Theoretical and Computational Photochemistry: Fundamentals, Methods, Applications and Synergy with Experimental Approaches 9780323917384

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Theoretical and Computational Photochemistry

Theoretical and Computational Photochemistry

Fundamentals, Methods, Applications and Synergy with Experimental Approaches

Edited by Cristina Garcı´a-Iriepa Universidad de Alcala´, Departamento de Quı´mica Analı´tica, Quı´mica Fı´sica e Ingenierı´a Quı´mica, Alcala´ de Henares, Madrid, Spain

Marco Marazzi Universidad de Alcala´, Departamento de Quı´mica Analı´tica, Quı´mica Fı´sica e Ingenierı´a Quı´mica, Alcala´ de Henares, Madrid, Spain

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 © 2023 Elsevier Inc. 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. ISBN: 978-0-323-91738-4 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals

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Contents Contributors Preface

xi xiii

Part I Fundamentals 1. Introduction to molecular photophysics

3

Alejandro Jodra, Luis Manuel Frutos, Cristina Garcı´a-Iriepa, and Marco Marazzi 1.1

Interaction between electromagnetic radiation and molecules 1.1.1 Electromagnetic radiation 1.1.2 Time-dependent perturbation theory: A tool for describing the matter-radiation interaction 1.1.3 Electric dipole transitions 1.1.4 Spontaneous emission 1.2 Quantization of energy 1.3 The Franck-Condon principle 1.4 Electronic absorption spectra 1.4.1 Transition energy 1.4.2 Transition intensity: The oscillator strength and the transition dipole moment 1.4.3 Absorption band shape: The dynamic effect and the vibronic coupling 1.4.4 Multiphotonic absorption 1.5 Fluorescence and phosphorescence emission 1.5.1 Fluorescence 1.5.2 Phosphorescence References

2. Theoretical grounds in molecular photochemistry

3 3

4 6 8 10 16 18 18

19

The Jablonski diagram

Potential energy surfaces and reaction paths 2.3 The Born-Oppenheimer approximation in detail: Adiabatic and diabatic representations 2.3.1 Separation of nuclei and electrons motion 2.3.2 Adiabatic representation 2.3.3 Diabatic representation 2.4 When potential energy surfaces do cross: Avoided crossings and conical intersections 2.5 Excited state molecular dynamics References

21 25 28 29 40 43

51

51

53

54 54 57 58

59 62 65

Part II Methods 3. Density-functional theory for electronic excited states

69

John M. Herbert 3.1 3.2

Alejandro Jodra, Cristina Garcı´a-Iriepa, and Marco Marazzi 2.1

2.2

Overview Linear-response (“time-dependent”) DFT 3.2.1 Theoretical formalism 3.2.2 Performance and practice 3.2.3 Systemic problems 3.3 Excited-state Kohn-Sham theory: The DSCF approach 3.3.1 Theory 3.3.2 Examples 3.4 Time-dependent Kohn-Sham theory: “Real-time” TDDFT 3.4.1 Theory 3.4.2 Examples Acknowledgment References

69 71 71 77 86 91 92 95 99 99 101 103 103

v

vi

Contents

4. Algebraic diagrammatic construction schemes for the simulation of electronic spectroscopies

6. Machine learning methods in photochemistry and photophysics 119

Andreas Dreuw and Adrian L. Dempwolff 4.1 4.2

Introduction Theoretical background 4.2.1 Intermediate-state representation 4.2.2 State properties and geometries 4.2.3 The physical meaning of ISR basis states and EE-ADC matrix elements 4.2.4 Relation of different ADC schemes 4.3 Comparison of ADC to configuration interaction and coupled cluster methods 4.4 ADC variants for excited states 4.4.1 Semiempirical EE-ADC schemes 4.4.2 EE-ADC schemes for multireference systems 4.4.3 EE-ADC methods for X-ray spectroscopies 4.5 Computational spectroscopy in complex environments with ADC 4.6 Computational photochemistry with ADC 4.7 Outlook and concluding remarks References

5. Multiconfigurational quantum chemistry: The CASPT2 method

119 120 121 122

123 124

124 125 126 126 127 129 130 131 131

135

Stefano Battaglia, Ignacio Fdez. Galva´n, and Roland Lindh 5.1 5.2 5.3

Introduction Prelude: CASSCF CASPT2 theory 5.3.1 CASPT2 fundamentals 5.3.2 The intruder state problem 5.3.3 Shift techniques 5.3.4 Alternative selection of the zeroth-order Hamiltonian 5.4 Multistate CASPT2 theory 5.5 Performance 5.6 Future developments 5.7 Summary and conclusions Acknowledgments References

163

Jingbai Li, Morgane Vacher, Pavlo O. Dral, and Steven A. Lopez 6.1 6.2

Introduction Machine learning models 6.2.1 Machine learning tasks 6.2.2 k-Nearest neighbor 6.2.3 Support vector machine 6.2.4 Kernel methods 6.2.5 Neural networks 6.3 Representations of molecules 6.3.1 Molecular strings and fingerprints 6.3.2 Molecular descriptors 6.3.3 Automatically generated descriptors 6.4 Training data for machine learning 6.4.1 Excited-state database across chemical space 6.4.2 Molecule-specific data generation 6.5 Applications of machine learning in photochemistry and photophysics 6.5.1 Machine learning-assisted highthroughput virtual screening 6.5.2 Machine learning-predicted electronic spectroscopy 6.5.3 Machine learning nonadiabatic molecular dynamics 6.5.4 Machine learning-extracted chemical insights from data 6.6 Summary References

7. Polaritonic chemistry

163 164 164 164 165 168 171 173 173 174 175 175 176 176 178 178 180 180 183 184 184

191

Jacopo Fregoni and Stefano Corni 135 136 138 138 142 144 145 153 155 157 159 159 159

7.1

Preliminary considerations on the electromagnetic field 7.2 Polaritonic eigenvalues and eigenstates 7.2.1 Cavity Born-Oppenheimer and vibrational strong coupling 7.2.2 Electronic strong coupling (ESC) 7.3 Polaritonic potential energy surfaces (PoPESs) 7.3.1 A didactical case: Azobenzene polaritonic potential energy surfaces (PoPESs) 7.3.2 Many molecules and dark states

193 194 195 197 199

199 202

Contents vii

7.4

Polariton dynamics and cavity losses 7.4.1 Nuclear dynamics in polaritonic systems: Full quantum vs semiclassical 7.5 Summary References

9.2 204

204 207 207

Part III Applications 8. First-principles modeling of dye-sensitized solar cells: From the optical properties of standalone dyes to the charge separation at 215 dye/TiO2 interfaces Valentin Diez-Cabanes, Simona Fantacci, and Mariachiara Pastore Introduction 215 Computational modeling of DSSCs: Methods, limitations, and practical strategies 219 8.2.1 Generalities 219 8.2.2 Electronic structure and optical properties of dyes in solution 220 8.2.3 Electronic structure and optical properties of semiconductor materials and dye-sensitized interfaces 222 8.2.4 Machine learning and semiempirical methods applied to DSSCs 223 8.3 Design rules for Ru(II) sensitizers: The role of spin-orbit coupling (SOC) 223 8.4 Modeling the photophysics of Fe(II) metal complexes: Tools and findings 226 8.5 Interfacial properties of Fe-NHC229 sensitized TiO2 8.6 Conclusions 234 References 234

8.1 8.2

9. Solar cells: Organic photovoltaic solar cells

247

Daniele Fazzi 9.1

Introduction 9.1.1 Organic photovoltaics 9.1.2 OPV materials 9.1.3 Models to describe charge generation in OSCs

247 247 248 249

Excitonic processes: Excited states at the donor/acceptor interfaces 251 9.2.1 Electronic structure methods to describe the excited states at D/A OPV interfaces 251 9.2.2 Analytical tools to characterize the excited-state wavefunction 253 9.2.3 Examples of polymer/fullerene OPV interfaces 253 9.3 Time-dependent processes: Excited-state dynamics in donor and donor/acceptor domains 256 9.3.1 Excited-state dynamics: Brief overview of nonadiabatic surface hopping and multiconfiguration time-dependent Hartree methods in the context of OPV 256 9.3.2 Excited-state dynamics of oligothiophenes as prototypes for P3HT 257 9.3.3 Examples of polythiophene-/ fullerene-based interfaces 259 9.4 Conclusions 260 References 261

10. Perovskite-based solar cells

265

Waldemar Kaiser and Edoardo Mosconi Introduction 265 First-principles modeling of perovskites 266 10.3 Point defects in perovskites 269 10.3.1 First-principles modeling of point defects 270 10.3.2 Ion migration in perovskite 270 10.3.3 Photochemistry of iodine Frenkel defects 272 10.4 Interfaces in perovskite solar cells 273 10.4.1 Understanding charge extraction at the perovskite/ETL interface 273 10.4.2 Chemical tuning of the perovskite/HTL interface 276 10.5 Degradation and passivation of metal-halide perovskites 280 10.5.1 Water-induced degradation of lead-halide perovskites 280 10.5.2 Instability of lead-free perovskites in water environment 282 10.5.3 Perovskite surface passivation 282 10.6 Summary 286 Acknowledgements 286 References 286 10.1 10.2

viii

Contents

11. Thermally activated delayed fluorescence

12.5

293

Leonardo Evaristo de Sousa and Piotr de Silva 11.1 11.2 11.3 11.4

Introduction Excited states calculations Condensed phase effects Role of charge transfer and local excited states 11.5 Vibronic effects and rate calculations 11.6 Synopsis References

12. DNA photostability

293 295 297 299 302 306 307

13. Fluorescent proteins

327 328 328

337

M.G. Khrenova and A.P. Savitsky 13.1 Introduction 13.2 Modeling of absorption spectra 13.3 F€ orster resonance energy transfer 13.4 Photochemical reactions 13.5 Concluding remarks Acknowledgment References

337 338 341 343 345 346 346

311 14. Chemi- and bioluminescence: A practical tutorial on computational chemiluminescence 351

Lara Martı´nez-Ferna´ndez and Antonio Franc
es-Monerris Photophysics of canonical nucleobases in the gas phase. Photostability 12.1.1 Absorption properties in the gas phase 12.1.2 Photophysical paths for purine nucleobases 12.1.3 Excited-state dynamics of purine nucleobases 12.1.4 Photophysical paths for pyrimidine nucleobases 12.1.5 Excited-state dynamics of pyrimidine nucleobases 12.2 Photophysics of canonical nucleobases in solution. Impact of the solvent effects into the photostability 12.2.1 Purine nucleobases 12.2.2 Pyrimidine nucleobases 12.3 Photophysics of modified nucleobases. Impact of the substitution effects into the photostability 12.3.1 Addition of external groups into the pyrimidine/purine core 12.3.2 Substitution of internal groups into the pyrimidine/purine core 12.4 Photophysics of canonical nucleobases in DNA/RNA environments. Photostability mechanisms 12.4.1 Single monomers embedded in a DNA/RNA environment 12.4.2 DNA/RNA light absorption and excited-state delocalization 12.4.3 Watson-Crick base pairing and interstrand charge transfer states. A doorway to proton transfer and photostability

Final remarks and future perspectives Acknowledgments References

12.1

Daniel Roca-Sanjua´n 312 312 312 314 314 315

316 316 317

319 319 321

322 322

14.1 14.2 14.3

Introduction Design of the methodology Identification of the molecule responsible for chemiexcitation 14.3.1 Walsh correlation diagrams 14.3.2 Reaction paths for the chemiexcitation of small models 14.3.3 “Activator”-“chemiluminophore” configuration 14.4 Reaction paths of the isolated system 14.4.1 Formation of the chemiluminophore 14.4.2 Chemiexcitation 14.4.3 Light emission 14.4.4 Identification of relevant parameters in challenging systems 14.5 Solvent effects 14.6 Dynamical aspects 14.7 A perspective on future research directions Acknowledgments References

15. Chemi- and bioluminescence: Bioluminescence

351 353 355 355

357 357 358 358 359 359

360 360 362 363 363 363

367

Isabelle Navizet 323

325

15.1 15.2

Introduction 367 Bioluminescence, a reaction scheme of a chemiluminescent system catalyzed by a protein: Challenges for theoretical and computational researchers 368

Contents

Overview of a bioluminescent process 15.2.2 Generation of HEI 15.2.3 Decomposition of HEI to the light emitter 15.2.4 Emission of light 15.3 Tools and choices of the theoretical chemist: Divide to conquer 15.3.1 Performing calculation of a small chemiluminescent model in vacuum 15.3.2 Performing calculation of a chemiluminescent model in the solvent 15.3.3 Performing calculation of a bioluminescent model in the protein 15.3.4 Modeling spectral shape 15.4 Modeling formation of HEI: Case of firefly bioluminescent system 15.4.1 From D-luciferin substrate to D-luciferyl adenylate intermediate 15.4.2 Approach of dioxygen to the D-luciferyl adenylate intermediate 15.4.3 Deciphering between reaction schemes for the reaction of dioxygen with the D-luciferyl adenylate intermediate 15.4.4 Formation of the dioxetanone ring: Addition-elimination mechanism? 15.5 Modeling decomposition of HEI leading to the light emitter in firefly 15.5.1 Failure of small models 15.5.2 Model in vacuum 15.5.3 Model in proteins 15.6 Modeling light emission 15.6.1 Challenges in modeling and experiments 15.6.2 Nature of the light emitter of firefly: The oxyluciferin 15.6.3 Use of analogs of firefly oxyluciferin 15.6.4 Influence of the protein on the emitted light color 15.6.5 Example of one mutation in luciferase 15.6.6 Different colors in different luciferases 15.6.7 Modeling emission spectra for substrate analogs 15.7 Conclusion References

16. Photocatalysis

15.2.1

368 368 369 369 369

370

372

372 373 374

374 374

374

374 377 377 377 378 378 378 379 380 380 381 382 382 383 383

ix

387

Xin-Ping Wu, Ming-Yu Yang, Zi-Jian Zhou, Zhao-Xue Luan, Lin Zhao, and Yi-Chun Chu 16.1

Introduction and historical overview 16.2 Fundamental mechanism of heterogeneous photocatalysis 16.2.1 Light absorption and photoexcitation 16.2.2 Charge migration and recombination 16.2.3 Photoredox reactions 16.3 Brief overview of computational methodologies 16.3.1 Kohn-Sham density functional theory (KS-DFT) 16.3.2 Multireference and multiconfigurational methods 16.3.3 Combined quantum mechanical and molecular mechanical (QM/MM) methods 16.4 Computational studies 16.4.1 TiO2 16.4.2 ZnO 16.4.3 MoS2 16.4.4 UiO-66 16.4.5 PCN-601 16.4.6 g-C3N4 16.5 Outlook Acknowledgments References

17. Nonlinear spectroscopies

387

388 388 389 389 390 390

391

391 392 392 395 397 399 401 402 404 405 405

417

Juliana Cu
ellar-Zuquin, Angelo Giussani, and Javier Segarra-Martı´ 17.1 17.2

Introduction Basic concepts 17.2.1 Introduction to the Liouville space 17.2.2 The displaced Brownian oscillator model 17.2.3 Including solvent effects through energy-gap correlation functions 17.3 Linear absorption 17.3.1 Linear spectroscopy in the Liouville space 17.3.2 The nuclear ensemble approach

417 419 419 420

422 422 423 428

x Contents

17.4

Nonlinear spectroscopy 17.4.1 Nonlinear spectroscopy in the Liouville space 17.4.2 Nonlinear spectroscopies within a static approximation 17.4.3 Spectral broadenings in nonlinear spectroscopies 17.5 Overview References

18. Mechano-photochemistry

430 431 432 434 439 440

447

Martina Nucci, Alejandro Jodra, and Luis Manuel Frutos 18.1 18.2 18.3

Introduction Mechanochemical models Methodology and models in mechano-photochemistry 18.3.1 Absorption and emission tuning 18.3.2 Triplet energy transfer modulation 18.3.3 Mechanical effect on conical intersections/avoided crossings 18.3.4 Non-adiabatic molecular dynamics with explicit inclusion of mechanical external forces 18.3.5 Substituent effect as mechanical entity 18.4 Mechanical control of molecular photophysics and photoreactivity 18.4.1 Excitation energy 18.4.2 Mechanochemical control of photochemical reactivity 18.5 Conclusions and perspectives References

447 448 449 449 450 452

454 455

456 456 462 469 470

Part IV Synergy with experimental approaches 19. Interplay between computations and experiments in photochemistry

475

Rau´l Losantos and Diego Sampedro Introduction 475 Correlation between the principal experimental techniques used in photochemistry and computational methods 477 19.2.1 UV-Vis spectroscopy 479 19.2.2 Photoluminescence spectroscopy, fluorescence, and phosphorescence 479 19.2.3 Time-resolved spectroscopy and transient absorption spectroscopy (TAS) 480 19.2.4 Other techniques 481 19.3 Different case studies combining theory and experiments 481 19.3.1 Design of new sunscreens by computational methods 481 19.3.2 Spectroscopic characterization of different families of photoswitches 483 19.3.3 Better understanding time-resolved spectroscopy 486 19.3.4 Two-photon absorption 489 19.4 Conclusions 491 References 491

19.1 19.2

Index

495

Contributors Numbers in parenthesis indicate the pages on which the authors’ contributions begin.

Stefano Battaglia (135), Department of Chemistry—BMC, Uppsala University, Uppsala, Sweden Yi-Chun Chu (387), Key Laboratory for Advanced Materials and Joint International Research Laboratory for Precision Chemistry and Molecular Engineering, Feringa Nobel Prize Scientist Joint Research Center, Centre for Computational Chemistry and Research Institute of Industrial Catalysis, School of Chemistry and Molecular Engineering, East China University of Science and Technology, Shanghai, People’s Republic of China Stefano Corni (191), Universita` degli Studi di Padova, Dipartimento di Scienze Chimiche, Padova, Italy Juliana Cuellar-Zuquin (417), Instituto de Ciencia Molecular, Universitat de Vale`ncia, Valencia, Spain

Simona Fantacci (215), Computational Laboratory for Hybrid/Organic Photovoltaics (CLHYO), Istituto CNR di Scienze e Tecnologie Chimiche “Giulio Natta” (CNR-SCITEC), Perugia, Italy Daniele Fazzi (247), Universita` di Bologna, Dipartimento di Chimica "Giacomo Ciamician", via F. Selmi, Bologna, Italy Ignacio Fdez. Galva´n (135), Department of Chemistry— BMC, Uppsala University, Uppsala, Sweden Antonio Frances-Monerris (311), Institut de Cie`ncia Molecular, Universitat de Vale`ncia, Valencia, Spain Jacopo Fregoni (191), Universidad Auto´noma de Madrid, Departamento de Fisica Teo´rica de la Materia Condensada, Madrid, Spain

Piotr de Silva (293), Department of Energy Conversion and Storage, Technical University of Denmark, Kongens Lyngby, Denmark

Luis Manuel Frutos (3,447), Universidad de Alcala´, Departamento de Quı´mica Analı´tica, Quı´mica Fı´sica e Ingenierı´a Quı´mica; Universidad de Alcala´, Instituto de Investigacio´n Quı´mica “Andres M. del Rı´o” (IQAR), Alcala´ de Henares, Madrid, Spain

Leonardo Evaristo de Sousa (293), Department of Energy Conversion and Storage, Technical University of Denmark, Kongens Lyngby, Denmark

Cristina Garcı´a-Iriepa (3,51), Universidad de Alcala´, Departamento de Quı´mica Analı´tica, Quı´mica Fı´sica e Ingenierı´a Quı´mica, Alcala´ de Henares, Madrid, Spain

Adrian L. Dempwolff (119), Interdisciplinary Center for Scientific Computing, Ruprecht-Karls University, Heidelberg, Germany

Angelo Giussani (417), Instituto de Ciencia Molecular, Universitat de Vale`ncia, Valencia, Spain

Valentin Diez-Cabanes (215), Laboratoire de Physique et Chimie Theoriques (LPCT); Laboratoire Lorrain de Chimie Moleculaire (L2CM), Universite de Lorraine & CNRS, Nancy, France Pavlo O. Dral (163), The State Key Laboratory of Physical Chemistry of Solid Surfaces, College of Chemistry and Chemical Engineering, Fujian Provincial Key Laboratory of Theoretical and Computational Chemistry, and Innovation Laboratory for Sciences and Technologies of Energy Materials of Fujian Province (IKKEM), Xiamen University, Xiamen, Fujian, China Andreas Dreuw (119), Interdisciplinary Center for Scientific Computing, Ruprecht-Karls University, Heidelberg, Germany

John M. Herbert (69), Department of Chemistry and Biochemistry, The Ohio State University, Columbus, OH, United States Alejandro Jodra (3,51,447), Universidad de Alcala´, Departamento de Quı´mica Analı´tica, Quı´mica Fı´sica e Ingenierı´a Quı´mica, Alcala´ de Henares, Madrid, Spain Waldemar Kaiser (265), Computational Laboratory for Hybrid/Organic Photovoltaics (CLHYO), Institute of Chemical Sciences and Technologies “Giulio Natta”, National Research Council (CNR-SCITEC), Perugia, Italy M.G. Khrenova (337), A.N. Bach Institute of Biochemistry, Research Centre of Biotechnology of the Russian Academy of Sciences; Department of Chemistry, Lomonosov Moscow State University, Moscow, Russia

xi

xii Contributors

Jingbai Li (163), Department of Chemistry and Chemical Biology, Northeastern University, Boston, MA, United States

Diego Sampedro (475), Universidad de La Rioja, Departamento de Quı´mica, Centro de Investigacio´n en Sı´ntesis Quı´mica, Logron˜o, Spain

Roland Lindh (135), Department of Chemistry—BMC, Uppsala University, Uppsala, Sweden

A.P. Savitsky (337), A.N. Bach Institute of Biochemistry, Research Centre of Biotechnology of the Russian Academy of Sciences; Department of Chemistry, Lomonosov Moscow State University, Moscow, Russia

Steven A. Lopez (163), Department of Chemistry and Chemical Biology, Northeastern University, Boston, MA, United States Rau´l Losantos (475), Universidad de La Rioja, Departamento de Quı´mica, Centro de Investigacio´n en Sı´ntesis Quı´mica, Logron˜o, Spain; Universite Paris Cite and CNRS, ITODYS, Paris, France Zhao-Xue Luan (387), Key Laboratory for Advanced Materials and Joint International Research Laboratory for Precision Chemistry and Molecular Engineering, Feringa Nobel Prize Scientist Joint Research Center, Centre for Computational Chemistry and Research Institute of Industrial Catalysis, School of Chemistry and Molecular Engineering, East China University of Science and Technology, Shanghai, People’s Republic of China Marco Marazzi (3,51), Universidad de Alcala´, Departamento de Quı´mica Analı´tica, Quı´mica Fı´sica e Ingenierı´a Quı´mica, Alcala´ de Henares, Madrid, Spain Lara Martı´nez-Ferna´ndez (311), Departamento de Quı´mica, Facultad de Ciencias and Institute for Advanced Research in Chemistry (IADCHEM), Campus de Excelencia UAM-CSIC, Universidad Auto´noma de Madrid, Madrid, Spain Edoardo Mosconi (265), Computational Laboratory for Hybrid/Organic Photovoltaics (CLHYO), Institute of Chemical Sciences and Technologies “Giulio Natta”, National Research Council (CNR-SCITEC), Perugia, Italy Isabelle Navizet (367), Professor at Laboratoire Modelisation et Simulation Multi Echelle, Universite Gustave Eiffel, France Martina Nucci (447), Universidad de Alcala´, Departamento de Quı´mica Analı´tica, Quı´mica Fı´sica e Ingenierı´a Quı´mica, Alcala´ de Henares, Madrid, Spain Mariachiara Pastore (215), Laboratoire de Physique et Chimie Theoriques (LPCT), Universite de Lorraine & CNRS, Nancy, France Daniel Roca-Sanjua´n (351), Instituto de Ciencia Molecular, Universitat de Vale`ncia, Vale`ncia, Spain

Javier Segarra-Martı´ (417), Instituto de Ciencia Molecular, Universitat de Vale`ncia, Valencia, Spain Morgane Vacher (163), Nantes Universite, CNRS, CEISAM UMR 6230, Nantes, France Xin-Ping Wu (387), Key Laboratory for Advanced Materials and Joint International Research Laboratory for Precision Chemistry and Molecular Engineering, Feringa Nobel Prize Scientist Joint Research Center, Centre for Computational Chemistry and Research Institute of Industrial Catalysis, School of Chemistry and Molecular Engineering, East China University of Science and Technology, Shanghai, People’s Republic of China Ming-Yu Yang (387), Key Laboratory for Advanced Materials and Joint International Research Laboratory for Precision Chemistry and Molecular Engineering, Feringa Nobel Prize Scientist Joint Research Center, Centre for Computational Chemistry and Research Institute of Industrial Catalysis, School of Chemistry and Molecular Engineering, East China University of Science and Technology, Shanghai, People’s Republic of China Lin Zhao (387), Key Laboratory for Advanced Materials and Joint International Research Laboratory for Precision Chemistry and Molecular Engineering, Feringa Nobel Prize Scientist Joint Research Center, Centre for Computational Chemistry and Research Institute of Industrial Catalysis, School of Chemistry and Molecular Engineering, East China University of Science and Technology, Shanghai, People’s Republic of China Zi-Jian Zhou (387), Key Laboratory for Advanced Materials and Joint International Research Laboratory for Precision Chemistry and Molecular Engineering, Feringa Nobel Prize Scientist Joint Research Center, Centre for Computational Chemistry and Research Institute of Industrial Catalysis, School of Chemistry and Molecular Engineering, East China University of Science and Technology, Shanghai, People’s Republic of China

Preface A year and a half ago, when we received the invitation to be guest editors for a book about theoretical and computational photochemistry, we did what most probably every scientist does in front of a new challenge: hesitate. In principle, our idea was (and finally is) to propose a book that could lead the reader from the early fundamentals of this topic, from a physical chemistry point of view, up to today’s applications, as a sort of journey passing through the required methods … but, how? Hence, we continued to act as typical scientists: both with a marker in our hand, in front of an apparently giant and completely empty whiteboard, thinking about how to organize such a book, and how to fill the different chapters. Some hours later we were filling each line of the future book’s outline with any sort of concept, approximation, method, and a pretty long list of diverse applications. Evidently, something was missing, similar to when a sport’s coach has clear ideas about every single player on the team … but no idea about how they should merge their skills to actually play on the court. Apparently, something important was missing, something nonscientific, something the lack of which was due to our total inexperience as book editors: which type of readers do we want to address? And then, how would they consider our book? Answering these questions, we found out that in the literature few books concerning photochemistry are devoted to experimentalists who would like to “gently” step into the “dark side” of science, and even the theoreticians who have already been living for a number of years in this “dark side,” many times found themselves absorbed in their own topic or subtopic, partially missing different computational approaches or points of view. The result of all this is a book divided into four main parts: the first part is dedicated to the fundamental concepts of photophysics and photochemistry, since we considered it impossible to address a photochemical study theoretically without an initial photophysical perspective. Like a textbook, this part aims at introducing qualitatively all main aspects and, moreover, allowing the interested reader to deepen their knowledge at a more profound level. The second part of the book includes the major past and present advances concerning photochemical methods, by which the third part can be understood in terms of applications in many relevant fields. Finally, the fourth part is exclusively dedicated to the synergy and interplay between experiment and theory, something definitely necessary if the final goal is a constant advance of photochemistry as a scientific branch. Of course, this huge effort could not be accomplished without the help of all the authors who decided to join this challenge, from many different countries and with so many different backgrounds. We hope that they, as well as the readers, enjoy the final outcome as much as we do. Cristina Garcı´a-Iriepa Marco Marazzi

xiii

Part I

Fundamentals

Chapter 1

Introduction to molecular photophysics Alejandro Jodraa, Luis Manuel Frutosa,b, Cristina Garcı´a-Iriepaa, and Marco Marazzia a Universidad de Alcala´, Departamento de Quı´mica Analı´tica, Quı´mica Fı´sica e Ingenierı´a Quı´mica, Alcala´ de Henares, Madrid, Spain, b Universidad de Alcala´, Instituto de Investigacio´n Quı´mica “Andr
es M. del Rı´o” (IQAR), Alcala´ de Henares, Madrid, Spain

Chapter outline 1.1 Interaction between electromagnetic radiation and molecules 1.1.1 Electromagnetic radiation 1.1.2 Time-dependent perturbation theory: A tool for describing the matter-radiation interaction 1.1.3 Electric dipole transitions 1.1.4 Spontaneous emission 1.2 Quantization of energy 1.3 The Franck-Condon principle 1.4 Electronic absorption spectra

3 3 4 6 8 10 16 18

1.4.1 Transition energy 1.4.2 Transition intensity: The oscillator strength and the transition dipole moment 1.4.3 Absorption band shape: The dynamic effect and the vibronic coupling 1.4.4 Multiphotonic absorption 1.5 Fluorescence and phosphorescence emission 1.5.1 Fluorescence 1.5.2 Phosphorescence References

18 19 21 25 28 29 40 43

1.1 Interaction between electromagnetic radiation and molecules 1.1.1 Electromagnetic radiation The electromagnetic radiation is composed by an oscillating electric (E) and magnetic (B) fields:   2pz E ¼ iℇx ¼ iℇ0x cos 2pnt  l   2pz B ¼ jBy ¼ jB0y cos 2pnt  l

(1.1) (1.2)

where ℇ0x and B0y are the amplitudes of the electric and magnetic fields (along the corresponding direction), and i and j are the unitary vectors along the x and y directions. This corresponds to a plane wave propagating along the z-axis and polarized (see Fig. 1.1). Several aspects define the characteristics of the electromagnetic radiation: the amplitude of the fields (related to the energy density), the wavelength (related to the distance between equivalent positions in the wave), and the frequency (related with the time required for the fields to completing a cycle). The fields are oscillating in space and time, where the frequency provides the number of oscillations of E (or B) per time unit: n¼

o 2p

(1.3)

l¼

2p k

(1.4)

And the wavelength is given by:

Theoretical and Computational Photochemistry. https://doi.org/10.1016/B978-0-323-91738-4.00017-8 Copyright © 2023 Elsevier Inc. All rights reserved.

3

4 PART

I Fundamentals

FIG. 1.1 Variation along the propagating direction z of the electric (E) and magnetic (B) fields in an electromagnetic plane wave. The Poynting vector S is also indicated. (Credit: Original.)

Both n and l are related by the velocity of propagation (v) of the electromagnetic wave: v ¼ ln. Usually, the propagation velocity is the speed of light in vacuum, c, therefore: c ¼ ln

(1.5)

Otherwise, this velocity should be corrected by taking into account the refraction index, n, being the expression: n1c ¼ ln. The energy of the electric and magnetic fields per volume unit is related to the square of the field vectors. More speE cifically, the density energy of the electric field is given by: uE ¼ 20 E2, while the density energy of the magnetic field is: uB ¼ 2m1 B2 , being E0 the vacuum permittivity, and m0 the vacuum permeability constant. The total energy density of the 0 electromagnetic field is therefore:   1 1 2 E0 E2 + u¼ B (1.6) 2 m0 This magnitude is related to the Poynting vector, S, which is defined as the flux of energy per unit of area and unit of time in the direction of the electromagnetic wave propagation (see Fig. 1.1): S¼

1 ðE  B Þ m0

(1.7)

Since E and B are perpendicular, the Poynting vector magnitude for an electromagnetic wave as considered in Fig. 1.1 reduces to its z component:   ℇx By ℇ0x 2 2pz 2 ¼ S ¼ Sz ¼ cos 2pnt  ¼ cu (1.8) m0 l cm0 where the relation between the electric and magnetic field: E ¼ cB has been taken into account. The Poynting vector divergence is related to the time variation of the energy density: —S¼

∂u ∂t

(1.9)

1.1.2 Time-dependent perturbation theory: A tool for describing the matter-radiation interaction The electric and magnetic fields of an electromagnetic radiation oscillate with time, not only the magnitude of the corresponding vector, but also its direction. The interaction energy of the radiation with an arrangement of charged particles, e.g., a molecular system, is dependent on these vectors, and therefore, it changes with time. If the intensity of the radiation is not extremely strong, this interaction can be studied as a perturbation of the system. In this regard, the time-dependent perturbation theory is a useful tool to predict the evolution with time of the perturbed system.

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The time-dependent Schr€ odinger equation is: iħ

∂CðrtÞ ¼ HðrtÞCðrtÞ ∂t

(1.10)

Let us assume that for the unperturbed system, the Hamiltonian is time-independent, and the perturbed system depends on time: HðrtÞ ¼ Hð0Þ ðrÞ + H0 ðrtÞ

(1.11)

ð0Þ

The unperturbed system, given by H ðrÞ , has a set of eigenfunctions, which are solution of the time-independent Schr€ odinger equation: Hð0Þ ðrÞcðn0Þ ðrÞ ¼ Eðn0Þ cðn0Þ ðrÞ

(1.12)

c(0) n (r)

where is the zeroth-order wave function for the “n” state, dependent only on the electronic coordinates (r). According to the superposition principle, the time-dependent wave function of the system solution of the Eq. (1.10), can be expressed as a superposition of the unperturbed: ð 0Þ X iEn (1.13) cn ðtÞcðn0Þ ðrÞe ħ t Cðr, tÞ ¼ n

Substituting into the Eq. (1.10), the following equation is obtained: iħ

ð0Þ ð 0Þ X ∂c ðtÞ X iEn iEn n cðn0Þ e ħ t ¼ cn ðtÞH0 cðn0Þ e ħ t ∂t n n

Multiplying by the left by the conjugate of c(0) m (x) and integrating over spatial coordinates, we get: E iEðn0Þ D E iEðn0Þ X ∂c ðtÞ D X n cðm0Þ jcðn0Þ e ħ t ¼ cn ðtÞ cðm0Þ jH0 jcðn0Þ e ħ t iħ ∂t n n where the perturbation energy matrix elements are: D E cðm0Þ jH0 jcðn0Þ ¼ H 0mn

Since the eigenfunctions of the unperturbed Hamiltonian define an orthonormal set: D E cðm0Þ jcðn0Þ ¼ dmn

(1.14)

(1.15)

(1.16)

(1.17)

The Eq. (1.15) can be simplified to:

iħ ð 0Þ

ð 0Þ

∂cm ðtÞ X ¼ cn ðtÞH 0mn eiomn t ∂t n

(1.18)

n . The above expression provides the set of equations that have to be solved in order to predict the evowith omn ¼ Em E ħ lution with time of the wave function (i.e., the set of coefficients {cn(t)}). Given a set of initial (t ¼ 0) values for the coefficients defining the wave function, the variation of the coefficient can be determined. In the case that the initial (t ¼ 0) state is defined by a pure state (e.g., cn(0) ¼ 1 and cm(0) ¼ 0 8 m 6¼ n), the following expression for the “m” states is obtained by integration: Z 1 t 0 iomn t0 0 cm ðt Þ ¼ H e dt (1.19) iħ 0 mn

where a first approach in the variation of the coefficients is assumed. The time-dependent transition probability from the initial state “n” and the final state “m” is given by the square of the corresponding coefficient, Pn!m(t) ¼ jcm(t)j2; therefore: 2 Z t c ðtÞ 2 ¼ 1 H 0 eiomn t0 dt0 (1.20) m mn 2 ħ 0

So finally, this expression provides the time-dependent transition probability (first-order approach) from an initial state “n” to a final state “m” where the perturbation energy term is Hmn0 .

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1.1.3 Electric dipole transitions In molecular photochemistry, the transitions between electronic states induced by electromagnetic radiation constitute a key step of the photochemical process. The absorbed energy is employed to further promote chemical reactions or being dissipated following other mechanism. The wavelength of the electromagnetic radiation necessary to induce electronic transitions typically falls in the range of hundreds of nanometers (i.e., from c. 200 nm for UV transitions to c. 700 nm in the red region of visible spectrum). This implies that molecules are much smaller than wavelengths exciting electronic transitions, and therefore, within a good approximation, we can consider the electric field constant for the molecule. In other words, the whole molecule feels a uniform electric field that only varies with time. Considering this approach, it is possible to determine the interaction energy between the electric field of the electromagnetic radiation and the arrangement of charged particles defining the molecule. This energy is just the scalar product between the electric dipole moment (m) and the electric field (E): V int ¼ m  E

(1.21)

where the dipole moment is defined as m¼

X

qi r i

(1.22)

i

being ri the charge position vectors and qi their respective charges. The corresponding interaction energy between the molecule and the electric field in an electromagnetic polarized radiation (see Fig. 1.1) is therefore: V int ¼ mx ℇ0x cosðotÞ

(1.23)

This energy can be understood as a perturbation of the molecular system, as it typically implies only a small fraction of the total energy of the molecule. Consequently, the perturbation term of the Hamiltonian is: mx ℇ0x cosðotÞ H0 ¼ ^

(1.24)

One of the most relevant interaction mechanisms between electromagnetic radiation and matter is that corresponding to electric dipole transitions, where the electric field of the electromagnetic radiation is constant, as discussed above, for the whole molecule (i.e., it only varies with time, but not in space). This defines the so-called dipolar approximation and permits to predict the transitions trough the electric dipole mechanism. The perturbation energy terms have therefore the following form: D E H0mn ðtÞ ¼ cðm0Þ jmx jcðn0Þ cosðotÞ (1.25)

Consequently, the probability transition according to Eq. (1.20) is: E 2 0 2 D ð0Þ 2 ℇ c jm jcð0Þ Z t x n x m 0 iomn t0 0 cosðot Þe dt Pn!m ðtÞ ¼ 2 ħ 0

(1.26)

where o is the angular frequency of the light source, and omn is the angular frequency associated with the n!m transition. By using the Euler’s formula for complex integrating the above expression, the following expression is obtained: E 2 0 2 D ð0Þ 2 ℇ c jm jcð0Þ iðomn +oÞt0 0 x n x m  1 eiðomn oÞt  1 e Pn!m ðtÞ ¼ + (1.27) o +o o o 4ħ2 mn

mn

The probability of transition between the two states is depending on several factors. On the one hand, it depends on the square of the electric field intensity; therefore, higher intensities are related to higher transition probabilities (e.g., laser light source is more D E efficient than conventional lamps). On the second hand, the dipole transition integral has to be nonzero: cðm0Þ jb mz jcðn0Þ 6¼ 0. This integral defines the selection rules for the type of transitions and stablishes which transitions are allowed or forbidden by dipole moment mechanism. The last term of the Eq. (1.27) can be analyzed to explore its dependence on the frequency of the light source and the evolution with time. If omn  o, then the first term in the expression can be ignored. In this case, the probability transition can be reduced to:

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FIG. 1.2 Transition probability as a function of time and frequency of the electric field. The maximum is reached for the resonant condition. (Credit: Original.)

E 2 0 2 D ð0Þ ℇ c jm jcð0Þ x n m x sin 2 ½ðomn  oÞt=2 Pn!m ðtÞ ¼ 2 ħ ðomn  oÞ2

(1.28)

The maximum probability is therefore reached when the following condition is fulfilled (see Fig. 1.2): Em  En ¼ hn. This situation corresponds with the stimulated absorption of a photon via electric dipole mechanism, and it is the most frequent mechanism of electronic excitation in atoms and molecules. In an analogous way, if we now consider omn  o in Eq. (1.28), the following probability of transition is obtained: E 2 0 2 D ð0Þ ℇ c jm jcð0Þ x n m x sin 2 ½ðomn + oÞt=2 (1.29) Pm!n ðtÞ ¼ ħ2 ðomn + oÞ2

where the maximum is reached for: En  Em ¼ hn. This corresponds to the stimulated emission following the electric dipole mechanism, where the interaction of the system with an oscillating electric filed induces the emission of a photon with similar wavelength (see Fig. 1.3). From Eq. (1.29), it is possible to calculate the transition probability by considering a light source where the intensity of the electric field is almost constant close to the resonance frequency, i.e., omn, therefore jE0xj2  constant for all values of o. In this case, it is possible to easily integrate over all the frequencies:

FIG. 1.3 The two mechanisms of electromagnetic field interaction with molecular systems. (left) Stimulated absorption, (right) stimulated emission. In both cases, the resonant condition is fulfilled. (Credit: Original.)

8 PART

I Fundamentals

Pn!m ðtÞ ¼

Z

∞ 0

Pn!m ðtoÞ ¼

D E 2 2rðomn Þ cðm0Þ jmx jcðn0Þ Z E 0 ħ2

∞

sin 2 ½ðomn  oÞt=2 ðomn  oÞ2

0

do

(1.30)

where r(omn) ¼ E0jE0xj2 is the energy density of the electromagnetic radiation. Integrating, the following probability transition is obtained:

Pn!m ðtÞ ¼

D E 2 rðnmn Þ cðm0Þ jmx jcðn0Þ 2E0 ħ2

t

(1.31)

The transition rate for the n ! m excitation is given by the transition probability per time unit: dPn!m ðtÞ ¼ dt

D E 2 rðnmn Þ cðm0Þ jmx jcðn0Þ 6E0 ħ2

(1.32)

So the number of atoms or molecules going from “n” to “m” energy levels per unit time is: dN n!m ðtÞ dP ðtÞ ¼ N n n!m ¼ N n dt dt

D E 2 rðnmn Þ cðm0Þ jmx jcðn0Þ 6E0 ħ2

(1.33)

For the stimulated emission, there is an equivalent expression. The number of molecules excited per unit time is:

where Bmn ¼

jh

ð 0Þ ð 0Þ cm jmjcn 2 6E0 ħ

dN abs n!m ðtÞ ¼ Bnm rðnmn ÞN n dt

2

ij

(1.34)

is the Einstein coefficient for absorption of light.

1.1.4 Spontaneous emission Apart of stimulated emission, spontaneous emission from excited states can take place in molecular systems. In fact, this mechanism of deactivation to the ground state is by far the most common radiative decay process in molecular photochemistry. The rate of photon emission for each one of the two possible mechanisms (i.e., stimulated emission and spontaneous emission) is: dN spe m!n ðtÞ ¼ Bmn rðnmn ÞN m dt

(1.35)

dN ste m!n ðtÞ ¼ Amn N m dt

(1.36)

where Amn, the so-called Einstein A coefficient or simply the rate of spontaneous emission, is a constant to be determined. For a system in thermal equilibrium, the absorbed and emitted energy are the same. Therefore, the rate of the three processes has to be equal: dN abs dN spe dN ste n!m ðtÞ m!n ðtÞ m!n ðtÞ ¼ + dt dt dt

(1.37)

Bnm rðnmn ÞN n ¼ Amn N m + Bmn rðnmn ÞN m

(1.38)

So:

Taking into account that thermal equilibrium is reached, the relative population of each state has to satisfy: Nm ¼ eðEm En Þ=kT ¼ ehnmn =kT Nn

(1.39)

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Determining the Nm/Nn ratio, from Eq. (1.38), we get: rðnmn Þ ¼

Amn ehnmn =kT Bnm  Bmn

(1.40)

The black-body is a model system in perfect thermodynamic equilibrium with the environment that absorbs and emits the same amount of energy. For this system, the following expression is fulfilled: rðnmn Þ ¼

8phn3 1 c3 ehnmn =kT  1

(1.41)

By comparing both expressions, it is straightforward to obtain the two following conditions: Bnm ¼ Bmn

(1.42)

and Amn ¼

8phn3mn Bmn c3

(1.43)

Therefore, the Einstein coefficient for spontaneous emission is: D E 2 16p3 n3mn cðm0Þ jmjcðn0Þ Amn ¼ 3E0 hc3

(1.44)

It is clear that the population of an excited state is decreasing exponentially by spontaneous emission, where the rate of emission is controlled by the Einstein coefficient Amn. The lifetime of the excited state is therefore inversely proportional to this coefficient, τ ¼ A1 so the population of the excited state decays exponentially: mn

N m ðtÞ ¼ N m ð0ÞeAmn t The decay rate constant depends mainly on two variables: the energy gap between states to the third power D E 2 transition dipole moment to the spare cðm0Þ jmjcðn0Þ (see Fig. 1.4).

(1.45) n3mn,

and the

FIG. 1.4 Variation of the spontaneous emission rate (Einstein coefficient Amn) with the energy gap between states and dipole moment transition. (Credit: Original.)

10

1.2

PART

I Fundamentals

Quantization of energy

For classical mechanics, the motion of a particle is governed by Newton’s laws. In particular, the second law for a particle moving along one dimension (x) tells us that F ¼ ma ¼ m

d2 x dt2

(1.46)

where F is the force acting on a particle, m is the mass, a is the acceleration, and t is the time. If we solve the above differential equation, we will find functions of the type: x ¼ f ðt, c1 , c2 Þ

(1.47)

where t is time, while c1 and c2 are the integration constants of the initial conditions. It follows that: v¼

d f ðt, c1 , c2 Þ dt

(1.48)

where v is the velocity. Specifically, in classical mechanics, for a certain system, we can know position, velocity, and forces acting on it, at every given time. These three variables determine the state of the system at an instant of time, giving us the possibility to predict how it will evolve and how it will be at any future instant. However, Werner Heisenberg, with his uncertainty principle, tells us that we cannot know the exact position and velocity for a given time, if the system is of quantum nature. Hence, with the advent of quantum mechanics, we have a new definition of the state of the system: there is a space- and time-dependent function known as the wave function, or state function (C), which contains all possible information that we can know about the state of the system. Similar to classical mechanics, which laws allow to know the future state based on the present one, also in quantum mechanics, we need an expression to predict the future state, knowing the wave function for the present state, i.e., an expression that tells us how the wave function will evolve over time: the time-dependent Schr€odinger equation: 

ħ ∂Cðr, tÞ ħ2 2 ¼ — Cðr, tÞ + V ðrÞCðr, tÞ i ∂t 2m

(1.49)

h and V(r) is the potential to which the system is subject and which for simplicity we assume to depend only on where ħ ≡ 2p the spatial coordinates. Solutions to Eq. (1.49) of the form:

Cðr, tÞ ¼ cðrÞeiEt=ħ

(1.50)

With E the energy of the system, are known as stationary wave functions. This type of states is characterized by a welldefined energy, having a time-independent probability density distribution: jCðr, tÞj2 ¼ C∗ ðr, tÞCðr, tÞ ¼ c∗ ðrÞcðrÞeiEt=ħ eiEt=ħ ¼ jcðrÞj2

(1.51)

Additionally, the spatial part of this type of wave functions is a solution of the time-independent Schr€odinger equation. This condition can be checked by substituting c(r) on Eq. (1.49): 

ħ2 2 — cðrÞ + V ðrÞcðrÞ ¼ EcðrÞ 2m

(1.52)

With the goal to simplify the following development and deductions, all mathematical operations will be performed along the x spatial coordinate, coming back to the three spatial coordinates r, when necessary. Hence, to solve Eq. (1.52), solutions are sought that satisfy the condition: Cðx, tÞ ¼ f ðtÞcðxÞ

(1.53)

Substituting in Eq. (1.49): cðxÞ

ħ df ðtÞ ħ2 d 2 cðxÞ ¼ f ðtÞ + V ðxÞf ðtÞcðxÞ i dt 2m dx2

(1.54)

By separating variables: 

1 ħ df ðtÞ 1 ħ2 d2 cðxÞ ¼ + V ðxÞ f ðtÞ i dt cðxÞ 2m dx2

(1.55)

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As it can be seen, both sides of the equation depend on different variables; hence, one side is independent on the other, leading to the following conclusion for the left-hand term of Eq. (1.55): 1 ħ df ðtÞ ¼E f ðtÞ i dt

(1.56)

df ðtÞ iE ¼ dt f ðt Þ ħ

(1.57)

where E is a constant. Reorganizing:

Integrating over t on both sides of the equation, we get: f ðtÞ ¼ eiEt=ħ

(1.58)

Now equaling the right-hand term to the constant E and multiplying both sides by c(x), we obtain: 

ħ2 d 2 cðxÞ + V ðxÞcðxÞ ¼ EcðxÞ 2m dx2

(1.59)

We have already stated, by recalling the Heisenberg uncertainty principle, that we cannot determine at the same time exact position and velocity of a quantum particle, but with the wave functions, we can define the probability of finding the particle in a given region of space. This is because Max Born postulated that the function jC(x, t)j2 is the density probability of finding the particle. If we exclude the time dependence, as it was done for developing the time-independent Schr€odinger equation (Eq. 1.3), we can simplify the expression of the density probability: jCðx, tÞj2 ¼ jcðxÞj2 so that the probability of finding the particle in a unidimensional Dx region is: Z b jcðxÞj2 dx P½ a  x  b ¼

(1.60)

(1.61)

a

If we integrate over the whole space, the probability of finding the particle is 1. This leads to the normalizing condition: Z +∞ Z +∞ P¼ c∗ ðxÞcðxÞdx ¼ 1 (1.62) jcðxÞj2 dx ¼ ∞

∞

∗

where c (x) is the complex conjugate of the wave function c(x). If we rearrange Eq. (1.59), we can write it as:   ħ2 d 2  + V ðxÞ cðxÞ ¼ EcðxÞ 2m dx2

(1.63)

where what multiplies the wave function in the left-hand term is an operator, which, as it can be seen, acts on one function, c(x), to transform it into another. When this other function obtained is the result of multiplying a constant by the function on which the operator acts, Ec(x), the equation is known as an eigenvalue equation. In general terms: Af ¼ cf

(1.64)

where A is an operator, f a function and c is a constant. For each operator, we will have a set of functions that will satisfy the eigenvalue equation, obtaining a characteristic constant for each of them. It should be noted that two functions can have equal constants. If a function satisfies the eigenvalue equation, it is known as the eigenfunction of the operator and the constant as the eigenvalue of the eigenfunction. We see therefore that the Schr€odinger equation is an eigenvalue equation, where the wave function is an eigenfunction of a given operator and has an associated eigenvalue. In the case we are looking at, the operator is known as the Hamiltonian operator, and the associated eigenvalue for the Hamiltonian is the energy of the state defined by the wave function. Since the values obtained are physical observables, the eigenvalues are real values. Therefore, since the operators are defined in the Hilbert space (belonging the wave functions to such space), and the eigenvalues are always real, these operators are known as Hermitic operators. Therefore, each of the operator’s eigenfunctions will represent a different state of the quantum system, ci(x), having a given eigenvalue associated with it, ai. Aci ðxÞ ¼ ai ci ðxÞ

(1.65)

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PART

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In general terms, this means that a quantum system has certain specific values for a given physical observable or, in other words, the observable can assume only quantized values. But, what happens if the wave function is not an eigenfunction of an operator associated with a physical observable. In this case, the measure of property B will give one of all possible values. These will be the eigenvalues associated with all the eigenfunctions of B. We can then define an average value of the measure of an observable B: X Pb b (1.66) h Bi ¼ b

where Pb is the probability of having the value b. Let’s consider the case of the average value of the particle position X Px x (1.67) hxi ¼ x

As we have seen before, the probability of finding the particle at a given position is determined by Eq. (1.61). Substituting its expression, we get: Z (1.68) hxi ¼ jcðxÞj2 xdx hxi ¼

Z

c∗ ðxÞxcðxÞdx

(1.69)

The same treatment applied to the position x can be used for any physical observable (B), leading to the general formulation: Z (1.70) hBi ¼ c∗ ðxÞBcðxÞdx If the wave function is the eigenfunction of the operator B, then combining Eqs. (1.62) and (1.70), we obtain: Z hBi ¼ b c∗ ðxÞcðxÞdx ¼ b

(1.71)

which means that the mean value is going to be the eigenvalue, since as we have said, when it is an eigenfunction this will be the only value that we can find if we measure B. Let us now consider the operator B which we have said to be Hermitic. For these operators, the following condition is satisfied: B ¼ B∗

(1.72)

Therefore, for two states cm an cn which are eigenfunctions of the operator B whose eigenvalues are m and n, respectively, it can be shown that: Z Z cm ∗ ðxÞBcn ðxÞdx ¼ cn ðxÞðBcm ðxÞÞ∗ dx (1.73) Substituting into the eigenvalue equation: Z Z n cm ∗ ðxÞcn ðxÞdx ¼ m cn ðxÞcm ∗ ðxÞdx

(1.74)

And rearranging: ðn  mÞ

Z

cm ∗ ðxÞcn ðxÞdx ¼ 0

(1.75)

We obtain: Z

cm ∗ ðxÞcn ðxÞdx ¼ 0

(1.76)

This equation tells us that two eigenfunctions of a Hermitic operator, whose eigenvalues are not equal, are orthogonal to each other. It can also be shown that for eigenfunctions with equal eigenvalues, they can be chosen to be orthogonal.

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With this orthogonality condition and the normalization condition (Eq. 1.62), we can conclude that the wave functions for the different states fulfill the following conditions: Z cm ∗ ðxÞcn ðxÞdx ¼ dmn ; dmn ¼ ½0, 1 (1.77) This means that a transition between state n and state m in a quantum system (e.g., a molecule) can be associated with a transition probability through an operator B and its eigenfunctions, which are cn and cm wave functions. The set of all eigenfunctions of a Hermitic operator representing a physical observable forms a complete set. X ci c i ðxÞ (1.78) fðxÞ ¼ i

where f(x) is a well-behaved function representing a quantum state. That is, any well-behaved state function that meets all the conditions set out above can be represented using the set of eigenfunctions as a linear combination of these. It should be noted that this eigenfunction f(x) does not have to be an eigenfunction of the operator. To calculate its mean value, we will substitute Eq. (1.78) into Eq. (1.70): Z X X ci c∗i ðxÞB ci ci ðxÞdx (1.79) h Bi ¼ i

i

Substituting into the eigenvalue equation: h Bi ¼

Z X

ci c∗i ðxÞ

i

X ci bi ci ðxÞdx

(1.80)

i

And using the orthonormalization conditions, we finally obtain: X 2 h Bi ¼ bi c i

(1.81)

i

Until now, we have considered our system as a single particle that can populate different states and move along a single spatial dimension (x). Nevertheless, since we are interested in photochemistry, we usually deal with molecules, i.e., with multiparticles’ systems. Such particles could interact in different ways, or not interact. For a system of noninteracting particles, the calculation of system properties is a simple problem. For example, in the calculation of the energy if the particles do not interact with each other, the Hamiltonian can be written as: H ¼ h1 ð r 1 Þ + h2 ð r 2 Þ + … + h n ð r n Þ

(1.82)

where hi is the Hamiltonian of the particle i and ri is the vector indicating its spatial coordinates with respect to the origin. Each of the members of the sum depends only on the coordinates of that particle, so the wave function can be written as: c ¼ ’1 ðr1 Þ’2 ðr2 Þ … ’n ðrn Þ

(1.83)

where ’i is the function describing the state of the particle i which depends on the coordinates’ vector ri. The Schr€odinger equation can be then formulated as follows: Hc ¼ h1 ðr1 Þ’1 ðr1 Þ’2 ðr2 Þ … ’n ðrn Þ + h2 ðr2 Þ’1 ðr1 Þ’2 ðr2 Þ … ’n ðrn Þ + hn ðrn Þ’1 ðr1 Þ’2 ðr2 Þ … ’n ðrn Þ ¼ ðe1 + e2 + … + en Þc

(1.84)

where ei is the energy of the particle i. As it can be seen, we have easily transformed an apparently complex n-particles problem into n problems, each constituted by a single particle. In this way, the total energy of the n-particles system (E) is simply the sum of the energies of the different particles: E ¼ e1 + e2 + … + en

(1.85)

The complexity raises when the particles of the system interact with each other, that is usually what can be expected by nuclei and electrons within a molecule. Especially, the interaction causes that some terms of the Hamiltonian now depend on the spatial coordinates of more than one particle, and therefore, we cannot separate the wave function as in Eq. (1.83). This opens several possible scenarios: depending on the type of particles and the interaction terms between them, different approximations can be proposed to express the wave function.

