Dynamics of Molecular Excitons: Theories and Applications (Nanophotonics) [1 ed.] 0081023359, 9780081023358

Table of contents :
Cover
DYNAMICS OF
MOLECULAR EXCITONS
Copyright
Dedication
Contents
About the author
Preface
Acknowledgments
1 Introduction
1.1 Motivation and objective
1.2 Frenkel and Wannier excitons
1.2.1 A brief overview
1.2.2 Model Hamiltonians for Frenkel excitons
1.2.2.1 Linear chain
1.2.2.2 Circular chain
1.2.3 Semiclassical model Hamiltonians for Wannier excitons
1.3 Disorder, fluctuations, and measure of delocalization
1.4 Utility and limitations of exciton models
2 Microscopic derivation of Frenkel exciton-bath Hamiltonian
2.1 Aggregates of chromophores
2.1.1 Preliminary step: Hamiltonian of each chromophore in the adiabatic basis
2.1.2 Inter-chromophore interaction Hamiltonian terms in the adiabatic basis of each chromophore
2.1.3 Single exciton Hamiltonian in the site excitation basis
2.1.3.1 General expression
2.1.3.2 Crude adiabatic approximation
2.1.3.3 Minimal Frenkel exciton-bath model
2.2 Aggregates of chromophores embedded in host media
2.2.1 General consideration
2.2.2 Linearly coupled harmonic oscillator bath model
2.3 Summary and additional remarks
3 Linear spectroscopy of molecular excitons
3.1 Absorption lineshape
3.1.1 General formalism
3.1.2 Diagonal approximation in the exciton basis
3.1.3 Projection operator formalism
3.1.4 Second order approximation for exciton-bath coupling
3.1.5 Linearly coupled harmonic oscillator bath
3.1.5.1 Diagonal approximation in the exciton basis
3.1.5.2 Second order QME lineshape
3.2 Stimulated emission lineshape
3.3 Model calculations
3.4 Summary and additional remarks
4 Exciton transfer rates and hopping dynamics
4.1 Transfer between two exciton states: Förster theory's and its generalizations
4.1.1 General rate expression
4.1.2 Constant resonance coupling and independent baths
4.1.3 Inelastic transfer and independent baths
4.2 Transfer between groups of exciton states
4.2.1 General rate expression
4.2.2 Constant resonance coupling and independent baths
4.3 Master equation approaches and long range exciton hopping dynamics
4.4 Summary and additional remarks
5 Quantum dynamics of molecular excitons
5.1 Projection operator formalism
5.1.1 Quantum master equations for reduced exciton density operator
5.1.1.1 Time nonlocal equation
5.1.1.2 Time local equation
5.1.2 Quantum master equations for populations
5.1.2.1 Time nonlocal equation
5.1.2.2 Time local equation
5.2 Second order approximations
5.2.1 Second order QMEs for reduced exciton density operator
5.2.1.1 Time nonlocal equation
5.2.1.2 Time local equation
5.2.2 Second order QMEs for populations
5.2.2.1 Time nonlocal equation
5.2.2.2 Time local equation
5.3 Fourth order approximations
5.3.1 QMEs for the reduced system density operator
5.3.1.1 Time nonlocal equation
5.3.1.2 Time local equation
5.4 Harmonic oscillator bath with linear coupling
5.4.1 Second order QMEs
5.4.2 Second order polaron transformed QME (PQME)
5.4.2.1 General formalism
5.4.2.2 Model calculations
5.5 Summary and additional remarks
6 Excitons and quantum light
6.1 Interaction of materials with quantum light
6.2 Microscopic derivation of Förster's spectral overlap expression
6.2.1 Spontaneous emission
6.2.2 Absorption
6.2.3 Förster's spectral overlap expression
6.3 Polariton
6.4 Summary and additional remarks
7 Time-resolved nonlinear spectroscopy of excitons
7.1 General assumption of material Hamiltonian
7.2 Two-pulse spectroscopy
7.2.1 General expression
7.2.2 Pump-probe spectroscopy
7.3 Four wave mixing spectroscopy
7.3.1 Response function formalism
7.4 Summary and additional remarks
8 Examples and applications
8.1 Excitons in natural light harvesting complexes
8.1.1 FMO complex of green sulfur bacteria
8.1.2 LH2 complex of purple bacteria
8.1.3 PBP of cryptophyte algae
8.2 Excitons for photovoltaic devices
8.3 Excitons for structural determination
8.3.1 FRET efficiency
8.3.2 Beyond FRET efficiency measurement
8.4 Summary and additional remarks
9 Summary and outlook
A Useful mathematical identities and solutions
A.1 Solution of eigenvalue problems for the simple Frenkel exciton models
A.2 Some identities for averages involving harmonic oscillator models
A.2.1 Average of the product of two exponential operators
A.2.2 Displaced harmonic oscillator and polaron transformation
B Interaction between matter and classical electromagnetic fields
B.1 Maxwell equations
B.1.1 Vector and scalar potentials
B.1.2 Electromagnetic fields in source-free space
B.2 Classical Hamiltonian for matter and radiation interaction
B.3 Quantum mechanical Hamiltonian for matter-radiation interaction in the weak field limit with Coulomb gauge
B.4 Interaction with a plane wave radiation and dipole approximation
Bibliography
Index
Back Cover

Citation preview

DYNAMICS OF MOLECULAR EXCITONS

DYNAMICS OF MOLECULAR EXCITONS SEOGJOO J. JANG Queens College, City University of New York Department of Chemistry and Biochemistry Queens, NY, United States

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2020 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-08-102335-8 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Matthew Deans Acquisitions Editor: Simon Holt Editorial Project Manager: Lindsay Lawrence Production Project Manager: Prasanna Kalyanaraman Designer: Greg Harris Typeset by VTeX

To my wife

Contents

vii

Contents About the author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 1.2 1.3 1.4

Motivation and objective. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Frenkel and Wannier excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Disorder, fluctuations, and measure of delocalization. . . . . . . . . . . . 16 Utility and limitations of exciton models . . . . . . . . . . . . . . . . . . . . . . . 18

Chapter 2 Microscopic derivation of Frenkel exciton-bath Hamiltonian. 21 2.1 Aggregates of chromophores. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Aggregates of chromophores embedded in host media . . . . . . . . . 45 2.3 Summary and additional remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Chapter 3 Linear spectroscopy of molecular excitons . . . . . . . . . . . . . . . . 53 3.1 3.2 3.3 3.4

Absorption lineshape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stimulated emission lineshape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary and additional remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 72 76 79

Chapter 4 Exciton transfer rates and hopping dynamics . . . . . . . . . . . . . . . 83 4.1 Transfer between two exciton states: Förster theory’s and its generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2 Transfer between groups of exciton states. . . . . . . . . . . . . . . . . . . . . . 93 4.3 Master equation approaches and long range exciton hopping dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.4 Summary and additional remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Chapter 5 Quantum dynamics of molecular excitons. . . . . . . . . . . . . . . . . 107 5.1 Projection operator formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2 Second order approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.3 Fourth order approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

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5.4

Harmonic oscillator bath with linear coupling. . . . . . . . . . . . . . . . . . 132

5.5

Summary and additional remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Chapter 6 Excitons and quantum light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.1

Interaction of materials with quantum light . . . . . . . . . . . . . . . . . . . . 150

6.2

Microscopic derivation of Förster’s spectral overlap expression . 152

6.3

Polariton. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.4

Summary and additional remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Chapter 7 Time-resolved nonlinear spectroscopy of excitons. . . . . . . . . 161 7.1 General assumption of material Hamiltonian . . . . . . . . . . . . . . . . . . 162 7.2 Two-pulse spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 7.3 Four wave mixing spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7.4 Summary and additional remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Chapter 8 Examples and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 8.1

Excitons in natural light harvesting complexes. . . . . . . . . . . . . . . . . 182

8.2

Excitons for photovoltaic devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

8.3

Excitons for structural determination. . . . . . . . . . . . . . . . . . . . . . . . . . 188

8.4

Summary and additional remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

Chapter 9 Summary and outlook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Appendix A Useful mathematical identities and solutions. . . . . . . . . . . . 199 A.1 Solution of eigenvalue problems for the simple Frenkel exciton models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 A.2 Some identities for averages involving harmonic oscillator models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Appendix B Interaction between matter and classical electromagnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 B.1 Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 B.2 Classical Hamiltonian for matter and radiation interaction . . . . . . 213 B.3 Quantum mechanical Hamiltonian for matter-radiation interaction in the weak field limit with Coulomb gauge . . . . . . . . . 214

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B.4 Interaction with a plane wave radiation and dipole approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

About the author

About the author Seogjoo J. Jang is a Professor of Chemistry at Queens College of the City University of New York (CUNY), and is a doctoral faculty of both Chemistry and Physics PhD programs at the Graduate Center of CUNY. He obtained his BS (1989) and MS (1993) degrees in Chemistry from Seoul National University, and a PhD degree (1999) in Chemistry from the University of Pennsylvania. He then worked as a postdoctoral associate at MIT (1999–2002) and as a Goldhaber Distinguished Fellow (2003–2005) at Brookhaven National Laboratory before starting his faculty position at Queens College, CUNY in 2005. His research expertise is in quantum dynamics theories and computational modeling. In particular, he has pioneered modern theories of resonance energy transfer that are now being incorporated into theoretical analyses of experimental data on complex molecular systems, and has made key contributions to understanding the role of delocalized excitons in photosynthetic light harvesting complexes. He is a recipient of the National Science Foundation CAREER Award (2009) and the Camille Dreyfus Teacher Scholar Award (2010).

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Preface

Preface

Writing a book on exciton is an act of significant courage given that there are already many authoritative classic books on this topic that are hard to match. However, when Prof. David Andrews invited me to write a book on molecular excitons as one of the Nanophotonics Series, I accepted the offer without much hesitation. I hoped writing such a book would be a great chance to organize the information and knowledge I have cumulated over about two decades of research. In addition, throughout my interactions with colleagues and students, I have indeed experienced the necessity for a new kind of monograph on excitons that can address current computational and experimental need, in particular concerning the dynamics. This book is intended for general readers who have completed standard graduate level quantum mechanics or quantum chemistry courses. I tried to make this book as a kind of self contained theoretical reference for those who try to understand theories of exciton dynamics. This book can help non-experts who would like to get into this field of research and also experts by offering new details and perspectives that are not often discussed explicitly. I tried to also address applications as much as possible, but I should admit that the account of applications in this book is quite limited. There are three major issues this book is particularly focused on. The first is to understand the assumptions and approximations implicit in conventional models for molecular excitons and their interactions with environments. Most of consideration is at formal level, which will form the basis for more detailed numerical investigations in the future. In addition, this part can be considered as an extension of standard molecular quantum mechanics, and can be used as supplemental reading for a quantum chemistry class. The second is to offer as comprehensive description as possible of most dynamics theories of molecular excitons. The main focus was on explaining the derivation of these theories and methods. The rate theories and quantum dynamics methods described here are in fact much more general and are applicable to other quantum dynamical processes. Finally, detailed theoretical description is provided for the spectroscopy of excitons. Again, the major focus is on theoretical aspects of these spectroscopic meth-

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Preface

ods and understanding their capability as a tool for probing the exciton dynamics. Seogjoo J. Jang New York December 6, 2019

Acknowledgments

Acknowledgments I thank the late Bob Silbey who taught me a great deal about molecular excitons when I was a postdoctoral researcher at MIT. He also continued inspiring me for many years afterwards as a colleague. Interactions with him were always special. He could easily explain important ideas and issues in a few minutes, which helped me understand their implications so quickly and pursue new ideas efficiently. Writing this book gave me another chance to interact with him indirectly through his publications, and I am still learning from works he has completed decades ago. Although not directly related to this topic, I am heavily indebted to Marshall Newton, one of the best critical thinker and theoretician I am aware of. His broad knowledge and expertise in electron transfer theories helped me envision a roadmap for developing theories of exciton transfer and dynamics, which have remained relatively unexplored in comparison to the field of electron transfer theories. Communications and interactions with him helped me evolve through my early formative period as an independent theoretician. There are also many others that I should not forget mentioning names for various contributions related to this book. I am indebted to my mentors at Seoul National University, Sangyoub Lee and Kook Joe Shin, who first introduced me to Förster theory and guided my first research on this topic, and to my PhD mentor Gregory Voth from whom I learned the fundamentals of quantum dynamics. I would like to thank Jeff Cina, David Reichman, Chris Bardeen, Yuan-Chung Cheng, Benedetta Mennucci, Joe Subotnik, Dorthe Eisele, Greg Scholes, Young Min Rhee, David Coker, Jianshu Cao, Jasper Knoester, Graham Fleming, and K. Birgitta Whaley for discussions and help at various stages leading to this work, and Frank Spano, Ari Chakraborty, Qiang Shi, and Andrew Marcus for allowing me use their images. Support from National Science Foundation, the Department of Energy, and Camille Dreyfus Foundation have made it possible to continue projects that constitute various parts of this book. I am grateful to the editor of the Nanophotonics Series, Prof. David Andrews, and the editorial office of Elsevier for support. Finally, I cannot thank enough my wife who has been truly supportive of writing this book. I am happy to dedicate this book to her. Seogjoo J. Jang January 28, 2020

xv

1 Introduction Contents 1.1 Motivation and objective 1 1.2 Frenkel and Wannier excitons 3 1.2.1 A brief overview 3 1.2.2 Model Hamiltonians for Frenkel excitons

6

1.2.2.1 Linear chain 7 1.2.2.2 Circular chain 11

1.2.3 Semiclassical model Hamiltonians for Wannier excitons 15 1.3 Disorder, fluctuations, and measure of delocalization 16 1.4 Utility and limitations of exciton models 18

1.1 Motivation and objective An exciton represents a quantum mechanical state of electronic excitation or a bound pair of an electron and its hole.1 Any electronic excitation of a molecule can be viewed as an exciton confined within the molecule. However, the real benefit of the concept of exciton is that it helps generalizing the excitation beyond molecular boundaries in a simple manner. For example, excitons can be formed across the confines of molecules and exist in quantum mechanical superposition states. These represent collective quantum states distinctive from independent sums of elementary molecular excitations, which can exhibit unique quantum mechanical properties of collective nature. One well-known example is the super-radiance, a collective emission phenomenon that applies to any atomic [1] or molecular systems [2–6]. Direct experimental confirmation of molecular excitons was difficult in early days of quantum mechanics. Although wellestablished conceptually, their practical significance was not clearly recognized either. Through decades of research and development, excitons have steadily become established as essential objects for solar light harvesting, imaging, sensing, lasing, and light-emitting devices. Applications of excitons for quantum information processing devices and novel photonic materials are also emerging areas of research. Thanks to these advances, excitons are now well supported by many computational and experimental 1 The hole itself is a quasi-particle and refers to the absence of electron. Dynamics of Molecular Excitons. https://doi.org/10.1016/B978-0-08-102335-8.00009-0 Copyright © 2020 Elsevier Ltd. All rights reserved.

1

2

Chapter 1 Introduction

data, and are perceived as important objects for a few key applications. However, while data informing the existence and contribution of excitons abound, accurate and complete information on excitons is still difficult to get in general. In particular, accurate measurement of both spatial patterns and temporal dynamics of excitons, which is crucial for satisfactory microscopic characterization, remains a challenging area of research even to date. The difficulty in detailed microscopic characterization of excitons stems in part from their indefinite spatial extents (in nanometer length scales) and transient dynamical nature (in time scales typically as short as tens of femtoseconds). Resolving this issue requires experimental tools that can simultaneously probe nanometer length scales at femtosecond time resolution, which remain challenging despite recent advances. The other reason, although less appreciated, is the fact that exciton has often been perceived by experimentalists as a loosely defined simple object of semi-empirical nature. This is in contrast to the current status of computational molecular quantum mechanics, which has come a long way of constructing sophisticated methods and rigorous microscopic formulations that can account for various many-body quantum effects. The gap between these modern practices of molecular quantum mechanics and the conceptual basis of most experimental investigation based on old exciton theories, is often the source of confusion and miscommunication. For both practical and fundamental reasons, there is increasing demand for accurate characterization and control of excitons in complex environments. However, for many interesting systems, the scale and accuracy of calculations necessary for reliable computation of excitonic properties is in general beyond the reach of rigorous computational methods at present. Significant theoretical progress is needed for more accurate and microscopic description of excitons as well as for further advances in experimental probe techniques. In promoting the theoretical progress, it is important to establish a general theoretical framework for excitons that is both rigorous and practical. This is one of the main motivations for this book. The first objective of this book is to help consolidating many versions and ideas of excitons into a single theoretical formalism. While this is an ambitious objective difficult to achieve at present, this book can set the stage for initiating a long term effort of such nature. In addition, many assumptions and approximations clarified in this book will help gain better understanding of the dynamics and spectroscopic data. The second objective is to offer comprehensive description of various theories of exciton dynamics along with their applications. There are multiple approaches

Chapter 1 Introduction

possible for achieving this goal. The approach taken in this book is to begin with molecular Hamiltonians. This approach makes it straightforward to clarify assumptions and approximations implicit in different forms of excitons from the outset. The formal foundation for this will be provided in the next chapter. In the rest of this chapter, for motivational purposes, a brief overview of Frenkel and Wannier excitons is presented. This is then followed by introduction of simple model Hamiltonians used for describing such excitons. A few words on the convention of notations. For exciton states and Hamiltonians, the first order quantization and the Dirac notation for states and operators will be used in most cases. The second quantization will be primarily reserved only for the description of electromagnetic radiation within the quantum electrodynamics formalism. When appropriate, semiclassical description of electromagnetic radiation will also be used for the description of matter-radiation interaction.

1.2 Frenkel and Wannier excitons 1.2.1 A brief overview Frenkel [7–9] and Wannier excitons [10,8] represent two wellknown forms of collective excitations that are applicable in two opposing limits and are amenable for simple quantum mechanical description. Frenkel exciton theory has been successful for describing the spectroscopy of molecular crystals and aggregates [8,11,12], where molecules are bound together by weak interactions. In this case, electronic properties of individual molecules remain mostly intact. Thus, collective excitation processes can be described still in terms of properties characterizing the excitation of individual molecules. Application of the Frenkel exciton theory has since been expanded to macromolecular and supramolecular systems. Most recently, Frenkel excitons were shown to play a major role in natural photosynthetic light harvesting complexes [13]. On the other hand, Wannier excitons [10,8] are appropriate for inorganic insulating or semiconducting solid materials where the band theory serves as a successful framework to describe delocalized electrons and holes. The concept of Wannier exciton played a key role in the development of inorganic photovoltaic and solid state laser devices. To a certain extent, Wannier excitons can be used to describe electronic excitation processes in highly confined systems such as quantum dots.

3

4

Chapter 1 Introduction

The basis set of quantum mechanical states used for Frenkel exciton is the direct product of individual molecular orbitals, which are typically assumed to be orthogonal to each other. To be more specific, consider an example of N aggregates of molecules, where the j th molecule has the ground electronic state |gj , and a single excited state |ej . Then, the ground electronic state of the aggregate is |g = |g1  ⊗ |g2  · · · ⊗ |gN  ,

(1.1)

where ⊗ represents direct product. One can then introduce N orthogonal states forming the basis of single exciton states, each of which is denoted as |sj . This represents a state where only the j th molecule is in the excited state |ej  while all others are in the ground electronic state. Thus, |sj  = |g1  ⊗ · · · |gj −1  ⊗ |ej  ⊗ |gj +1  · · · ⊗ |gN  .

(1.2)

Higher order excitons such as double and triple excitons can also be defined in a similar manner. The original formulation of Frenkel exciton [7,9] used a single orbital to define each state |ej . However, this is not in fact an essential assumption. Each electronic state at molecule j can be any general molecular electronic state accounting for all many-body electronic correlations within the molecule as has been suggested by Agranovich [14]. The only assumption necessary for the construction of Frenkel exciton basis, as will become clear in the next chapter, is that ej |ek  = 0 for j = k. Consequently, sj |sk  = 0, for j = k .

(1.3)

Despite the assumption of weak interactions implicit in Eq. (1.3), the electronic excitation |ej  at the molecule j can still cause perturbation of electrons in the ground electronic state |gk  of another molecule k. Such interaction is dominated by exchange interaction at short range and becomes mainly Coulombic at moderate to long distances. These interactions couple the excitation of electrons at the j th molecule with that of electrons at the kth molecule, and are often called resonance interactions. In other words, given the electronic Hamiltonian of the aggregate Hˆ e , there is a non-vanishing resonance term Jj k = sj |Hˆ e |sk , for j = k .

(1.4)

These resonance interactions, Jj k ’s, cause eigenstates of Hˆ e to be coherent superpositions of |sj ’s. The resulting N eigenstates form

Chapter 1 Introduction

the exciton basis, which is different from the site excitation basis. The exciton states are fully delocalized and also form a band of virtually continuous states in the limit of large N . This exciton band is however constructed out of two electron processes and is different from the conduction and valance band of the single electron picture that form the basis of the Wannier excitons as described below. The concept of Wannier exciton is built on the band theory of electronic states in semiconducting/insulating solid materials, which represent almost continuum states of delocalized electrons (conduction band) and holes (valance band). Namely, it is based on the following effective electronic Hamiltonian:   Hˆ e = Ec (k)ek† eˆk + Ev (k)hˆ †k hk k

+

1  † V (p, k, k )eˆk+p hˆ †p−k hˆ p−k eˆp+k , Nt 

(1.5)

p,k,k

where Ec (k) and Ev (k) are energies of conduction and valance bands corresponding to wave vector k, eˆk† (eˆk ) and hˆ †k (hˆ k ) are creation (annihilation) operators of an electron and hole, respectively. Nt is the total number of lattice sites and V (p, k, k ) is the Coulomb potential between electron and hole represented in the momentum space. For electrons and holes localized near the band edges, semiclassical wave packets consisting of the superposition of the nearby band states can provide reasonable description of their properties. These wave packets behave in almost classical manner with effective masses reflecting the dispersion relationships of respective bands. Given that the distance between the centers of electron and hole wave packets is much larger than lattice spacings of the solids, their effective interaction can be modeled by simple Coulomb attraction screened by the average dielectric constant of the medium. Within this picture, the energy levels of excitons (in the simplest case) have the same structure as the electronic energy of the hydrogen atom, but with effective charges and masses that are different from bare ones. In the presence of confinement or boundary effects, the effective potential deviates significantly from a simple Coulombic behavior. Accurate theoretical understanding of such deviation is necessary for the determination of the exciton binding energy. The manner of localization/delocalization is different for Frenkel and Wannier excitons. In the former case, the delocalization occurs only through quantum mechanical superposition

5

6

Chapter 1 Introduction

Figure 1.1. Illustration of the formation of a Frenkel exciton. Each ellipse with + and − represents intramolecular excitation, and the whole exciton state (black line) is formed as a linear combination of these localized excitations. C1 , · · · , CN are complex coefficients such that |C1 |2 + |C2 |2 + · · · + |CN |2 = 1.

Figure 1.2. Illustration of the formation of Wannier exciton. Localized wavepackets of electron in the conduction band and hole in the valance band are formed and are bound to each other via Coulomb interaction.

of originally localized excitation states, namely, formation of delocalized exciton states. Fig. 1.1 provides illustration of such states. On the other hand, in the latter case, formation of excitons amount to partial localization of band states, which are fully delocalized effective single electron and hole states. Fig. 1.2 depicts this process. The sources of localization can vary but are most likely the combination of disorder, dynamic relaxation, and thermalization effect. The Coulombic binding potential between electron and hole also contributes to the localization as well.

1.2.2 Model Hamiltonians for Frenkel excitons For more concrete understanding, it is instructive to consider two well-known models of Frenkel excitons formed by N identical molecular units. One is a linear chain with only nearest neighbor interactions. The other is a circular chain with arbitrary non-

Chapter 1 Introduction

nearest neighbor interactions satisfying cyclic symmetry. In the absence of any disorder or dynamical fluctuations, these models have simple analytic expressions for their eigenvalues and eigenstates. Detailed consideration of these states can be used to illustrate key concepts and languages of molecular excitons. In the two models described below, it is assumed that the level spacings of excited states within each molecular unit are much larger than the resonance coupling. Under this condition, one can focus on only one excited state of each molecule with energy E as far as single exciton is concerned.

1.2.2.1 Linear chain Consider a linear chain of N identical molecules, with only the nearest neighbor resonance interaction denoted as J . Then, the single exciton Hamiltonian in the basis of |sj , with j = 1, · · · , N , can be represented as Hˆ e = E

N  j =1

|sj sj | + J

N−1 

 |sj sj +1 | + |sj +1 sj | .



(1.6)

j =1

The eigenstates and eigenvalues of this Hamiltonian are as follows:   pπ , (1.7) Ep = E + 2J cos N +1   1/2   N 2 pnπ |ϕp  = sin (1.8) |sj  , N +1 N +1 n=1

where p = 1, · · · , N . Detailed mathematical steps leading to these expressions are provided in Appendix A. For J < 0, the lowest value of the exciton energy and its eigenstate correspond to p = 1 in Eqs. (1.7) and (1.8) as follows:   π E1 = E + 2J cos , (1.9) N +1   1/2   N 2 nπ sin (1.10) |sj  . |ϕ1  = N +1 N +1 n=1

On the other hand, the highest value of the exciton energy and the corresponding eigenstate correspond to p = N as follows:   π EN = E − 2J cos , (1.11) N +1

7

8

Chapter 1 Introduction

 |ϕN  =

2 N +1

1/2  N

 sin

n=1

 nπ (−1)n+1 |sj  . (1.12) N +1

Therefore, the energies of all the exciton states are within a band of the following width:   = 4|J | cos

π N +1

 ≤ 4|J | ,

(1.13)

which are distributed symmetrically2 around E. Thus, the exciton band width is characteristic of the nearest neighbor coupling. For J > 0, the order of eigenstates is reversed. In other words, Eq. (1.11) becomes the lowest exciton energy and Eq. (1.9) the highest exciton energy. Except for this, other features remain the same as in the case of J < 0. Let us now consider a more concrete model for transition dipoles. Assume that the j th molecular unit has the following transition dipole moment:   μj = μ sin θj cos φj , sin θj sin φj , cos θj ,

(1.14)

where it is assumed that the linear chain is along the z-direction. Thus, θj is the axial angle and φj is the azimuthal angle of the direction of the transition dipole μj . Let us also assume that J is given by a sum of constant nearest neighbor term and an effective transition dipole interaction as follows: J = J0 +

 μ2  sin θj sin θj +1 cos(φj − φj +1 ) − 2 cos θj cos θj +1 , 3 d (1.15)

where  is an effective optical dielectric constant and d is the distance between nearest neighbors. This assumption is reasonable if the distance d is large enough such that μ2 /(8d 3 ) 0 becomes Eq. (3.4) with the exciton-radiation Hamiltonian, Eq. (3.5). The time evolution operator for this Hamiltonian is    i t  ˆ  dt HT (t ) , UˆT (t) = exp(+) −  0

(3.7)

where (+) represents chronological time ordering. Accordingly, the density operator of the material at time t is given by ρ(t) ˆ = UˆT (t)ρ(0) ˆ UˆT† (t) .

(3.8)

The probability to find the exciton at time is then given by Pe (t) =

Nc 

  T rb sj |UˆT (t)ρ(0) ˆ UˆT† (t)|sj  .

(3.9)

j =1

The lowest order terms contributing to Pe (t) can be found by expanding the time evolution operator up to the first order of Hˆ er (t) as follows:  i t  −i Hˆ m (t−t  )/ ˆ ˆ  (1) −i Hˆ m t/ ˆ UT (t) ≈ e − dt e Her (t  )e−i Hm t / . (3.10)  0 When the above approximation and its Hermitian conjugated are used for the propagators in Eq. (3.9), its approximation up to the

57

58

Chapter 3 Linear spectroscopy of molecular excitons

second order of Hˆ er (t) becomes Pe (t) ≈

1 2 ×



t

dt 



0

Nc 

t

dt 

0



ˆ

ˆ





Trb sj |e−i Hm (t−t )/ Hˆ er (t  )e−i Hm t / |gg|ρˆb

j =1 ˆ

× e i Hm t

 /

  ˆ Hˆ er (t  )ei Hm (t−t )/ |sj  .

