic, 137, 153, 155, 159, 197, 198, 201–203, 205, 206, 213, 234, 464, 588, 590 Transition monopole, 382, 386, 387, 400, 424–428 Triclinic, 173, 174, 176 Trigonal, 173–175 Triplet state, 22, 25, 69–76, 194–195, 206, 226, 233, 265, 276, 288, 295–298, 309, 311, 352, 396, 398, 500 Tryptophan, 6, 7, 9, 20, 21, 23, 32, 66, 162, 183, 213, 222, 249, 250, 267, 269–271, 282, 301, 304, 306, 361, 437, 469, 602, 614 Tunneling, 117, 545 Two-dimensional spectroscopy IR (2D-IR), 556 UV, 558 Tyrosine emission spectrum, 21, 271 quenching, 271

U Ulexite, 174, 176 Ulstrup-Jortner, 303, 304

650 Uncertainty principle, 58, 80, 81, 83, 84 Ungerade, 56, 606

V Vacuum, 4, 11, 62, 87–89, 94, 96, 97, 99–101, 109, 110, 115, 116, 120 Vibrational absorption, 26, 56, 470, 519, 546, 556, 561, 562, 566, 571, 591, 592, 596–600, 604, 614 circular dichroism (VCD), 362, 470–473 mode, 118, 471, 517, 519, 556, 557, 561, 564–571, 573, 583, 584, 591–593, 596–604, 614 Vibronic coupling, 164, 206, 236, 565 Vibronic state, 196, 203, 228, 566, 567, 589, 591 Violaxanthin, 401, 402 Vision, 546 Visual pigments, 208–210, 213 van der Waals, 336, 337, 339, 355, 434 von Neumann equation, 491, 495, 533

W Wagging, 472 Warshel, A.A., 298, 337 Water, 7, 19, 20, 102, 117, 118, 448, 468, 499, 605, 613 Wave equation, 94, 96, 106, 108 Wavefunction adiabatic, 67 amplitudes, 36, 53, 61, 63, 67, 81, 113, 147, 150, 567, 587, 597 antisymmetric, 62, 63, 65, 71–74, 84, 148 basis, 65, 71, 77, 487 classical, 35, 50, 57, 110 d, 63, 65, 67, 111 diabatic, 66 dimer, 377, 410 electronic, 60, 65, 66, 71, 74, 75, 193, 195, 196, 200, 201, 212, 225, 450, 471, 561, 567, 587, 597, 598 free particle, 51 harmonic oscillator, 57

Index hydrogen atom, 57, 61, 65, 67 interference, 49, 110 lifetime, 81, 82 oligomer, 420, 421 p, 35, 567 particle in a box, 52, 147 s, 71 spatial, 36, 37, 57, 71–73, 75, 76, 84, 487, 561 spin, 65, 71, 72, 74–76, 84 spin-orbital, 73 symmetric, 62, 63, 71, 72, 74, 606 symmetry, 606 time dependent, 148, 487, 570, 597 vibrational, 65, 471, 472, 561, 562, 567, 570, 597, 598 Wavenumber, 4, 5, 26, 27, 31, 43, 44, 51, 56, 104, 220, 584, 585, 604 Wavepacket vibrational, 53, 561–565, 568, 569, 571, 587, 596, 597, 600 Wavevector, 97, 454, 531, 532, 543, 544, 547–549, 554, 558, 573, 603, 609 Wien displacement law, 104 Wiener-Khinchin theorem, 506 Wilson, 35 Wollaston, W.H. (1766–1828), 4 Work function, 10, 31

Y Yttrium aluminum garnet, 29, 175

Z Zeaxanthin, 401, 402 Zeeman effect, 70, 473 Zeiger, H.J., 357 Zero-field splitting, 73 Zero-phonon hole (ZPH), 205–207, 236, 519 Zero-point energy, 56, 109, 508 level, 109, 592, 598, 600 ZnSe, 275 ZnTe, 275