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Especially, if we have a molecular system, we can write the corresponding Schr€odinger equation as follows: HCðRrÞ ¼ ECðRrÞ

(1.86)

where H is the Hamiltonian operator, C is the wave function, that depends on nuclear R and electronic r coordinates, and E is the total energy of the system. It would be therefore highly desirable to separate both R and r coordinates using the noninteracting particle approximation, as in Eq. (1.83). However, this is not possible, as nuclei and electrons interact among them. One of the most used approximations to separate the motion of electrons and nuclei is the Born-Oppenheimer approximation (1): the wave functions of atomic nuclei and electrons in a molecule can be treated separately, since nuclei are much heavier than electrons. Such separation makes possible to solve an equation for the nuclear part and another equation for the electronic part. When we have a system of n particles, where every particle interacts with all the others, these interactions depend consequently on the relative coordinates between each couple of particles i and j: rij ¼ rj  ri

(1.87)

These interactions make the system stay “together” behaving in a certain way like a unique body. We can then calculate a specific position, known as the center of masses. X mi ri i Rcm ¼ X

mi

(1.88)

i

where mi is the mass of the particle i. Rcm will describe the motion of the system as if all the masses were concentrated at this point and moving as a whole. Let us consider the simplest possible example: a system composed by two particles interacting one with the other. The total energy is: 1 1 E ¼ m1 jr_1 j2 + m2 jr_2 j2 + V ðr Þ 2 2

(1.89)

where r is the distance between particles i and j, while the dot above the coordinates’ vector indicates the time derivative, i.e., the velocity. As proposed in Eqs. (1.87) and (1.88), our bi-particle system can be described by the center of mass and one relative coordinate, respectively: Rcm ¼

m 1 r1 + m 2 r2 m1 + m2

r12 ¼ r2  r1 defining a system of two equations with two unknowns, which can be readily solved: m2 r1 ¼ Rcm  r m1 + m2 12 r2 ¼ Rcm +

m1 r m1 + m2 12

(1.90) (1.91)

(1.92) (1.93)

Substituting the expressions of r1 and r2 in Eq. (1.89): 2 1 m 1 m 2 1 E ¼ ðm1 + m2 Þ R_cm + jr_ j2 + V ðr12 Þ 2 2 m1 + m2 12

(1.94)

We can then construct the operators and consequently the Hamiltonian:

H ¼ T ðRcm Þ + T ðr12 Þ + V ðr12 Þ

(1.95)

where T takes into account the kinetic energy and V the potential energy. This allows to separate the Hamiltonian in two terms, each of them depending on a single spatial variable: H ¼ HM ðRcm Þ + Hm ðr12 Þ

(1.96)

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where HM ðRcm Þ ¼ T ðRcm Þ;

Hm ðr12 Þ ¼ T ðr12 Þ + V ðr12 Þ

(1.97)

Thus, Eq. (1.96) has the same form as Eq. (1.82). Therefore, we can now treat mathematically a system of two interacting particles of mass m1 and m2 at positions r1 and r2, respectively, in terms of a system of two noninteracting particles of mass M ¼ m1 + m2 and m ¼ mm1+mm2 at positions Rcm and r12, respectively. Hence, the Schr€odinger equation can be written as 1 2 follows:   Hcðr1 r2 Þ ¼ HM ðRcm Þ’M ðRcm Þ + Hm ðr12 Þ’m ðr12 Þ ¼ eM + em c (1.98) Conveniently separating the two particles:

T ðRcm Þ’M ðRcm Þ ¼ eM ’M ðRÞ

(1.99)

½T ðr12 Þ + V ðr12 Þ’m ðr12 Þ ¼ em ’m ðr12 Þ

(1.100)

By solving Eqs. (1.99) and (1.100), we will then find the total energy of the system: E ¼ eM + em

(1.101)

On one hand, if we look at Eq. (1.99), we can easily deduce that it is describing is the motion of a free particle, whose mass M is the sum of the masses of all the particles within the system, and which is not subject to any potential. Therefore, we can realize that this equation describes the motion of the system as a whole, which is not subject to any interaction, thus obtaining the translational energy of the system moving as a whole. On the other hand, Eq. (1.100) describes a particle which position is a function of the relative coordinates, being this time subject to a potential V(r12), giving as solution the internal energy. If we look at the Hamiltonian terms of Eq. (1.98) and the expression of the total energy of the system (Eq. 1.101), we can finally conclude that the energy of the system will therefore be the sum of the kinetic energy due to the change in the vector r12 both in magnitude and orientation, plus the interaction potential. In conclusion, we can say that for a system of n particles, defining the center of mass, we can separate the total energy of the system as the sum of the translational energy (eM) and the relative energy (em). We can solve Eq. (1.99) and find the translational energy of the system without any problem, using the free particle model. However, solving Eq. (1.100) is not a straightforward task. This equation is a function of the relative coordinates (rij) or, in other words, of the vector joining particles i and j. Let us consider again the example of a bi-particle system. If we notice that the interaction potential is spherically symmetric, it depends only on a single parameter, the distance between the particles. It can be therefore considered as a system evolving under the influence of a central force field, also called a central force problem. The most common way to deal with these problems is to use spherical coordinates (R, y, f). In this way, the vector r12 can be expressed as a function of these coordinates. By making the corresponding coordinates change, we can then write Eq. (1.100) as a function of the spherical coordinates:      ħ2 ∂ 2 2 ∂ 1 ∂2 ∂ 1 ∂2 + + (1.102) + cot y + + V ð R Þ ’m ðR, y, fÞ ¼ em ’m ðR, y, fÞ  R ∂R R2 ∂y2 ∂y 2m ∂R2 sin 2 y ∂f2 Eq. (1.102) can be then regrouped in terms of coordinates dependence and applying the Laplacian operator in spherical coordinates: ½T ðRÞ + f ðRÞT ðy, fÞ + V ðRÞ’m ðR, y, fÞ ¼ em ’m ðR, y, fÞ

(1.103)

Finally, Eq. (1.103) can be separated in two expressions: ½T ðRÞ + V ðRÞPðRÞ ¼ eR PðRÞ

(1.104)

½f ðRÞT ðy, fÞY ðy, fÞ ¼ f ðRÞey,f Y ðy, fÞ

(1.105)

where Y(y, f) are the eigenfunctions of T(y, f) known as spherical harmonics and P(R) are the eigenfunctions that depend on the potential V(R). If we look at Eq. (1.105), we see that f(R) behaves as a parameter. These spherical harmonics depend only on the angles (y, f); therefore, they will describe the kinetic energy due to the change of orientation of the relative vector r12, i.e., the rotational motion, being ey, f the rotational energy. On the other hand, we have Eq. (1.104), which depends on the type of interaction between the particles. Consequently, to describe the nuclear motion, we must consider V(R) as the nuclear interaction potential U(R), i.e., V(R) ≡ U(R), which is deduced from the Born-Oppenheimer approximation aforementioned.

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Thus, Eq. (1.104) can be explicitly given as:     ħ2 ∂ 2 2 ∂ + + U ð R Þ PðRÞ ¼ eR PðRÞ  2m ∂R2 R ∂R

(1.106)

which can be simplified by applying: FðRÞ ¼ RPðRÞ

(1.107)

 ħ2 ∂ 2 + U ð RÞ Fð RÞ ¼ e R F ð RÞ  2m ∂R2

(1.108)

thus resulting in: 

There are many methods to know the expression of the function U(R). The simplest way is the Taylor series expansion around the break-even point of the potential energy function: 2      1   (1.109) U ðRÞ ¼ U Req + U0 Req R  Req + U00 Req R  Req 2 where Req is the equilibrium position, therefore the first derivative U0 (Req) ¼ 0. By changing the variable R  Req ¼ x and renaming the second derivative U00 (Re) ¼ keq, we obtain:   1 UðRÞ ¼ U Req + keq x2 (1.110) 2 By substitution in Eq. (1.108):

 2 2    ħ ∂ 1 2 + keq x SðxÞ ¼ ðeR  U Req ÞSðxÞ  2 2 2m ∂R

(1.111)

ET ¼ UðRÞ + evib + ey,f + eM

(1.112)

where S(x) are the solutions of the harmonic oscillator, corresponding to eR as the sum of the potential energy obtained by solving the Schr€ odinger equation for the electronic and vibrational parts, referred to the spheric harmonic model. Therefore, by using different approximations, we have shown how to convert an interacting particles’ problem into a noninteracting particles’ one. Such separation of variables can be applied to a molecular system, obtaining in this way the different contributions to the total energy that, as a consequence, can be approximated as:

1.3

The Franck-Condon principle

As explained in the previous two sections, the most common way to deal with the problem of the interaction between light and matter is to resort to perturbation theory. Having in mind that, to describe the complete behavior of the system, it is necessary to solve the Schr€ odinger equation that was previously expressed in general terms for a molecule (Eq. 1.86). If our molecular system is exposed to electromagnetic radiation, the total Hamiltonian is composed by two terms: the Hamiltonian of the isolated molecule (H0) and the contribution arising from the interaction between an oscillating electromagnetic field with the charged particles of the molecule (H0 ): H ¼ H0 + H0

(1.113)

odinger H0 can then be seen as a perturbation of the initial operator H0, thus resulting in the following formulation of the Schr€ equation: HCðR, r, tÞ ¼ ECðR, r, tÞ

(1.114)

where the eigenfunction now depends also on time t, since the perturbation depends on time, due to the fact that the electromagnetic field oscillates over time. Mathematically, the eigenfunction can be expanded in terms of a weighted (through the coefficient ck) linear combination of each kth stationary wave function, Ck, describing the system: X cn ðtÞCn ðR, rÞ (1.115) CðR, r, tÞ ¼ n

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In effect, the temporal dependence can be totally included in the cn expansion coefficients, thus allowing Cn to be the same wave functions as in the isolated system: Hence, at an initial time (t ¼ 0), we can state that the system properties are solely determined by the eigenfunction of the unperturbed system, while for t > 0, the total wave function will be temporally dependent on the expansion coefficients cn. Then, the probability of finding an electron in a state m 6¼ n will be determined by: D E 2 (1.116) P∝ Ck0 j jmjCik where m is the state-transition dipole moment operator, defined as the sum of the electronic (me) and the nuclear (mN) ones: X X m ¼ me + mN ¼ e ri + e Zj Rj (1.117) i

j

where e is the elementary charge and Zj is the quantized value of electric charge of the nuclei. At this point, we need to recall once again the Born-Oppenheimer approximation, i.e., the assumption that electronic and nuclear motions can be separated. In the mathematical point of view, this means that nuclear and electronic wave functions can be accordingly separated. In the physical point of view, this means that the electronic wave function can be computed for each infinitesimally changing nuclear geometry, a condition that is reminiscent of the adiabatic theorem, and by a matter of fact explains the procedure to obtain a potential energy surface by directly applying the adiabatic approximation (see next chapter for the difference between diabatic and adiabatic representations of potential energy surfaces). Here, the Born-Oppenheimer approximation serves as a manner to separate electronic and nuclear coordinates of both Cn and Cm in Eq. (1.116). Especially, for each fixed nuclear configuration Rj, it corresponds movement of all electrons to new positions ri: * +    X     X 

 cm ri ; Rj wm Rj jmj cn ri ; Rj wn Rj Cm jmjCn ¼ (1.118) i, j i, j For convenience, summations can be eliminated, leaving 



Cm jmjCn ¼ cm ðr; RÞwm ðRÞjmjcn ðr; RÞwn ðRÞ

Separating each of the m terms that depend on the electronic (me) and nuclear (mN) coordinates, we obtain:





 Cm jmjCn ¼ cm ðr; RÞwm ðRÞjme jcn ðr; RÞwn ðRÞ + cm ðr; RÞwm ðRÞjmN jcn ðr; RÞwn ðRÞ

Since me depends only on the electronic coordinates and mN on the nuclear coordinates, we can simplify to: 









Cm jmjCn ¼ wm ðRÞjwn ðRÞ cm ðr; RÞjme jcn ðr; RÞ + cm ðr; RÞjcn ðr; RÞ wm ðRÞjmN jwn ðRÞ

(1.119)

(1.120)

(1.121)

By definition, the second term of the sum is 0 since the condition of orthonormality of the electronic states makes that hcm(r; R)j cn(r; R)i ¼ 0. Thus, we are left with





 (1.122) Cm jmjCn ¼ wm ðRÞjwn ðRÞ cm ðr; RÞjme jcn ðr; RÞ We can further separate the electronic states according to the spatial coordinates (R, r) and spin coordinates (s), thus including an additional factorial term:







 (1.123) Cm jmjCn ¼ wm ðRÞjwn ðRÞ Sm ðsÞjSn ðsÞ ’m ðr; RÞjme j’n ðr; RÞ

Therefore, the probability of encountering an electron interacting with an electromagnetic field in a state m state, different from the initial state n, is determined by three terms. The first of these is known as the Frank-Condon factor, and it will be discussed in more detail below. The second term is the spin overlap integral, which is a function of the spin coordinates and whose value depends on the initial and final spin states of the promoted electron. Finally, the third term is called the electronic transition dipole moment, which depends on the electronic coordinates and ultimately on the overlap between the spatial functions of the two electronic states. The Frank-Condon factor is at the basis of what is known as the Franck-Condon Principle, initially proposed by Franck (2) and later mathematically developed by Condon (3). As we have explained above, the n ! m transition probability due to an applied electromagnetic radiation depends on the Franck-Condon factor, which is a function of the nuclear coordinates. More in detail, this term itself indicates the

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FIG. 1.5 Vertical transitions among vibrational levels (u ! u0 ) of two different electronic states (n ! m), each of them sketched as a unidimensional potential energy (V) surface, when fulfilling (A) or not fulfilling (B) orthogonality among the Franck-Condon terms. (Credit: Original.)

overlap between the nuclear wave functions. These wave functions, as we will see in the next section, are vibrational functions and indicate the vibrational level at which the molecule is located. Therefore, the Franck-Condon factor gives us information about the extent to which the transition between the different vibrational levels of two electronic states (n and m) is allowed. Let us assume that the terms dependent on both the electronic and spin coordinates are equal to unity, leaving the transition probability as a function of a single term dependent on the nuclear coordinates, i.e., the Franck-Condon factor.

 2 P∝ wm ðRÞjwn ðRÞ (1.124)

Let us first assume an ideal case, in which the nuclear wave functions of both states are equal. This situation is indeed possible, but it implies a double requirement: the equilibrium structure of both states has to be the same, and the curvature along the nuclear coordinates also has to be the same (Fig. 1.5A). This ensures that the only difference between the two potential energy surfaces defining the two electronic states is a constant value of the potential energy surface (V). The set of nuclear wave functions being then equal for both m and n states, since it is an orthonormal set, the Franck-Condon factor results in a constant value:

 wm ðRÞjwn ðRÞ ¼ dmn (1.125)

The consequence is that the only possible transitions would be those in which there is no change in vibrational level (see blue and red arrows (dark gray and gray in print version) in Fig. 1.5A, as examples of transitions among the first and second vibrational levels of initial and final electronic state, respectively). On the other hand, if the surfaces do not match in equilibrium structure and curvature (as it would be expected for the vast majority of cases), then the Franck-Condon terms would have to be evaluated one by one, depending on the nuclear coordinates R at which vertical transition takes place, since they are now not orthogonal to each other, i.e., Eq. (1.125) is not anymore valid (Fig. 1.5B). This reasoning about vertical transitions is right underneath the Franck-Condon principle, which, in more general terms, affirms that electron rearrangements occur at a so rapid timescale that nuclei, during an electronic transition from an initial to a final state, can be considered as fixed until the electronic rearrangement in the final state is complete.

1.4

Electronic absorption spectra

1.4.1 Transition energy The Franck-Condon principle introduced in the previous section constitutes the basic principle to explain and simulate the electronic absorption spectra. Indeed, it appears now straightforward that the process of electronic excitation is referred to the vertical transition by which, at a certain time t, a molecular system is promoted from an electronic state n to a different

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electronic state m. Such electronic transition corresponds to a photon absorption by a molecule (a chromophore) at a determined nuclear geometry. Since there are photons of different energy (ultraviolet-visible solar irradiation, laser pulses, light emitting diodes, X-rays), the first condition that has to be met to obtain a spectroscopic signature related to electronic absorption, is that the photon’s energy needs to be at least equal to the energy difference among k and k0 electronic states at the instant when the photon is absorbed. Although n and m can be, in principle, whatever electronic state, when we refer to photon absorption spectroscopy, we usually mean a linear process by which a molecular system is promoted from its electronic ground state (n, the initial state) to an upper-in-energy electronic excited state (m, the final state). The reader interested in the vast and complex world of nonlinear spectroscopies should read the dedicated chapter in this book (Chapter 17). Concerning linear one-photon absorption, the aforementioned concepts can drive us to a practical simulation scheme: after having optimized and characterized the chromophore at the level of theory that we consider accurate and feasible, we can calculate the electronic excited states corresponding to that structure. In this way, if we consider the ground state of singlet nature (as it usually happens for organic molecules), we can compute all energy differences corresponding to electronic excitations that are spin-allowed, that is, conserving the spin multiplicity: S0!Sm>0 (see Section 1.5 concerning phosphorescence spectra for transitions among states with different multiplicity). With this simple procedure, we obtain a “lines spectrum,” where each line position corresponds to each energy difference. Nevertheless, one fundamental information is missing: the height of each line or, in other words, the intensity of each peak.

1.4.2 Transition intensity: The oscillator strength and the transition dipole moment Experimentally, if we direct a monochromatic light through a solution with an incident intensity I0 and a transmitted intensity I, this information can be easily obtained by using a spectrophotometer and by applying the Lambert-Beer law, as initially observed by Bouguer (4), then formally developed in a different way first by Lambert (5), secondly by Beer (6), and then combined into its final formulation (7):   I A ¼ log 10 0 ¼  log 10 T ¼ ecl (1.126) I where the experimentally measured absorbance A is logarithmically dependent on the ratio I0=I (that can also be defined as transmittance, T) and, at the same time, is linearly proportional with the molar absorptivity, or molar absorption coefficient, e, the concentration of the solution, c, and the length of the cuvette through which the light passes, l (Fig. 1.6A). Of practical use for experiments, the Beer-Lambert law has no direct counterparts in theoretical chemistry, due to the obviousness of a simulation (usually) limited to one single chromophore, eventually coupled with a solvent model. Theoreticians, on the other hand, can calculate the electric dipole oscillator strength, f, within their quantum mechanical model, although the concept arises from a classical electromagnetic model of the photon absorption. Indeed, in classical electromagnetics, fi is used as a statistical weight indicating the relative number of oscillators bound at each i resonant frequency, ni. In quantum mechanics, instead, fnm corresponds to the relative strength of the n!m electronic transition within our molecule, promoted by a photon of energy E ¼ hn. Hence, for our purpose, the oscillator strength is a nondimensional measure of the intensity of each of the calculated S0!Sm>0 transitions. Within our “lines spectrum” representation, it would therefore correspond to the height of each line (see Fig. 1.6B). Much more than that, the oscillator strength may be used to calculate the Einstein’s coefficients A (proportional to the rate of spontaneous emission of photons) and B (proportional to the absorption and stimulated emission of photons), the transition dipole matrix elements, and the transition probabilities. There is not a unique formal definition for f itself. Actually, Chandrasekhar has shown in 1945 (8) that different equivalent formulas can be drawn to calculate f, being two of them of most common use nowadays: the “dipole-length” (fLnm) and the “dipole-velocity” (fVnm) form (9):

 2 (1.127) f Lnm ¼ 2DE Cn jmL jCm f Vnm ¼

 2 2

Cn jmV jCm DE

(1.128)

where DE ¼ E(Cm)  E(Cn) corresponds to the energy difference between the two electronic states k and k0 , being common for both expressions. The difference lies instead in the definition of the transition dipole moment, m: mL ≡

K X k¼1

zk

(1.129)

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I Fundamentals

(A)

(B)

Incident intensity

Transmitted intensity

I

I0

t

en

m eri

p

Lambert-Beer

Ex

fs0os2

Relative intensity

20

fs0os1

ls0os1

ls0os2

l (nm)

Electronic states +

ory

e Th

u = 0 ou⬘ = 0

(D)

Molecular conformations

(C)

Vibronic resolution +

Relative intensity

Relative intensity

S0 oS2

l (nm)

S0 oS1 Conformation 1

Conformation 2

l (nm)

u = 0 ou⬘ = 1 FIG. 1.6 Scheme showing the different strategies to simulate theoretically a UV-Vis absorption spectrum. Experimentally, following the Lambert-Beer law, the absorbance can be measured for a wavelengths’ window, considering incident and transmitted intensity of the radiation through a solution-filled cuvette within a spectrophotometer (A). Increasing complexity, the simulation can include (B) only electronic states transitions energies and intensities, eventually reproducing the bands’ broadening through Gaussians’ convolution, (C) electronic states and conformational information, (D) electronic states, conformational, and vibrational resolution. (Credit: Original.)

mV ≡

K X ∂ ∂z k k¼1

(1.130)

where zk is the coordinate of the kth electron, and N is the number of electrons. The Hartree atomic units’ system is used (ħ ¼ m ¼ e ¼ 1). Which is the most accurate way to calculate f ? The “dipole-length” or the “dipole-velocity” form? It can be clearly seen by the equations’ formulation (Eq. 1.127–1.130) that f Lnm ¼ f Vnm when Cn and Cm are exact wave functions. The problem is that, for most of real molecular systems’ problems, approximate wave functions need to be employed. In these cases, f Lnm and/or f Vnm can be much different from the “true” f, i.e., the f value calculated if the exact wave function would be available, that hence stays unknown. This means that the previous question needs to be reformulated: which is the most accurate way to calculate f, when approximate wave functions are employed? Over the past 80 years, many scientists tried to answer, attempting to give a basis for deciding which form is the more accurate for a given molecular study. One of the criteria relies on the wave function changes during the vertical transition: spatial regions distant from the nuclei, and thus less important to the energy, determine an unexpected contribution from f Lnm, while the wave function optimization based (usually) on an energy criterium should be more accurately described by f Vnm (10,11). Nevertheless, what above stated was checked only for transitions involving a change in the principal quantum number, while for the rest of transitions it was argued that f Lnm is more accurate (12). In other cases, the attention was rather focused on the quantum mechanical method used to calculate the wave functions, concluding that f Lnm is the “physically

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correct” choice for a certain class of variational methods including configuration interaction and Hartree-Fock (13). Due to the evident lack in finding a general answer, a mixed approach was proposed by Roginsky and Klapisch (14), based on a weighted linear combination of mL and mV (amL + bmV) that can be practically afforded by modifying just one of the two wave functions, Cn or Cm, into Cn(a, b) or Cm(a, b). Which one of the two should be selected for applying this procedure? Roginsky and Klapisch suggest using the one more similar to the initial wave function. An alternative approach was developed by Jennings and Wilson (15) to calculate positive and negative “error bounds” for each f value, in order to choose the one having the highest relative accuracy. In the more general extension proposed by Anderson and Weinholdt (9), a procedure was finally established for each individual transition, taking into account the principal quantum number of the transition, the relative size of the transition moments, the transition energy, and the quality of the wave function, being able to propose a ratio between the “dipole-length” based-error (DL) and the “dipole-velocity” based-error (DV) that can finally determine, for each case study, when it is preferred one or the other f formulation.

1.4.3 Absorption band shape: The dynamic effect and the vibronic coupling Therefore, although with some technical issues, we have now the required knowledge to complete a “lines spectrum,” each line starting on the x-axis at the corresponding transition energy, and with a corresponding line height given as an f value. Nevertheless, at a certain temperature (e.g., room temperature), an absorption spectrum usually assumes the shape of one or multiple bands, in some cases well-defined bands, and in some other cases partially overlapping bands, that can be considered as broadening of the “lines spectrum.” Hence, the next questions that we should answer are: Which is the origin of each band? And what is the meaning of the shape of each band? In other words, while an experimentalist usually measures an absorption spectrum and, a posteriori, tries to assign each peak—or shoulder among peaks—to a certain characteristic electronic transition of some species in its sample (making a sort of deconvolution effort), the theoretician first computes the transition energies and relative intensities presumably corresponding to the maximum of each peak, and only after chooses the method to simulate the shape of each band associated to each peak (making a sort of convolution effort). Several times, by working in close collaboration for the same case study, only a coupled experimental-theoretical effort made possible to disclose the complete and rational assignment of an absorption spectrum. Many examples can be found in literature, especially concerning organic chromophores in liquid solvents or biological media, some of which (far away from being a complete set) could be found in the following Refs. (16–21). Until now, we have considered that a single global minimum structure (the Franck-Condon structure) is the responsible for the totality of the absorption spectrum. Nevertheless, organic compounds, metal-organic complexes, and especially macromolecules as peptides, proteins, and more in general polymers are characterized by a landscape of many different conformations that could be in principle all of them relevant in terms of population of the electronic ground state. If more than one conformation is representative of the ground state, then vertical transitions corresponding to each conformer should be considered (Fig. 1.6C). This can be accomplished in different ways, depending on the conformational landscape and computational approach: for little molecules of no more than 40–50 atoms, a full conformational search could be performed, based on all possible combinations of their dihedral angles defined by the internal coordinates (in the case that more dihedrals than the ones defined by the internal coordinates are necessary, one can always resort on redundant internal coordinates, a procedure that is indeed applied by most of nowadays quantum chemistry codes). Each optimized structure confirmed as a potential energy minimum will have a certain ground state energy, and therefore, a Boltzmann distribution could be applied a posteriori to assign to each conformer a certain weight of the total ground state population. Hence, the intensity of the vertical transitions of each conformation can be normalized by a factor corresponding to the relative ground-state population, following the principle that the most populated conformer will have more opportunities to absorb a photon, and therefore to show its spectroscopic signature. Such a deterministic approach is not feasible for macromolecular entities that, most of the times, have a large number of dihedral angles, hence determining a really high number of local minima, in some cases separated by just few kcal/mol and, hence, all of them representing almost evenly the ground state potential energy landscape. What to do in these cases? Since macromolecular systems can be hardly described (in their full complexity) by quantum mechanical models, one has to apply classical molecular mechanics, i.e., mathematical formulas determining the vibration of each covalent bond, angle and dihedral around their respective equilibrium value, thus determining (together with classic formulations of noncovalent interactions) the so-called force field of the macromolecule. There are specific force fields for each type of macromolecule (proteins (22), lipids that form biological membrane (23), nucleic acids (24), artificial polymers (25–28), etc.) and for most of the solvent molecules and ions that are explicitly included in these classical molecular simulations. This allows to perform classical molecular dynamics (i.e., integrating Newton’s laws) that can reach several microseconds of simulation time. Also, several replica trajectories can be run starting from different initial conditions (i.e., different nuclei positions and

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FIG. 1.7 Scheme showing methodologies to include the dynamical effects in the simulation of the UV-Vis absorption spectrum, in this example limited to the S0 ! S1absorption band: A potential energy surface can be built around the ground state minimum energy structure (qS0eq) through quadratic approximation, followed by dynamics on the S0 analytic surface or Wigner distribution to properly sample the S0 phase space (A, B). Once the S0 structures are extracted, the relative S0 ! S1excitation energies can be obtained by (A) energy difference with the S1 analytic surface, also built by quadratic expansion, or (B) single-point excited state calculations. For large molecular systems (e.g., biomolecules, polymers, etc.) a convenient and affordable strategy is to perform a priori classical “force field”-based molecular dynamics on the ground state, followed by selection of snapshots (qS0) and single-point excited state calculations (C). (Credit: Original.)

different associated velocities). By these extensive classic approaches, it can be noted, especially by an a posteriori analysis of the root-mean square-deviation (RMSD) how, along each trajectory, the macromolecule can assume different conformations. By properly sampling the system along the simulations, we can extract n system structures and, for each of them, perform an absorption spectrum calculation (Fig. 1.7C). Of course, this requires a quantum mechanical treatment, as aforementioned, and usually a hybrid quantum mechanics/molecular mechanics (QM/MM) treatment can serve as a solution: the chromophoric part of the macromolecule, susceptible of being in the energy window to absorb a photon, is treated by an appropriate quantum mechanical method, while the rest of the macromolecules and the environment can be treated by a classical molecular mechanics potential, since it is not expected to interact with the incoming electromagnetic radiation. This means that, through classical molecular dynamics, the conformational space of a macromolecule can be satisfactorily explored if (1) the force field is accurate enough, (2) the sampling time is high enough, and (3) the amount and type of initial conditions are adequate. Although all the three criteria can be reasonably fulfilled, the dynamic approach is conceptually different from the static (potential energy optimization) approach since the total energy is inevitably the sum of potential and kinetic energy. On the macromolecular side, this difference can be eliminated by optimizing the potential energy of a certain number of trajectory snapshots (thus eliminating the dynamic contribution) prior to calculation of the electronic absorption spectrum. This is also a viable strategy to identify the number and type of macromolecular conformations effectively populated by our system, since different groups of snapshot structures, once geometrically optimized, could fall within the same conformer potential well (21). But what about the opposite? What if we would like to include the dynamical effects into the fully optimized conformers of our small molecule? This can be performed with different strategies, of which maybe the most common one is the use of the information retrieved form the quadratic expansion that gives us the force constants applicable to each normal mode of our molecule. On one hand, in terms of computational approach, this means that the shape of the potential energy surface around the energy minimum of each conformer can be obtained through the calculation of the respective frequencies, all of them being positive real values. On the other hand, in terms of theoretical description, this means that: EGS ðDqÞ ¼ EGS ðqeq Þ +

1 T Dq HGS Dq 2

(1.131)

where EGS is the ground state energy, qeq is the ground state equilibrium geometry, HGS is the Hessian matrix calculated for this geometry, and Dq ¼ q  qeq is the vector describing the displacement from the equilibrium geometry. This approximation relies, therefore, on the assumption that a quadratic approximation of the energy around the minimum geometry is enough to correctly describe the dynamic effects. If this assumption does not hold, then additional terms should be added to Eq. (1.131), in order to include the required anharmonicity to our harmonic model. In most of the cases, the quadratic approximation is a convenient and reliable choice to build our analytic potential energy surface and, based on that, two main techniques are commonly applied to account for dynamics effects: (1) molecular dynamics on the analytically built surface or (2) Wigner distribution. Classical molecular dynamics can be indeed performed on the analytic surface, e.g., selecting the temperature for sampling the phase space offered by the quadratic approximation. In this way, a certain number of snapshots can be collected

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along the trajectory, each snapshot containing the displaced geometry q with respect to the original equilibrium geometry qeq. For each of the snapshots’ geometry, a single-point excited state calculation can be performed, finally obtaining a “lines spectrum” for each of the sampled geometries (Fig. 1.7B). A posteriori Gaussian convolution and intensity normalization (usually based on Boltzmann distribution to assign a population weight to each conformer on the ground state, based on its Gibbs free energy) of all the “lines spectra” will provide a simulation, at a certain temperature, of the electronic absorption spectrum, including its shape. This approach is computationally highly efficient concerning the classical molecular dynamics, since it is performed on an analytical surface; nevertheless, the bottleneck lies in the amount of quantum mechanical single-point calculations that has to be run a posteriori. A possible solution to avoid this bottleneck is to build a quadratic analytical surface also for the electronic excited states of interest (EES), also around qeq. In this case, not only the Hessian matrix but also the gradient contribution (gES) will appear in the energy equation, since for qeq the forces (and hence the gradient of the energy) will not be null at any excited state surface: EES ðDqÞ ¼ EES ðqeq Þ + DqT gES +

1 T Dq HES Dq 2

(1.132)

This more sophisticated approach allows to obtain the absorption spectrum by calculating only the excitation energies corresponding to the qeq geometry, hence not necessitating the a posteriori single-point calculation for each of the extracted snapshots (Fig. 1.7A). Indeed, in this case, for each structure along the dynamics, EGS and EES values can be analytically given (Eqs. 1.131 and 1.132), thus being the electronic excitation energy: Eexc ¼ EES ðDqÞ  EGS ðDqÞ

(1.133)

Also in this case, a final convolution of Gaussian functions is required to simulate the absorption shape. This strategy was successfully applied to simulate the UV-Vis absorption spectrum of S-nitrosothiols (29), azobenzene (30) and, within the quadratic approximation, it can be in principle applied to any molecular system. Recently, this same strategy was applied to elucidate phosphorescence (i.e., singlet-triplet energy gaps, see Section 1.5) spectra of porphyrin systems (31). Although highly reliable, due to the construction of an analytic ground state potential energy surface, would it be possible to include dynamic effects in absorption spectra simulations, without performing molecular dynamics at all? It seems a contradictive question, but the answer is actually yes: again, based on the Hessian matrix (quadratic expansion) built for each equilibrium structure (or minimum energy conformer), a Wigner distribution can be then performed. It is not in the scope of this book to give a detailed description of the Wigner distribution (also called Wigner-Ville distribution); hence, we will briefly describe it as a quasiprobability distribution that allows to connect mathematically the probability distribution in a portion of phase space to C, the wave function of the Schr€odinger’s equation (see Sections 1.1 and 1.2): Z W ðq, pÞdp ¼ jCðqÞj2 (1.134) where p is the momentum and W(q, p) is the Wigner distribution:     Z 1 q q ðipq ℏ Þ C∗ e W ðq, pÞ ¼ C q + dq q  2 2 ð2pℏÞ3

(1.135)

Computationally wise, the application of the Wigner distribution results in the approximation of the quantum oscillator and provides a desired set of geometries and velocities (momenta) around qeq, which is the same type of outcome obtained by molecular dynamics following the quadratic approximation but avoiding an explicit molecular dynamic run. For the rest, the same applies concerning the need of single-point excited state calculations for each displaced geometry, as well as Gaussian convolution to obtain the simulated absorption spectrum. This strategy was profitably applied to simulate the UV-Vis electronic absorption spectra of a certain number of organic molecules, including chlorin-e6 (32), boron containing arenes (33), and different thiophene derivatives (34,35). Until now, we have considered the dynamics’ effect acting on each of the molecule conformers as the only source of the absorption spectrum shape. In other words, we have briefly introduced theoretical strategies and techniques to correctly describe, within certain approximations, the nuclear displacements that the molecule is undergoing around qeq. This is indeed true if we limit our vision to the purely electronic structure theory. Nevertheless, in a broader photophysicochemical point of view, we have already introduced the concept of vibrational levels defining the depth of the potential well for each electronic state of our system (see Fig. 1.5 in Section 1.3). According to this concept, we can therefore define an electronic absorption spectrum not only as the simple photon absorption that causes population of an electronically excited state, but as the vertical transition from some vibrational levels of the electronic ground state

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to other vibrational levels of the electronic excited state (see arrows in Fig. 1.5). In the molecular point of view, this means that the incoming electromagnetic radiation involves the interaction between electronic and nuclear vibrational motions of different states. More in detail, the absorption spectrum recorded through an experiment, results from the absorption of several photons, that correspond to multiple vertical energy transitions from different vibrational levels of the electronic ground state, accessible at a certain temperature, to various vibrational levels of the electronic excited state/s. Such intrinsic relation between the electronic and vibrational nature in molecules is referred as vibronic coupling, and the magnitude of such coupling, proportional to the overlap between ground and excited vibrational wave functions for a particular vertical transition, has a direct influence on the absorption spectrum shape. Although, as we will see in Chapter 2, vibronic couplings are fundamental to the understanding of nonadiabatic processes of paramount importance in photochemistry (i.e., conical intersections), the underlying fact that the Born-Oppenheimer approximation neglects the vibronic coupling, makes it difficult and hence uncommon to be directly calculated. However, in the following, we will see how it is possible to calculate vibrationally resolved electronic spectra, definitely helping in assigning the spectrum shape to a certain vibrational fingerprint of a given molecule (Fig. 1.6D). To figure out in more detail vibrationally resolved electronic spectra, we have to take a step back, understanding more in-depth the origin of line intensities in absorption spectra: from Eqs. (1.127) and (1.128), we already know that such intensities are proportional to the square of the transition dipole moment hCnj mj Cmi. Now, considering a vibronic framework within the Born-Oppenheimer approximation, we can assume the total molecular wave function as the product of an electronic (e) and a nuclear (N) wave function, for both initial (n) and final (m) electronic states: Cn ¼ ’en ’Nm ; Cm ¼ ’em ’Nm The transition dipole moment can then be formulated as follows:  



Cn jmjCm ¼ ’en ’Nn jme j’em ’Nm + ’en ’Nn jmn j’em ’Nm

(1.136)

(1.137)

where the second term of the right-hand side of Eq. (1.137) vanishes since every electronic wave function is orthogonal to each other. Moreover, the nuclear wave function can be approximately separated in three terms: translational (tr) that can be completely discarded for single-molecule studies; rotational (rot), related to energies small enough to be considered negligible; vibrational (u), thus resulting in the prominent term: ’N ¼ ’tr ’rot ’u  ’u Eq. (1.137) can be therefore approximated as:





 Cn jmjCm ¼ ’en ’Nn jme j’em ’Nm  ’en ’un jme j’em ’um

(1.138)

(1.139)

Rearranging:

 



Cn jmjCm  ’un ’en jme j’em ’um ¼ ’un jmenm j’um

where the purely electronic transition dipole moment

(menm)

(1.140)

is:



menm ¼ ’en jme j’em



(1.141)

As for the only electronic methods described above, also the vibrationally resolved methods can resort, at least at a first instance, to the harmonic approximation. This means that the vibrational wave function of the molecule (’un for the ground state), of intrinsic multidimensional nature, can be formulated as a product of monodimensional wave functions: ’un ¼

K Y

’uni ðQi Þ

(1.142)

i

where Qi is the normal coordinate defining the ith molecular vibration, with the molecule being described by K normal coordinates. Nevertheless, the problem to solve Eq. (1.140) relies in the unknown analytic form of the electronic transition dipole moment (Eq. 1.141). To overcome this obstacle, the Franck-Condon principle (Section 1.3) needs to be invoked: following the Franck-Condon principle, the highest probability for the vertical transition to occur corresponds to the highest overlap among the wave function of the vibrational level at electronic ground state (i.e., the initial state) and the wave function of the vibrational level at the electronic excited state (i.e., the final state). A convenient visual description of such vibrational mode overlaps between initial and final electronic states can be given by the Duschinsky matrix (36), a matrix composed by the indexes of the vibrational modes of the initial electronic state on the x-axis and of the final electronic state on the y-axis, with each matrix element being an overlap value from 0 to 1, or 100%. The more the structural difference

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between electronic state energy minima of initial and final states (qS0eq and qS1eq in Fig. 1.7A), the more probable event after photon absorption is that an excited vibrational level of the final state (i.e., u0 > 0) is populated, then followed by vibrational relaxation until reaching its ground vibrational level (u0 ¼ 0). From there, different photophysical and photochemical phenomena could take place: fluorescence emission (Section 1.4), intersystem crossing and phosphorescence emission (Section 1.5), population of crossing regions among electronic states of the same multiplicity to obtain one or more photoproducts (Chapter 2). The most important consequence underlying the application of the Franck-Condon principle is the assumption that menm stays constant during the transition (2,3). Since this is physically not strictly true but, at the same time, the nuclear displacements within the molecule are supposed to be rather small during the electronic transition, it can be proposed to express menm by a Taylor series around qeq m , that is, the equilibrium geometry of the final electronic state: !eq !eq K0 K X K 2 e e X X
∂m 1 ∂ m nm nm qm,i + qm,i qm,j + … (1.143) menm ðqm Þ ffi menm qeq m + 2 i¼1 j¼1 ∂qm,i ∂qm,j ∂qm,i i¼1 where i and j are indexes running over the N normal modes. According to the above-mentioned Franck-Condon principle, if we consider only the 0th order term of Eq. (1.143), i.e., the first term of the right-hand side, we are assuming a constant transition dipole moment, thus simplifying Eq. (1.143) as follows:
menm ðqm Þ ffi menm qeq (1.144) m

For vibrationally resolved spectra, this approximation is therefore called Franck-Condon (FC) approximation, that is expected to be in good agreement with the experiment if jmenm(qeq m )j ≫ 0, i.e., when the electronic transitions are completely allowed (optically bright transitions). Nevertheless, if the electronic transitions are partially allowed (optically dark), then e eq jmenm(qeq m )j 0, and for strictly dipole-forbidden transitions jmnm(qm )j ¼ 0. In these latter cases, the FC approximation does not offer a correct description of the spectrum, because of the lack of the more intense vibronic transitions. Hence, the simplification of a constant transition dipole moment should be abandoned in favor of its variation during the transition. Especially, the 1st order term of Eq. (1.143), i.e., the second term of the right-hand side, takes into account a linear variation of the transition dipole moment along the normal coordinates, ending up in the Herzberg-Teller (HT) approximation (37): !eq K X ∂menm e mnm ðqm Þ ffi qm,i (1.145) ∂qm,i i¼1 Both FC and HT approximations can be considered at the time, resulting in the FCHT approximation, thus resulting in the sum of the first two terms of the right-hand side of the general Eq. (1.143): !eq K
X ∂menm e e eq qm,i (1.146) mnm ðqm Þ ffi mnm qm + ∂qm,i i¼1

All in all, although vibrationally resolved electronic absorption spectra are evidently computationally more expensive to be computed, compared to the usual only electronic methods aforementioned, several organic chromophores of small and medium size were successfully treated by the FC, HT, or FCHT approximations, in some cases resulting in valuable studies to assign the shape of the absorption spectrum to a certain vibronic transition (or group of vibronic transitions), instead of different geometrical conformations (17,38–41).

1.4.4 Multiphotonic absorption Until now, we have focused on the most studied and applied electronic absorption mechanism: one-photon absorption (1PA), belonging to the so-called linear spectroscopy, where linear means that the attenuation of a beam of light by an absorbing material (the absorbance) is linearly dependent on the intensity of the light beam (see the Lambert-Beer law expressed in Eq. (1.126) and Fig. 1.5A). At first physicists in the past, and recently also chemists and biologists interested in enhanced properties for their materials or tissues, draw the attention on multiphotonic absorption. Although, in principle, the absorption of a number k of photons is possible (k > 1), due to experimental setup complexity and structure-to-property nontrivial prediction, most of the technology is nowadays developed for two-photon absorption (2PA), to serve the fields of high-precision lithography (42–45) and photomedicine (46–48). A valuable niche investigating three-photon absorption

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(3PA) (49) is nevertheless recently active in metal coordination chemistry (50,51) and, at a more applied level, metalorganic frameworks (52,53) and metal nanoparticles (54). Also, some general assessment regarding 4PA and 5PA was established through experimental state-of-the-art photonic studies and analytical theories to characterize such multiphoton absorbers (55–57). All kPA (k > 1) absorption mechanisms can be collected under the name of nonlinear absorption spectroscopy. Indeed, among the possible multiphotonic absorption processes, in 2PA the intensity (I) can be expressed as follows: ∂I ¼ rM a2PA I 2 ∂z

(1.147)

where z is the direction toward which the light is polarized (see Fig. 1.1), rM is the molecular density (i.e., the number of molecules per unit of volume), and a2PA is the 2PA molecular coefficient. The latter takes the form of: s a2PA ¼ 2PA (1.148) hn where s2PA is the molecular 2PA cross-section and n ¼ n1 ¼ n2 is the frequency of the incident lasers. This corresponds to the most used degenerate 2PA setup where, indeed, the two laser beams have the same frequency. In more general terms, the two light sources can have different frequency (n1 6¼ n2), resulting in nondegenerate 2PA (see Fig. 1.8). Substituting Eq. (1.148) in Eq. (1.147): ∂I s ¼ rM 2PA I 2 ¼ rM s2PA FI ∂z hn

(1.149)

where the photon flux F ¼ I=hn. In any case, by Eqs. (1.147) and (1.149), it appears now clear why 2PA is a nonlinear process: the absorbance does not depend anymore linearly on the light intensity, but quadratically. Both the facts that two laser pulses are required and that nonlinear absorption is expected make of 2PA a highly desirable technology because of the following reasons (58): (i) The simultaneous absorption of two photons to populate an electronic excited state allows, if degenerate 2PA is considered, to divide the required excitation energy by halves. Since energy, E, and photon wavelength, l, are inversely related (E ¼ hn ¼ h lc , where c is the speed of light and h is the Planck’s constant; see Sections 1.1 and 1.5.1.1), this means that the half of l can be used to absorb the same photon energy that would be absorbed through a 1PA process, thus remarkably red-shifting the necessary incoming photon energy (also called bathochromic shift). If the therapeutic spectral window (approx. 650–1350 nm) can be reached, it can be ensured the highest possible biological tissue penetration, thus constituting a highly desirable goal in photomedicine. Indeed, if l < 650 nm, the tissue penetration (or the engraving lithography thickness) is much lower and, at the same time, if l > 1350 nm, water infrared absorption becomes predominant (59,60).

FIG. 1.8 Electronic energy levels’ scheme (vibrational levels are omitted for clarity) summarizing |ni ! | mi one-photon absorption, 1PA; |ni ! | ii ! | mi two-photon absorption, 2PA, including degenerate (hn1 ¼ hn2) and nondegenerate (hn1 6¼ hn2) types; |ni ! | ii ! | i + 1i ! | mi three-photon absorption, 3PA, with hn1 ¼ hn2 ¼ hn3, i.e., degenerate type. In all cases, the same eventual stimulated fluorescence is expected. (Credit: Original.)

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(ii) Following the reasoning in (i), high-energy excited states could be populated by adding the energy of the two simultaneously incoming photons, thus opening potential paths toward unexplored photochemistry (61–66), in terms of both photochemical quantum yields’ modification and generation of novel photoproducts. (iii) As aforementioned, the absorbance (and therefore the probability) for the two incoming photons to be absorbed simultaneously, is proportional to the square of the laser intensity. This means that, compared with 1PA, 2PA is intrinsically more precise in terms of space: when leaving the laser focal point, nonlinear absorption processes decrease in intensity much faster than linear ones. This apparent drawback is, on the other hand, an important advantage for any application requiring high spatial precision, e.g., skin tumors that have to be cured in limited surface regions, avoiding side effects in the nearby skin region (67,68), or 3D laser lithography for nanofabrication (69–71). As shown in Fig. 1.8, whatever nonlinear kPA absorption process (i.e., k > 1) requires v ¼ k  1 virtual states and i ¼ k  1 intermediate states for its description, apart from initial (n) and final (m) states, already introduced in linear 1PA absorption (see Fig. 1.3). If we focus on 2PA absorption, its physical modeling requires therefore three quantum states (|ni, |ii, and |mi) and one v state. On one hand, the v state can be simply seen as the virtual state placed hn1 energy above the |ni energy and hn2 energy below the |mi energy. On the other hand, the real |ii state, evidently of transient nature, is somewhere placed between |ni and |mi energies. More in detail, if a first incoming photon is lower in energy than the n ! m transition energy but different (higher or lower) than the n ! i transition enegy (i.e., hn1 < E|mi|ni; hn1 6¼ E|ii|ni), than the uncertainty principle—initially proposed by Heisenberg (72) and then mathematically derived first by Kennard (73) and then by Weyl (74)—allows |ii to exist for a lifetime based on the energy detuning between hn1 and E|ii|ni (usually 1–5 fs (75)). Hence, based on this criterium of resonant frequency, the more hn1 approaches E|ii|ni, the higher the |ii lifetime. If, within this time, a second photon interacts with the molecule, the electron (initially promoted from |ni to |ii by the first photon) could be excited until populating the |mi final state, where it stays during the lifetime allowed by that electronic excited state. Although the concept of vibronic coupling introduced in Section 1.4.3 does certainly apply also for multiphotonic absorption, for the sake of simplicity, we will neglect it all along the description of the 2PA process. As of now, we have only named s2PA as the molecular 2PA cross-section in Eq. (1.148). In the following, we will understand why and how s2PA can be considered a measure of absolute intensity in 2PA spectra. First, we should focus on the quadratic proportionality between s2PA and the modulus of the transition dipole moment tensor, Snm: 2 (1.150) s2PA ¼ ab Snm ¼ abd2PA where d2PA ¼ jSnmj2 and is therefore defined as the 2PA transition probability (or transition strength), a is a multiplicative factor, and b is a normalized line shape function: a ¼ ð2peÞ4 n1 n2 ðchÞ2

(1.151)

b ¼ gð n 1 + n 2 Þ

(1.152)

By sum-over-states analysis, Snm can be expressed as: Snm ¼

ðl1  mn!i Þðmi!m  l2 Þ ðl2  mn!i Þðmi!m  l1 Þ + E|ii|ni  hn1 E|ii|ni  hn2

(1.153)

where l1 ¼ (l1, x, l1, y, l1, z) is the polarization vector of the first photon, l2 is the polarization vector of the second photon, mn!i is the transition dipole moment from initial to intermediate state, and mi!m is the transition dipole moment from intermediate to final state. The sum-over-states approach allows to conveniently generalize Eq. (1.153) for kPA absorption processes when k > 2, and hence, more than one |ii state is involved (3PA, 4PA, …): Snm ¼

ðl1  mn!i Þðmi!i+1  l2 Þ ðl2  mn!i Þðmi!i+1  l1 Þ + E|ii|ni  hn1 E|ii|ni  hn2

X ðl  m ðl  m Þðmi+1!i+2  li+1 Þ i+1 i!i+1 Þðmi+1!i+2  li+2 Þ + i+2 i!i+1 + E|i+1i|ii  hni+1 E|i+1i|ii  hni+2 i +

!

(1.154)

ðli+2  mi+1!i+2 Þðmi+2!m  lk Þ ðlk  mi+1!i+2 Þðmi+2!m  li+2 Þ + E|mi|i+2i  hni+2 E|mi|i+2i  hnk l1

l2

li+1

li+2

where the first term refers to the |ni ! |ii ! |i+ 1i starting steps, the middle term to the |ii ! |i+ 1i ! |i+ 2i transitions li+2 lk among intermediate states, and finally, the third term to the |i+ 1i ! |i+ 2i ! |mi conclusive steps.

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Back to 2PA, in disordered liquids (i.e., all usual chemical solutions), the molecular cross-section is directly proportional to the transition probability averaged over all spatial orientations: s2PA ∝ hjSnmj2i. The polarization vectors l1and l2 are evidently under the control of the experimentalist, with the only physical requirement for them to be of unit length: l1  l1 ∗ ¼ l1   l2 ∗ ¼ 1. Concerning their mathematical definition and physical nature, both polarization vectors are real in case of applying linearly polarized light (see Fig. 1.1), while they are complex in case of circularly or elliptically polarized light. The resulting Snm is therefore a 3  3 tensor spanned over the Cartesian axes that can be conveniently diagonalized for calculation purposes: " # xy Snm ¼

Sxx nm Syx nm zx Snm

Snm Syy nm Szy nm

Sxz nm Syz nm zz Snm

(1.155)

where the first column corresponds to a vector describing the transition dipole moment contribution along the x-axis, yx zx y z Sxnm ¼ (Sxx nm, Snm, Snm), while second and third columns are Snm and Snm, respectively. Experimentally, the multiphotonic absorption cross-section is usually given in G€oppert-Meyer units (1 GM ¼ 1050 cm4 s molecules1 photons1), as 2PA was first predicted in 1931 by Maria G€oppert-Meyer (76), and only 30 years later observed thanks to the discovery of lasers (77). As it can be easily evinced by the theoretical description, 2PA experiments do need more sophisticated setups compared with 1PA. Indeed, in the case of 2PA, we need to measure the laser’s energy, the laser’s pulse width, and the laser’s spatial distribution due to its propagation through the sample (78). Two main experimental techniques were developed, depending on whether direct or indirect 2PA measurements are performed: 2PA direct absorption measurements are needed for the “z-scan technique,” where z stands for the axis defined as the linear distance between the laser beam and the detector (79); indirect 2PA absorption measurements emerged later (80,81), and require the 1PA and 2PA absorption spectra of a reference compound (e.g., rhodamine B for organic compounds). Then, a comparison between 1PA and 2PA induced fluorescence spectra of reference and sample compounds allows to cancel out many of the required variables for a direct 2PA measurement, finally retrieving the sample 2PA absorption spectrum. As drawbacks, this latter technique needs a proper reference compound and sample fluorescence, hence resulting in a more affordable but less general technique compared to the direct 2PA absorption measurement. After having introduced the main concepts, we can attempt to find molecular structure-to-property relationships. As first, let us split chromophores between centrosymmetric and noncentrosymmetric ones: for an organic dye to absorb light in the UV-Vis spectral range, a p-conjugated chemical structure is required, with the addition of donor (D) and acceptor (A) groups acting as substituents. This can lead to the design of D-p-D and A-p-A centrosymmetric molecules or to D-p-A noncentrosymmetric molecules. Indeed, although not formally explained, in the recent past, most of the attention to raise 2PA absorption cross-section values was focused on centrosymmetric molecules carrying strong D or A substituents, as they were found to generate a more persistent and energetically convenient intermediate state (82,83). Nevertheless, an increasing interest in multiphotonic absorption during these last years, mainly due to its attractive potential applications, has helped in designing novel noncentrosymmetric 2PA active absorbers characterized by a relevant s2PA value (32,33,35,84).

1.5

Fluorescence and phosphorescence emission

After discussing the absorption process of one or multiple photons as vertical transitions to energetically higher electronic excited states, in this section, we address the emission processes, fluorescence, and phosphorescence. Both can be defined as luminescent processes, in the sense that both are characterized by the emission of a photon (spanning from ultraviolet to visible and infrared spectral ranges) from an electronically excited state to the ground state. In particular, fluorescence and phosphorescence are photoluminescent processes as the excited species are generated after photon absorption, in contrast to chemiluminescent or bioluminescent processes, where the emissive state is populated by thermal energy due to a chemical reaction (see Chapter 12). Although the photophysics of fluorescence and phosphorescence are similarly described (photon emission from an excited state populated after photon absorption), their difference arises from the multiplicity of the electronic state involved in the radiative process. In fluorescence, the radiative process takes place between an excited and a ground state of the same spin multiplicity, whereas when states of different spin multiplicity are involved then phosphorescence occurs. This feature determines other differential properties between fluorescence and phosphorescence. For instance, fluorescence is strongly spin-allowed, leading to intense and short-lived emission. However, phosphorescence is weakly spin-allowed, giving less intense and long-lived emission. In the following, we are going to discuss more in detail these two radiative processes.

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1.5.1 Fluorescence As aforementioned, this type of emission takes place between electronic states of the same spin multiplicity. In most cases, the emission occurs from the lowest-energy singlet excited state (S1) to the ground singlet state (S0), apart from a few exceptions belonging to the anti-Kasha fluorescence (85–89). Fluorescence is a process largely used in several applications mainly due to the high sensitivity of the emission detection (fluorometric techniques), fluorescence specificity and spatial/temporal resolution. In particular, these processes have been largely used in the biological field for medical diagnostics, genetic analysis, DNA sequencing, cellular and molecular imaging, etc. (90–94). Moreover, through fluorescence studies, molecular information of a given compound can be retrieved. For instance, information about the compound surrounding environment can be obtained as its emission could be largely sensitive to it, a process known as solvatochromic fluorescence (95,96). This finding has been exploited by using these compounds as fluorescent probes to get insights into the electrostatic nature of the environment (i.e., hydrophobic protein pockets and water clusters around a reaction center (97–99)). Highly solvent dependent fluorophores were also engineered with the green fluorescent protein (GFP) as building block, making possible to obtain a wide palette of fluorescence colors (100). It should be remarked that the fluorescence spectrum is specific of each compound, being a characteristic feature of it. For this reason, understanding the definition of this spectrum and which information can be retrieved from it is crucial.