(3.11)

Employing Eq. (3.5) for Hˆ er (t) in the above expression, Pe (t) can then be expressed as Pe (t) ≈

|A|2 2 ×



t



dt 

t

0 0 N N N c c c  



dt  e−iωr (t −t

 )

(μj  · er )(μj  · e∗r )Trb

j =1 j  =1 j  =1



ˆ

ˆ





× sj |e−i Hm (t−t )/ |sj  g|e−i Hm t / |g   ˆ  ˆ × g|ρˆb ei Hm t / |gsj  |ei Hm (t−t )/ |sj  .

(3.12)

For the exciton Hamiltonian given by Eq. (3.2), the generic form considered in this chapter, there is no coupling between |g and |sj  except for those due to the radiation. Thus, the following relation holds. ˆ

ˆ

g|e−i Hm t/ |g = e−iEg t/ e−i Hb t/ .

(3.13)

When this expression and its complex conjugate are inserted into Eq. (3.12), the probability to find the exciton at time t can be expressed as Pe (t) ≈

|A|2 2 ×

 0

t

dt 



t



dt  e−i(ωr +Eg /)(t −t

 )

0

Nc Nc  Nc  

(μj  · er )(μj  · e∗r )

j =1 j  =1 j  =1

  ˆ ˆ  × Trb sj |e−i Hm (t−t )/ |sj  e−i Hb t /   ˆ  ˆ × ρˆb ei Hb t / sj  |ei Hm (t−t )/ |sj  .

(3.14)

Chapter 3 Linear spectroscopy of molecular excitons

Figure 3.2. Diagram of the material Hamiltonian Hˆ m and its division into different components.

Let us introduce Hˆ mex as the projection of the material Hamiltonian onto the exciton manifold as follows:

Hˆ mex =

Nc Nc  

|sj sj |Hˆ m |sj  sj  |

j =1 j  =1

= Hˆ es + Hˆ eb + Hˆ b ,

(3.15)

where Hˆ es is the exciton Hamiltonian excluding the ground electronic state and is thus expressed as

Hˆ es =

Nc 

Ej |sj sj | +

Nc Nc  

Jj k |sj sk | .

(3.16)

j =1 k=j

j =1

Fig. 3.2 illustrates various components constituting Hˆ m that have been defined above. Let us also introduce the following sum of site excitation states weighted by transition dipoles: Nc  (μj · er )|sj  . |Der  = j =1

(3.17)

59

60

Chapter 3 Linear spectroscopy of molecular excitons

Then, Eq. (3.14) can be expressed as Pe (t) ≈

|A|2 2 ×



t

dt 

0

Nc 



t



dt  e−i(ωr +Eg /)(t −t

 )

0



ˆ ex



Trb sj |e−i Hm (t−t )/ |Der Der |

j =1 ˆ

× ρˆb ei Hb (t

 −t  )/

ˆ ex

ei Hm (t−t

 )/

 |sj  .

(3.18)

Since all the operators within the trace of the above equation are defined in the subspace formed by the bath and the exciton states,  c for which N j =1 |sj sj | forms an effective electronic identity operator, Eq. (3.18) can be rearranged and simplified as Pe (t) ≈

|A|2 2



t

dt 

0



× Trex e



t



dt  e−i(ωr +Eg /)(t −t

 )

0 −i Hˆ mex (t  −t  )/

ˆ

|Der Der |ρˆb ei Hb (t

 −t  )/

 ,

(3.19)

where Trex represents trace over the bath and all the exciton states. The ideal absorption line shape (without averaging over the ensemble of disorder) can then be expressed as the steady state limit of the time derivative of Eq. (3.18), divided by a normalization factor, which is 2 /(2π |A|2 ). Thus, 2 lim Pe (t) 2π|A|2 t→∞  ∞   1 ˆ ˆ ex = Re dtei(ωr +Eg /)t Trex ei Hb t/ e−i Hm t/ |Der Der |ρˆb , π 0 (3.20)

I (ωr ) =

where the cyclic invariance of the trace operation has been employed. This lineshape expression is general and applicable for any form of Hˆ mex , which is assumed to be time independent.

3.1.2 Diagonal approximation in the exciton basis Let us denote the eigenvalue and eigenstate of Hˆ es defined by Eq. (3.2) as Ep and |ϕp  with p = 1, · · · , Nc . Examples of these eigenstates and eigenvalues for linear and circular chains of chromophores were provided in Chap. 1. The transformation between |ϕp ’s and |sj ’s is determined by the unitary matrix with the following element: Ujp = sj |ϕp . Then, |Der  defined by Eq. (3.17) can

Chapter 3 Linear spectroscopy of molecular excitons

be expressed as |Der  =

Nc 

˜ p · er )|ϕp  , (μ

(3.21)

p=1



∗ . Inserting the identity operator defined in μj Ujp  the electronic exciton subspace, 1ˆ e = p |ϕp ϕp |, into Eq. (3.20), the absorption lineshape can be expressed as  ∞ 1 I (ωr ) = Re dtei(ωr +Eg /)t π 0    ˆ ˆ ex ˜ p · er )Der |ρˆb . × Trex ei Hb t/ |ϕp ϕp |e−i Hm t/ |ϕp (μ

˜p = where μ

p

j

p

(3.22) Then, taking the trace over the exciton states |ϕp ’s and then using the fact that ρˆb is independent of these, the above expression can be simplified to  ∞ 1 dtei(ωr +Eg /)t I (ωr ) = Re π 0    ˆ ˆ ex ˜ p · er )(μ ˜ p · er )∗ ρˆb . × Trb ei Hb t/ ϕp |e−i Hm t/ |ϕp (μ p

p

(3.23) Let us now divide Hˆ mex given by Eq. (3.15) into two terms, one diagonal and the other off-diagonal in the exciton basis as follows: Hˆ mex =

Nc 

|ϕp ϕp |Hˆ mex |ϕp ϕp | +

|ϕp ϕp |Hˆ mex |ϕp ϕp |

p=1 p  =p

p=1 ex ≡ Hˆ m,0

Nc  

ex + Hˆ m,1

(3.24)

,

where the terms in the second line are respectively defined by the ex is diagonal in the exciton expressions in the first line. Thus, Hˆ m,0 ex is off-diagonal in the exciton basis. For the material basis and Hˆ m,1 Hamiltonian given by Eq. (3.1), note that these can be expressed as ex Hˆ m,0 =

Nc 

Ep + Bˆpp + Hˆ b |ϕp ϕp | ,

(3.25)

p=1 ex Hˆ m,1 =

Nc   p=1 p  =p

Bˆpp |ϕp ϕp | ,

(3.26)

61

62

Chapter 3 Linear spectroscopy of molecular excitons

where Bˆpp = ϕp |Hˆ eb |ϕp  =

Nc Nc  

∗ Ujp Ukp Bˆ j k .

(3.27)

j =1 k=1 ex is negligible compared to Hˆ ex , Given that the contribution of Hˆ m,1 m,0 one can simply ignore the former in Eq. (3.22), which leads to the following diagonal-bath (in the exciton basis) approximation:  1  ∞ Ie−db (ωr ) = Re dteiωr t+i(Eg −Ep )t/ π 0 p   ˆ ˆ ˆ ˜ p · er |2 ρˆb . × Trb ei Hb t/ e−i(Bpp +Hb )t/ |μ (3.28)

The above expression amounts to the simplest approximation accounting for dynamical broadening (or dephasing) of each exciton state due to interactions with the bath. As yet, using this approximation is physically more satisfactory than simply dressing each exciton state with a phenomenological lineshape function because it can account for different extents of broadening of different peaks based on clear physical basis. Eq. (3.28) is applicable to any kind of Hˆ b and Bˆpp . In addition, it is also applicable to the ˜ p depends on the bath degrees case where the transition dipole μ of freedom. It is possible to systematically improve Eq. (3.23) by including ex . Most notable of such an approximation the contributions of Hˆ m,1 is the so called modified Redfield equation, for which readers can refer to recent theoretical works. Another alternative method of calculating the lineshape is to employ a quantum master equation (QME) approach, which treats the entire Hˆ eb as a perturbation term. The projection operator formalism offers a compact way to develop this approach, which will be described in the next section.

3.1.3 Projection operator formalism The calculation of the lineshape expression given by Eq. (3.20), which is exact, needs only the information on the following coherence term of the total density operator: ˆ

ˆ ex

ρˆeg (t) = ei Hb t/ e−i Hm t/ |Der Der |ρˆb .

(3.29)

The initial condition of this is ρˆeg (0) = |Der Der |ρˆb . One can also ˆ define a similar coherence term by moving ei Hb t/ to the righthand end side of Eq. (3.29), which is equivalent to Eq. (3.29) in calculating the lineshape due to the cyclic invariance of the trace

Chapter 3 Linear spectroscopy of molecular excitons

operation. For solving the time evolution equation, the form given by Eq. (3.29) is somewhat more convenient. Taking the time derivative of Eq. (3.29) and utilizing the fact that Hˆ es commutes with Hˆ b , one can obtain the following time evolution equation:

d i ˆ ˆ ex ρˆeg (t) = ei Hb t/ Hˆ b − Hˆ es − Hˆ eb − Hˆ b e−i Hm t/ |Der Der |ρˆb dt 

i ˆ ˆ = − Hˆ es + ei Hb t/ Hˆ eb e−Hb t/ ρˆeg (t) . (3.30)  Solving the above time evolution equation for the full degrees of freedom is very challenging if not impossible. Thus, reduction of the degrees of freedom is needed to make the calculation feasible. Let us assume that exact calculation of the lineshape by Eq. (3.20) is possible through Pρeg (t), where P is a projection super-operator. The detailed form of P is not important at this stage except for a couple of key generic properties. Note that P applies to the entire product of operators on its right-hand side and that P 2 = 1 by definition. Let us also assume that the Hamiltonian in Eq. (3.30) can be divided as follows: ˆ ˆ Hˆ es + ei Hb t/ Hˆ eb e−Hb t/ = Hˆ 0 (t) + Hˆ 1 (t) ,

(3.31)

where the form of Hˆ 0 (t) and Hˆ 1 (t) can be arbitrary except that P Hˆ 0 (t) = Hˆ 0 (t)P and P Hˆ 1 (t)P = 0. Depending on the form of the projection operator, the manner of this division can be different. The only condition being required here is that it is easy to calculate the propagator for Hˆ 0 (t) either analytically or numerically. With this in mind, now let us introduce the following (partial) interaction picture operator: t

ˆ

i dτ H0 (τ )/ ˆ G(t) = e(−)0 ρˆeg (t) .

(3.32)

The time evolution equation of this operator is as follows: 



t d ˆ i i t dτ Hˆ 0 (τ )/ ˆ i dτ Hˆ 0 (τ )/ d G(t) = e(−)0 ρˆeg (t) H0 (t)ρˆeg (t) + e(−)0 dt  dt i ˆ , (3.33) = − Hˆ 1,I (t)G(t) 

where Eqs. (3.30) and (3.31) have been used in obtaining the second equality and Hˆ 1,I (t) is the following interaction picture operator: t

ˆ

t

ˆ

i dτ H0 (τ )/ −i dτ H0 (τ )/ . Hˆ 1,I (t) = e(−)0 Hˆ 1 (t)e(+) 0

(3.34)

63

64

Chapter 3 Linear spectroscopy of molecular excitons

The remaining procedure now is a standard projection operator technique applied to Eq. (3.33). With the application of P and Q = 1 − P, Eq. (3.32) can be decomposed into the following two ˆ coupled equations for the projected and unprojected parts of G(t): d ˆ i ˆ , P G(t) = − P Hˆ 1,I (t)QG(t) dt  d ˆ i ˆ . QG(t) = − QHˆ 1,I (t) (Q + P) G(t) dt 

(3.35) (3.36)

In Eq. (3.35), the fact that P Hˆ 1,I (t)P = 0 has been used. The formal solution of Eq. (3.36) is as follows:

 i t ˆ ˆ QG(t) = exp(+) − dτ QHˆ 1,I (τ ) QG(0)  0

  i t i t  ˆ ˆ ). − dτ exp(+) − dτ QH1,I (τ  ) P G(τ  0  τ

(3.37)

Inserting this into Eq. (3.35), we obtain the following equation:

 d ˆ i ˆ i t ˆ ˆ dτ QH1,I (τ ) QG(0) P G(t) = − P H1,I (t) exp(+) − dt   0  1 t − 2 dτ P Hˆ 1,I (t)  0

 i t  ˆ  ˆ ). dτ QH1,I (τ ) QHˆ 1,I (τ )P G(τ × exp(+) −  τ (3.38) Except for the inhomogeneous term, this is a closed form equation ˆ involving only P G(t). Employing Eq. (3.32), this can be converted to the time evolution equation for the projected part of ρˆeg (t) as follows: d i P ρˆeg (t) = − Hˆ 0 (t)P ρˆeg (t) dt 

  i −i 0t dτ Hˆ 0 (τ )/ ˆ i t ˆ − e(+) P H1,I (t) exp(+) − dτ QH1,I (τ ) Qρˆeg (0)   0

  t t 1 i t  ˆ −i 0 dτ Hˆ 0 (τ )/  ˆ − 2 dτ e(+) P H1,I (t) exp(+) − dτ QH1,I (τ )  τ  0 τ

i dτ × QHˆ 1,I (τ )e(−)0

 Hˆ

0 (τ

 )/

P ρˆeg (τ ) .

(3.39)

The fact that P Hˆ 0 (t) = Hˆ 0 (t)P has been used in obtaining the above equation.

Chapter 3 Linear spectroscopy of molecular excitons

Although Eq. (3.39) appears to be much more complicated than Eq. (3.29), it makes a systematic approximation for the latter easier by singling out unprojected part, for which various perturbative expansions can be made. For example, when expanded up to the second order of Hˆ 1 (t), Eq. (3.39) can be approximated as d i P ρˆeg (t) ≈ − Hˆ 0 (t)P ρˆeg (t) dt  t i −i dτ Hˆ 0 (τ )/ − P Hˆ 1 (t)e(+) 0 Qρˆeg (0)   t τ  t  1 −i dτ Hˆ 0 (τ  )/ −i dτ Hˆ 0 (τ  )/ Hˆ 1 (τ )e(+) 0 dτ P Hˆ 1 (t)e(+) τ − 2  0 × Qρˆeg (0)  t 1 t −i dτ  Hˆ 0 (τ  )/ Hˆ 1 (τ )P ρˆeg (τ ) . − 2 dτ P Hˆ 1 (t)e(+) τ  0 (3.40)   t In obtaining the above equation, exp(+) − i 0 dτ QHˆ 1,I (τ ) has been expanded up to the first and zeroth orders of QHˆ 1,I (τ ) in the second and third lines of Eq. (3.39), respectively. Since  −i

t

dτ  Hˆ 0 (τ  )/

e(+) τ can be determined easily at least numerically, solving the above equation is much more practicable than the original equation. More detailed account of this will be provided in the next subsection.

3.1.4 Second order approximation for exciton-bath coupling Let us consider a well-known example of the projection operator, which involves pre-averaging over the bath degrees of freedom as follows: P(·) = ρb Trb {(·)} .

(3.41)

For this definition, Hˆ 0 (t) = Hˆ es , which is time independent, and ˆ ˆ Hˆ 1 (t) = ei Hb t/ Hˆ eb e−i Hb t/ . In addition, Qρˆeg (0) = |Der Der |Qρˆb = 0. As a result, Eq. (3.40) simplifies to d i P ρˆeg (t) ≈ − Hˆ es P ρˆeg (t) dt   1 t ˆ ˆ ˆ − 2 dτ Pei Hb t/ Hˆ eb e−i(t−τ )Hes / e−i(t−τ )Hb /  0 ˆ

× Hˆ eb e−i Hb τ/ P ρˆeg (τ ) .

(3.42)

65

66

Chapter 3 Linear spectroscopy of molecular excitons

The above equation can easily be converted to the lineshape expression. For this, let us take trace of Eq. (3.42) over the bath degrees of freedom, and multiply both sides of the resulting equation with ei(ωr +Eg /)t . Then, integration of this equation over t leads to the following expression:  ∞ ei(ωr +Eg /)t T rb ρˆeg (t)   ∞0   − i(ωr + Eg /) dtei(ωr +Eg /)t T rb ρˆeg (t) 0  ∞    i ˆ (2) dtei(ωr +Eg /)t T rb ρˆeg (t) , (3.43) ≈ − Hes − Kˆ eb (ωr )  0 where 1 (2) Kˆ eb (ωr ) = 2 



∞

dtei(ωr +Eg /)t

0

  ˆ ˆ ˆ × Trb ei Hb t/ Hˆ eb e−i Hb t/ e−i Hes t/ Hˆ eb ρˆb .

(3.44)

  In Eq. (3.43), the fact that ei(ωr +Eg /)t T rb ρˆeg (t) approaches zero as t → ∞ was used. This is generally true if lifetime decay is also taken into consideration. Eq. (3.43) can then be rearranged as follows:  ∞  −1   (2) dtei(ωr +Eg /)t T rb ρˆeg (t) ≈ i ωr + Eg / − Hˆ es / + i Kˆ eb (ωr ) 0

× |Der Der | , (3.45)     where the fact that T rb ρˆeg (0) = T rb |Der Der |ρˆb = |Der Der | has been used. The above expression can then used in Eq. (3.20) to obtain the following 2nd order QME expression for the absorption lineshape: Nc  −1  1 (2) (2) Ieb (ωr ) = − Im sj | ωr + Eg / − Hˆ es / + i Kˆ eb (ωr ) π j =1

× |Der Der |sj   −1 1 (2) = − Im Der | ωr + Eg / − Hˆ es / + i Kˆ eb (ωr ) |Der  , π (3.46)  c where the fact that N j =1 |sj sj | forms an effective identity operator in the exciton manifold has been used in the second equality. The lineshape expression given by Eq. (3.46) can be calculated easily because it only involves inversion of a complex matrix defined in the exciton subspace. This expression can be used for any

Chapter 3 Linear spectroscopy of molecular excitons

kind of bath and exciton-bath Hamiltonians. However, due to the assumption of weak exciton-bath coupling, it fails to account for multiphonon contributions. Therefore, vibrational progressions cannot be represented accurately by this expression. Another subtle issue is that the lineshape calculated by this expression is not guaranteed to be positive definite. Although the evaluation of Eq. (3.46) does not require a specific choice of basis, it is often useful to express it in the exciton basis, namely, in the basis of |ϕp ’s. In this basis, the transition dipole state |Der  is given by Eq. (3.21). The exciton-bath Hamiltonian can also be expressed as Hˆ eb =

Nc  Nc  Nc Nc  

∗ ˆ Ujp Bj k Ukq |ϕp ϕq | =

p=1 q=1 j =1 k=1

Nc Nc  

Bˆpq |ϕp ϕq | ,

p=1 q=1

(3.47) where Bˆpq has been defined by Eq. (3.27). Employing this expres(2) sion, Kˆ eb (ωr ) defined by Eq. (3.44) can be shown to be  ∞ (2) dtei(ωr +(Eg −Ep )/)t Kˆ eb (ωr ) = p

p



q

0

 × Trb Bˆpp (t)Bˆp q ρˆb |ϕp ϕq | ,

(3.48)

ˆ ˆ where Bˆpp (t) = ei Hb t/ Bˆpp e−i Hb t/ . One well-known approximation that simplifies the calculation of Eq. (3.46) is the secular approximation, which amounts to as(2) suming that Kˆ eb (ωr ) is diagonal in the exciton basis as follows:  (2) (2) Keb,p (ωr )|ϕp ϕp | , (3.49) Kˆ eb (ωr ) ≈ p

where (2) Keb,p (ωr ) =

 p

∞

  dtei(ωr +(Eg −Ep )/)t Trb Bˆpp (t)Bˆp p ρˆb . (3.50)

0

(2) (2),R (ωr ) as Keb,p (ω) and Denoting the real and imaginary parts of Keb,p (2),I

iKeb,p (ω), Eq. (3.46) within the approximation of Eq. (3.49) simplifies to (2) (ωr ) ≈ Ieb

(2),R Nc ˜ p · er |2 Keb,p |μ (ω) 1 . 2  π (2),I (2),R p=1 ωr + (Eg − Ep )/ − Keb,p (ωr ) + Keb,p (ωr )2

(3.51)

67

68

Chapter 3 Linear spectroscopy of molecular excitons

(2),I

In the above expression, Keb,p (ωr ) represents the shift of the ex(2),R

citation energy due to interaction with the bath and Keb,p (ωr ) represents the broadening due to bath relaxation. Eqs. (3.46) and (3.51) can be improved further by including higher order approximations with respect to Hˆ eb . For example, as shown recently by the author, a general 4th order expression can be derived for the lineshape expressions [42]. In principle, it is possible to go even beyond the 4th order approximation although the resulting expressions are expected to be much more complicated let alone the difficulty of numerical evaluation.

3.1.5 Linearly coupled harmonic oscillator bath For aggregates of chromophores that remain structurally stable upon excitation, whether self assembled or embedded in matrices, the molecular vibrations and environmental degrees of freedom interacting with excitons can be modeled as a set of coupled harmonic oscillators. In general, the number of the modes is large enough to be assumed as being infinite. After normal mode transformation of all those oscillators, the bath Hamiltonian can then be expressed as a sum of independent modes, resulting in Eq. (2.88) for the bath Hamiltonian. Given that the major exciton-bath interactions can be approximated only by terms linear in the displacement of the oscillators, they can be represented by linear combinations of the normal mode coordinates, each of which is proportional to (bˆn + bˆn† ). Although not essential, these exciton-bath couplings are typically assumed to be diagonal in the basis of site excitation states. This represents the conventional physical situation of aggregates where the major excitation-vibration couplings come from within and around each chromophore unit. Thus, in this section, it will be assumed that the exciton-bath coupling is diagonal in the site excitation basis and that exciton-bath Hamiltonian is given by Eq. (2.89). Even with this simplification, a rich class of physical situations can be realized by different forms of the bath spectral density, Eq. (2.90). When expressed in the exciton basis |ϕp ’s, the bath coupling terms defined by Eq. (3.27) for the exciton-bath Hamiltonian Eq. (2.89) has the following form: Bˆpq =

 j

∗ ωn gj n Ujp Uj q (bˆn + bˆn† ) .

(3.52)

n

More detailed lineshape expressions will be provided for this case.

Chapter 3 Linear spectroscopy of molecular excitons

3.1.5.1 Diagonal approximation in the exciton basis The expression for the lineshape expression Eq. (3.28), which is based on the assumption that off-diagonal component of the exciton-bath coupling in the exciton basis can be ignored, will be presented here for the bath model of linearly coupled harmonic ˜ p · er |2 is independent of the oscillators. Here, it is assumed that |μ bath degrees of freedom. Then, Eq. (3.28) can be expressed as follows:  ∞ 1  ˜ p · er |2 |μ dteiωr t+i(Eg −Ep )t/ Ie−db (ωr ) = Re π 0 p   ˆ ˆ ˆ × Trb eit Hb / e−it (Bpp +Hb )/ ρˆb . (3.53) In the above equation, the trace over the bath can be expressed as    ˆ ˆ ˆ ˆ ˆ ˆ Trb eit Hb / e−it (Bpp +Hb )/ ρˆb = eit Hb,n / e−it (Bpp,n +Hb,n )/  , n

(3.54) where Hˆ b,n = ωn (bˆn† bˆn + 1/2), Bˆpp,n = ˆ

 j

ωn gj n |Ujp |2 (bˆn + bˆn† ), ˆ

and · · ·  denotes averaging over e−β Hb,n /T r{e−β Hb,n }. Each average in Eq. (3.54) corresponds to C(t) defined by Eq. (A.34) in Appendix A for the specific Hˆ b,n and Bˆpp,n . Thus, employing the final expression, Eq. (A.39), for each component of the product in Eq. (3.54) with appropriate change of notation, it can be expressed as follows:   ˆ ˆ ˆ Trb eit Hb / e−it (Bpp +Hb )/ ρˆb ⎧ ⎨   = ei n λp,n t/ exp − |Ujp |2 |Ukp |2 gj n gkn ⎩ n j k     βωn × coth , (3.55) (1 − cos(ωn t)) + i sin(ωn t) 2 where λp,n = ωn

 j

|Ujp |2 |Ukp |2 gj n gkn .

(3.56)

k

In Eq. (3.55), the sum over n can be replaced with integral over ω employing the spectral density Jj k (ω) defined by Eq. (2.90). Similarly, the sum of λp,n can also be expressed in terms of this spectral

69

70

Chapter 3 Linear spectroscopy of molecular excitons

density as follows: λp =

 n

λp,n =

 j

|Ujp |2 |Ukp |2

k

1 π

 0

∞

Jj k (ω) . ω

(3.57)

Thus, inserting Eq. (3.55), with the replacements by the spectral density as noted above, into Eq. (3.53), one can obtain the following expression for the absorption lineshape:  ∞ 1  ˜ p · er |2 |μ dteiωr t+i(Eg −Ep +λp )t/ Ie−db (ωr ) = Re π 0 p ⎧  ∞ ⎨ 1  Jj k (ω) × exp − |Ujp |2 |Ukp |2 dω ⎩ π ω2 0 j k     βω × coth (1 − cos(ωt)) + i sin(ωt) . 2 (3.58) For the case where there are no correlation between the coupling to different sites such that Jj k (ω) = δj k Jjj (ω), the above expression simplifies to  ∞ 1  ˜ p · er |2 |μ dteiωr t+i(Eg −Ep +λp )t/ Ie−db (ωr ) = Re π 0 p ⎧  ∞ ⎨ 1  Jjj (ω) × exp − |Ujp |4 dω ⎩ π ω2 0 j      βω × coth (1 − cos(ωt)) + i sin(ωt) . 2 (3.59) If the spectral density for each site is identical,  the relative broadening of each exciton is determined by j |Ujp |4 , which corresponds to the participation ratio of the exciton state |ϕp . The measure of delocalization defined as inverse participation ratio has direct physical implication in this simple limit.

3.1.5.2 Second order QME lineshape For the calculation of the lineshape based on the second order QME, Eq. (3.46), the bath correlation function super-operator, Eq. (3.48), needs to be specified in more detail. For the present model, this can be calculated by inserting Eq. (3.52) into Eq. (3.48) ˆ ˆ and using the fact that ei Hb t/ (bˆn + bˆn† )e−i Hb t/ = bˆn e−iωn t + bˆn† eiωn t .