1.5.1.1 Fluorescence spectrum A fluorescence spectrum can be defined as a distribution in energy of the emitted photons. In other words, for each photon, a value of the fluorescence intensity (y-axis) corresponds to a certain energy (x-axis). Although the x-axis should be therefore expressed in energy units (eV are usually a convenient choice for phenomena ranging visible and ultraviolet windows), it is nowadays a typical choice of experimentalists to use a wavelength (nm) that is inversely related to the photon energy by the Planck’s equation: c E ¼ hn ¼ h (1.156) l where E is the photon’s energy, h is the Planck’s constant, n is the frequency, c is the speed of light, and l is the corresponding wavelength. As a whole, i.e., when considering all photons emitted by a certain species, the fluorescence spectrum can be understood as a probability distribution of the emission from the lowest vibrational level of S1 to the differently available vibrational levels of S0. If the absorption and fluorescence spectra of a given compound are compared, we can observe some special features. First, although the 0-0 energy (energy difference between the lowest vibrational levels of the excited and ground states) is the same for absorption and fluorescence, the absorption spectrum is located at shorter wavelengths (higher energies) than the fluorescence spectrum (101). If we limit our study to S0!S1 absorption, this finding can be explained by an energy loss following S1 population, mainly through intramolecular vibrational relaxation from high to low vibrational levels of the excited state (Fig. 1.9). This feature is usually observed but, there are some cases in which the absorption and fluorescence spectra partially overlap. This means that some emitted photons have larger energy than some of the absorbed photons, which is in clear contradiction with the previous reasoning. This finding can be easily explained if we consider that at a given temperature, some molecules can access vibrational levels higher than the zero one in the ground state. Hence, as long as the temperature decreases, the overlap of the absorption-fluorescence spectra should decrease. As a consequence of this energy loss after absorption and prior to emission, the lowest-energy absorption (S0!S1) and emission (S0 S1) bands usually do not overlap. From this fact arises another characteristic feature of a given compound, known as Stokes shift, firstly observed by Sir George Gabriel Stokes in 1852 (101). This shift is defined as the difference (in energy or wavelength) between the maxima of S0$S1 0-0 absorption and emission bands, where 0-0 refers to vertical transitions among zeroth vibrational levels (Fig. 1.10). It is important to note that the Stokes shift can be sometimes quite sensible to solvent effects (102–105), having different values for the same compound in different solvents. Indeed, through the analysis of the Stokes shift values measured in different solvents, insights into the electronic nature of the absorption and emission processes can be revealed. For instance, the Stokes shift increases with the solvent polarity when the dipole moment of the molecule is larger in the excited state than in the ground state (106,107). Moreover, if excited state reactions, energy transfer to or from the media, and complex formation occur, it could also influence the Stokes shift values for a given compound (108–111). These experimental observations would be described in detail in Section 1.5.1.5.

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FIG. 1.9 Schematic representation of the vibrational relaxation (VR) from the populated vibrational level of the S1 excited state after absorption (blue arrow, dark gray in print version) to its lowest vibrational level. Emission from the S1 lowest vibrational level to the ground state (S0) is depicted as a green line (gray in print version), showing its lower energy compared with the absorption process. (Credit: Original.)

FIG. 1.10 Absorption (blue, dark gray in print version) and emission (green, gray in print version) spectra showing the transitions between the different vibrational levels of ground and excited state, leading to the “mirror image rule.” The Stokes shift is also depicted. (Credit: Original.)

Additionally, the fluorescence spectrum could resemble in shape the absorption one, a finding known as the “mirror image rule” (112–114), although it has exceptions: three requirements need to be fulfilled to understand if this rule applies. The first one is that emission should take place from the lowest vibrational level of the S1 excited state to high-in-energy vibrational levels of the S0 ground state. Secondly, emission should happen at a similar molecular structure with respect to the Franck-Condon structure. Thirdly, the energy difference between the vibrational levels should be similar for both

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ground and excited state. If all these requirements are met, the “mirror image rule” applies (Fig. 1.10). However, as the reader could easily evince, this rule mainly applies only to rigid molecules, for which ground and excited state minima are similar, and this is generally not the case. The effect of solvent, usually a liquid solution, can also break the “mirror image rule,” due to the fact that a much higher vibrational broadening is expected for fluorescence, ending up in a broader S0 S1 emission band when compared to the S0!S1 absorption band. Other deviations from the “mirror image rule” can arise when reactions in the excited state take place after photon absorption, for instance, excited state proton transfer reactions, formation of charge-transfer complexes, etc. (115–118).

1.5.1.2 Kasha’s rule Unlike absorption, one characteristic feature of fluorescence is that, whatever the selected excitation wavelength is, the same fluorescence spectrum would be recorded. This fact is known as the Kasha’s rule (119) and could be explained considering the following: after excitation to higher in energy excited states (Sn, n > 1) or vibrational levels (n > 0), a fast energy relaxation to the zeroth vibrational level of the lowest-in-energy electronic excited state (S1) takes place. This energy relaxation, called internal conversion, is much faster than photon emission from upper-in-energy (Sn, n > 1) excited states, as the energy difference between electronic excited states is significantly lower than the energy difference between ground state and the upper-in-energy excited states. Only when the system is in S1, then internal conversion to Sn (n > 1) is competitive with emission to the ground state. Hence, the emission of a photon would generally occur from the zeroth vibrational level of S1 to a manifold of vibrational levels in S0, giving the same fluorescence emission spectrum, irrespective of the absorption wavelength. Some exceptions to the Kasha’s rule can happen when the energy separation among electronic states is somewhat peculiar, as it was already reported S0 S2 emission for some given compounds (120–122). In such cases, the “mirror image rule” applies if considering S0!S2 absorption band and S0 S2 emission band. This could happen if: (i) the energy difference between S2 and S1 is too large, making S1 S2 internal conversion slower than S0 S2 emission. Fluorescence from the S2 state would be experimentally observed only if the oscillator strength for this transition is relatively high (Fig. 1.11A) (85–87). (ii) S2 is close in energy to S1, allowing a fast S1 S2 internal conversion, but the oscillator strength of the S0 S1 emission is nearly null, then not allowing to observe fluorescence from this state. Hence, if the oscillator strength of the S0 S2 emission is much higher, then fluorescence from this state could be measured if the S1 lifetime is long enough to significantly populate back S2, through a reversed S1!S2 internal conversion mechanism (Fig. 1.11B) (88,89).

FIG. 1.11 Schematic representation of two Kasha’s rule exceptions. (A) Large energy difference between S2 and S1 favoring emission from S2 against its relaxation to S1 and further emission from this state. (B) Small energy difference between S2 and S1 and really low oscillator strength of S1 emission, hence observing mainly emission from S2. (Credit: Original.)

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1.5.1.3 Fluorescence lifetime and quantum yield As explained in the previous section, after electronic excitation the system relaxes to the lowest vibrational level of the lowest-in-energy excited state (Kasha’s rule (119)). Once the system is in n ¼ 0 of S1, it can decay to the ground state (S0) through both fluorescence emission or nonradiative processes. The most common nonradiative processes are intersystem crossing, internal conversion, dissociation, etc. (123–125). Before completing this decay, the system stays in S1 an average time that is well known as excited state lifetime. More rigorously, excited state lifetime is defined as the average time the system spends in S1 before decaying to S0. The more general equation to define the excited state lifetime τ considers both radiative and nonradiative decay processes (123–125): τ¼

1 kr + knr

(1.157)

where kr and knr are the rates of the radiative and nonradiative processes, respectively. If nonradiative decay processes are negligible (knr 0), then the excited state lifetime only depends on the rate of fluorescence, and it takes the name of intrinsic or natural lifetime (τn). In most cases, τn and τ do not match, being τn larger than τ, due to the significant nonradiative decay rate compared to fluorescence rate. Hence, τ approaches τn only when the nonradiative processes are much slower than fluorescence (knr ≪ kr). If we define both radiative and nonradiative decay processes as first-order reactions, then we can make use of chemical kinetics to define the velocity of S1 disappearance, by considering that the S1 excited reactant can be deactivated on a twochannels competitive basis: 

d ½ S1  ¼ ðkr + knr Þ½S1  dt

(1.158)

By solving this differential equation through integration from [S1]0 (just after absorption) to [S1]t (at a later time), an exponential decay of the concentration of molecules in S1 during time can be set: ½S1 t ¼ ½S1 0 eðτÞ t

(1.159)

As a special case, when t ¼ τ: ½S1 t ¼ ½S1 0 eð1Þ ¼

½ S1  0 e

(1.160)

Hence, the S1 lifetime can be also defined as the time needed to decrease the concentration of molecules in S1 by a fraction 1/e from its initial value (at t ¼ 0). As the concentration of molecules in S1 is proportional to the fluorescence intensity, then the S1 lifetime can be experimentally obtained by monitoring the emission intensity decay along time. When the fluorescence intensity decreases a factor of 1/e from the initial value, this time can be recorded as the S1 lifetime. It should be stressed that fluorescence is a spontaneous and random process hence, not all the molecules will decay to the ground state at the same time. For this reason, the excited state lifetime is defined as an average time. Few molecules will decay at a time equal to the lifetime but others would decay at times smaller or larger than the lifetime. Lifetime is also a characteristic of a given compound, and it gives valuable information about the excited state, as during this time the system can interact with the environment or diffuse. Moreover, during this time solvent molecules can reorient around the excited state dipole, making emission quite sensitive to solvent polarity (126). Another important property of fluorescence, which is directly related to the excited state lifetime, is the fluorescence quantum yield (’f) that can be defined as the ratio between the emitted and the absorbed photons. In other words, ’f is the fraction of molecules initially populating S1 decaying to the ground state through fluorescence (123–125). In mathematical terms: ’f ¼

kr kr + knr

(1.161)

By substitution of Eq. (1.157) in Eq. (1.161), the fluorescence quantum yield can be expressed as: ’f ¼ k r τ Alternatively, considering the natural lifetime as the inverse of the radiative reaction rate: τ ’f ¼ τn

(1.162)

(1.163)

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Eqs. (1.161)–(1.163) make clear that fluorescence quantum yield values range between 0 and 1. A value of 1 indicates that nonradiative rates are negligible compared with fluorescence rates. The other way round, a value of 0 indicates that only nonradiative processes are taking place. Prediction of the fluorescence quantum yield is not a trivial task. Of course, if we have knowledge of the order of magnitude of the radiative and nonradiative decay rates, the quantum yield could be roughly estimated a priori. Nevertheless, there are some structural criteria for the specific compound to meet, as a sort of strategy for the researcher interested to ensure a significantly high fluorescence quantum yield (127–130). Indeed, all these criteria are focused on diminishing the nonradiative decay rates: (1) Compounds consisting in a rigid structure usually minimize nonradiative decays that are on the other hand favored if the structure is flexible, allowing rotations and vibrations associated with normal modes described by low-force constants (131). (2) In addition, compounds with low spin-orbit coupling values, decreasing the probability of intersystem crossing, would ensure higher fluorescence quantum yields (e.g., avoiding heavy atoms in the structure, as well as mixing of orbitals of different nature (132–134)). Regarding experimental measurement of the fluorescence quantum yield (135), the easiest approach is to use a standard solution with a ’f already known (measured with high precision) (136). To properly apply this method, first an accurate selection of the standard should be done. Attention should be paid when preparing the samples as both, the compound and the standard samples should have the same absorption at the excitation wavelength to ensure that both absorb the same number of photons. It is also recommended choosing a standard which emission spectrum covers a similar spectral range as the compound emission. Moreover, concentration effects should be checked and other experimental parameters, such as temperature, should be kept equal for both measurements. To apply this method, the fluorescence spectra of both the compound and the standards should be recorded for several samples at different concentrations. Only the samples within the linear regime would be considered. Then, the integrated fluorescence intensity (the area under the fluorescence spectrum) is calculated. Finally, the integrated fluorescence intensity vs the absorption of each sample for both, the compound and the standard, is plotted, retrieving the slope of each plot. The fluorescence quantum yield of the compound can be calculated as: ’compound ¼ ’standard f f

mcompound 2compound mstandard 2standard

(1.164)

where mcompound and mstandard are the slopes of the aforementioned plots for the compound and the standard, respectively and compound and sample are the refractive indices for the solvent used for the compound and standard sample preparation.

1.5.1.4 Fluorescence quenching It has been experimentally observed that in some cases, the fluorescence lifetime, intensity, and quantum yield of a given compound can be decreased by the presence of other molecules. This observation is commonly known as fluorescence quenching and the molecule responsible for it is called quencher (130,137–139). A wide variety of quenchers have been described being molecular oxygen one of the most known (140). For this reason, to correctly measure the fluorescence of given compounds, it is necessary to remove the dissolved oxygen from the sample. Within the same molecule, substituents can act as quenchers, as aromatic and aliphatic amines, mainly affecting the fluorescence of aromatic hydrocarbons (141– 143). In addition, compounds containing heavy atoms such as halogenated derivatives are known fluorescence quenchers (132). Although a wide variety of fluorescence quenchers have been already reported, not all are able to quench the emission of whatever compound. To evaluate and/or predict if the fluorescence of a given compound could be quenched by a certain molecule, then the mechanism behind the quenching process should be proposed/understood. Mainly, two different types of fluorescence quenching mechanism can be observed (124,144). The first one is known as collisional quenching (also known as dynamic quenching), based on S1 nonradiative deactivation after collision between the compound and the quencher (Fig. 1.12A). This type of quenching reduces both, the fluorescence quantum yield and lifetime of a given compound. The second type is the static quenching, which arises from the formation in the ground state of compound-quencher complexes which are nonfluorescent (Fig. 1.12B). Hence, only the molecules free from forming the complex in the ground state would emit after photon absorption. For this reason, static quenching does not modify the S1 decay, thereby not modifying the fluorescence lifetime of the compound. Hence, lifetime measurements can get insights into the quenching type that takes place, depending on its value being influenced or not by the presence of the quencher. In the following, we are going to describe in detail these two types of fluorescence quenching mechanisms.

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FIG. 1.12 Mechanistic representation of the two discussed types of fluorescence quenching: (A) Collisional or dynamic quenching and (B) static quenching (C, compound potentially fluorescent; Q, quencher). (Credit: Original.)

In collisional quenching, the ratio between the fluorescence intensity in presence (F) and absence (F0) of the quencher can be related to the ratio of their respective decay rates: F g ¼ F0 g + kq ½Q

(1.165)

where g denotes the decay rate in absence of the quencher, kq is the bimolecular collision constant and [Q] is the quencher molar concentration. Quenching is an additional process that decreases the population of the excited state. For this reason, the term kq[Q] should be added to consider the rate of all the processes depopulating the excited state (denominator in Eq. 1.165). It should be stressed that the viscosity of the medium can influence the rate of compound-quencher collision, as this results from diffusive encounters. Since the excited state decay rate corresponds to the inverse of the lifetime in absence of quencher (g ¼ τ1 0 ), Eq. (1.165) can be rewritten as: F 1 1 ¼ ¼ F0 1 + kq g1 ½Q 1 + kq τ0 ½Q

(1.166)

This relation between the fluorescence intensities and the decay rates (in absence/presence of the quencher) is well known as the Stern-Volmer equation (145,146) in the form: F0 ¼ 1 + kq τ0 ½Q ¼ 1 + K D ½Q F

(1.167)

where KD is usually known as the Stern-Volmer quenching constant. By analyzing Eq. (1.167), it can be concluded that F the F0/F ratio is expected to depend linearly on the quencher molar concentration. Hence, by easily plotting F0 vs [Q], the Stern-Volmer quenching constant can be retrieved from the representation slope. Knowing the value of KD is of special F interest as its inverse (K1 D ) corresponds to the quencher concentration halving F . Coming back to Eq. (1.165), the right side 0 of the equation corresponds to the ratio of the excited state lifetime in presence and absence of the quencher since, as we 1 know, g ¼ τ1 0 and, additionally, (g + kq[Q]) ¼ τ : F τ ¼ F0 τ 0

(1.168)

This relation explains a characteristic feature of collisional quenching: the decrease in the fluorescence intensity is equivalent to the decrease in the excited state lifetime. Moreover, the observed decrease in the fluorescence quantum yield in presence of a quencher can be easily explained by the fact that this process competes with fluorescence, being a nonradiative process. As previously introduced, fluorescence intensity can be also decreased by static quenching. While the collisional (dynamic) quenching is a time-dependent process, according to diffusion of both compound and quencher, the static quenching is not time-dependent, hence its name. Static quenching is governed by the formation of “compound-quencher complex” in the electronic ground state, prior to photon absorption. With the term compound-quencher complex, we define the supramolecular assembly, sometimes called supermolecule as introduced by Karl Lothar Wolf et al. in 1937 (147),

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formed by (at least) one compound molecule and one quencher molecule, not covalently bonded. In this case, the decrease in fluorescence intensity relies on the fact that the formed complex is not fluorescent and therefore, after photon absorption, nonradiative decay to the ground state is the only possible deactivation channel. The remaining fluorescence intensity, if any, in the presence of the quencher is due to free compound, not complexed with the quencher and hence still able to emit after photon absorption. To get insights into the ratio of the fluorescence intensity in presence and absence of the quencher, we start from the compound-quencher complexation process. The association constant (K) for the complex formation can be defined as: K¼

½C⋯Q ½C½Q

(1.169)

where [C⋯Q], [C], and [Q] are the molar concentrations at the equilibrium of the compound-quencher complex, of the free compound, and of the free quencher, respectively. The complex concentration can be defined as the difference between the initial concentration and the concentration of free compound: ½C⋯Q ¼ ½C0  ½C

(1.170)

After substitution of Eq. (1.170) in Eq. (1.169), the association constant assumes the form: K¼

½C0  ½C ½C0 1  ¼ ½C½Q ½C½Q ½Q

(1.171)

Considering that the fluorescent intensity is proportional to the free compound concentration, we can write Eq. (1.171) as a function of the fluorescence intensity instead of the free compound concentration:   F0 1 F0 1 ¼ (1.172) K¼  1 ½Q F½Q ½Q F From this equation, we can obtain an expression that relates the fluorescence intensity ratio as a function of the association constant and the free quencher concentration: F0 ¼ 1 + K ½Q ¼ 1 + K D ½Q F

(1.173)

By comparing Eqs. (1.173) and (1.167), we can conclude that both the collisional and static quenching are ruled by a linear dependency of the fluorescence intensity ratio with the quencher concentration. Hence, this is not a differential feature and the quenching type cannot be discerned through this data. It should be noted that while for collisional quenching the slope of the representation yields the Stern-Volmer constant (kqτ0), in the case of static quenching this term corresponds to the association constant (K). As aforementioned, the main differential property of static quenching is that the excited state lifetime in presence of quencher remains unchanged, being the same as in absence of quencher. The free compound, responsible for the emission, is the same in presence and in absence of quencher and so, its emission properties as the fluorescence lifetime. In some cases, both collisional and static quenching take place for a given compound in presence of a quencher (148,149). In these more complex cases, the representation of the F0/F versus the quencher concentration does not result in a linear trend, since it appears an upward curvature, concave toward the y-axis.

1.5.1.5 Factors influencing fluorescence Apart from the presence of a quencher, other factors can alter significantly the fluorescence properties of a given compound. These factors can be intrinsic to the compound or external such as the environment or other experimental parameters. Regarding to the intrinsic factors, the intensity of the emission is largely related to the electronic nature of the lowest-inenergy singlet excited state (S1) and to the structure rigidity (150). In general, the fluorescence of compounds with a S1 state characterized by a (p,p*) transition presents a stronger emission and larger fluorescence quantum yield than compounds which S1 state is described by a (n,p*) transition. This feature can be explained by the shorter lifetimes and slower nonradiative decay rates of S1(p,p*) states. For S1(n,p*) states, nonradiative decay processes such as intersystem crossing with a triplet state are usually fast, mainly due to the smaller singlet-triplet energy difference and enhanced spin-orbit coupling involving this type of states, e.g., 1(n,p*)!3(p,p*). Another characteristic of the compound that can affect its own fluorescence is its rigidity. The more rigid the structure is, the more intense its fluorescence (131). Hence, compounds with free to rotate bonds (e.g., single carbon-carbon bonds) or easy to bend bonds should be characterized by low emission intensity

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as this flexibility could promote nonradiative decay until reaching the ground state, thus disadvantaging fluorescence. Moreover, the fluorescence of bromine or iodine compounds is in general not quite intense, due to efficient intersystem crossing as a nonradiative decay process, depopulating the excited state and competing with emission (132). This general trend is justified by the relatively enhanced spin-orbit coupling values of these types of compounds, hence favoring the aforementioned nonradiative process. Concerning the external parameters, both the environment properties and experimental parameters should be considered. For instance, a temperature increase causes in general a decrease in the fluorescence quantum yield and lifetime (151,152): as long as the temperature increases, the rate of nonradiative processes due to thermal agitation (i.e., rotations, vibrations, collisions) increases, competing with the fluorescence emission. Also, when it comes to external parameters, the rigidity of the medium can play a role. For instance, samples prepared in highly viscous solvents or using rigid materials such as glass would favor fluorescence emission (153–155). A key external parameter that could significantly influence fluorescence is solvent polarity (126,156). Polar compounds exhibit fluorescence sensitivity to the solvent polarity (i.e., solvatochromism), whereas the fluorescence of nonpolar compounds is usually not influenced by the solvent polarity. This effect is pronounced when S1, the excited state responsible for fluorescence, is a charge transfer transition. Indeed, studies of the fluorescence energy dependence on the solvent polarity are quite useful to get insights into the polarity of a complex environment surrounding the chromophore, for instance within a protein or other biomolecules (96,98,99,157). In general, polar solvents are expected to red-shift (i.e., decrease in energy) the fluorescence of polar compounds. As already discussed in this section, the emission energy is always lower than the absorption energy due to energy loss through vibrational relaxation. Moreover, further stabilization may arise from polar solvent effects: usually, the excited state dipole moment is larger than the one in the ground state. Hence, after excitation, the solvent molecules have time to reorganize around the excited chromophore, depending on the nature of the excited state dipole moment (Fig. 1.13). The solvent reorganization is the responsible of the further stabilization of the excited state energy. So, depending on the solvent polarity, the emission energy can be modulated and so its fluorescence spectrum. In general, for polar compounds, the higher is the solvent polarity, the lower is the emission energy, and hence, the emission band is shifted to longer wavelengths (Fig. 1.13). It should be noted that solvent relaxation is significantly faster (10–100 ps) than fluorescence (1–10 ns). However, this solvent orientation does not take place during absorption, as it is slower than photon absorption. This is the reason why fluorescence is more sensitive to solvent polarity than absorption. So, for a given compound, the Stokes shift (i.e., the energy difference between the maxima of the first absorption and emission bands) can vary depending on the solvent polarity. In order to rationalize the interdependence among emission spectrum, Stokes shift and solvent effect, the theory of general solvent effects (156) can be applied. Within this framework, some approximations should be appraised. For instance, the chromophore is assumed to be spherical and considered as a dipole in a continuous medium with a given dielectric constant. That is, explicit interactions such as hydrogen-bond interactions between the solvent and the compound are not considered. By evaluating the chromophore-solvent interactions and the excited state dipole moment change upon

FIG. 1.13 Representation of the solvent relaxation process after absorption, stabilizing the first singlet excited state responsible for emission. The polarity effect on the further energy stabilization is depicted. (Credit: Original.)

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absorption, an expression to predict the energy difference between the absorption and emission can be obtained, known as the Lippert-Mataga equation (158–160):     m m 2 e g 2 e1 n2  1  nabs  nem ¼ +d (1.174) hc 2e + 1 2n2 + 1 a3

where nabs and nem are the absorption and emission wavenumbers (cm1), respectively; h is the Planck’s constant; c is the speed of light; e is the solvent dielectric constant and n the solvent refractive index; me and mg are the dipole moments in the electronic excited and ground state, respectively. Finally, a is the Onsager cavity radius of the fluorophore (i.e., the emissive chromophore) in that solvent, and d is an additive constant term accounting for the energy loss due to vibrational relaxation in the electronic excited state after photon absorption, as discussed in Section 1.5.1.1. The first term of the Lippert-Mataga equation can be separated in three factors: a first constant factor ð2=hcÞ, the solvent orientation polarizability factor, and a third factor which is a function of the fluorophore structure and excited state dipole moment change upon photon absorption. More in detail, the solvent orientation polarizability factor can be further sepa
rated in a term depending only on the dielectric constant 2ee1 , accounting for the effect on the Stokes shift of both, the +1 reorientation of the solvent dipoles and the redistribution of the electrons of the solvent molecules  2  according to the fluorophore excited state dipole moment, and a term depending only on the refractive index 2nn 21 + 1 , accounting only for the redistribution of the electrons of the solvent molecules. Since the overall orientation polarizability is the difference between these two terms, then only the reorientation of the solvent dipoles after photon absorption happen to influence the Stokes shift as the solvent electronic redistribution cancels out. Physically, this can be explained by the instantaneous redistribution of the solvent electrons in both ground and excited state, hence stabilizing equally both electronic states and therefore, not affecting the Stokes shift. Although the Lippert-Mataga equation gives a general and simple expression to predict the Stokes shift modulation by the solvent polarity, deviations from the theory of general solvent effects have been reported (161–163). Evidently, the solvent dependence of the fluorescence emission is a quite complex topic, as it refers to different parameters regarding the solvent itself as well as fluorophore-solvent interactions. As aforementioned, the Lippert equation does not consider explicit fluorophore-solvent interactions, hence reliable correlations could be obtained only in case of weak hydrogen-bond interactions. Indeed, when strong hydrogen-bond interactions are present between fluorophore and the surrounding solvent molecules, the experimentally recorded emission energy will be further red-shifted but the Lippert-Mataga model cannot take it into proper account (163). Moreover, depending on the solvent polarity, the electronic nature of the S1 emitting state be different, leading to different emission spectra. For instance, if high polar solvents are used, charge transfer excited states are likely to be largely stabilized in energy, possibly resulting in the S1 state. However, if nonpolar solvents are used, charge transfer excited states are usually relegated to high-in-energy electronic states, above locally excited states. Therefore, the relative stability of the excited states could be reverted depending on the solvent polarity. Apart from the Lippert-Mataga model, which is dated back to 1957, alternative solvent polarity functions were hence proposed to increase the precision of the predictions, as the Bakhshiev (164), the Bilot-Kawski (165–167), and the Chamma-Viallet (168) models, all of them developed during the 1960s until reaching the early 1970s. More recently, single- and multiparameter analyses were introduced, in order to use experimental solvatochromic data to quantitatively predict the contribution of different types of interaction on the fluorescence specific properties (169). Among them, some multiparameter analyses were proven to provide reliable quantitative descriptions of solvatochromic shifts due to specific and nonspecific fluorophore-solvent interactions: the Kamlet-Taft and the Catalan models (170), with the latter offering the advantage over the former of a separation of the nonspecific solvent effects in the two terms of polarity and polarizability.

1.5.1.6 Steady-state vs time-resolved fluorescence Along the previous sections, we have presented diverse characteristics of fluorescence emission as well as factors affecting it. In those discussions, it has been assumed that recorded emission takes place after continuous sample illumination with a continuous beam of light. Under continuous illumination, it can be considered that the concentration of the excited chromophore is constant, and so this concentration reaches the steady state. In other words, the rate of change in concentration of the excited chromophore is equal to zero, as the excited specie is formed after photon absorption, but it is also eliminated through decay processes leading to the ground state. For this reason, the emission measurements performed under these conditions are commonly known as steady-state measurements and the recorded spectra are known as steady-state spectra. This is the most common type of fluorescence measurement. The experimental results can be assumed as an average of the processes occurring after absorption and while excited state relaxation is taking place. In the simple cases in which, after absorption, only a fast vibrational relaxation takes place (no chemical reactions or conformational changes are occurring),

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the steady-state spectrum is enough to characterize the system. However, in more complex cases, quite valuable information is lost during the time averaging process. In those cases, emission could be observed prior or during alternative relaxation processes or excited state reactions are taking place. Hence, information about these processes could be retrieved from fluorescence data taken at different illumination times. These experiments are more complex than steady-state measurements and require expensive instrumentation. In particular, the sample is illuminated with a pulse of light, which width is in general shorter than the decay time of the sample. Then, the intensity decay is recorded with a high-speed detector, which allows having the fluorescence decay evolution at the ns timescale, after illumination. These experiments are commonly known as time-resolved emission experiments (171–174). Recording the emission spectra at discrete times following the pulse excitation would lead to the time-resolved emission spectra. In the cases where an excited state reaction or conformational equilibrium takes place after absorption, the steady-state spectrum would contain contributions from each form (if both are fluorescent) but no kinetic information of these processes can be obtained. For this reason, time-resolved fluorescence measurements are crucial to properly characterize the excited state nature and eventual photo-reactivity of a given compound. Moreover, the time-resolved data would allow the determination of the excited state decay lifetime due to each process. Hence, the time-dependent spectral changes have to be analyzed in detail. Two main types of spectral changes can be observed. The first one is characterized by a shift of the emission spectra but keeping the spectral shape, commonly known as continuous spectral shift (Fig. 1.14A) (174). A process driving to this type of spectral change is solvent relaxation, from absorption upon emission (175–178). This spectral change would be more evident as long as the solvent organization rate decreases, being its rate comparable with the emission rate. Hence, solvent reorganization could take place during emission. Different spectral changes are observed when a chemical reaction, a charge transfer or a conformational equilibrium occurs in the excited state after absorption. In this case, not only a spectral shift is observed but also the spectral shape can be

(A)

(B)

hnabs

Normalized emission intensity

Q= 2 Q= 1 Q= 0

S1 VR

hnem

Q= 2 Q= 1 Q= 0

Solvent relaxation

Q= 2 Q= 1 Q= 0

t2

t3

Wavelength

t4

1 treact

1 tprod

prod

hnem

S0

S0

t1

react hnem

1 t

S0

Ch e rea mica ctio l n

VR

hnabs

hnem

1 t

S1

Normalized emission intensity

Q= 2 Q= 1 Q= 0

S0

t4

t1

t2

t3

t3

t2

Wavelength

FIG. 1.14 Scheme of the mechanism (top) and the time-resolved spectra (bottom) recorded for (A) the continuous spectral shift and (B) the excited state reaction model or the two-state model. (Credit: Original.)

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modified (179–181). This type of change is known as the excited state reaction model or the two-state model (Fig. 1.14B). In these cases, different excited molecular species can be relevant, considering the initial specie and the ones formed after absorption, which could emit or not. If more than one excited specie would emit, their fluorescence emission could be different as their electronic states or structure are different. If the rate of the excited state process is slow compared with the fluorescence decay of the initial specie, then only its fluorescence would be recorded. However, if the excited state process is significantly faster than the emission decay of the initial specie, only the fluorescence of the formed excited specie would be measured, if it is fluorescent. An intermediate picture arises when the rate of the excited states processes is comparable to the emission decays of the initial and formed species. In this case, the emission of both species could be observed depending on the discrete time after excitation at which the fluorescence is recorded.

1.5.1.7 Anti-Stokes photon emission Until now, applying the restriction due to the law of energy conservation, we have considered that the energy of the emitted photon has to be lower than that of the excitation photon or, in other words, fluorescence is expected always at longer wavelengths. This energy difference is at the basis of the Stokes shift defined above. But, what would happen if somehow fluorescence is more energetic than absorption? We will be in presence of anti-Stokes luminescence. Of course, since the overall energy still has to be conserved, some different mechanisms need to be introduced, all of them relying on nonlinear processes. In particular, three processes can be invoked: two-photon absorption (see Section 1.4.4), photon upconversion, and hot band absorption (Fig. 1.15) (182). Anti-Stokes luminescence can be indeed a rather straightforward consequence of two-photon absorption: the populated excited state has an energy corresponding to the sum of the two photon frequencies. Even considering some possible excited state vibrational relaxation, if nonradiative decay channels are absent, the fluorescence would result in emission of a higher frequency photon (Fig. 1.15A). This process is used for the corresponding fluorescence imaging technique called twophoton excitation microscopy, of special interest for in situ biological measurements (183,184). An apparently similar scheme can be proposed by photon upconversion, since they share the final outcome: population of a luminescent excited state at higher energy than the sum of two previously absorbed photons. Nevertheless, the main difference lies in the nature of the excited state population: while two-photon absorption calls for a two-(electronic)states (ground and excited) model separated by a virtual state between them, upconversion technologies require a three-(electronic)state model, thus including a real (nonvirtual) intermediate state. This can be achieved in two different ways: by lanthanide (Ln)-based and triplet-triplet annihilation-based upconversion (Fig. 1.15B). On one hand, lanthanide ions can have long-lived (hence relatively stable) excited states functioning as intermediate states (185). If coupled with ad

FIG. 1.15 Energy level representation of the different anti-Stokes mechanisms: (A) Two-photon absorption (TPA)-based fluorescence; (B) lanthanide (Ln)-based and triplet-triplet (T-T) annihilation-based upconversion; (C) hot (vibrational) band absorption-based fluorescence applied to Coherent anti-Stokes Raman Scattering (CARS). hnabs,s, electronic absorption within the sensitizer; hnA-fl, anti-Stokes fluorescence; IC, internal conversion; ISC, intersystem crossing (the last two terms will be explained in the following section). (Credit: Original.)

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hoc selected sensitizers, such intermediate states can be in turn excited toward the final state, from where anti-Stokes emission can follow (186,187). On the other hand, triplet sensitizers are at the core of the triplet-triplet annihilation-based upconversion: triplet-triplet energy transfer from the sensitizer to the emitter makes possible, if the emitter ensures a longlived triplet state, that two emitter triplets do encounter, favoring their annihilation to generate a higher-energy excited singlet state, from where anti-Stokes emission is observed (188–190). This latter upconversion technology is especially attractive for increasing the efficiency of solar cells (191,192). Hot band absorption could be also used to obtain anti-Stokes luminescence (Fig. 1.15C). In this case, we go back to a twostates model, as in two-photon absorption, but only one incoming photon is required, obtaining the required “extra” energy from the vibrational frequencies of the electronic ground state. Specifically, if our compound can be stably prepared at a high-energy vibrational mode (e.g., by heating) the required photon energy to promote population of the lowest lying electronic excited state will be lower with respect to a cold compound, that is vibrating through low-energy vibrational modes. The resulting fluorescence can be therefore at higher energy than absorption if the cold band is populated by vertical emission (193). Hence, in principle, hot band absorption is the less convenient process to gain energy for an anti-Stokes emission (194). Nevertheless, this particularity of being precise over the ground state vibrational modes, makes it of high appeal for characterization techniques. Indeed, Coherent anti-Stokes Raman Scattering (CARS) microscopy allows the combination of spectroscopy and microscopy to get specific signals of studied compounds (195). In this case, an ultrafast (fs to ps) pump-probe experimental setup is required, since the laser pump is applied to firstly produce the hot band by a Stokes emission, followed by a laser probe (usually at the same frequency of the pump) to populate the electronic excited state responsible of anti-Stokes emission.

1.5.2 Phosphorescence Photon emission (luminescence) from an electronic excited state to the electronic ground state could be also due to phosphorescence. The main difference between the two photoluminescent processes, fluorescence and phosphorescence, is that in phosphorescence the radiative decay involves states of different spin multiplicity, m: m ¼ 2S + 1

(1.175)

where S is the total spin quantum number or total spin angular momentum. Most of the reported phosphorescent processes are due to radiative decay from the lowest-in-energy triplet state (T1, m ¼ 3, S ¼ 1) and the singlet ground state (S0, m ¼ 1, S ¼ 0). With this respect, a little digression should be made: although most of the times chromophores do have T1 higher in energy with respect to S0, we cannot formally speak of T1 as of an electronic excited state. For sure, T1 usually corresponds to an energetically higher state than S0. Nevertheless, in the pure electronic point of view, T1 is the electronic ground state with m ¼ 3 (by convention, T0 does not exist). Also, please note that in some molecular systems T1 happens to be lower in energy than S0, corresponding to the global electronic ground state, i.e., the lowest-in-energy electronic ground state of any spin multiplicity. A typical example is molecular oxygen, O2, which is present in the terrestrial atmosphere as inert triplet oxygen, while the eventual production, usually through energy transfer, of singlet oxygen, 3O2!1O2, converts it into a higher energetic molecule treated as a real danger for biological processes, including DNA, proteins and, more in general, cellular damage. In the laboratory, the presence of singlet oxygen could be therefore detected by phosphorescence from hn singlet oxygen, through a S0!T1 (1O2!3O2) luminescent process (196–198). Coming back to the origins of phosphorescence, it was in 1944 when G.N. Lewis and M. Kasha explained phosphorescence as the T1!S0 radiative decay (199). In addition, they stated that the phosphorescence emission spectrum of a given compound is a signature of it, and hence, it can be used for its identification and spectroscopic characterization. It should be remarked that whatever emission between electronic states of different spin multiplicity is a spin forbidden hn transition, therefore expecting it to be kinetically slower and less intense than fluorescence. Regarding the overall T1! S0 phosphorescence process, prior to T1 radiative decay, this triplet state should be populated. But, if the transition between states of different multiplicity is a spin-forbidden transition, how can T1 be populated from a (usually) singlet electronic ground state? Two possible solutions do arise: (a) S0!T1 thermal processes or due to energy transfer, thus not involving formally any electronic excited state or (b) triplet population through electronic excited states. The former solution does not apply to many real cases, although a highly discussed system corresponds to the paradigmatic nonvertical (non-FranckCondon) triplet-triplet energy transfer involving cis-stilbene as acceptor molecule (200–203). Hence, the latter solution is the one most found in practice for organic compounds: the first step corresponds to a vertical (Franck-Condon) excitation from the singlet ground state (S0) to an upper-in-energy singlet excited state (Sn, n 1). In the simplest case, S1 is populated and vibrational relaxation can lead the system to the S1 lower vibrational level. If an upper-in-energy singlet excited state is populated (Sn, n > 1), the system needs to relax through both, vibrational relaxation within each singlet excited state and

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internal conversion from higher to lower singlet excited states, until reaching, also in this case, the S1 lower vibrational level. The two, vibrational relaxation and internal conversion, are relatively fast relaxation processes taking place right after photon absorption. One has to note that, if the system relaxes in S1 prior to population of the triplet manifold, fluorescence emission can also take place on a competitive basis with phosphorescence. In general, any nonradiative transition from any singlet/triplet electronic state is known as intersystem crossing, constituting in principle, as aforementioned, a spin forbidden transition. However, it may happen that the physico-chemical characters of a certain couple of singlet and triplet states are mixed, due to the overlap of their vibrational levels. This mixing is defined as spin-orbit coupling and it favors intersystem crossing, together with energy proximity of the two interacting states. Once T1 is finally populated, relaxation to the lowest vibrational level takes place, from where photon emission to the singlet ground state occurs, leading to phosphorescence. An overall scheme of the possible mechanisms is shown in Fig. 1.16. For most of the chromophores, and as pictorially shown in Fig. 1.16B, T1 lies lower-in-energy than S1. Hence, for a certain compound, the phosphorescence spectrum is usually red-shifted compared with the fluorescence spectrum, as hn hn hn the T1!S0 energy transition is lower than the S1!S0 transition. Moreover, as the T1!S0 transition is spin forbidden, phosphorescence is characterized by a lower emission intensity, smaller rate constant, and longer lifetime compared with fluorescence. By analyzing the general scheme of phosphorescence, it can be easily concluded that it would be favored by increasing singlet-to-triplet intersystem crossing rate constants. The factors influencing these rate constants have been already introduced in the previous section when discussing fluorescence emission, as this process competes normally with

FIG. 1.16 Diagrammatic representation of the phosphorescence process. (A) Reactive scheme for nonvertical and vertical induced phosphorescence, including only the possible electronic states involved. Both vertical and nonvertical paths are shown, including internal conversion (IC) and intersystem crossing (ISC). (B) One-dimensional potential energy surfaces of the vertical (Franck-Condon) mechanism, including vibrational relaxation (VR): after excitation (hnabs) to the excited singlet manifold (S1 or Sn>1), fast VR and IC phenomena take place, possibly followed by fluorescence (hnfl). If ISC occurs, then the triplet manifold is populated and after relaxing through VR and IC to the lowest T1 vibrational level, phosphorescence takes place (hnph). (Credit: Original.)

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fluorescence. As a reminder, the presence of heavy atoms in the structure would increase the intersystem crossing rate constant, decreasing the fluorescence but favoring phosphorescence. hn In addition, the longer T1 lifetime, compared with S1, as a consequence of the T1!S0 spin forbidden transition, makes phosphorescence more susceptible to quenching than fluorescence. In fact, the probability of collisional deactivation of T1 by solvent molecules, impurities or other molecules present in the sample is higher, decreasing phosphorescence emission. As a rule of thumb, phosphorescence of a molecule in solution and at room temperature is rare and not quite intense. Although there are clear differential characteristics between fluorescence and phosphorescence, the general concepts and basics explained in the previous section for fluorescence can be generally applied also to phosphorescence. For instance, T1 lifetime can be defined as the average time the system spends in T1 before decaying to the ground state (S0). Hence, the rate constants of the processes, both radiative and nonradiative, competing with this decay should be considered, similarly to Eq. (1.161). Regarding the phosphorescence quantum yield, a particular consideration has to be made compared with fluorescence quantum yield (150). In the case of phosphorescence, a small fraction of the absorbed photons would populate the emissive T1 state, in contrast to fluorescence where most of the photons populate S1 (if we exclude other photochemical channels). Hence, to define the phosphorescence quantum yield as the fraction of photons emitted from T1 after singlet excited state population, the quantum yield of triplet population (’T) has to be defined first, as the fraction of photons resulting in triplet states population from an excited singlet state, undergoing intersystem crossing: ’T ¼

kISCðST Þ kISCðST Þ + kIC + kf

(1.176)

where kISC(ST) is the rate constant for singlet-triplet intersystem crossing, kIC is the rate constant for singlet and triplet internal conversion, and kf is the fluorescence rate constant. Once populated T1, not all molecules will decay through phosphorescence to the ground state. This can be taken into account by the phosphorescence quantum efficiency (yph) defined as the fraction of molecules in T1 leading to phosphoresce: yph ¼

kph kph + knr

(1.177)

where kph is the phosphorescence rate constant and knr is the rate of all the nonradiative processes starting from T1 and competing with phosphorescence. The phosphorescence quantum yield (’ph) can then be defined as the product of the triplet quantum yield and the phosphorescence quantum efficiency, representing the fraction of photons emitted from T1 after absorption: ! ! kISCðST Þ kph ’f ¼ (1.178) ¼ ’T yph kph + knr kISCðST Þ + kIC + kf As for fluorescence, diverse parameters could influence phosphorescence such as temperature, molecular structure, and solvent. These parameters could modify not only the phosphorescence emission energy and its intensity but also the phosphorescence quantum yield and triplet lifetime. Both structural and environmental factors increasing the singlet-triplet intersystem crossing rates and decreasing the triplet nonradiative decay rates would, in general, favor phosphorescence emission. On one hand, the increase in singlet-triplet intersystem crossing rates can be achieved by including heavy atoms (i.e., halogens) in the molecular structure (132). On the other hand, hindering the collisional deactivations of T1 with solvent molecules, or other species present in the sample, would decrease the T1 nonradiative decay rates. This reasoning is analogous to the one already presented for fluorescence, aimed to decrease the S1 nonradiative decay rates to favor fluorescence. Hence, the factors already discussed for fluorescence apply to phosphorescence, such as molecular structural rigidity, temperature, and media viscosity. It should also be noted that, when triplet and excited singlet manifolds are nearly degenerated in energy and nonradiative decay channels are negligible, several intersystem crossings may happen and therefore, singlet-to-triplet processes can be followed by reversed triplet-to-singlet processes, also called reverse intersystem crossing. Once back on the singlet manifold, the system can emit light by fluorescence, but the overall excited state lifetime would be larger than in conventional fluorescence, and therefore, the term delayed fluorescence is applied. A typical field of application of reverse intersystem crossing is thermally activated delayed fluorescence (TADF), of particular interest in solid state materials for the design of new generation’s organic light-emitting diodes (OLEDs) (204–207). In solution, benzophenone could be considered the prototypical organic compound showing, by multiple-reverse intersystem crossings, delayed fluorescence (208,209).

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FIG. 1.17 Scheme representing the explicit electronic configurations associated with singlet-triplet and doublet-quadruplet phosphorescence (HOMO, highest occupied molecular orbital; LUMO, lowest unoccupied molecular orbital; SOMO, singly occupied molecular orbital). (Credit: Original.)

1.5.2.1 Phosphorescence from doublet and quartet states In the most majority of compounds, as we have explained in the previous section, phosphorescence takes place by triplet-tohn hn singlet radiative emission, T1!S0, although some compounds show the reverse emission, S0!T1 (as in the discussed case of molecular oxygen), since it is usually, but not always, S0 the lowest-in-energy electronic ground state. These two spin multiplicities are the ones usually invoked for phosphorescence since, in the orbitals point of view, just by promoting one electron from a closed-shell doubly occupied valence orbital (singlet) to an unoccupied virtual one will generate two singly occupied orbitals (triplet). Nevertheless, it may happen that chemical species have an odd number of electrons (ions, radicals), therefore having an open-shell characterized by a singly occupied highest-energy orbital (doublet). In this case, the promotion of one electron results in three unpaired electrons (quartet), leading to possible phosphorescence generated by quadruplet-to-doublet or doublet-to-quadruplet spin forbidden radiative decays (Fig. 1.17). Since singlet and triplet electronic configurations are usually typical of compounds characterized by higher chemical stability compared with the species relying on doublet and quartet electronic configurations, this latter phosphorescence mechanism is much less documented. Nevertheless, already in 1973 inorganic chemistry offered the opportunity to interpret the anomalous phosphorescence observed for Cr(III) complexes to doublet-to-quadruplet radiative emission (210). Much more recently, in 2022, the detected phosphorescence from the phosphaethynyl radical (C^P) was assigned to quadruplet-to-doublet emission (211).

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188. Singh-Rachford, T. N.; Castellano, F. N. Photon Upconversion Based on Sensitized Triplet–Triplet Annihilation. Coord. Chem. Rev. 2010, 254 (21–22), 2560–2573. 189. Zhao, J.; Ji, S.; Guo, H. Triplet–Triplet Annihilation Based Upconversion: From Triplet Sensitizers and Triplet Acceptors to Upconversion Quantum Yields. RSC Adv. 2011, 1 (6), 937–950. 190. Simon, Y. C.; Weder, C. Low-Power Photon Upconversion through Triplet–Triplet Annihilation in Polymers. J. Mater. Chem. 2012, 22 (39), 20817–20830. 191. Gao, C.; Wong, W. W. H.; Qin, Z.; Lo, S.; Namdas, E. B.; Dong, H.; Hu, W. Application of Triplet–Triplet Annihilation Upconversion in Organic Optoelectronic Devices: Advances and Perspectives. Adv. Mater. 2021, 33 (45), 2100704. 192. Gray, V.; Dzebo, D.; Abrahamsson, M.; Albinsson, B.; Moth-Poulsen, K. Triplet–Triplet Annihilation Photon-Upconversion: Towards Solar Energy Applications. Phys. Chem. Chem. Phys. 2014, 16 (22), 10345–10352. 193. Liu, Y.; Su, Q.; Zou, X.; Chen, M.; Feng, W.; Shi, Y.; Li, F. Near-Infrared In Vivo Bioimaging Using a Molecular Upconversion Probe. Chem. Commun. 2016, 52 (47), 7466–7469. 194. Drobizhev, M.; Karotki, A.; Kruk, M.; Krivokapic, A.; Anderson, H. L.; Rebane, A. Photon Energy Upconversion in Porphyrins: One-Photon HotBand Absorption Versus Two-Photon Absorption. Chem. Phys. Lett. 2003, 370 (5–6), 690–699. 195. M€ uller, M.; Zumbusch, A. Coherent Anti-Stokes Raman Scattering Microscopy. ChemPhysChem 2007, 8 (15), 2156–2170. 196. Davies, M. J. Singlet Oxygen-Mediated Damage to Proteins and Its Consequences. Biochem. Biophys. Res. Commun. 2003, 305 (3), 761–770. 197. Agnez-Lima, L. F.; Melo, J. T. A.; Silva, A. E.; Oliveira, A. H. S.; Timoteo, A. R. S.; Lima-Bessa, K. M.; Martinez, G. R.; Medeiros, M. H. G.; Di Mascio, P.; Galhardo, R. S. DNA Damage by Singlet Oxygen and Cellular Protective Mechanisms. Mutat. Res. Rev. Mutat. Res. 2012, 751 (1), 15–28. 198. Sies, H.; Menck, C. F. M. Singlet Oxygen Induced DNA Damage. Mutat. Res./DNAging 1992, 275 (3–6), 367–375. 199. Lewis, G. N.; Kasha, M. Phosphorescence and the Triplet State. J. Am. Chem. Soc. 1944, 66 (12), 2100–2116. 200. Saltiel, J.; Hammond, G. S. Mechanisms of Photochemical Reactions in Solution. XVII. Cis-Trans Isomerization of the Stilbenes by Excitation Transfer From Low Energy Sensitizers. J. Am. Chem. Soc. 1963, 85 (16), 2515–2516. 201. Saltiel, J.; Marchand, G. R.; Kirkor-Kaminska, E.; Smothers, W. K.; Mueller, W. B.; Charlton, J. L. Nonvertical Triplet Excitation Transfer to Cis-and Trans-Stilbene. J. Am. Chem. Soc. 1984, 106 (11), 3144–3151. 202. Forward, P. J.; Gorman, A. A.; Hamblett, I. ‘Nonvertical’ Triplet Energy Transfer to Cyclooctatetraene: Support for the Single-Bond Torsion Mechanism. J. Chem. Soc. Chem. Commun. 1993, (3), 250–251. 203. Zapata, F.; Marazzi, M.; Castan˜o, O.; Acun˜a, A. U.; Frutos, L. M. Definition and Determination of the Triplet-Triplet Energy Transfer Reaction Coordinate. J. Chem. Phys. 2014, 140 (3), 34102. 204. Noda, H.; Nakanotani, H.; Adachi, C. Excited State Engineering for Efficient Reverse Intersystem Crossing. Sci. Adv. 2018, 4 (6), eaao6910. 205. Goushi, K.; Yoshida, K.; Sato, K.; Adachi, C. Organic Light-Emitting Diodes Employing Efficient Reverse Intersystem Crossing for Triplet-toSinglet State Conversion. Nat. Photonics 2012, 6 (4), 253–258. 206. Gibson, J.; Monkman, A. P.; Penfold, T. J. The Importance of Vibronic Coupling for Efficient Reverse Intersystem Crossing in Thermally Activated Delayed Fluorescence Molecules. ChemPhysChem 2016, 17 (19), 2956–2961. 207. Wada, Y.; Nakagawa, H.; Matsumoto, S.; Wakisaka, Y.; Kaji, H. Organic Light Emitters Exhibiting Very Fast Reverse Intersystem Crossing. Nat. Photonics 2020, 14 (10), 643–649. 208. Turek, A. M.; Krishnamoorthy, G.; Phipps, K.; Saltiel, J. Resolution of Benzophenone Delayed Fluorescence and Phosphorescence With Compensation for Thermal Broadening. J. Phys. Chem. A 2002, 106 (25), 6044–6052. 209. Marazzi, M.; Mai, S.; Roca-Sanjua´n, D.; Delcey, M. G.; Lindh, R.; Gonza´lez, L.; Monari, A. Benzophenone Ultrafast Triplet Population: Revisiting the Kinetic Model by Surface-Hopping Dynamics. J. Phys. Chem. Lett. 2016, 7 (4), 622–626. 210. Schmidtke, H.-H.; Hoggard, P. E. Anamalous Phosphorescence Spectra of Cr (III) Complexes and Their Theoretical Interpretation. Chem. Phys. Lett. 1973, 20 (1), 119–123. 211. Ganesan, E.; Custer, T.; Lawzer, A.; Guillemin, J.; Kołos, R. Unusual Quartet-Doublet Phosphorescence From the Phosphaethynyl Radical, CP. Angew. Chem. 2022, 134 (43), e202210521.

Chapter 2

Theoretical grounds in molecular photochemistry Alejandro Jodra, Cristina Garcı´a-Iriepa, and Marco Marazzi Universidad de Alcala´, Departamento de Quı´mica Analı´tica, Quı´mica Fı´sica e Ingenierı´a Quı´mica, Alcala´ de Henares, Madrid, Spain

Chapter outline 2.1 The Jablonski diagram 2.2 Potential energy surfaces and reaction paths 2.3 The Born-Oppenheimer approximation in detail: Adiabatic and diabatic representations 2.3.1 Separation of nuclei and electrons motion 2.3.2 Adiabatic representation 2.3.3 Diabatic representation

51 53 54 54 57 58

2.4 When potential energy surfaces do cross: Avoided crossings and conical intersections 2.5 Excited state molecular dynamics References

59 62 65

2.1 The Jablonski diagram Until now, we have focused our attention on photophysics. In more detail, we have formally and computationally explained (as well as compared to experimental outcomes) the methods and techniques by which vertical transitions among molecular electronic states can be described. Such approaches led us to different ways of predicting radiative processes, namely electronic absorption and emission, including fluorescence and phosphorescence. During the years between the two World Wars, in order to show graphically the ensemble of all possible photophysical processes, the Polish physicist Aleksander Jablonski proposed to use specific energy levels diagrams, which would be later called after him the “Jablonski diagrams” (1,2). This diagram is characterized by horizontal lines describing the energy levels of the considered electronic states. For instance, the lower-in-energy electronic state corresponds to the ground state which commonly is a singlet state for organic molecules, denoted by S0. Upper-in-energy singlet excited states (S1, S2, etc.) are represented by upper-in-energy horizontal lines. In addition, triplet states could be also relevant in some excited state processes (denoted as T1, T2, etc.) and they are also represented by horizontal lines whose position depends on their relative energy. Apart from the illustration of electronic states, the Jablonski diagram also accounts for vibrational levels, again depicted as horizontal lines and usually denoted as u ¼ 0, 1, …, for each electronic state. As aforementioned, the Jablonski diagram is used to describe excited state processes. Especially, photophysical processes are illustrated by vertical arrows. For instance, absorbance is depicted as a straight vertical line starting from the ground singlet state (populated at room temperature) and finishing in the singlet excited state populated after photon absorption, depending on the photon energy (red arrow in Fig. 2.1). This vertical line considers that the geometry is kept invariant during this process. In fact, photon absorption is an almost instantaneous process (ca. 10 15 s), much faster than the nuclei displacement, as stated by the Franck-Condon principle, introduced in Chapter 1. Usually, an excited state vibrational level, higher in energy than the fundamental one, u0 > 0, is populated (see also Fig. 1.5 in Chapter 1). After the population of a given vibrational excited singlet state, the system usually relaxes rapidly (10 12–10 10 s timescale) to the lowest vibrational level of that singlet excited state, through the process commonly known as vibrational relaxation (black wavy arrow in Fig. 2.1). This relaxation usually takes place due to collision between the excited species with other molecules or solvent molecules, dissipating the collision energy as heat. It should be therefore noted that in Jablonski

Theoretical and Computational Photochemistry. https://doi.org/10.1016/B978-0-323-91738-4.00019-1 Copyright © 2023 Elsevier Inc. All rights reserved.

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FIG. 2.1 Jablonski diagram for a given molecule, depicting photon absorption and the possible photophysical processes taking place afterward.

diagrams, radiative processes (absorption and emission) are depicted by straight vertical arrows while nonradiative processes are depicted by wavy arrows. If a singlet excited state upper in energy than S1 is populated (e.g., S2), then the molecule can relax to a lower-in-energy singlet excited state through the internal conversion process (e.g., S2!S1). As this is a nonradiative process, it would be depicted in the Jablonski diagram as a wavy arrow (purple wavy arrow in Fig. 2.1). Internal conversion occurs between states of the same spin multiplicity, hence resulting in singlet-to-singlet or triplet-to-triplet processes. If considering the vibrational resolution for both states involved in internal conversion (i.e., vibronic states, as explained in Chapter 1), three possibilities arise: (1) the initial vibrational level is higher in energy than the vibrational level to be finally populated, (2) the opposite or (3) the two initial and final vibrational levels are isoenergetic. This last possibility corresponds to the most efficient channel for internal conversion to take place, corresponding to the resonant condition, with its efficiency decreasing the more the two energy levels differ. This process is characterized by an ultra-short timescale when involving excited states (10 14–10 11 s), therefore being usually the first happening after photon absorption, while other radiative (fluorescence, phosphorescence) and nonradiative (see Section 2.4) processes take place only after internal conversion ends. It should be noted that internal conversion from S1 to S0 is a slower process characterized by a timescale of 10 9–10 7 s, due to their typically larger energy difference (compared with energy differences among excited states). Hence, considering that both internal conversion and excited state vibrational relaxation are ultrafast processes of comparable timescales, after photon absorption the system usually relaxes in a rapid fashion to the lowest-in-energy vibrational level of the singlet excited state S1, as stated by the Kasha’s rule introduced in Chapter 1. Another process that can take place after excitation is intersystem crossing, which is defined as the transition between states of different multiplicity (e.g., S1!T1). This process is a spin-forbidden transition, with maximum efficiency in resonant conditions (i.e., when S1 and T1 are degenerate in energy) and it is therefore depicted as a wavy arrow in the Jablonski diagram, since it is associated with a nonradiative decay process (orange wavy arrow in Fig. 2.1). Due to the short timescale of vibrational relaxation after excitation, compared with intersystem crossing timescale (10 11–10 6 s), this process usually takes place from the lower-in-energy vibrational level of the initial state to a “vibrationally hot” level of the final state. The larger the energy overlap between these two vibrational levels, the larger the intersystem crossing probability. It is well known that relatively large intersystem crossing rates do involve molecules with heavy atoms in their structures (e.g., iodine or bromine atoms) or, more in general, do involve a change of molecular orbital type, e.g., 1(n p*) ! 3(p p*), according to the El-Sayed rule (3). After excitation, apart from nonradiative processes, also radiative decay to the ground state can occur, such as fluorescence or phosphorescence. As already described in Chapter 1, both are photoluminescent processes, as they are characterized by photon emission. The main difference between them is that fluorescence is due to the photon emission between states of the same spin multiplicity (usually the singlet manifold), whereas in phosphorescence this radiative decay involves states of different spin multiplicity (usually from triplet to singlet manifolds). An overview about these two

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processes can be found in Chapter 1. Since both are radiative transitions, they are represented by straight arrows in the Jablonski diagram (grey arrow for fluorescence and brown arrow for phosphorescence, in Fig. 2.1).

2.2 Potential energy surfaces and reaction paths After having introduced in Chapter 1 the interaction between light and matter, in order to explain the basic concepts of photophysics, we can now focus the attention on the subsequent evolution of matter, in molecular terms. Overall, we have seen in the previous section how the Jablonski diagram allows for a schematic representation of all the possible relaxation processes that occur in the excited state after excitation, at both nonradiative and radiative levels. Although this diagram aids evidently the understanding of the photophysics of a given molecular system, no information about its photochemistry (i.e., when a chemical reaction takes place in the excited state) can be obtained, since no geometrical evolution of the nuclei can be accounted for. Such evolution can be mathematically and pictorially described by introducing the concept of potential energy surface (PES) for each of the electronic state involved. As we have seen in Chapter 1, for each spin multiplicity, an electronic ground state and at least one electronic excited state need to be considered in order to define a photochemical process. Especially, following the Born-Oppenheimer approximation, for each nuclear arrangement of the molecular system under study (i.e., a defined spatial disposition of the atomic nuclei), it corresponds a certain potential energy of electronic ground and excited states. If we consider, by a classical approach, that only one of such electronic states can be populated, the molecule will evolve depending on the actual potential energy expressed by such electronic state, defining a set of molecular geometries that, taken all together, are part of a surface. How many physical dimensions have such PES? It depends on the number of atoms belonging to the molecular system. In more detail, after subtracting the rotational (3) and translational (3) coordinates defined for the molecule as a whole (6 coordinates in total), it results that a PES has 3N 6 dimensions, where N is the total number of nuclei. An exception should be made for linear molecules that are described with 3N 5 dimensions, since it cannot be observed the rotation around its molecular axis. Because of the intrinsic multidimensional nature of a PES, it can be also defined as a potential energy hypersurface. A complete PES is, therefore, a collection of all possible molecular geometries and corresponding energies, given a certain electronic state. Nevertheless, for the sake of simplicity, a PES is usually depicted by a bidimensional figure (the energy as a function of one representative coordinate, e.g., a distance, an angle, or a torsion) or by a tridimensional figure (the energy as a function of two representative coordinates). More in general, by mapping the potential energy along different nuclear coordinates, the related PES can be retrieved. By only considering the electronic ground state PES, corresponding to the lowest potential energy among the possible spin multiplicities, we obtain a reliable molecular description of the possible thermal reaction paths (Fig. 2.2A), within the Born-Oppenheimer approximation briefly introduced in Chapter 1. If we would like to understand the possible

FIG. 2.2 Schematic description of thermal chemistry (A) and photochemistry (B), in terms of potential energy surfaces’ energy as a function of a representative reaction coordinate. For thermal chemistry, a common reactant (R) to product (P) reaction is sketched, with the relative thermal barriers determined by a transition state (TS) connecting R and P. For photochemistry, different possible scenarios are depicted, starting from R photon absorption to either S1 or S2 electronic states (hnabs), followed by fluorescence (hnfl), phosphorescence (hnph), or nonradiative production of P1 and P2, through eventual previous singlet-to-triplet intersystem crossing or pathway branching at a singlet crossing region, respectively.

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photochemical reaction paths, i.e., the evolution of the molecular system after photon absorption and population of an electronic excited state, then the required excited state PESs need to be added to the ground state one, each of the PES being described by the same 3N 6 dimensions. Such physico-mathematical description of the “potential force fields” applicable to a molecular system when interacting with light, makes possible to explain a much larger variety of possible photochemical reaction paths, compared with the limited thermal paths. Indeed, while the vast majority of thermal paths can be reduced to the localization of a transition state separating two different isomers—in the case of conformational change—or a reactant and a product—in the case of a chemical reaction (excepting minor cases involving bifurcations or transition states connecting another transition state), photochemical paths can lead to a much higher number of scenarios: radiative decay as fluorescence or phosphorescence (explained in Chapter 1), but also different nonradiative decay channels starting from the same reactant, eventually leading to various photoproducts (photoreactivity) and/or internal conversion (photostability), as shown in Fig. 2.2B. Hence, analyzing the shape of these PESs aids the understanding of both photophysical processes and photochemical reaction paths of a given molecule (4,5). Regarding photochemical reactions, it is of crucial importance, from a computational perspective, the location of the critical points that determine type and feasibility of the photochemical process, as well as nuclear geometrical rearrangement within the molecular system. This task can be accomplished by first selecting the appropriate level of theory (see Part 2) and then, conveniently optimizing all relevant stationary points through first and second derivatives of the potential energy with respect to the nuclear coordinates, finally retrieving molecular forces and the Hessian matrix. Especially, two different types of stationary points need to be found, both requiring null forces: (i) energy minima, further characterized by a Hessian matrix with all positive eigenvalues, that is, all positive frequencies; (ii) saddle points, further characterized by a Hessian matrix with n > 0 negative eigenvalues, also referred to as n > 0 imaginary frequencies. When only one negative eigenvalue (or imaginary frequency) is present, then the saddle point takes the name of transition state, due to the development of the transition state theory (6,7), by which rate constants of chemical reactions can be calculated. As aforementioned, in order to build complete PESs, a large number of calculations to determine the energy at the different nuclear coordinates are necessary, usually resulting in scans of the energy along the different molecular coordinates of interest. Apart from the computational requirements, which are more and more affordable thanks to the continuous improvement of informatics hardware and dedicated software, a conceptual issue can be raised by mapping PESs along specific molecular coordinates: the aforementioned critical points could be missed, in terms of both geometrical structure and related potential energy. Hence, a different methodological strategy could be used, although not excluding the previous one: the calculation of the minimum energy path (MEP (8)). Once again, this can be defined for both thermal and photochemical reactions, although it was first formulated for thermal chemistry: once located a transition state that is thought to be the responsible of connecting reactant and product, the intrinsic reaction coordinate (IRC (9,10)) is calculated along both directions indicated by the imaginary frequency vector, finally resulting in a series of molecular structures and corresponding energies, connecting reactant to transition state, and this to the product. Since the energy barrier between the characterized reactant and transition state is defined as the lowest possible energy barrier to be overcome to obtain the product, the final outcome takes the name of MEP. The same IRC methodology can be applied to characterize photochemical paths (11), although in this case care has to be taken when electronic states do cross (see Section 2.4), providing essential information on the spectroscopy, energetics, and mechanism of the possible photochemical reactions.

2.3 The Born-Oppenheimer approximation in detail: Adiabatic and diabatic representations 2.3.1 Separation of nuclei and electrons motion Before introducing the different types of electronic state crossings, which can probably be considered as the most characteristic elements of the photochemical reactivity, allowing ultrafast nonradiative photoproducts formation, it is useful to describe in detail the basics of the mentioned Born-Oppenheimer approximation. Indeed, as we will see in Section 2.4, electronic state crossings challenge the PES definition based on the Born-Oppenheimer approximation. The separation of the system in terms of coordinates is a type of approximation which was already introduced in Chapter 1, with the Born-Oppenheimer approximation falling within this group: the motion of nuclei and electrons can be separated, since the motion of the electrons is much faster due to the much larger mass of the nuclei. Hence, within the electrons motion timscale, we can approximate the nuclei as static compared with the electrons. This is one of the most widely used approximation in quantum chemistry and allows us to solve the electron-coordinate-dependent Schr€ odinger

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equation. One of the major consequences of this approach, as already evinced in Chapter 1 and further discussed in this chapter, is the emergence of molecular electronic states. The total Hamiltonian defining a system of N nuclei and n electrons is: H ¼ T N + T e + V Ne + V ee + V NN

(2.1)

where TN and Te are the kinetic energy operators of the nuclei and electrons, respectively, VNe is the potential energy between nuclei and electrons, Vee is the potential energy between electrons, and VNN is the potential energy between nuclei. His therefore an operator spanning both electronic and nuclear spaces, since it depends on the coordinates of both electrons and nuclei. Solving the resulting Schr€ odinger equation by applying the Hamiltonian to the total wave function C(R, r), already presented in Eq. (1.86), retrieves the energy of our system. To solve the Schr€odinger equation, we could resort to the separation of the variables, so that the total wave function can be expressed by the product of two separate wave functions: CðR, rÞ ¼ cðrÞwðRÞ

(2.2)

where c(r) is the electronic wave function, which depends only on the electronic coordinates, and w(R) is the nuclear wave function, which depends only on the nuclear coordinates. However, the total Hamiltonian H (Eq. 2.1) does not allow for such a separation, since the term VNe depends on both nuclear and electronic coordinates. Then, the wave function separation needs to be rearranged as follows: CðR, rÞ ¼ cðr; RÞwðRÞ

(2.3)

where c(r; R) is the electronic wave function in which the semicolon indicates a parametric dependence on the nuclear coordinates. This separation is possible only if we consider electrons in motion around static nuclei. Therefore, for a given nuclear configuration R, we will have a given state function c(r; R), which defines the electronic system. Extracting then TN from Eq. (2.1), the electronic Hamiltonian Hel can be defined, which depends only on the evolution of the electronic coordinates r, with R being a parameter. The eigenfunctions of this operator will then be the electronic wave functions seen above: Hel ck ðr; RÞ ¼ V k ðRÞck ðr; RÞ

(2.4)

where Vk(R) is the eigenvalue of the electronic Hamiltonian for a given nuclear configuration R and corresponds to the electronic energy for a given electronic state, ck(r; R). These functions, being characteristic of a Hermitian operator, form a complete set in the electronic space. Any function contained in the electronic space can be expressed as a linear combination of these functions: X ck ðRÞck ðr; RÞ (2.5) cðr; RÞ ¼ k

Hence, we have shown how the total wave function can be expanded by applying the Born-Oppenheimer approximation: X CðR, rÞ ¼ ck ðr; RÞwk ðRÞ (2.6) k

By substituting this expression of the total wave function in Eq. (1.86) in Chapter 1, the Schr€odinger equation can be formulated as: X X H ck ðr; RÞwk ðRÞ ¼ E ck ðr; RÞwk ðRÞ (2.7) k

k

Now considering, as aforementioned, the total Hamiltonian H as the sum of the nuclear kinetic energy operator TN and the electronic Hamiltonian Hel, Eq. (2.7) can be rewritten as: X  X T N + Hel ck ðr; RÞwk ðRÞ ¼ E ck ðr; RÞwk ðRÞ (2.8) k

k

Developing Eq. (2.8), we obtain: X X X TN ck ðr; RÞwk ðRÞ + wk ðRÞHel ck ðr; RÞ ¼ E ck ðr; RÞwk ðRÞ k

k

k

(2.9)

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The nuclear kinetic energy operator can be expressed in the nuclear space as a summation over a nuclei: TN ¼

X ħ2 2 —a a 2Ma

(2.10)

where Ma is the mass of the nucleus a and —a is the gradient with respect to the nuclear coordinates of the nucleus a. Substituting Eq. (2.10) into Eq. (2.9) and applying the Leibniz rule, we obtain: ( )  ħ2 X 1 X 2 2 wk ðRÞ—a ck ðr; RÞ + ck ðr; RÞ—a wk ðRÞ + 2—a ck ðr; RÞ—a wk ðRÞ 2 a Ma k X X + wk ðRÞHel ck ðr; RÞ ¼ E ck ðr; RÞwk ðRÞ (2.11) k

k

c∗k0 (r; R)

on the left and integrating over the electronic coordinates r, the result is: )      2  ħ2 X 1 X   2 ck0 —a ck wk + ck0 jck —a wk + 2 ck0 j—a jck —a wk 2 a Ma k X X   + ck0 jck wk ck0 jHel jck wk ¼ E

Multiplying in Eq. (2.11) by

(

(2.12)

k

k

We consider that the electronic functions ck(r; R) are well-behaved. Fulfilling then the orthonormality condition: D E (2.13) ci jcj ¼ dij Eq. (2.12) leaves us with: ( !) X X      ħ2 X 1 2 2 ck0 jHel jck wk ¼ Ewk0 — a wk 0 + ck0 j—a jck wk + 2 ck0 j—a jck —a wk + 2 a Ma k k

(2.14)

To simplify Eq. (2.14), we can write the kinetic energy operator as: TN ¼

ħ2 —∙— 2M

(2.15)

thus, eliminating the summation, with M being the weighted mass and — a vector in the nuclear space. Eq. (2.14) can therefore be rearranged as follows: ( ) X X      ħ2 2 2 — wk 0 + (2.16) ck0 jHel jck wk ¼ Ewk0 ck0 j— jck wk + 2 ck0 j—jck —wk + 2M k k We can write Eq. (2.16) as a matrix equation, by calculating values for every c∗k0 (r; R): ½HN ðRÞ + He ðRފxðRÞ ¼ ExðRÞ

(2.17)

where x(R) is a column vector containing the nuclear wave functions wk(R) (k ¼ 1, 2…. K), HN(R) is the matrix containing the energy contributions due to the nuclear Hamiltonian, He(R) is the matrix containing the energy contributions due to the electronic Hamiltonian, and E is the total energy of the system. With the above definition, we can define the elements of the matrix HN(R): ðHN ðRÞÞk0 k 5 that can be simplified as follows:

    ħ2 X 1   ck0 j—2a jck + dk0 k —2a + 2 ck0 j—a jck —a 2 a Ma

ðHN ðRÞÞk0 k 5

    ħ2   ck0 j—2 jck + dk0 k —2 + 2 ck0 j—jck — 2M

(2.18)

(2.19)

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By defining the following G(R) and F(R) matrices:   ðGðRÞÞk0 k ¼ ck0 j—2 jck   ðFðRÞÞk0 k ¼ ck0 j—jck

(2.20) (2.21)

we can rewrite Eq. (2.19) as a sum over three terms: ðHN ðRÞÞk0 k ¼

 ħ2  Gk0 k + dk0 k —2 + 2Fk0 k ∙— 2M

(2.22)

where the elements of the G matrix are scalars known as nonadiabatic scalar couplings, while the elements of the F matrix are vectors known as nonadiabatic derivative couplings. Using these definitions, Eq. (2.16) assumes the form: 

 ħ2  GðRÞ + —2 + 2FðRÞ∙— + He ðRÞ 2M

E xð R Þ ¼ 0

(2.23)

Eq. (2.23) corresponds to the most general form of the Born-Oppenheimer approximation. Nevertheless, this equation is complex to solve, as there are costly elements to be obtained. Hence, adiabatic and diabatic approximations need to be introduced to find simpler solutions.

2.3.2 Adiabatic representation Looking at the general equation derived from the Born-Oppenheimer approximation (Eq. 2.23), we can easily understand that the different elements of the equation depend on the chosen basis functions, ck(r; R). The most commonly used basis functions are adiabatic functions. These are functions of the electronic Hamiltonian (Hel), i.e.: Hel ck ðr; RÞ5V k ðRÞck ðr; RÞ

(2.24)

where Vk(R) is the electronic energy for the state ck(r; R). Therefore, we are now interested in developing the term He(R), introduced in Eq. (2.17):     (2.25) ðHe ðRÞÞk0 k 5 ck0 jH el jck ¼ ck0 jck V k ðRÞ

Since these functions are eigenfunctions of the electronic operator, which is a Hermitian operator, they form a complete orthonormal set and satisfy Eq. (2.13): ðHe ðRÞÞk0 k 5dk0 k V k ðRÞ

(2.26)

Hence, He(R) is a diagonal matrix, whose elements are the energies of the different electronic states. It remains to realize the fate of the matrices G(R) and F(R), introduced by Eqs. (2.20) and (2.21), respectively. Indeed, for adiabatic basis functions, these matrices do not have a diagonal form. On one hand, this problem can be solved by showing that F(R) is an anti-Hermitian matrix, and therefore, by selecting real basis functions that are adiabatic basis functions, the diagonal terms become zero since the F(R) matrix becomes antisymmetric. On the other hand, the diagonal terms of the G(R) matrix are usually negligible, justifying that they can be neglected. Hence, we are left with only the off-diagonal terms of both G(R) and F(R) matrices and, comparing them, G(R) off-diagonal terms are smaller in magnitude than those of F(R), since they contain second derivatives instead of first derivatives. Therefore, G(R) off-diagonal terms can be also neglected, letting only the F(R) off-diagonal terms. Multiplying Eq. (2.24) by —, we obtain: —Hel ck + Hel —ck 5ck —V k + V k —ck Multiplying from the left by coordinates, we get:

c∗k0

(2.27)

0

(where k 6¼ k, since we care about off-diagonal terms) and integrating over the electronic

        ck0 j—Hel jck + —ck jHel jck0 5 ck0 jck —V k + V k ck0 j—jck

(2.28)

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Simplifying:   V k0 ck0 j—jck

    ck0 j—Hel jck 5V k ck0 j—jck

and rearranging, we reach a convenient expression of the F(R) off-diagonal terms:   ck0 j—Hel jck ðFðRÞÞk0 k 5 V k0 ðRÞ2V k ðRÞ

(2.29)

(2.30)

Looking at the denominator of Eq. (2.30), we can easily realize that the F(R) off-diagonal terms are non-negligible values if Vk0 (R)  Vk(R). Finally, the general form of the Born-Oppenheimer approximation (Eq. 2.23) can be rewritten as follows, by using adiabatic basis functions:  2 ħ —2 + He ðRÞ E xðRÞ ¼ 0 (2.31) 2M This gives rise to what is known as the adiabatic approximation. In areas where two electronic states are close in energy (i.e., in the nearby of an electronic states crossing), this approximation is not correct, due to relevant (F(R))k0 k values.

2.3.3 Diabatic representation In order to deal with those areas in which the adiabatic approximation is not fulfilled, there is another way to simplify Eq. (2.23). As we have already mentioned, its terms depend on the chosen basis functions. For the adiabatic approximation, some functions are chosen that make the He(R) matrix to be a diagonal matrix. This time, we will choose basis functions that make the off-diagonal terms of the HN(R) matrix equal to zero. These functions are known as diabatic basis functions (since they will give rise to the diabatic approximation) and can be taken as real functions, so that the diagonal terms are zero: Fkk 50

(2.32)

In addition, the G(R) diagonal terms can be neglected, so that the diabatic Born-Oppenheimer approximation can be defined by the same Eq. (2.31) developed for the adiabatic approximation but, in this case, the He(R) off-diagonal terms are non-zero. As mentioned above, the basis functions’ space can be expanded using the set of adiabatic wave functions. Hence, the diabatic functions can be expressed as a combination of adiabatic functions. In fact, diabatic functions are usually obtained by an orthogonal transformation of adiabatic functions. This process is known as adiabatic-to-diabatic transformation (ADT). To illustrate this process, we will consider two adiabatic electronic states, c1 and c2, which present some degeneracy in a defined nuclear subspace, thus causing the adiabatic approximation to be not valid. As aforementioned, we can then propose some diabatic functions, which are the result of an orthogonal transformation of the adiabatic functions:

 cos aðRÞ sin aðRÞ SðRÞ5 (2.33) sin aðRÞ cos aðRÞ f1 c1 T ð R Þ ¼ S (2.34) f c 2

2

where f1 and f2 are the diabatic functions and a(R) is the diabatic angle. Applying these functions, the HN(R) matrix has a diagonal form: ðHN ðRÞÞk0 k 5

 ħ2  G 0 + dk0 k —2 + 2Fk0 k ∙— ¼ 0 2M k k

(2.35)

When k0 6¼ k (i.e., considering the off-diagonal terms), it follows that: Gk0 k 522Fk0 k ∙—

(2.36)

being Eq. (2.36) true only if Fk0 k ¼ 0.Therefore, being F(R) an antisymmetric matrix of 22 size: F12 ¼

F21 5hf1 j—jf2 i ¼ 0

(2.37)

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sin aðRÞc2 j—j cos aðRÞc2 + sin aðRÞc1 i

(2.38)

Substituting Eq. (2.33) into Eq. (2.37): hf1 j—jf2 i ¼ h cos aðRÞc1 Developing Eq. (2.38): hf1 j—jf2 i ¼ h cos aðRÞc1 j—j cos aðRÞc2 i + h cos aðRÞc1 j—j sin aðRÞc1 i h sin aðRÞc2 j—j sin aðRÞc1 i

h sin aðRÞc2 j—j cos aðRÞc2 i (2.39)

and applying the Leibniz rule: hf1 j—jf2 i ¼ cos aðRÞð— cos aðRÞhc1 jc2 i + cos aðRÞhc1 j—jc2 iÞ + cos aðRÞð— sin aðRÞhc1 jc1 i + sin aðRÞhc1 j—jc1 i sin aðRÞð— cos aðRÞhc2 jc2 i + cos aðRÞhc2 j—jc2 iÞ sin aðRÞð— sin aðRÞhc2 jc1 i + sin aðRÞhc2 j—jc1 iÞ

(2.40)

Since ck are real functions, the terms hckj — j cki are 0 and, applying the orthonormality condition, we obtain: f1 j —j f2 ¼ cos 2 aðRÞc1 j —j c2 + cos aðRÞ— sin aðRÞ

sin aðRÞ— cos aðRÞ

2

sin aðRÞc2 j —j c1

(2.41)

As mentioned above, the matrix F(R) is antisymmetric, meaning that: hc1 j—jc2 i ¼

hc2 j—jc1 i

(2.42)

Finally, by applying the rules of derivation for sine and cosine, we get: hf1 j—jf2 i ¼ hc1 j—jc2 i + —aðRÞ

(2.43)

2.4 When potential energy surfaces do cross: Avoided crossings and conical intersections As we have introduced in Section 2.2, photochemical multi-channeling scenarios allow for, in principle, a richer and more unpredictable chemistry compared with thermal chemistry, i.e., chemistry that can be described taken into account only the electronic ground state. In this section, we will explore qualitatively and then formally, based on the adiabatic and diabatic representations detailed in Section 2.3, the interaction between PESs that makes possible the opening of photochemical channels. Especially, both adiabatic and diabatic representations are not just mathematical methods to solve the BornOppenheimer approximation, but they can be considered two different physical representations of particular interest when two PESs are close to energy degeneracy. Let us consider a general example: a molecule with a singlet closed-shell electronic configuration in its ground state which, after photon absorption, populates the lowest-lying excited state (S1) characterized by intramolecular charge transfer, formally defining a zwitterion, evidently maintaining the singlet spin multiplicity, in order to be a spin-allowed vertical transition. If, due to the specific molecule under study, the potential energy of the excited state decreases, it can result in (i) population of an S1 excited state minimum, from where radiative decay to the ground state occurs, i.e., fluorescence, or (ii) almost-degeneracy with the ground state. If the former happens, the Born-Oppenheimer approximation still holds. But, if the latter happens, the S1/S0 energy degeneracy region can be described in two ways: on one hand, the diabatic approximation (following the Born-Oppenheimer approximation) could be invoked if there is no coupling between the intersecting states (or it is negligible), resulting in a specific molecular geometry corresponding to the formal energy degeneracy point between closed-shell and charge-transfer states. On the other hand, if the intersecting diabatic electronic states are significantly coupled, we should apply the adiabatic representation, by determining an avoided crossing region near the expected energy degeneracy, i.e., no geometry corresponding to the actual energy degeneracy can be defined. More in detail, the splitting in energy between S1 and S0 PESs, due to the avoided crossing, is twice the coupling between the intersecting diabatic states, i.e., the higher the adiabatic coupling, the higher the diabatic energies’ splitting or repulsion. If we limit our example to a one-dimensional problem, as it is the case for diatomic molecules, avoided crossings are the only adiabatic singularities that can be described. Nevertheless, for larger-size molecules of more general application, it is possible to cross also for two adiabatic states. For this to happen, two requirements need to be fulfilled: (i) energy degeneracy of the two adiabatic states and (ii) at the molecular geometry corresponding to the energy degeneracy fulfillment, the coupling must vanish. To ensure both requirements, two independent nuclear coordinates are needed, which allow to tune

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independently the adiabatic energy splitting and the adiabatic coupling. If fulfilled, the two requirements lead to a defined molecular structure known as conical intersection (in general, there can be more conical intersections for a certain molecular system). Such nuclear coordinates can be expressed in terms of vectors, specifically the “gradient difference” and the “derivative coupling” vectors, pointing from the conical intersection outward, and together called the nonadiabatic coupling vectors. While the “gradient difference” vector measures the distortion of the molecular system to reach the highest energy difference between the two electronic states, the “derivative coupling” vector measures the molecular distortion leading to the maximum coupling among the two electronic states. In all the remaining 3N 8 dimensions, the energy degeneracy is maintained or, explained the other way round, the two dimensions defined by “gradient difference” and “derivative coupling” vectors are the only ones that allow leaving the energy degeneracy region, and therefore define the so-called branching plane (12–14). In a computational point of view, nowadays several theoretical chemistry-dedicated software can compute, at different levels of theory, the nonadiabatic coupling vectors. Since, as aforementioned, they are nuclear vectors pointing from the geometry corresponding to the PESs crossing, the angle formed between them can indicate the nature of the crossing: parallel vectors diminish the bi-dimensionality of the branching plane to one dimension, thus defining an avoided crossing; nonparallel vectors (i.e., whatever angle different from 0 and 180 degrees) inherently define a two-dimensional space corresponding to the branching plane, thus indicating the presence of a conical intersection. The presence or absence of a branching plane leads to a completely different photochemistry behavior: avoided crossings, being limited to a single direction, act in a transition state fashion, connecting only two possible photoproducts on the lower-energy conical intersections can be seen as excited state funnels, by which two or more photoproducts can be generated. On one hand, the inherent energy splitting of the near-crossing PESs (due to the specific adiabatic coupling value) characterizes the avoided crossing as a kinetically slow channel, when reached from an upper-in-energy excited state (see Fig. 2.3A): dynamically, we expect the molecule to vibrate several times around the avoided crossing point, before actually populate the lower-in-energy state. On the other hand, trajectories experiencing conical intersections do usually require (nearly) a single molecular vibration to "hop" from the higher- to the lower-in-energy state (see Section 2.5 for a brief description of excited state molecular dynamics). Indeed, conical intersections are responsible of literally any ultrafast nonradiative photochemical transformation or process at the molecular level, although the final photochemical quantum efficiency largely depends on the specific topology of the conical intersection (15). Keeping the discussion at a qualitative level (for a physico-chemical insight, we suggest ref. (16)), we should at least mention the existence of the main conical intersection topologies: peaked and sloped (or tipped), according to Ruedenberg’s terminology (13) (Fig. 2.3B and C). The main consequence over the expected photoreactivity is that only a peaked topology could ensure the population of two or more photoproducts, while a sloped topology could in principle only give rise to one photoproduct. Hence, while the former case is usually more appealing for photochemical studies, especially when the conical intersection is located between the lowest-lying excited state and the ground state (usually a S1/S0 conical intersection), the latter case can describe a highly efficient photostability process, if the only achievable photoproduct is the reactant itself. Finally, we should remind that, although of less importance in the chemical point of view since it is a low probability event, conical intersections can in principle be found also among three PESs, defining three-state (electronic) conical intersections. They can be found, e.g., in nucleic acid bases exposed to light (17–22). Also, it can be possible to find entire regions of intersections (i.e., energy degeneracy regions) between two PESs, giving rise to so-called “crossing seams.” Although the topological description is possibly the most important one from a chemical and qualitative standpoint, we should note that only physical and mathematical considerations made possible the explanation of the existence of conical intersections. Indeed, several scientists worked on this issue during the 1960s and 1970s, including Baer, Herzberg, Michl, Robb, Truhlar, Yarkony, and Zimmerman (5,12,14,23–35). This requires studying the interaction of the two (potentially) intersecting electronic states 1 and 2, represented by the diabatic functions f1 and f2, through an electronic Hamiltonian, Hel, defined as:

H 12 ðRÞ 11 ðRÞ Hel ¼ H (2.44) H ðRÞ H ðRÞ 21

22

where the matrix elements, all dependent on the nuclear coordinates R, are: D E H ij ðRÞ ¼ fi jHel jfj ; i, j ¼ 1, 2

(2.45)

3 Pictorial view of (A) an avoided crossing and (B) a conical intersection between two crossing potential energy surfaces in 3D view, i.e., as a function of two internal coordinates (q1 and q2 ew, along a single q internal coordinate. A zoom into the peaked conical intersection topology is shown in (C), from both perspectives of the multiple channels offered to the excited reactant ( P3, R) and of the branching plane defined by the nonadiabatic “derivative coupling” and “gradient difference” (x1 and x2) vectors. The other main conical intersection topology (sloped) i describing in this case a photostability (R* to R) pathway. Adapted from Marazzi, M.; Sancho, U.; Castano, O.; Frutos, L. M. First Principles Study of Photostability Within Hydrogen-Bo Acids. Phys. Chem. Chem. Phys. 2011, 13 (17), 7805–7811 and Boggio-Pasqua, M. Computational Mechanistic Photochemistry: The Central Role of Conical Intersections. Universit e Toulou

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and where the two off-diagonal elements correspond to the coupling between the approaching electronic states, H12(R) 5 H21(R). By diagonalizing the 22 matrix of Eq. (2.44), we can obtain the two possible adiabatic energies for a certain molecular geometry R, corresponding to the eigenvalues of Hel: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi

2 DH (2.46) + H 12 2 E1,2 ¼ H  2 where H is the average among H11 and H22, while DH is its difference: H¼

H 11 + H22 2

DH ¼ H 11

H 22

(2.47) (2.48)

For a conical intersection to exist, the two adiabatic energies need to be degenerate, and for that to happen, at least two independent molecular coordinates have to fulfill the following two independent conditions: H 11 ¼ H 22

(2.49)

H 12 ¼ 0

(2.50)

This means that in a diatomic molecule, that can be represented, by definition, by only one independent coordinate (i.e., the interatomic distance), two electronic states sharing the same symmetry cannot be energetically degenerate, meaning that no conical intersection is allowed, since only Eq. (2.49) can be satisfied. This is known since 1929 as the “non-crossing rule” (36). Nevertheless, if the two electronic states differ in spatial or spin symmetry, then also Eq. (2.50) can be satisfied, and a conical intersection can therefore exist even in a diatomic molecule. Nowadays, in most of our (theoretical and experimental) studies, we are interested in molecules of much larger size than diatomic ones. Hence, usually more than one independent coordinate is at hand, and therefore, there can exist two of them satisfying both Eqs. (2.49) and (2.50), even if the two electronic states belong to the same spatial or spin symmetry. Such couple of independent coordinates was indeed aforementioned as the nonadiabatic coordinates that, within the space, can be visualized as vectors and defined as:   ∂H x1 ¼ c1 j el jc2 (2.51) ∂R x2 ¼

∂ðE1 E2 Þ ∂R

(2.52)

where x1 and x2 are the “derivative coupling” and “gradient difference” vectors, respectively (see Fig. 2.3C). So far, we have focused, first qualitatively and now on a formal basis, on the topology of the electronic states crossing. Finally, we can also introduce the vector usually indicated as h12, which module corresponds to the magnitude of the adiabatic states’ (c1 and c2) coupling, as a function of the molecular geometry R: D E ∂H   c1 j ∂Rel jc2 ∂c2 x1 h12 ¼ c1 j ¼ (2.53) ¼ E2 E1 ∂R E2 E1 It can be therefore evinced that the interstate coupling is higher the lower is the difference in energy between the electronic states, until reaching an infinitely high value at the conical intersection geometry, where E1 ¼ E2.

2.5

Excited state molecular dynamics

As we have seen in Section 2.4, mechanistic photochemistry can give us fundamental information about the fate of a photochemical reaction (37), based on the topology of crossings among electronic states, along with the feasibility of reaching them. However, a theoretical and computational approach based “only” on the location of PESs’ stationary points and singularities (in terms of both nuclear coordinates and energy landscape) lacks some highly desirable data: the timescale of the studied photoprocess, the quantum yield of each of the possible photochemical channels, or the simulation of time-resolved spectroscopic experiments. Such additional information can be retrieved only by excited state molecular dynamics, being fundamental to improve the synergy between theory and experiment (38,39).

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Indeed, a mechanistic study based on the calculation and interpretation of the minimum energy paths does not consider the kinetic energy of the system, since only the potential energy is taken into account, hence ending up in potential energy surfaces. In a vibrational point of view, we could therefore affirm that mechanistic studies describe the behavior of vibrationally cold molecular systems, while including the finite kinetic energy of a vibrationally hot molecule makes possible to study its dynamical behavior. Moreover, due to excess vibrational energy usually provided by a laser beam or by sun irradiation (i.e., the photon0 s energy can populate excited vibrational levels of the excited electronic state), such dynamical behavior could deviate, at least partially, from the expected minimum energy path, thus visiting unforeseen regions of the potential energy landscape. In some cases, it is therefore possible to reliably describe the photochemistry of a certain system only if excited state molecular dynamics are performed. In a nontrivial attempt to classify all possible types of molecular dynamics, we can suggest the following three main categories: classical dynamics, quantum dynamics, and mixed quantum-classical dynamics. Classical dynamics are based on a classical force field, that is, a mathematical expression to describe the vibration of each bond length, angle, and dihedral, as well as the electrostatic and Van der Waals interactions. The analytical nature of such expressions makes of high computational feasibility the evaluation of the total (potential and kinetic) energy at each step of the trajectory. Also, the propagation of the same trajectory at each step (i.e., spatial position and velocity of all atoms of the system) can be easily obtained by applying the Newton’s laws. Classical dynamics found much application for extensive ground state studies of large systems (several thousands of atoms) with the goal of understanding their conformational stability and eventual changes. Examples are biologically relevant systems that can be studied at all-atomistic resolution (proteins, DNA, cell membranes), with trajectories reaching nowadays timescales of several microseconds. Nevertheless, such classical force fields have the clear limitation of being inappropriate to describe chemical reactions, since a certain connectivity among the atoms need to be chosen a priori, based on the scientist’s knowledge of the molecular system, therefore limiting the types of processes that can be investigated. In principle, so-called reactive force fields (ReaxFF) were proposed, considering particular criteria (e.g., distance among atoms) based on which a chemical reaction can happen as an instantaneous event, followed by the required modification of the classical force field to continue running the classical dynamics with the new atomic connectivity (40–43). Although attractive for specific applications, we still cannot address the relevance of ReaxFFs on a general basis, due to the limited amount of studies performed. Another serious limitation of classical dynamics refers to the treatment of electronic excited states, since such states are rarely not inducing chemical reactions. Proper parametrization of molecular force fields can be indeed performed to set up specific excited state force fields, which could be anyway valid only to describe the behavior of excited state minima, e.g., the S1 minimum region from where a molecule is expected to fluoresce on the microsecond timescale (44–47). As we have explained in the above sections of this chapter, the description of a photochemical nonradiative reaction requires the presence of crossings among PESs, i.e., topological points where the Born-Oppenheimer approximation is not anymore valid, therefore in principle needing a quantum mechanical description of both nuclei and electrons. This means that quantum dynamics would be required as the most proper approach. In other terms, we would need time-integration of the time-dependent Schr€ odinger equation. In contrast to classical dynamics, quantum dynamics hence results in a highly expensive computational treatment, since a nuclear wave packet needs to be built on the electronic excited state where the photon is absorbed, followed by wave packet splitting at each PES crossing, until such propagation reaches all possible photoproducts in a certain ratio (Fig. 2.4A). The main restriction of this approach is that only a limited number of atoms or internal coordinates can be considered to reach exact numerical results through, e.g., the multiconfigurational timedependent Hartree (MCTDH) method (48). Nevertheless, advances were recently reported on organometallic complexes (49,50). In order to merge the pros of quantum and classical dynamics (highly feasible implementation of Newton’s laws for classical dynamics; proper treatment of surface crossings for quantum dynamics) while downing the cons (severe limitations to study photochemistry for classical dynamics; limitation in the system size for quantum dynamics), several mixed quantum-classical dynamics methods were proposed, also called semiclassical dynamics methods, due to their hybrid nature. They can be mainly divided into (i) Ehrenfest dynamics and (ii) trajectory surface-hopping dynamics. On one hand, Ehrenfest dynamics are based on the mean-field concept: trajectories classically propagate on a timedependent average of all the electronic (quantum) states included in the study, weighted by the corresponding populations (Fig. 2.4B). Different implementations of this general concept, of interest for photoisomerization studies, are the semiclassical electron-radiation-ion dynamics (SERID) (51) and the ab initio multiple cloning—multiconfigurational Ehrenfest (AIMC-MCE) method (52). On the other hand, trajectory surface-hopping dynamics are based on the concept that each excited state trajectory propagates along a certain electronic state and, whenever a PESs’ crossing region is reached, hopping events among the crossing electronic state can happen depending (at each trajectory step) on its calculated probability (Fig. 2.4C).

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FIG. 2.4 Schematic example to graphically depict the basics underlying excited state quantum dynamics (A), and two of the main mixed quantumclassical dynamics approaches: Ehrenfest dynamics (B) and trajectory surface-hopping dynamics (C).

Therefore, this approach requires the run of a certain finite number of trajectories, each of them needing defined initial conditions (as previously discussed in Chapter 1 for simulating the absorption spectrum; see Fig. 1.7), in order to obtain statistically relevant results. But how many trajectories are needed? This question does not have an univocal answer and mainly depends on the computational cost for each trajectory and on the number of accessible physical processors within the computing cluster used by the researcher. Anyway, apart from software and hardware requirements, we could also set scientific criteria: on one hand, if some experimental observables are known (e.g., excited state lifetime, quantum yield of a photoproduct), they should be reproduced within a certain margin of error. On the other hand, if no experimental data are available, the attention could be focused on the time required to hop on a certain electronic state, considering that the standard deviation associated to the average time need to be statistically meaningful. For example, by studying the cistrans photoisomerization of molecular switches (e.g., retinal within Rhodopsin or azobenzene in liquid solvent), we can expect a defined behavior: once in the excited state, the photon0 s energy is initially dissipated through vibrational relaxation in this state, by redistributing single and double carbon-carbon bonds (bond length alternation) on a sub-150 fs timescale, before starting the isomerization process. After hopping on the ground state, one pathway leads back to the reactants, recovering the initial conformation, and another pathway finally allows the photoisomer formation, on a 400–600 fs timescale (53,54). By definition, the trajectory surface-hopping dynamics relies most of its efficiency in the definition and implementation of the hopping probability between crossing PESs, that should be conveniently evaluated at every step of the trajectory. It mainly depends on the coupling between the crossing states that, as it was defined in Eq. (2.53), increases the more the surfaces are near in energy (typically, lower than 10 kcal mol 1). The most used technique is based on Tully’s fewest-

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switches surface-hopping algorithm, mainly due to the feasibility of its implementation and to the shown success in reproducing experimental data (55). It is worth bearing in mind that a decoherence correction has to be added to the initially developed Tully’s algorithm, otherwise, the trajectory can show unphysical coherence among the quantum coefficients of the crossing states, particularly when extensive simulation times are needed (56).

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37. Farfan, C. A.; Turner, D. B. A Systematic Model Study Quantifying How Conical Intersection Topography Modulates Photochemical Reactions. Phys. Chem. Chem. Phys. 2020, 22 (36), 20265–20283. 38. Marazzi, M.; Mai, S.; Roca-Sanjua´n, D.; Delcey, M. G.; Lindh, R.; Gonza´lez, L.; Monari, A. Benzophenone Ultrafast Triplet Population: Revisiting the Kinetic Model by Surface-Hopping Dynamics. J. Phys. Chem. Lett. 2016, 7 (4), 622–626. 39. Polli, D.; Altoe`, P.; Weingart, O.; Spillane, K. M.; Manzoni, C.; Brida, D.; Tomasello, G.; Orlandi, G.; Kukura, P.; Mathies, R. A.; Garavelli, M.; Cerullo, G. Conical Intersection Dynamics of the Primary Photoisomerization Event in Vision. Nature 2010, 467 (7314), 440–443. 40. Islam, M. M.; Zou, C.; Van Duin, A. C. T.; Raman, S. Interactions of Hydrogen with the Iron and Iron Carbide Interfaces: A ReaxFF Molecular Dynamics Study. Phys. Chem. Chem. Phys. 2016, 18 (2), 761–771. 41. Qiu, Y.; Zhong, W.; Shao, Y.; Yu, A. Reactive Force Field Molecular Dynamics (ReaxFF MD) Simulation of Coal Oxy-Fuel Combustion. Powder Technol. 2020, 361, 337–348. 42. Dasgupta, N.; Yilmaz, D. E.; Van Duin, A. Simulations of the Biodegradation of Citrate-Based Polymers for Artificial Scaffolds Using Accelerated Reactive Molecular Dynamics. J. Phys. Chem. B 2020, 124 (25), 5311–5322. 43. Wang, Y.; Liu, Y.; Tu, Y.; Wang, Q. Reductive Decomposition of Solvents and Additives toward Solid-Electrolyte Interphase Formation in LithiumIon Battery. J. Phys. Chem. C 2020, 124 (17), 9099–9108. 44. Garcı´a-Iriepa, C.; Gosset, P.; Berraud-Pache, R.; Zemmouche, M.; Taupier, G.; Dorkenoo, K. D.; Didier, P.; Leonard, J.; Ferre, N.; Navizet, I. Simulation and Analysis of the Spectroscopic Properties of Oxyluciferin and Its Analogues in Water. J. Chem. Theory Comput. 2018, 14 (4), 2117–2126. 45. Garcia-Iriepa, C.; Navizet, I.; Garcia-Iriepa, C.; Navizet, I. Effect of Protein Conformation and AMP Protonation State on Fireflies’ Bioluminescent Emission. Molecules 2019, 24 (8), 1565. 46. Garcia´-Iriepa, C.; Zemmouche, M.; Ponce-Vargas, M.; Navizet, I. The Role of Solvation Models on the Computed Absorption and Emission Spectra: The Case of Fireflies Oxyluciferin. Phys. Chem. Chem. Phys. 2019, 21 (8), 4613–4623. 47. Zemmouche, M.; Garcı´a-Iriepa, C.; Navizet, I. Light Emission Colour Modulation Study of Oxyluciferin Synthetic Analogues via QM and QM/MM Approaches. Phys. Chem. Chem. Phys. 2020, 22 (1), 82–91. 48. Meyer, H.-D.; Gatti, F.; Worth, G. A. Multidimensional Quantum Dynamics: MCTDH Theory and Applications; John Wiley & Sons, 2009. 49. Fumanal, M.; Daniel, C.; Gindensperger, E. Excited-State Dynamics of [Mn (Im)(CO) 3 (Phen)]+: PhotoCORM, Catalyst, Luminescent Probe? J. Chem. Phys. 2021, 154 (15), 154102. 50. Fumanal, M.; Gindensperger, E.; Daniel, C. Ultrafast Excited-State Decays in [Re (CO) 3 (N, N)(L)] N+: Nonadiabatic Quantum Dynamics. J. Chem. Theory Comput. 2017, 13 (3), 1293–1306. 51. Dou, Y.; Torralva, B. R.; Allen, R. E. Semiclassical Electron-Radiation-Ion Dynamics (SERID) and Cis-Trans Photoisomerization of Butadiene. J. Mod. Opt. 2003, 50 (15–17), 2615–2643. 52. Makhov, D. V.; Symonds, C.; Fernandez-Alberti, S.; Shalashilin, D. V. Ab Initio Quantum Direct Dynamics Simulations of Ultrafast Photochemistry with Multiconfigurational Ehrenfest Approach. Chem. Phys. 2017, 493, 200–218. 53. Garcı´a-Iriepa, C.; Marazzi, M.; Frutos, L. M.; Sampedro, D. E/Z Photochemical Switches: Syntheses, Properties and Applications. RSC Adv. 2013, 3 (18), 6241–6266. 54. Marazzi, M.; Garcia-Iriepa, C. Retinal Inspired Photoswitches: From the Isomerization Mechanisms Toward Recent Applications in Photoisomerization: Causes, Behavior and Effects; Nova Science Publishers: New York, 2019; pp. 189–238. 55. Tully, J. C. Molecular Dynamics with Electronic Transitions. J. Chem. Phys. 1990, 93 (2), 1061–1071. 56. Subotnik, J. E.; Jain, A.; Landry, B.; Petit, A.; Ouyang, W.; Bellonzi, N. Understanding the Surface Hopping View of Electronic Transitions and Decoherence. Annu. Rev. Phys. Chem. 2016, 67, 387–417.

Part II

Methods

Chapter 3

Density-functional theory for electronic excited states John M. Herbert Department of Chemistry and Biochemistry, The Ohio State University, Columbus, OH, United States

Chapter outline 3.1 Overview 3.2 Linear-response (“time-dependent”) DFT 3.2.1 Theoretical formalism 3.2.2 Performance and practice 3.2.3 Systemic problems 3.3 Excited-state Kohn-Sham theory: The DSCF approach 3.3.1 Theory 3.3.2 Examples

69 71 71 77 86 91 92 95

3.4 Time-dependent Kohn-Sham theory: “Real-time” TDDFT 3.4.1 Theory 3.4.2 Examples Acknowledgment References

99 99 101 103 103

3.1 Overview Following its implementation in molecular quantum chemistry codes in the early 1990s (1–6), density-functional theory (DFT) quickly emerged as the most popular tool for ground-state electronic structure calculations due to its favorable balance of relatively low cost with reasonable accuracy for thermochemistry. The first excited-state implementations quickly followed (7–12), based on a linear-response (LR) formalism (13–15) that mirrors much earlier work on time-dependent Hartree-Fock (TDHF) theory (16). The historical development of TDDFT has been summarized elsewhere (17). The LR formulation is now known almost universally as “time-dependent” (TD-)DFT, despite its frequency-domain formulation and implementation. In its most pedestrian applications, LR-TDDFT produces vertical excitation energies for closed-shell molecules at ground-state geometries to within a statistical accuracy of 0.3 eV (18), at a cost that is often only a few times more than the cost of the ground-state self-consistent field (SCF) calculation and possessing the same formal scaling (19). This is a useful accuracy for electronic absorption spectra. In view of its low cost, LR-TDDFT has become the de facto method for computing electronic excitation spectra of finite molecular systems, although some fundamental problems remain in its application to periodic materials (20–22). LR-TDDFT is also becoming increasingly popular for photochemical applications (23–25), despite some problems with the description of conical intersections (26–28). In part, this popularity is due to a growing recognition that complete active-space (CAS-)SCF methods cannot be considered quantitative approaches for excited-state dynamics (29–32), due to a lack of dynamical electron correlation. This chapter provides an overview of TDDFT and other DFT-based methods for computing excitation spectra, excitedstate properties, and for simulating photochemical reactions, emphasizing theory rather than applications but with some molecular examples to motivate the discussion. For those unfamiliar with the formal underpinnings of TDDFT, a natural question to ask is “what does time have to do with excitation energies?” In fact, one knows from basic quantum mechanics that the time evolution of a nonstationary wave function encodes the system’s excitation energies via the Bohr frequencies, ojk ¼ ðEj Ek Þ=ħ, therefore the time evolution of a quantum system can be used to extract excitation energies. The existence of a time-dependent extension of DFT is formally justified by the Runge-Gross theorem (33–39), which provides something akin to a time-dependent extension of the first Hohenberg-Kohn theorem for the ground state (40), that is, a density-to-potential mapping. In the time-dependent case, there are important caveats about initial-state dependence and memory effects (41–43). Those issues have yet to be fully resolved in a computationally feasible way, but this has not stymied the practical development and application of TDDFT. Theoretical and Computational Photochemistry. https://doi.org/10.1016/B978-0-323-91738-4.00005-1 Copyright © 2023 Elsevier Inc. All rights reserved.

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Following a perturbation to the ground state, which creates a superposition of energy eigenstates, the Fourier components of the time-dependent dipole moment are precisely the Bohr frequencies. A Fourier transform of the dipole moment function is itself an absorption spectrum (44), Z 1 +∞ (3.1) hmð0Þ
mðtÞie iot dt IðoÞ ¼ 2p ∞ Excitation energies are also encoded in the isotropic frequency-dependent polarizability, a(o), which has a sum-over-states expression aðoÞ ¼ where me is the electron mass, on0 ¼ ðEn

e2 X f 0n me n>0 o2n0 o2

E0 Þ=ħ, and   2me on0 ^ 2 jh0jmjnij f 0n ¼ 3e2 ħ

(3.2)

(3.3)

is the dimensionless dipole oscillator strength for the j0i!jni transition (44). The poles of response function a(o) therefore encode excitation energies, with residues that encode oscillator strengths (19). In the early days of quantum chemistry, Eq. (3.2) was actually used to compute excitation energies for atoms and atomic ions (45, 46), by computing a(o) as the response to an applied field, and a version of this approach would eventually reappear in the form of “real-time” TDDFT (47, 48). The poles of the Kohn-Sham response function also serve this purpose (9, 49, 50), and the LR formalism applied to the Kohn-Sham ground state turns this idea into a robust computational paradigm, in the form of an eigenvalue-type problem for the excitation energies (13–15). Although the LR formulation exists strictly in the frequency or energy domain, the time-dependent origins of the phenomenology are suggested in the name “TDDFT.” Despite its overwhelming popularity, LR-TDDFT excitation energies do tend to be more sensitive to the details of the exchange-correlation (XC) functional as compared to ground-state properties computed with DFT. In some sense, the statistical accuracy of 0.3 eV that is quoted above should therefore be interpreted as representative of the best-case scenario with state-of-the-art functionals, and assuming that certain systemic pathologies can be avoided. LR-TDDFT may not be the theory that we want, but it remains the best theory that we have for excited states of large and even medium-sized molecules. This theory is introduced formally in Section 3.2 and that discussion constitutes the most substantial part of this chapter, just as LR-TDDFT occupies the most significant place among excited-state DFT methods. It holds that position because it is easy to use, not significantly more expensive than ground-state DFT, and provides a slew of excited states in an automated way, starting from a ground-state SCF solution. While the accuracy of LR-TDDFT is often quite reasonable, certain systematic problems have been identified, and excited-state Kohn-Sham procedures have been developed to circumvent these. Rather than applying LR to the ground state, these methods look for an excited-state (non-aufbau) solution to the SCF equations, and for this reason the excited-state Kohn-Sham approach is often called a “DSCF” method. Although not formally justified by the Runge-Gross theorem, the DSCF approach has an admirable record of rectifying the deficiencies of LR-TDDFT, again at a cost comparable to that of a ground-state DFT calculation. What is lost in the DSCF approach is the ability to compute a whole spectrum of states at once, making the state-specific DSCF procedure much more labor intensive for the user. This approach is described in Section 3.3. Finally, it is possible to take the time dependence in TDDFT at face value and to propagate Kohn-Sham molecular orbitals (MOs) in time, following a perturbation applied to the ground-state density. This is accomplished by solving the time-dependent Kohn-Sham (TDKS) equation, iħ

d c ðr, tÞ ¼ F^s cks ðr, tÞ dt ks

(3.4)

which is a one-electron analog of the time-dependent Schr€odinger equation. (Here, s {a, b} is a spin index.) The oneelectron effective Hamiltonian in Eq. (3.4) is the Fock operator F^s that comes from the ground-state Kohn-Sham eigenvalue problem that determines the MOs: F^s cks ðrÞ ¼ ek cks ðrÞ

(3.5)

The “real-time” approach to TDDFT (51, 52), which is described in Section 3.4, consists in solving Eq. (3.4) by propagating the MOs in time following a perturbation to the ground state that creates a time-evolving density,

Density-functional theory for electronic excited states Chapter

rs ðr, tÞ ¼

occ X k

jcks ðr, tÞj2

3

71

(3.6)

expressed here for s-spin electrons. (The total charge density is r ¼ ra + rb.) This approach can be used to simulate strongfield electron dynamics (53), a topic of contemporary interest in attosecond molecular science (54–57). It also provides a route to broadband spectra via Fourier transform of the time-dependent dipole moment function, in a direct realization of Eq. (3.1). This chapter assumes a basis familiarity with ground-state DFT, as represented by the SCF eigenvalue problem in Eq. (3.5), which will serve as our starting point. It should therefore be familiar that the Fock operator takes the form F^s ¼

1 ^2 r + vext + vH + v^sxc 2

(3.7)

in atomic units. The quantities vext, vH, and v^sxc are known as the external, Hartree, and XC potentials, respectively. In the field-free case, the external potential is simply the interaction of the electrons with the nuclei (58), X ZA vext ðrÞ ¼ (3.8) k r RA k A More generally, vext(r) might also contain a field-dependent contribution such as Eðr, tÞ
r in the presence of an electric field Eðr, tÞ. The Hartree (or Coulomb) potential vH(r) describes self-repulsion of the electrons (58), equivalent to what is often called “J” in Hartree-Fock theory (40, 59). It is a functional of the density, given by Z rðr, r0 Þ dr0 (3.9) vH ½rŠðrÞ ¼ k r r0 k

The final component of F^s is v^sxc ¼ dExc =drs , the XC operator for s-spin electrons. In “pure” Kohn-Sham theory, this quantity should be a local potential vsxc(r) rather than an operator, but herein we allow the possibility for mixing some nonlocal Hartree-Fock exchange (HFX), as is done in the hybrid density functionals that are most useful in molecular DFT. For hybrid functionals, v^sxc is a nonlocal operator and this scenario is sometimes called generalized Kohn-Sham theory (39). Although inconsistent with the original Kohn-Sham paradigm, the use of hybrid functionals can no longer be considered exotic in contemporary molecular DFT. The textbook by Koch and Holthausen (40) is a good resource for ground-state DFT (though not for TDDFT), as are several literature overviews (58, 60). Updated ground-state benchmarks, relative to the rather dated ones in Ref. (40), can be found elsewhere (61, 62). For TDDFT, the textbook by Ullrich (63), or else overviews by Gross and coworkers (36, 64, 65), provide the rigorous foundations of the theory, which are mostly omitted here. Several other reviews cover LR-TDDFT in a pedagogical way (66–68). For overviews of molecular applications of LR-TDDFT, see reviews by Jacquemin and coworkers (69–73), who have also reviewed accuracy benchmarks (18) and functional selection (74).