Chapter 3 Linear spectroscopy of molecular excitons

The resulting expression is as follows:  (2) ∗ ∗ (ωr ) = ωn2 gj n gj  n Ujp Ujp Ukp Kˆ eb  Ukq |ϕp ϕq | p

p ∞



×

q

j

k

n

  dteiωr t+i(Eg −Ep )t/ Nˆ n + 1e−iωn t + Nˆ n eiωn t ,

0

(3.60) where Nˆ n  = bn† bn  = 1/(eβωn − 1), which is the thermal average of the vibrational quanta of mode n. Once again the sum over n in the above equation can be replaced with an integration over ω employing the spectral density Jj k (ω) defined by Eq. (2.90). The resulting expression is as follows: 1  ∗ (2) ∗ (ωr ) = Ujp Ujp Ukp Kˆ eb  Ukq |ϕp ϕq | π p  q j k p  ∞ × dteiωr t+i(Eg −Ep )t/ 0      ∞ βω × dωJj k (ω) coth cos(ωt) − i sin(ωt) . 2 0 (3.61) While the above expression involves the bath spectral density in a similar manner as Eq. (3.58), the latter has an additional factor, ω−2 . This additional factor results from the contribution of all the multi-phonon terms, which is absent in the second order QME approximation. For Ohmic or sub-Ohmic spectral density, this causes significant qualitative difference between the two approximations. For the secular approximation of the 2nd order lineshape, only (2) (2) the diagonal component Keb,p (ωr ) = ϕp |Kˆ eb (ωr )|ϕp  is needed, which is expressed as  ∞ 1  ∗ (2) ∗ Keb,p (ωr ) = Ujp Ujp Ukp U dteiωr t+i(Eg −Ep )t/  kp π  0 j k p      ∞ βω × dωJj k (ω) coth cos(ωt) − i sin(ωt) . 2 0 (3.62) The lineshape expression Eq. (3.51) can then be calculated by using the real and imaginary parts of the above bath correlation function. In the simple case where there is no correlation in the spectral density between different sites, namely,

71

72

Chapter 3 Linear spectroscopy of molecular excitons

Jj k (ω) = Jjj (ω)δj k , Eq. (3.62) simplifies to (2) Keb,p (ωr ) =

 ∞ 1  2 2 |Ujp | |Ujp | dteiωr t+i(Eg −Ep )t/ π  0 j p      ∞ βω × dωJjj (ω) coth cos(ωt) − i sin(ωt) . 2 0 (3.63)

Note that this bath correlation does not reduce to that involving only the participation ratio, unlike Eq. (3.59), even in the limit where Jjj (ω) is independent of j .

3.2 Stimulated emission lineshape Stimulated emission is similar to the absorption in that it is caused by the linear matter radiation interaction as represented by Eq. (3.5). The difference is that the initial condition of the material is an excited state, which is typically created by another absorption of photons. While the detailed nature of the excited state can be different depending on the specific mode of preparation, if the material has reached the steady state limit and can be represented by the canonical density operator for Hˆ mex defined by Eq. (3.15) (see Fig. 3.2), the initial density operator can be expressed as ˆ ex

ρˆex (0) =

e−β Hm

T r{e−β Hˆ m } ex

(3.64)

.

For the above initial density operator, the probability to find the material in the ground electronic state, up to the second order of the matter-radiation interaction, is

Pg (t) ≈

|A|2 2



t

dt 



0



× Trb g|e

t



dt  eiωr (t −t

 )

0

ˆ

(μj  · e∗r )(μj  · er )

j  =1 j  =1

−i Hˆ m (t−t  )/

× ρˆex (0)ei Hm t

Nc Nc  

 /

ˆ



|gsj  |e−i Hm t / ˆ

|sj  g|ei Hm (t−t

 )/

 |g .

(3.65)

This can be obtained following a procedure similar to that of deriving Eq. (3.12). Employing Eq. (3.13) and the definition Eq. (3.17),

Chapter 3 Linear spectroscopy of molecular excitons

this can be expressed as   |A|2 t  t  i(ωr +Eg /)(t  −t  ) Pg (t) ≈ 2 dt dt e  0 0   ˆ   ˆ ex   × Trex e−i Hb (t −t )/ |Der Der |ρˆex (0)ei Hm (t −t )/ . (3.66) Then, as in the case of absorption, the lineshape for the stimulated emission can be defined as the steady state limit of the time derivative of Eq. (3.66), divided by 2 /(2π |A|2 ). Thus, 2 lim Pg (t) 2π|A|2 t→∞  ∞ 1 = Re dte−i(ωr +Eg /)t π 0   ex ˆ ˆ × Trex ei Hm t/ e−i Hb t/ |Der Der |ρˆex (0) .

E(ωr ) =

(3.67)

Inserting the identity operator in the electronic exciton subspace,  ˆ ex 1ˆ e = p |ϕp ϕp |, before and after ei Hb t/ , and taking trace over the exciton states first, one can obtain  ∞ 1 dte−i(ωr +Eg /)t E(ωr ) = Re π 0   ˆ ex ˆ × Trex ϕp |ei Hm t/ |ϕp e−i Hb t/ p

p

p 

 ˜ p · er )(μ ˜ p · er )∗ ϕp |ρˆex (0)|ϕp  . × (μ (3.68) The above expression is similar to Eq. (3.23) except for difference in sign and the fact that ϕp |ρˆex (0)|ϕp  is nonzero in general even when p  = p. At the simplest level, one can make diagonal approximation for ˆ ex both ei Hm t/ and ρˆex (0) in Eq. (3.68). The resulting lineshape expression is as follows.  1  e−β Ep ∞ dte−iωr t−i(Eg −Ep )t/ Ee−db (ωr ) = Re π Z eb 0 p   ˆ ˆ ˆ ˆ ˆ ˜ p · er |2 e−β(Bpp +Hb ) , × Trb eit (Bpp +Hb )/ e−it Hb / |μ (3.69) where Zeb =

 p

ˆ

ˆ

e−β Ep T rb {e−β(Bpp +Hb ) } .

(3.70)

73

74

Chapter 3 Linear spectroscopy of molecular excitons

For the model of linearly coupled harmonic oscillator bath, for which Bˆpq is given by Eq. (3.52), the above expression can be calculated explicitly in a way similar to the absorption lineshape. As˜ p · er |2 is independent of bath and dividing the trace suming that |μ over the bath into different modes, Eq. (3.69) can be expressed as −β Ep  ∞ 1  2e ˜ p · er | Ee−db (ωr ) = Re |μ dte−iωr t−i(Eg −Ep )t/ π Z eb 0 p    ˆ ˆ ˆ ˆ ˆ × Trbn eit (Bpp,n +Hb,n )/ e−it Hb,n / e−β(Bpp,n +Hb,n ) , n

(3.71)  where Hˆ b,n = ωn (bˆn† bˆn + 1/2), and Bˆpp,n = j ωn gj n |Ujp |2 (bˆn + bˆn† ). Each trace over the bath in Eq. (3.71) corresponds to F (t) defined by Eq. (A.40) in Appendix A for each mode. Employing the final expression, Eq. (A.44), for each component of the product in Eq. (3.71) with appropriate change of notation, the product of traces can be expressed as follows:    ˆ ˆ ˆ ˆ ˆ Trbn eit (Bpp,n +Hb,n )/ e−it Hb,n / e−β(Bpp,n +Hb,n ) n

 = e−iλp t/ eβλp

 n

e−βωn /2 (1 − e−βωn )



⎧ ⎨  × exp − |Ujp |2 |Ukp |2 gj n gkn ⎩ n j k     βωn × coth , (1 − cos(ωn t)) + i sin(ωn t) 2

(3.72)

where λp has been defined by Eq. (3.57). On the other hand, Zeb =



e−β(Ep −λp )

p

 n

e−βωn /2 . (1 − e−βωn )

(3.73)

Inserting Eqs. (3.72) and (3.73) into Eq. (3.71), it can be expressed as  e−β(Ep −λp ) ∞ −iωr t−i(Eg −Ep +λp )t/ 1  ˜ p · er |2 |μ dte Ee−db (ωr ) = Re π Ze 0 p ⎧  ∞ ⎨ 1  Jj k (ω) 2 2 × exp − |Ujp | |Ukp | dω ⎩ π ω2 0 j

k

Chapter 3 Linear spectroscopy of molecular excitons

    βω × coth (1 − cos(ωt)) + i sin(ωt) , 2 where Ze =



e−β(Ep −λp ) .

(3.74)

(3.75)

p

For the case where there is no correlation in the spectral density between different chromophores, namely, Jj k (ω) = δj k Jjj (ω), Eq. (3.74) simplifies to the following expression: Ee−db (ωr ) =

 e−β(Ep −λp ) ∞ −iωr t−i(Eg −Ep +λp )t/ 1  ˜ p · er |2 |μ dte Re π Ze 0 p ⎧  ∞ ⎨ 1  Jjj (ω) 4 × exp − |Ujp | dω ⎩ π ω2 0 j     βω × coth (1 − cos(ωt)) + i sin(ωt) . (3.76) 2

It is also possible to employ a projection operator formalism, as in Sec. 3.1.3, to develop a QME formulation for the emission lineshape. For this, it is convenient to rewrite Eq. (3.67) as follows:  ∞   1 dte−it (ωr +Eg /) Trex ρˆge (t) , (3.77) E(ωr ) = Re π 0 where ˆ ex

ˆ

ρˆge (t) = |Der Der |ρˆex (0)eit Hm / e−it Hb / .

(3.78)

Taking the time derivative of the above operator, one can find the following time evolution equation: i d ˆ ex ρˆge (t) = |Der Der |ρˆex (0)ei Hm t/ dt 

ˆ × Hˆ es + Hˆ eb + Hˆ b − Hˆ b e−i Hb t/

i ˆ ˆ = ρˆge (t) Hˆ es + ei Hb t/ Hˆ eb e−i Hb t/ . 

(3.79)

A projection operator (that applies from the righthand side) can be defined to decompose the time evolution into a zeroth order and first order parts, and to derive a formally exact QME expression analogous to Eq. (3.40) [43]. Then, a second order approximation with respect to Hˆ eb can be made to derive a lineshape expression analogous to Eq. (3.46). One major difference here is that there are inhomogeneous terms due to the entanglement of the exciton and

75

76

Chapter 3 Linear spectroscopy of molecular excitons

bath degrees of freedom that cannot be removed from the initial condition of the density operator ρˆex (0). The evaluation of these terms is nontrivial and they add to the sensitivity of the emission lineshape on details of the spectral density. A theory of emission lineshape of this kind has been implemented for the LH2 complex of purple bacteria [43], which can be applied to other kinds of models as well.

3.3 Model calculations The two approximations for absorption lineshape considered in Secs. 3.1.5.1 and 3.1.5.2 and the emission lineshape expression considered in Sec. 3.2 are further illustrated here for a simple model. For this, the spectral density is assumed to be uniform for all sites. Thus, Jj k (ω) = δj k J (ω). Under this assumption and also incorporating the effect of fast modulating time dependent fluctuation with rate f l , the diagonal bath approximation for the absorption lineshape, Eq. (3.59), can be expressed as  ∞ 2 1   ˜ p  Re Ie−db (ωr ) ≈ er · μ dt ei(ωr +i f l )t+i(Eg −Ep +fp λ)t/ π p 0   ∞ fp J (ω) × exp − dω 2 π 0 ω     βω × coth (1 − cos(ωt)) + i sin(ωt) , (3.80) 2 where fp is the participation ratio (or fraction) of the exciton state p defined as fp =

Nc  j =1

|Ujp |4 =

Nc 

|sj |ϕp |4 ,

(3.81)

j =1

and λ is the overall reorganization energy given by  1 ∞ J (ω) λ= dω . π 0 ω

(3.82)

Similarly, the emission lineshape within the diagonal approximation in the exciton basis, Eq. (3.76), can be expressed as follows. Ee−db (ωr ) ≈

2 1  e−β(Ep −fp λ)  ˜ p e·μ π p Ze  ∞ × Re dt e−i(ωr −i f l )t−i(Eg −Ep +fp λ)t/ 0

Chapter 3 Linear spectroscopy of molecular excitons

  fp ∞ J (ω) × exp − dω 2 π 0 ω     βω × coth (1 − cos(ωt)) + i sin(ωt) . (3.83) 2 On the other hand, the 2nd-order QME approximation for the absorption lineshape, also including the contribution of f l , has the following expression [42]: ⎧ ⎨  1 (2) ˜ · er )∗ (μ ˜ p · er ) (ωr ) = − Im (μ Ieb ⎩ p  p π p   −1 (2) ×ϕp | ωr + i f l + (Eg − Hˆ es )/ + i Kˆ eb (ωr + i f l ) |ϕp  . (3.84) (2) More detailed expression for Kˆ eb (ωr + i f l ) can be found in Ref. [42]. All the lineshape expressions given above, Eqs. (3.80), (3.83), and (3.84), are applicable for any form of the bath spectral density. Model calculations in this section will be focused on the following form of the spectral density:

Jn (ω) = πη

ωn ωcn−1

e−ω/ωc .

(3.85)

For n = 1, this corresponds to Ohmic density in the limit of ωc → ∞. However, representation of any realistic environments requires choosing a finite value for ωc . The model calculations being presented below will be for ωc = 100 cm−1 . For the exciton Hamiltonian Hˆ es , the model of a linear chain of chromophores similar to that introduced in Sec. 1.2.2.1 of Chap. 1 will be considered. For this case, the coordinates of chromophores can be represented as r j = (0, 0, j d), where j = 1, · · · , Nc and d is the spacing between the nearest neighbor chromophores. The transition dipole of each chromophore has the same form as Eq. (1.14) with θj = θ0 and φj = φ0 + j δφ. On the other hand, the electronic coupling between chromophores j and k are assumed to include all the non-nearest neighbor transition-dipole interactions in addition to the nearest neighbor term, which has additional contributions of higher order Coulomb and exchange interactions. Thus, the electronic coupling between chromophores j and k is given by dp

Jj k = J0 δj k±1 + Jj k ,

(3.86)

77

78

Chapter 3 Linear spectroscopy of molecular excitons

Figure 3.3. Lineshapes for Case A, θ0 = δφ = 0°. Absorption lineshapes were calculated by Eqs. (3.84) (QME2) and (3.80) (d-eb), and emission lineshape calculated by Eq. (3.83). The labels I, II, and III respectively denote the value of n = 1, 2, and 3 in the spectral density, Eq. (3.85). f l = 10 cm−1 was chosen.

where J0 is the contribution of higher order multipolar and nondp Coulomb interactions, and Jj k is the transition dipole interaction with the following expression for the present model: dp

Jj k = μ 2

sin2 θ0 cos[δφ(j − k)] − 2 cos2 θ0 . (d(j − k))3

(3.87)

This model represents a large class of linear aggregates of dye molecules and also conjugated oligomers/polymers to some extent. Even at the level of simple model as described above, the dimension of the parameter space is too large to consider all representative cases. Here, calculation results are illustrated for one specific choice of values, J0 = −500 and μ = 10, in the Heaviside– Lorentz units ( = c = 1) and assuming d as the unit length. Let us consider two specific choices of angle parameters, one (Case A) with θ0 = δφ = 0° and the other (Case B) with θ0 = 90°, δφ = 180°. Fig. 3.3 shows lineshapes for the former case and Fig. 3.4 shows that for the latter. The results in both cases are for Nc = 5, but the lineshape expressions being tested here can be applied to fairly large values of Nc on the order or hundreds and thousands easily.

Chapter 3 Linear spectroscopy of molecular excitons

Figure 3.4. Lineshapes for Case B, θ0 = 90°, δφ = 180°. Other parameters and conditions are the same as in Fig. 3.3. The intensities of emission lineshapes in the present have been multiplied by a factor of 10 to make them more visible.

As can be seen from Fig. 3.3, Case A corresponds to J -aggregate [4,16] and Case B to an H -aggregate [4,16]. Note that the intensity of the emission lineshape for Case B is extremely small. Thus, these states will appear almost dark. For the case of moderate exciton-bath coupling (η = 1), the two different approximations for the absorption lineshape, Eqs. (3.80) and (3.84), are fairly close to each other. On the other hand, for stronger exciton-bath coupling (η = 5), qualitative difference of the two is clear.

3.4 Summary and additional remarks This chapter has provided a theoretical description of linear absorption and stimulated emission spectroscopic lineshapes of molecular excitons, and has derived simple lineshape expressions applicable to general cases. For example, Eqs. (3.28), (3.46), and (3.69) can be used for general exciton-bath and bath Hamiltonians. The theoretical formulations of lineshapes provided here are based on the standard assumption that the rate of exciton creation or destruction is equal to that of photon absorption or emission. Thus, they are examples of FGR. The explicit derivations of these expressions presented here, although somewhat redundant, will

79

80

Chapter 3 Linear spectroscopy of molecular excitons

be useful in identifying generalizations of these expressions for more complex situations such as the case where there are time dependent fluctuations in addition to exciton-bath couplings. Eq. (3.28) and its example for linearly coupled harmonic oscillator bath, Eq. (3.58), are exact for the cases where exciton-bath couplings are diagonal in the exciton basis. In general, this is not true because couplings between electronic and vibrational degrees of freedom tend to be spatially localized whereas excitons are generally delocalized through electronic couplings. The noncommutativity of the exciton-bath Hamiltonian and the exciton Hamiltonian causes transitions between different exciton states, leading to additional broadening mechanisms. To the extent that these are less significant than those due to dephasing of each exciton state due to coupling to the bath, the diagonal exciton-bath approximation in the exciton basis can serve as reasonable approximations. On the other hand, Eq. (3.46), for which a detailed expression (2) for Keb (ωr ) for linearly coupled harmonic oscillator bath is provided in Eq. (3.61), is applicable to the general case of excitonbath coupling as long as it is weak compared to other parameters. This expression is not capable of describing the broadening due to multiphonon effects (see Figs. 3.3 and 3.4), but is expected to be a reasonable approximation for highly delocalized excitons with weak enough effective exciton-bath couplings due to exchange narrowing effect. A major disadvantage of this expression is that it still requires numerical inversion of the complex matrix and that the resulting lineshape is not guaranteed to be positive definite. The secular approximation, Eq. (3.51), serves as a practical solution for these issues in that it does not require numerical inversion and is guaranteed to be positive definite, but at the expense of ignoring off-diagonal terms in the exciton basis. The emission lineshape of exciton is analogous to the absorption lineshape except for additional complication. There is entanglement in general between the exciton and bath degrees of freedom in the quasi-equilibrium distribution in the exciton manifold, which makes it difficult to express it in a simple manner. Within the simplest diagonal approximation in the exciton basis, a compact expression for the lineshape, Eq. (3.69) for general case and Eq. (3.74) for linearly coupled harmonic oscillator bath, can be obtained. However, beyond that, even at the level of the second order QME, the entangled initial state results in additional terms that are not necessarily simple to calculate. The lineshape expressions derived and demonstrated in this section are based on non-trivial but the simplest level of approximations. These can be improved further in many different ways,

Chapter 3 Linear spectroscopy of molecular excitons

81

Figure 3.5. Absorption lineshapes of linear aggregates with Nc = 6 calculated based on two-particle approximation (TPA) and one-particle approximation (OPA). Adapted with permission from Ref. [44]. © (2015) AIP Publishing.

for which significant progress has been made for the past two decades. There are now a range of higher order approximations that are both practicable and effective. An important class of approaches is called modified Redfield equation [3,45–48], where off-diagonal correction terms are incorporated into the diagonal approximation either employing cumulant approximations or quantum master equation approaches. Because of reasonable accuracy and efficiency of this modified Redfield equation approach, it has gained popularity in recent years. There has also been significant effort to assess its accuracy and to make further improvement [49–51]. Finally, it is also important to note recent success in the development of virtually exact methods designed for relatively simple class of models. These include hierarchical equations of motion (HEOM) approach and path integral methods, which in combination with the response function formalism that involves the dipole correlation function formalism, can be used for the calculation of lineshapes. While these methods are numerically expensive and thus limited to relatively small systems, new theoretical efforts have made it possible to improve the efficiency of these methods through novel approximations. For example, Shi and coworkers [44] combined two particle approximations with the HEOM approach, and developed fairly accurate approximation methods (see Fig. 3.5).

4 Exciton transfer rates and hopping dynamics Contents 4.1 Transfer between two exciton states: Förster theory’s and its generalizations 84 4.1.1 General rate expression 85 4.1.2 Constant resonance coupling and independent baths 87 4.1.3 Inelastic transfer and independent baths 91 4.2 Transfer between groups of exciton states 93 4.2.1 General rate expression 94 4.2.2 Constant resonance coupling and independent baths 98 4.3 Master equation approaches and long range exciton hopping dynamics 101 4.4 Summary and additional remarks 105 The dynamics of molecular excitons can be described as rate processes in many circumstances, especially when the transfer occurs in time scales much longer than those of molecular vibrations. However, direct experimental investigation of such processes had remained difficult for a long time because of the limitation in experimental time resolution. On the other hand, theoretical advances were much ahead of such experimental capability. Förster [52–54] and Dexter [55] explained long time ago major mechanisms of rate processes due to transition dipole, multipole, and exchange interactions between localized centers of excitons, and derived general rate expressions that have turned out to be extremely important for quantitative understanding of exciton transfer processes. In particular, Förster’s spectral overlap expression [52–54] has played a central role in quantitative modeling of exciton transfer processes employing frequency domain spectroscopic data. These rate theories have also been instrumental for the modeling of exciton hopping dynamics as transition probabilities of master equations governing the time evolution of exciton populations. Major exciton transfer rates and hopping dynamics as noted above have already been reviewed extensively multiple times in well-known reviews and books [56,12,57,58]. However, considerDynamics of Molecular Excitons. https://doi.org/10.1016/B978-0-08-102335-8.00012-0 Copyright © 2020 Elsevier Ltd. All rights reserved.

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84

Chapter 4 Exciton transfer rates and hopping dynamics

ing the significant historical gap between these past accounts and recent theoretical and experimental interests accompanying new theoretical capability and experimental evidence, it is worthwhile to offer a new kind of review providing details that are of interest in modern context. This is important because the old exciton transfer rate theories continue serving as fundamental basis for our understanding of exciton dynamics at present and can be refined further for more complex situations. The primary objective of this chapter is to offer a self-contained description of exciton transfer rate theories employing the formalism and notations constructed in previous chapters. These include Förster’s resonance energy transfer (FRET) theory and its modern generalizations by the author, which have recently been reviewed in different context [59,60]. The description provided here is far from being a reliable historical account and is not meant to be comprehensive. The major point of this chapter is to clarify that all of these exciton transfer rate theories, being based on the weakness of electronic couplings, can be considered as examples of Fermi’s golden rule (FGR). Thus, these are simple examples of incoherent quantum dynamics. The description of these rate theories will then be followed by a short account of earlier works that employed rate theories in master equations for the simulation of long range exciton hopping dynamics.

4.1 Transfer between two exciton states: Förster theory’s and its generalizations Consider the transfer of single excitons between two centers, which are labeled as D and A. The two single exciton states where these are excited are respectively denoted as |D and |A. The effective Hamiltonian representing the two single excitons and their coupling to environments can be expressed as Hˆ = Eg |gg| + Hˆ es + Hˆ DA + Hˆ eb + Hˆ b ,

(4.1)

Hˆ es = ED |DD| + EA |AA| ,

(4.2)

Hˆ DA = Jˆ(|DA| + |AD|) ,

(4.3)

Hˆ eb = Bˆ D |DD| + Bˆ A |AA| .

(4.4)

where

All the parameters shown above follow the same convention as those in previous chapters. Thus, their definitions should be obvious. The two exciton states given above can be either localized

Chapter 4 Exciton transfer rates and hopping dynamics

or delocalized. The only assumption that is required here is that |D and |A are the only electronic states involved with the exciton transfer mechanism and that the coupling between them Jˆ, which in general is an operator depending on bath degrees of freedom, are small compared to other terms. This statement is somewhat ambiguous, and its ultimate justification can be made only through comparison with higher level quantum dynamics calculations. It is worthwhile to note again that the Hamiltonians specified by Eqs. (4.1)–(4.4) are much simpler than the general version provided in Chap. 2, which contains various nonadiabatic terms and employ full adiabatic states of chromophores. Development of exciton rate theories for such general exciton-bath Hamiltonian remains an important issue that requires further theoretical investigation.

4.1.1 General rate expression Let us consider the simplest situation where an impulsive but narrow enough pulse excites D selectively at time t = 0. Let us assume that the bath remains in its thermal equilibrium with respect to the ground electronic state |g during the short duration of excitation. Then, the corresponding density operator at t = 0+, after the excitation, is ˆ

ρ(0) ˆ = |DD|e−β Hb /Zb ,

(4.5)

ˆ

where Zb = T rb {e−β Hb }. The probability that the exciton has transferred from D to A at time t, following such excitation, is then defined by   ˆ i Hˆ t/ PA (t) = T rb A|e−i H t/ ρ(0)e ˆ |A   1 ˆ ˆ ˆ = T rb A|e−i H t/ |De−β Hb D|ei H t/ |A . Zb

(4.6)

For short enough time compared to /||Jˆ||, where ||Jˆ|| is a measure of the magnitude of Jˆ, one can make the first order perturbaˆ tive expansion of A|e−i H t/ |D with respect to Hˆ DA as follows: ˆ

A|e−i H t/ |D  i t  −iEA (t−t  ) −i(Hˆ b +Bˆ A )(t−t  )/ ˆ −iED t  −i(Hˆ b +Bˆ D )t  / ≈− dt e e e . Je  0 (4.7)

85

86

Chapter 4 Exciton transfer rates and hopping dynamics

Inserting this approximation and its Hermitian conjugate into Eq. (4.6), one can obtain the following second order approximation (with respect to Jˆ) for the acceptor population: 1 PA (t) ≈ 2 



t

dt 0





t



dt  ei(EA −ED )(t −t

 )/

0

 1    ˆ ˆ ˆ ˆ ˆ × T rb ei(Hb +BA )(t −t )/ Jˆe−i(Hb +BD )t / e−β Hb Zb   ˆ ˆ × ei(Hb +BD )t / Jˆ† , (4.8)

where the cyclic invariance of the trace operation with respect to  ˆ ˆ the bath has been employed in moving ei(Hb +BA )(t−t )/ to the first place within the trace operation. This results in cancellation of terms depending on t. Let us now define a time dependent rate k(t) of exciton transfer as the derivative of PA (t) as follows: d PA (t) dt  t 1  ≈ 2 dt  ei(EA −ED )(t −t)/  0  1   ˆ ˆ ˆ ˆ ˆ × T rb ei(Hb +BA )(t −t)/ Jˆe−i(Hb +BD )t / e−β Hb Zb  ˆ ˆ × ei(Hb +BD )t/ Jˆ†  t  + dt  ei(EA −ED )(t−t )/

k(t) ≡

0

 1  ˆ ˆ ˆ ˆ ˆ × T rb ei(Hb +BA )(t−t )/ Jˆe−i(Hb +BD )t/ e−β Hb Zb   ˆ ˆ × ei(Hb +BD )t / Jˆ† .

(4.9)

In the above expression, it is easy to confirm that the two integrals are complex conjugates of each other by replacing the integrand t  in the second term with t  and employing the cyclic invariance of trace operation. Using this fact and replacing the time integrand t  with t − t  in the first integral, k(t) given above can be expressed as  t 2  k(t) ≈ 2 Re dt  ei(ED −EA )t /  0   1 ˆ i(Hb +Bˆ D )t/ ˆ† −i(Hˆ b +Bˆ A )t  / ˆ −i(Hˆ b +Bˆ D )(t−t  )/ −β Hˆ b × T rb e e . J e Je Zb (4.10)

Chapter 4 Exciton transfer rates and hopping dynamics

The above expression can be simplified by introducing ˆ

ˆ



ρˆbD (t − t  ) = e−i(Hb +BD )(t−t )/

ˆ

e−β Hb i(Hˆ b +Bˆ D )(t−t  )/ e , Zb

(4.11)

which represents a nonequilibrium time evolution of the bath dynamics for exciton in the state |D, but starting from a thermal equilibrium state with respect to |g. Inserting Eq. (4.11) into Eq. (4.10), the time dependent rate can be expressed as  t 2  k(t) ≈ 2 Re dt  ei(ED −EA )t /   0  ˆ ˆ ˆ ˆ  × T rb ei(Hb +BD )t / Jˆ† e−i(Hb +BA )t / JˆρˆbD (t − t  ) . (4.12) If the bath dynamics in the state |D is ergodic, ρˆbD (t − t  ) is expected to approach the following limit: lim ρˆbD (t − t  ) = 

t−t →∞

ˆ

ˆ

e−β(Hb +BD ) D ≡ ρˆb,s , ZbexD

(4.13)

 ˆ ˆ where ZbexD = T rb e−β(Hb +BD ) . Given that the above limit is achieved much quicker than the exciton transfer time, ρˆbD (t − t  ) D . Within this approximation in Eq. (4.12) can be replaced with ρˆb,s and in the steady steady state limit, Eq. (4.12) approaches the following time independent rate expression:  ∞ 2 kF G = 2 Re dtei(ED −EA )t/  0   ˆ ˆ ˆ ˆ D × T rb ei(Hb +BD )t/ Jˆ† e−i(Hb +BA )t/ Jˆρˆb,s . (4.14) The above rate expression, as indicated by the subscript, is an example of FGR. No assumption has yet been made regarding the details of Bˆ D , Bˆ A , and Jˆ. Thus, Eq. (4.14) and its time dependent version Eq. (4.12) can be applied to much more general situation than the conventional limit of FRET. Thus, these can be used as general starting points in developing rate theories that go beyond the FRET theory.