3.2 Linear-response (“time-dependent”) DFT This section describes the formalism and application of LR-TDDFT, commonly known simply as “TDDFT.” The starting point is the TDKS equation (3.4) that describes how the ground-state MOs cks evolve in time following a perturbation that is applied at t ¼ 0. If that perturbation is taken to be a time-oscillating field at frequency o, 1 VðtÞ ¼ ðEe 2

iot

+ E * e+iot Þ

(3.10)

then in the weak-field limit (E ! 0), the response of the ground state can be computed exactly using first-order perturbation theory (15). Formally, one ought to show that the poles of the frequency-dependent response function can be obtained from those of the independent-particle (Kohn-Sham) response function (9), but for that exercise the reader is referred to reviews by Marques and Gross (64, 65). For a derivation of LR-TDDFT based on a variational principle, see Ref. (75).

3.2.1 Theoretical formalism The derivation from perturbation theory starts from the equivalent Liouville-von Neumann (LvN) form of the TDKS equation, which is

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iħ

d P^s ¼ F^s P^s dt

P^s F^s

(3.11)

where P^s ðtÞ ¼

occ X k

jcks ðtÞihcks ðtÞj

(3.12)

is the time-evolving one-electron density operator for s-spin electrons. Expanding Eq. (3.11) to first order in the perturbed Fock and density matrices, in the presence of the perturbation V(t), one obtains the unperturbed LvN equation at zeroth order. This is equivalent to the ground-state Kohn-Sham eigenvalue problem in Eq. (3.5). Working equations for LR-TDDFT are obtained by equating the first-order terms (11, 15, 66), as elaborated below.

3.2.1.1 Linear-response theory To consider this in more detail, recognize that the perturbation V(t) in Eq. (3.10) is a one-electron operator whose spatial part can be expanded in the MO basis, leaving the time dependence to be carried by eiot. Introducing a set of unknown coefficients zpqs and zqps , representing real and imaginary parts of the first-order response, the first-order perturbed density matrix can be expressed as 1 ð0Þ ð1Þ ð0Þ Ppqs ðtÞ ¼ Ppqs + Ppqs ðtÞ ¼ Ppqs + ðzpqs e 2

iot

+ zqps eiot Þ

(3.13)

ð0Þ where Ppqs is the unperturbed density matrix at t ¼ 0. This change in the density matrix is accompanied by a corresponding change in the Fock matrix. Through first order, the Fock matrix is (11) X∂Fpqs  ð0Þ Fpqs ðtÞ ¼ Fpqs + V pq + (3.14) Pð1Þ rsτ ðtÞ ∂Prsτ rsτ ð0Þ

where the unperturbed Fock operator F^s has the form given in Eq. (3.7). The first-order response of the density matrix is thus coupled to a term of the form (11)  2     d Exc  ∂Fpqs  s r ¼ ðps qs jsτ rτ Þ + ps qs  drs drτ  τ τ ∂Prsτ (3.15) sτ ^ ¼ ðpqjsrÞ + ðp q jf js r Þ s s xc

τ τ

2 The first term, (psqsjsτrτ) ¼ (pqjsr), is a Coulomb integral expressed in Mulliken notation (59), while f^sτ xc ¼ d Exc =drs drτ . The latter quantity is discussed in more detail below. So far, the MO indices p, q, r, s are arbitrary and could refer either to occupied or virtual orbitals. In fact, the idempo2 tency condition P^s ¼ P^s imposes restrictions. Through first order, the idempotency condition is X ð0Þ ð1Þ ð1Þ ð0Þ ð1Þ ðPprs Prqs + Pprs Prqs Þ ¼ Ppqs (3.16) r

ð0Þ ð0Þ ð0Þ since P^s P^s ¼ P^s . As a matrix, Psð0Þ contains only occupied-occupied and virtual-virtual blocks because the occupiedvirtual block vanishes as a condition of SCF convergence (76). Using i, j, … to index occupied MOs and a, b, … for virtual ð0Þ ð0Þ ð1Þ MOs, this means Pias ¼ 0 ¼ Pais, so the constraint in Eq. (3.16) implies that the only nonvanishing coefficients in Ppqs are zias and zais (11, 66). Conventional LR-TDDFT notation is obtained by relabeling these coefficients as

xias ¼ zais

(3.17a)

yias ¼ zias

(3.17b)

Collecting these unknowns into vectors x and y, one may rewrite the first-order terms in the LvN equation in matrix form as (66–68)   ðnÞ !   ðnÞ ! A B 1 0 x x ¼ on (3.18) ðnÞ B* A* 0 1 y yðnÞ

Density-functional theory for electronic excited states Chapter

ðnÞ

3

73

ðnÞ

This represents a system of equations for the excitation energies on and the amplitudes xias and yias , and constitutes the basic working equation of LR-TDDFT. (The index n, which enumerates excited states, will usually be omitted for compactness.) The system in Eq. (3.18) is often called the Casida equation (13, 14), although it is formally identical to the equations of TDHF theory (16). The matrices A and B are known as orbital Hessians (19), for reasons that are discussed below, and they originate in the derivative of F^s with respect to P^τ in Eq. (3.15). In the canonical MO basis, the matrix elements of A and B are (19, 66) Aias, jbτ ¼ ðeas

eis Þdij dab dsτ + ðiajjbÞ

Bias, jbτ ¼ ðiajbjÞ

ahfx ðijjabÞdsτ + ð1

ahfx ðibjajÞdsτ + ð1

ahfx Þðiaj^ k sτ xc jjbÞ

ahfx Þðiaj^ k sτ xc jbjÞ

(3.19a) (3.19b)

^sτ a ðd2 E =dr dr Þ. The quantity ahfx will be used throughout this chapter to mean the coefficient of ^ sτ where k xc ¼ f xc hfx HFX s τ HFX (often called “exact exchange”) contained in the functional Exc[r], with 0  ahfx  1. For example, ahfx ¼ 0.2 for B3LYP, meaning 20% HFX and 80% semilocal exchange. We have chosen to separate the HFX terms in Eq. (3.19), as these ^ sτ can be expressed in terms of electron repulsion integrals (ijjab) and (ibjaj), leaving k xc as the second functional derivative of the semilocal contribution Exc EHFX. The solution (x, y) of Eq. (3.18) parameterizes the transition density matrix for the excitation in question. In real space, this quantity is (10, 15, 19) X Tðr,r0 Þ ¼ ½xias cas ðrÞcis ðr0 Þ + yias cis ðrÞcas ðr0 ފ (3.20) ias

The unknowns x and y satisfy a bi-orthogonal normalization condition (15), X ðx2ias y2ias Þ ¼ 1

(3.21)

ias

which is also a feature of the much older TDHF theory (16). For historical reasons, TDHF is also known as the randomphase approximation (RPA) (77, 78), because it can be derived within an equation-of-motion formalism for the singleparticle excitation operators (68), similar to the historical RPA (77). However, TDHF/RPA can also be considered as a ^ sτ special case of LR-TDDFT corresponding to the Hartree-Fock functional, that is, ahfx ¼ 1 and k xc ≡0. The number of unknown amplitudes in Eq. (3.18) is 2noccnvir, hence to solve this equation for all of the excitation energies o would incur sixth-order cost, Oðn3occ n3vir Þ. Because matrix-vector products such as Ax or By can be computed with only fourthorder cost, in practice Eq. (3.18) is solved iteratively for just the lowest few (nroots) eigenvalues (7, 10, 79–82). The cost of that calculation scales as nroots  Oðn2occ n2vir Þ (19), which is typically not significantly greater than the cost of the ground-state SCF calculation if nroots  10. Therefore, if ground-state DFT is feasible then LR-TDDFT is probably tractable also, at least for the lowest few excited states. It is worth noting, however, that the memory footprint to solve Eq. (3.18) is nroots  Oðnocc nvir Þ, which is significantly more than the ground-state memory requirement. This can become a problem for large systems if a large number of excited states is desired, for example, in models of semiconductors, where a band structure is emerging (83). For such applications, the real-time approach that is described in Section 3.4 offers a low-memory alternative to LR-TDDFT. Some alternative forms of the basic LR-TDDFT equation are also worth considering. We first note that the matrices A and B in Eq. (3.19) can be rewritten as Aias,jbτ ¼ ðeas

eis Þdij dab dsτ + K ias,jbτ

Bias,jbτ ¼ K ias,bjτ where K is a coupling matrix (13, 84),   ZZ 1 sτ 0 K ias,jbτ ¼ cis ðrÞcas ðrÞ + f xc ðr, r Þ cjτ ðr0 Þcbτ ðr0 Þdrdr0 k r r0 k

(3.22a) (3.22b)

(3.23)

with a Hartree-XC kernel (67). In practice, this looks like the energy-transfer coupling (85) between transition densities rias(r) ¼ cis(r)cas(r) and rbjτ(r0 ) ¼ cbτ(r0 )cjτ(r0 ). One can therefore consider that solution of the LR equations reveals how the zeroth-order, independent-particle excitations cis ! cas are coupled to obtain excited states of the interacting system.

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Assuming that the orbitals are real, so that A* ¼ A and B* ¼ B, then Eq. (3.18) is equivalent to a pair of equations ðA  BÞðx  yÞ ¼ oðx  yÞ

(3.24)

which makes it clear that Eq. (3.18) is not a conventional eigenvalue problem. However, upon introducing new variables pffiffiffiffi z ¼ oðA  BÞ 1=2 ðx  yÞ (3.25)

which satisfy the more conventional normalization condition z{ z ¼ 1 (86, 87), the LR-TDDFT equation can be transformed into either of two equivalent, Hermitian eigenvalue problems (81, 86–89). These are V z ¼ o 2 z

(3.26)

V ¼ ðA  BÞ1=2 ðA  BÞðA  BÞ1=2

(3.27)

where

The V+ version of Eq. (3.26) is especially convenient for semilocal functionals (ahfx ¼ 0), because in that case A B is diagonal and one obtains a Hermitian eigenvalue problem with half the dimension of the original pseudo-eigenvalue problem in Eq. (3.18). For a closed-shell (spin-restricted) ground state, another important transformation is pffiffiffi x (3.28a) ia ¼ ðxiaa  xiab Þ= 2 p ffiffi ffi y (3.28b) ia ¼ ðyiaa  yiab Þ= 2 which affords amplitudes for singlet (+) and triplet ( ) spin functions. Making use of the unitary transformation (78)       A+B 0 1 A B 1 1 1 1 ¼ (3.29) 2 0 A B 1 1 B A 1 1

in addition to Eq. (3.28), one obtains singlet and triplet versions of A B that function as stability matrices (7, 8, 86). In other words, these are Hessian matrices whose eigenvalues characterize whether the ground state is stable with respect to orbital rotations. For example, the singlet stability matrix is (86) ðA+ + B+ Þia, jb ¼ ðea

^bb ei Þdij dab + 4ðiajjbÞ + 2ðiajðf^aa xc + f xc ÞjjbÞ

(3.30)

A negative eigenvalue in A+ + B+ indicates an instability, which manifests as a negative excitation energy from the standpoint of LR-TDDFT. This is a consequence of the Thouless theorem (90), which states that orbital rotations (and therefore orbital relaxation) can always be parameterized as single excitations. Along similar lines, eigenvalues of the triplet instability matrix (86) ðA +B Þia, jb ¼ ðea

ei Þdij dab + 2ðiajðf^aa xc

f^ab xc ÞjjbÞ

(3.31)

indicate whether the ground-state solution is stable with respect to spin-symmetry breaking (restricted ! unrestricted relaxation) (91). In the presence of an unstable reference state, the transformation in Eq. (3.25) may become problematic, which can lead to failure of certain iterative LR-TDDFT algorithms.

3.2.1.2 Adiabatic approximation We have not yet discussed the key ingredient in the orbital Hessian matrices that makes LR-TDDFT different from TDHF/ RPA, namely, the exchange-correlation kernel, f^sτ xc . A more careful application of LR theory would note that the quantities A(o) and B(o) are themselves functions of the excitation energy o (13–15). In wave function terms, this can be understood based on the fact that any exact theory of many-electron excitation energies that is formulated as an effective single-particle theory must ultimately invoke an effective Hamiltonian that is energy dependent, in order to encapsulate the effects of higher-order excitations (14, 92, 93). (In many-body theory, this energy-dependent contribution is sometimes called sτ the “self-energy” (94).) Proof-of-concept models for an energy-dependent kernel fxc (r, r0 , o) have been put forward (42, 43, 93, 95), which have close connections to many-body perturbation theory and the Bethe-Salpeter equation (92, 93). However, there are no production models for molecular Hamiltonians at present.

Density-functional theory for electronic excited states Chapter

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To appreciate the nature of the approximation that is made by neglecting the energy dependence of f^sτ xc , consider that this quantity arises in Eq. (3.15) as the second functional derivative of the XC energy with respect to the density, or the first derivative of the XC potential. For a time-evolving density rs(r, t), this means (41) 0 f sτ xc ðr, r , t

t0 Þ ¼

d^ vsxc ½rŠðr, tÞ rτ ðr0 , t0 Þ

(3.32)

(This expression leaves v^sxc in the form of an operator, which technically makes this an example of generalized Kohn-Sham sτ theory (39).) The time dependence in fxc (r, r0 , t t0 ) means that this quantity depends on the whole history of the time sτ (r, r0 , o). For practical evolution of the density (41–43), imparting a frequency dependence upon Fourier transformation: fxc purposes, it is basically a requirement to invoke the adiabatic approximation (13, 68, 96), which assumes locality in time and therefore differentiates with respect to the instantaneous density (13): d^ vsxc ½rŠðr, tÞ  dðt rτ ðr0 , t0 Þ

t0 Þ

d^ vxc ½rŠðrÞ rτ ðr0 Þ

(3.33)

The “memory” of the kernel is thereby neglected, tantamount to assuming that v^xc ½rŠðr, tÞ can be approximated using a conventional ground-state energy functional Exc[r], whose functional derivative is evaluated using the time-evolving density (96):  dExc ½rŠ s (3.34) v^xc ðr, tÞ ¼ drs ðrÞ rs ðrÞ¼rs ðr,tÞ

Time dependence is thus carried entirely by the density and not by the functional. The frequency dependence of f^sτ xc disappears and conventional (ground-state) density functionals are all that is required for LR-TDDFT within the adiabatic approximation. One immediate ramification of this approximation is that the LR-TDDFT equation has precisely 2noccnvir solutions for the excitation energy o, coinciding with the number of unknown amplitudes xias and yias. In wave function language, these are the “one-particle, one-hole” (1p1h) states, as in conventional configuration interaction with single excitations (CIS). States with significant double-excitation character (2p2h states) are either absent altogether (95–99), or at best severely shifted (98). The latter are therefore generally considered to be out of reach within the adiabatic approximation to LR-TDDFT that is ubiquitous in practical calculations (99).

3.2.1.3 Tamm-Dancoff approximation Given a ground-state functional Exc[r], all that is required for LR-TDDFT within the adiabatic approximation are second functional derivatives 0 f^sτ xc ðr, r Þ ¼

d2 Exc drs ðrÞdrτ ðr0 Þ

(3.35)

from which the matrix elements of A and B can be evaluated. Upon solution of Eq. (3.18) or one of its equivalent forms, it is often found that the amplitudes yias are 102–103 times smaller than the largest xias. Invoking the approximation yias  0, one obtains a conventional Hermitian eigenvalue problem Ax ¼ ox

(3.36)

whose dimension is half that of the original LR-TDDFT pseudo-eigenvalue problem, and where the matrix B does not appear. The basis for this approximation can be understood from the fact that the matrix elements of A are typically much larger (at least along the diagonal) as compared to the matrix elements of B, because the leading contribution to A is a difference in one-particle energy levels (Aias,ias ¼ eas eis + ⋯ ). For historical reasons that are related to a similar approximation that is made in nuclear physics (63), neglect of y is known as the Tamm-Dancoff approximation (TDA). For hybrid functionals, the reduction in dimension leads to a concomitant reduction in cost although for semilocal functionals the same reduction in dimensionality can be achieved using the V+ version of Eq. (3.26). For the Hartree-Fock functional (ahfx ¼ 1 and no correlation), Eq. (3.36) is equivalent to the CIS eigenvalue equation (100). Excited-state wave functions in CIS are linear combinations of singly excited Slater determinants jCas is i, jCi ¼

occ X vir X X i

a

s

xias jCas is i

(3.37)

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and for this reason we identify the variables xias as excitation amplitudes. The neglected amplitudes yias represent deexcitation, insofar as TDHF/RPA was originally introduced in the nuclear and many-body physics literature as a means to add correlation to the ground state (16, 77). In fact, LR-TDDFT in the form of Eq. (3.18) was introduced in molecular quantum chemistry as the “DFT random-phase approximation” (101, 102). For that reason, solution of Eq. (3.18) or its equivalents, without invoking the TDA, is sometimes called “RPA.” However, in view of a resurgence of interest in using the RPA formalism as a means for correlating the ground state (103–110), it is better to refer to Eq. (3.18) as “full” LR-TDDFT, if there is a need to distinguish it from the TDA version in Eq. (3.36). Quantitatively, the impact of the TDA on excitation energies is often 0

This constraint is satisfied by the full LR-TDDFT approach, at least in the complete-basis limit (13, 15, 111). Some smallmolecule tests suggest that errors incurred by the TDA are relatively mild ( 0. Note that triplet instabilities are associated with spin-symmetry breaking, that is, with the emergence of an unrestricted solution that is lower in energy than the restricted solution. Where they appear, these instabilities cause significant artifacts in potential energy surfaces computed using LR-TDDFT, including the appearance of spurious cusps (84, 114–117). In contrast, the variational nature of the CIS-type equation, as opposed to the pseudoeigenvalue problem that characterizes full LR-TDDFT, prevents this from happening within the TDA (68). Along similar lines, it has been appreciated for a long time that TDHF specifically is prone to triplet instabilities (90, 91, 118–124). In fact, the appearance of imaginary excitation energies at equilibrium geometries of small molecules led Furche and Ahlrichs to conclude that this method is “rather useless... for the investigation of excited potential energy surfaces” (125). In contrast, spin-symmetry breaking near the equilibrium geometry is often significantly mitigated when DFT is substituted for Hartree-Fock theory (126). Because most molecular LR-TDDFT calculations use hybrid functionals that include some fraction of HFX, it can be expected that problems with triplet instabilities may increase as that fraction increases, which is precisely what is found in practice (127–131). Similarly, in calculations using range-separated functionals, which incorporate HFX at long range in the Coulomb potential, these instabilities are sensitive to the length scale on which that mixing is introduced (132–137). Invoking the TDA thus improves the accuracy of triplet excitation energies (134–136). For photochemical problems, where exploration of excited-state potential energy surfaces is paramount, Casida et al. suggest that the TDA is effectively mandatory (138), in order to avoid excitation energies that drop to zero (and then become imaginary) as the system moves through a Coulson-Fischer point where spin-symmetry breaking occurs. Furthermore, instabilities appear to proliferate as one moves away from the ground-state geometry on an excited potential surface (138–140). For example, in the photochemical ring-opening reaction of oxirane (C2H4O), 51% of configuration space is estimated to exhibit an instability with semilocal DFT, as compared to 93% of space with B3LYP (138).

3.2.1.4 Analytic gradients Photochemical simulations require analytic excited-state gradients. That formalism, which is closely connected to response theory for optical properties (141), is not discussed here but can be found elsewhere (19, 125, 142–145). Nonadiabatic or “derivative coupling” vectors between excited states (27, 146), which are needed for nonadiabatic molecular dynamics simulations (23–27), have also been formulated (147–154). Evaluation of the nonadiabatic couplings has the same formal complexity as evaluation of the excited-state gradient (25, 147). The gradient formalism also bears on static properties of the excited state, such as the dipole moment or atomic population analysis. The density matrix for the excited state can be written as Prlx ¼ Punrlx + Z ¼ P0 + DP + Z

(3.39)

which is sometimes called the “relaxed” density matrix, with Punrlx ¼P0 + DP as the "unrelaxed" contribution. Here, P0 represents the ground-state density matrix and the unrelaxed change upon excitation (DP) can be obtained from the amplitudes x and y. The remaining contribution (Z) represents orbital relaxation (19, 144, 145). The unrelaxed density change DP can be separated into particle (electron) and hole contributions,

Density-functional theory for electronic excited states Chapter

DP ¼ DPelec + DPhole

3

77

(3.40)

which are given by (125, 145, 155, 156) 1 DPelec ¼ ½ðx + yÞ{ ðx + yÞ+ðx yÞ{ ðx yފ 2 1 ½ðx + yÞðx + yÞ{ + ðx yÞðx yÞ{ Š DPhole ¼ 2

(3.41a) (3.41b)

These expressions can be rearranged to afford (155) ðDPelec Þabs ¼

X

ðDPhole Þijs ¼

ðx⁎ias xibs + y⁎ias yibs Þ

(3.42a)

X ðxias x⁎jas + yias y⁎jas Þ

(3.42b)

i

a

Note that DP only contains occupied-occupied (DPhole) and virtual-virtual (DPelec) contributions to the excited-state density matrix, not occupied-virtual contributions. The latter are contained in Z (19, 125), the evaluation of which requires solution of so-called Z-vector equations (157). This has the same formal complexity as an excited-state gradient calculation. Excited-state properties should be computed using the relaxed density matrix Prlx, because DP and Z make similar contributions (100, 158, 159). Especially when the change in density is large, as for an excitation with significant charge-transfer (CT) character, the use of the unrelaxed density matrix may lead to unacceptable errors in excited-state properties. For example, excited states of p-nitroaniline involving intramolecular CT character exhibit relaxed and unrelaxed dipole moments (computed using Prlx vs. Punrlx, respectively) that differ by more than 10 D in some cases (158)!

3.2.2 Performance and practice As discussed earlier, the formal scaling of LR-TDDFT is nroots  Oðn4basis Þ for hybrid functionals (19). In practical terms, where only a few low-lying excited states are desired, this means that LR-TDDFT is generally feasible if the corresponding ground-state calculation is practical, perhaps up to about 400 atoms for single-point calculations or 150–250 atoms for excited-state geometry optimization (70, 71), with more severe limitations for excited-state frequency calculations (71, 160). These estimates are appropriate where basis sets of double-z quality are used, which is generally adequate. Triple-z basis sets may be considered to be converged (67). This section provides an overview of other practical considerations in LR-TDDFT calculations, including selection of an XC functional. Techniques for visualizing and understanding the excited states are also discussed.

3.2.2.1 Restriction of the excitation manifold Significant cost reduction in LR-TDDFT calculations for large molecules can be achieved by neglecting some of the amplitudes xias, in addition to neglecting all of the amplitudes yias. Fig. 3.1 shows examples of excitation spectra computed for C60 and for C119H154ClN21O40, performed by excluding over 70% of the virtual orbitals (based on orbital energies eas), without adverse effects on the spectral envelope (161). Similar truncations of the excitation manifold can be used to access core-excited states (162–165). There is significant interest in core excitations in contemporary quantum chemistry (164–169), driven by the recent availability of tabletop laser sources with femtosecond time resolution (170–174). However, core-to-valence excitations lie embedded in an ionization continuum and, at a practical level, lie above all of the valence-excited states, such that the use of iterative eigensolvers is prohibitively expensive if the spectrum must be computed starting from the lowest excitation energies. By retaining only those amplitudes xias for which i is a core orbital on the atom of interest, core-excited states emerge as the smallest eigenvalues and can be computed directly. This “frozen-valence occupied” approximation has historically been called corevalence separation (175–177), and it introduces negligible error for K-edge excitations where cis is a 1s orbital (177). Another strategy to access core-level excitations is energy windowing (162, 178), in which the amplitudes xias are excluded unless eas eis lies within the window of interest.

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(a) intensity

(b)

2

3

4

5 6 energy (eV)

7

8 3

4

5 6 energy (eV)

7

8

FIG. 3.1 Electronic absorption spectra of (A) C60 (PBE/6-31G* level) and (B) the antibiotic ramoplanin (271 atoms and 2483 basis functions, CAMB3LYP/6-31G* level). Thin curves represent the experimental spectra and thick curves are computed from LR-TDDFT/TDA excitation energies with 0.2 eV Gaussian broadening. Core orbitals and 70% of virtual orbitals are excluded from each calculation. (Adapted from Hanson-Heine, M. W. D.; George, M. W.; Besley, N. A. Assessment of Time-Dependent Density Functional Theory Calculations With the Restricted Space Approximation for Excited State Calculations of Large Systems. Mol. Phys. 2018, 116, 1452–1459, under a CC BY License.)

3.2.2.2 Exchange-correlation functionals Before considering the accuracy of LR-TDDFT, it is useful to introduce a paradigm for classifying various densityfunctional approximations, for which we use the taxonomy of “Jacob’s ladder” (179–181). At each rung on this metaphorical ladder, the functional dependence of Exc grows more intricate, incorporating more sophisticated functionality ^ ,r ^ 2 r , τ ,fc gŠ. In a statistical sense (and only depending on the density, its gradients, the Laplacian, etc.: Exc ½rs , rr s s s as in a statistical sense), it is true that the best functionals on the higher rungs of the ladder outperform the best functionals on ^ ,r ^ 2 r ,… as follows. the lower rungs (61, 182). These rungs map onto various inputs rs , rr s s l

l

l

Rung 1: Local density approximation (LDA). The baseline LDA functional comes from the uniform electron-gas model in which Exc is a functional of r(r) only, or of ra(r) and rb(r) if the system is spin-polarized. This approach does not afford useful accuracy for molecular quantum chemistry, with errors of 60–100 kcal/mol for atomization energies (61, 183) and 20 kcal/mol for barrier heights (61). Rung 2: Generalized gradient approximations (GGAs). This class of functionals includes a dependence on the density ^ ðrÞ. These are often called “semilocal” approximations, to distinguish them from LDA while acknowlgradients rr s edging that in their mathematical form, GGAs remain local in the sense that vsxc(r) is a multiplicative potential. GGA functionals significantly improve thermochemistry relative to LDA; typical errors are 10–20 kcal/mol for atomization energies (61, 183) and 5–15 kcal/mol for barrier heights (61, 182). Rung 3: Meta-GGAs (mGGAs). These functionals are also semilocal but incorporate additional derivatives including ^ 2 r and the kinetic energy density, r s τs ðrÞ ¼

l

l

occ X i

^ ðrÞk2 k rc is

(3.43)

^ 2 r it can be used to express The function τs(r) is related to the electron localization function (184), and together with r s the noninteracting kinetic energy (185). The best mGGA functionals improve upon GGA thermochemistry, with errors of 5–10 kcal/mol for atomization energies (61) and 3–6 kcal/mol for barrier heights (61, 182). It is worth noting that some mGGAs introduce a considerable number of parameters (61), and exhibit basis-set and grid sensitivities suggesting that they may be overfitted (186–189). Rung 4: Hyper-GGAs. As originally defined by Perdew et al. (179, 180), this category consists of functionals that incorporate “exact exchange and compatible correlation” (179). A few genuine hyper-GGAs have been put forward (190, 191), but it has proven difficult to construct correlation functionals that work well with 100% HFX. As such, the fourth rung on Jacob’s ladder has effectively been redefined to mean hybrid functionals (181), which incorporate some fraction of HFX (0 < ahfx < 1), in conjunction with a fraction 1 ahfx of semilocal exchange. These functionals are sometimes further categorized as either hybrid GGAs or hybrid mGGAs, depending on the nature of the semilocal contribution. The best hybrid functionals exhibit errors 0) functional, the corresponding LRC functional is SR LR ELRC xc ¼ ahfx Ex,HF + Ex,HF + ð1

GGA ahfx ÞESR x,GGA + Ec

(3.48)

The parameter m in Eq. (3.44) still controls the length scale on which LR-HFX is activated, but a + b ¼ 1 is satisfied by construction and therefore vsxc(r)  r 1 for any m > 0. The LRC strategy is thus to graft correct asymptotic behavior onto an existing semilocal XC functional, while doing the least possible damage to that functional at short range. Nonempirical adjustment (or “tuning”) of the parameter m is often employed in this context, especially where CT states are involved. See Section 3.2.3 for additional discussion of this topic. LRC functionals require modification of the semilocal GGA exchange functional in order to use an attenuated Coulomb potential. (HFX integrals can be modified once and for all to separate them into LR and SR contributions (229).) There are several routes to modify Ex,GGA. The first of these, originally introduced by Hirao and coworkers (200–202), modifies the exchange inhomogeneity factor that multiplies the electron-gas exchange energy density. The present author has suggested that these functionals should be denoted as LRC-mGGA (230, 231), where “GGA” indicates the semilocal parent functional, for example, GGA ¼ BLYP or PBE. Note that “LC” is another common abbreviation for long-range correction so that functionals such as LC-BLYP (200) might more descriptively be called LRC-mBLYP, in order to emphasize which SR-GGA exchange function (mBLYP) is being used. For semilocal exchange functionals such as PBE that are based on a model for the exchange hole (232, 233), an alternative strategy is to combine that model with an attenuated Coulomb potential in order to obtain ESR x,GGA (228, 233). To distinguish this from the LRC-mPBE functional constructed using Hirao’s approach, the present author has suggested the nomenclature LRC-oPBE for the model based on the PBE exchange hole (230, 231), which comports with the notation for the range-separation parameter (o) that was introduced in Ref. (233). The term LC-oPBE is synonymous with LRC-o PBE, and LR-oPBEh is sometimes used to indicate a short-range hybrid (ahfx > 0). In contrast, oPBE refers only to the SR modified exchange functional, ESR x,PBE, and should not be used to mean the LRC functional because Ex,PBE is used in other capacities. For example, the HSE functional (234), sometimes called HSE06 (235), uses oPBE in conjunction with SR-HFX to construct a hybrid functional that is efficient for periodic calculations.

3.2.2.3 Accuracy for vertical excitation energies There have been numerous systematic surveys of the accuracy of various XC functionals for use in LR-TDDFT (236–246), enough to have spawned a metareview of the benchmark studies themselves (18). Two of these studies are highlighted here, to provide some sense for how various categories of functionals can be expected to perform. As usual in DFT (and even more so in LR-TDDFT), for any given molecule it is likely that one could find some XC functional that outperforms the statistically best approach. As such, it is only by understanding trends among functionals (and likely trying the same calculation with more than one functional) that results can be taken seriously. Both of the studies highlighted here compare vertical excitation energies to experimental data, and while that has the advantage of being a direct comparison against numbers that one might hope to simulate, it has the disadvantage that vertical excitation energies are not strictly measurable quantities and various effects including solvatochromatic shifts and vibrational averaging are folded into the comparison. Other studies have compared LR-TDDFT excitation energies against correlated wave function benchmarks (236–239), which make for a much more straightforward test of the theory although unfortunately such comparisons sometimes get little traction outside of quantum chemistry circles, where comparison of theory against theory is often viewed with derision. Fortunately, the trends that are highlighted herein are reasonably similar to those obtained by comparing against ab initio benchmarks. In assessing the performance of various functionals, we will use the taxonomy of Jacob’s ladder as an organizing principle. The accuracy of the best-performing LR-TDDFT functionals follows this paradigm reasonably well, with the caveat that existing hybrid mGGA functionals do not consistently outperform hybrid GGAs (239).

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A first set of benchmarks is depicted in Fig. 3.3, for a set of 101 transitions in 14 gas-phase molecules (244). Error statistics are grouped and color coded by category, including (GH-)GGAs and (GH-)mGGAs but not RSH functionals. Errors are further separated into singlet excitations, triplet excitations, np*, pp*, and Rydberg excitations. Examining these data, it quickly becomes apparent that the GH functionals significantly outperform the semilocal ones, across all types of data, although it is less clear whether GH-mGGA functionals are categorically superior to GH-GGAs. Perhaps surprisingly, the PBE0 and B3LYP functionals outperform most other functionals, including much newer mGGAs and some GH-mGGAs of the Minnesota type (247), although M06-2X does exhibit slightly smaller errors. The B3LYP and PBE0 functionals, which for many years have served as the closest there is to a “default” setting in molecular DFT, continue to outshine many other functionals for vertical excitation energies. Other benchmarks give a slight advantage to oB97X-D (239).

(b)

(a) O

H 2O water

ethylene

H

H

formaldehyde N

N

N

pyridine

pyrazine

benzene

N

N

N

N

s-tetrazene

H N

O

pyrrole

furan

cyclopentadiene

methylenecyclopropene

O O

O

H

E-butadiene

s-transacrolein

s-transglyoxal

M06-2X M06-HF M05 TPSSh M08-HX M08-SO M05-2X M06 M06-L VSXC revTPSS TPSS TPSSm PKZB PBE0 B3LYP BH&HLYP X3LYP PBE PW91 BLYP OLYP LDA

GH-mGGA

mGGA

GH-GGA

GGA LDA

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

mean absolute error (eV)

(d)

(c) M06-2X M06-HF M05 TPSSh M08-HX M08-SO M05-2X M06 M06-L VSXC revTPSS TPSS TPSSm PKZB PBE0 B3LYP BH&HLYP X3LYP PBE PW91 BLYP OLYP LDA

triplet singlet all

valence n * valence * all GH-mGGA

mGGA

GH-GGA

GGA LDA

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

mean absolute error (eV)

M06-2X M06-HF M05 TPSSh M08-HX M08-SO M05-2X M06 M06-L VSXC revTPSS TPSS TPSSm PKZB PBE0 B3LYP BH&HLYP X3LYP PBE PW91 BLYP OLYP LDA

Rydberg valence all GH-mGGA

mGGA

GH-GGA

GGA LDA

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

mean absolute error (eV)

FIG. 3.3 Errors in TDDFT/6-311++G(3df,3pd) vertical excitation energies, versus experiment. (A) Molecular data set, including 63 singlets (15 1pp*, 14 1np*, 3 1ns*, 1 1sp*, and 30 Rydberg excitations) and 38 triplets (15 3pp*, 12 3np*, and 11 Rydberg excitations). Error statistics are then plotted for (B) singlet versus triplet excitation energies, (C) np* versus pp* excitation energies, and (D) Rydberg versus valence excitation energies. Functional names are grouped according to the taxonomy of Jacob’s ladder: global hybrids, meta-GGAs (mGGAs), and GGAs. The global hybrids are further categorized according to whether they are based on GGAs (GH-GGAs) or mGGAs (GH-mGGAs). Within a given category, the functionals are ordered according to the overall MAEs for the entire data set. For ease of comparison, the horizontal scale is the same in each panel. (Adapted with permission from Leang, S. S.; Zahariev, F.; Gordon, M. S. Benchmarking the Performance of Time-Dependent Density Functional Methods. J. Chem. Phys. 2012, 136, 104101; copyright 2012 American Institute of Physics.)

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The best-performing functionals (PBE0, B3LYP, and M06-2X) exhibit mean absolute errors (MAEs) of 0.3 eV for the entire data set. Unlike other functionals examined in Fig. 3.3, these ones do not seem to be systematically worse for np* states as compared to pp* states. In contrast, none of the GGA functionals has an MAE below 0.5 eV and the semilocal mGGAs also have MAEs ≳0.4 eV, with M06-L as the best performer among the latter. All of the semilocal functionals perform significantly worse for np* excitations than they do for pp* excitations. The comparison between Rydberg and valence excitations in Fig. 3.3D warrants special attention. With few exceptions, errors are significantly larger for the Rydberg excitations. Significant errors in Rydberg excitation energies were noted in the early molecular applications of LR-TDDFT (248), leading to the understanding that these excitation energies are quite sensitive to the long-range behavior of the XC potential. That behavior is incorrect for almost all of the functionals evaluated in Fig. 3.3. Later this analysis was extended to CT excitation energies in general (203), of which Rydberg excitations can be considered a special case insofar as these states involve excitation into a diffuse orbital, relatively far from the molecular core. This observation eventually led to the understanding that HFX is the only component of modern functional construction that exhibits the proper asymptotic behavior for a charge-separated state, whereas semilocal XC potentials fall off much too rapidly with distance and thus significantly underestimate both Rydberg and CT excitation energies (84, 115, 248–252). It is therefore no accident that the only functionals in Fig. 3.3D for which the valence excitation error is larger than the Rydberg excitation error are precisely the ones with the largest fractions of HFX: M06-2X (ahfx ¼ 0.54) (253), M06-HF (ahfx ¼ 1.0) (253), PBE0 (ahfx ¼ 0.25) (254), and BH&HLYP (ahfx ¼ 0.5) (255). A second statistical survey is presented in Fig. 3.4, taken from one of the largest statistical assessments of LR-TDDFT to date (240): 614 singlet excitation energies in 483 solution-phase organic molecules. Vertical excitation energies have been corrected for solvent effects and compared to experimental band maxima. (For a discussion of dielectric continuum solvation models and their application to LR-TDDFT, see Ref. (256).) Functionals are once again grouped by category and this larger data set makes it clear that the GH functionals generally outperform the semilocal mGGA functionals, which themselves outperform the semilocal GGAs. For most of the GH functionals, MAEs are 0.2–0.3 eV as compared to 0.4–0.5 eV for the semilocal functionals, but the mean signed errors (Fig. 3.4A) are much smaller for the GH functionals. Signed errors are nearly zero for PBE0 and B3LYP, indicating no systematic error in these cases. In contrast, errors are much larger for GH functionals containing a large fraction of HFX, including BMK (ahfx ¼ 0.42) (257), M05-2X (ahfx ¼ 0.56) (258), and BH&HLYP (ahfx ¼ 0.5) (255). These large-ahfx functionals exhibit bias toward overestimation of excitation energies, whereas semilocal functionals consistently underestimate them. Also included in Fig. 3.4 are error statistics for a set of RSH functionals. MAEs for these functionals span a wide range from 0.2 to 0.5 eV and in that sense are not better than the GH functionals. Furthermore, whereas semilocal functionals HF LC- PBE LC-TPSS LC- HCTH LC-PBE LC-OLYP LC-BLYP CAM-B3LYP LC- PBE(20) M05-2X BH&HLYP BMK M05 mPW1PW91 PBE0 B98 X3LYP B3LYP HCTHhyb O3LYP TPSSh VSXC TPSS HCTH PBE OLYP BP86 BLYP LDA

RangeSeparated Hybrids

Global Hybrids

meta-GGAs LDA / GGAs –0.4

(A)

HF LC- PBE LC-TPSS LC- HCTH LC-PBE LC-OLYP LC-BLYP CAM-B3LYP LC- PBE(20) M05-2X BH&HLYP BMK M05 mPW1PW91 PBE0 B98 X3LYP B3LYP HCTHhyb O3LYP TPSSh VSXC TPSS HCTH PBE OLYP BP86 BLYP LDA

–0.2

0.0

0.2

mean error (eV)

0.4

0.8

0.0

(B)

RangeSeparated Hybrids

Global Hybrids

meta-GGAs LDA / GGAs 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8 0.9

mean absolute error (eV)

FIG. 3.4 (A) Mean errors and (B) mean absolute errors for 614 singlet excitation energies of 483 molecules, comparing LR-TDDFT/6-311+G(2d,p) vertical excitation energies (with solvent corrections) to experimental absorption maxima, using data from Ref. (240). (Adapted with permission from Laurent, A. D.; Jacquemin, D. TD-DFT Benchmarks: A Review. Int. J. Quantum Chem. 2013, 113, 2019–2039; copyright 2013 John Wiley & Sons.)

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systematically underestimate excitation energies, GH functionals are largely free of that bias except when ahfx > 0.4. Finally, RSH functionals systematically overestimate excitation energies, which has also been observed in more recent benchmarks for biochromophores (236). Putting these observations together, it seems that some HFX is optimal, perhaps ahfx  0.20–0.25, with excitation energies that are too low for smaller values and too high for larger ones. Included the latter category are many LRC functionals that use ahfx ¼ 1 in the asymptotic limit. With that in mind, it is interesting to compare error statistics for LC-oPBE and LC-oPBE(20) in Fig. 3.4. The former uses a range-separation parameter o ¼ 0.4 bohr 1 that was optimized for ground-state properties (259), whereas in LC-oPBE(20) that parameter is set to o ¼ 0.2 bohr 1, leading to significant reduction in the errors. Attempts to fit o using both ground-state properties as well as excitation energies typically lead to values in the range o ¼ 0.2–0.3 bohr 1, depending on whether short-range HFX is present or not (226, 228, 260). This is consistent with the revised choice in LC-oPBE(20).

3.2.2.4 Visualization Having computed an excitation energy, there are a variety of tools available to visualize the excited state in question. One could simply examine each pair of occupied and virtual MOs for which the amplitude xias is large, but this is often tedious due to significant configuration mixing, especially in the virtual space. At the CIS level, it is easy to understand why the canonical MOs are not a good basis for visualization purposes, since Koopmans’ theorem implies that the virtual MOs are reasonable orbitals for electron attachment, not excitation (59). The Hartree-Fock virtual MOs feel the full repulsive potential of the N-electron charge density, whereas the occupied MOs feel only N 1 electrons, and this makes the virtual MOs significantly more diffuse than the occupied MOs. Often, the Hartree-Fock virtual MOs are simply unbound and therefore represent discretized continuum states (261), whose shapes are sensitive to small changes in basis set (262). Significant configuration mixing is therefore necessary in order to obtain a localized valence excitation. In principle, exact Kohn-Sham MOs are a much better basis for excitations (262–264), since both occupied and virtual MOs are subject to the same N-electron potential, and in practice it is often the case that the first few Kohn-Sham virtual orbitals are bound (eas < 0). Hybrid functionals, however, push the virtual orbitals and their eigenvalues back toward the Hartree-Fock picture and even 20%–25% HFX can be enough to engender significant configuration mixing due simply to the diffuseness of the virtual MOs. This type of configuration mixing is artificial, in the sense that it can be removed via orbital rotation and therefore does not represent true multiconfigurational character in the excited state. The relevant transformation of the canonical occupied MOs is a unitary matrix Uo that diagonalizes DPelec in Eq. (3.42a): 1 0 2 l1 0 0 ⋯ C B B0 0 ⋯ C l22 2 elec { C B (3.49) Uo ðDP ÞUo ¼ L ¼ B ⋱ 0 C A @ 0 ⋯ 0 l2nocc The nocc  nocc diagonal matrix L2 contains the eigenvalues, which are strictly nonnegative and are normalized such that P 2 i li ¼ 1. The corresponding transformation of the canonical virtual MOs is ! L2 0 hole { Uv ðDP ÞUv ¼ (3.50) 0 0

These two transformations define the natural transition orbitals (NTOs) (265–268), which are the natural orbitals (eigenfunctions) of the excited-state density matrix (268). They can equivalently be defined based on a singular value decomposition of the nocc  nvir matrix of amplitudes, x + y (266, 267):   L 0 { Uo ðx + yÞUv ¼ (3.51) 0 0 This form demonstrates that no more than nocc of the singular values {li} are nonzero. These eigenvalues appear in pairs (l2i ) when DPelec and DPhole are diagonalized, because the natural occupation numbers of the excited-state density matrix are (268)

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8 2 > < 1 lr , nr ¼ l2r , > : 0,

1  r  nocc nocc < r  2nocc

(3.52)

r > 2nocc

The matrices Uo and Uv transform the canonical occupied and virtual MOs, respectively, into a set of “hole” orbitals fchole g i along with corresponding “particle” (or “electron”) orbitals, fcelec g. These are the NTOs, and their diminishing importance i for describing the excitation in question is quantified by the values l21 > l22 > l23 > ⋯ . NTOs provide a much more compact description of the wave function as compared to the canonical MOs, yet one that preserves the phase and nodal structure that can be helpful in qualitatively characterizing the nature of the excitation. This is illustrated in Fig. 3.5 for the S0 !S2 excitation of a five-unit polyfluorene molecule in which a carbonyl defect in one of the terminal fluorene monomers serves to localize the excitation. That localization, however, is not obvious from the canonical MOs, which are delocalized across the length of the molecule, but arises from a coherent superposition of four different occupied MOs. Upon transformation to the NTO basis, there is only one significant singular value, with l21 ¼ 0:96. The in Fig. 3.5B, thus paints a picture that is 96% complete. ! celec principle NTO pair, chole 1 1 There is an unfortunate tendency in the literature to refer to chole as the “highest-occupied NTO” (HONTO), with celec 1 1 then deemed the “lowest-unoccupied NTO” (LUNTO). This terminology is incorrect insofar as “highest” and “lowest” (as in HOMO and LUMO) refer to orbital energies, which are not well defined for the NTOs because the Fock matrix is not diagonal in that representation. The HONTO/LUNTO terminology should therefore be avoided so as not to conflate visual depictions of NTOs with qualitative arguments that might be based on one-electron energy levels, which are only well defined in the canonical MO basis. The term principle transition orbitals (or perhaps principle NTOs) is suggested instead, to refer to the pair of orbitals corresponding to the largest li. One might therefore describe a sequence of principle NTOs (pNTOs): pNTO, pNTO 1, pNTO 2, … for l21 > l22 > l23 > ⋯ . Another common tool to visualize an excitation is the density difference as compared to the ground state. The unrelaxed density difference DrðrÞ ¼ Drelec ðrÞ+Drhole ðrÞ has particle and hole components that are the real-space analogs of the density matrices DP Using the NTOs, the particle and hole densities may be expressed as

(3.53) elec

and DP

hole

in Eq. (3.41).

(a) canonical molecular orbitals

HOMO–1

20% 26%

HOMO–2

21%

LUMO

HOMO–3

20%

HOMO–4

(b) natural transition orbitals 96% FIG. 3.5 (A) Canonical MO representation (with weights x2ia expressed as percentages) and (B) principle NTO pair (with weight l2i ) for S0 !S2 excitation of a five-unit, fluorenone-terminated polyfluorene molecule in which the leftmost monomer contains a carbonyl defect that localizes the excitation. LR-TDDFT calculations were performed at the CAM-B3LYP/3-21G* level within the TDA and the unrelaxed density is analyzed.

Density-functional theory for electronic excited states Chapter

Drelec ðrÞ ¼

nocc X i¼1

2 l2i jcelec i ðrÞj

nocc X

Drhole ðrÞ ¼

i¼1

l2i jchole ðrÞj2 i

3

85

(3.54a)

(3.54b)

Note that Drelec(r) is positive definite and Drhole(r) is negative definite, consistent with Eq. (3.41). Because the NTOs are defined by a singular value decomposition, which distills the nocc  nvir matrix x + y into the fewest number of nonzero parameters, the densities in Eq. (3.54) are often dominated by the principle NTO pair. Although it is not widely recognized, the quantities Drelec(r) and Drhole(r) are precisely the attachment density and the detachment density, respectively, which have long been used to visualize excited states (269–271). (These were originally introduced in a different way (269), based on eigenvectors of DP that afford positive or negative eigenvalues, respectively.) In the author’s view, NTOs are still the preferable description since phase information is lost upon squaring the orbitals in Eq. (3.54). Fig. 3.6 illustrates these densities for the same S0 !S2 excitation of polyfluorene that was examined in Fig. 3.5. Because l21  1, the particle and hole (or attachment and detachment) densities have the same information content as the principle NTO pair in Fig. 3.5B. Also shown in Fig. 3.6 is the transition density T(r) ≡ T(r, r), where T(r, r0 ) is defined in Eq. (3.20). For an excitation jC0i!jCi, the general definition of this quantity is (272) Z 0 Tðr, r Þ ¼ N C*0 ðr0 , r2 , …, rN ÞCðr, r2 , …, rN Þdr2 …drN (3.55) and for LR-TDDFT in the NTO representation it is Tðr, r0 Þ ¼

X hole 0 * li celec ðr ފ i ðrÞ½ci

(3.56)

i

Thus, the NTOs distill the content of the transition density into the smallest possible number of particle-hole pairs. In that well-defined sense, the NTOs are the best orbitals for visualization purposes, and detection of more than one significant singular value li indicates unresolvable multideterminant character in the excited state. For the excitation depicted in hole Fig. 3.6, there is little such character, and T(r)  celec 1 (r)c1 (r) is well described by the principle NTO pair. The nature 2 of this product accounts for the somewhat more complicated nodal structure as compared to Drelec(r) jcelec 1 (r)j or 2 hole Drhole(r) jc1 (r)j .

particle (attachment) density

hole (detachment) density

difference density

transition density FIG. 3.6 Visualization of the S0 !S2 excitation of the fluorenone-terminated polyfluorene whose orbital depiction is given in Fig. 3.5, represented here in terms of difference densities. These include the particle density (or attachment density) Drelec(r), the hole density (or detachment density) Drhole(r), the unrelaxed difference density Dr(r) ¼ Drelec(r) + Drhole(r), and the transition density T(r). Each isosurface encompasses 95%–97% of the density in question.

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3.2.3 Systemic problems The utility of LR-TDDFT lies in its combination of low cost, which facilitates calculations on systems such as C119H154ClN21O40 (Fig. 3.1B) or conjugated polymers (Figs. 3.5 and 3.6), along with an accuracy of 0.3 eV for localized valence excitations. That level of accuracy requires a treatment of dynamical correlation effects, as seen from the CIS errors in Fig. 3.4 that exceed 0.8 eV, comparable to the 1 eV of correlation energy for a pair of electrons. That is the good news. In this section, we discuss some of the bad news, namely, systematic errors that make certain types of problems extremely challenging for LR-TDDFT. Of these, the most widely discussed is severe underestimation of excitation energies for states with substantial CT character, ultimately manifesting as an explosion of spurious CT states in a sufficiently large system. A second problem concerns the topology of conical intersections that involve the ground state, which presents a problem for ab initio photochemical simulations of internal conversion following photoexcitation.

3.2.3.1 Description of charge transfer Problems with the description of long-range CT excitations manifests in small, gas-phase molecules as Rydberg excitation energies that are systematically too low (248), even when reasonable accuracy is obtained for valence excitations. This was noticed in the early studies of LR-TDDFT and was quickly diagnosed as a symptom of incorrect asymptotic decay of the XC potential in GGA functionals that existed up to that point (249–252). The same problem was quickly recognized to affect CT excitation energies (84, 115). Both CT and Rydberg excitations are sensitive to the long-range behavior of the potential, which should be vsxc(r)  r 1 for a charge-neutral molecule (273–277). This asymptotic behavior ought to be borne by the exchange potential because correlation dies off more quickly (274, 278), but in practice so does semilocal exchange. Consider the form of the LR-TDDFT pseudo-eigenvalue problem for an excitation between MOs cis and cas that are well separated in space, such that cis(r)cas(r)  0 everywhere. A semilocal expression for vsxc(r) affords a semilocal XC kernel, such that the matrix elements (iajksτ xcjjb) in A and B vanish in such a situation, for all j and b. Ignoring spin by setting s ¼ τ in Eq. (3.19), this leaves Aia,jb  ðea

ei Þdij dab

ahfx ðijjabÞ

(3.57)

Only the integral (ijjab), which comes from the HFX term, survives to provide distance dependence for the i ! a excitation. A pictorial illustration is provided in Fig. 3.7, which plots the distance dependence of the lowest CT excitation energy (oCT) in the ðC2 H4 Þ⋯ðC2 F4 Þdimer as a function of intermolecular separation (203). Only Hartree-Fock theory affords the correct distance dependence for oCT(R), which varies according to the Mulliken formula (66, 279, 280), oCT ðRÞ ¼ IEdonor + EAacceptor

3

2

H

H

H

H

e– R

1 R

(3.58)

F

F

Hartree-Fock (αhfx = 1.0) F

F

BH&H-LYP (αhfx = 0.5) 1

0

B3LYP (αhfx = 0.2)

4

5

6

7

8

9

LDA (αhfx = 0.0) 10

R/Å FIG. 3.7 Distance dependence for the lowest intermolecular CT excitation in ðC2 H4 Þ⋯ðC2 F4 Þ computed using functionals with various fractions of ˚ in order to emphasize the distance dependence of oCT(R), which varies asympHFX, as indicated. The curves are shifted to a common origin at R ¼ 4 A totical as ahfx/R. (Adapted with permission from Dreuw, A.; Weisman, J. L.; Head-Gordon, M. Long-Range Charge-Transfer Excited States in TimeDependent Density Functional Theory Require Non-Local Exchange. J. Chem. Phys. 2003, 119, 2943–2946; copyright 2003 American Institute of Physics.)

Density-functional theory for electronic excited states Chapter

3

87

in atomic units. For hybrid functionals, the last term becomes ahfx/R rather than 1/R, leading to a too-small penalty for long-range CT. For semilocal functionals where ahfx ¼ 0, the CT excitation energy has no distance dependence whatsoever once the donor and acceptor moieties are sufficiently far apart such that their orbitals do not overlap. This is reflected in the flat oCT(R) profile for the LDA functional in Fig. 3.7. As a result, long-range CT excitation energies are almost invariably too small in LR-TDDFT unless the functional contains 100% HFX, which it usually does not because fully nonlocal exchange is somewhat unbalanced given the local nature of existing correlation functionals. The M06-HF functional is an example that does use 100% HFX, leading to reasonable performance for Rydberg states but larger errors for valence excitations (Fig. 3.3D). Where small-molecule benchmarks are available, errors in CT excitation energies can exceed 3 eV (228), but the problem gets worse in larger molecules so that value is likely limited only by the size of the benchmark systems for which reliable ab initio results are available. A consequence of this severe underestimation of CT excitation energies is the appearance of completely spurious CT excited states in large systems, especially solvated chromophores (281–288) but also large molecules (159, 289, 290). When the system size is sufficiently large, there are inevitably well separated occupied and virtual MOs such that the orbital energy gap ea ei is small. For ahfx  0, the electron-hole interaction vanishes and the diagonally dominant A matrix consists of weakly coupled blocks corresponding to these spurious CT transs (r, r0 ) lacks the long-range exchange (or a derivative discontinuity (63, 68, 280), or frequency sitions. The kernel fxc dependence (291)) that is needed to provide an energetic penalty for CT, and an upshift to oCT  ea ei as in Eq. (3.57). A physical example is shown in Fig. 3.8 for a model of aqueously solvated uracil (287). Whereas this system ought to have only a 1np* and a 1pp* state below 6 eV (227), a hybrid LR-TDDFT calculation using the PBE0 functional results in numerous low-energy solvent-to-chromophore CT states, including 27 states below 6 eV for the (uracil)(H2O)25 cluster that is shown in Fig. 3.8 and additional states as the size of the water cluster grows (227, 287). Many of these states are accidentally near degenerate with the optically bright 1pp* state and as a result these nominally dark CT states can acquire intensity from the bright state, which diminishes the intensity of the latter because total oscillator strength is conserved by the Thomas-Reiche-Kuhn sum rule, Eq. (3.38). The state o9 in Fig. 3.8 exhibits the largest degree of pp* character (287), yet due to spurious intensity borrowing it does not exhibit the largest oscillator strength and itself contains some contribution from solvent-to-chromophore CT. Fortunately, the same sum rule can be used to argue that the overall spectral envelope may still be valid upon ensemble averaging and broadening, even if some fraction of the oscillator strength has been ported onto spurious CT excitations.

= 5.05 eV f8 = 0.032

8

= 5.05 eV f7 = 0.064

7

= 5.08 eV f9 = 0.019

9

= 5.16 eV f10 = 0.018

10

= 5.20 eV f11 = 0.015

11

FIG. 3.8 Selected detachment (hole) and attachment (particle) densities, for excited states of (uracil)(H2O)25 computed using LR-TDDFT at the PBE0/631+G* level. These states exhibit spurious solvent-to-chromophore CT in the spectral vicinity of the 1pp* state at o  5.1 eV. Excitation energies on and oscillator strengths f0n are shown, illustrating intensity borrowing by the spurious CT states. (Reprinted with permission from Lange, A.; Herbert, J. M. Simple Methods to Reduce Charge-Transfer Contamination in Time-Dependent Density-Functional Calculations of Clusters and Liquids. J. Chem. Theory Comput. 2007, 3, 1680–1690; copyright 2007 American Chemical Society.)

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In large chromophores, such as conjugated polymers, spurious low-energy CT excitations can manifest as artificial delocalization of the excitation across the length of the chromophore (159, 292, 293), whereas the CIS method predicts that exciton size eventually saturates even as conjugation length increases (293). As such, there is a need to develop a metric for whether a particular excited state has too much CT character for its excitation energy to be trusted. The first such CT metric to see widespread use was the quantity L defined by (294)

L¼

X ðxias + yias Þ2 Oias ias

X jbτ

(3.59)

ðxjbτ + yjbτ Þ2

where Oias ¼

Z

jcis ðrÞj
jcas ðrÞjdr

(3.60)

measures the overlap of jcis(r)j and jcas(r)j. (Absolute values are required since the occupied and virtual MOs are orthogonal.) This overlap is then weighted by the LR-TDDFT amplitudes and normalized such that 0  L  1. For calculations that do not invoke the TDA, however, the denominator in Eq. (3.59) is an odd choice, given the normalization condition in Eq. (3.21), and this inconsistency has propagated into other CT metrics used in LR-TDDFT (295, 296). Regarding the metric in Eq. (3.59), an early benchmark study concluded that 0.45  L  0.89 for localized valence excitations, making values in this range “safe” for LR-TDDFT, whereas 0.08  L  0.27 for Rydberg excitations, which are unsafe (294). It was suggested that excitation energies for which L≲0:3 0:4 (depending on the functional) should not be trusted. Various LR-TDDFT errors have been rationalized by appeal to L or similar metrics (135, 244, 297, 298). The point at which CT character becomes a problem is dependent on the manner in which it is quantified (299), and several alternative CT metrics have been suggested (294–296, 300–306). Ciofini and coworkers introduced a widely used “DCT metric” (300), originally defined in a rather complicated way but which ultimately measures the distance between the centroids of Drelec(r) and Drhole(r). The centroid of Drelec(r) is Z hrelec i ¼ rDrelec ðrÞdr (3.61) with an analogous definition for Drhole(r). If one defines d elec=hole ¼k hrelec i  hrhole i k

(3.62) d+elec=hole

then the distance between centroids of the electron and the hole is d elec=hole, whereas is the average position of the center of mass of the exciton. The quantity d elec=hole is equivalent to the DCT metric but is more directly connected to the physics of the excitation. Other similar descriptors can be envisaged (307, 308). For example, by defining the root-meansquare size of the electron and the hole, selec ¼ ðhrelec
relec i

hrelec i
hrelec iÞ1=2

(3.63a)

shole ¼ ðhrhole
rhole i

hrhole i
hrhole iÞ1=2

(3.63b)

1 ðs + shole Þ 2 elec

(3.64)

one may define a charge-displacement distance, dCD ¼ delec=hole

The quantity dCD connects directly to the properties of the exciton and is a more physically motivated version of the “electron displacement” metric introduced by Adamo and coworkers (296), and one that avoids the incorrect normalization in Eq. (3.59) and is thus rigorously invariant to orbital rotations even when the TDA is not invoked. To the best of our knowledge, dCD is introduced here for the first time but we suggest that delec=hole and dCD should replace alternative CT metrics that serve essentially the same purpose. To combat the long-range CT problem without going beyond the adiabatic approximation, LRC functionals are used to provide an XC potential with correct asymptotic behavior for an electron-hole pair. The LRC modification was introduced in Eq. (3.48) and contains an additional parameter that controls the separation between semilocal GGA exchange at short range and nonlocal HFX at long range. Fig. 3.9 shows an example of how these functionals can be used to mitigate the

Density-functional theory for electronic excited states Chapter

(a)

O

3

89

p CT–

S

O

(b)

–

CIS

LRCωPBEh LRCZPBE

B3LYP PBE

BLYP

(c)

PBE

BLYP

B3LYP LRC- PBE

CIS

FIG. 3.9 (A) The chromophore trans-thiophenyl-p-coumarate (pCT ), along with (B) a plot of the lowest TDDFT/6-31G excitation energy for pCT (aq) as a function of the number of water molecules included in the calculation, and (C) the number of TDDFT states below 3 eV in this calculation. (Adapted with permission from Isborn, C. M.; Mar, B. D.; Curchod, B. F. E.; Tavernelli, I.; Martı´nez, T. J. The Charge Transfer Problem in Density Functional Theory Calculations of Aqueously Solvated Molecules. J. Phys. Chem. B 2013, 117, 12189–12201; copyright 2013 American Chemical Society.)

growth in spurious CT states around a chromophore in aqueous solution (288). Whereas the number of CT states increases extremely rapidly as water molecules are added around the system, and hybrid functionals such as B3LYP only partially forestall this increase, the functionals LRC-oPBE (228) and LRC-oPBEh (260) control this growth completely. The LRCoPBEh functional is a short-range hybrid with ahfx ¼ 0.2, whereas LRC-oPBE is semilocal at short range (ahfx ¼ 0), but both functionals employ 100% HFX in the long-range limit. This should be contrasted with functionals such as CAMB3LYP (225), which use range separation but sacrifice proper asymptotic behavior in an effort to obtain more accurate excitation energies for localized valence transitions. Although RSH functionals such as CAM-B3LYP and oB97X-D are good choices in many respects for valence excitations, neither improves the accuracy of LR-TDDFT for CT excitations (309). Standard double-hybrid functionals contain only a fraction of HFX and thus do not improve the situation for CT states either (310), unless the LRC scheme employed (311). In the early development of LRC functionals, the range-separation parameter was often fit to minimize error in some benchmark thermochemical or excitation energy data (200, 202, 226, 228, 259). However, excitation energies were found to be quite sensitive to this parameter (226, 227, 260), especially for states with CT character (227, 260). More recently, the community has increasingly turned to a more theoretically well-grounded “optimal tuning” strategy (231, 312–315), based on the ionization energy (IE) theorem of exact DFT (316, 317). That theorem simply states that IE ¼ eHOMO for the exact Kohn-Sham functional, consistent with the fact that the IE is set by the asymptotic decay of the wave function (261). This condition is violated badly by common GGA and even hybrid functionals, often by several electron volts (318, 319). The optimal tuning (or “IE tuning”) procedure consists in enforcing this condition for an approximate XC functional, by adjusting the range-separation parameter m such that eHOMO ðN, mÞ ¼ EðN 1, mÞ EðN, mÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} IEðN , mÞ

(3.65)

Here, IE(N, m) represents the DSCF value of the IE for the N-electron molecule, computed using an LRC functional with range-separation parameter m. Alternatively, one might try to find the value of m that comes closest to satisfying Eq. (3.65) for both the N-electron molecule and its (N + 1)-electron anion, representing donor and acceptor for electron transfer. That procedure has been shown to reproduce not only CT excitation energies but also to afford Kohn-Sham gaps (eLUMO eHOMO) in good agreement with fundamental gaps (IE EA) (197). The optimally tuned value of m does exhibit a strong dependence on system size, however (320–323). Strategies to mitigate this dependence have been suggested (323–325).

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3.2.3.2 Conical intersections A different systemic problem with LR-TDDFT, which is relevant in the context of computational photochemistry, is that it predicts the wrong topology around any conical intersection that involves the ground state (146, 326). The TDHF method suffers from the same deficiency, which is not a DFT artifact per se but rather a linear response artifact, arising from an unbalanced description of the ground (reference) state and the excited (response) states (27). The result is that the branching space around a conical seam that involves the two lowest electronic states is necessarily one-dimensional rather than twodimensional. (For examples, see Ref. (146) or (326).) Even the CIS method can exhibit erratic behavior when the ground state becomes quasidegenerate with the first excited state (26, 147), because in the absence of double excitations the ground- and excited-state eigenvalue problems are decoupled (according to Brillouin’s theorem) (59), leading to an unbalanced description (27, 326). This is not a problem for conical intersections between two excited states because those states are coupled by the matrix A, in both CIS and LR-TDDFT. An example of a conical intersection involving the ground state is Jahn-Teller symmetry lowering from D3h to C2v, which is illustrated for the H3 radical in Fig. 3.10 (147). In the vicinity of the D3h conical intersection, the upper-state potential surface exhibits erratic behavior at both CIS and LR-TDDFT levels of theory. This warping of the potential surface around a conical intersection has consequences in nonadiabatic molecular dynamics simulations, including SCF convergence difficulties (327) and incorrect internal conversion timescales (26). As a result, nonadiabatic trajectory surface-hopping calculations based on LR-TDDFT should probably be terminated prior to internal conversion to the ground state (26).

(a) RO-CIS

(b) LR-TDDFT / TDA (B3LYP)

b (Å)

b (Å)

1.10 1.15

1.05

–1.44

energy (Ha)

energy (Ha)

1.05

–1.46 –1.48 –1.50 55

60

(deg)

1.10 1.15

–1.54

–1.56

–1.58 55

60

65

65

(deg)

b (c) SF-CIS

(d) SF-TDDFT (BH&HLYP) b (Å)

b (Å) 1.35 1.40

energy (Ha)

energy (Ha)

1.30

1.05

–1.50 –1.51 –1.52 55

–1.46 –1.47 –1.48 –1.49 55

60

(deg)

65

1.10 1.15

60

(deg)

65

FIG. 3.10 Potential energy surfaces for the lowest two doublet states of H3 radical along a bond-length coordinate b and a bond-angle coordinate y, illustrating Jahn-Teller symmetry lowering D3h ! C2v. The methods are (A) CIS based on a restricted open-shell (RO) reference state, (B) LR-TDDFT/TDA using unrestricted B3LYP, (C) SF-CIS, and (D) SF-TDDFT using BH&HLYP. (Reprinted with permission from Zhang, X.; Herbert, J. M. Analytic Derivative Couplings for Spin-Flip Configuration Interaction Singles and Spin-Flip Time-Dependent Density Functional Theory. J. Chem. Phys. 2014, 141, 064104; copyright 2014 American Institute of Physics.)

Density-functional theory for electronic excited states Chapter

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The “spin-flip” (SF) variant of LR-TDDFT (328) has been suggested as a way to overcome this problem, as discussed in detail in Ref. (27). Briefly, SF-TDDFT uses a sacrificial reference state that is not the ground state of interest, but rather a state with higher spin multiplicity S + 1, for target states with total spin S. By combining single excitations with a single a ! b spin flip, SF-TDDFT generates both ground and excited states of multiplicity 2S + 1 as spin-flipping excitations, meaning that both are obtained as solutions to a common eigenvalue problem. This eliminates the imbalance and restores correct topology to conical intersections involving the ground state, as seen for H3 in Fig. 3.10C and D. Functionals with 50% HFX perform well in the context of SF-TDDFT (27, 329), and the Becke “half-and-half” functional BH&HLYP (with ahfx ¼ 0.5) has become the de facto standard for SF-TDDFT (27). An unfortunate side effect of SF-TDDFT is that it tends to exacerbate spin contamination (27, 330), especially as one moves away from the Franck-Condon point on the potential surface and starts to enter regions of photochemical interest. This necessitates the use of state-tracking algorithms to maintain a consistent spin multiplicity (330, 331). There have been various attempts to find a more theoretically appealing solution to this conundrum by adding additional determinants to the 2 excitation space in order to restore S^ symmetry (27). Methods developed along these lines include a fully spin complete 2 version of SF-TDDFT (330), which adds the minimal number of additional determinants needed to obtain S^ eigenstates (based on an equation-of-motion formalism) (77), and also a “mixed-reference” spin-flip (MRSF) approach, which uses a combination of high-spin and low-spin S + 1 reference states to generate target states with spin S (332–337). Although the MRSF-TDDFT excitation manifold is not formally spin-complete, in practice the spin contamination is very small (332). The analytic gradient (335) and nonadiabatic derivative couplings (336) for MRSF-TDDFT have recently been formulated, facilitating nonadiabatic molecular dynamics simulations.

3.3 Excited-state Kohn-Sham theory: The DSCF approach For periodic DFT calculations, LR-TDDFT is theoretically ill-posed if semilocal functionals are used within the adiabatic approximation (20–22). Specifically, the too-rapid asymptotic decay of vsxc(r) causes the lowest LR-TDDFT excitation energy to collapse to the Kohn-Sham gap, ħo ¼ eLUMO eHOMO (20, 21). Semilocal LR-TDDFT also does not produce bound excitons in periodic systems (22), and in large (but finite) conjugated polymers, the exciton delocalization length typically extends to the length of the entire molecule (293, 338). This observation can be conceptualized as incomplete cancellation of self-interaction that grows worse with system size, and infinitely worse under periodic boundary conditions (20). Equivalently, it is a manifestation of the systematic underestimation of CT energies that were discussed in Section 3.2.3. In recognition of these and other systemic problems exhibited by LR-TDDFT, there has been growing interest in “DSCF” approaches that attempt to determine excited-state solutions to the Kohn-Sham SCF equation (339, 340). Having found such a solution, the excitation energy is computed simply as the difference relative to the ground-state energy, hence “DSCF.” In contrast to the well-automated machinery of LR-TDDFT, these methods are less “black box,” involving more effort and finesse on the part of the user, because each excited state requires a separate calculation. On the other hand, the DSCF approach can exploit ground-state gradient technology for geometry optimizations and vibrational frequency calculations (341). For this reason, the DSCF procedure has sometimes been called excited-state Kohn-Sham theory (339, 341). In cases where LR-TDDFT exhibits known deficiencies, the DSCF approach may be more accurate and more reliable even if the formal justification (based on the Hohenberg-Kohn theorems (40, 58)) is absent because the system is not in its ground state. The method therefore rests upon the assumption that the description of short-range dynamical correlation depends upon the local environment of an electron and can be ported to a “non-aufbau” solution of the SCF equations, in which an electron has been promoted into a virtual MO. Such a state does not formally satisfy the noninteracting v-representability requirement of ground-state DFT (17, 58, 68, 342). Excited-state SCF solutions do contain full orbital relaxation, yet these solutions are inherently unstable because they are saddle points rather than local minima in the space of orbital rotations. Attempts to locate these non-aufbau solutions, each characterized by a virtual (empty) level that is lower in energy than the HOMO level, may suffer “variational collapse” to the ground state or to a lower-lying SCF solution. It is up to the user to determine that the SCF solution corresponds to the state of interest; if not, then the search must begin anew, using a different SCF convergence algorithm or a different initial guess. Several modified SCF algorithms have been developed to try to locate non-aufbau solutions, based on overlaps with a set of user-specified MOs (343–346) or else based on direct search (347–349). These algorithms are described in Section 3.3.1. Examples of the DSCF methodology in action are presented in Section 3.3.2.

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3.3.1 Theory 3.3.1.1 General considerations A flowchart for the SCF procedure is illustrated in Fig. 3.11. At each iteration, the occupied MOs {cis} are used to construct the Fock matrix Fs, which is then diagonalized to obtain new MOs. Notably, diagonalization results in nbasis ¼ nocc + nvir MOs and one must decide how to choose the occupied set. Ordinarily, the lowest eigenvalues ers are selected (aufbau principle), resulting in the ground-state determinant upon SCF convergence. To locate an excited-state SCF solution instead, one seeds the procedure with initial-guess MOs from a ground-state calculation but with non-aufbau occupancies, promoting an electron from HOMO to LUMO, for example. This makes the LUMO into an occupied level and the HOMO into a virtual level, resulting in a “hole,” or in other words a virtual level whose energy lies below that of the highestoccupied level. When the Fock matrix is constructed from this new set of occupied MOs and then diagonalized, the question becomes which of the new MOs should be the occupied ones, since energy levels may have shifted. The SCF procedure therefore deviates from the usual one only when it comes to selecting the occupied subset from among the nbasis MOs. Several different options have been explored, as discussed below. Before reviewing algorithms for locating non-aufbau SCF solutions, however, it is important to note some properties of those solutions that are different from ground-state properties. First, because the effective Hamiltonian F^s ½fcis gŠ depends on the MOs themselves, the ground- and excited-state Slater determinants are eigenfunctions of different Hamiltonians and are therefore not orthogonal. One consequence is that the formula for oscillator strengths in terms of transition dipole matrix elements (Eq. 3.3) is not strictly valid (350), as that formula is derived using the assumption that the eigenfunctions of the Hamiltonian form a complete orthonormal set (44). In small-molecule tests, however, overlap integrals between groundand excited-state determinants are found to be ≲0.1 (343). Another general concern is that excited states are always open-shell species, even if the ground state is closed-shell, so any single-determinant approximation is certain to be spin contaminated, perhaps badly so. Indeed, single-determinant 2 approximations for open-shell singlet states are often characterized by hS^ i  1 (in atomic units of ħ2 ), which is equal to the average of pure-state singlet and triplet values. A similar phenomenon occurs, for similar reasons, in the case of the spin-unrestricted Hartree-Fock wave function in the separated-atom limit (59), because a spin-pure state with two half-filled orbitals can be described using a minimum of two Slater determinants. The same two determinants (with different relative signs) are needed to describe both singlet spin-coupling (total S ¼ 0) as well as the MS ¼ 0 component of triplet spin-coupling (S ¼ 1). In practice, DSCF excitation energies for open-shell singlet excited states are often surprisingly accurate despite significant spin contamination (343, 345, 351), although there are exceptions. One such exception is the 1B1u state of ethylene, whose underestimation by almost 2 eV is attributed to severe spin contamination (343). Yamaguchi and coworkers have developed spin-projection techniques that can be used to recover spin-pure states in such cases (352–355), and approximate spin purification is often used as a practical workaround in “broken-symmetry” DFT calculations of transition metal complexes (354–356). For an open-shell singlet, the most common approach is to approximate the singlet energy as Esinglet  2Emix

guess MOs

Etriplet

construct F from the MOs

(3.66)

SCF procedure

diagonalize F to get new MOs

select occupancies for new MOs

no

converged? ([F,P] = 0?)

yes

done

FIG. 3.11 Flowchart illustration of the SCF algorithm. In the usual approach, occupancy selection is done according to the aufbau criterion, with the lowest-energy MOs chosen as the occupied set. For DSCF calculations, a different choice is required.

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Here, Emix is the energy of the contaminated (mixed-spin or broken-symmetry) state that is obtained in searching for a singlet solution, whereas Etriplet is the triplet energy for the same system, for which spin contamination is typically less severe. This procedure has a long history (356, 357), and Eq. (3.66) can be viewed as an approximate form of spin projection, generalizable to cases where the target state has spin S > 0 (353). The formula in Eq. (3.66) is sometimes implemented in a self-consistent way, that is, using Eq. (3.66) as the ansatz and minimizing with respect to orbital rotations. That method is known as restricted open-shell Kohn-Sham (ROKS) theory (358–360), and it affords a common set of orbitals for both multiplicites. More often, however, Eq. (3.66) is used as an a posteriori correction scheme (341, 361–367). Even then, Eq. (3.66) can easily be used in geometry optimizations (at the cost of two energy and gradient evaluations per step) and in vibrational frequency calculations (341). For the aforementioned 1B1u state of C2H4, application of Eq. (3.66) reduces the DSCF error (as compared to experiment) from 1.8 to 0.3 eV (343).

3.3.1.2 Orbital-optimized non-aufbau SCF solutions The simplest means to construct a non-aufbau occupied set is known as the maximum overlap method (MOM) (343–346). Starting from an initial guess corresponding to non-aufbau occupation of the ground-state MOs, this approach uses an overlap criterion to identify the new occupied MOs at each subsequent SCF iteration. To do this, one must compute ðnÞ at the nth iteration onto a reference set of MOs. The reference set might be the the projections prs of the MOs crs MOs at the previous iteration, in which case !1=2 occ X ðn 1Þ ðnÞ 2 prs ¼ hcis jcrs i (3.67) i

ð0Þ

or else it could be the initial set of ground-state MOs, fcis g: prs ¼

occ X i

ð0Þ ðnÞ 2 hcis jcrs i

!1=2

(3.68)

The first choice (Eq. 3.67) represents the original version of the algorithm (343), whereas Eq. (3.68) has been called the “initial MOM” (IMOM) algorithm and tends to have better success at converging orbital-relaxed non-aufbau states (345). The signature of success is a “hole below the Fermi level,” that is, a virtual MO whose energy is lower than the HOMO energy. The MOM and IMOM algorithms consist simply in replacing the aufbau selection of occupied MOs with a selection based on the nocc largest values of the overlaps prs. All other aspects of the SCF algorithm remain the same. This approach exhibits the same cost per SCF iteration as ground-state DFT and when it succeeds, the rate of convergence (measured by the number of SCF cycles) is typically on par with a conventional ground-state calculation. There are certainly cases where MOM and IMOM fail (348, 349), however, typically resulting in variational collapse to the ground-state SCF solution. In such cases, more robust SCF convergence algorithms are required. One such approach is the “s-SCF” method (347), which is based on minimizing the functional s2o ½CŠ ¼ hCjðo

^ 2 jCi FÞ

(3.69)

^ for a specified energy, o. This idea stems from recognizing that eigenstates FjCi ¼ ojCi satisfy the zero-variance con2 2 ^ ^ dition hF i ¼ hFi . The s-SCF approach avoids variational collapse by solving a proper minimization problem, but the 2

appearance of F^ means that four-particle operators are required and the requisite transformations endow this method with Oðn5basis Þ scaling (347). This makes the s-SCF approach much more expensive than conventional SCF theory. An alternative approach with the same formal scaling as the ground-state SCF problem is squared-gradient minimization (SGM) (348). Here, the idea is to convert an inherently unstable saddle-point optimization into a search for a local minimum by optimizing an objective function equal to the squared gradient of the energy with respect to orbital rotations. A local minimum can always be converged (if slowly), whereas a saddle point can be missed, and this makes SGM more robust as compared to MOM or IMOM. While the cost remains Oðn3basis Þ, it is 2–3 times more expensive per SCF iteration as compared to a conventional SCF calculation, due to the cost of constructing the objective function (348). It is also known 2 ^ that the squared gradient k rVðxÞk of a function V(x) may contain minima that do not correspond to stationary points of the original function (368–371). From the standpoint of trying to locate an orbital-relaxed excited-state Slater determinant, these are spurious solutions.

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A middle way between MOM and SGM is state-targeted energy projection (STEP) (349), which does not increase the cost per SCF iteration yet shows much more robust convergence behavior as compared to MOM or IMOM. The STEP approach constructs a projection operator onto the virtual space, ^ ¼ Q s

vir X a

jcas ihcas j

(3.70)

^ is Q ¼ C C{ , where Cs where the summation runs over some or all of the virtual MOs. The matrix representation of Q s s s s consists of column vectors corresponding to whichever MOs are included in Eq. (3.70). The Fock matrix is then modified according to F0s ¼ Fs + SQs S

(3.71)

where S is the atomic orbital (AO) overlap matrix. The effect of the additional term is to shift all of the orbitals that are included in Eq. (3.70) by an energy . By preselecting a virtual MO from the ground-state calculation that will be occupied in the first iteration of STEP, one can modify the Fock matrix to shift other virtual orbitals (including a lower-energy one that was occupied in the ground state but whose electron was promoted) to energies above the non-aufbau orbital. For example, upon HOMO ! LUMO promotion, the original HOMO is unoccupied and should be included in Eq. (3.70), ^ . The STEP algorithm is a form of level shifting whereas the LUMO becomes occupied and should be excluded from Q s that tends to ensure that the SCF algorithm converges to the “closest” stationary point in the space of MO coefficients, which therefore resemble the initial guess (349). Like MOM and IMOM, STEP can be used in conjunction with ground-state gradient technology to perform geometry optimizations and vibrational frequency calculations.

3.3.1.3 Transition potential methods The DSCF methods described so far each involve state-specific orbital optimization, meaning that the SCF procedure must be iterated to convergence separately for each excited state of interest. This has the advantage of including full orbital relaxation effects (beyond linear response), but the disadvantage that there is no guarantee that an excited state resembling the one of interest can actually be found. A simpler (if cruder) approach was devised long ago by Slater (372, 373), and forms the basis of several popular techniques for estimating X-ray excitation energies from Kohn-Sham eigenvalues (374–380). To understand Slater’s method, imagine that E({ni}) is the energy of a single-determinant wave function with orbital occupation numbers {ni}, some of which might be fractional. Expanding the energy as a Taylor series around a reference energy E0 ¼ Eðfn0i gÞ, keeping the orbitals fixed, one obtains E ¼ E0 +

X i

ðni

n0i Þ

∂E 1X + ðn ∂ni 2 i, j i

n0i Þðnj

n0j Þ

∂2 E +⋯ ∂ni ∂nj

(3.72)

According to the Slater-Janak theorem (381), the first derivative is an orbital eigenvalue: ei ¼ ∂E/∂ni. Now consider promotion of one electron from an occupied MO to a virtual MO. It suffices to deal with just a pair of occupancies (ni, na), in terms of which the transition in question can be abbreviated as (1, 0) ! (0, 1). If a fractional-occupancy state with ni ¼ 1/2 ¼ na is used for the reference state fn0i g, then using Eq. (3.72) to compute the excitation energy DE ¼ E(0, 1) E(1, 0) leads to an estimate     1 1 1 1 ei , (3.73) DESTM  ea , 2 2 2 2 3

where the approximation neglects terms of order ðnj n0j Þ (375). This forms the basis of the Slater transition method (STM), wherein an SCF calculation is carried out for the fractional-occupancy state (ni ¼ 1/2, na ¼ 1/2) and then the energy difference DE ¼ ea ei affords an estimate of the excitation energy. Variants of STM have historically been popular for X-ray spectroscopy (374–380), particularly in the context of periodic DFT calculations for which LR-TDDFT with semilocal functionals is problematic (21). In principle, this method requires a separate SCF calculation for each excitation of interest, and while it is generally easy to converge the X-ray “edge” in this way (i.e., a core ! LUMO transition), higher-lying states will require a convergence algorithm that can avoid variational collapse. Moreover, this state-by-state approach leads to nonorthogonal MOs and therefore exhibits the same ambiguities regarding oscillator strengths as the DSCF method (350). For these reasons, it is common to omit the 1/2 electron in the virtual space (with only pragmatic justification), leaving ni ¼ 1/2 in the core-excited MO. This variant

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of the procedure has been called the transition potential (TP) method (378, 380,382–385). By neglecting to occupy any core-excited virtual states at all, this approach sidesteps the issue of nonorthogonality, at least for a given choice of ni. ^ ai ij2 constructed from Oscillator strengths can be computed in a straightforward way from matrix elements jhC0 jmjC the orbitals obtained from the fractional-occupancy SCF calculation. Modifications to the formula in Eq. (3.73) have also been proposed (378, 386–388), sometimes involving more than one SCF calculation with differing fractional occupancies, or by combining eigenvalues of both the neutral molecule and its cation or anion (389, 390). These modifications represent attempts to eliminate higher-order errors in Eq. (3.72). An example is the “generalized STM” (gSTM) method (378, 380,386), which replaces Eq. (3.73) with      1 3 1 2 1 2 DEgSTM ¼ ½ea ð1, 0Þ ei ð1, 0ފ+ ea , ei , (3.74) 4 4 3 3 3 3 This is based on an alternative approximation for the integral Z 0 dEðni ¼ x, na ¼ 1 xÞ dx DE ¼ dx 1 Z 0 ¼ ½ei ðni ¼ x, na ¼ 1 xÞ ea ðni ¼ x, na ¼ 1

(3.75) xފdx

1

The original STM in Eq. (3.73) corresponds to a midpoint approximation for this integral (378, 379,386). The gSTM approach requires two separate SCF calculations, one with (ni ¼ 1, na ¼ 0) and the other with (ni ¼ 1/3, na ¼ 2/3). Variants that set ni ¼ 0 (removing the entirety of the core electron) have also been suggested and are sometimes called full core hole (FCH) methods (374–378). The TP approach is then a half core hole (HCH) method. Although the FCH approach deviates significantly from Slater’s original idea, it can be conceptualized as an attempt to restore charge balance, once the 1/2 electron in the virtual space has been abandoned for reasons of convenience. The excited core hole (XCH) approach (391) is yet another variant that creates a charge-neutral state (which is important for periodic DFT calculations) by placing the excited electron in the LUMO and using the full virtual spectrum from that calculation (378, 380,391): DEXCH ¼ ea ðni ¼ 0, nLUMO ¼ 1Þ

ei ðni ¼ 0, nLUMO ¼ 1Þ

(3.76)

Together, these STM- and TP-type procedures are known as occupancy-constrained DSCF methods. In that context, there has been some discussion of “many-electron” effects on oscillator strengths for X-ray transitions (392, 393). What “manyelectron” means in this context is multideterminant character in the final state, which is of course included automatically in a LR-TDDFT calculation.

3.3.2 Examples The primary purpose of this chapter is to survey methods rather than applications but we will highlight a few recent applications of the DSCF approach in order demonstrate that it can be an elegant and low-cost alternative in cases where LR-TDDFT performs poorly, such as for CT states (339, 394). Whereas LR-TDDFT systematically (and sometimes dramatically) underestimates CT excitation energies, the same excitation energies are systematically overestimated by the uncorrelated CIS method (395). At the CIS level, a long-range excitation uses up the one occupied ! virtual excitation that is included in the ansatz and leaves no excitations to facilitate orbital relaxation around either the electron or the hole, hence the overestimation. LR-TDDFT and CIS may therefore bracket the correct answer for a CT state but these upper and lower bounds can be several electron volts apart (260)! The DSCF approach includes full orbital relaxation and is also less sensitive to the asymptotic behavior of the XC potential. There has also been some preliminary work on the description of conical intersections and nonadiabatic dynamics using DSCF methods (340, 396). States with double-excitation character represent another categorical failure of LR-TDDFT within the adiabatic approximation (99), with the most famous example being the optically dark S1(21Ag ) state in carotenoids (397–399), or the analogous 21Ag state in butadiene and other conjugated polyenes (400–404). Doubly excited states can be captured accurately using DSCF methods (339, 348, 349), as shown for a few examples in Fig. 3.12. For these challenging cases, taken from a benchmark data set of double excitations (405), several mGGA and hybrid functionals prove to be significantly more accurate than the CC3 method, which includes triple excitations and is generally close to CCSD(T) in quality (406), with similar scaling (407). For the full data set from Ref. (405), the hybrid GGA functional oB97X-V achieves a mean absolute

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FIG. 3.12 Errors in doubly excited states at the DSCF/aug-cc-pVTZ level vs. benchmarks from Ref. (405). CC3 values are also provided, for comparison. (Reprinted with permission from Hait, D.; Head-Gordon, M. Orbital Optimized Density Functional Theory for Electronic Excited States. J. Phys. Chem. Lett. 2021, 12, 4517–4529; copyright 2021 American Chemical Society.)

error (MAE) of 0.6 eV and a maximum error of 1.1 eV, whereas for CC3 the MAE is 1.0 eV and the maximum error is 1.8 eV (349). The mGGA functional B97M-V does even better, with an MAE of 0.15 eV and a maximum error of 0.46 eV (349). The DSCF methodology can also be used to compute an electronic absorption spectrum, although this must be done one state at a time by converging a sequence of non-aufbau determinants representing individual excited states, and there is no guarantee that some states are not accidentally omitted. A successful example is shown in Fig. 3.13, reproducing the absorption spectrum of the chlorophyll a molecule that was only recently measured in the gas phase (408, 409). Using a STEP-based DSCF procedure, the major peaks in that spectrum can be identified with transitions among the frontier MOs (349), confirming the basic picture of Gouterman’s four-orbital model (410). LR-TDDFT calculations of the same molecule require twice as many states in order to resolve the spectrum up to 300 nm. Many of these states have near-zero oscillator strengths (408), suggesting possible contamination by spurious CT states. Core-valence excitation energies are fertile ground for DSCF techniques. These states appear at photon energies ħo > 200 eV and therefore it is not feasible to reach them by iterative solution of an eigenvalue problem starting from the lowest excitation energies. The frozen-valence approximation is one way to reach these states in LR-TDDFT, which is very accurate for K-edge transitions (177) but may be questionable for L- or M-edge excitations. Fortunately, core-toLUMO excitations are relatively easy to locate using MOM (344). Table 3.1 shows some error statistics for a benchmark set

Experiment ΔSCF (B97M-V)

intensiity (arb. units)

(a)

(b) LUMO+2 LUMO

HOMO HOMO–1 300

350

400

450

500

550

500

650

700

wavelength (nm) FIG. 3.13 (A) Absorption spectrum of chlorophyll a computed via STEP-based DSCF calculations at the B97M-V/def2-TZVP level, spin-purified according to Eq. (3.66) and superimposed on a gas-phase experimental spectrum from Ref. (408). (B) Pictorial representation of Gouterman’s four-orbital model. (Reprinted with permission from Carter-Fenk, K.; Herbert, J.M. State-Targeted Energy Projection: A Simple and Robust Approach to Orbital Relaxation of Non-Aufbau Self-Consistent Field Solutions. J. Chem. Theory Comput. 2020, 16, 5067–5082; copyright 2020 American Chemical Society.)

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TABLE 3.1 Error statistics (in eV) for ROKS calculations of core-level excitation energies, including relativistic corrections. K edgea Functional

Mean

L2,3 edgesb RMSE

LDA

4.3

4.4

PBE

0.9

0.9

B97M-V

1.8

1.8

SCAN

0.1

0.2

PBE0

0.6

0.6

oB97X-V

0.3

0.4

Mean

RMSE

0.1

0.2

0.2

0.4

Data from Ref. (411), using the SGM algorithm and aug-cc-pCVTZ basis set. a Data set includes 40 transitions for C, N, O, and F atoms. Error is defined with respect to experiment, with atom-specific scalar relativistic effects included in the calculation. b Including spin-orbit effects.

of K- and L-edge transitions (411). Except for the LDA functional, all of the errors are 0 for all data to ensure they are on the correct side of the boundary. The SVM algorithm optimizes a hyperplane (Fig. 6.2) that separates the two classes with a maximum margin, which is the closest distance between the data point and the hyperplane.

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FIG. 6.2 Illustration of a hyperplane separating binary data. The solution on the left has a smaller margin than the one on the right. A smaller margin may indicate that the SVM model is overfitted toward the training data. The solution on the right prevents overfitting and is more robust to classify new data.

The closest margin r is given by r¼

minðjyf ðw, x, bÞjÞ kw k

(6.6)

We can rescale w with an appropriate factor k such that the minimum value of yf(w, x, b) equals 1 simplifying the margin to 1 kwk. The rescaling does not change the margin distance as the factor k cancels out when f(w, x, b) is divided by kwk. Thus, maximizing the margin is equivalent to minimizing an objective function kwk or 12 kwk2 with respect to the constraint yf(w, x, b) 1. It is advisable to optimize 12 kwk2 because it corresponds to a standard quadratic programming problem. The quadratic objective is subject to linear constraints, known as the primal problem in convex optimization. We can solve the primal problem by using the Lagrange multiplier to derive a dual problem. Let the dual variable a (i.e., the Lagrange multiplier) be an n-dimensional vector and an ≥ 0. The generalized Lagrangian L(w, b, a) is given as N X

1 Lðw, b, aÞ ¼ kwk2 2 1 Lðw, b, aÞ ¼ kwk2 2

N X

n

   an yn wT xn + b N X

an yn wT xn

an y n b +

n

n

1



(6.7)

N X

an

(6.8)

n

Eq. (6.8) has a minimum when the partial derivatives with respect to w and b are zero, which gives N X

∂Lðw, b, aÞ ¼w ∂w

an y n xn

(6.9)

n

N X ∂Lðw, b, aÞ ¼2 an y n ∂b n

(6.10)

and therefore w¼

N X

an y n xn

(6.11)

n

0¼

N X

an yn

(6.12)

n

Inserting Eqs. (6.11) and (6.12) into (6.8) leads to the dual form of the objective Lðw, b, aÞ ¼

N N X 1X ai aj yi yj xTi xj 2 i j ¼

N N X X i

ai aj yi yj xTi xj

j

N N X N X 1X ai aj yi yj xTi xj + an # 2 i n j

0+

N X

an

(6.13)

n

(6.14)

Machine learning methods in photochemistry and photophysics Chapter

Eq. (6.14), subject to the constraint

N P n

6

167

an yn (Eq. 6.12) and an 1, can be maximized with a quadratic problem solver, such

as the sequential minimal optimization (SMO) algorithm (23). The solution of an is used to compute w (Eq. 6.11). According to Eq. (6.6), the hyperplane is determined by the closest margin distance. Thus, the constraints yn(wxn + b) > 1 do not affect the hyperplane, which lead to an ¼ 0. The points lying on the decision boundary are called support vectors denoted by S, satisfying yn(wxn + b) ¼ 1. The prediction function of the SVM model (Eq. 6.3) now becomes f ðw, x, bÞ ¼ wT x + b ¼

N X

an yn xTn x + b

(6.15)

n

where the constant b can be solved by averaging over all support vectors S N  1 X y b¼ jS j n n

T

w xn



N 1 X ¼ yn N n

N X

am ym xTm xn

m

!

(6.16)

Unfortunately, experiments and computations in photochemistry do not produce data that can be simply classified with a linear decision boundary. It is generally unable to find an acceptable solution that satisfies the hard constraints, yn f(w, x, b) 1for all data. To circumvent this problem, we allow the SVM models to misclassify a number of data and keep the margin as wide as possible so that the majority of the data points can still be classified correctly. For each data point, we use a slack variable, xn 0, to determine the margin distance to the corresponding class. It gives a soft margin constraint ynf(w, x, b) 1 xn. The value of xn will be large if the data are classified to the wrong side of the margin, otherwise zero. The number of misclassifications introduces a hyper-parameter C 0 and the new objective becomes min

N X 1 xn kwk2 + C 2 n

  s:t:xn  0, yn wT xn + b  1

xn

(6.17)

When C ¼ 0, the training only focuses on maximizing the margin ignoring all misclassifications, whereas when C ! ∞ it avoids all misclassification, thus recovering the small and unregularized hard-margin classifier. The corresponding Lagrangian and the dual form for the soft margin classifier become N X 1 Lðw, b, a, j, mÞ ¼ kwk2 + C xn 2 n

LðaÞ ¼

N X

N X n

   an yn wT xn + b

N N X 1X ai aj yi yj xTi xj 2 i j

an

n

0  an  C N X

an y n ¼ 0

1 + xn



N X

mn xn

(6.18)

n

(6.19) (6.20) (6.21)

n

Eq. (6.18) is identical to Eq. (6.14) in the hard margin case. However, the constraints are different (Eq. 6.20). When an ¼ 0, the point is inside the margin, thus ignored. When 0 < an < C, the point lies on the margin, so xn ¼ 0. When an ¼ C, the point is classified inside the correct marge if xn  1, or misclassified if xn > 1. Finally, the constant b is computed by the average of the biases in all support vectors M having 0 < an < C. ! N  N N X  1 X 1 X T T y w xn ¼ yn a m y m xm xn (6.22) b¼ dMe n n N n m

Alternatively, the “Kernel Trick” can circumvent the linear inseparable problem. The dual problem (Eq. 6.19) depends on the inner product of vectors xm and xn. We can substitute the inner product with a positive definite kernel function K(xm, xn). Choosing an appropriate kernel function will enable SVM to optimize a nonlinear classifier and generalize SVM to a kernel method. The details kernel methods are described in Section 6.2.4. The SVM models have been used in bioinformatics (24) (e.g., classification of genes, proteins, and DNA (25)) and chemoinformatic research (26), such as drug discovery. It shows promising performance on analyzing the fluorescence

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spectra data (21), predicting the maximum absorption (27), and fluorescence quantum yields (3). In addition, the SVM models provide an effective tool to assess the outcomes (e.g., geometry optimization) of the high-throughput computations and identify the unsuccessful simulations, enabling efficient chemical space exploration for material discovery (28,29).

6.2.4 Kernel methods Kernel methods (30) are widely used to learn the photophysical properties and nonadiabatic molecular dynamics trajectories based on the similarity between data points, making flexible predictions beyond a linear classification or regression. Kernel methods provide an efficient tool to predict atomistic properties and construct ML potential (10,11,31–33). Many studies have trained kernel-based models to predict the molecular absorptions (5,33) and simulate excited-state molecular dynamics (34,35). In Section 6.2.3, we have shown that the linear SVM classifier learns the structure-property relationship from the inner products of the molecular vector features xTi xj (Eq. 6.14). The “Kernel Trick” offers a convenient tool to learn the linearly inseparable data. It maps the original data in higher dimensional space by replacing the inner product with a nonlinear kernel function K(xi, xj). Fig. 6.3 shows an example of a set of 2D nonlinearly separable data that can be classified in 3D space using a nonlinear kernel function. The kernel function K(x, y) must be positive definite symmetric to guarantee the convexity of the optimization problem (Eq. 6.14), ensuring convergence to a global minimum in minimizing the prediction errors (Eq. 6.7). Fig. 6.4 illustrates a variety of kernel functions. The “Kernel Trick” makes a convenient regressor for learning photophysical and photodynamics simulation data. The kernel functions measure the similarity between the representations of the ensemble of molecular geometries in the training set. The property y of the molecule x in a given electronic state can be predicted by the sum of kernel functions K(x, xi) weighted with the regression coefficients ai as y¼

N X

ai K ðx, xi Þ

(6.23)

i

FIG. 6.3 Classification of a set of 2D nonlinear separable data (left) in 3D space using K(x, y) ¼

(x2 + y2) function (right).

FIG. 6.4 Illustration of the polynomial, sigmoid, Gaussian, and Laplacian kernel functions in 2D feature representation.

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The kernel function K(x, xi) yields an n  n symmetric positive semidefinite matrix K (i.e., kernel matrix) over the n-points training set. Eq. (6.23) has a more compact form as Ypred ¼ Ka

(6.24)

where Ypred gives a column matrix of all predicted values, Ypred 5 [y1, y1, …, yn]T and a is a column matrix of all regression coefficients, a 5 [a1, a1, …, an]T. In a simple least square regression problem, the optimization objective minimizes the prediction error given by LðaÞ ¼ kKa2Yk2

(6.25)

Here Y is a column matrix of reference data of all compounds in the training set. The function L(ɑ) (Eq. 6.25) describes the square error between the predicted and the reference data. It estimates the loss of the model using the current set of coefficients compared to the ground-truth values. Thus, L(ɑ) is called the loss function. By minimizing the loss function, we find an optimal set of coefficients that best fit the reference data. The minimum of the loss function is obtained by setting its firstorder derivative(s) with respect to a to zero, which gives dLðaÞ ¼ 2KðKa2YÞ ¼ 0 da

(6.26)

Ka ¼ Y

(6.27)

Solving Eq. (6.27) requires the kernel matrix K to be an invertible matrix (i.e., the determinant is not equal to zero). The inverse of K gives a unique solution to Eq. (6.27), a 5 (K) 1Y. However, this condition is not always guaranteed because of the arbitrary choice of kernel functions. To ensure that Eq. (6.27) always has a unique solution, we can combine kernel methods with ridge regression, called kernel ridge regression (KRR). It includes an additional penalty term in the loss function. LðaÞ ¼ kKa2Yk2 + laT Ka

(6.28)

The minimum of the new loss function is obtained by 2KðKa

YÞ + 2lKa ¼ 0

2K½ðK + lIÞa

(6.29)

YŠ ¼ 0

(6.30)

ðK + lIÞa ¼ Y

(6.31)

Matrix I is an identity matrix. Thus, the matrix K + lI is always invertible, and the solution to ɑ can be conveniently obtained by a ¼ ðK + lIÞ 1 Y

(6.32) 1

The KRR learns the coefficients a from the product of the inverse kernel matrix, (K + lI) and reference data Y (Eq. 6.32). Then, it applies the learned coefficients to predict a new value by evaluating the kernel function between the representations of the new and reference molecules (Eq. 6.23). The use of kernel function can take advantage of the joint probabilistic distribution of training data. Suppose that the training data follow a multivariate normal distribution, the kernel function K becomes a covariance matrix, which informs the joint variability of the two data points. A positive covariance suggests that both data tend to be high or low at the same time; a negative covariance means that when one data is high, the other tends to be low. In this case, the KRR model becomes the Gaussian process regression (GPR) model. To understand the connection between KRR and GPR, consider a training set D ¼ [(x1, y1), (x2, y2), …, (xn, yn)]. We assume a function f(xn) that projects x to high-dimensional space and returns yn ¼ f(xn) + en at xn, where en is an independently, identically distributed Gaussian noise with zero mean and variance s2. The Gaussian process (GP) (36) is to find a multivariate Gaussian distribution among the training data, such as   yx  N mx , Kx,x + s2 I (6.33) where yx is the output vector, yx ¼ [y1, y2, …yn], mx is the mean function of yx, mx 5½y1 , y2 , …yn Š, and Kx, x is the covariance matrix, Kx,x 5 (yx 2mx)(yx 2mx)T. It is desirable for GP to predict outputs y for a set of new inputs x, which may not be included in the training set. The definition of GP gives the joint distribution as

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yx y∗



N



 mx m∗ ,

Kx,x + s2 I Kx, ∗ KTx, ∗ K∗, ∗

!!

(6.34)

where y∗ ¼ [y1∗, y2∗, …, yn∗], mx 5½y1∗ , y2∗ , …, yn∗ Š, and Kx,x is a n  n covariance matrix for training data, Kx,∗ is the n  n∗ covariance matrix that connects training data and new data and K∗,∗ is the n∗  n∗ covariance matrix for new data. According to the standard rules (14) for conditioning Gaussian distribution, Eq. (6.34) gives the posterior predictive density of y∗ at a set of test points x∗

pðy∗ jD, x∗ Þ ¼ N y∗ jm∗|x , S∗|x (6.35)   1 m∗|x ¼ m∗ + KTx,∗ Kx,x + s2 I ðyx S∗|x ¼ K∗,∗



KTx,∗ Kx,x + s2 I



1

mx Þ

Kx,∗

(6.36)

(6.37)

where m∗|x and S∗|x are the posterior mean and variance of y∗. If the output data yx and y∗ are standardized to zero mean, we can write the posterior mean as   1 (6.38) m∗|x ¼ KTx,∗ Kx,x + s2 I yx

The matrix (Kx,x + s2I) 1yx is equivalent to Eq. (6.32) by noticing l ¼ s2. Inserting Eq. (6.32) into Eq. (6.38) gives m∗|x ¼ KTx,∗ ax

(6.39)

which has the same form as Eq. (6.24). KTx,∗ is an n∗  n matrix that connects the dimensions of the training data and test data. Therefore, the GPR can be considered as a special case of KRR when choosing the same kernel function and setting l ¼ s2. The GPR prediction is always associated with an intrinsic covariance (Eq. 6.38), which conveniently quantifies the uncertainty about the prediction. The advantage of the kernel methods is that they only need to tune the internal hyper-parameters such as a parameter controlling the width in the Gaussian kernel functions and the regularization factor l. The latter introduces noise to prevent overfitting. Otherwise, the model may predict a significant error for data not included in the training set. The hyper-parameters optimization for the KRR model is not trivial and often requires grid or random search techniques. In contrast, the hyper-parameters in the GPR model are assumed to follow Gaussian distributions. The likelihood of hyper-parameters can be estimated by a Gaussian function depending on the training data and learned coefficients (37). The optimal hyper-parameters are solved by maximizing the likelihood function. When training a kernel model, the data set is randomly split into training and test sets. The training set is used to optimize hyper-parameters of the kernel method models without the bias from the test data. The test set assesses the learning performance of the KRR model by computing the mean absolute error (MAE) and root-mean-square error (RMSE). With a limited number of training data, k-fold cross-validation is a convenient and effective approach to find the hyper-parameters without expanding expensive training data. In the k-fold cross-validation, the training set is further split into k parts. The k 1 parts of the training set are used to train the models, and the last part is used to validate the model. This procedure is carried out k-times as each part of training data is used once for validation. The average validation errors indicate the accuracy of the chosen hyper-parameters. Once the optimal hyper-parameters are found, the accuracy of the final model is measured by computing the prediction errors on the test set. The optimal hyper-parameters may be defined as such hyper-parameters which give the lowest error in the entire training set (as in the cross-validation all points of the training set are used for validation) (13) or the lowest average error for the validation splits (38). After the hyper-parameters are found, the model is trained with them on the entire training set. In the end, the accuracy of the final model is measured by computing the prediction errors on the test set. A major drawback of the kernel methods is that the size of the kernel matrix is compounded by the number of training data scaling as O(N2) and the training timescales as O(N3) which is the major computation overhead. As training sets are rapidly expanding nowadays, it poses a problem for training models and special solutions are required (31). In addition, usually, kernel methods are single-value predictors and one often trains multiple models to predict the molecular properties containing multiple values, such as the energies of multiple electronic states (11), but it is possible to use extended versions for learning multiple values (34,36).

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FIG. 6.5 Schematic representation of a feed-forward multilayer perceptron (a multilayer perceptron-feed-forward, fully connected neural network). f is an activation function.

6.2.5 Neural networks Neural networks (NN) represent another popular class of ML models for learning photophysics and photodynamics simulation data. The complexity of NN depends on the architecture, including the number of hidden layers and the number of nodes per layer. Thus, NN is often referred to as parametric methods, where the architectures are independent of the training set. This feature makes NN suitable to predict atomistic properties and construct ML potential for vast data sets rather than kernel methods (10,11,32). The basic unit of neural network is a perceptron that consists of an input value, weights, and bias. It predicts an output value by transforming the sums of the weighted input values with a nonlinear activation function. A typical NN architecture is built upon multilayer perceptron (MLP); more specifically, MLP is a so-called feedforward, fully-connected NN. The first layer of MLP is the input layer receiving the molecular feature vectors. The subsequent layers are several hidden layers transforming the features. The last layer is the output layer predicting the new data (Fig. 6.5). One of the advantages of NN is that the output layer has no limit on the number of nodes, thus supporting prediction of multiple properties simultaneously. Suppose layer n 1 has N nodes and layer n has M nodes. A mapping function connects them in the following form: yn ¼ f ðwn yn

1

+ bn Þ

(6.40)

yn 1 gathers the input features of layer n, an N  1 matrix from layer n 1. yn is the output of layer n in the shape of M  1, wn is the M  N weights matrix, and bn is an M  1 matrix containing the bias on each node in layer n. f is a nonlinear activation, desirably differentiable, activation function, enabling NN to fit untrivial nonlinear problems. Fig. 6.6 illustrates a variety of commonly used activation functions. The objective of training NN is the same as the kernel methods: it is to minimize a loss function defined by  2   LðwÞ ¼ ypred 2yref  lkw kn (6.41)

l is a smaller factor of the penalty term (i.e., weight decay) in the loss function, and n is an exponential factor to control the type of regularization. For example, n ¼ 1 leads to Lasso regularization, called L1, and n ¼ 2 gives the ridge regularization, called L2. Alternatively, one can set l ¼ 0 and use other regularization techniques including early stopping and dropout (39,40). Because of the complex functional form of NN (Eq. 6.40), the analytical solution to Eq. (6.41) is often unavailable in contrast to kernel methods. A numerical solution can be obtained by stochastic gradient descent methods, such as Adam (41). The NN first computes the difference between the predicted and reference data in the training set, passing the input features to the output layer. Then it updates the weights of NN from the output layer back to the input layer by computing the gradients at each node using the chain rule, also known as backpropagation. The gradient descent update of weights is given by wk+1 ¼ wk

l∙

dLðwÞ dw

(6.42)

where k is the number of training steps, and l is the learning rate, which controls the step size of the gradient descent. In practical training, the gradient decent is done by fitting a batch of the training data instead of the full set. It reduces the computer memory usage and speeds up the training. A full swap over all batches is called an epoch.

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FIG. 6.6 Illustration of several commonly used activation functions.

The number of layers, nodes, choice of regularizer, regularization factor, and the learning rate are the hyper-parameters of NNs. As such, the use of NN requires a comprehensive hyper-parameter search such as a grid or random search (40). It usually needs to train a few hundred NNs with the combinations of hyper-parameters and compares the error of their validation set to narrow the hyper-parameters space. This procedure repeats until the optimal NN has a satisfactory validation error. NN is prone to overfitting. Overfitting can be prevented by using the regularization in the form shown before (weight decay, Eq. 6.41). However, the simplest and one of the most widely used ways to prevent overfitting is a procedure known as early stopping (39,40). It splits the training data into two parts, e.g., in 9:1, where the first part is used to train the NN and the other part for validation. The training procedure will stop when the validation error increases while the training error decreases. Another useful technique to minimize overfitting is called dropout (39,40). It randomly ignores the outputs from some number of nodes to prevent complex co-adaptation of nodes within the layer, making a more robust prediction. The main drawback of the MLP model is that the input dimensionality is fixed. It requires canonicalized molecular features for different molecules, which is not always satisfied when the molecular properties depend on the number of atoms, such as energies. A few advanced NN architectures are designed to circumvent the limitations compounded to the number of atoms. Behler introduced a high-dimensional NN potential (HDNNP) based on training multiple NNs for each type of atom (42). The individual NN learns the local environments around each atom center, such as the bond distances, bending angles, and torsion angles to the neighboring atoms. Therefore, the input feature does not depend on the total number of atoms. The HDNNP predicts the molecular potential energy by summing up the NN outputs from each atom. Another example is the ANAKIN-ME (Accurate NeurAl networK engINe for Molecular Energies) model or ANI for short (43,44). Convolutional NN (CNN) paves another avenue toward a generalized model for learning molecular properties. CNN is different from conventional NN by performing a convolution operation on a grid, which effectively extracts the patterns of features while substantially reducing the feature dimensionality. Fig. 6.7A illustrates the forward procedures in a CNN. When learning molecular properties, it is preferable to adopt the convolutional operation to a molecular graph, for instance, the bond connectivity, bond types, and atomic hybridizations. The convolution will embed an atomistic contribution for each atom by summing the information obtained from neighboring atoms, resembling a feature filter to generate local

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FIG. 6.7 Illustration of (A) convolutional and (B) graph convolutional neural networks.

atomic environment representations. The generated representation can be directly used or integrated into an end-to-end architecture to fit molecular properties. Since the convolution procedure proceeds with a specific molecule graph, this type of CNN is also called graph convolutional NN (GCNN), shown in Fig. 6.7B. Recent developments on GCNNs are message passing neural networks (MPNN) (45), deep tensor neural networks (DTNN) (46), continuous-filter convolutional neural networks, SchNet (47), hierarchically interacting particle neural networks (HIP-NN) (48), and PhysNet (49).