4.1.2 Constant resonance coupling and independent baths Let us consider the case where Jˆ becomes a parameter independent of the bath degrees of freedom and the bath Hamiltonian

87

88

Chapter 4 Exciton transfer rates and hopping dynamics

consists of two independent terms for D and A, respectively, as follows: Hˆ b = Hˆ bD + Hˆ bA .

(4.15)

Then, Jˆ and Jˆ† in Eqs. (4.12) and (4.14) can be taken out of the trace operation, and the bath density operator can be divided into two components as follows: 1 −β Hˆ g g = ρˆbD ρˆbA , e Z ˆ

ˆ

(4.16) ˆ

ˆ

where ρˆbD = e−β HbD /T rbD {e−β HbD } and ρˆbA = e−β HbA /T rbA {e−β HbA }. Since the bath degrees of freedom constituting Hˆ bD are independent of those constituting Hˆ bA , Bˆ D commutes with Hˆ bA and vice versa for Bˆ A . In this case, Eq. (4.11) becomes g

g

D (t − t  )ρˆbA , ρˆbD (t − t  ) = ρˆbD g

(4.17)

where ˆ

ˆ



ˆ

ˆ



D (t − t  ) = e−i(HbD +BD )(t−t )/ ρˆbD ei(HbD +BD )(t−t )/ . ρˆbD g

(4.18)

Similarly, the propagators in Eq. (4.12) can be divided into a product of D and A terms as follows: ˆ

ˆ



ˆ

ˆ



ˆ



ei(Hb +BD )t / = ei(HbD +BD )t / ei HbA t / , ˆ

ˆ

e−i(Hb +BA

)t  /

ˆ

= e−i HbD

t  /

ˆ

ˆ

e−i(HbA +BA

)t  /

(4.19) .

(4.20)

Inserting these identities and Eq. (4.17) into Eq. (4.12), the time dependent rate can be expressed as follows [61]:  t   2|J |2  ˆ  ˆ ˆ  g k(t) = 2 Re dt  ei(ED −EA )t / T rbA ei HbA t / e−i(HbA +BA )t / ρˆbA    0   ˆ ˆ ˆ D (t − t  ) . (4.21) ×T rbD ei(HbD +BD )t / e−i HbD t / ρˆbD Under the same condition validating Eq. (4.14), the above expression approaches the following limit: kF G =

 ∞   2|J |2 ˆ ˆ ˆ g Re dtei(ED −EA )t/ T rbA ei HbA t/ e−i(HbA +BA )t/ ρˆbA 2  0

  i(Hˆ bD +Bˆ D )t/ −i Hˆ bD t/ D × T rbD e e ρˆbD,s , (4.22)

Chapter 4 Exciton transfer rates and hopping dynamics

where ˆ

D ρˆbD,s ≡

ˆ

e−β(HbD +BD ) T rbD {e−β(Hˆ bD +Bˆ D ) }

.

(4.23)

The rate expressions, Eqs. (4.21) and (4.22), can be expressed in terms of lineshape functions as will be described below. For this, let us first consider the following absorption lineshape function of the acceptor:  IA (ω) =

∞

−∞

  ˆ ˆ ˆ g dt eiω−i(EA −Eg )t/ T rbA ei HbA t/ e−i(HbA +BA )t/ ρˆbA . (4.24)

Note that this lineshape function is equivalent to Eq. (3.20) in Chap. 3 except for the normalization factor of 2π . Then, the average over the acceptor bath degrees of freedom in Eqs. (4.21) and (4.22) can be replaced  ∞ with the inverse Fourier transform of the above expression, −∞ dω e−iωt IA (ω)/(2π). First, the time dependent rate can be expressed as follows [61]: |J |2 k(t) = π2



∞

−∞

dω IA (ω)ED (t, ω) ,

(4.25)

where the fact that IA (ω) is a real function has been used and ED (t, ω) is a time dependent stimulated emission lineshape function defined by 

t





dt  e−iωt ei(ED −Eg )t / 0     ˆ ˆ ˆ D × T rbD ei(HbD +BD )t / e−i HbD t / ρˆbD (t − t  ) . (4.26)

ED (t, ω) = Re

Note that this expression cannot be simplified further because  ˆ ˆ D (t − t  ) in general. Simei(HbD +BD )t / does not commute with ρˆbD ilarly, Eq. (4.22) can be expressed as kF G =

|J |2 π2



× Re

∞

dω IA (ω)

−∞  ∞

dte−iωt ei(ED −Eg )t/

0

  ˆ ˆ ˆ D × T rbD ei(HbD +BD )t/ e−i HbD t/ ρˆbD,s .

(4.27)

89

90

Chapter 4 Exciton transfer rates and hopping dynamics

ˆ

ˆ

D , this expression can be Since ei(HbD +BD )t/ commutes with ρˆbD,s simplified further as follows:  ∞ |J |2 kF G = dω IA (ω)ED (ω) , (4.28) 2π 2 −∞

where ED (ω) is the stimulated emission lineshape function defined as  ∞ ED (ω) = dω e−iωt ei(ED −Eg )t/ −∞   ˆ ˆ ˆ D × T rbD ei(HbD +BD )t/ e−i HbD t/ ρˆbD,s . (4.29) This is equivalent to the emission lineshape expression defined by Eq. (3.67) in Chap. 3. When expressed in terms of the wavenumber, ν˜ = ω/(2πc), Eq. (4.28) can be expressed as  ∞ c kF G = 2 |J |2 d ν˜ IA (2πcν˜ )ED (2πcν˜ ) . (4.30)  −∞ For the case where the coupling between |D and |A occurs only through the transition dipole interaction, J is given by the following expression: J=

1 κDA |μgD ||μAg | , 3 r RDA

(4.31)

where RDA is the distance between D and A, r is the optical dielectric constant, and κDA is the orientational factor defined as κDA =

2 μgD · μAg − 3(μgD · RDA )(μAg · RDA )/RDA

|μgD ||μAg |

(4.32)

.

When this is inserted into Eq. (4.30), it becomes the following FRET rate expression: kF RET =

1 τD



RF RDA

6 (4.33)

,

where τD is the lifetime of the excited D and RF is the Förster radius given by  RF = cτD



κDA |μD ||μA | r 

2 

∞

−∞

1/6 d ν˜ IA (2πcν˜ )ED (2πcν˜ )

. (4.34)

Chapter 4 Exciton transfer rates and hopping dynamics

This does not appear to be the same as the original Förster’s spectral overlap expression, but the equivalence of the two can be shown by replacing IA (2πcν˜ ) with molar extinction coefficient and ED (2πcν˜ ) with normalized spontaneous emission profile. Detailed derivation of these relationships will be provided in Chap. 6.

4.1.3 Inelastic transfer and independent baths In general, the donor-acceptor electronic coupling is a dynamical quantity depending on the bath degrees of freedom as suggested from the general expression in Chap 2. The operator form assumed for Jˆ in Eq. (4.3) also reflects this. For the cases where the influence of the bath on the electronic coupling can be modeled as a fluctuation independent of the dynamics of excitons, the energy transfer rate can be calculated by averaging the population or the rate over the ensemble of such fluctuations. Otherwise, the quantum dynamical nature of Jˆ needs to be taken into consideration [62]. This is expected especially when the donor and the acceptor moieties are joined by a bridging molecular unit or if the environments they are locked in are soft and flexible. One important physical process in this case is that some of the transferring energy can be lost to or gained from the quantum degrees of freedom coupled to Jˆ, which makes the exciton transfer inelastic. The general rate expressions, Eqs. (4.12) and (4.14), are valid even in such case and can be used as the starting point. As the simplest case, consider the situation where the bath degrees of freedom coupled to Jˆ are independent of those coupled to |D and |A. Further simplification of the rate expression is possible in this limit. To be more specific, let us assume that Hˆ b can be decomposed into three components as follows: Hˆ b = Hˆ bD + Hˆ bA + Hˆ bJ ,

(4.35)

where Hˆ bJ is the Hamiltonian governing all the bath modes coupled to Jˆ. The assumption that the donor-bath and the acceptorbath are independent of each other is kept the same as in the previous subsection. The condition that Hˆ bJ is independent of these donor and acceptor bath modes can be specified by the following conditions: [Hˆ bJ , Bˆ D ] = [Hˆ bJ , Hˆ bD ] = 0 ,

(4.36)

[Hˆ bJ , Bˆ A ] = [Hˆ bJ , Hˆ bA ] = 0 .

(4.37)

Then, the total equilibrium bath density operator for the ground electronic state can be decomposed into the product of three

91

92

Chapter 4 Exciton transfer rates and hopping dynamics

components as follows: ˆ

e−β Hb g g = ρˆbD ρˆbA ρˆbJ , Zb g

(4.38)

g

where ρˆbD and ρˆbA have been defined below Eq. (4.16) and ρˆbJ = ˆ

ˆ

e−β HbJ /T rbJ {e−β HbJ }. Inserting Eq. (4.38) into Eq. (4.11) and using the conditions, Eqs. (4.36) and (4.37), one can obtain D ρˆbD (t − t  ) = ρˆbD (t − t  )ρˆbA ρˆbJ , g

(4.39)

D (t −t  ) has been defined by Eq. (4.18). Inserting Eq. (4.39) where ρˆbD into Eq. (4.12), decoupling the bath operators in the propagators, and taking the trace operations for the three types of bath separately, one can simplify the time dependent rate expression as follows [62]:

k(t) =



   ˆ  ˆ  dt  ei(ED −EA )t / T rbJ ei HbJ t / Jˆe−i HbJ t / Jˆ† ρˆbJ 0  ˆ  ˆ ˆ  g × T rbA ei HbA t / e−i(HbA +BA )t / ρˆbA     ˆ ˆ ˆ D (t − t  ) . (4.40) × T rbD ei(HbD +BD )t / e−i HbD t / ρˆbD 2 Re 2

t

In the steady state limit and under the same condition that leads to Eq. (4.14), the above time dependent rate approaches the following FGR rate: 

  ˆ dt ei(ED −EA )t/ T rbJ eiHbJ t/ Jˆe−i HbJ t/ Jˆ† ρˆbJ 0  ˆ ˆ ˆ g × T rbA ei HbA t/ e−i(HbA +BA )t/ ρˆbA   ˆ ˆ ˆ D . (4.41) ×T rbD ei(HbD +BD )t/ e−i HbD t/ ρˆbD,s

2 kF G = 2 Re 

∞

This can also be expressed in terms of the overlap of spectral densities. First, the terms involving acceptor and donor baths can be replaced with IA (ω) and ED (ω), which are defined respectively by Eqs. (4.24) and (4.29). This results in the following expression:  ∞  ∞ 1 kF G = Re dω dω ED (ω)IA (ω ) 2π 2 2 −∞ −∞  ∞   i(ω−ω )t i Hˆ bJ t/ ˆ −i Hˆ bJ t/ ˆ† × dt e T rbJ e . (4.42) Je J ρˆbJ 0

Chapter 4 Exciton transfer rates and hopping dynamics

Since the integrands ED (ω) and IA (ω) are real valued, the above expression can be further rearranged as follows [62]: kF G =

1 2π2





∞

−∞

dω

∞

−∞

dω ED (ω)IA (ω )KJ (ω − ω ) ,

(4.43)

where 1 KJ (ω) = Re π



∞

dt e 0

iωt

 T rbJ e

i Hˆ bJ t/

ˆ Jˆe−i HbJ t/ Jˆ† ρˆbJ

 . (4.44)

For the case where there is no bath mode coupled to Jˆ, the above expression becomes a delta function. Thus, Eq. (4.43) reduces to the original FRET rate expression Eq. (4.28) in this limit. The physical implication of Eq. (4.43) is clear. It shows that exciton transfer rate can be significant even when the spectral overlap between the donor and acceptor lineshape functions is zero as long as KJ (ω) can fill the gap in-between. In case KJ (ω) represents dynamic modulation of the D-A distance, it is also easy to show that Eq. (4.43) results in difference distance than the original FRET even when the electronic coupling originates entirely from the transition dipole interaction. Finally, the temperature dependence of Eq. (4.43) is expected to be different from that of the original FRET. While Eq. (4.43) is applicable for fairly general condition, it is based on the assumption that the bath degrees of freedom coupled to Jˆ are independent of those for the donor and the acceptor. In case this assumption is not applicable, one needs to use Eq. (4.14) instead. For the case where the bath modes can be approximated as linearly coupled harmonic oscillators, more explicit rate expressions can be derived even in these general cases.

4.2 Transfer between groups of exciton states The transfer of excitons between groups of chromophores (multichromophores) can be described as rate process if the electronic couplings between the two groups’ exciton states are small compared to the electronic and exciton-bath couplings within each group. It is straightforward to extend the derivation of rate expressions presented in the previous section to this case [63].

93

94

Chapter 4 Exciton transfer rates and hopping dynamics

4.2.1 General rate expression Consider a multichromophoric D consisting of ND site excitation states |Dj  with j = 1, · · · , ND . The exciton Hamiltonian for D is assumed to be Hˆ e,D =

ND 

EDj |Dj Dj | +

ND   j =j 

j =1

 † JˆD,jj  |Dj Dj  | + JˆD,jj ,  |Dj  Dj | (4.45)

where JˆD,jj  is the electronic coupling between |Dj  and |Dj  . As the notation suggests, JˆD,jj  is in general an operator depending on the bath degrees of freedom. Similarly, for a multichromophoric A, there are NA site excitation states |Ak  with k = 1, · · · , NA . The exciton Hamiltonian for A is also assumed to have the following form: Hˆ e,A =

NA 

EAk |Ak Ak | +

NA   k =k 

k=1

 † JˆA,kk  |Ak Ak  | + JˆA,kk ,  |Ak  Ak | (4.46)

where JˆA,kk  is the electronic coupling between |Ak  and |Ak   and is in general an operator depending on the bath degrees of freedom. On the other hand, the electronic coupling Hamiltonian between D and A is given by Hˆ DA =

NA  ND    Jˆj k |Dj Ak | + Jˆj†k |Ak Dj | ,

(4.47)

j =1 k=1

where Jˆj k is the electronic coupling between |Dj  and |Ak  and is assumed to be in general an operator depending on the bath degrees of freedom. The total Hamiltonian being considered in this section consists of the three terms introduced above and the bath degrees of freedom as follows: Hˆ = Eg |gg| + Hˆ e,D + Hˆ e,A + Hˆ DA + Hˆ eb + Hˆ b ,

(4.48)

where Hˆ b is the bath Hamiltonian, and Hˆ eb is the exciton-bath coupling given by Hˆ eb =

 j

Bˆ Dj |Dj Dj | +

 k

Bˆ Ak |Ak Ak | ≡ Hˆ eb,D + Hˆ eb,A . (4.49)

Chapter 4 Exciton transfer rates and hopping dynamics

In the above expression, the second equality serves as the definitions of Hˆ eb,D and Hˆ eb,A , respectively. Note that the above terms include only the exciton-bath coupling terms that are diagonal in the site excitation states. All the off-diagonal bath couplings have already been incorporated into JˆD,jj  , JˆA,kk  , and Jˆj k . For the formulation to be presented below, it is convenient to introduce the following zeroth order Hamiltonian: Hˆ 0 = Hˆ e,D + Hˆ e,A + Hˆ b .

(4.50)

Accordingly, the total Hamiltonian Eq. (4.48) can be expressed as Hˆ = Eg |gg| + Hˆ 0 + Hˆ DA + Hˆ eb,D + Hˆ eb,A .

(4.51)

The initial condition assumed in this rate description is the donor exciton state created at D by an impulsive excitation at time t = 0. Before and during this excitation, the bath degrees of freedom are assumed to be in the canonical density operator for the ground electronic state. Thus, it is assumed that the total density operator at t = 0 right after the excitation is given by ˜ D| ˜ ρˆ g , ρ(0) ˆ = |D b

(4.52)

    ˜ = j pj |Dj , with pj = eˆ eˆ · μj /( j eˆ |eˆ · μj |2 )1/2 . where |D This initial condition corresponds to the case where the width of pulse (in frequency domain) is broad enough to cover all the excitation energies of the donor chromophores. Then, the probability for the exciton has transferred to A at time t is given by PA (t) =

NA 

  ˆ i Hˆ t/ T rb Ak |e−i H t/ ρ(0)e ˆ |Ak  .

(4.53)

k=1

For short enough time compared to /max(||Jˆj k ||), the above population probability can be approximated by a second order perturbative expansion with respect to Hˆ DA . This approximation requires only making up to the first order approximation of the time evolution operator e−iH t/ and its Hermitian conjugate. It is straightforward to find that the matrix element for the first order ˆ approximation of e−i H t/ is as follows: Ak |e

−i Hˆ t/

˜ ≈ |D

Nd  NA   j  =1 k  =1 0

t

 ˆ ˆ ˆ dt  Ak |e−i(He,A +Hb +Heb,A )(t−t ) |Ak  Jˆj† k 

ˆ

ˆ

ˆ



˜ . × Dj  |e−i(He,D +Hb +Heb,D )t / |D

(4.54)

95

96

Chapter 4 Exciton transfer rates and hopping dynamics

Inserting this expression and its Hermitian conjugate into Eq. (4.53), the population probability for the acceptor can be approximated as PA (t) ≈

 t  1  t  dt dt  2  0 0 j  j  k  k  k   ˆ ˆ ˆ × T rb Ak |e−i(He,A +Hb +Heb,A )(t−t )/ |Ak  Jˆj† k   ˆ ˆ ˆ ˜ × Dj  |e−i(He,D +Hb +Heb,D )t / |D

ˆ

ˆ

ˆ

˜ i(He,D +Hb +Heb,D )t × ρˆb D|e g

ˆ

ˆ

ˆ

×Ak  |ei(He,A +Hb +Heb,A

 /

|Dj  Jˆj  k   |Ak  .

)(t−t  )/

ˆ

ˆ

(4.55)

ˆ



In the above expression, the term Ak  |e−i(He,A +Hb +Heb,A )(t−t )/ |Ak  can be moved to the front position using the cyclic  invariance of trace operation. Then, employing the fact that k |Ak Ak | forms an effective identity operator in the electronic subspace constituting Hˆ e,A + Hˆ b + Hˆ eb,A , one can simplify Eq. (4.55) as follows: PA (t) ≈

 t  1  t  dt dt  2  0 0 j  j  k  k     ˆ ˆ ˆ × T rb Ak  |ei(He,A +Hb +Heb,A )(t −t )/ |Ak  Jˆj† k  ˆ

ˆ

ˆ



˜ × Dj  |e−i(He,D +Hb +Heb,D )t / |D g ˜ i(Hˆ e,D +Hˆ b +Hˆ eb,D )t  / |Dj  Jˆj  k  × ρˆb D|e

 .

(4.56)

As in the case of the single chromophoric exciton transfer described in the previous section, the time derivative of PA (t) expressed above can be used as the exciton transfer rate as follows: k(t) ≡ ≈

d PA (t) dt   1  j  j  k  k 

2

0 ˆ

t

  ˆ ˆ ˆ dt  T rb Ak  |ei(He,A +Hb +Heb,A )(t −t)/ |Ak  Jˆj† k  ˆ

ˆ



ˆ

ˆ

ˆ

˜ ρˆ D|e ˜ i(He,D +Hb +Heb,D )t/ |Dj   × Dj  |e−i(He,D +Hb +Heb,D )t / |D b  × Jˆj  k   t   ˆ ˆ ˆ + dt  T rb Ak  |ei(He,A +Hb +Heb,A )(t−t )/ |Ak  Jˆ†  0

g

jk

Chapter 4 Exciton transfer rates and hopping dynamics

ˆ ˆ ˆ ˜ ρˆ g D|e ˜ i(Hˆ e,D +Hˆ b +Hˆ eb,D )t  / |Dj   × Dj  |e−i(He,D +Hb +Heb,D )t/ |D b

 ˆ . (4.57) × Jj  k 

The two integrands in the above expression are Hermitian conjugates of each other. Then, replacing t  with t − t  in the first term, the expression for k(t) can be simplified as follows:    t 1  ˆ ˆ ˆ dt  T rb Ak  |e−i(He,A +Hb +Heb,A )t / |Ak  Jˆj† k  k(t) ≈ 2 Re      0 jj kk ˆ

ˆ

ˆ



˜ × Dj  |e−i(He,D +Hb +Heb,D )(t−t )/ |D g ˜ i(Hˆ e,D +Hˆ b +Hˆ eb,D )t/ |Dj  Jˆj  k  × ρˆb D|e



(4.58)

.

In the above expression, one can identify a time evolving part of the donor exciton-bath density operator defined as ˆ

ˆ

ˆ



˜ ρˆbD (t − t  ) = e−i(He,D +Hb +Heb,D )(t−t )/ |D ˆ

ˆ

ˆ



˜ i(He,D +Hb +Heb,D )(t−t )/ . × ρˆb D|e g

ˆ

ˆ

(4.59) ˆ

Then, dividing the time evolution operator ei(He,D +Hb +Heb,D )t/ in Eq. (4.58) into the product of the same one for (t − t  ) and t  and employing the above definition, one can obtain the following expression:   t 1 k(t) = 2 Re dt   0     jj kk   ˆ ˆ ˆ × T rb Jˆj  k  Ak  |e−i(He,A +Hb +Heb,A )t / |Ak  Jˆj† k 

 ˆ e,D +Hˆ b +Hˆ eb,D )t  / D  i( H ×Dj  |ρˆb (t − t )e |Dj   . (4.60) Let us assume that the exciton bath dynamics in the subspace of donor chromophores is ergodic such that the following assumption holds. lim ρˆbD (t − t  ) = 

t−t →∞

ˆ

ˆ

ˆ

e−β(He,D +Heb,D +Hb ) D ≡ ρˆb,s , ZD

(4.61)

 ˆ ˆ ˆ where ZD = T r e−β(He,D +Heb,D +Hb ) . This is a multichromophoric generalization of Eq. (4.13). Given that the above limit is achieved

97

98

Chapter 4 Exciton transfer rates and hopping dynamics

much quicker than the transfer time, ρˆbD (t − t  ) in Eq. (4.60) can D for all time. Within this approximation and be replaced with ρˆb,s in the steady state limit, k(t) given by Eq. (4.60) approaches the following time independent rate:    ∞ 1 kF G = 2 Re dt  jj  j  k  k  0  ˆ ˆ ˆ × T rb Jˆj  k  Ak  |e−i(He,A +Hb +Heb,A )t/ |Ak  Jˆj† k 

 ˆ e,D +Hˆ b +Hˆ eb,D )t/ D i( H × Dj  |ρˆb,s e |Dj   . (4.62) The above expression is once again an application of FGR. Except for the smallness of Hˆ DA , no other approximations have been made yet. Thus, this can be used for general exciton Hamiltonians and exciton-bath couplings. The next section will provide further refinement of this expression for the simple case where Jˆj k becomes a parameter independent of the bath degrees of freedom and the whole bath can be divided into an independent sum of those coupled to the donor and the acceptor, respectively.

4.2.2 Constant resonance coupling and independent baths Let us consider the case where the bath Hamiltonian consists of two independent terms, one for the group of donor chromophores and the other for acceptor chromophores as follows: Hˆ b = Hˆ bD + Hˆ bA .

(4.63)

In addition, let us assume that the bath terms constituting Hˆ eb,D are independent of Hˆ bA and vice versa. This can be specified by the following conditions: [Hˆ bD , Bˆ Ak ] = [Hˆ bA , Bˆ Dj ] = 0 .

(4.64)

Under the above conditions and the assumption that the donoracceptor couplings Jˆj k ’s are simple parameters independent of nuclear degrees of freedom, one can show that Eq. (4.60) reduces to the following expression:   t  1 k(t) = 2 Re Jj  k  Jj∗ k  dt   0 jj  j  k  k    ˆ  ˆ ˆ ˆ × T rb ei HbA t / Ak  |e−i(He,A +HbA +Heb,A )t / |Ak  

Chapter 4 Exciton transfer rates and hopping dynamics

×e

−i Hˆ bD t  /

Dj  |ρˆbD (t



− t )e

i(Hˆ e,D +Hˆ bD +Hˆ eb,D )t  /

|Dj   . (4.65)

Under the same assumption of independent baths, ρˆbD (t − t  ) shown above can be expressed as the product of two terms as follows: D ρˆbD (t − t  ) = ρˆbD (t − t  )ρˆbA , g

(4.66)

where ˆ

ˆ

ˆ



D ˜ ρˆbD (t − t  ) = e−i(He,D +HbD +Heb,D )(t−t )/ |D g ˜ i(Hˆ e,D +Hˆ bD +Hˆ eb,D )(t−t  )/ . × ρˆbD D|e

(4.67)

Inserting Eq. (4.66) into Eq. (4.65) and decoupling the trace over the bath into those for the donor and acceptor terms, Eq. (4.65) can be expressed as follows [63]:   t  1 Jj  k  Jj∗ k  dt  k(t) = 2 Re  0 j  j  k  k      ˆ ˆ ˆ ˆ g × T rbA ei HbA t / Ak  |e−i(He,A +HbA +Heb,A )t / |Ak  ρˆbA

  ˆ e,D +Hˆ b +Hˆ eb,D )t  / i( H −i Hˆ bD t  / D  D × T rbD e Dj  |ρˆbD (t − t )e |Dj   . (4.68) As in the case of single chromophoric exciton transfer, the time dependent rate given above can be expressed in terms of lineshapes functions defined in the frequency domain. For this, let us introduce the following absorption lineshape matrix element of the acceptor:  ∞ IA,k  k  (ω) ≡ dteiωt −∞   ˆ ˆ ˆ ˆ g × T rbA ei HbA t/ Ak  |e−i(He,A +HbA +Heb,A )t/ ρˆbA |Ak   . (4.69) Inserting the inverse Fourier transform of this lineshape function into Eq. (4.68), the time dependent rate can be expressed as   ∞  1 ∗   Jj k Jj  k  dωIA,k  k  (ω) k(t) = 2 Re  −∞     jj kk

99

100

Chapter 4 Exciton transfer rates and hopping dynamics



   ˆ dt  e−iωt T rbD e−i HbD t / Dj  |ρˆbDD (t − t  ) 0

 i(Hˆ e,D +Hˆ bD +Hˆ eb,D )t  / |Dj   . ×e

×

t

(4.70)

∗ From Eq. (4.69), it is easy to confirm that IA,k  k  (ω) = IA,k  k  (ω). Employing this identity and interchanging the indices between j  and j  and between k  and k  , Eq. (4.68) can be expressed as follows:

k(t) =

 ∞ 1  ∗  k  J   J dω IA,k  k  (ω)Dj  |Eˆ D (t, ω)|Dj   , j jk 2     −∞ jj kk

(4.71) where Eˆ D (t, ω) is an operator defined in the excitonic subspace of D and is given by  t 1  Eˆ D (t, ω) = dt  e−iωt 2 0     ˆ ˆ ˆ ˆ × T rbD e−i HbD t / ρˆbDD (t − t  )ei(He,D +HbD +Heb,D )t /

 t   −i(Hˆ e,D +Hˆ bD +Hˆ eb,D )t  / D  iωt   i Hˆ bD t  / + dt e T rbD e ρˆbD (t − t )e . 0

(4.72) This represents the overall time dependent stimulated emission profile of donor chromophores. Under the assumption that D (t − t  ) becomes the following equilibrium density fast enough, ρˆbD ˆ

ρˆbDD ,s =

ˆ

ˆ

e−β(He,D +Hb +Heb,D )  , T r e−β(Hˆ e,D +Hˆ b +Hˆ eb,D )

(4.73)

and in the steady state limit (t → ∞), Eq. (4.72) can be approximated by the following steady state stimulated emission lineshape operator of the donor chromophores. Eˆ D,s (ω) =



∞

−∞

  ˆ ˆ ˆ ˆ D dte−iωt T rbD e−i HbD t/ ρˆbD,s ei(He,D +HbD +Heb,D )t/ . (4.74)

Chapter 4 Exciton transfer rates and hopping dynamics

Thus, in the steady state limit, k(t) of Eq. (4.70) reduces to the following time independent rate [63]:  ∞ 1  ∗ kF G = 2 Jj  k  Jj  k  dω IA,k  k  (ω)Dj  |Eˆ D,s (ω)|Dj   .      −∞ jj kk

(4.75) This expression is again an application of FGR for multichromophoric (MC) donor and acceptor, and has been termed MCFRET rate. Sumi [64] was the first to recognize this rate expression. Later, the author, along with Newton and Silbey, provided an independent and detailed derivation of this rate expression [63].