6.3 Representations of molecules Cartesian coordinates are the most straightforward representation to uniquely define molecular structures in QC calculations for computing molecular properties. However, such representations are not optimal for ML models due to the translations or rotations of molecules breaking the mapping between the input and output data. It introduces significant and unnecessary computational costs to fit the underlying models. Therefore, a machine-learnable representation for molecules should be translationally and rotationally invariant: requirements that are satisfied by internal coordinates and many other representations specifically designed/tested for ML (50). The permutational invariance is often also needed for ML tasks involving arbitrary atom orders (50). Differentiable representations help reduce the computational costs to learn the vectorial properties such as atomic force by taking advantage of the analytical gradients of the ML models (11). The ML representation can encode more than just molecular structural information. Integrating the physics laws and approximations thereof (e.g., many-body interactions) with ML can facilitate the regression problem and achieve the highest possible accuracy (46,51). The following section will describe popular molecular representations, such as the string-based descriptors, molecular fingerprints, 3D molecular descriptors, and automatically generated descriptors.

6.3.1 Molecular strings and fingerprints Simplified molecular-input line-entry system (SMILES) was invented to address the arising challenges of molecular representation and identification (52). SMILES is a simple string-based representation following the principles of molecular graph theory. The molecular structure specification of SMILES has straightforward rules. Since invented, it has become a standard tool for string-based representing molecular information in computational chemistry and cheminformatics. For discovering new molecules, the SMILES strings introduce a substantial problem as a significant amount of the resulting SMILES strings does not correspond to valid molecules. They are either syntactically invalid, i.e., do not even correspond to a molecular graph, or violate basic chemical rules, such as the maximum number of valence bonds between atoms. DeepSMILES overcomes most syntactic issues to generate graphs. However, it does not deal with semantic constraints

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that are introduced by the specific domain (53). Recently, a new string-based representation, SELF-referencIng Embedded Strings (SELFIES), was designed to ensure a 100% robust generation of molecular graphs (54). SELFIES has shown simplicity and efficiency in forming the chemical spaces and obtaining chemical paths for discovering a vast amount of molecules (55). Molecular fingerprints are numerical and discrete representations that dictate the information present or absent for training ML models. They are essential cheminformatics tools for virtual screening and representations of vast chemical spaces. The central topic is the development of general and readily applicable representations to allow for algorithm-based pattern recognition in molecular structures (56). These fingerprints are originally designed in bit vectors, such as the onehot encoding, for substructure and similarity searches (57–59). Modern implementations such as Morgan fingerprints (60), extended-connectivity fingerprints (ECFPs) (61), and MinHashed fingerprints (MHFPs) (60) are generated based on the molecular graph using the string-based representations (e.g., SMILES and SELFIES). They have been validated as suitable inputs for machine learning models. These fingerprints are predictive of the chemical properties of small-to-medium-sized organic molecules by perceiving the presence of specific circular substructures around each atom in a molecule. However, they are unable to describe the global 3D features of large molecules. Atom-pair fingerprints (APFPs) tackle the limitations by encoding atomic invariants combined with their bond distances (62). The original APFPs do not encode detailed molecular structures, thus perform poorly in small molecules. A recent work has combined the MHFPs and APFPs to develop new fingerprints suitable for small and large molecules, called Minhashed atom-pair fingerprints (MAPFPs), which outperform other fingerprints in extended benchmarks on small molecules, biomolecules, and the metabolome database (63). Another conformer-specific fingerprint is the extended three-dimensional fingerprints (E3FP). It extends the logic of ECFPs to encode the structural information by iterating several radial shells from a target atom (64).

6.3.2 Molecular descriptors Molecular descriptors are chemically relevant representations of molecular structures. They can be internal coordinates (defined by bond distances, bond angles, and torsional angles) or other representations reflecting 3D molecular geometries. The distance matrix is one of the simplest descriptors that preserves translation and rotational invariance. The inverse of the distance matrix is commonly used to inform the inverse decay of Coulomb force between atom i and j. Dij ¼ r i

1

rj

(6.43)

Using an inverse distance matrix to represent molecules in ML models usually flattens the upper or lower triangular matrix ignoring the ill-defined diagonal elements. One way to define the diagonal elements is to introduce a point (nuclear) charge for each atom. Then, the inverse distance matrix becomes the Coulomb matrix (65) defined by 8 > 0:5Z2 , i ¼ j > < (6.44) Cij ¼ Zi Zj , i 6¼ j > > : r i r j

The distance-based descriptors are widely used in ML studies to accurately predict photophysical properties and photochemical reaction outcomes (11). However, the distance-based descriptors have a systematic issue with permutational variance because their construction depends on arbitrary atom indexing. This issue can be partially mitigated if the training sets are preorganized by a specific sorting rule, which considerably decreases the transferability of the ML models. Sorting techniques (e.g., bag-of-bonds (66)) are introduced to build permutationally invariant representations but can lead to discontinuous potential energy surfaces due to the substantial dependence of energy on molecular structure (33). Permutationally invariant polynomials (PIP) (67) are introduced to rigorously handle atom permutations in NN (68–70). The PIP descriptor is obtained by applying a symmetrization operator S to Morse-like functions of internuclear distances ri, j and a hyper-parameter c: Y PIP ¼ S e crij (6.45)

Since c needs additional hyperparameter optimization, the PIP representation is not uniquely defined. It should be noted that the number of permutations quickly increases with the system size. Thus, the PIP descriptor is limited to small molecules.

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Data augmentation is another solution without modifying the descriptor. One can expand the training set to learn the permutation invariance using randomly sorted Coulomb matrices (38) or symmetry-adapted permutations (8). On the other hand, the permutational invariance can be incorporated into the ML algorithms using a permutationally invariant kernel function, such as the permutationally invariant KRR with RE descriptor and Gaussian kernel (pKREG) (33), symmetrized gradient-domain machine learning (sGDML) (71), and reproducing kernel Hilbert space (RKHS) interpolation (72). The above descriptors are global descriptors that provide complete information on molecular structures. One can unambiguously reconstruct the molecular structures, ensuring a learnable mapping between representation and molecular properties. However, the drawback of global descriptors is that they are compounded to the molecular system in a fixed size. Therefore, ML models trained with different input sizes are not transferable. Although we can extend the input dimension to match the largest systems, it would yield an unnecessarily large input for small systems containing many zero values. In addition, the dimensionality of global descriptors grows polynomially with the increasing molecular size, which quickly becomes infeasible for current computer memory for large molecules with thousands of atoms, which urges the development of more flexible molecular descriptors. The atom-centered representations allow one to train ML models for molecules of varying sizes. These descriptors are also called local descriptors, as they usually employ a cutoff function that defines a sphere around each atom to encode the local chemical environment. The fitting of molecular properties using local descriptors follows the concept of energy decomposition, where the total energy is expressed as the sum of atomic energy, including the interatomic interactions. Thus, the construction of local descriptors is permutationally invariant and size-extensive. A well-known example of local descriptors is the Behler-Parrinello’s atomic centered symmetry functions (BP-ACSF) (73). It employs a set of basis functions to combine the atomic radial and angular distributions in a given cutoff sphere. Other local descriptors under development are smooth overlap of atomic positions (SOAP) (74), atomic spectrum of London and Axilrod-Teller-Muto (aSLATM) (75), Faber-Christensen-Huang-Lilienfeld (FCHL) (76), Gaussian moments (77), spherical Bessel functions (78,79), and the atomic descriptors for deep potential molecular dynamics (DPMD) (80).

6.3.3 Automatically generated descriptors Choosing and tuning optimal molecular descriptors are untrivial and require expert knowledge in physics, chemistry, and data science. To simplify the design of molecular descriptors, one can incorporate the descriptor parameters inside the neural network architectures. As such, the descriptors can be automatically fitted to the optimal values during training the model. The message-passing neural networks demonstrate a straightforward implementation of such descriptors (45). The message passing is a function that encodes atomic information by convoluting the neighbor information of the node features (i.e., atom type) and edge features (i.e., pairwise nuclear interactions) within a local environment according to a molecular connectivity graph. The encoded messages are refined by an update function and then used to fit the molecular properties, ensuring high accuracy if the NN model is appropriately trained. Following such concepts, more sophisticated convolutional layers extract the atomic features acting as a pattern filter for the local atomic environment. This type of features was implemented in Deep tensor NN (DTNN) (46) and its descendant SchNet (47), which uses Gaussian functions to learn the internuclear distance for fitting molecular potential and force field. Similar NNs are PhysNet (49), HIP-NN (48), and DeepPot-SE (81). Recently, the automatically generated features are used to learn electronic wave functions as demonstrated in SchNOrb (82) and SpookyNet models (83).

6.4 Training data for machine learning Experimental data sets are usually limited to a class of compounds (e.g., dyes (84)), which correspond to a narrow region of chemical space. When the experimental data are unavailable or insufficient, QC calculations can be used to generate datasets via massive and comprehensive QC calculations. The QC calculations can produce consistent and noise-free data for well-studied thermal reactions and thermodynamic properties (i.e., free energies). However, it becomes more challenging to prevent the noise in excited-state data (6,85). In the case of excitation energy, the continuity of the energy as a function of nuclear configurations might be interrupted because of the insufficient number of states under consideration, where higher states are needed to describe smooth energy changes through state crossings. The chosen QC method might have an intrinsic limitation, i.e., the selection of active space, thus cannot avoid the discontinuous energy point when the initial setting (e.g., the active space of a reactant molecule) is no longer valid for other nuclear configurations (e.g., conical intersections). There are two types of training set most often used for applying ML in predicting photophysics properties and photochemical reaction outcomes. One is a large set of data corresponding to the excited-state properties of various compounds

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obtained from the optimized ground- and/or excited-state geometries. Such a database provides a sparse chemical space for ML models to explore their excited-state properties and design new molecules. The other is a molecule-specific dataset, which only includes a molecule or a class of similar molecules, but their excited-state properties (e.g., energies and forces) are computed at various geometries (i.e., snapshots of molecular dynamics or interpolations). This type of data set offers the reference to explicitly learn how structural changes control the excited-state properties, which is suitable for predicting molecular dynamics.

6.4.1 Excited-state database across chemical space Many existing QC databases have recently included the excited-state properties for the scientific community to apply ML in photochemistry. QM7 (65), QM7b (86), and QM8 (87), are the most prominent databases that have been extensively used to benchmark ML models in many publications (47,76,88–94). These databases are the subset of the chemical universe GDB database, which was designed for searching lead compounds in drug design (95). The QM7b set is one of the first databases that provide excited-state properties. It contains more than 14,000 molecular geometries from QM7 (7165 molecules) and an additional 7211 molecules with atoms C, N, O, H, S, and Cl. The lowest excitation energies were computed with TDDFT (86). The QM8 set provides the lowest excitation energy and the corresponding oscillator strengths of 20,000 small molecules with atoms C, O, N, F, which enables accurate prediction of electronic spectra at the CC2 level (87). 4000 geometries were randomly selected from the larger QM9 (96) set of small and drug-like molecules to construct the QMspin set for studying the structural and electronic relationship in singlet and triplet carbene, where the electronic energies were computed with the MR-CI method (88). Other excited-state databases are PubChemQC, which includes the lowest 10 excitation energies for 2 million molecules computed at TDDFT/B3LYP/6-31+G* (97), and QM-symex, which includes the lowest 10 singlet and triplet transitions for 143,000 molecules at TD-DFT/B3LYP/6-31G(d,p) (98). While the databases mentioned above support a general-purpose exploration in the chemical space formed by small molecules with common elements in living organisms, specific photochemical problems could involve different elements, larger numbers of atoms, and excited-state properties beyond excitation energies. Therefore, we need to construct specialized databases regarding the problems under investigation in such cases. For example, Lopez and co-workers published the Virtual Excited State Reference for the Discovery of Electronic materials Database (VERDE materials DB), which currently has the ground- and excited-state geometries, redox properties, dipole moments, and frontier molecular orbitals of 4300 light-responsive p-conjugated organic molecules. This number continues to increase as we continually add more results to VERDE materials DB (99). The VERDE database employs an automatic workflow to generate diverse molecular structures from the predefined cores, spacers, and terminals expressed by SMILES strings, as shown in Fig. 6.8.

6.4.2 Molecule-specific data generation The QC databases described in the previous section are not suitable for predicting dynamical processes in photochemical reactions because they only contain data for nuclear configurations at the equilibrium geometry. In molecular dynamics simulations, the kinetic energy distributed on atoms immediately drives the molecule into under-sampled nonequilibrium geometries. Therefore, the data set must sufficiently sample the configuration space of chemical reactions. However, expanding the data set by covering all different geometries for each molecule increases the chemical space significantly, resulting in a computationally prohibitive number of calculations. In practical applications, these training sets are usually molecule-specific for the reaction being studied. The generation of molecule-specific data sets for predicting the outcome of photochemical reactions is different from thermal reactions. Thermal reaction data are accessible from reaction path interpolation techniques (e.g., nudged elastic band (100)), normal modes sampling (6,44,85), and molecular dynamics (MD) based sampling (e.g., umbrella sampling (101), trajectory-guided sampling (102), enhanced sampling (103), and metadynamics (104)) with inexpensive semiempirical electronic structure methods, such as GFN2-xTB (32,105) or force field. It has been proven that the ML models (e.g., ANI (44)) trained with the above data generation techniques can be transferable for predicting accurate energy and forces for various molecules. However, these techniques become ineffective for generating the photochemical reaction dataset because the actual excited-state MD usually involves substantially different geometries from those sampled from the ground state MD. As a result, the ML models trained on the nuclear configurations obtained from the ground-state simulations are still undertrained for excited states. Sampling excited-state geometries from nonadiabatic molecular dynamics (NAMD) may mitigate the problem (35,85,106). However, it should be noted that the reliability of NAMD data is heavily dependent on chosen QC methods. It is well known that TDDFT and other single reference methods predict incorrect dimensionality of branching space at conical intersections, which could lead to significant errors in excited-state

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FIG. 6.8 Illustration of the combinatorial molecular structure generation scheme and the excited-state calculation workflow. (Reproduced from the reference Abreha, B. G.; Agarwal, S.; Foster, I.; Blaiszik, B.; Lopez, S. A., Virtual Excited State Reference for the Discovery of Electronic Materials Database: An Open-Access Resource for Ground and Excited State Properties of Organic Molecules. J. Phys. Chem. Lett. 2019, 10 (21), 6835–6841 with permission. Copyright 2019 American Chemical Society.)

dynamics compared to multireference methods (107,108). Therefore, it is not recommended to use single reference methods to sample excited-state data unless they are calibrated to multiconfigurational QC methods. Due to the computational cost of ab initio multireference methods, the sampling time approaches an upper limit of 1 ps (typical wall clock time of 20–30 days), which may still not provide sufficient data to train ML models. A more efficient strategy is active learning, where we let the ML models determine the requisite training data. It first trains a preliminary ML model using an initial training set, and then applies it for NAMD simulations to locate undersampled geometries on excited-state potential energy surfaces. The collected data will be recomputed with a multiconfigurational QC method to expand the training set and retrain the model. This automatic procedure, also known as adaptive sampling (6,7), will iterate until reaching a predetermined threshold following the same concept of self-consistent field calculations. A commonly used threshold can be that the number of under-sampled data is no more than a certain value, where the ideal value is zero. We consider no preliminary data of the underlying photochemical reaction to be available to eliminate human biases; this enables a maximally meaningful initial training set. To achieve this goal, some of us proposed a composite structural sampling protocol (7). It combines the Wigner sampling, which samples the local geometries near reactant and product(s), and the geodesic interpolation (109) from the reactant to a minimum energy crossing point (MECI) and the product, which informs a hypothetical photochemical reaction path. The Wigner sampled geometries generate atom-centered distortions compared to the optimized geometry. We can apply these geometrical distortions to the interpolated geometries, further expanding the set of the sampled geometries along the interpolated reaction path. Our structural sampling protocol only requires three geometry optimizations (i.e., reactant, MECI, and product) and one frequency calculation for reactant, which can quickly generate thousands of geometries for training data calculations. Although the

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FIG. 6.9 (A) Workflow of adaptive sampling in PyRAI2MD. (B) The active learning progress in training NNs for photo-induced cis-trans isomerization of hexafluorobut-2-ene. Black dots are the data in the initial train set. Collected data are colored according to the adaptive sampling iteration.

MECI and the interpolated path might deviate from the visited regions during the NAMD simulations, they provide a baseline for the ML model to learn the structure and excited-state properties, which the iterative adaptive sampling can improve. The ML model needs to monitor the uncertainty of on-the-fly prediction in energy, forces, and other requisite properties in adaptive sampling to identify the under-sampled data. The uncertainty metric is naturally available in the GPR model as it always returns the prediction’s covariance. However, a single NN model does not have an uncertainty measure. Instead, we follow the concept of the query by committee (110), where we independently train two NN models and use the standard deviation between their predictions to measure the uncertainty. Fig. 6.9A illustrates the workflow of adaptive sampling implemented in the Python Rapid Artificial Intelligence Ab Initio Molecular Dynamics (PyRAI2MD) code (7). In our recent works, we have seen the power of active learning in the self(data)-driven exploration when learning photochemical reactions. One of our studies on the photo-induced cis-trans isomerization of hexafluorobut-2-ene has shown that with exclusive information about the cis and trans-configurations, the active learning rediscovers the unseen structures involved in the H-migration mechanism, reproducing the reference QC results (Fig. 6.9B) (7).

6.5

Applications of machine learning in photochemistry and photophysics

Machine learning is revolutionizing our perspectives and paradigms in investigating and understanding photochemistry. The ML applications provide us a more efficient tool to assist the QC methods calculations and help us to gain new chemical insight from the data for elucidating the complex mechanism of photochemical reactions. This section will overview the most recent works that demonstrated the significance of ML in studying photochemistry.

6.5.1 Machine learning-assisted high-throughput virtual screening Photoswitches are a class of molecules (e.g., azobenzenes) that absorb light and reversibly interconvert between their thermodynamically stable and metastable forms at photostationary states. Azobenzenes undergo efficient isomerization from their thermodynamically stable E-form to their metastable Z-form (i.e., Z) under ultraviolet radiation (314 nm), and the Z-form has a thermal half-life (t1/2) of 2 days (111). The ultraviolet absorption may promote undesired side reactions competing with the isomerization of azobenzenes and causing DNA damage (e.g., [2 + 2]-dimerization of thymine). It also has a

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limited tissue penetration depth (0.1 mm) (112), thus limiting the therapeutic potential of photoswitches in photopharmacology (113). Ideal photoswitches suited for photopharmacology should feature a long absorption wavelength and long t1/2. Optimization of these parameters was conventionally carried out by high throughput virtual screen based on density functional theory (DFT) calculations (114,115). While recent research demonstrated that the functionalization of azobenzene and introducing heteroaryl rings could lead to red-shifted absorption wavelength and elongated thermal half-life (116–122), it creates a substantially extended chemical space of azoarenes compared to azobenzene approaching 106 compounds. Even with high-performance computing and efficient quantum chemistry codes, exploring such a diverse chemical space with DFT calculations is formidable. Machine learning techniques have shown a promising ability to leverage the high-throughput virtual screening. Previous work by Ju and co-workers developed an ML algorithm to predict fluorescence wavelength and quantum yield based on 4300 experimental data (3). Thawani and co-workers curated a dataset of 6142 photoswitches with azobenzene and azoarene derivatives to predict the absorption wavelengths (123). More recently, Lopez and co-workers published an active search approach to efficiently select 717 desired candidates with absorption wavelengths of 451–602 nm from a vast chemical space of 255,991 azoarenes, which only requires 2117 DFT calculations as 1% of the chemical space (2). The active search algorithm is summarized in Fig. 6.10A. The chemical space was generated using the combinatorial scheme with 29 azoarene cores and 11 terminals. Of the 255,991 azoarenes, a subset of 1436 azoarenes was used to compose the initial training data associated with the absorption wavelength computed with oB97XD/6-31 + G(d,p)// M06/6-31 + G** method. The azoarenes were represented by Morgan fingerprints (60) and labeled in True or False regarding a threshold of the absorption wavelength > 450 nm. Of 1436, 981 azoarenes had absorption

FIG. 6.10 (A) Illustration of the workflow of the active search algorithm. The automatic computational workflow utilizes the VERDE structural generation, and DFT calculations demonstrated in Fig. 6.8. (B) The subset of cores was searched after 20 iterations with at least one substituted molecule with a lmax > 450 nm. (Reproduced from the reference Mukadum, F.; Nguyen, Q.; Adrion, D. M.; Appleby, G.; Chen, R.; Dang, H.; Chang, R.; Garnett, R.; Lopez, S. A., Efficient Discovery of Visible Light-Activated Azoarene Photoswitches with Long Half-Lives Using Active Search. J. Chem. Inf. Model. 2021, 61 (11), 5524–5534 with permission. Copyright 2021 American Chemical Society.)

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wavelength > 450 nm (68% True). The labeled data were used to train a kNN classification model based on the Tanimoto similarity coefficients (124). The trained kNN model was then used to search new unlabeled azoarenes with each type of core that is highly probable to be in the True class. Based on the kNN predictions, the active search algorithm collects 50 recommended candidates, whose absorption wavelengths were computed with the TDDFT calculations to update the training set. Then the kNN model was retrained using the updated training set and used to search for another 50 candidates for the subsequent iteration. As more azoarenes are labeled with True in the training set, the active search drives the kNN model concentrating on the chemical space containing more prospective azoarenes (Fig. 6.10B). The active search algorithm has shown superior performance compared to random search. In the demonstration, the random search found 11 out of the 87 (13%) molecules with absorption wavelength (lmax) > 450 nm, and none have lmax > 550 nm (2). The active search illustrated an increasing ratio from 12% to 35% in the first 20 iterations. In the subsequent 20 refinement iterations, the percentage constantly varies from 44% to 56%. The overall average ratio in 40 iterations is 37%, which is triple that of the random search. The longest wavelength was found to be in 550–600 nm. Based on the active search-assisted high-throughput virtual screening, Lopez and co-workers successfully discovered four types of azoarene cores that generally favor a wavelength of longer than 450 nm and a thermal barrier higher than 23 kcal mol 1 paving the way toward further optimization of azoarene photoswitches.

6.5.2 Machine learning-predicted electronic spectroscopy The time-resolved spectroscopy techniques provide a powerful tool to probe the ultrafast electronic and nuclear motions in many photophysical and photochemical phenomena. However, in molecules with complex structures and strong vibronic coupling, the spectroscopic data are less trivial, thus hard to interpret. Highly accurate QC calculations thus become a crucial instrument in chemists’ toolbox to disentangle the elusive spectroscopic features and shed insights on the electronic transition characters. However, computing the photophysics of molecular excited states requires tremendously expensive QC calculations for solving the electronic structures. The highly accurate QC methods (e.g., MR-CISD) often scale exponentially with the number of electronic states and atoms (125,126). Moreover, reproducing the experimental spectra usually needs to sample many different molecular configurations (i.e., nuclear ensemble) to generate a statistical distribution comparable to the experiment (127). It further limits the applicability of the nuclear ensemble approach to resolve the spectroscopic features. ML accelerates the prediction of electronic state energies (4,6,11,128), transition dipole moments (4,6,11,128), and oscillator strengths (87,129). A ML-nuclear ensemble approach based on learning excitation energies and oscillator strengths with the KREG model successfully achieved a cost-efficient, semi-automatic generation of the precise spectra with the MLatom package for small- and medium-sized molecules and dozens of excitations (5,33). In contrast, fitting the transition dipole moments for many configurations of the target molecule is challenging because the sign resulting from two different electronic states is arbitrary due to the arbitrary phase of the wave function (6,130), and the vectorial form must maintain its rotational covariance. Thus, the training ML model for transition dipole moments requires phase corrections (6,128,131). Marquetand and co-workers extended their SchNarc model to enable a simultaneous prediction of permanent and transition dipole moment vectors of a given number of electronic states (128), which was originally developed to learn excited-state potentials, forces, and couplings (85) based on the convolutional NNs, SchNet (47). In addition, they have confirmed, at least qualitatively, the transferability of the SchNarc model trained on a data set including CH2NH+2 and C2H4 by predicting the absorption spectra from the lowest-lying excitations of some similar molecules, CH2NH, CHNH2, and C2H+5 , of which the data were not used to train the model (Fig. 6.11). Recently, Westermayr and Maurer developed a modified architecture of SchNet (SchNet+H) to predict the absorption and emission spectra for a range of large and complex organic molecules across the chemical composition space (4). They introduced latent Hamiltonian matrix representation to describe multiple resonances simultaneously. As a result, the predicted orbital and quasiparticle energies were highly accurate. Furthermore, the SchNet+H model clearly showed the ability to differentiate functional groups and predict trends as a function of molecule size in conjugated systems, demonstrating the model’s transferability and scalability.

6.5.3 Machine learning nonadiabatic molecular dynamics Photochemical reactions undergo complex mechanisms upon photoexcitation. For example, the excited-state lifetime, radiative and nonradiative decay channels, and branching ratio through conical intersections are often controlled by dynamical effects depending on the instantaneous momentum and local topography of the potential energy surface (PES) (133). When electronic states become energetically degenerate, the Born-Oppenheimer approximation breaks down due to the

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FIG. 6.11 UV/visible absorption spectra computed from 500 Wigner sampled conformations with MR-CISD/aug-cc-pVDZ on the left, and from 20,000 Wigner sampled conformations with SchNarc on the right. The molecules (A) CH2NH+2 and (B) C2H4 are included in the training set, and the performance of the ML model trained on one (solid lines) and both molecules (dashed lines) is compared, while (C) CH2NH and (D) CHNH2 are not included in the training set, and the ML model trained on both molecules is used for the prediction. In panel (B), the experimental UV/visible absorption spectrum (abbreviated as “Exp.”) is plotted, where the data are taken from the reference (132). (Reproduced from the reference Westermayr, J.; Marquetand, P., Deep Learning for UV Absorption Spectra with SchNarc: First Steps Toward Transferability in Chemical Compound Space. J. Chem. Phys. 2020, 153 (15), 154112 under CC-BY.)

dramatically rising nonadiabatic coupling along with the nuclear motions. Therefore, characterizing photochemical reactions mechanisms relies on NAMD simulations. The NAMD simulations are generally resource-intensive, comprising a significant amount of consecutive QC calculations used to parametrize the excited-state PESs for full quantum dynamics, including the coupled nuclear and electronic wave functions, e.g., multiconfiguration time-dependent Hartree (MCTDH) (134), or to compute the classic atomic force with nonadiabatic coupling for propagating mixed quantum-classical trajectories, e.g., trajectories surface hopping (TSH) (135). Due to the complexity of coupled electronic and nuclear wave functions, full quantum dynamic methods are only applicable for model systems or within a truncated dimensionality of the nuclear degree of freedom (136). The TSH methods are more popular as they can be directly applied to medium-sized molecules with all nuclear degrees of freedom available (135,137). However, even for a small molecule like CH2NH+2 , a 100 fs trajectories using highly accurate QC calculations (MRCISD/aug-cc-PVDZ) would take 20 h on a single CPU, which resulting in 4.5 years of CPU time to compute an ensemble of trajectories satisfying the 95% confidence interval (11). In the past years, machine learning techniques have emerged as a prominent accelerator for NAMD simulations, which enables modeling complex photophysical and photochemical processes in a long length and time scale (11). We have seen a rapid growth of the ML-NAMD methodologies and applications to study photochemical reactions. In 2018, Lan and co-workers reported NAMD simulations for a 12-atom molecule, 6-aminopyrimidine, based on a KRR model, trained with 65,316 data points (35). Dral, Barbatti, and Thiel published a KRR model for a 33-dimensional spin-boson system using 1000 and 10,000 data points (138). Cui and co-workers trained the NN-based DPMD model for CH2NH using 90,000 data points (106). Later, they reported another DPMD model for 5 water molecules as the environment for the QM/ML NAMD of CH3N¼NCH3 with more than 30,000 data points (139). In 2019, Marquetand and co-

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workers demonstrated a feed-forward NN model for CH2NH+2 with a training set of 4000 data points (6). Their works achieved the first nanosecond-scale ML-accelerated photodynamics simulation. In 2020, Marquetand and co-workers successfully developed and used their SchNarc model for NAMD simulation of CH2NH+2 and CSH2 with 3000 and 4703 data points, respectively (85). Meanwhile, Brorsen and co-workers reported a KRR model for a 3-dimensional spin-boson model system with 4913 data points (140). In 2021, Lopez and co-workers published a feed-forward NN model for trans-hexafluoro-2-butene (12 atoms) and norbornyl cyclohexadiene (25 atoms) trained with 6207 and 6267 data points, respectively. The NAMD simulations for trans-hexafluoro-2-butene demonstrated that a 10 ns ML photodynamics only spent 2 days of human time (7). The NAMD simulations for norbornyl cyclohexadiene explained the regioselectivity of the 4p-electrocyclic ring-closing reaction in excellent agreement with the experiment. Moreover, it uncovered a thermal interconversion from an experimentally unobserved cis,trans-cyclohexadiene isomer to the reactant in 1 ns, which resolved the discrepancy between the CASSCF results and the experiment. It necessitates the consideration of nanosecond-scale NAMD in mechanistic studies in photochemistry. In 2022, we applied the ML-NAMD approach to investigate the substituent effects on the [2 + 2]-photocycloaddition toward cubane (Fig. 6.12) (8). The model systems are octamethylcubane (44 atoms), octatrifluorocubane (44 atoms), and octacyclopropylcubane (72 atoms). With the help of active learning, the final training sets are compact with 3835,

FIG. 6.12 (A) Three geometrical parameters describe the reaction space of the substituted [3]-ladderdienes. Trajectory plots of the machine learning nonadiabatic molecular dynamics simulations for the [2 + 2]-photocycloaddition reactions of [3]-ladderdiene substituted by (B) octamethyl, (C) octatrifluoromethyl, and (D) octacyclopropyl groups. Each plot contains 400 randomly sampled trajectories from the overall 3835, 3259, and 3122 trajectories of octamethyl [3]-ladderdiene (4), octatrifluoromethyl [3]-ladderdiene (7), and octacyclopropryl [3]-ladderdiene (10), respectively. The averaged starting points, surface hopping points, and products are marked with red dots, black circles, and black dots. The inset legends illustrate the snapshot of the last trajectory point as the product with predicted quantum yields. The predicted quantum yield of octamethyl cubane (5), octatrifluoromethyl cubane (8), and octacyclopropyl cubane (11) are 1%, 14%, and 20%, suggesting the steric effects of a bulky group increase the preference of [2 + 2]-photocycloaddition and brings down the ring-opening process. (The figure is reproduced from the reference Li, J.; Stein, R.; Adrion, D. M.; Lopez, S. A., Machine-Learning Photodynamics Simulations Uncover the Role of Substituent Effects on the Photochemical Formation of Cubanes. J. Am. Chem. Soc. 2021, 143 (48), 20166–20175. Copyright 2021 American Chemical Society.)

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3259, and 3122 data points, respectively. It is evident to see substantial leverage of the ML-NAMD methods in feasibility and robustness for studying larger systems and more complex photochemical reactions. Currently, the major drawback of ML-NAMD is the need for an accurate model for nonadiabatic coupling (NAC) prediction. The NAC has a phase factor as it depends on two electronic state wave functions, whose phase factors are randomly initialized in QC programs. Thus, the NACs as a function of nuclear configurations may encounter an interruption if no phase correction is carried out (6). Marquetand and co-workers proposed a phase-corrected loss function in NN models, which helps predict NACs in a unified phase (85). Another issue in predicting NACs is the singularity at a conical intersection while the magnitudes are close to zero elsewhere. Therefore, it is advisable to fit an ML model only for the numerator of the NACs and compute the NACs with the predicted state energy gaps. The remaining challenge is to predict the NAC vectors satisfying the rotational covariance. This requirement can be fulfilled by fitting the NACs to the first-order derivative of a virtual potential (85). However, it was shown that the over-fitting of NACs persisted when a molecule has >10 atoms (7,8,141). The over-fitting may have many possible causes. One may accuse the unphysical virtual potential, which has no training data to minimize the corresponding loss function, or criticize the generally accepted chemical accuracy (1 kcal mol 1), over-estimating the energy gap near the conical intersections. As a compromise, the ML-NAMD can choose Zhu-Nakamura theory (142,143) to compute the surface hopping probability, which uses diabatized forces to approximate NACs. Alternatively, the NACs can be approximated with a time-dependent Baeck-An (TD-BA) approach based on the energy gap and second-order derivatives (144,145). In the most recent implementation, the TD-BA equation was represented based on time-dependent derivative couplings corresponding to the curvature of the PES, which only requires the prediction of excited-state energy (146).

6.5.4 Machine learning-extracted chemical insights from data As illustrated in the previous sections, ML can be used to predict accurately and efficiently diverse molecular properties of excited states (e.g., absorption wavelengths and transition dipole moments) and photochemical reaction outcomes. Here, we will illustrate how machine learning trained on data obtained using QC methods can be used to gain chemical and physical insights. The increase in computational resources and algorithm efficiency has led to the generation of large amounts of data. The application of ML to reduce the cost of ab initio calculations is expected to accelerate the amount of generated and interpreted data. But the gain in understanding becomes obscured, simple lessons are often lost, and novel tools are needed. One of the six challenges identified for the simulation of matter in the 21st century is to allow machines to provide a source of inspiration to humans to elaborate on new concepts in chemistry (147). In the following, we describe several examples that used ML to interpret QC simulations and understand fundamental chemical underpinnings. Time-resolved photoluminescence experiments are standard techniques to study the optical properties of materials and devices. The interpretation of the recorded data is, however, far from trivial. One often tries to fit the data with a model decay rate function based on physical assumptions about the mechanism of the studied process. The choice of model decay rate function could bias the interpretation of the measurements. Wood and co-workers provided a machine learning code to analyze time-resolved photoluminescence data without a priori assumptions on the physics of the process (148). They demonstrated their approach on a computational time-resolved photoluminescence data set. The learned decay rate distributions were used to develop an unbiased physical model of the luminescence dynamics, which could be extended to transient absorption data. Measuring the intrinsic structural disorder in organic optoelectronics design is challenging. The conventional computational approach derives a physical model for describing the electronic property based on, for instance, absorption spectra. Aspuru-Guzik and co-workers advanced an approach based on Bayesian machine learning models to correlate electronic absorption spectra of nanoaggregates of oligomers with the strength of intermolecular electronic couplings (149). Their findings suggest that ML models can identify physical correlations between the measurable electronic absorption spectra and the strength of intermolecular electronic couplings, which in turn determine the charge transport. Moreover, the application of ML has shed deeper insight into the important light-driven processes. Chemiluminescence is the emission of light due to a thermally activated chemical reaction during which a nonadiabatic transition occurs from the electronic ground state to an excited state. Lindh and co-workers demonstrated that the chemiexcitation yield of dioxetane molecules (i.e., the production yield of electronic excited states) is related to the timescale of the corresponding dissociation reaction (150). Longer dissociations were shown to be due to “frustrated” events at some specific geometries (151). In a later work, Vacher and co-workers trained Bayesian neural networks to predict the dissociation time from a particular molecular structure (152). By analyzing and using the machine learning models, they could identify important nuclear coordinates (Fig. 6.13) and evidence empirical rules that are, today, part of the common chemical knowledge: octet

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FIG. 6.13 Coefficient magnitude distributions of the input features (geometry coordinates in normal modes) of the trained Bayesian neural network. (Figure reproduced from the reference Hase, F.; Fdez Galvan, I.; Aspuru-Guzik, A.; Lindh, R.; Vacher, M., How Machine Learning Can Assist the Interpretation of ab Initio Molecular Dynamics Simulations and Conceptual Understanding of Chemistry. Chem. Sci. 2019, 10 (8), 2298–2307 with permission from the Royal Society of Chemistry.)

rule, orbital hybridization, and valence shell electron pair repulsion (VSEPR) model. They also re-visited the “frustrated” dissociation events using machine learning classifiers and could identify a key nuclear coordinate for the dissociation reaction (153).

6.6

Summary

ML is revolutionizing modern quantum chemical methodologies with an ever-growing number of applications in studying photochemistry. Diverse ML techniques offer rather accurate predictions of molecular excited-state properties, such as excitation energies, transition dipole moments, and oscillator strengths, with a marginal computational cost compared to conventional electronic structure calculations. The success of applying ML in photochemistry is built upon versatile molecular descriptors that encode a machine learnable representation and incorporate the physical meaning behind the molecular structures. Of the numerous choices of ML techniques, the k-nearest neighbor, kernel methods, and neural networks are among the most popular models providing convenient and flexible classification and regression ability to learn the elusive structure-property relationship. The ML models trained on massive QC calculation data also infer deep physical insights that are nontrivial and elusive based on direct data interpretations. Recent works have demonstrated that ML techniques can significantly accelerate the high throughput screening for novel photoswitch discovery. The ML-assisted spectroscopy simulations have become a powerful tool to disentangle the spectroscopic data enabling the investigations of complex absorption and emission phenomena in photochemistry. Further applications such as ML-accelerated NAMD simulation have substantially leveraged the computational power to explore photochemical reactions in long length and timescales. As practitioners of quantum chemistry at the age of ML, we are witnessing the increasing feasibility and robustness of ML techniques in photochemistry.

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Chapter 7

Polaritonic chemistry Jacopo Fregonia and Stefano Cornib a

Universidad Auto´noma de Madrid, Departamento de Fisica Teo´rica de la Materia Condensada, Madrid, Spain, b Universita` degli Studi di Padova,

Dipartimento di Scienze Chimiche, Padova, Italy

Chapter outline 7.1 Preliminary considerations on the electromagnetic field 7.2 Polaritonic eigenvalues and eigenstates 7.2.1 Cavity Born-Oppenheimer and vibrational strong coupling 7.2.2 Electronic strong coupling (ESC) 7.3 Polaritonic potential energy surfaces (PoPESs)

193 194 195 197 199

7.3.1 A didactical case: Azobenzene polaritonic potential energy surfaces (PoPESs) 7.3.2 Many molecules and dark states 7.4 Polariton dynamics and cavity losses 7.4.1 Nuclear dynamics in polaritonic systems: Full quantum vs semiclassical 7.5 Summary References

199 202 204 204 207 207

The strong coupling between a photon and any other excitation of the matter results in the so-called “polaritons.” With matter excitation, one can either refer to excitations of small molecular systems (electronic, vibrational, and spin excitations) or to their collective excitation counterparts in extended systems described as quasi-particles, i.e., plasmons, phonons, and magnons. Historically, polaritons have been studied in nano-optics and nano-photonics communities as a powerful toolbox for controlling the properties of light in nano-photonics technologies (1). However, it was only in the last decade that polaritons have been used to actively modify the properties of molecular systems, thus resulting in a new branch of chemistry which goes under the name of polaritonic chemistry (2,3). Another common name for this emerging discipline is molecular polaritonics (4) albeit the view the two different names entail is not the same. While the accent in molecular polaritonics is place on how to exploit molecules to control the conditions in which polaritons are obtained, polaritonic chemistry makes use of polaritons to modify the chemical reactivity, and even enable new reactions to occur (5,6). The conditions for forming polaritons and the dynamics of polaritonic systems drastically depend on the system’s subcomponents. As a rule of thumb, we can identify polaritons as the eigenstates of a system where the interaction between light and matter is too large that the system cannot be treated by perturbative approaches (7). Namely, one condition on the coupling strength between light and matter (g) must be fulfilled to achieve the strong coupling regime: the coupling strength shall be larger than the individual decay channels of the system (k, g) as shown in Fig. 7.1A. A convenient estimate to quantify the coupling strength (g) between two resonant transitions is given by (8) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi ħo g ¼ N jEjm
l with jEj ¼ (7.1) 2e0 er V eff where m identifies the transition dipole associated with the molecular transition, N is the number of (typically dipole-like) emitters involved in the matter excitation, l is the polarization of the electric field, and |E| is the electric field strength, which in turn depends on the frequency of the electric field o and the effective mode volume Veff. The dependence of the electric field on the number of emitters, the frequency, and the effective mode volume already suggests how reaching strong coupling at different wavelengths requires very different setups. A common strategy to obtain small effective mode volumes is to confine light in optical cavities. The most popular optical cavities make use of metallic mirrors or distributed Bragg reflectors to confine the electromagnetic field down to volumes of several mm3. In such volumes, the optical cavities accommodate a number of molecules ranging from

Theoretical and Computational Photochemistry. https://doi.org/10.1016/B978-0-323-91738-4.00004-X Copyright © 2023 Elsevier Inc. All rights reserved.

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FIG. 7.1 Polaritonic states and setups. Panel A shows the formation of polaritonic states upon resonance of a molecular (matter) transition and a quantized electromagnetic field mode. Polaritonic states can be obtained via collective coupling of molecular ensembles in optical cavities (Panel B), where the electromagnetic mode is confined between two metallic mirrors, or in plasmonic nano-cavities (Panel C) where few emitters and the electromagnetic field are confined in nano-metric gaps via plasmonic platforms. (Credit: No permission required.)

107 to 1012. Due to the large number of emitters accommodated by optical cavities, organic dyes can be efficiently exploited to achieve an incredibly strong interaction (0.5 eV) and bring the system up to the ultra-strong coupling regime. So far, polaritons involving molecules in optical cavities have been observed and exploited in two distinct frequency ranges: infrared and visible. The former is the realm of vibropolaritonic chemistry, which targets the manipulation of ground state molecular reactivity. Vibrational polaritons are obtained by coupling vibrational (IR-active) excitations of a large number of emitters to infrared cavity modes. This very active field is not covered in the current chapter, so we invite an interested reader to consult the recent reviews and perspectives in the field (9–11). Instead, we shall focus on the circumstance where cavity modes in the visible/UV strongly couple to electronic excitations, also known as electronic strong coupling (ESC). The use of optical cavities to influence photochemical events stems from the experiments by Ebbesen and collaborators in the early 2010s (12), where they demonstrated that the rate of photoisomerization reactions can be modified by confining light and molecules. The largely delocalized nature of these polaritons, i.e., the long-range interaction between molecules mediated by the electromagnetic field confined in the cavity, also inspired an intense research activity aimed at improving energy transfer (13–18), charge transfer (19–21) and processes between different spin multiplets in organic photoactive systems, like singlet fission (22,23) and triplet harvesting (24). The very same reason why optical cavities are very efficient for energy transfer processes is also the major challenge to its modeling: the electromagnetic mode is delocalized over all the cavity volume, allowing the interaction between 107 and 1010 excited molecules at very large distances. As such, the polaritonic states are delocalized over all the molecular ensemble (Fig. 7.1B), limiting the description to either effective excitonic models or to classical electromagnetism. A major breakthrough for the applicability of ESC in photochemistry, however, came by replacing the cavity’s metallic mirrors with plasmonic nanoparticles (25). The implementation of plasmonic cavities enables the confinement of the plasmon-enhanced electromagnetic field down to nano-metric volumes, ensuring an effective mode volume in the range of 10–100 nm3 (26,27), possibly down to 1 nm3 (28–30). According to Eq. (7.1), the decrease of the effective mode volume by several orders of magnitude opens up the possibility to achieve strong coupling with a very limited number of emitters, to the extent that ESC has been observed in one-to-few molecules at room temperature. The most alluring consequence is that polaritons can be used to modify and potentially control photochemical reactions in single-to-few molecules. However, the development of predictive and accurate theoretical methods for these systems faces several challenges on different footing depending on the system adopted. On the other hand, plasmonic nano-cavities face a completely different challenge: due to the extremely small volumes where one-to-few molecules are confined, the electromagnetic field in such systems is extremely inhomogeneous. The ideal model should then account for multiple quantized plasmonic modes coupled to a quantum chemical description of the molecules (31,32), including geometrical effects of the inhomogeneous electromagnetic field and quantum effects like electron tunneling from the nanoparticles to the molecules (Fig. 7.1C). Due to the typically large dimensions of the nanoparticles, a full quantum and atomistic description of such systems is not achievable, hence a lot of theoretical effort has been directed to develop methods where the plasmon modes are quantized, starting from their classical electromagnetic description (33–38). Given the introduction above, the modeling of polaritonic systems is inherently multiscale and multidisciplinary, as it requires an extremely extended collection of methods borrowed from quantum optics, nano-photonics, quantum electrodynamics, material sciences, and quantum chemistry. As such, an accurate theoretical description of each individual subcomponent of the system would be enough to fill a book per se. Thus, it is the role of the initial part of this chapter to discuss briefly the main approximations and assumptions which shall be kept in mind when approaching systems under ESC.

Polaritonic chemistry Chapter

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193

Bearing in mind the initial general considerations, we extend the theoretical (quantum chemistry) tools, typically adopted to simulate photochemical processes, to the case of ESC.

7.1 Preliminary considerations on the electromagnetic field We start by discussing the minimal number of concepts concerning the more exotic subcomponent of the system: the cavity. bEM) interacting with a system of charges (H bch) is directly borrowed The quantum description of the electromagnetic field (H from nonrelativistic quantum electrodynamics, commonly formulated as a coupled Hamiltonian in the Coulomb gauge: b ¼ Hbch + HbEM + Hbch H

EM

(7.2)

In free space and in the absence of external sources of radiation, the interaction (Hbch EM) represents the coupling between charges and the quantised electromagnetic field, initially in its vacuum state. The interaction is typically weak and accounts for the spontaneous emission of molecular excited states. The quantum chemistry strategy to describe the eigenstates and dynamics of this problem is to focus on the system of charges (molecules), while the spontaneous emission is either ignored (for short dynamics) or treated perturbatively (39). However, the electromagnetic field and its coupling with the system of charges become stronger and more complex upon confinement in cavities according to Eq. (7.1), so that the perturbative treatment is no longer effective. The role of cavities in this electromagnetic field enhancement is subtle and controversial: on one hand, the cavity can be described as an abstract tool which directly modifies the properties of the vacuum electromagnetic field. Within this conceptual framework, the cavity electromagnetic field is then described as if it is in free space, with the only difference that it is stronger and yields a stronger interaction. This description brings naturally to the inclusion of a dipolar self-energy term in Eq. (7.2) (40–42), resulting in the well-known Pauli-Fierz Hamiltonian. Such a term arises naturally when working with a truncated number of electromagnetic modes (43,44) and it is reminiscent of the electromagnetic description in vacuum. The quantization of the electromagnetic field modes in this framework relies on the free-space description, and it is treated in most quantum electrodynamics and quantum optics books (39,45). On the other hand, the cavities can be modeled as concrete systems of charges (metallic mirrors or plasmonic platforms), so that the cavity modes naturally originate from material excitations of the system. More precisely, a free-space electromagnetic field interacts with the charges of the medium constituting the cavity, resulting in medium-assisted electromagnetic modes. Within this picture, the mode distribution of the electromagnetic field does not resemble the free-space electromagnetic modes anymore, as it typically results in a series of interacting Lorentzian modes embedded in a flat continuum. This description accounts for the losses of the cavity, which can occur by nonradiative relaxation (described by the Lorentzian broadening) or by leakage of the cavity excitation to other radiative modes of the medium supporting the electromagnetic field propagation (described by the interaction between Lorentzians). In contrast to what happens in free-space, where the transverse electric field interacts with the molecules, the electromagnetic field propagating through a material system of charges gains a dominant longitudinal component in the near-field (34) associated to the polarization of the material system. In this description, the selfdipole term does not show up in the interaction as a consequence of the longitudinal nature of the electric field. Several methods in the literature are devoted to computing and quantizing the electromagnetic modes of such complex environments (27,31,46), so we point the interested reader to more thorough discussions on the topic (44). For the purpose of this chapter, we limit ourselves to acknowledging that—whichever cavity description is adopted—the quantized electromagnetic field associated with the cavity can be mapped to a set of bosonic modes (harmonic oscillators)   X { 1 bcav ¼ H ħok bbk bbk + (7.3) 2 k {

where bbk, bbk are the bosonic creation and annihilation operators, respectively. For the rest of the chapter, we will address the excitation of the quantized electromagnetic field as either “cavity mode” or “photonic mode” interchangeably. With this in mind, we write the most general Hamiltonian of a molecule (or an ensemble) coupled to a cavity: bpol ¼ H bmol + H bcav + H bint H

(7.4)

bmol is either a single molecule or many-molecules Hamiltonian, H bcav describes the excitation of the cavity, and H bint where H describes the interaction between the two subcomponents of the system. Such interaction is typically formulated in the dipolar approximation, although some recent works treat the light-matter interaction beyond such limit by resorting to multipolar expansion (47,48) or reformulating the interaction in terms of molecular transition density methods (31,32). Within the dipolar formulation, the light-molecule interaction Hamiltonian in the Coulomb gauge reads (39)

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bint ¼ H

N ph N em X X i¼1

k¼1

 {  b i
Ek bbk + bbk m

(7.5)

where the electric field strength for the k-th photon mode is defined in Eq. (7.1). The index i runs over the number of molecular emitters, where each intramolecular transition between two states n, n0 is described by the dipole operator N st P ^i ¼ m jn mi,nn0 n0 j. Passing now to a wave-function formalism, the time-evolution of the polaritonic wave function n, n0 ¼1 Cpol(r, R, q, t) is then governed by the polaritonic time-dependent Schr€odinger equation (TDSE) iħ

∂Cpol ðr, R, q, tÞ bpol Cpol ðr, R, q, tÞ ¼H ∂t

(7.6)

where r is the set of electronic coordinates, R collects the nuclear coordinates, and q denotes the explicit dependence of the polaritonic wave function on the photonic coordinates. To familiarize with the meaning of the photonic coordinates, we briefly summarize the procedure for canonically quantizing the EM field in free space (39,49,50). The idea behind the canonical quanb tization of the electromagnetic field comes from substituting the canonically conjugated magnetic and electric field operators B b with a different set of canonically conjugated variables, namely, the photonic momentum and position operators, pb and and E, k bEM) is then represented as a sum of k harmonic oscillators, expressed in terms of qbk. The electromagnetic field Hamiltonian (H their momentum and position operators pbk and qbk . The Hamiltonian associated to the field reads Z    X 2 b2 + E b 2 dr ¼ 1 bEM ¼ 1 H pbk + o2k qb2k , (7.7) B 8p 2 k

where the integral denotes the total electromagnetic energy in space expressed as a sum of the transverse magnetic and electric field operators, whereas the second term represents the same energy as a sum of harmonic oscillators. By expressing b pk in terms of creation and annihilation operators, qk and b qffiffiffiffiffiffiffiffiffiffiffi {   pffiffiffiffiffiffiffiffi { bqk ¼ 1= 2ok bbk + bbk , bpk ¼ i ok =2 bbk bbk (7.8)

we retrieve the simple Hamiltonian form of Eq (7.3). We can then see how the coordinates q assume the meaning of a proper photonic position, namely, the displacement of the harmonic oscillator with respect to its equilibrium position qk ¼ 0. Now that we introduced all the subcomponents of a polaritonic system; in the next section, we shall delve deeper into the hybrid light-matter nature of polaritons: in particular, we will examine the contributions of electrons, nuclei and photons to the polaritonic wavefunction, together with discussing the most common approximations to treat it.

7.2

Polaritonic eigenvalues and eigenstates

To introduce the conceptual toolbox to tackle polaritonic chemistry problems, we shall start by focusing on the timeindependent problem. In particular, we shall see how the eigenvalues and eigenstates of a polaritonic system differ with respect to the barely molecular ones. On a very general ground, the polaritonic problem is formulated as finding the eigenvalues Ei and eigenstates Ci(r, R, q) of a generalized correlated electron-nuclear-photon Hamiltonian (49), bpol ¼ Hbe ðr Þ + H bn ðRÞ + H ben ðr, RÞ + Hbp ðqÞ + H bep ðr, qÞ + H bnp ðR, qÞ + H benp ðr, RÞ, H

(7.9)

Hbpol ci ðr, R, qÞ ¼ Ei ci ðr, R, qÞ:

(7.10)

bmol ¼ Tbe ðr Þ + Tbn ðRÞ + W b nn ðRÞ + W b ee ðr Þ + W b en ðr, RÞ; H

(7.11)

by solving the time-independent polaritonic Schr€odinger equation

In Eq. (7.9), the subscripts, e, n, p, respectively denote the electronic, nuclear, and photonic contributions to the polaritonic Hambmol ¼ H be ðr Þ + H bn ðRÞ + H ben ðr, RÞ, the iltonian. By comparing Eqs. (7.4) and (7.9), we can identify the molecular contribution H bcav ¼ H bp ðqÞ, and the light-matter interaction term H bint ¼ H bep ðr, qÞ + Hbnp ðR, qÞ + H benp ðr, RÞ. cavity contribution H b The terms appearing in the molecular Hamiltonian are the standard kinetic energy Tb and Coulomb interaction W operators for each subcomponent of the system. the cavity Hamiltonian as a function of the photonic coordinates, Hbp ðqÞ, is defined in Eq. (7.7).

Polaritonic chemistry Chapter

The light-matter interaction terms are expressed in the dipolar approximation as X   bep ðr, qÞ ¼ b e ðr Þ , H ok qbk lk
m

7

195

(7.12)

k

Hbnp ðR, qÞ ¼

X k

bepn ðr, RÞ ¼ 1 H 2

b n ðRÞÞ, ok qbk ðlk
m

X k

b ðr, RÞÞ2 : ð lk
m

(7.13)

(7.14)

pffiffiffiffiffiffiffiffiffiffiffi In the above equations, lk ¼ 2=ok Ek is related to the electric field strength defined in Eq. (7.1). The dipole operators b n ðRÞ, and m b ðr, RÞ, respectively, denote the electronic, the nuclear, and the total dipole operators acting on the b e ðr Þ, m m molecule. The contribution in Eq. (7.14) is the so-called dipole self-energy. Making use of the argument from the preliminary discussion on the electromagnetic field, we will not include it for the rest of the chapter. Without any approximation, we can then expand the polaritonic eigenstate in two terms (49), evaluated at fixed sets of (R, q): the first is the mixed nuclear-photonic wavefunction wji(R, q), whereas the second term is a mixed electron-nuclear-photonic state Fj(r, R, q) X wji ðR, qÞFj ðr, R, qÞ: (7.15) Ci ðr, R, qÞ ¼ j

Such expansion is also an extension of the Born-Huang expansion (51) to the case of polaritons. We note here that the BornHuang expansion of the polaritonic wave function is not the only strategy: by adopting a time-dependent perspective, it is possible to resort to the exact factorization approach. Exact factorization has been recently proposed as an alternative and applied to solve polaritonic problems (52–55). However, for the purpose of this chapter, we shall opt for a more traditional formalism. To solve the eigenvalue problem (Eq. 7.10) with the Hamiltonian of Eq. (7.9), we are now left with two choices: the first one treats on the same footing the nuclear and photonic degrees of freedom as amplitudes wn+p ji (R, q), such that the states Fej (r; R, q) are purely electronic. The advantage of this representation is to leave the electronic states formally unaltered, as they just gain a dependence on additional (photonic) degrees of freedom on the same footing as the nuclear ones X n+p Ci ðr, R, qÞ ¼ wji ðR, qÞFej ðr; R, qÞ: (7.16) j

Conversely, the second one is to fix the set of nuclear coordinates R, while treating the photonic degrees of freedom on the same footing as the electronic ones Fe+p j (r, q; R), X Ci ðr, R, qÞ ¼ wnji ðRÞFje+p ðr, q; RÞ: (7.17) j

Here, the electronic states are mixed with the photonic ones, and they depend only parametrically on the nuclear coordinates (2,56–58). In both cases, we note that a semicolon (;) appears as a separator in the arguments of the two functions Fej (r; R, q) and Fe+p j (r, q; R). The semicolon denotes a parametric dependence of the function on the arguments following the (;) sign. In other words, the polaritonic wave function ci(r, R, q) can be obtained by either solving the purely electronic problem (Fej (r; R, q)) for each fixed couple of coordinates (R, q) or by solving the hybrid electronic + photonic problem (Fe+p j (r, q; R)) at each fixed nuclear position R. We stress that the two approaches are formally identical till approximations are introduced in order to describe the polaritonic systems. In fact, the two approaches entitle to very different approximations, by effect of which they serve as a natural ground for describing vibrational strong coupling and electronic strong coupling, respectively.