4.3 Master equation approaches and long range exciton hopping dynamics For a sparse distribution of exciton centers with large enough center-to-center distances, regardless of whether they are single chromophores or multichromophores, the rate theories described in previous sections can be used to represent the elementary steps constituting the whole exciton hopping dynamics across the distribution. If time scale separation justifies assuming that all the rates are independent of time, such hopping dynamics can be described at the level of a simple Pauli Master equation (PME), as will be described below. Let us consider the case where there are two types of exciton centers, donors (D) and acceptors (A), embedded in a host matrix. Note that each D or A here can be either a single chromophore or multichromophore. In other words, D here represents a molecule or macromolecule that gets excited first and A represents the one that receives the exciton from D. The only assumption here is that a well-defined time independent rate can be determined for a given pair of exciton centers once their configurations are known, for which any type of FGR rate expression such as FRET or MCFRET can be used. Let us assume that the number of D’s is ND and that the number of A’s is NA , and define a single population vector with component pn (t) such that n = 1, · · · , ND denotes D and n = ND + 1, · · · , ND + NA denotes A. In addition, let assume that each exciton center has a well-defined lifetime, τn = τD or τA . Then, in the limit of dilute exciton density such that the probability for a given exciton encounters another exciton is virtually zero, the time evolution of exciton populations can be described by the follow-

101

102

Chapter 4 Exciton transfer rates and hopping dynamics

ing PME: ND +NA 1 d {Wm→n pm (t) − Wn→m pn (t)} , (4.76) pn (t) = − pn (t) + dt τn m=1

where Wn→m is the rate of exciton transfer from the exciton center n to m. Without losing any generality, one can assume that pn (0) = δn1 . Namely, n = 1 denotes the initial site where the exciton is created at time t = 0. There have been numerous theoretical and experimental works for the types of exciton dynamics that can be described by Eq. (4.76). In particular, the case where D serves as both initiator of the exciton and fluorophore whereas A serves as a quencher has been the subject of various studies. This represents a large class of luminescent materials that are controlled through substitution of appropriate fluorescing and quenching agents, and has had practical applications. The exciton transfer rate from a given quencher to others is typically assumed to be zero, which corresponds to Wn→m = 0 for n > ND . In the limiting case where ND = 1, the solution of Eq. (4.76) for the population of D is simple and can be expressed as p1 (t) = e−t/τ1

N A +1

(1 − e−W1→n t ) .

(4.77)

n=2

The average population of exciton in D is then calculated by averaging the above expression over the distribution of acceptors. For random distribution of acceptors and for the case where the rate is given by the FRET rate as follows: W1→n =

1 τ1



RFDA R1n

6 (4.78)

,

where R1n is the distance between site 1 and n and RFDA is the Förster radius between the donor and the acceptor which has been defined by Eq. (4.34) (note that an additional subscript DA was used here to distinguish this with other types of Förster radius), it is easy to show that the average of Eq. (4.77) results in p1 (t) = e−t/τ1 −γ (t/τ1 )

1/2

,

(4.79)

where γ = 4π 3/2 RF3 DA NA /(3V) with V being the volume of the system. This expression remains valid for ND k j3

× Hˆ er (τ2 )e

j4 −i Hˆ m (τ2 −τ3 )/

|dj2 k 

ˆ × dj2 k |Hˆ er (t3 )|sj3 sj3 |e−i Hm (t3 −t4 )/ |sj4 sj4 |Hˆ er (t4 )|g  ˆ

ˆ

× e−i Hb t4 / ρˆb ei Hb t/ .

(7.12)

(4)

Second, the term ρˆb (t) given by Eq. (7.10) and its Hermitian conjugate have third order interactions with the radiation on one side (ket or bra) whereas first order interactions with the radiation on the other side (bra or ket). Since the bra side ends up on the single exciton states, ρˆb(4) (t) contributes only to the populations in the single exciton space. Thus, for the calculation of populations that originate from this term, one only needs to calculate the following elements in the single exciton subspace:   τ1  τ2  t 1 t (4) sj |ρˆb (t)|sj   = − 4 dτ1 dt2 dt3 dτ4  0 0 0 0   ˆ ˆ sj |e−i Hm (t−τ1 )/ Hˆ er (t1 )|ge−i Hb (τ1 −τ2 )/ g|Hˆ er (t2 )|sj1  j1

j2

j3

165

166

Chapter 7 Time-resolved nonlinear spectroscopy of excitons

ˆ ˆ ˆ × sj1 |e−i Hm (τ2 −τ3 )/ |sj2 sj2 |Hˆ er (τ3 )|ge−i Hb τ3 / ρˆb ei Hb τ4 / ˆ

× g|Hˆ er (τ4 )|sj3 sj3 |ei Hm (t−τ4 ) |sj      ˆ sj |e−i Hm (t−τ1 )/ |sj1 sj1 |Hˆ er (τ1 )|dj1 k1 dj1 k1 | + (j1 ,k1 ) (j2 ,k2 ) j3

×e

j4

−i Hˆ m (τ1 −τ2 )/

|dj2 k2  ˆ

× dj2 ,k2 |Hˆ er (τ2 )e−i Hm (τ2 −τ3 )/ |sj3 sj3 |Hˆ er (τ3 )|g ˆ

ˆ

× e−i Hb τ3 / ρˆb ei Hb τ4 /



× g|Hˆ er (τ4 )|sj4 sj4 |e

−i Hˆ m (t−τ4 )/

|sj   ,

(7.13)

where the first term involves coherence between single exciton states and the ground electronic state, and the third term involves coherence between the single and double exciton states. (4) Finally, ρˆc (t) given by Eq. (7.11) and its Hermitian conjugate involve double interactions with the radiation on both sides. This term produces populations in both the ground electronic state and double exciton states. Thus, ρˆc(4) (t) = 

 j1

1 4





t

dτ1



t

dτ2 0

0



τ1

τ4

dτ4 0

dτ3 0

ˆ

ˆ

|ge−i Hb (t−τ1 )/ g|Hˆ er (τ1 )e−i Hm (τ1 −τ2 )/ |sj1 

j2

ˆ × sj1 |Hˆ er (τ2 )|ge−i Hb τ2 / ρˆb ˆ ˆ ˆ × ei Hb τ3 / g|Hˆ er (τ3 )ei Hm (τ4 −τ3 )/ |sj2 sj2 |Hˆ er (τ4 )|gei Hb (t−τ4 )/ g|    ˆ e−i Hm (t−τ1 )/ |dj1 k1 dj1 k1 |Hˆ er (τ1 ) + (j1 ,k1 ) j2

×e

j3 (j4 ,k4 )

−i Hˆ m (τ1 −τ2 )/

|sj2 sj2 |Hˆ er (τ2 )|g

ˆ ˆ ˆ × e−i Hb τ2 / ρˆb ei Hb τ3 / g|Hˆ er (τ3 )ei Hm (τ4 −τ3 )/ |sj3   i Hˆ m (t−τ4 )/ ˆ × sj3 |Her (τ4 )|dj4 k4 dj4 k4 |e

+ ··· ,

(7.14)

where the first term corresponds to stimulated emission from the single exciton to the ground electronic state and the second term corresponds to excited state absorption from the single to double exciton states. The remaining terms denoted as · · · are all coher-

Chapter 7 Time-resolved nonlinear spectroscopy of excitons

ence terms involving two out of the ground electronic state, blocks of single exciton states, and double exciton states. Although rather complicated, the expressions provided above, Eqs. (7.12)–(7.14), serve as important starting expressions for analyzing various two-pulse spectroscopic experiments. As one example, further calculations are provided for pump-probe spectroscopy in the next subsection.

7.2.2 Pump-probe spectroscopy The major feature of the pump-probe spectroscopy is that the radiation consists of short pump pulse followed by a probe pulse with controllable time delay T . The matter-radiation interaction Hamiltonian, Eq. (7.15), for this case is expressed as follows: Hˆ er (t) = −



 (μj · e1 )|sj g|A1 (t)eik1 ·r−iω1 t

j



+ (μj · e∗1 )|gsj |A∗1 (t)e−ik1 ·r+iω1 t −



 (μj · e2 )|sj g|A2 (t − T )eik2 ·r−iω2 t

j



+ (μj · e∗2 )|gsj |A∗2 (t −



− T )e

−ik2 ·r+iω2 t

 (μk · e2 )|dj k sj |A2 (t − T )eik2 ·r−iω2 t

(j,k)

+ (μk · e∗2 )|sj dj k |A∗2 (t − T )e−ik2 ·r+iω2 t

 , (7.15)

where A1 (t) is a narrowly peaked function around t = 0 and A2 (t) is typically a much more smooth function than A1 (t) with vanishing magnitude for t < 0. The rotating wave approximation has also been assumed in the above expression. Let us assume that T is large enough for the coherence terms to fully disappear during this interval and that both pump and probe pulses are in the weak field limit. The latter assumption allows considering only the second order interactions for each remaining term. Under these assumptions, only the third component given

167

168

Chapter 7 Time-resolved nonlinear spectroscopy of excitons

by Eq. (7.14) survives, which can be expressed as ρˆc(4) (t) =



1 4



t

dτ2 0



t



t

dτ1 0

τ2 iω1 (τ3 −τ2 )

t

dτ3

dτ4 τ3

× A1 (τ2 )A∗1 (τ3 )e A2 (τ1 − T )A∗2 (τ4 − T )   (μj1 · e2 )(μj1 · e1 )(μj2 · e1 )(μj2 · e2 )|gg| × eiω(τ1 −τ4 ) j1

j1

j2

j2

ˆ

ˆ

ˆ

× e−i Hb (t−τ1 )/ sj1 |e−i Hm (τ1 −τ2 )/ |sj1 e−i Hb (τ2 −τ3 )/ ˆ

ˆ

× ρˆb sj2 |ei Hm (τ4 −τ3 )/ |sj2 ei Hb (t−τ4 )/    + e−iω(τ1 −τ4 ) (δj1 j2 (μj1 · e2 ) + δk1 j2 (μk1 · e2 )) j2

(j1 ,k1 ) j2

j3

j3 (j4 ,k4 )

× (μj 2 · e1 )(μj 3 · e1 )(δj4 j3 (μj4 · e2 ) ˆ

+ δk4 j  (μk4 · e2 ))e−i Hm (t−τ1 )/ |dj1 k1  3

ˆ

ˆ

ˆ

× sj2 |e−i Hm (τ1 −τ2 )/ |sj2 e−i Hb (τ2 −τ3 )/ ρˆb sj3 |ei Hm (τ4 −τ3 )/ |sj3   ˆ

× dj4 k4 |ei Hm (t−τ4 )/ .

(7.16)

In the above expression, let us make the replacement of the following time variables: τ1 − T → τ1 and τ4 − T → τ4 . In addition, let us assume that t >> T for which A1 (T ) ≈ 0. Then, Eq. (7.16) can be approximated as   t−T  ∞ 1 t−T dτ1 A2 (τ1 ) dτ4 A∗2 (τ4 ) dτ2 ρˆc(4) (t) ≈ 4  −T −T 0  ∞ × dτ3 A1 (τ2 )A∗1 (τ3 )eiω1 (τ3 −τ2 ) 0   × eiω(τ1 −τ4 ) (μj1 · e2 )(μj1 · e1 )(μj2 · e1 )(μj2 · e2 )|gg| j1

j1

j2

j2

ˆ

ˆ

ˆ

× e−i Hb (t−τ1 −T )/ sj1 |e−i Hm (τ1 +T )/ |sj1 e−i Hb (τ2 −τ3 )/ ˆ

ˆ

× ρˆb sj2 |ei Hm (τ4 +T )/ |sj2 ei Hb (t−τ4 −T )/    + e−iω(τ1 −τ4 ) (δj1 j2 (μj1 · e2 ) + δk1 j2 (μk1 · e2 )) (j1 ,k1 ) j2

j2

j3

j3 (j4 ,k4 )

× (μj 2 · e1 )(μj 3 · e1 )(δj4 j3 (μj4 · e2 ) + δk4 j3 (μk4 · e2 )) ˆ

× e−i Hm (t−τ1 −T )/ |dj1 k1 

Chapter 7 Time-resolved nonlinear spectroscopy of excitons

ˆ

ˆ

ˆ

× sj2 |e−i Hm (τ1 +T )/ |sj2 e−i Hb (τ2 −τ3 )/ ρˆb sj3 |ei Hm (τ4 +T )/ |sj3   ˆ

× dj4 k4 |ei Hm (t−τ4 −T )/ .

(7.17)

Let us now define ˆ s   (t  , t  ) G jj  ∞  = dτ2 0

×e

∞

0 −i Hˆ m t  /

dτ3 A1 (τ2 )A∗1 (τ3 )eiω1 (τ3 −τ2 ) (μj  · e1 )(μj  · e∗1 ) ˆ

ˆ



|sj  e−i Hb (τ2 −τ3 )/ ρˆb sj  |ei Hm t / .

(7.18)

In addition, let us assume that A2 (τ ) is a step function with unit amplitude A2 for τ > 0. Then,   t−T |A2 |2 t−T (4) ρˆc (t) ≈ 4 dτ1 dτ4  0 0   × eiω(τ1 −τ4 ) (μj1 · e2 )(μj2 · e2 )|gg| j1

j1

j2

j2

ˆ ˆ s (τ1 + T , τ4 + T )|sj  ei Hˆ b (t−τ4 −T )/ × e−i Hb (t−τ1 −T )/ sj1 |G j1 j2 2    −iω(τ1 −τ4 ) +e (δj1 j  (μj1 · e2 ) + δk1 j  (μk1 · e2 )) (j1 ,k1 ) j2

j2

j3

2

j3 (j4 ,k4 )

2

ˆ

× (δj4 j3 (μj4 · e2 ) + δk4 j3 (μk4 · e2 ))e−i Hm (t−τ1 −T )/ |dj1 k1   ˆ m (t−τ4 −T )/ s i H ˆ × sj  |G (τ1 + T , τ4 + T )|sj  dj4 k4 |e . 2

j2 j3

3

(7.19)

One can then calculate (unnormalized) stimulated emission rate from the above expression as follows:   d ks→g (t) = T rb g|ρˆc(4) (t)|g dt t−T  |A2 |2 dτ4 eiω(t−T −τ4 ) (μj1 · e2 )(μj2 · e2 )|gg| = 4  0 j1 j1 j2 j2   ˆ s (t, τ4 + T )|sj  ei Hˆ b (t−τ4 −T )/ × T rb sj1 |G j1 j2 2  t−T  + dτ1 eiω(τ1 −t+T ) (μj1 · e2 )(μj2 · e2 )|gg| 0

j1

j1

j2

j2

  −i Hˆ b (t−τ1 −T )/ s ˆ × T rb e sj1 |G . (7.20) j1 j2 (τ1 + T , t)|sj2 

169

170

Chapter 7 Time-resolved nonlinear spectroscopy of excitons

Replacing the integrand t − T − τ4 with τ in the first term of the above expression and making a similar replacement,  t−T  |A2 |2 dτ eiωτ (μj  · e2 )(μj  · e2 ) ks→g (t) = 4 Re 1 2  0 j1 j1 j2 j2

  ˆ ˆ s (t, t − τ )|sj  ei Hb τ/ . (7.21) × |gg|T rb sj  |G 1

j1 j2

2

Similarly, the unnormalized excited state absorption rate from Eq. (7.19) can be calculated as follows:   d  T rb dj k |ρˆc(4) (t)|dj k  dt

ks→d (t) =

(j,k)



=

t−T |A2 |2 dτ4 e−iω(t−T −τ4 ) 4 0    × (δjj2 (μj · e2 ) + δkj2 (μk · e2 )) j2

(j,k) j2

j3

j3 (j4 ,k4 )

× (δj4 j3 (μj4 · e2 ) + δk4 j3 (μk4 · e2 ))   ˆ s (t, τ4 + T )|sj  dj4 k4 |ei Hˆ m (t−τ4 −T )/ |dj k  × T rb sj2 |G j2 j3 3  t−T + dτ1 e−iω(τ1 −t+T ) 0    × (δj1 j2 (μj1 · e2 ) + δk1 j2 (μk1 · e2 )) (j1 ,k1 ) j2

j2

j3

j3 (j,k)

× (μj 2 · e1 )(μj 3 · e1 )(δjj3 (μj · e2 ) + δkj3 (μk · e2 )) 

× T rb dj k |e

−i Hˆ m (t−τ1 −T )/

ˆ s (τ1 |dj1 k1 sj2 |G j2 j3

+ T , t)|sj3 



.

(7.22) Replacing the integrand t − T − τ4 with τ in the first term of the above expression and also making similar replacement in the second term, one can obtain the following expression  t−T |A2 |2 dτ4 e−iωτ ks→d (t) = 4 Re  0    (δjj2 (μj · e2 ) + δkj2 (μk · e2 )) (j,k) j2

j2

j3

j3 (j4 ,k4 )

× (δj4 j3 (μj4 · e2 ) + δk4 j3 (μk4 · e2 ))

Chapter 7 Time-resolved nonlinear spectroscopy of excitons



ˆ s (t, t × T rb sj2 |G j2 j3

− τ )|sj3 dj4 k4 |e

i Hˆ m τ/

|dj k 



.

(7.23) Eqs. (7.21) and (7.23) are final results of this section, which can be employed for the calculation time dependent excited state absorption and stimulated profiles in pump-probe spectroscopy of excitons that can be represented by the Hamiltonian, Eq. (7.1).

7.3 Four wave mixing spectroscopy 7.3.1 Response function formalism The most well-known formulation of four wave mixing spectroscopy (FWMS) is the response function formalism [3,168–172], where the major physical observable of material is the transient polarization vector of the material due to interactions with applied fields. For the case of FWMS, this amounts to the following third order polarization: P(3) (t) ≡ T r{Pˆ ρˆ (3) (t)} ,

(7.24)

where ρˆ (3) (t) is the third order expansion of the density operator with respect to interactions with three incoming pulses. The polarization is then considered as the source of signal field in purely classical electromagnetic description. A little more detailed description is provided below. In the absence of any charge density and in the Coulomb gauge, Eq. (B.27) (in the Gaussian units) reduces to: −∇ 2 As (r, t) +

r μr ∂ 2 4πμr As (r, t) = Js (r, t) , 2 2 c c ∂t

(7.25)

where the fact that = constant has been used. Assuming that the contribution of magnetization to the current is negligible and that the only source is the third order polarization given by Eq. (7.24), Js (r, t) =

∂ (3) P (r, t) . ∂t

(7.26)

Since =constant, Eq. (B.16) (in the Gaussian units) reduces to Es (r, t) = −

1 ∂ As (r, t) . c ∂t

(7.27)

171

172

Chapter 7 Time-resolved nonlinear spectroscopy of excitons

Combination of Eqs. (7.25), (7.26), and (7.27) leads to the following equation for the signal electric field: ∇ 2 Es (r, t) −

r μ r ∂ 2 4πμr ∂ 2 (3) Es (r, t) = 2 P (r, t) , 2 2 c ∂t c ∂t 2

(7.28)

where appropriate choices for r and μr need to be made depending on the nature of the media and experimental setup. Solving the above partial different equation is far from being trivial and is a complicating factor in the interpretation of any FWMS data. Typically, instead of solving the above equation, it is assumed that [168] Es (r, t) is proportional to P(3) (r, t). There are two types of detection in FWMS. A homodyne detection corresponds to the measurement of the intensity of Es (r, t). While this is straightforward and requires less experimental set up, it does not provide full phase information of the response function. The other type is called heterodyne detection, where the signal field interferes with a local oscillator field (the fourth pulse that does not go through the material but has well defined phase relationship with the incoming three pulses). The measurement of the intensity of the resulting interfered signal can provide separate information on the real and imaginary parts of the third order polarization. Now, let us consider in more detail the calculation of the third order polarization. The exciton-radiation interaction Hamiltonian in this case becomes Eq. (7.6) with ν = 3. With rotating wave approximation, it can be expressed as Hˆ er (t) =

3 

Aν (t − tν )Pˆν eikν ·r−iων t + H.c. ,

(7.29)

ν=1

where Aν (t −tν ) is the envelope function for the pulse around t = tν with t3 ≥ t2 ≥ t1 > 0, and   eν · μj |sj g| + eν · μk |dj k sj | Pˆν = j

= Pˆν,gs + Pˆν,sd .

(j,k)

(7.30)

In the above expression, eν is the polarization vector of the νth pulse, Pˆν,gs is the term corresponding to the transition from the ground to the single exciton states, and Pˆν,sd is the term corresponding to the single to double exciton states. It is useful to express these polarization operators in the basis of exciton states. Let us define the eigenstates of the single exciton states as |ϕp(1)  (2) and those for double exciton states as |ϕq . Then, denoting the

Chapter 7 Time-resolved nonlinear spectroscopy of excitons

(1)

(1)

(2)

(2)

transformation matrices as Ujp = sj |ϕp  and Uj k,q = dj k |ϕq , they can be expressed as  (1) ˜ (1) eν · μ (7.31) Pˆν,gs = p |ϕp g| , p

Pˆν,sd =

 p

(2) (1) ˜ (2) eν · μ pq |ϕq ϕp | ,

(7.32)

q

 (1)∗ (2)∗ (1) ˜ (2) Ujp μj and μ pq = (j,k) Uj k,q Ujp μk . Expanding the time evolution operator for Hˆ T (t) and its Hermitian conjugate with respect to Hˆ er (t) and collecting all the third order terms, one can find that ˜ (1) where μ p =



j

(3)

(3)†

ρˆ (3) (t) = ρˆa(3) (t) + ρˆa(3)† (t) + ρˆb (t) + ρˆb where i ρˆa(3) (t) = − 3 





t

τ

dτ 0

dτ



0



t

(7.33)

(t) ,

 ˆ ˆ dτ  e−i Hm (t−τ )/ Hˆ er (τ )e−i Hm (τ −τ )/

0

 ˆ  ˆ i Hˆ m τ  / ˆ Her (τ  )ei Hm (t−τ )/ , × Hˆ er (τ  )e−i Hm τ / ρ(0)e ˆ

ρˆb(3) (t) = −

i 3





t

dτ 0

τ

dτ 

(7.34)



τ

ˆ

ˆ

i Hm t dτ  e−i Hm tp / ρ(0)e ˆ

 /

0

0

 ˆ   ˆ ˆ × Hˆ er (t  )ei Hm (t −t )/ Hˆ er (t  )ei Hm (t−t )/ Hˆ er (t)ei Hm (tp −t)/ . (7.35)

(3)

In the above expression for ρˆa (t), Eq. (7.34), the integration over τ  can be split into three regions, 0 < τ  < τ  , τ  < τ  < τ , and τ < τ  < t. Relabeling the dummy time integration variables in each region such that τ ≥ τ  ≥ τ  , ρˆa(3) (t) can be divided into three terms corresponding to the three regions. Each term has the same time integration regions and boundaries as those in Eq. (7.35) for (3) ρˆb (t). As a result, the third order components can be expressed as ρˆ

(3)

i (t) = − 3 





t

τ

dτ 0

dτ





τ

dτ 

0

0

4 

Tˆj (t, τ, τ  , τ  ) + H.c. ,

j =1

(7.36) where    ˆ ˆ Tˆ1 (t, τ, τ  , τ  ) ≡ e−i Hm (t−τ )/ Hˆ er (τ  )e−i Hm (τ −τ )/ Hˆ er (τ  )

ˆ

× e−i Hm τ

 /

ˆ

ˆ

i Hm τ/ ˆ ρ(0)e ˆ Her (τ )ei Hm (t−τ )/ , (7.37)

173

174

Chapter 7 Time-resolved nonlinear spectroscopy of excitons

 ˆ ˆ ˆ  Tˆ2 (tp , τ, τ  , τ  ) ≡ e−i Hm (t−τ )/ Hˆ er (τ )e−i Hm (t−τ )/ Hˆ er (τ  )e−i Hm τ /

ˆ

i Hm τ × ρ(0)e ˆ

 /

 ˆ Hˆ er (τ  )ei Hm (t−τ )/ ,

ˆ ˆ Tˆ3 (t, τ, τ  , τ  ) ≡ e−i Hm (t−τ )/ Hˆ er (τ )e−i Hm ˆ

i Hm τ × ρ(0)e ˆ

Tˆ4 (t, τ, τ , τ ) ≡ e 



−i Hˆ m t/

 /

ρ(0)e ˆ

ˆ

× ei Hm (τ −τ

 )/

(τ −τ  )/

(7.38)

ˆ  Hˆ er (τ  )e−i Hm τ /

 ˆ Hˆ er (τ  )ei Hm (t−τ )/ ,

i Hˆ m τ  /

Hˆ er (τ )e 

(7.39)

i Hˆ m (τ  −τ  )/

Hˆ er (τ  )

ˆ Hˆ er (τ )ei Hm (t−τ )/ .

(7.40)

Let us assume that the three pulses are well separated in time, each with narrow peak width, and that the interaction of each peak is at most once. Then, when Eq. (7.29) is inserted, the above operators can be expressed as follows:   ˆ Tˆ1 (t, τ, τ  , τ  ) ≡ e−i Hm (t−τ )/ (A2 (τ  − t2 )Pˆ2,sd eik2 ·r−iω2 τ 

† + A∗2 (τ  − t2 )Pˆ2,gs e−ik2 ·r+iω2 τ ) ˆ

× e−i Hm (τ

 −τ  )/

ˆ



A1 (τ  − t1 )Pˆ1,gs eik1 ·r−iω1 τ e−i(Hb +Eg )τ

ˆ

 /

ˆ

† e−ik3 ·r+iω3 τ ei Hm (t−τ )/ , × ei(Hb +Eg )τ/ A∗3 (τ − t3 )Pˆ3,gs

ρ(0) ˆ

(7.41)

ˆ Tˆ2 (t, τ, τ  , τ  ) ≡ e−i Hm (t−τ )/ (A3 (τ − t3 )Pˆ3,sd eik3 ·r−iω3 τ † + A∗3 (τ − t3 )Pˆ3,gs e−ik3 ·r+iω3 τ )

× e−i Hm (τ −τ

ˆ

 )/

ˆ

 /

× ei(Hb +Eg )τ

  ˆ A1 (τ  − t1 )Pˆ1,gs eik1 ·r−iω1 τ e−i(Hb +Eg )τ / ρ(0) ˆ

  ˆ † A∗2 (τ  − t2 )Pˆ2,gs e−ik2 ·r+iω2 τ ei Hm (t−τ )/ , (7.42)

ˆ Tˆ3 (t, τ, τ  , τ  ) ≡ e−i Hm (t−τ )/ (A3 (τ − t3 )Pˆ3,sd eik3 ·r−iω3 τ † + A∗3 (τ − t3 )Pˆ3,gs e−ik3 ·r+iω3 τ ) ˆ

× e−i Hm (τ −τ ×e

 )/

i(Hˆ b +Eg )τ  /

ˆ



A2 (τ  − t2 )Pˆ2,gs eik2 ·r−iω2 τ e−i(Hb +Eg )τ τ 

† A∗1 (τ  − t1 )Pˆ1,gs e−ik1 ·r+iω1 e

 /

i Hˆ m (t−τ  )/

ρ(0) ˆ , (7.43)

ˆ † i Hˆ b τ  / ∗  ˆ A1 (τ − t1 )Pˆ1,gs Tˆ4 (t, τ, τ  , τ  ) ≡ e−i(Hb +Eg )t/ ρ(0)e 

ˆ

× e−ik1 ·r+iω1 τ ei Hm (τ

 −τ  )/ 

ˆ

† × (A∗2 (τ  − t2 )Pˆ2,sd e−ik2 ·r+iω2 τ ei Hm (τ −τ

 )/

A3 (τ − t3 )Pˆ3,sd eik3 ·r−iω3 τ

Chapter 7 Time-resolved nonlinear spectroscopy of excitons

  ˆ † + A2 (τ  − t2 )Pˆ2,gs eik2 ·r−iω2 τ ei Hm (τ −τ )/ A∗3 (τ − t3 )Pˆ3,gs e−ik3 ·r+iω3 τ )

ˆ

× ei Hm (t−τ )/ .

(7.44)

In the above expressions, the fact that ρ(0) ˆ = |gg|ρˆb has been used in expressing the propagator next to it in terms of Hˆ b + Eg . Note that the third order polarization at time t can be expressed as P(3) (t) = T r{Pˆ ρˆ (3) (t)}  τ  τ 4  t    2 dτ dτ  dτ  T r Pˆ Tˆj (t, τ, τ  , τ  ) . = 3 Im  0 0 0 j =1

(7.45) Then, employing Eqs. (7.41)–(7.44), more detailed expressions for the components constituting the third order polarization can be found. The term for j = 1 in Eq. (7.45) can be expressed as 



t

τ

dτ 0

0

dτ 



τ

  dτ  T r Pˆ Tˆ1 (t, τ, τ  , τ  )

0

= e−i(k3 −k2 −k1 )·r ei(ω3 t3 −ω2 t2 −ω1 t1 )





t

τ

dτ 0

0

dτ 



 0

τ

  dτ  χ (1) a (t, τ, τ , τ ) 

× A∗3 (τ − t3 )eiω3 (τ −t3 ) A2 (τ  − t2 )e−iω2 (τ −t2 ) A1 (τ  − t1 )e−iω(τ −t1 )  t  τ  τ   + e−i(k3 +k2 −k1 )·r ei(ω3 t3 +ω2 t2 −ω1 t1 ) dτ dτ  dτ  χ (1) e (t, τ, τ , τ ) 0

× A∗3 (τ

− t3 )e

iω3 (τ −t3 )

A∗2 (τ 

− t2 )e

0 iω2 (τ  −t2 )

0

A1 (τ  − t1 )e−iω(τ

 −t ) 1

, (7.46)

where the first term represents an excited state absorption as its final stage. On the other hand, the second term represents a stimulated emission. This becomes clear by the material response vectors that are defined as follows: 

  i(τ −τ )Eg / χ (1) a (t, τ, τ , τ ) = e  ˆ −i Hˆ m,d (t−τ  )/ Pˆ2,sd e−i(τ  −τ  )Hˆ m,s / Pˆ1,gs × T r Pe  ˆ  ˆ ˆ † ×e−i Hb τ / ρˆb ei Hb τ/ Pˆ3,gs ei(t−τ )Hm,s / , (7.47) 



  i(τ −τ )Eg / −i(t−τ )Eg / e χ (1) e (t, τ, τ , τ ) = e  ˆ −i Hˆ b (t−τ  )/ Pˆ † e−i(τ  −τ  )Hˆ m,s / Pˆ1,gs × T r Pe 2,gs   ˆ ˆ ˆ † ×e−i Hb τ / ρˆb ei Hb τ/ Pˆ3,gs ei(t−τ )Hm,s / . (7.48)

175

176

Chapter 7 Time-resolved nonlinear spectroscopy of excitons

In the above expressions, the material Hamiltonian within the propagator is denoted with additional subscript s or d so as to make it explicit whether the time evolution is in the single exciton (1) or double exciton subspace. In χ a (t, τ, τ  , τ  ) the polarization operator Pˆ couples double exciton and single exciton states. On the   ˆ other hand, in χ (1) e (t, τ, τ , τ ), the polarization operator P couples the ground electronic state and single exciton states. Similarly, the term with j = 2 in Eq. (7.45) can be expressed as 



t

t

dτ 0

dτ 



τ

  dτ  es · T r Pˆ Tˆ2 (t, τ, τ  , τ  )

0

0

= ei(k3 −k2 +k1 )·r e−i(ω3 t3 −ω2 t2 +ω1 t1 )





t

τ

dτ 0

0

dτ 



τ

0



  dτ  χ (2) a (t, τ, τ , τ ) 

× A3 (τ − t3 )e−iω3 (τ −t3 ) A∗2 (τ  − t2 )eiω2 (τ −t2 ) A1 (τ  − t1 )e−iω1 (τ −t1 )  t  τ  τ   + e−i(k3 +k2 −k1 )·r ei(ω3 t3 +ω2 t2 −ω1 t1 ) dτ dτ  dτ  χ (2) e (t, τ, τ , τ ) 0

× A∗3 (t

− t3 )e

iω3 (τ −t3 )

A∗2 (τ 

− t2 )e

0 iω2 (τ  −t2 )

0

A1 (τ  − t1 )e−iω1 (τ

 −t ) 1

, (7.49)

where   i(τ χ (2) a (t, τ, τ , τ ) = e

 −τ  )E

g /



ˆ −i Hˆ m,d (t−τ )/ Pˆ3,sd e−i(τ  −τ  )Hˆ m,s / Pˆ1,gs × T r Pe   ˆ ˆ  ˆ  † ×e−i Hb τ / ρˆb ei Hb τ / Pˆ2,gs ei(t−τ )Hm,s / , (7.50)   i(τ χ (2) e (t, τ, τ , τ ) = e

 −τ  )E



g /

e−i(t−τ )Eg /

ˆ −i Hˆ b (t−τ )/ Pˆ † e−i(τ −τ  )Hˆ m,1 / Pˆ1,gs × T r Pe 3,gs     ˆ ˆ ˆ † ×e−i Hb τ / ρˆb ei Hb τ / Pˆ2,gs ei(t−τ )Hm,s / . (7.51) (2) In χ a (t, τ, τ  , τ  ), the polarization operator Pˆ couples double exciton and single exciton states. On the other hand, in (2) χ e (t, τ, τ  , τ  ), the polarization operator Pˆ couples the ground electronic state and single exciton states. Next, the term with j = 3 in Eq. (7.45) can be expressed as





t

τ

dτ 0

0

dτ 



τ

  dτ  T r Pˆ Tˆ3 (t, τ, τ  , τ  )

0

= ei(k3 +k2 −k1 )·r e−i(ω3 t3 +ω2 t2 −ω1 t1 )





t

τ

dτ 0

dτ 

0

× A3 (τ − t3 )e−iω3 (τ −t3 ) A2 (τ  − t2 )e−iω2 (τ



 −t ) 2

0

τ

  dτ  χ (3) a (t, τ, τ , τ )

A∗1 (τ  − t1 )eiω1 (τ

 −t ) 1

Chapter 7 Time-resolved nonlinear spectroscopy of excitons

+e

−i(k3 −k2 +k1 )·r i(ω3 t3 −ω2 t2 +ω1 t1 )





t

e

τ

dτ 0

× A∗3 (τ − t3 )eiω3 (τ −t3 ) A2 (τ  − t2 )e

dτ



0 −iω2 (τ  −t2 )



τ

  dτ  χ (3) e (t, τ, τ , τ )

0 ∗  A1 (τ

− t1 )eiω1 (τ

 −t ) 1

, (7.52)

where   i(τ χ (3) a (t, τ, τ , τ ) = e

 −τ  )E

g /



ˆ −i Hˆ m,d (t−τ )/ Pˆ3,sd e−i(τ −τ  )Hˆ m,s / Pˆ2,gs × T r Pe   ˆ ˆ  ˆ  † ×e−i Hb τ / ρˆb ei Hb τ / Pˆ1,gs ei(t−τ )Hm,s / , (7.53)   i(τ χ (3) e (t, τ, τ , τ ) = e

 −τ  )E



g /

e−i(t−τ )Eg /

ˆ −i Hˆ b (t−τ )/ Pˆ † e−i(τ −τ  )Hˆ m,s / Pˆ2,gs × T r Pe 3,gs     ˆ ˆ ˆ † ×e−i Hb τ / ρˆb ei Hb τ / Pˆ1,gs ei(t−τ )Hm / . (7.54) In χ a (t, τ, τ  , τ  ), the polarization operator Pˆ couples double exciton and single exciton states. On the other hand, in (3) χ e (t, τ, τ  , τ  ), the polarization operator Pˆ couples the ground electronic state and single exciton states. Finally, the term with j = 4 in Eq. (7.45) can be expressed as (3)





t

dτ 0

τ

dτ 0





τ

  dτ  T r Pˆ Tˆ4 (t, τ, τ  , τ  )

0

= ei(k3 −k2 −k1 )·r e−i(ω3 t3 −ω2 t2 −ω1 t1 )





t

τ

dτ 



τ

  dτ  χ (4) a (t, τ, τ , τ ) 0 0   × A3 (τ − t3 )e−iω3 (τ −t3 ) A2 (τ  − t2 )∗ eiω2 (τ −t2 ) A∗1 (τ  − t1 )eiω1 (τ −t1 )  t  τ  τ   + e−i(k3 −k2 +k1 )·r ei(ω3 t3 −ω2 t2 +ω1 t1 ) dτ dτ  dτ  χ (4) e (t, τ, τ , τ ) 0 0 0   × A∗3 (τ − t3 )eiω3 (τ −t3 ) A2 (τ  − t2 )∗ e−iω2 (τ −t2 ) A∗1 (τ  − t1 )eiω1 (τ −t1 ) ,

dτ

0

(7.55) where   i(τ χ (4) a (t, τ, τ , τ ) = e

 −t)E / g

 ˆ −i Hˆ b t/ ρˆb ei Hˆ b τ  / Pˆ † ei Hˆ m,s (τ  −τ  )/ × T r Pe 1,gs   ˆ ˆ † ×Pˆ2,sd ei(τ −τ )Hm,d / Pˆ3,sd ei(t−τ )Hm,s / , (7.56)

177

178

Chapter 7 Time-resolved nonlinear spectroscopy of excitons

  i(τ χ (4) e (t, τ, τ , τ ) = e

 −t)E / g



ei(τ −τ

 )E

g /

ˆ −i Hˆ b t/ ρˆb ei Hˆ b τ  / Pˆ † ei Hˆ m,s (τ  −τ  )/ × T r Pe 1,gs   )Hˆ / † i(τ −τ i(t−τ )Hˆ m,s / b Pˆ3,gs e ×Pˆ2,gs e . (7.57) In both χ a (t, τ, τ  , τ  ) and χ e (t, τ, τ  , τ  ), the polarization operator Pˆ couples the ground and single exciton states. For the case where the bath Hamiltonian can be modeled as a sum of harmonic oscillators and the exciton-bath coupling approximated as being linear in the displacements of oscillators and diagonal in the exciton basis, it is possible to find simple closed-form expressions for all the response functions defined above. This can be done employing a generalized cumulant approach [173], or, alternatively, a sequence of polaron transformations [174]. Even in this simple case, it is important to note that the resulting response functions cannot be simplified into products of linear-spectroscopic lineshape functions as long as there are common bath modes coupled to different exciton states. For delocalized excitons, it is in general difficult to find the case where bath modes coupled to different exciton states are independent of each other. Thus, this situation is expected to occur more often than not. Second, there are terms oscillatory with respect to the population time T = τ − τ  even in the absence of inter-exciton systembath coupling. This is because transition to different exciton states is possible during the interaction with the pulse. This way, coherence between different exciton states can be maintained even in the absence of bath-mediated inter-exciton couplings. These are factors that can complicate the interpretation of experimental results. For more general case but under the assumption of weak exciton-bath couplings, it is possible to generalize the QME formulation and derive a set coupled time evolution equations [174]. However, numerical feasibility and practical utility of this approach is still to be seen. (4)

(3)

7.4 Summary and additional remarks This chapter has provided formal descriptions of time resolved nonlinear spectroscopies that have significant implications for exciton dynamics. The first description was two-pulse spectroscopy. This involved expansion of the total density operator up to second orders with respect to pump and probe pulses, respectively. For the case of a pump-probe spectroscopy, general expressions

Chapter 7 Time-resolved nonlinear spectroscopy of excitons

were derived for the population transition rates from single exciton to double exciton states (excited state absorption) and to the ground electronic state (stimulated emission). The second description was the response function formalism of the four wave mixing spectroscopy, which is a framework widely employed at present. The major focus of this formulation was to express all the terms in the electronic state space explicitly (not in Liouville space) while accounting for all the excited state absorption and stimulated emission components. Although the formalisms presented here are not entirely new, they have a few merits that are helpful for future investigation of exciton dynamics. The expressions derived here have taken advantage of the information on the exciton-bath Hamiltonian as explicitly as possible, and represent detailed information on how different exciton states and their couplings to environments contribute to response functions. The derived expressions also make application of general quantum dynamics methods more straightforward. As should be clear from the expressions derived in this chapter, calculation of physical observables of the time resolved nonlinear spectroscopic observables is more challenging than simple lineshape calculations. Indeed, information on lineshape function alone is not sufficient for the description of either population dynamics in two-pulse spectroscopy or response functions in four wave mixing spectroscopy. On the other hand, quantities necessary to model these physical observables are not necessarily all important for the description of exciton dynamics under minimally perturbed situations such as under incoherent sun light [150]. From the viewpoint of the consistent history interpretation of quantum mechanics [175], different types of nonlinear spectroscopy amount to creating different histories of “measurement.” Thus, all of these different types are not necessarily consistent with others. While it is believed to be possible to construct the whole dynamical picture of quantum systems through combination of all compatible histories, it is not in general possible to predict outcomes for one type of measurements from the other when they involve multiple interactions with radiations and are not necessarily compatible with each other. In this sense, it is true that observing coherent exciton dynamics from any nonlinear spectroscopy does not necessarily imply that such coherent dynamics also occur under any other conditions. Thus, care should be taken in proper understanding of implications of nonlinear spectroscopic data. While it is important to take this point into consideration, it is still true that time resolved nonlinear spectroscopies

179

180

Chapter 7 Time-resolved nonlinear spectroscopy of excitons

remain as powerful and unique tools for investigating the real time dynamics of excitons.

8 Examples and applications Contents 8.1 Excitons in natural light harvesting complexes 182 8.1.1 FMO complex of green sulfur bacteria 182 8.1.2 LH2 complex of purple bacteria 183 8.1.3 PBP of cryptophyte algae 185 8.2 Excitons for photovoltaic devices 186 8.3 Excitons for structural determination 188 8.3.1 FRET efficiency 188 8.3.2 Beyond FRET efficiency measurement 191 8.4 Summary and additional remarks 192 Excitons are material forms of energy created at the interface of the interaction with the light. Thus, their characteristics can be detrimental to the manner of conversion between the light and material energy. In this context, understanding detailed characteristics of excitons have significant implications for the development of solar energy conversion and optoelectronic devices. In addition, intrinsic quantum nature of excitons makes them unique objects for demonstrating and testing fundamental quantum principles in complex molecular environments, opening new opportunities for imaging, sensing, and information processing. This chapter provides a brief account of three examples/applications of molecular excitons, and addresses issues that are important in the context of the dynamics of molecular excitons and their characterization. The first set of examples are photosynthetic light harvesting complexes. Although they are not historically the first kind of systems investigated for excitons, decades of structural, spectroscopic, and computational studies make them some of the most well characterized excitonic systems. Then, brief accounts will be provided for excitons for photovoltaic devices. One important area of application of excitons is for structural determination, imaging and sensing. The success of these techniques being used to date stems mostly from the sensitivity of FRET or other forms of long-range RET (LRET) on the distance between exciton donor and acceptor [176–180]. These serve as unique tools that allow fairly accurate distance determination at nanometer length scales through spectroscopic means, and there Dynamics of Molecular Excitons. https://doi.org/10.1016/B978-0-08-102335-8.00016-8 Copyright © 2020 Elsevier Ltd. All rights reserved.

181

182

Chapter 8 Examples and applications

is good possibility for further improvement and generalization of these techniques. Thus, it is important to clarify the theoretical basis of the FRET technique and examine how this can be generalized further.

8.1 Excitons in natural light harvesting complexes Photosynthetic organisms cumulate their energy for life by employing various kinds of light harvesting complexes (LHCs) [13, 181], which collect light energy in the form of excitons and transfer these over about 100 nm length scale distances. They have limited time scales and resources to construct light harvesting complexes. For example, no crystalline materials can be formed in the soft environments of these organisms, excluding the possibility of Wannier-type excitons, and the types of pigment molecules that can be utilized and recycled are fairly limited. A practical solution that became available for these photosynthetic organisms was to maximally utilize select pigment molecules so as to create Frenkel type molecular excitons. Except for the case of chlorosome, which is a closely packed assembly of typically more than 100,000 chlorophyll molecules, all the light harvesting complexes consist of ∼10–100 pigment molecules embedded in protein matrices, which then constitute larger networks of complexes along with reaction centers. The kinds of pigment molecules in LHCs are surprisingly few and are mostly chlorophylls, bacteriochlorophylls, carotenoids, and billins [181]. The nearest neighbor distances of these pigment molecules are ∼1–2 nm, and structures of these complexes in general do not have apparent high symmetry except for light harvesting complexes of purple bacteria [182,183]. However, even for these cases, due to variations of local environments and fluctuations, there is significant energetic disorder and the actual exciton Hamiltonian does not have the high symmetry expected from the X-ray crystallography structure. Description of excitons in three well-known LHCs, for which an extensive review has been made recently [13], is provided below.

8.1.1 FMO complex of green sulfur bacteria The most well-known example of LHC is Fenna–Matthews– Olson (FMO) complex of green sulfur bacteria, although it is not the best one representing all others. It was the first LHC with its structure determined by the X-ray crystallography [184,185]. The

Chapter 8 Examples and applications

full FMO complex is a trimer with each unit consisting of 8 (often approximated as having 7) bacteriochlorophylls (BChls). Thus, the exciton Hamiltonian in the single exciton space for each monomer of the FMO complex has the form of Eq. (3.16) with Nc = 8 [13]. Although all the BChls in a given FMO complex are the same molecules, their local protein environments are different, causing their excitation energies to be different. The resulting variations in site excitation energy, |Ej − Ek |’s are in the range of ∼100–500 cm−1 . On the other hand, the magnitudes of electronic couplings, |Jj k |’s, are in the range of ∼1–100 cm−1 . Since the electronic couplings are less than the energy differences among different sites, the excitons in this complex are not expected to be delocalized significantly, with likely delocalization sizes at most 2–3 BChls. This implies that the dynamics of excitons in this system should be mostly hopping-like, an obvious fact that may not have required high level quantum dynamics calculations and nonlinear spectroscopic measurements to confirm. Although oscillatory population dynamics were observed, they are confined mostly within 2–3 BChls and are not extended over the entire complex. The exciton-bath coupling in this system is rather weak, with the expected reorganization energy less than 50 cm−1 and the range of the major bath spectral density (within the range of single exciton subspace) about 100 cm−1 . This means that the spectral overlap between different sites are rather marginal, and the dynamics are mainly driven by the funneling structure of the excitation energies. As a result, the exciton dynamics are not particularly fast. Indeed, the time scale of the full population dynamics within a monomer of the FMO complex is about 5 ps, whereas it is ∼20–30 ps across the entire trimer. Thus, considering the size and time scale of the exciton dynamics, the FMO complex is far from being a quantum wire. Rather, it is more like a classical-like wire with significant modulating function of the exciton dynamics so as not to saturate the reaction center.

8.1.2 LH2 complex of purple bacteria Purple bacteria have two LHCs [182,183]. One is light harvesting 2 (LH2) complex, which serves as the primary harvester and relayer of the exciton, and the other is light harvesting 1 (LH1) complex, which functions as the final destination of the exciton before it hops to the reaction center enclosed in it. The structure of LH2 complex is shown in Fig. 1.5B, and has N -fold circular symmetry, with N = 8–10 depending on the species of the bacteria. There are three BChls within each symmetry unit. These are

183

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Chapter 8 Examples and applications

the same molecules, but are positioned differently and bound to different residues of protein scaffolds constituting the complex. Denoting these as α, β, and γ , one can express the Hamiltonian representing the single excitons formed by the 3N BChls, in the absence of disorder, as follows [13,19,186,187]: 0 Hˆ LH 2=

N   Eα |αj αj | + Eβ |βj βj | + Eγ |γj γj | j =1

+ Jαβ (0)(|αj βj | + |βj αj |) + Jαγ (0)(|αj γj | + |γj αj |)  +Jβγ (0)(|βj γj | + |γj βj |) +

N N  



Jss  (j − k)|sj sk | ,

(8.1)

j =1 k=j s,s  =α,β,γ

where |sj , with s = α, β, or γ , represents the excited state where only sj BChl is excited with its energy Es , and Jss  (j − k) is the electronic coupling between |sj  and |sk . The α and β BChls form concentric rings on the same plane with radii about 2.6 nm and 2.7 nm, respectively, according to their X-ray crystal structure [182, 183,188,189]. The γ BChls also form a ring with its radius about 3.1 nm, and the center of the ring displaced towards the cytoplasmic side by about 1.7 nm. More detailed structural information can be found in Ref. [187]. The nearest neighbor electronic couplings in the unit consisting of α and β BChls are estimated to be in the range of 250–450 cm−1 , with the exact value as yet difficult to determine precisely. On the other hand, the values of Eα and Eβ are believed to be almost the same. Thus, this unit consisting of α and β BChls, which is normally called B850 unit because their major single exciton band is in the 850 nm wavelength region, is expected to have much more delocalized excitons than those in the FMO complex. While the exciton in this B850 unit should be fully delocalized over the entire ring if the symmetry of the X-ray crystal structure is maintained, the disorder and fluctuations in real membrane environments cause Eα and Eβ to be different for each site. According to various spectroscopic modelings, the magnitude of the disorder in the excitation energy is ∼250 cm−1 . On the other hand, the exciton-bath coupling for each site is moderate, with the reorganization energy about 90 cm−1 and the range of the bath spectral density ∼170 cm−1 . Thus, the exciton in this B850 unit is truly in the intermediate regime with all energetic parameters similar to room temperature thermal energy. The exciton dynamics within this B850 unit is believed to equilibrate within about 200 fs. On the other hand, the exciton transfer from the

Chapter 8 Examples and applications

unit consisting of γ -BChls (called B800 unit because its absorption maximum occurs at about 800 nm region) to B850 occurs in the range of 0.7–1 ps, and that between LH2 complexes in the range of 4–25 ps. The quantum delocalization of exciton within the B850 unit was shown to play a major role in achieving these transfer rates [63,186,76,190].

8.1.3 PBP of cryptophyte algae Cryptophyte algae also has interesting peripheral LHCs normally called phycobiliproteins (PBPs). Each PBP has 8 pigment molecules called billins, non-cyclic analogue of chlorophyll, bound to protein residues by covalent bonding. Two types of PBPs called PE5454 and PC645 have received significant attention of researchers since their X-ray structures were determined in 1999–2004 [191,192] and 2014 [193], respectively. Both of these can be expressed commonly by abbreviating the two pigments with the highest excitation energies as H1 and H2 , the two lowest ones as L1 and L2 , and the four intermediate ones as Mj , j = 1, · · · , 4. Thus, the exciton Hamiltonian representing the single exciton electronic space can be expressed as [13] Hˆ P BP =

2  {EHj |sHj sHj | + ELj |sLj sLj |} j =1

+

4  j =1

EMj |sMj sMj | +



Jj k |sj sk | ,

(8.2)

j k=j

where j and k in the last sum run over all possible sites. The electronic couplings between H1 and H2 are estimated to be ∼150–300 cm−1 . All other electronic coupling constants between pigment molecules are less than 50 cm−1 . On the other hand, the energy differences between different groups are large, spanning almost 2000 cm−1 range. Nonetheless, the exciton transfer from the dimers of the highest energies to those of lowest energies occur quickly, in about 0.6–0.7 ps. This becomes possible through the mediation of high frequency vibrational modes that make the spectral overlap between two exciton bands significant. Thus, this is a good example where a combination of funneling structure and moderately strong exciton-bath couplings ensure efficient collection and down conversion of excitons. Although some coherent features have been observed in this complex as well, it is most likely that it is an outcome of two oscillations between two strongly coupled states, unless it has vibrational origin.

185

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Chapter 8 Examples and applications

8.2 Excitons for photovoltaic devices There are at least five key properties desired for excitons in creating successful photovoltaic devices as listed below. (1) The exciton band has to match the solar spectrum as closely as possible. (2) The yield of exciton upon irradiation has to be as high as possible. (3) Excitons have to be readily available for charge separation. This means excitons have to be either nearby or move fast enough to charge separation regions. (4) Exciton binding energies have to be as low as possible in order to minimize the energy loss during their separation. At the same time, the probability for the bound electron and hole pair to recombine as excitons has to be as low as possible. (5) Materials and manufacturing have to be cost effective. Excitons created in inorganic solar cells (ISC) that are mostly based on silicon and/or Ga/As materials are Wannier-type, which meet the criteria (3) and (4) easily. Significant advances have been made over 60 years in improving (1) and (2), and the best efficiency solar cells are all these ISCs [194]. However, there is fundamental limit in meeting the criteria (5). Although significant advances have been made in bringing the cost down, there remains great need for more cost effective and easily processable solar energy conversion devices. Organic solar cells (OSC) developed initially based on conjugated polymers meet the condition (5), but were not able to satisfy other criteria [195]. Due to the dismal efficiency of the first generation of OSCs, there had not been significant efforts to push forward for about two decades. The notion that excitons in conjugated polymers are strongly bound and are difficult to separate was also contributing factor to the lack of significant development. However, advances in computational modeling [196,197] and the development of much more diverse organic molecules including well-known derivatives of thiophenes, phenylene-vinylenes, and fullerenes brought new possibilities. Most importantly, the development of bulk hetero-junction morphology has led to competitive OSCs that soon reached efficiencies hovering over 10% within a decade [198–200]. While these were outcomes of some improvements concerning the criteria (1)–(3), these have not yet been satisfactory, hampering further improvement of the efficiency. Excitons in conjugated organic molecules are typically in intermediate regime between Frenkel and Wannier type excitons.

Chapter 8 Examples and applications

Thus, higher level ab initio calculations are needed for accurate determination of exciton levels, which are still limited to fairly small systems. Significant efforts have been made to characterize the delocalization lengths of excitons at both single molecule and aggregate levels, and also their dynamic modulations. As yet, the extent of excitons and key factors determining their dynamics, in particular, in heterogeneous and disordered environments typical for actual OSC materials are not well understood. There are two major directions to improve OSCs. One is to lower the exciton-binding energy and the other is to improve the mobility of excitons. For a long time, it was difficult to lower the exciton binding energy in OSC below 0.3 eV. However, new experimental evidences and theoretical results have demonstrated that further reduction of the binding energy is possible through proper choice of molecules and environments [201,202]. Likewise, there remain much room for further enhancement of the mobility of excitons in conjugated organic materials. This could make it possible to utilize a different morphology that are less sensitive to the charge mobility, which is a key bottleneck that limits the efficiency of OSC. While a recent breakthrough [203] improved the efficiency mainly through increase in the charge mobility, enhancing the mobility of excitons in OSC remains another key factor that can be used to improve the efficiency. It can help develop alternative morphologies for which limited charge transport property is less of an issue. In recent years, lead halide perovskites have emerged as promising materials and have attracted great interest across diverse communities of researchers. The success of these perovskite materials stems from the fact that they meet the criteria (3) and (4) easily. The nature of excitons and the magnitudes of exciton binding energies in these systems were not well understood initially, but now it is relatively well established that the binding energies of excitons in these systems are very low. They are estimated to be in the range of 2–20 meV [204,205]. The efficiency of three dimensional perovskites has already reached beyond 25% and is expected to improve further. Despite this efficiency, the instability of three dimensional perovskites and the toxicity of lead remain critical issues. Two dimensional and layered perovskites can be used to overcome the issue of instability, but at the expense of lower efficiency. The exciton binding energies in two dimensional perovskites are believed to be in hundreds of meV due to confinement effects. There is also evidence for significant polaronic effect associated with the coupling of low frequency vibrational modes [206]. As yet, confined perovskites have much promise even for light emitting diodes [207]. Time resolved spectroscopic studies

187

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Chapter 8 Examples and applications

on novel confined perovskites [208,209] have significant implications in that respect. Future studies of similar kind will help clarify the nature exciton dynamics in these perovskite systems.

8.3 Excitons for structural determination Considering the sensitivity of the exciton dynamics on the configuration and arrangement of chromophores, it is natural to ask whether such sensitivity can be utilized to gain a structural information of underlying chromophoric systems. The answer to this question depends on the availability of a theory providing a clear relationship between controllable features of the exciton dynamics and meaningful structural information. The well known FRET technique, often referred to as a spectroscopic ruler, is the simplest of this kind, and is based on the sensitivity of FRET efficiency on the distance between two chromophores transferring an exciton from one to the other. While this technique was originally developed between dye molecules with well-characterized electronic excitations, applications of the FRET technique have now expanded to systems with delocalized excitons such as quantum dots and graphene surface. The success of the FRET technique and its expanding applications suggest the possibility of more advanced methods utilizing more general class of exciton dynamics. In this sense, it is important to review first the theoretical basis of the FRET efficiency measurement. The author has provided such an analysis in a recent work [210], some of which are incorporated into the description presented below.

8.3.1 FRET efficiency For a pair of donor (D) and acceptor (A) chromophores interacting via transition dipole interactions that are weak compared to other energetic factors, the exciton transfer from D to A can be described by the FRET rate that depends on the inverse sixth power of the D − A distance, RDA . The FRET rate is given by Eq. (4.33), where the Förster radius RF can be defined as Eq. (4.34). If the conventional Förster’s rate expression Eq. (6.32) is used, the Förster radius has the following familiar form:  RF =

9000(ln 10)κ 2 128π 5 NA n4r

 d ν˜

fD (˜ν )A (˜ν ) ν˜

1/6 .

(8.3)

In the above equation, κ is the orientational factor determined by relative orientations of two transition dipoles and hardly remains constant in real situation. Therefore, it is a common practice to

Chapter 8 Examples and applications

use an average value, κ 2 . However, care should be taken in relying on such expression because the original FRET theory was not derived for the case where κ fluctuates in time or is influenced by quantum mechanical modulation. This ambiguity in κ 2 is often viewed as primary source of error in determining the distance dependence, but there are other important factors as well. The success of FRET as a spectroscopic ruler relies on the following expression for FRET efficiency: E=

1 1 + (RDA /RF )6

.

(8.4)

A simple derivation of this FRET efficiency is provided below. Let us consider an ensemble of donors subject to steady irradiation that selectively excites D. In the absence of FRET, the concentration of the excited state donor is denoted as [D ∗ ]0 (t). Then, its time derivative is determined by d 1 [D ∗ ]0 (t) , [D ∗ ]0 (t) = Ir [D] − dt τD

(8.5)

where Ir is the rate of incident photons causing excitation, and [D] is the concentration of ground state D. For sufficiently small Ir , [D] can be assumed to be virtually constant. Let us denote the concentration of the excited state donor in the presence of FRET as [D ∗ ]. Then, under the assumption that the reverse process from the excited acceptor can be neglected,   d 1 ∗ + kF RET [D ∗ ](t) . [D ](t) = Ir [D] − dt τD

(8.6)

Assuming steady state limits where the rate of concentration change of excited state donors becomes virtually zero, the corresponding steady state concentrations can be obtained from Eqs. (8.5) and (8.6) as follows: Ir [D] , 1/τD Ir [D] [D ∗ ]s = . kF RET + 1/τD

[D ∗ ]0,s =

(8.7) (8.8)

Therefore, the FRET efficiency can be defined as E =1−

kF RET [D ∗ ]s = . ∗ [D ]0,s kF RET + 1/τD

(8.9)

189

190

Chapter 8 Examples and applications

Inserting Eq. (4.33) into the above expression, one can easily show that E=

(RF /RDA )6 1 + (RF /RDA )6

=

1 1 + (RDA /RF )6

.

(8.10)

The second expression for E in Eq. (8.9) makes it possible to determine the efficiency in terms of lifetime measurements in the absence and presence of FRET. Let us define the lifetime of the excited donor in the presence of FRET as τD,F , which is given by 1 1 = + kF . τD,F τD

(8.11)

Then, then efficiency can also be expressed as E =1−

τD,F . τD

(8.12)

The expressions for the efficiency derived in this section are based on the assumption that the spontaneous lifetime of the excited state donor remains the same in the presence of FRET. Thus, care should be taken in using them if the new environment for FRET introduces a new non-radiative process or local field corrections. The derivation of the FRET efficiency needs to be modified if the reverse transfer from excited acceptor is non-negligible, for which the following rate equations should be used.   1 d + kF [D ∗ ](t) + kFr [A∗ ](t) , [D ∗ ](t) = Ir [D] − dt τD   d ∗ 1 + kFr [A∗ ](t) , [A ](t) = kF [D ∗ ](t) − dt τA

(8.13) (8.14)

where kFr is the FRET rate for the reverse process from A to D. Once again, assuming steady state and weak irradiation, kF [D ∗ ]s , kFr + 1/τA Ir [D] . [D ∗ ]s = 1/τD + kF /(kFr τA + 1)

[A∗ ]s =

(8.15) (8.16)

For this case, the efficiency is given by the following expression: E=

kF τD . kF τD + kFr τA + 1

(8.17)

Chapter 8 Examples and applications

Assuming that the reverse process also follows the same FRET process but with different FRET radius RFr , the efficiency in this case is expressed as E=

1 . + (RDA /RF )6

(8.18)

1 + (RFr /RF )6

This is similar to Eq. 8.4, but amounts to having a different effective Förster radius, RF (1 + (RFr /RF )6 )1/6 . The maximum of efficiency is also smaller than unity.

8.3.2 Beyond FRET efficiency measurement Let us consider a different excitation process where a short excitation pulse creates [D ∗ ](0) (with [A∗ ](0) = 0) at time t = 0. Once the excitation pulse becomes inactive, the time dependent concentrations of D ∗ and A∗ are governed by the following equations: d dt



  −1/τD − kF [D ∗ ](t) = kF [A∗ ](t)

kFr −1/τA − kFr



[D ∗ ](t) [A∗ ](t)

 .

(8.19)

The two eigenvalues of the above matrix equation are as follows:    kr τ D + τA 1 1+ F + λ ± = kF − 2 kF kF τ D τ A 

1/2 ⎫  ⎬ kFr 1 τA − τD 2 kFr ± + + 1− . (8.20) ⎭ 4 kF kF τ D τ A kF The corresponding time dependent concentrations are as follows:  [D ∗ ](0)  (−A + B) eλ+ t + (A + B) eλ− t , 2B  [D ∗ ](0)  λ+ t e − eλ− t , [A∗ ](t) = 2B

[D ∗ ](t) =

(8.21) (8.22)

where   kr τ − τD 1 1− F + A , 2 kF kF τ D τ A  kr B = A2 + F . kF

A=

(8.23) (8.24)

The above expression can be used for fitting experimentalpump probe data by bi-exponential curves, from which both rates

191

192

Chapter 8 Examples and applications

kF and kFr can be determined. This can provide more accurate determination of the distance RDA and possibly the orientation factor. Although FRET rate was assumed in obtaining the above expressions, Eqs. (8.20)–(8.22), they are valid even when kF and kFr are replaced with other kinds of expressions. For example, the inelastic and MC-FRET rate expressions derived in Chap. 4 can be employed, for which the dependences of rates on structural data are deducible as long as donor and acceptor and their interactions with environments are well characterized. Applications of these rates or other types of rate expressions involving different electronic coupling mechanisms are interesting possibilities to explore for structure determination by spectroscopic means. Another important direction that has ample potential is to utilize quantum dynamics of excitons and time resolved spectroscopy for more intricate exciton systems. Already some successful attempts to utilize polarization information of 2-d electronic spectroscopy data has been used for the FMO complex. Many new ideas developed through applications of quantum information processing to excitons in LHCs, may be tested and developed further to this end.

8.4 Summary and additional remarks This chapter has provided brief overviews of three examples/applications of molecular excitons. For the first topic, excitons in LHCs, exciton Hamiltonians for FMO complex, LH2 complex, and PBPs were introduced, and major features and time scales of exciton dynamics were explained. For the excitons for photovoltaic devices, five important criteria that excitons need to meet were laid out, and the features of excitons in organic photovoltaic devices and lead halide perovskites were discussed in this context. Finally, the FRET technique used as a spectroscopic ruler was reviewed theoretically, followed by a few suggestions for utilization of other exciton dynamics. There are many examples of molecular excitons that have not been considered here [211,212,4,213,16,214–217,216,218–221]. These include excitons in self-assembled nanotubes, DNAchromophore complexes [222,223], and quantum dots and their aggregates. The range of current and potential applications of excitons is also broad, most of which have not been covered here. These include applications for light emitting diodes and lasing, new efforts for driving chemical reactions [224,225], optical switches [226], and information processing [227]. Finally, the unique capability of excitons to form entanglement and coherent

Chapter 8 Examples and applications

delocalization offers new opportunities for quantum technology such as quantum information processing and quantum computation [228–230].

193

9 Summary and outlook The primary focus of this book has been the dynamics of molecular excitons. Due to the fact that most quantum dynamics studies of molecular excitons rely on simplified model Hamiltonians, it was first necessary to understand and examine the assumptions and approximations involved in such model Hamiltonians. On the basis of this formal clarification, a comprehensive description of various theories of exciton dynamics were offered. This was then followed by theoretical description of time resolved nonlinear spectroscopy, and some account of examples and applications of excitons. Below are more detailed summaries of all the chapters. Chapter 1 provided an overview of excitons in general and brief description of physical assumptions behind two major types of excitons, Frenkel and Wannier excitons. Simple models were introduced to illustrate major concepts, and key issues concerning delocalization of excitons and disorder were discussed. Chapter 2 provided a general derivation of the exciton-bath Hamiltonians starting from the standard molecular Hamiltonian, and demonstrated the existence of various nonadiabatic terms at formal level. Then, assumptions involved in conventional Frenkel type exciton-bath Hamiltonian were clarified. These include crude adiabatic approximation for the electronic state of each chromophore, and neglecting most intrachromophore nonadiabatic terms and some interchromophore nonadiabatic terms. Chapter 3 provided a theoretical description of linear absorption and stimulated emission spectroscopic lineshapes of molecular excitons, and simple approximations applicable to general cases. For example, Eqs. (3.28), (3.46), and (3.69) can be used for general exciton-bath and bath Hamiltonians. Then, more detailed expressions were provided for the model of linearly coupled harmonic oscillator bath. Chapter 4 provided a description of exciton transfer rate theories that are based on FGR level approximations. These include the original FRET rate, inelastic FRET, and multichromophoric FRET. Some works on exciton hopping dynamics that employ the FRET rate as transition probabilities were also described. Dynamics of Molecular Excitons. https://doi.org/10.1016/B978-0-08-102335-8.00017-X Copyright © 2020 Elsevier Ltd. All rights reserved.

195

196

Chapter 9 Summary and outlook

Chapter 5 provided derivations of QME approaches for the exciton dynamics applicable to general exciton-bath models, and full expressions for kernels of the QME for the model of linearly coupled harmonic oscillator bath. On the basis of exact projection operator formalism, approximate expression valid up to the fourth or the second order were derived. For the linearly coupled harmonic oscillator bath model, expressions for both standard second order QME and polaron transformed QME approach were fully derived. Chapter 6 provided an account of the quantized electromagnetic field and its interaction with excitons. This was used to provide a microscopic derivation of Förster’s spectral overlap expression. A brief overview of the concept of polaritons was provided as well. Chapter 7 provided formal descriptions of time resolved nonlinear spectroscopies. Detailed expressions were provided for twopulse pump-probe spectroscopy based on rate description. General expressions for the four wave mixing spectroscopy within the response function formalism were also derived. Chapter 8 provided overviews of three primary examples/applications, excitons in light harvesting complexes, excitons for photovoltaic devices, and employing exciton dynamics for structural determination. Quantitative description of the exciton dynamics in condensed and complex molecular environments has remained challenging for a long time. This is because of (i) the difficulty of conducting reliable quantum dynamics calculations for large scale systems that are coupled to environments of virtually infinite degrees of freedom, (ii) the difficulty of calculating excited state properties in general including the electronic coupling constants, and (iii) the lack of satisfactory experimental information that can offer reliable picture of the spatio-temporal dynamics of excitons. Many advances resolving some of these issues have been made during the past two decades. For example, for the case of LHCs, the combination of calculations, simulations, and time resolved spectroscopies have made it possible to develop consensus on the types and mechanisms of exciton dynamics. While there are still many details that need to be clarified and confirmed, it is now becoming clear that LHCs are prime examples for moderately delocalized Frenkel-type excitons (with the exception of chlorosomes). Many standard quantum mechanical theories and calculation methods developed for decades apply to these systems nicely. While there are no surprising or dramatic quantum effects expected, many hidden quantum effects underneath seemingly classical behavior can be identified

Chapter 9 Summary and outlook

for these systems. One important lesson from the decades of efforts for these systems is the importance of the availability of reliable Hamiltonians capturing major features of exciton dynamics while being amenable for dynamics calculations. From the theoretical point of view, there still remain various issues to be resolve for reliable description of exciton dynamics for a broader class of systems. These include going beyond simple harmonic oscillator bath models for the environments, and detailed analysis of various nonadiabatic terms that are typically ignored in simplified descriptions, including conical intersections. How to go beyond crude adiabatic approximations and develop efficient methods for describing the intermediate situations between Frenkel and Wannier excitons are also challenging issues.

197

A Useful mathematical identities and solutions A.1 Solution of eigenvalue problems for the simple Frenkel exciton models For Hˆ e with dimension N defined by Eq. (1.6), introduce hˆ e = Hˆ e /E and a = J /E. Then, the determinant, |hˆ e − λ|, can be expressed as    1−λ a 0 ··· 0 0     a 1−λ a ··· 0 0     0 a 1 − λ ··· 0 0   (A.1) dN =  . ..  . .. .. .. .. .. ..  .. .  . ... . .   0 0 ··· 1 − λ a    0 0 0 ··· a 1−λ  It is easy to show that the above determinant satisfies the following recursion relationship: dN = (1 − λ)dN−1 − a 2 dN−2

(A.2)

Let 1 − λ = α + β and a 2 = αβ. Then, dN = (α + β)dN−1 − αβdN−2

(A.3)

This can be rearranged to dN − αdN−1 = β (dN−1 − αdN−2 ) dN − βdN−1 = α (dN−1 − βdN−2 )

(A.4) (A.5)

For N = 2, it is easy to show that d2 − αd1 = β 2

(A.6)

d2 − βd1 = α

(A.7)

2

Therefore, dN − αdN−1 = β N

(A.8)

dN − βdN−1 = α

(A.9)

N

Dynamics of Molecular Excitons. https://doi.org/10.1016/B978-0-08-102335-8.00018-1 Copyright © 2020 Elsevier Ltd. All rights reserved.

199

200

Appendix A Useful mathematical identities and solutions

Combination of these leads to the following solution β N+1 − α N+1 β −α     N−1 N β β + + ··· + 1 = αN α α

dN =

(A.10)

The solutions of dN are complex valued in general and are given by β = e2πik/(N+1) , k = 1, · · · , N α

(A.11)

Since α + β and αβ are real, α and β are complex conjugates of each other and are thus equal to α = ae−iπk/(N +1) β = ae

(A.12)

iπk/(N +1)

(A.13)

Therefore,  λ = 1 − (α + β) = 1 − 2a cos

kπ N +1

 , k = 1, · · · , N

(A.14)

Let’s reorder the indices and define     (N + 1 − k)π kπ λk = 1 − 2a cos = 1 + 2a cos , k = 1, · · · , N N +1 N +1 (A.15) Then, denoting the eigenvector for λk as ck with its transpose given by cTk = (ck,1 , · · · , ck,N ), it is easy to show that  ck,j −1 − 2 cos

 kπ ck,j + ck,j +1 = 0 N +1

(A.16)

with the boundary conditions ck,0 = ck,N+1 = 0. The above recursion relation is of the same type as Eq. (A.3) and can be solved by the same method by introducing α = e−ikπ/(N +1) and β = eikπ/(N +1) . Employing these and following the same procedure as finding the expression for dN , one can show that ck,j =

β j − αj sin(kj π/(N + 1)) ck,1 = ck,1 β −α sin(kπ/(N + 1))

(A.17)

Appendix A Useful mathematical identities and solutions

Then, 2 2 2 + ck,2 + · · · + ck,N ck,1

=



2 ck,1

4 sin2 (kπ/(N + 1))

2 − e2ikπ/(N+1) − e−2ikπ/(N+1) + 2 − e4ikπ/(N+1) − e−4ikπ/(N+1) + ··· +2 − e2iN kπ/(N+1) − e−2iN kπ/(N+1)

=

2 ck,1

2 sin2 (kπ/(N + 1))

Therefore,

(N + 1) = 1

 ck,j =

2 N +1

(A.18)



1/2 sin



kj π N +1

 (A.19)

A.2 Some identities for averages involving harmonic oscillator models Identities and some relationship for harmonic oscillator system are derived here for the following harmonic oscillator Hamiltonian and linear displacement operator:   1 , (A.20) Hˆ ω = ω bˆ † bˆ + 2 Bˆ = ωgω (bˆ + bˆ † ) , (A.21) where gω is an arbitrary real number. In this section, the average of any operator Aˆ is defined by

ˆ −β Hˆ ω T r Ae ˆ =

. (A.22) A T r e−β Hˆ ω

A.2.1 Average of the product of two exponential operators For a harmonic oscillator with angular frequency ω, let us consider the following trace of the product of two exponential operators with unnormalized canonical density operator:

∗ † ˆ (A.23) I (α, γ ) = T r e−αb eγ b e−β Hω ,

201

202

Appendix A Useful mathematical identities and solutions

where α and γ are arbitrary complex numbers, β = 1/(kB T ), and T r represents trace over all the eigenstates of the harmonic oscillator. Expanding the exponential operators with respect to α and γ within the trace of the above expression and taking trace over the eigenstates in (A.23), it can be expressed as I (α, γ ) =

∞ ∞   1 (−αγ ∗ )m k|bˆ m bˆ †m |ke−βω(k+1/2) . m!2

(A.24)

k=0 m=0

Since bˆ †m |k = I (α, γ ) =

√ (k + m)!/k!|k + m, this can be expressed as

∞ ∞   1 (k + m)!  −βω k+1/2 . (−αγ ∗ )m e m! k!m!

m=0

(A.25)

k=0

The sums over k and m in the above expression can be done easily, resulting in the following expression: ∞  1 (−αγ ∗ )m (1 − e−βω )−(m+1) e−βω/2 m! m=0  m ∞ e−βω/2  1 αγ ∗ = − m! 1 − e−βω 1 − e−βω m=0

e−βω/2 αγ ∗ = exp − . 1 − e−βω 1 − e−βω

I (α, γ ) =

ˆ

(A.26)

∗ ˆ†

Then, the thermal average of e−α b eγ b over the canonical density operator of the harmonic oscillator can be expressed as

I (α, γ ) αγ ∗ ˆ ∗ ˆ† e−αb eγ b  = = exp − . (A.27) I (0, 0) 1 − e−βω For the case where α = γ , this becomes the following expression: ˆ

e−αb eα

∗ bˆ †

 = exp −

|α|2 1 − e−βω

.

(A.28)

A.2.2 Displaced harmonic oscillator and polaron transformation For the linear displacement operator given by Eq. (A.21), let us consider the following generator of the displacement operator: ˆ ˆ† θˆ = egω (b−b ) .

(A.29)

Appendix A Useful mathematical identities and solutions

Then, employing the Baker–Campbell–Hausdorff (BCH) identity, it is easy to show that the unitary transformation of the harmonic oscillator Hamiltonian Eq. (A.20) with the above generator of displacement operator becomes ˆ bˆ † b] ˆ θˆ † Hˆ ω θˆ = Hˆ ω + ωgω (bˆ † − b), 1 ˆ [(bˆ † − b), ˆ bˆ † b]] ˆ + ωgω2 [(bˆ † − b), 2 = Hˆ ω − Bˆ + ωgω2 ,

(A.30)

where Bˆ is defined by Eq. (A.21). On the other hand, application of BCH to Bˆ leads to ˆ (bˆ † + b)] ˆ = Bˆ − 2ωgω2 θˆ † Bˆ θˆ = Bˆ + gω2 [(bˆ † − b),

(A.31)

Combining the two, we find that ˆ θˆ = Hˆ ω − λω , θˆ † (Hˆ ω + B)

(A.32)

λω = ωgω2 .

(A.33)

where

Eq. (A.32) can now be used to evaluate the following time correlation function: ˆ

ˆ

ˆ

C(t) = ei Hω t/ e−i(Hω +B)t/  .

(A.34)

Inserting 1 = θˆ θˆ † in the above expression before and after ˆ ˆ ˆ ˆ ˆ ˆ ˆ† ˆ e−i(Hω +B)t/ and noting that θˆ † e−i(Hω +B)t/ θˆ = e−i θ (Hω +B)θt/ , one can express it as follows: ˆ

ˆ

ˆ −i(Hω −λω )t/ θˆ †  = eiλω t θˆ (t)θˆ †  , C(t) = ei Hω t/ θe

(A.35)

ˆ is defined by where Eq. (A.32) has been used and θ(t) ˆ ˆ ˆ −iωt ˆ † iωt θˆ (t) = ei Hω t/ θˆ e−i Hω t/ = egω (be −b e ) .

(A.36)

In the above expression, θˆ (t)θˆ † can be rearranged as follows:  

ˆ −iωt − 1) − bˆ † (eiωt − 1) ˆ θˆ † = e−igω2 sin(ωt) exp gω b(e θ(t)

2 2 ˆ −iωt − 1) = e−igω sin(ωt) egω (1−cos(ωt)) exp gω b(e

(A.37) × exp −gω bˆ † (eiωt − 1) ,

203

204

Appendix A Useful mathematical identities and solutions

where the following identity: ˆ

ˆ

ˆ

ˆ

ˆ ˆ

eA eB = eA+B e[A,B]/2 ,

(A.38)

which is valid for any Aˆ and Bˆ that satisfy the condition of ˆ [A, ˆ B]] ˆ = [B, ˆ [A, ˆ B]] ˆ = 0 has been used repeatedly. Inserting [A, Eq. (A.37) into Eq. (A.35) and employing the identity, Eq. (A.27), the correlation function can be expressed as follows:

(1 − cos(ωt)) 2 2 C(t) = eiλω t/ e−igω sin(ωn t) egω (1−cos(ωt)) exp −2gω2 (1 − e−βω )     βω = eiλω t/ exp −gω2 coth (1 − cos(ωt)) + i sin(ωt) . 2 (A.39) For the calculation of the emission lineshape, it is useful consider the following time correlation function:

ˆ ˆ ˆ ˆ ˆ F (t) = T r ei(Hω +B)t/ e−i Hω t/ e−β(Hω +B)

(A.40)

Inserting θˆ θˆ † in the above expression and employing Eq. (A.32), it can be expressed as

ˆ ˆ ˆ ˆ ˆ F (t) = T r θˆ θˆ † ei(Hω +B)t/ θˆ θˆ † e−i Hω t/ θˆ θˆ † e−β(Hω +B)

ˆ ˆ ˆ ˆ = e−iλω t/ T r ei Hω t/ θˆ † e−i Hω t/ θˆ θˆ † e−β(Hω +B) θˆ

ˆ (A.41) = e−iλω t/ eβλω T r θˆ † (t)θˆ e−β Hω , where ˆ

ˆ

ˆ † eiωt −be ˆ −iωt )

θ † (t) = ei Hω t/ θˆ † e−i Hω t/ = egω (b

.

(A.42)

In Eq. (A.41),  

2 ˆ −iωt − 1) θˆ † (t)θˆ = e−igω sin(ωt) exp gω bˆ † (eiωt − 1) − b(e

2 2 ˆ −iωt − 1) = e−igω sin(ωt) egω (1−cos(ωt)) exp −gω b(e

(A.43) × exp gω bˆ † (eiωt − 1) .

Appendix A Useful mathematical identities and solutions

Inserting Eq. (A.43) into Eq. (A.41) and employing the identity of Eq. (A.26) lead to the following expression: eβ(λω −ω/2) −igω2 sin(ωn t) gω2 (1−cos(ωt)) e e F (t) = e−iλω t/ (1 − e−βω )

(1 − cos(ωt)) × exp −2gω2 (1 − e−βω ) eβ(λω −ω/2) = e−iλω t/ (1 − e−βω )     βω (1 − cos(ωt)) + i sin(ωt) . × exp −gω2 coth 2 (A.44)

205

B Interaction between matter and classical electromagnetic fields This chapter provides a short review of electromagnetic fields [231] and their interaction with matter.

B.1 Maxwell equations The electromagnetic phenomena in vacuum are fully described by specifying the electric field E, the magnetic field B, the charge density ρ, and the flux J. For the case where the fields are in certain media, these correspond only to the “microscopic” fields, and are governed by the following Maxwell equations: ρ , 0 ∂E C2 0 μ 0 = C2 μ 0 J , ∇×B− C1 ∂t ∂B ∇ × E + C3 =0, ∂t ∇·B=0, ∇ · E = C1

(B.1) (B.2) (B.3) (B.4)

where C1 , C2 , and C3 are constant factors depending on the choice of units and can be related to the parameters in the Appendix of Ref. [231], which provide detailed description of unit conversion. The constant factors C1 , C2 , and C3 were introduced in favor of rationalized units such as SI units. Namely, in rationalized units, C1 = C2 = C3 = 1, and 0 μ0 = 1/c2 , where c is the speed of light. In Gaussian units, 0 = μ0 = 1, C1 = 4π , C2 = 4π/c, and C3 = 1/c. In Heaviside–Lorentz units, 0 = μ0 = C1 = 1, and C2 = C3 = 1/c. In a medium, there are virtually infinite number of sources for charge and current, atoms and molecules. Thus, the electromagnetic fields in its full microscopic picture are complicated. Still, “macroscopic” fields can be defined, which result from smoothening (averaging) the “microscopic” electric and magnetic fields over length scales larger than those of atoms and molecules. One thus can define “macroscopic” electric field and magnetic induction, E and B. In addition, one needs to define “macroscopic” electric Dynamics of Molecular Excitons. https://doi.org/10.1016/B978-0-08-102335-8.00019-3 Copyright © 2020 Elsevier Ltd. All rights reserved.

207

208

Appendix B Interaction between matter and classical electromagnetic fields

displacement D and magnetic field H. These four fields satisfy the following Maxwell equations: ∇ · D = C1 ρ , C2 ∂D ∇×H− = C2 J , C1 ∂t ∂B ∇ × E + C3 =0, ∂t ∇·B=0,

(B.5) (B.6) (B.7) (B.8)

where ρ and J are now macroscopic charge density and current density. D and H are defined by D = 0 E + C1 P 1 H= B − C1 M μ0

(B.9) (B.10)

where P is the electric polarization and M is the magnetization of the medium, which is equal to the magnetic induction minus the magnetization of the medium. There are general relations called constitutive relations, which relate D and H to E and B. The detailed constitutive relations depend on the nature of the materials. The simplest are the following linear constitutive relations: D = r 0 E = E ,

(B.11)

B = μr μ0 H = μH ,

(B.12)

where r is the dielectric constant, μr is magnetic permeability, which are dimensionless, and  = 0 r and μ = μr μ0 . Note that  = r and μ = μr for Gaussian and Heaviside–Lorentz units. Inserting these into Eqs. (B.5) and (B.6), one can obtain C1 ρ,  C2 ∂E = μC2 J . ∇ × B − μ C1 ∂t

∇·E=

(B.13) (B.14)

Eqs. (B.13) and (B.14), along with Eqs. (B.7) and (B.8), are the most commonly used equations for macroscopic electric and magnetic fields.

B.1.1 Vector and scalar potentials The Maxwell equations involve six unknowns, three components of the electric field and three components of the magnetic

Appendix B Interaction between matter and classical electromagnetic fields

field. The number of unknown functions can be reduced by introducing vector and scalar potentials, which in total involve four unknown functions. First, one can introduce a vector potential such that B=∇×A.

(B.15)

This definition automatically satisfies Eq. (B.8) because curl (∇×) of any vector has zero divergence. Second, one can define a scalar potential  such that E + C3

∂A = −∇ . ∂t

(B.16)

This definition satisfies Eq. (B.7) because curl of any gradient is equal to zero, and ∇ × C3

∂A ∂ ∂B = C3 (∇ × A) = C3 , ∂t ∂t ∂t

(B.17)

where Eq. (B.15) has been used. Inserting Eq. (B.16) into Eq. (B.13), one can find that −

C1 1 ∂ ρ. (∇ · A) − ∇ 2  = c ∂t 

(B.18)

Inserting Eqs. (B.15) and (B.16) into Eq. (B.14), one can find that   ∂A C2 ∂ 2 (B.19) −C3 − ∇ = μC2 J , ∇ (∇ · A) − ∇ A − μ C1 ∂t ∂t where the following identity has been used: ∇ × (∇ × A) = ∇ (∇ · A) − ∇ 2 A .

(B.20)

Eqs. (B.18) and (B.19) can be simplified further by noticing that there is a great degree of freedom in choosing the vector and scalar potentials. Assume the following replacement: A = A + ∇f , ∂f  =  − C3 . ∂t

(B.21) (B.22)

Then, because the curl of a gradient is zero, one can show that A and  serve as another set of potentials that produce the same electric and magnetic fields. That is, B = ∇ × A , E + C3

∂A ∂t

= −∇ .

(B.23) (B.24)

209

210

Appendix B Interaction between matter and classical electromagnetic fields

Eqs. (B.21) and (B.22) represent a well-known gauge transformation, which provides ways to choose appropriate vector and scalar potentials that are easy to work with while not changing the electric and magnetic fields. One of the most widely used gauge is the Coulomb gauge, which is characterized by the following condition: ∇·A=0.

(B.25)

The gauge transformation, Eqs. (B.21) and (B.22), makes it always possible to find a vector potential satisfying the above property. Let us assume that we happen to know a pair of A and  corresponding to given electric and magnetic fields, but that ∇ · A = 0. Then, we can choose a function f in Eq. (B.21) such that ∇ · A = ∇ 2 f . The resulting A satisfies the Coulomb gauge condition, Eq. (B.25). Inserting Eq. (B.25) into Eqs. (B.18) and (B.19), one can find that C1 ρ,  C 2 C3 ∂ 2 A C2 ∂ − ∇ 2 A + μ + μ ∇ = μC2 J . C1 ∂t 2 C1 ∂t

− ∇ 2 =

(B.26) (B.27)

Solution of these equations and the use of definitions, Eqs. (B.15) and (B.16), provide the electric and magnetic fields for a given set of charge and current densities.

B.1.2 Electromagnetic fields in source-free space Assume that there are no sources of charge and current. Then, ρ = 0 and J = 0. Under this situation, Eq. (B.26) becomes ∇ 2 = 0 .

(B.28)

A solution of this, which satisfies the boundary condition of the infinite space, is that  = 0. That is, in the source-free space, the scalar potential can be assumed to be zero (in the Coulomb gauge). With this assumption, Eq. (B.27) now simplifies to ∇ 2A −

r μr ∂ 2 A =0, c2 ∂t 2

(B.29)

where the fact that 0 μ0 C2 C3 /C1 = 1/c2 in all units has been used. Many solutions are possible for this partial differential equation. One of the simplest solutions is the following plane wave solution: A(r, t) = A0 e cos(k · r − ωt) =

 A0  −iωt ik·r e + eiωt e−ik·r , (B.30) e e 2

Appendix B Interaction between matter and classical electromagnetic fields

where e is the unit vector along the direction of the electric field and k is the wave vector of the radiation with its magnitude k (this should not be confused with the integer summation index for denoting site excitation states) satisfying the following condition: k 2 = |k|2 =

 r μr 2 ω . c2

(B.31)

Assuming that both r and μr are real and positive (nonabsorptive medium), the refractive index is defined as √ n r = r μ r . (B.32) Thus, in a medium with refractive index nr , the plane wave travels with a speed v = ω/k = c/nr . Inserting Eq. (B.30) into the condition Eq. (B.25), one can find that k·e=0.

(B.33)

That is, e is perpendicular to the direction of the propagation of the vector potential. Inserting Eq. (B.30) into Eq. (B.15), B(r, t) = A0 (k × e) sin(ωt − k · r) = A0 ke sin(ωt − k · r) ,

(B.34)

where e = k × e/k. Inserting Eq. (B.30) into Eq. (B.24) with  = 0, E(r, t) = C3 A0 ωe sin(ωt − k · r) = C3

A0 ck e sin(ωt − k · r) . (B.35) nr

The normalization constant A0 can be determined by calculating the energy density corresponding to the electromagnetic waves given above. According to the Poynting’s theorem [231], the energy density of the electromagnetic field is given by Ef =

1 (E · D + B · H) . 2C1

(B.36)

Assuming the linear constitutive relationship, Eqs. (B.11) and (B.12), this can be expressed as   1 1 |E|2 + |B|2 . (B.37) Ef = 2C1 μ Employing Eqs. (B.34) and (B.35), the above expression for a single plane wave can be shown to be Ef =

(0 C32 c2 + 1/μ0 ) A20 k 2 2 sin (ωt − k · r) . C1 2μr

(B.38)

211

212

Appendix B Interaction between matter and classical electromagnetic fields

We can use this expression to determine the amplitude for the minimum quantum of the energy density, ω/V, where V is the volume of the space. Since the time average of sin2 (ωt − k · r) is equal to 1/2, this means that (0 C32 c2 + 1/μ0 ) A20 k 2 ω = . C1 4μr V

(B.39)

From the above expression and using Eqs. (B.31) and (B.32), one can obtain the following expression for the minimum quantum of the energy density:  A0 =

4C1 μr ω 2 (0 C3 c2 + 1/μ0 )k 2 V

1/2

 =

2C1 μ0 nr c r kV

1/2 .

(B.40)

In the above expression, the fact that 0 μ0 C32 c2 = 1 in all units has been used. Now, using this value and the fact that the vector potential can in general be expressed as the sum of many modes with different wave vector k and polarization λ, it can be expressed as follows: A(r, t) =



 ek,λ

k,λ

C1 μ0 nr c 2r kV

1/2   e−iωk t eik·r + eiωk t e−ik·r , (B.41)

where ωk = ck/nr . Note that C1 μ0 = 1/(0 c2 ) in the rationalized units and C1 μ0 = 4π in the Gaussian units. Thus, for nr = r = 1, the above expression reduces to the familiar expression in the vacuum. Accordingly, the electric and magnetic fields can also be expressed as

E(r, t) = i



 ek,λ

k,λ

C1 C32 μ0 c3 k 2r nr V 

B(r, t) = i

1/2

  e−iωk t eik·r − eiωk t e−ik·r ,

 C1 μ0 nr ck (uk × ek,λ ) 2r V

1/2 

(B.42)  e−iωk t eik·r − eiωk t e−ik·r ,

k,λ

(B.43) where uk = k/k.

Appendix B Interaction between matter and classical electromagnetic fields

B.2 Classical Hamiltonian for matter and radiation interaction Consider a classical particle of charge q moving with velocity v in the presence of electric and magnetic fields, E and B. Then, the force on the particle will be   dr d 2r (B.44) m 2 = q E(r, t) + C3 × B(r, t) − ∇V (r) , dt dt where V (r) is the external potential independent of the electromagnetic fields. The Hamiltonian corresponding to this force can be shown to be H=

1 (p − C3 qA(r, t))2 + q(r, t) + V (r) . 2m

(B.45)

The validity of this Hamiltonian can be confirmed as detailed below. In showing this, it is useful to note the following vector identity [231]: ∇(a · b) = (a · ∇)b + (b · ∇)a + a × (∇ × b) + b × (∇ × a) . (B.46) First, taking gradient of the Hamiltonian with respect to the momentum,

1 dr = ∇p H = (p − C3 qA(r, t)) · ∇ p (p − C3 qA(r, t)) dt m

1 + (p − C3 qA(r, t)) × ∇ p × (p − C3 qA(r, t)) m 1 = (p − C3 qA(r, t)) , (B.47) m where ∇ p is gradient with respect to p = (px , py , pz ). The fact that ∇ p p = p and ∇ p × p = 0 has been used in obtaining the second equality of the above equation. Second, taking gradient with respect to the position, 1 dp = −∇H = − {(p − C3 qA(r, t)) · ∇} (p − C3 qA(r, t)) dt m 1 − (p − C3 qA(r, t)) × {∇ × (p − C3 qA(r, t))} m − q∇(r, t) − ∇V (r)   dr dr · ∇ A(r, t) + C3 q × (∇ × A(r, t)) = C3 q dt dt − q∇(r, t) − ∇V (r) , (B.48)

213

214

Appendix B Interaction between matter and classical electromagnetic fields

where Eq. (B.47) has been used in obtaining the second equality. Taking time derivative of Eq. (B.47), m

d 2 r dp A(r, t) = − C3 q , dt dt 2  dt dr ∂A(r, t) dp = − C3 q · ∇ A(r, t) − C3 q . dt dt ∂t

(B.49)

Inserting Eq. (B.48) into Eq. (B.49), we find that  ∂A(r, t) dr d 2r − ∇(r, t) + C3 q × {∇ × A(r, t)} m 2 = q −C3 ∂t dt dt − ∇V (r) . (B.50) The above equation can be shown to be equivalent to Eq. (B.44) by using that relationship between potentials and fields, Eqs. (B.15) and (B.16). For materials consisting of N charged particles, Eq. (B.44) can easily be generalized to the following form: H=

N   2 1 ˆ j , t) + V (r1 , · · · , rˆ N ) . pj − C3 qj A(rj , t) + qj (r 2mj j =1

(B.51)

B.3 Quantum mechanical Hamiltonian for matter-radiation interaction in the weak field limit with Coulomb gauge For quantum materials interacting with classical electromagnetic field, the corresponding Hamiltonian operator can be obtained by quantizing the material parts in Eq. (B.45) as follows: Hˆ =

N  2  1  ˆ rˆ j , t) + qj ( ˆ rˆ j , t) + Vˆ (rˆ 1 , · · · , rˆ N ) , ˆpj − C3 qj A( 2mj j =1

(B.52) where pˆ and rˆ are momentum and position operators. Expanding the square in Eq. (B.52),

2 pˆ j − C3 qj A(rˆ j , t) = pˆ 2j − C3 qj pˆ j · A(rˆ j , t) − C3 qj A(rˆ j , t) · pˆ j + C32 qj2 A(rˆ j , t)2 .

(B.53)

Appendix B Interaction between matter and classical electromagnetic fields

Employing the following identity:  pˆ j · A(rˆ j , t) = A(rˆ j , t) · pˆ j + ∇j · A(rˆ j , t) , i

(B.54)

Eq. (B.52) can be reexpressed as Hˆ =

N p  ˆ 2j j =1

+

2mj

+ Vˆ (rˆ 1 , · · · , rˆ N )

N  

ˆ rˆ j , t) − C3 qj (

j =1

+

C32

qj2 2mj

qj ˆ qj ˆ rˆ j , t) A(rˆ j , t) · pˆ j − C3 ∇ · A( mj i

 ˆ rˆ j , t) A(

2

(B.55)

.

Employing the fact that ∇ · A(rˆ , t) = 0 in the Coulomb gauge, and assuming weak enough field strength, for which the quadratic term A(rˆ , t)2 can be neglected, one can simplify Eq. (B.52) as follows: Hˆ (t) = Hˆ 0 +

N  

qj (rˆ j , t) − C3

j =1

qj ˆ A(rˆ j , t) · pˆ j mj

,

(B.56)

where Hˆ 0 is the Hamiltonian in the absence of the electromagnetic radiation and is given by Hˆ 0 =

N p  ˆ 2j j =1

2mj

+ Vˆ (rˆ 1 , · · · , rˆ N ) .

(B.57)

B.4 Interaction with a plane wave radiation and dipole approximation Let us consider the case where the particle is subject to electromagnetic radiation represented as a sum of plane waves and that (rˆ , t) = 0. Then, inserting Eq. (B.41) into Eq. (B.56), one can find the following matter-radiation interaction Hamiltonian: Hˆ (t) = Hˆ 0 + Hˆ 1 (t) ,

(B.58)

215

216

Appendix B Interaction between matter and classical electromagnetic fields

where 1/2  N   C1 C32 μ0 nr c qj Hˆ 1 (t) = − 2r kV mj j =1 k,λ   × e−iωk t eik·ˆrj + eiωk t e−ik·ˆrj ek,λ · pˆ j .

(B.59)

Given that the wavelength of the radiation is much larger than the molecular length scale, one can make an approximation that k · (rj − r0 ) ≈ 0, where r0 is the reference position vector of the entire material, for which the center of mass coordinate can be used. This is known as the dipole approximation for a reason to be clear below. Then, the matter radiation interaction can be approximated as 1/2  N    C1 C32 μ0 nr c qj  −iωk t ik·r0 Hˆ 1 (t) ≈ − e + eiωk t e−ik·r0 e 2r kV mj j =1 k,λ

× ek,λ · pˆ j ,

(B.60)

where the momentum operator pˆ j can be expressed in terms of the position operator employing the following relation: pˆ j =

 imj  Hˆ 0 , rˆ j . 

(B.61)

Then, Eq. (B.60) can be expressed as Hˆ 1 (t) = −i

1/2   1  C1 C 2 μ0 nr c 3 e−iωk t eik·r0 + eiωk t e−ik·r0 2r kV  k,λ

ˆ , × [Hˆ 0 , ek,λ · μ]

(B.62)



qj rˆ j , the dipole operator. Let us introduce the eigenstates of Hˆ 0 as |En ’s and insert the identity operator 1ˆ =  |E E | in both sides of the commutator of the above equation. n n n Then,

ˆ = where μ

j

1/2      C1 C 2 μ0 nr c 3 Hˆ 1 (t) = −i e−iωk t eik·r0 + eiωk t e−ik·r0 2r kV  n,n k,λ

×

(En − En ) ˆ n En | . |En En |ek,λ · μ|E 

(B.63)

Appendix B Interaction between matter and classical electromagnetic fields

Using the condition of energy conservation En − En = ±ωk for e∓iωk t , this can be converted into the following form:  1/2     C1 C 2 μ0 c3 k 3 e−iωk t eik·r0 − eiωk t e−ik·r0 Hˆ 1 (t) = −i 2r nr V  n,n k,λ

ˆ n En | × |En En |ek,λ · μ|E ˆ . = −E(r0 , t) · μ

(B.64)

This is the interaction Hamiltonian between electric field with the dipole operator of the material. In the above expression, C1 C32 μ0 c3 = 1/0 in the rationalized unit and C1 C32 μ0 c3 = 4πc in the Gaussian unit.

217

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Index

223

Index A Absorption intensity, 54 lineshape, 57, 60, 61, 66, 70, 74, 76, 79, 80, 89, 99, 152, 159 measurement, 156 process, 56, 155 rate, 155, 170 Acceptor bath degrees, 89 bath modes, 91 chromophores, 98 lineshape functions, 93 moieties, 91 population, 86 Adiabatic electronic operator, 25 Adiabatic electronic states, 27, 51 Adiabatic states, 28, 32, 42 chromophore, 41, 85 Aggregates, 3, 11, 40, 41, 45, 68, 145, 151, 192 chromophore, 23, 35, 45, 50, 68 molecule, 4

B Bacteriochlorophylls (BChls), 183, 184 Bath coordinates, 48 correlation, 72, 125 correlation function, 70, 71, 124, 137 coupling, 68 degrees, 48–50, 56, 62, 65, 66, 69, 76, 80, 85, 87, 88, 91, 93, 94, 115, 146, 154, 158, 164 density operator, 88, 113 dynamics, 87, 108 Hamiltonian, 48, 55, 68, 79, 87, 94, 98, 112, 121, 152, 162, 178

Markovian, 108, 146 modes, 22, 93 operators, 92, 115, 119, 158 parts, 47 quantum, 146 relaxation, 68 spectral density, 49, 50, 68, 71, 77, 141, 183, 184 time correlation, 146

C Chlorophyll molecules, 182 Chromophore adiabatic states, 41, 85 aggregates, 23, 35, 45, 50, 68 exciton states, 46, 47 ground electronic states, 37, 39 Hamiltonian, 45 linear chain, 77 nuclear coordinates, 47, 56 units, 24, 35, 45, 68 Completely positive (CP), 146 Crude adiabatic approximation, 21, 22, 41, 47, 51, 56, 147, 195, 197 Crystalline environments, 15, 105, 159 Cyclic invariance, 60, 62, 86, 96

D Degrees bath, 48–50, 56, 62, 65, 66, 69, 76, 80, 85, 87, 88, 91, 93, 94, 115, 146, 154, 158, 164 electronic, 23, 30, 32, 47 environmental, 46, 47, 53, 68, 107, 108, 162 nuclear, 26, 27, 40, 42, 43, 45–48, 98, 147, 162 radiation, 153

Delocalization, 5, 16, 17, 70 exciton, 195 length, 17, 18 Delocalized electrons, 3, 5 exciton, 51, 178, 184, 188 exciton states, 6 Density operator, 32, 57, 72, 76, 85, 95, 105, 107, 110, 116, 123, 124, 136, 164, 165, 171 bath, 88, 113 exciton, 145 Dilute exciton density, 101 Donor bath, 92 chromophores, 95, 97, 98, 100 exciton, 181 Double exciton, 163, 176, 177 states, 51, 163, 166, 167, 172 subspace, 176 Dye molecules, 188 linear aggregates, 78 Dynamics bath, 87, 108 calculations, 197 exciton, 2, 20, 22, 26, 51, 53, 84, 91, 102–104, 107, 108, 133, 145, 159, 161, 162, 178, 179, 183, 184, 188, 195, 196 exciton transfer, 143 quantum, 20, 32, 51, 84, 85, 108, 145, 147, 161, 179, 183, 195, 196

E Electromagnetic radiation, 56 Electronic basis, 33 correlation effects, 51 coupling, 52, 77, 84, 91, 93, 94, 183–185, 192 coupling Hamiltonian, 94 degrees, 23, 30, 32, 47

224

Index

energy, 26 excitations, 23, 188 exciton subspace, 61, 73 Hamiltonian, 4 resonance interactions, 35, 44, 48 single exciton, 185 spectroscopy, 53, 54, 161, 192 states, 5, 21, 25, 32–35, 38, 39, 41, 44, 51, 53, 85, 147, 195 structure, 35, 45, 51 transition, 35, 54 transition densities, 33 wavefunction, 29 Electrons, delocalized, 3, 5 Emission lineshape, 75, 76, 79 exciton, 80 expression, 76, 90 Energetic disorder, 103, 104, 182 Environmental decoherence, 32 degrees, 46, 47, 53, 68, 107, 108, 162 effects, 22, 47 Hamiltonian, 46 molecules, 49 Equilibrium bath density operator, 91 Exciton abound, 2 amount, 6 band, 5, 185, 186 band width, 8 basis, 5, 60–62, 67–69, 76, 80, 178 bath dynamics, 97 binding energies, 5, 186, 187 centers, 101, 102, 104 creation, 79 delocalization, 195 density, 105 density operator, 145 diffusion constant, 103, 104 donor, 181 dynamics, 2, 20, 22, 26, 51, 53, 84, 91, 102–104, 107, 108, 133, 145, 159, 161, 162, 178, 179, 183, 184, 188, 195, 196 emission lineshape, 80

energy, 7, 8, 11, 53, 54 energy level, 9, 14 enhanced exciton mobility, 187 for photovoltaic devices, 181, 186, 192, 196 Hamiltonian, 11, 12, 16, 21, 22, 32, 36, 45, 58, 59, 77, 80, 94, 98, 112, 115, 158, 182, 183, 185, 192 hopping dynamics, 83, 84, 101, 105, 106, 195 in LHCs, 192 interaction, 107, 149 interfacial, 16 levels, 187 manifold, 53, 59, 66, 80 models, 18, 26 operators, 159 populations, 83, 101, 145 processes, 19 quantum dynamics, 192 rate, 85 single, 7, 55, 84, 163, 165, 166, 176, 179, 183 space electronic Hamiltonian, 141 states, 3, 5, 8–11, 51, 54, 55, 60–62, 84, 93, 151, 161, 163, 165, 166 subspace, 66, 119, 132 subspace electronic Hamiltonian, 134 temporal dynamics, 2 transfer, 54, 86, 87, 102, 184, 185, 188 dynamics, 143 inelastic, 91 mechanism, 85 processes, 83, 105 rate, 83, 84, 93, 96, 102, 104, 105, 195 Excitonic polaron, 133 properties, 2 state, 133 systems, 181 transition energies, 54

F Fenna–Matthews–Olson (FMO) complex, 147, 182–184, 192 Fermi’s golden rule (FGR) rate, 92, 155, 156 rate expression, 101 Förster radius, 90, 102 Förster theory, 84 Förster’s resonance energy transfer (FRET), 84, 87, 101, 105, 159, 181, 189, 190 efficiency, 188–190 efficiency measurement, 188 process, 191 radius, 191 rate, 90, 102, 103, 105, 156, 157, 188, 190, 192, 195 technique, 182, 188, 192 theory, 87 Four wave mixing spectroscopy (FWMS), 171, 179, 196 Frenkel exciton, 3–6, 16–18, 21, 22, 35, 186, 195, 197 Hamiltonian, 44

G Gaussian units, 150, 151, 171 Generalized master equation (GME), 104 Ground electronic state, 4, 35, 38, 39, 41–43, 57, 59, 72, 85, 91, 95, 138, 162–166 chromophore, 37

H Hamiltonian bath, 48, 55, 68, 79, 87, 94, 98, 112, 121, 152, 162, 178 chromophore, 45 electronic, 4 environmental, 46 exciton, 11, 12, 16, 21, 22, 32, 36, 45, 58, 59, 77, 80, 94, 98, 112, 115, 158, 182, 183 interactions, 23, 162 model, 159 nuclear, 43 operator, 31 radiation, 152

Index

single exciton, 7, 35 zeroth order, 95, 115, 135 Harmonic oscillator bath, 48–50, 57, 68, 74, 80, 106, 108, 109, 132, 141, 145, 195–197 Hermitian conjugate, 86, 95–97, 119, 124–126, 131, 134, 153, 165, 166, 173 Hierarchical equations of motion (HEOM), 81, 108 Hopping dynamics, 83, 101, 141–143, 145 exciton, 83, 84, 101, 105, 106, 195

I Inelastic exciton transfer, 157 Inorganic solar cells (ISC), 186 Interactions Hamiltonian, 23, 46, 162 interchromophore, 44, 48 nonadiabatic, 27 potential term, 24 term, 46 Interchromophore interaction Hamiltonian, 37 interactions, 44, 48 nonadiabatic, 51, 195 Intrachromophore component, 36 nonadiabatic, 51, 195 Inverse participation ratio (IPR), 17

L Light harvesting complex (LHC), 182, 196 Lindblad equation, 145, 146 Linear chain, 6–8, 13, 17 chromophore, 77 Lineshape absorption, 57, 61, 66, 70, 74, 76, 79, 80, 89, 99, 152, 159 expression, 57, 60, 62, 66, 68, 69, 71, 75, 77–79 function, 89, 99 Lorentzian, 54 spectral, 54

stimulated emission, 72, 89, 90, 100, 159 theory, 54, 55, 109, 162

M Microscopic Hamiltonian, 145 molecular Hamiltonian, 19, 50, 55 Molar extinction coefficient, 91, 150, 152, 156, 159 Molecule aggregates, 4 conformational dynamics, 53 forming transition dipoles, 12 Multichromophores, 93, 101 Multichromophoric donor, 105 FRET, 105

N Nearest neighbor chromophores, 77 electronic couplings, 184 interactions, 6 resonance interaction, 7 Nonadiabatic component, 37 effects, 32, 51 interactions, 27 interchromophore, 51, 195 term, 33, 42 Nonlinear spectroscopy, 51, 161, 162, 178, 179, 196 Nuclear charges, 33 coordinates, 25, 41, 152 coordinates for chromophore, 25 degrees, 26, 27, 40, 42, 43, 45–48, 98, 147, 162 Hamiltonian, 43 position, 36 position state, 25 potential energy, 26, 45 spin dynamics, 108 spin relaxation, 108 state, 30 wave function, 27, 29

225

O Optoelectronic devices, 181 Organic aggregates, 54 Organic molecules, 186 Organic solar cells (OSC), 186

P Pauli master equation (PME), 101 Phenomenological lineshape function, 62 Photon absorption, 79, 155 Photovoltaic devices, 186 Pigment molecules, 23, 182, 185 Polaritons, 150, 157–159, 196 Polarization operator, 176–178 Polaron transformed QME (PQME), 133, 143, 145 dynamics, 145 Population dynamics, 143, 179, 183 excitons, 104 Projection operator (PO), 109 Protein environments, 183 Purple bacteria, 76, 182, 183

Q Quantized electromagnetic field, 150, 159, 196 Quantum bath, 146 dynamics, 20, 32, 51, 84, 85, 108, 145, 147, 161, 179, 183, 195, 196 electrodynamics, 3 Quantum master equation (QME), 62, 108 approaches, 105, 108, 109, 145, 196 lineshape, 70

R Radiation degrees, 153 field, 56 Hamiltonian, 152 Resonance interactions, 4 electronic, 35, 44, 48

226

Index

S Single chromophores, 101 chromophoric exciton transfer, 96, 99, 104 exciton, 7, 55, 84, 163, 165, 166, 176, 179, 183 band, 184 electronic, 185 Hamiltonian, 7, 35 states, 4, 10, 36, 38–40, 44, 48, 51, 84, 151, 162, 163, 165–167, 172, 176–178 subspace, 163, 165, 183 Spectral density, 49, 50, 69–71, 75–77, 141 lineshape, 54

overlap, 83, 91, 93, 150, 152, 156, 157, 159, 183, 185, 196 Spectroscopy, 20, 26, 50, 161, 162 electronic, 53, 54, 161, 192 Spontaneous emission, 56, 91, 150, 152, 153, 155, 157, 159 Static disorder, 16, 17 Stimulated emission, 72, 73, 100, 152, 166, 175, 179, 195 lineshape, 72, 89, 90, 100, 159 rate, 169 spectroscopic lineshapes, 79 Stochastic Liouville equation (SLE), 145 Symmetric Hamiltonian, 16

T Time nonlocal equation, 114, 117, 119, 121, 124

Transition dipole, 8–11, 56, 59, 62, 67, 77, 83, 158, 163, 188 interaction, 8, 11, 78, 90, 93, 157, 188 moment, 8 Triplet exciton dynamics, 108

W Wannier exciton, 3, 5, 15–18, 186, 195, 197 delocalization, 18 Weak interactions, 3, 4

Z Zeroth order approximation, 42 Hamiltonian, 95, 115, 135 term, 114