7.2.1 Cavity Born-Oppenheimer and vibrational strong coupling By substituting Eq. (7.16) into Eq. (7.10), we obtain a set of coupled equations: h i b0 e ðr; RÞ + H bep ðr, qÞ Fe ðr; R, qÞ ¼ V e ðR, qÞFe ðr; R, qÞ; H i i i h

∞ h i i X e bn ðRÞ + H bn ðRÞ + Tbp ðqÞjFe ðr ; R, qÞi wn+p ðR, qÞ bp ðqÞ + H bpn ðR, qÞ + V e ðR, qÞ wn+p ðR, qÞ + H hF ð r; R, q Þj T ij k j k ik

(7.18)

j¼1

n+p ¼ Ei wik ðR, qÞ:

(7.19)

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Eq. (7.18) is the electronic time-independent Schr€ odinger equation, and it is solved for each fixed couple of (R, q) which are b0 e ðr; RÞ ¼ H be ðr Þ + Hben ðr; RÞ. accounted as classical parameters. In Eq. (7.18), we define H Eq. (7.19) instead shows that the cavity modes are treated on the same footing as nuclear degrees of freedom. This is convenient when the energy associated with a cavity mode is comparable to the one of vibrational transitions (typically in the infrared), that is, when vibrational strong coupling conditions can be realized (59,60). Considering the energy of the cavity mode in the IR allows us to greatly reduce the complexity of the problem, by noting that the electronic transition and the cavity mode are strongly detuned. By observing Eq. (7.18) and Eq. (7.19), one realizes the similarity with the conditions leading to the Born-Oppenheimer approximation for calculating the electronic wave function and the adiabatic potential energy surfaces. Here, we identify the nonadiabatic coupling terms as the action of both the photonic and nuclear kinetic energy operators (Tbn ðRÞ+Tbp ðqÞ) on the purely electronic states. As usual, we consider the effect of the nuclear nonadiabatic couplings to be small when considering well-separated electronic potential energy surfaces. By similar considerations, we can also assume the photonic term to be negligible when the electronic states and photonic states are largely detuned. Neglecting the nonadiabatic couplings results in the polaritonic counterpart of the so-called Born-Oppenheimer approximation: the cavity Born-Oppenheimer (cBO) (49,61,62). As for its purely molecular counterpart, cBO allows to decouple the electronic motion with respect to the nuclear plus photonic one. In practice, it assumes that electrons are able to instantaneously adapt (i.e., adiabatically follow) to both the position of the nuclei and the values of the photonic coordinates (in turn related to the electric field). To visualize the physics entailed by the cBO approach, we consider a molecule (depicted in Fig. 7.2A) described by a Shin-Metiu model (63) in the presence of a cavity, realized after Refs. (7, 61). The molecular electronic ground state and first excited state assume the well-known double-well shape along the nuclear coordinate R, as depicted in Panel B of Fig. 7.2. The Hamiltonian associated with such system is ! 2 X pb2 X b q 2 0 k k b ¼ Tbn ðRÞ + H b e ðr; RÞ + b ðr; RÞ, H + ok (7.20) ok qbk lk
m + 2 2 k k

where the cavity Hamiltonian and the light-matter interaction term can be recast into the more familiar form of Eq. (7.5) by   pffiffiffiffiffiffiffiffiffiffiffi { pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi { considering that qb ¼ 1= 2o bb + bb , pb ¼ i o =2 bb bb and l ¼ 2=o E . k

k

k

k

k

k

k

k

k

k

k

Here, pbk and qbk denote the momentum and position operators associated with the k-th cavity mode, while lk takes into account the electric field strength (Eq. 7.1) and the polarization associated with the k-th cavity mode. The dipole operator b ðr; RÞcollects the electronic and nuclear dipole moments. We can promptly obtain the electronic potential energy surfaces m in the cBO framework (displayed in Fig. 7.2) by diagonalizing the new electronic Hamiltonian

a.

c.

b. Photonic Displacement +q

Bare molecule excited state

r R

Photonic Displacement +q

Bare molecule ground state

FIG. 7.2 Cavity Born Oppenheimer electronic potential energy surfaces. Panel A is a sketch of the Shin-Metiu model, where three nuclei (yellow—light gray color in print version) are displaced with respect to the center of mass by the nuclear coordinate R. The red (gray color in print version) sphere b and the cavity mode represents a single electron displaced by the center of mass position of r. The total (nuclei + electron) transition dipole moment m polarization l are directed along the longitudinal axis of the molecule. Panel B shows a 1D bare molecular potential along the nuclear coordinate, together with the projection of the curves obtained by increasing the photonic coordinate q in the cBO framework. Panel C shows the ground and first excited electronic potential energy surfaces in the cavity Born-Oppenheimer framework, where the nuclear and photonic degrees of freedom are treated alike. (Credit: No permission required.)

Polaritonic chemistry Chapter

b00

b H e ðr; R, qÞ ¼ H

X pb2 k 2 k

!

Tbn ,

7

197

(7.21)

b00 e ðr; R, qÞ on q, setting where we are also replacing each operator qbk with a number, due to the parametric dependence of H up the time-independent electronic Schr€ odinger equation (Eq. 7.12) b00 e ðr; R, qÞFe ðr; R, qÞ ¼ V e ðR, qÞFe ðr; R, qÞ: H i i i

(7.22)

In Fig. 7.2, it becomes evident how both the electronic ground state and the first excited state potential energy surfaces assume a (parametric) parabolic dependence with respect to the harmonic photonic coordinate. Conversely, it is not evident the contribution of the hybridization between nuclear + photonic sublevels included on different electronic surfaces. This is mostly a problem when investigating the photochemical properties of molecules, as they are heavily determined by nonadiabatic events between distinct electronic states. In this sense, the photochemical properties of polaritonic systems make no exception, as they are determined by nonadiabatic events between hybrid light-matter states (64). In addition to the typical cases where the Born-Oppenheimer approximation breaks down, i.e., when the electronic potential energy surfaces become degenerate, cases of failure of the cBO approximation have been studied close to light-induced conical intersections (64–69), namely, when the photon-electron nonadiabatic couplings become strong (62). We then understand why the cavity Born-Oppenheimer approach is best exploited to describe polaritonic reactions occurring on the same electronic state, namely, it is best suited for polaritonic reactions occurring in the electronic ground state. As such, cBO serves as a natural framework for the vibrational strong coupling (VSC) regime. In this regime, the (nuclear + photon) polaritonic dynamics takes place on a single high-dimensional PES, which is purely electronic and conceptually similar to out-ofcavity ground state reactions, as the one obtained for the Shin-Metiu model. An intense research activity aims at exploring the capability of cavities to influence and control ground-state reactions (70). To this aim, several techniques based on semiclassical (nuclear and photonic) descriptions have been devised (53,71,72), which have been shown to grasp the main physics of vibrational strong coupling. Yet, there is still discrepancy between the observed VSC phenomenology and the theoretical models, with many fundamental questions still opened (11,73,74). While VSC is an intriguing perspective to control chemistry, we shall focus on the photochemical properties of molecule under electronic strong coupling for the remainder of the chapter. However, we invite the interested reader to consult the recent discussions which cover thoroughly the VSC capabilities (4,9,10,75,76).

7.2.2 Electronic strong coupling (ESC) In the previous section, we discussed how factorizing the wave function according to Eq. (7.16) opens up the cBO framework, and how it serves as a natural framework for describing polaritonic reactions in the ground state, namely, in vibrational strong coupling. In this section, instead, we shall see how the factorization in Eq. (7.17) serves as a natural framework for electronic strong coupling, where the hybridization occurs between electronic and photonic degrees of freedom. After an initial effort to compute and interpret the polaritonic potential energy surfaces, we will realize that this second approach allows a straightforward extension of all the conceptual and computational tools adopted in photochemistry: the diabatic and adiabatic representation of the states, the conical intersections, the semiclassical and full-quantum propagation of the nuclear wave packet on potential energy surfaces. In analogy with the previous section, we begin by substituting Eq. (7.17) into Eq. (7.6) to get a set of coupled equations h i be ðr Þ + H ben ðr; RÞ + W b nn ðRÞ + H bp ðqÞ + H bep ðr, qÞ + Hbnp ðq; RÞ Fe+p ðr, q; RÞ ¼ V e+p ðRÞFe+p ðr, q; RÞ; H (7.23) i i i h

∞ h i i X e+p n bn ðRÞjFe+p ðr , q; RÞi wn ðRÞ ¼ Ei wn ðRÞ: Tbn ðRÞ + V e+p hF ð r, q; R Þj T ð R Þ w ð R Þ + j i ij ik ik k

(7.24)

j¼1

Eq. (7.23) is now the polaritonic time-independent Schr€odinger equation, and it is solved for each set of nuclear coordinates ben ðr; RÞand W b nn ðRÞbecome parametric (R), which are accounted as parameters. In contrast with Eq. (7.18), also the terms H in R, thus they are embedded in the polaritonic Hamiltonian for electronic strong coupling (Eq. 7.23). In contrast with the cBO framework, most of the complexity deriving by the inclusion of the photonic degrees of freedom is embedded in Eq. (7.23), whereas Eq. (7.24) becomes purely nuclear. We stress that this factorization is ideal when the photon can be best treated on the same footing as the electronic degrees of freedom, that is, when the cavity mode is resonant with an electronic excitation in the visible or in the UV. This implies that the vibrational transitions and the cavity excitation

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are strongly detuned. The polaritonic states and energies can then be obtained as solutions of the time-independent polaritonic Schr€ odinger equation in a more compact form h i b0 e ðr; RÞ + H bp ðqÞ + H bint ðr , q; RÞ Fe+p ðr, q; RÞ ¼ V e+p ðRÞFe+p ðr, q; RÞ, H (7.25) i i i

b0 e ðr; RÞ ¼ H be ðr Þ + Hben ðr; RÞ + W b nn ðRÞ, while the light-molecule interaction is where the new electronic Hamiltonian is H b b b H int ðr, q; RÞ ¼ H ep ðr, qÞ + Hnp ðq; RÞ. The interpretation of the coupled Hamiltonian in Eq. (7.25) becomes clear by anabint ðr, q; RÞ. Starting with the electronic part H b0 e , several lyzing each subcomponent of the system: Hb0 e ðr Þ, Hbp ðqÞ and H

methods to compute electronic excited states and energies were presented in other chapters. Therefore, we can assume that we have already solved the time-independent electronic problem for each set of fixed nuclear coordinates R, b0 e ðr; RÞFe ðr; RÞ ¼ U n ðRÞYe ðr; RÞ, H n n

(7.26)

Yen(r; R)

and obtained a set of n electronically adiabatic states and their associated electronic energies Un(R). As a cavity Hamiltonian, we make use of the compact form as a sum of harmonic oscillators, as in Eq. (7.3). By solving the eigenvalue equation for the cavity modes, !     2 2 { b b 1 1 p q p p p 2 k k b b b H p ðqÞYk ðqÞ ¼ + ok (7.27) Yk ðqÞ ¼ ħok f k + Ypk ðqÞ, Yk ðqÞ ¼ ħok bk bk + 2 2 2 2

we obtain a set of eigenstates of cavity eigenstates Ypk (q). Here, the eigenvalues are composed by multiples of the mode frequency ok, which is multiplied by the occupation number of the photon mode k. We note that the cavity spans a Hilbert space of k modes, each represented by a Fock state with occupation number fk. We already see that the dimension of the problem for many modes can become computationally cumbersome even if we were to treat only the photonic states. Luckily, most of plasmonic nanocavities’ resonances can be decently approximated by single-to-few pseudomodes (37,77,78). Other possibilities to effectively represent the quantized electromagnetic field with a limited number of modes are available in literature. Such approaches are based on different refinements of the Huttner-Barnett model (79), where the molecule directly interacts with a single effective mode, which in turn is embedded in a flat thermal bath. The effect of the bath is to add an effective decay rate to the electromagnetic excitation (33–35,44,46,80). Considering the noninteracting b0 e ðr; RÞ + H bp ðqÞ, we build the basis of uncoupled states as the tensor product between the eleclight-matter Hamiltonian H tronic states space and the cavity space: Ye+p ðr, q; RÞ ¼ Ye ðr; RÞ Yp ðqÞ:

(7.28)

As we shall see in the example presented later on, the uncoupled basis is particularly insightful to interpret the polaritonic phenomenology, as the projection of the polaritonic states C(r, q; R) on the basis Ye+p(r, q; R) carries the information whether, at a given nuclear coordinate, the energy associated to a polaritonic excitation is mostly stored in the molecule or as a free photon in the cavity. In the followings, we shall indicate the states represented by Ye+p(r, q; R) as {j n, Fi}, where n denotes the electronic state n ¼ {S0, S1, …, Sn}, while F collects the photonic number states associated to the k-th cavity mode F ¼ {j f0i j fi … j fki}. The most common approximation in polaritonic chemistry is to truncate the cavity occupation to one for each photonic mode, since electronic strong coupling occurs by exchange of a single photon between molecule and cavity. Very few works go beyond this assumption; however, the description does not seem to largely improve by including multiple occupations of the cavity modes for what concerns the study of photochemical reactivity. The light-matter interaction is defined in its dipolar form in Eqs. (7.12) and (7.13), and by making use of the definitions of Eq. (7.8), we can recast it in the more compact form:  {  X bint ðr, q; RÞ ¼ b ðr; RÞ bbk + bbk : H Ek
m (7.29) k

Eq. (7.29) encloses the essence of electronic strong coupling: we have reduced the polaritonic problem to computing the interaction between electronic states and the cavity modes. Despite its apparently compact form, the operator in Eq. (7.29) embodies distinct physical effects:

l

The most relevant is by far the creation/destruction of the k-th photonic mode while D the opposite process E occurs between 0 0 0 b b jn , f k + 1 (with n > n0 ), two distinct electronic states n and n (n 6¼ n ). Put differently, the terms n, f k jbk m D E { b jn0 , f k 1 (with n < n0 ), respectively, describe the molecular absorption and emission of a single photon. n, f k jbbk m As already extensively discussed in the introduction of the present chapter, such rate is largely enhanced in cavities. The

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absorbed/emitted photon is taken by/given to the cavity mode fk and the maximum effect is achieved when the cavity frequency matches the difference in energy between the electronic states n and n0 . We can then interpret the electronic strong coupling and the hybridization between electronic and photonic degrees of freedom as an ultrafast exchange of photon between the molecules and the cavity. This interaction in the strong coupling regime typically results in the splitting between the energies of polaritonic states, commonly named Rabi splitting. D E D E { 1 can be understood as the modification of an The terms where n ¼ n0 , that is, m f jbb jf + 1 and m f jbb jf nn

k

k

k

nn

k

k

k

electronic state energy upon removal/addition of a photon to the molecular system. We stress that in the present case, the photon is not absorbed nor emitted, whereas it can be thought as an electrostatic environment in which the molecule is embedded. In this sense, such terms represent the polarization of the photon field due to the presence of the molecule. These terms are usually disregarded or considered constant and the same for each electronic state along the reactive coordinate. However, when a substantial change of configuration is involved in a reaction (e.g., in photoisomerizations), this is not necessarily a good approximation, as this contribution is expected to largely differ between different electronic states and different values of the reaction coordinates. The creation/destruction of the k-th photonic mode while the same process occurs for electronic states. More practically, it is a simultaneous creation or annihilation of both the electronic excitation and the cavity mode, respectively D E D E { b jn0 , f b jn0 , f + 1 (with n < n0 ) and n, f jbb m 1 (with n > n0 ). These terms are typically named “counter n, f jbb m k

k

k

k

k

k

rotating terms” after the “rotating wave approximation,” which is the approximation which allows to discard them due to their minor contribution. We note here that the counter rotating terms can become very relevant in the ultra-strong coupling regime (81–84) and when the electronic description of the molecules includes more than two states (58).

All of the effects described above spur from the effect of the photonic field on the electronic degrees of freedom. As such, we have shown how the second factorization serves as a natural framework for electronic strong coupling. In the current treatment, we fully focused on solving the polaritonic time-independent Schr€odinger equation. After having computed the polaritonic states, we are left to solve The only exotic D the nuclear problem. E terms we encounter in Eq. (7.24) are the polari b e+p e+p tonic nonadiabatic couplings Gkj ¼ F ðr, q; RÞ T n ðRÞ F ðr, qjRÞ . By treating the photonic degrees of freedom on the k

j

same footing as the electronic ones, the evaluation of nonadiabatic couplings is reduced to evaluating the effect of the nuclear kinetic energy operator on the polaritonic states. As in the typical Born-Oppenheimer approximation for electronic states, nonadiabatic couplings are negligible when the energy separation between the electronic states is sufficiently large. On the same ground, the nonadiabatic couplings can be safely neglected when the energy separation between polaritonic states (Rabi splitting) is sufficiently large. However, in most systems, the lower bound of Rabi splitting is few tens of meV, which is comparable to the kinetic energy stored in nuclear coordinates. As such, the magnitude of nonadiabatic couplings becomes extremely important in determining the dynamics of the system. The next section will put to use the general concepts explained in this section. By building the polaritonic quantities for a specific molecule, we shall discuss some of the most iconic features of photochemical reactions in the polaritonic context. Through the example, we hope to grant the reader with a more visual and practical intuition about how ESC influences the photochemical properties of molecules.

7.3 Polaritonic potential energy surfaces (PoPESs) 7.3.1 A didactical case: Azobenzene polaritonic potential energy surfaces (PoPESs) For illustrative purposes, we shall restrict our example by assuming all of the approximations discussed previously. More specifically, we will consider a single-cavity mode, neglect the polarization terms, and adopt a dipolar interaction in the rotating wave approximation. We consider an azobenzene molecule in a cavity, depicted in Fig. 7.3A. Azobenzene is a prototypical system for photochemistry, as it presents all the features of a complex trans-cis photoisomerization reaction despite its reduced dimensionality. The trans-cis photoisomerization of azobenzene occurs upon UV irradiation, whereas the backward reaction is triggered either via visible light irradiation or thermally, since it corresponds to the higher-energy (i.e., less stable) isomer. The photoisomerization reaction involves chiefly two reaction coordinates: the main coordinate, i.e., the torsion of the CNNC dihedral around the NdN bond, and a secondary coordinate, i.e., the pyramidalization of the NNC angles (85). Along the reactive coordinate (at CNNC ¼ 90 degrees), azobenzene features an electronic conical intersection between the states S1 and S0, which is responsible for its fast nonradiative relaxation from S1 to S0 (86–88). The detailed description of the conical intersection shape is of the utmost importance, as minor modifications can severely affect the reaction quantum yields of both the trans-cis and cis-trans reactions. The nonradiative pathway is then the main relaxation mechanism in azobenzene, as it occurs on a timescale of few hundreds of femtoseconds

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a.

b.

Vis,

Trans

c.

Cis

d.

Cavity Mode Polarisation

UV

|S2 , 1 |S2 , 0 |S1 , 1 |S1 , 0

|S0 , 1

|S0 , 0

e.

Cav

Mol FIG. 7.3 Polaritonic potential energy surfaces in the azobenzene molecule. Panel A shows schematically the two azobenzene isomers in a cavity (atom color code: white ¼ hydrogen, light blue (light gray color in print version) ¼ carbon, dark blue (gray color in print version) ¼ nitrogen). The polarization along the cavity mode is directed along the longitudinal axis of the molecule. In panel B, we display the uncoupled states of the cavity + molecule system, as in Eq. (7.28). The electronic states of the bare molecule are reported in blue (gray color in print version) and labeled as |n0 , 0i states, whereas the cavity modes are colored in orange (light gray color in print version) and labeled as |n0 , 1i. Here, n and n0 denote the singlet electronic states S0, S1 and S2. The electronic conical intersection is located at R ¼ 90 degrees (R: CNNC torsion angle). Panel C shows the polaritonic states, obtained after diagonalizing the polaritonic Hamiltonian in Eq. (7.25). The molecular transition dipole varies along the nuclear coordinate (so does the strength of the light-matter interaction), as indicated by the purple lines in panels B and C. Panels D and E, respectively, show the zoom on the two regions of the PoPESs: the cis-azobenzene (B) is under strong coupling, where the signature polaritonic avoided crossing (Rabi splitting) is shown; the trans-azobenzene (E) is a region of zero coupling (LiCI), where the two PoPESs become degenerate and the two PoPESs become indistinguishable with respect to the uncoupled basis, as described by a light-induced (or polaritonic) conical intersection (LiCI). (Credit: No permission required.)

(85). As a starting point, let us assume that we already diagonalized its electronic Hamiltonian as in Eq. (7.26) along the reactive coordinate; therefore, we obtained the electronic singlet states of the bare molecule, depicted in Fig. 7.3. To simplify the molecular description and to focus on the polaritonic effects, the electronic states are shown along the CNNC torsional coordinate: the left basin of Fig. 7.3B corresponds to the cis configuration, whereas the right basin is the trans. The purely electronic states are indicated, according to the notation defined in this section, as n ¼ {S0, S1, …, Sn}. Since we are working in the uncoupled basis, which is a tensor product between the electronic and photonic Hilbert spaces, the bare molecular states are associated to a cavity photon occupation number of zero. Through the inclusion of the cavity states with a photon occupation number p ¼ 1 in the uncoupled representation, we obtain a replica of the bare electronic states shifted by the cavity frequency ok. These states are displayed in Fig. 7.3B as orange lines (light gray color in print version), and they are labeled as |n0 , 1i. By resorting to the uncoupled representation, we can easily identify in which region of the energy landscape the energy is expected to be stored as electronic energy (blue—gray color in print version) or as free photon in the cavity (orange—light gray color in print version). We can recall the textbook definition of diabatic electronic states, where two states of the same symmetry but different electronic configuration cross at some nuclear coordinate. In this sense, we can identify the uncoupled states in the light-molecule space as the diabatic representation of polaritonic states. To visualize all the features of electronic strong coupling, we include the interaction between the uncoupled states by considering a transition dipole moment as a sigmoid function of the nuclear coordinates (for realistic simulations, the transition dipole moments should be properly calculated by the electronic states). Such function is displayed as the purple dotted line in Fig. 7.3B and C. Following the same philosophy adopted so far, we aim at providing an interpretative framework for electronic strong coupling. To this aim, we sacrifice the accuracy of the molecular description and assume the same transition dipole function for each couple of electronic states mS0S1(R) ¼ mS1S2(R) ¼ mS0S2(R), and discard the permanent dipole moments. This

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is obviously not a faithful description of a realistic molecule. By diagonalizing the electronic strong coupling Hamiltonian in Eq. (7.25), we obtain the polaritonic potential energy surfaces (PoPESs). From a visual inspection of the curves, it becomes immediately apparent how much the PoPESs differ with respect to purely electronic ones. Several considerations can be made by coloring the PoPESs according to the projection of each polaritonic state on the uncoupled basis. In particular, we shall assign to the regions of the PoPESs: the blue color (gray color in print version) where the dominant contribution of the polariton is molecular, the orange color (light gray color in print version) where the dominant contribution is the cavity mode, and gray where the cavity and the electronic state are strongly mixed. This representation is helpful to understand the polaritonic states as the adiabatic analogue of the uncoupled states basis. As for the electronic adiabatic states, the polaritons manifest a distinctive avoided crossing (at CNNC ¼ 50 degrees, Fig. 7.3D) where the interaction between the uncoupled states (diabatic analogue of the electronic states) becomes strong enough. Proceeding along the torsional coordinate toward the trans basin (high CNNC values), the transition dipole moment between the electronic states approaches to zero. Consequently, the Rabi splitting is drastically reduced, to such an extent that the PoPESs present almost no difference with respect to the uncoupled basis. As a consequence of this reduction, the PoPESs approach each other and the avoided crossing results in a light-induced crossing seam. Depending on the interaction’s magnitude profile along the nuclear coordinates, the polaritonic crossing seam can assume the shape of a singular point and be thought as the polaritonic analogue to the conical intersections, Fig. 7.3E. This feature is known in literature as light-induced conical intersection (64,68) (LiCI) or polaritonic conical intersection (3,69), and they are also observed when realistic molecular calculations are performed (57,62,66,89). In analogy to the case where the Born-Oppenheimer approximation breaks down in the purely electronic states, the polaritonic nonadiabatic coupling terms contribution becomes dominant along the crossing seams, meaning that they need to be taken into account to describe the dynamics of the system. Additional qualitative considerations on the dynamics can be inferred from the PoPESs in Fig. 7.3C. In particular, simulating the full dynamics of the system means to study the motion of an excited nuclear wave packet traveling on the PoPESs. To make such considerations, we shall recall that the shape of the PESs yields the forces acting on the nuclei, and the quantum yield associated with a photochemical reaction is strongly influenced by the shape of such PESs pathway and by nonadiabatic events. By making use of the example adopted so far, in Fig. 7.4A, we show a sketch of the motion of the nuclear wave packet (WP) associated with the cis-trans isomerization of azobenzene in absence of strong coupling. The quantum yield of the FIG. 7.4 Nuclear wavepacket dynamics on polaritonic potential energy surfaces. Panel A shows the nuclear wave packet dynamics at an electronic conical intersection on purely electronic potential energy surfaces. Panels B–D represent the three possibilities for a nuclear dynamics occurring at polaritonic crossing seams. In particular, panel B shows the case polaritonic adiabatic nuclear dynamics achieved when the lightmolecule coupling is extremely strong; panel C shows the case of a polaritonic diabatic dynamics, where the nuclear dynamics on purely electronic surfaces is retrieved; panel D shows the intermediate case, where the light-molecule coupling originates different branchings of the nuclear wave packet, giving rise to polaritonic chemistry phenomena. (Credit: No permission required.)

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reaction is determined by the branching probability of the WP at the electronic conical intersection (eCoIn) (90). This branching occurs according to the magnitude and orientation of the nonadiabatic couplings vector. From the slope of the electronic potential energy surfaces in the cis and trans basins, we can already qualitatively predict that the cis-trans yield will be greater than the trans-cis one, since the slope of the PESs yields information on the kinetic energy accumulated in the nuclear degrees of freedom. As such, the higher cis-trans slope along the CNNC torsion suggests that the WP would approach the electronic conical intersection with larger momentum (corresponding to a nonadiabatic coupling vector pointing toward the trans basin), associated with a greater probability of the WP to cross the eCoIN toward the trans basin. As for its electronic counterpart, the dynamics on polaritonic states can be thought as the nuclear wave-packet traveling on polaritonic potential energy surfaces. In analogy to the purely electronic case for a bare molecule, the quantum yields of a polaritonic reaction is governed by the shape of the PoPESs and by the nonadiabatic events. By tailoring the shape of the potential energy surfaces with strong coupling, we can then understand the importance of polaritonic chemistry in controlling photochemical processes. However, we shall remark that, in the polaritonic case, the nuclear wave-packet traveling along a polaritonic potential energy surfaces entails a more complex phenomenology with respect to the purely electronic case. Such complexity can be summarized in three distinct behaviors at polaritonic crossing seams, respectively, sketched in Fig. 7.4: Fully adiabatic (polaritonic): when the Rabi splitting is strong enough, the nuclear motion becomes decoupled with respect to the polaritonic states. As such, a WP traveling on the polaritonic state Fig. 7.4B repeatedly bumps back and forth between electronic and photonic regions. As such, the population of the polaritonic state remains constant; however, the population of the uncoupled states oscillates between purely molecular and purely photonic case. This regime is often associated to suppression of photochemical reactions, while it shines for applications where only the energetics of the electronic states is relevant (the nuclear motion does not induce transitions between adiabatic states) and many molecules are involved, e.g., for energy transfer applications. Fully diabatic (uncoupled): it is indeed the most intuitive case. When the light-molecule coupling approaches zero at a crossing seam, it means that the molecular dynamics is weakly perturbed by the presence of the cavity mode. As such, we expect the wave packet to fully follow the molecular uncoupled states. Since we work in the polaritonic basis, in this case, the polaritonic nonadiabatic coupling vector points toward the molecular branch with high intensity, promoting a smooth propagation of the WP along the purely electronic state (Fig, 7.4C). The dynamics in this case can be interpreted as a molecule driven by a weak electromagnetic field. This is the field of traditional photochemistry. Intermediate: here, the Rabi splitting is somewhat comparable to the nuclear kinetic energy along some excited coordinate (order of few tens of meV). This particular regime is where the electronic strong coupling exhibits its most exotic properties at the level of photochemistry for single-to-few molecules. The delicate interplay between nuclear dynamics, polaritonic behavior, and light-induced nonadiabatic effects allows multiple branchings of the WP (Fig. 7.4D). As a consequence, new reaction pathways may open to achieve effects such enhanced photoisomerization or photoprotection. By discussing the PoPESs composition and the nonadiabatic couplings in the time-independent framework, we have somehow exhausted the qualitative information about the capability of electronic strong coupling to act on single-tofew molecules (nano-cavities). Additional insight can be granted by switching to a time-dependent perspective, where the cavity will be included. Yet, before switching to such perspective, the time-independent perspective can still yield information on the capability of strong coupling to influence the reactivity in molecular ensembles.

7.3.2 Many molecules and dark states The case of many molecules largely differs from the treatment of a single molecule, as the polaritonic excitation involves an electronic collective effect. The collectivity arises from a large spatial delocalization of the electromagnetic field in the cavity, by the effect of which (approximately) the cavity state interacts with all the molecules. To represent such systems, we shall switch to an excitonic representation. We consider an ensemble of Nmol identical azobenzene molecules randomly oriented in space, diluted enough to neglect their direct dipole-dipole interactions. Although this approximation is quite common in the polaritonic literature, the purpose of this approximation is merely didactical: by neglecting the direct dipoledipole interactions, we are able to observe that the electromagnetic modes can mediate long-range interactions between (otherwise noninteracting) molecules. Such interaction occurs via the exchange of a photon between molecules and the cavity mode. Within our approximation, the excitonic wave function is represented as a tensor product between the Hilbert spaces spanned by each molecular eigenvector:

(7.30) Cexc ðr; RÞ ¼ijn1 i jn2 i … nNmol

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The Hamiltonian associated with the noninteracting molecular ensemble is the simple sum of the electronic Hamiltonians of the individual molecules, conveniently written on the basis of the excited states jnii for each molecule, where we consider the state where each molecule is in its equilibrium configuration in the trans ground state as the zero in energy. We write the molecular excitonic Hamiltonian as XNmol XNst

 n E ðR Þ n , bexc ðr; RÞ ¼ H (7.31) i,n i i i i¼1 n¼1

where the first sum runs over n, i.e., the number of the excited state included for each molecule and i runs over the number of molecules. We note that we did not drop the dependence of the molecular electronic energy from the nuclear coordinates of the i-th molecule. We consider the intermolecular distances in the ensemble to be large enough to neglect their direct dipoledipole interactions. As a result, the potential energy surfaces associated with the Nmol noninteracting molecules are simply the Nmol-times degenerate electronic PESs of azobenzene (blue lines—gray color in print version in Fig. 7.3B). Similarly, the excited cavity + molecule PESs obtained by building the uncoupled Hamiltonian are Nmol-fold degenerate (orange lines—light gray color in print version in Fig. 7.3). As discussed before, we consider the cavity mode as fully delocalized, meaning that it interacts independently with each randomly oriented molecule, and for the sake of simplicity, we assume the field associated to the mode to be position-independent. While the form of the cavity Hamiltonian is the same as in Eq. (7.3) (although we remind the reader that we are considering a single mode), we can adapt the interaction between such mode and the ensemble of molecules as   XNmol XNst

n E
m 0 ðR Þhn0 | bb{ + bb , bint ðr, q; RÞ ¼ H (7.32) i i i,nn i 0 i¼1 n,n ¼1 The results of the diagonalization of the polaritonic Hamiltonian is shown in Fig. 7.5, respectively, for a large value (Panel A) and small value (Panel B) of the interaction. In both cases, we can clearly identify two distinct set of states: the two polaritons (thick lines), which correspond to the two bright linear combinations of the molecules, and N 1 quasi-dark states (thin lines), which are homogeneously broadened due to the scalar product between the electromagnetic field and the randomly oriented transition dipole. The two bright polaritons are formed through the collective interaction of all molecules promoted by the cavity field: despite the fact that the starting point is to assume Nmol noninteracting molecules, we note that the molecular degeneracy is lifted by the effect of the interaction with the cavity mode. In other words, the cavity mode allows all the molecules under its influence to interact, forming collective molecular excitations. The range at which this interaction occurs is of the order of the (typically micrometric) cavity length, meaning that molecules which are distant in space can still interact via the cavity field. This idea has led strong coupling to become popular for long-range applications, such as remote chemistry and long-range energy transfer. In addition, this very same idea nourishes the perspective to realize organic optical devices. However, the presence of the dark state manifold is often detrimental for polaritonic chemistry. Indeed, their presence is the main reason why many realizations fail at modifying chemical reactivity with strong coupling.

a.

b.

FIG. 7.5 Polaritonic states in molecular ensembles: the problem of the dark states. Panel A shows the polaritonic avoided crossing (thin line) in the case of Nmol molecules randomly oriented in space. The nuclear coordinate R denotes a collective coordinate, where the geometry of each molecule is changed. The results of Panel A refer to a case of a strong collective interaction, which produces a Rabi splitting larger than the homogeneous broadening of the quasi-dark states. The limit of a weak collective interaction in the same system is depicted in Panel B, where the dark state broadening embeds the polaritonic states. (Credit: No permission required.)

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The dark states mostly correspond to a localization of the excitation on a single molecule, meaning that the linewidth of the dark states can be roughly approximated as the linewidth of the bare molecular absorption, which is typically subject to inhomogeneous broadening due to the interactions with the environment. When driving polaritonic states with external light, a collective excitation between molecules is created, and the bright polaritonic states are populated. If the polaritonic bands are not overlapped with the manifold of dark states (Fig. 7.5A), the system exhibits a purely polaritonic behavior, and the dynamics of the system is governed by collective polaritons. This condition is the one sought to influence photochemistry with cavities. On the contrary, when the dark states embed the two polaritons (Rabi splitting smaller than the bare molecular absorption linewidth Fig. 7.5B), a direct channel opens for the polaritonic excitation to collapse on the electronic excitations localized on single molecules. As a consequence, the collective behavior excitation is lost with a lifetime inversely proportional to the overlap between polaritons and dark states. In this case, the excited system evolves in time as an individual excited molecule. As a rule of thumb, it is then convenient to compare the Rabi splitting to the absorption line width of the bare molecules. In more realistic simulations where also inhomogeneous broadening is taken into account, this comparison provides a qualitative estimate to know a priori whether polaritons could be able to modify the reactivity of a system. While the lack of overlap between polaritons and dark states is a necessary condition to polaritonic reactivity (91), we note that it is not a sufficient condition, as many other decay channels can cause the collective excitation to decohere (decoherence to dark states, cavity radiative and nonradiative losses, nonradiative relaxations).

7.4

Polariton dynamics and cavity losses

Many formulations of the time-dependent problem are discussed in literature, and they range from semiclassical descriptions to full-quantum treatments. In the former case, the choice of which degrees of freedom to treat classically is arbitrary, and different formulations of the polaritonic problem lead to methods which treat classically either the nuclear degrees of freedom, the photonic degrees of freedom or a mix of them. The major problem of classical photonic treatments is the inclusion of the cavity losses, i.e., this second approach is best suited to describe optical cavities, whose lifetime is on the order of several picoseconds. Conversely, the plasmonic nano-cavities typically operate at visible frequencies and their lifetime is of the order of tens of femtoseconds. In this section, we will note that working in the Born-Huang framework allows a straightforward implementation of the most common nonadiabatic dynamics techniques. Until now, master equation approaches, on-the-fly methods and PESfitting methods have been adapted to polaritonic systems. Among them, we mention the polaritonic versions of surface hopping algorithms like Tully’s Fewest Switches (FSSH) (92) and Direct Trajectories (57,58,89) (DTSH), multitrajectory Ehrenfest dynamics (93), multiconfigurational time-dependent Hartree (94,95) (MTDH), Finite-Elements methods coupled to a Lindblad master equation (96,97), and ring-polymer molecular dynamics (98). The nonadiabatic dynamics methods applied to the polaritonic case inherit the same advantages and disadvantages as their electronic counterpart. For a complete review of such techniques, we point the interested reader to additional material (99–101), specifying that the strengths and weaknesses of each method for electronic states are directly transferable to the polaritonic case.

7.4.1 Nuclear dynamics in polaritonic systems: Full quantum vs semiclassical The solution of the polaritonic time-dependent Schr€odinger equation (Eq. 7.6) yields the full dynamics of the polaritonic system. On a par with the time-independent case, the factorization of the polaritonic wave function is analogous to the molecular case (100,102,103), with the additional problem of the embedding of photonic coordinates. As such, the time-dependent polaritonic wave function (Cpol(r, R, q, t)) can either be factorised in the Born-Huang picture (described in the previous sections) or in an exact factorization picture (XF) (55). A thorough discussion on the XF approach is out of focus with respect to the current chapter, however, we remark that different methods building on XF for both full-quantum and semiclassical simulations of polaritonic systems have been developed (52–55). For the remainder of the chapter, we will adopt the Born-Huang framework, specifically the factorization (Eq. 7.11) which leads to the electronic strong coupling, X cpol (7.33) wji ðR, tÞFe+p i ðr, q, R, tÞ ¼ j ðr, q; RÞ j

By substituting Eq. (7.33) in the polaritonic time-dependent Schr€odinger equation (Eq. 7.6), one has access to the polaritonic wave function in the full-quantum framework,

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 ∂wji ðR, tÞ X ¼ Tbn + H pol iħ kj ∂t k







b e+p Fe+p k T n F j



7



 ∂ e+p ∂R F w ðR, tÞ: ∂R j ∂t ki

Fe+p k

205

(7.34)

bpol ¼ H be + H bp + H be+p and its respective eigenvectors For compactness, we define the polaritonic Hamiltonian as H e+p ci (r, q; R), according to Eq. (7.25). We also dropped the explicit dependence of the operators on the photonics and electronic degreesE of freedom. In the time-dependent formulation, two nonadiabatic terms emerge: the first one, D e+p b F jT n jFe+p , is reminiscent of the time independent formulation (Eq. 7.18), whereas the second one emerges as a conk

j

sequence of the time-dependent problem. In particular, the term



e+p ∂ e+p ∂R e+p ∂ e+p ¼ Fk Fj Fk Fj , ∂R ∂t ∂t

(7.35)

entirely depends on the time-evolution of the polaritonic wave function. The full-quantum methods are extremely accurate in simulating the full dynamics of the system; however, their most accurate formulations (i.e., QEDFT (42,50,104), XF (52,54,55), QED-CC (105,106)) typically include explicitly the cavity coordinate (q). There, the photon is included as a displacement coordinate, and not as Fock state. This representation of the photons suffers a strong drawback, as the inclusion of cavity losses becomes a technical limitation. Luckily, some full-quantum methods adopt a Fock state description of the cavity: such methods (FEDVR (77), MCTDH (94), Table 7.1a) rely on precomputing the PoPESs along a selected number of degrees of freedom, as the inclusion of all the degrees of freedom is computationally not achievable. As a second step, they represent the nuclear wave function dynamics on the PoPESs as a set of basis functions on a grid. The main advantage of these formulations is that they are quantitative even for the case of extremely complex potential energy landscapes (several conical intersections and polaritonic states). However, their applicability is severely limited by three factors:

l

As the reactive degrees of freedom need to be preselected, they build on an extensive preexisting knowledge of the system. As such, they often fail at identifying new reactive pathways along other unexpected nuclear coordinates.

TABLE 7.1 Nonadiabatic dynamics. Method

a. Quantum wavepacket dynamics (QWD):

Strengths

Weaknesses

Most suited for

Quantitative on complex surfaces

Few pre selected degrees of freedom

Small known molecules, few diatomic molecules in model cavities

Quantum e ects (tunneling)

Di cult inclusion of the environment

Exact lossy dynamics (Lindblad, non-Hermitian)

Extensive computational power

Large number of degrees of freedom

Fails for many degenerate states (dark states)

QM/MM interface for the chemical environment

Requires ad-hoc corrections for quantum e ects

Stochastic losses (quantum jump)

Speci c implementation for each algorithm

Collective e ects are well-described

Sacri ces the local description of the mechanism

QM/MM interface for the chemical environment

The mean- eld can be unphysical

Inclusion of average deexcitation channels

Only qualitative description of collective e ects

MCTDH, FEDVR

b. Trajectory methods: Surface-hopping algorithms

c. Mean-field dynamics: Ehrenfest algorithms

Single molecules embedded in complex environments

Large ensembles of molecules in optical cavities

Summary of the most common nonadiabatic dynamic techniques building on the Born-Huang factorization (100). The different rows present the main strengths and weaknesses for each different class of algorithms, together with suggesting the most suitable application to polaritonic systems. (Credit: No permission required.)

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The effect of the chemical environment (solvent, other species in solution) is typically disregarded, as it is difficult to include in precomputed quantities such as the PoPESs. They are suitable to investigate an extremely limited number of degrees of freedom; hence, they are suitable for small molecules or few diatomic molecules at the best.

On the bright side, grid methods allow to include the cavity losses as a non-Hermitian formalism or via a Lindblad master equation. In this framework, the losses are included based on the assumption that the cavity mode is weakly coupled to a Markovian bath. Following the standard open-quantum system theory, the time-evolution of the polaritonic density matrix is accounted through the Lindblad master equation h i   ∂r b bb{ g bb{ br b + rbb{ bb , b r + gbr ¼ i H, (7.36) ∂t 2

b bb{ are the usual annihilation and creation operators for the cavity mode. The where g denotes the photon decay rate and b, b bb{ is the one responsible for the losses, namely, its effect is to induce an incoherent transition |n, pi ! | n, p 1i in term br the uncoupled representation of the system. Such transition corresponds to the loss of a single photon, and it occurs with a probability gjCfj2 (where Cf is the total photon contribution to the polaritonic wave function). Once more, we can then estimate whether the dynamics is governed by photon losses effects by inspecting the PoPESs in Fig. 7.4: when the wavepacket travels in the photonic areas of the PoPESs, the probability that the photon is lost from the system is maximized. As they both rely on exact formalisms, quantum wave-packet dynamics and open-quantum system techniques match well for the inclusion of lossy channels. As such, they provide robust benchmark for other approximate semiclassical methods, which provide the opportunity to avoid the time propagation of the full-nuclear wave function. To take the semiclassical limit of Eq. (7.34) means to consider classical nuclei evolving in the quantum potential of the polaritonic states. As a consequence, the time dependence of the polaritonic wave function is entirely embedded in the expansion coefficients Cj (which no longer represent the nuclear wave functions wji(R, t) of Eq. 7.33), X ð r, q, R, t Þ ¼ Cj ðR, tÞFje+p ðr, q; RÞ: (7.37) cpol i j

In this framework, the time-dependent polaritonic Schr€odinger equation represents the evolution of the electron + photon wave function, in the remarkably simplified form

  ∂ ∂Cj ðR, tÞ X pol ∂R ¼ Fke+p Fe+p Hkj iħ (7.38) C ðtÞ: j ∂t ∂R ∂t k k The inclusion of the nuclear motion in the classical case is achieved by evaluating the polaritonic forces acting on the nuclei F, i.e., the gradient of the i-th polaritonic potential energy surface (Epol i ) via the Hellmann-Feynman theorem, F¼

∂Epol i ¼ ∂R







b pol e+p Fpol i rR Η Fi

:

(7.39)

is the i-th polaritonic state. Different algorithms to solve the Here, R denotes each nuclear degree of freedom and Fe+p i semiclassical polaritonic dynamics are available in literature, both for the case of single (57,92) and many molecules (17,107,108), as well as for the case of multiconfigurational wave functions (58,89). Surface Hopping (SH) algorithms allow to formulate the time-dependent problem both at the on-the-fly level (where the polaritonic observables, including energy and forces, are computed at each time step) and at the PES-fitting level (where the PoPES are prebuilt and the classical nuclei evolve on those). The main strength of this class of approaches is the computational affordability with respect to. fully quantum descriptions. Even the requirement to produce a large number of classical nuclear trajectories propagating stochastically to retrieve the statistics of a nuclear wave-packet dynamics (Table 7.1b) is not too cumbersome computationally, since the trajectories are independent, and therefore, this part of the approach is embarrassingly parallel. In addition, they allow to include a classical chemical environment (e.g., solvent and metallic surfaces) via QM/MM interfaces, opening up the possibility to simulate extremely complex polaritonic systems with a large number of nuclear degrees of freedom. SH drawback is that the basic algorithms are often not accurate enough due to their semiclassical formulation, which results in an overestimate of the nuclear coherences and a lack of the tunneling effects (101,109,110), namely, the so-called “momentum jump.” Anyway, to partially account for quantum effects, one can resort to ad-hoc corrections that may indeed lead to reliable methods (111–113). A technical limitation to SH algorithms is the study of systems which present many degenerate states, as it becomes difficult to retrieve accurate probabilities for

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207

nonadiabatic events. As such, SH is not ideal for the study of polaritons in molecular ensembles due to the presence of a dense manifold of quasi-degenerate dark states. An alternative to SH methods for polaritonic molecular ensembles is the Ehrenfest dynamics (Table 7.1c), where the nuclei propagate on a single effective potential energy surface obtained by averaging of polaritonic states (114). This procedure becomes ideal when most of the states involved are quasi-degenerate, as the effective potential energy surface would closely resemble the manifold of “realistic” ones. We can then see why this approach works best for molecular ensembles, where the polaritonic system and the dark states manifold are wellrepresented by a single effective PoPES resembling the dark states (17,107,108). In trajectory-based methods, the cavity losses are included by exploiting the stochastic nature of the trajectories, that is, via a quantum jump algorithm. The evaluation of the probability to lose the cavity excitation (Pdec(t)) is evaluated for each trajectory and at each time step as Pdec(t) ¼ gjCpj2Dt. At time step of the propagation, a random number between [0,1] is generated. If the number falls in the interval [0, Pdec], the cavity excitation is lost from  the system and the polaritonic wave function is projected to the p 1 states via the action of the decay operator Pb ¼ I el bb . Indeed, the action of this projector is to induce the incoherent transition jn, pi ! | n, p 1i in the uncoupled basis representation. The dynamics of the polaritonic system in a lossy cavity is then obtained by averaging time-wise the large number of propagating trajectories.

7.5 Summary In this chapter, we introduced the main theoretical tools to describe polaritonic systems from a chemical point of view. Our focus has been to solve the correlated electron-photon-nuclei problem introducing two separate approaches to factorize the polaritonic wave function: the first one is to treat the photonic degrees of freedom on the same footing as the nuclei, while the second embeds the photonic degrees of freedom in the electronic ones. By analyzing the time-independent problem of the two approaches, we have shown that the former serves as a natural framework to describe Vibrational Strong Coupling (VSC) in a cavity Born-Oppenheimer (cBO) framework, whereas the latter is best suited to describe Electronic Strong Coupling (ESC), which is the most relevant for photochemical applications. In ESC, we examined the structure of the wave function and the polaritonic potential energy surfaces (PoPESs). By a thorough discussion of the PoPESs in a didactical case, we extended the typical concepts of computational photochemistry to the polaritonic case. In particular, we have defined the polaritonic analogue to the diabatic and adiabatic potential energy surfaces and examined the meaning of nonadiabatic couplings in polaritons in the case of single molecules interacting with a cavity. At the same level, we have analyzed the case of collective excitations for many molecules interacting with a cavity: we have discussed the birth of long-range interactions mediated by the cavity electromagnetic field and the emergence of dark states, which provide a fast decoherence channel for the collective excitation. The final part of the chapter is devoted to the discussion on the nuclear motion in polaritonic systems, with a critical comparison of the most common nonadiabatic techniques applied to polaritonic chemistry.

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Sci. 2020, 11 (5), 1290–1298. 70. Garcia-Vidal, F. J.; Ciuti, C.; Ebbesen, T. W. Manipulating Matter by Strong Coupling to Vacuum Fields. Science 2021, 373 (6551), eabd0336. 71. Li, T. E.; Nitzan, A.; Subotnik, J. E. Cavity Molecular Dynamics Simulations of Vibrational Polariton-Enhanced Molecular Nonlinear Absorption. J. Chem. Phys. 2021, 154 (9), 094124. 72. Li, T. E.; Nitzan, A.; Subotnik, J. E. Collective Vibrational Strong Coupling Effects on Molecular Vibrational Relaxation and Energy Transfer: Numerical Insights via Cavity Molecular Dynamics Simulations. Angew. Chem. Int. Ed. 2021, 60 (28), 15533–15540. 73. Climent, C.; Feist, J. On the SN2 Reactions Modified in Vibrational Strong Coupling Experiments: Reaction Mechanisms and Vibrational Mode Assignments. Phys. Chem. Chem. Phys. 2020, 22 (41), 23545–23552.

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74. Dunkelberger, A. D.; Simpkins, B. S.; Vurgaftman, I.; Owrutsky, J. C. Vibration-Cavity Polariton Chemistry and Dynamics. Annu. Rev. Phys. Chem. 2022, 73 (1). 75. Climent, C.; Garcia-Vidal, F. J.; Feist, J. Chapter 10: Cavity-Modified Chemistry: Towards Vacuum-Field Catalysis. In Effects of Electric Fields on Structure and Reactivity; Shaik, S., Stuyver, T., Eds.; Royal Society of Chemistry, 2021; pp. 343–393. 76. Yuen-Zhou, J.; Xiong, W.; Shegai, T. Polariton Chemistry: Molecules in Cavities and Plasmonic Media. J. Chem. Phys. 2022, 156 (3), 030401. 77. Felicetti, S.; Fregoni, J.; Schnappinger, T.; Reiter, S.; de Vivie-Riedle, R.; Feist, J. Photoprotecting Uracil by Coupling with Lossy Nanocavities. J. Phys. Chem. Lett. 2020, 11 (20), 8810–8818. 78. Hughes, S.; Richter, M.; Knorr, A. Quantized Pseudomodes for Plasmonic Cavity QED. Opt. Lett. 2018, 43 (8), 1834–1837. 79. Huttner, B.; Barnett, S. M. Quantization of the Electromagnetic Field in Dielectrics. Phys. Rev. A 1992, 46 (7), 4306–4322. 80. Buhmann, S. Y.; Welsch, D.-G. Casimir-Polder Forces on Excited Atoms in the Strong Atom-Field Coupling Regime. Phys. Rev. A 2008, 77 (1), 012110. 81. Garziano, L.; Stassi, R.; Macrı`, V.; Kockum, A. F.; Savasta, S.; Nori, F. Multiphoton Quantum Rabi Oscillations in Ultrastrong Cavity QED. Phys. Rev. A 2015, 92 (6), 063830. 82. Macrı`, V.; Minganti, F.; Kockum, A. F.; Ridolfo, A.; Savasta, S.; Nori, F. Revealing Higher-Order Light and Matter Energy Exchanges Using Quantum Trajectories in Ultrastrong Coupling. Phys. Rev. A 2022, 105 (2), 023720. 83. Sa´nchez Mun˜oz, C.; Frisk Kockum, A.; Miranowicz, A.; Nori, F. Simulating Ultrastrong-Coupling Processes Breaking Parity Conservation in Jaynes-Cummings Systems. Phys. Rev. A 2020, 102 (3), 033716. 84. Frisk Kockum, A.; Miranowicz, A.; De Liberato, S.; Savasta, S.; Nori, F. Ultrastrong Coupling between Light and Matter. Nat. Rev. Phys. 2019, 1 (1), 19–40. 85. Ciminelli, C.; Granucci, G.; Persico, M. The Photoisomerization Mechanism of Azobenzene: A Semiclassical Simulation of Nonadiabatic Dynamics. Chem. A Eur. J. 2004, 10 (9), 2327–2341. 86. Cattaneo, P.; Persico, M. An Abinitio Study of the Photochemistry of Azobenzene. Phys. Chem. Chem. Phys. 1999, 1 (20), 4739–4743. 87. Ishikawa, T.; Noro, T.; Shoda, T. Theoretical Study on the Photoisomerization of Azobenzene. J. Chem. Phys. 2001, 115 (16), 7503–7512. 88. Cembran, A.; Bernardi, F.; Garavelli, M.; Gagliardi, L.; Orlandi, G. On the Mechanism of the Cis Trans Isomerization in the Lowest Electronic States of Azobenzene: S0, S1, and T1. J. Am. Chem. Soc. 2004, 126 (10), 3234–3243. 89. Fregoni, J.; Corni, S.; Persico, M.; Granucci, G. Photochemistry in the Strong Coupling Regime: A Trajectory Surface Hopping Scheme. J. Comput. Chem. 2020, 41 (23), 2033–2044. 90. Persico, M.; Granucci, G. Wavepacket Dynamics and Geometrical Relaxation. In Photochemistry: A Modern Theoretical Perspective; Persico, M., Granucci, G., Eds.; Theoretical Chemistry and Computational Modelling; Springer International Publishing: Cham, 2018; pp. 119–140. 91. Mony, J.; Climent, C.; Petersen, A. U.; Moth-Poulsen, K.; Feist, J.; B€orjesson, K. Photoisomerization Efficiency of a Solar Thermal Fuel in the Strong Coupling Regime. Adv. Funct. Mater. 2021, 31 (21), 2010737. 92. Antoniou, P.; Suchanek, F.; Varner, J. F.; Foley, J. J. Role of Cavity Losses on Nonadiabatic Couplings and Dynamics in Polaritonic Chemistry. J. Phys. Chem. Lett. 2020, 11 (21), 9063–9069. 93. Hoffmann, N. M.; Sch€afer, C.; Rubio, A.; Kelly, A.; Appel, H. Capturing Vacuum Fluctuations and Photon Correlations in Cavity Quantum Electrodynamics with Multitrajectory Ehrenfest Dynamics. Phys. Rev. A 2019, 99 (6), 063819. 94. Vendrell, O. Coherent Dynamics in Cavity Femtochemistry: Application of the Multi-Configuration Time-Dependent Hartree Method. Chem. 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Part III

Applications

Chapter 8

First-principles modeling of dye-sensitized solar cells: From the optical properties of standalone dyes to the charge separation at dye/TiO2 interfaces Valentin Diez-Cabanesa,b, Simona Fantaccic, and Mariachiara Pastorea a

Laboratoire de Physique et Chimie Th
eoriques (LPCT), Universit
e de Lorraine & CNRS, Nancy, France, b Laboratoire Lorrain de Chimie Mol
eculaire

(L2CM), Universit
e de Lorraine & CNRS, Nancy, France, c Computational Laboratory for Hybrid/Organic Photovoltaics (CLHYO), Istituto CNR di Scienze e Tecnologie Chimiche “Giulio Natta” (CNR-SCITEC), Perugia, Italy

Chapter outline 8.1 Introduction 8.2 Computational modeling of DSSCs: Methods, limitations, and practical strategies 8.2.1 Generalities 8.2.2 Electronic structure and optical properties of dyes in solution 8.2.3 Electronic structure and optical properties of semiconductor materials and dye-sensitized interfaces 8.2.4 Machine learning and semiempirical methods applied to DSSCs

215 219 219 220

8.3 Design rules for Ru(II) sensitizers: The role of spin-orbit coupling (SOC) 8.4 Modeling the photophysics of Fe(II) metal complexes: Tools and findings 8.5 Interfacial properties of Fe-NHC-sensitized TiO2 8.6 Conclusions References

223 226 229 234 234

222 223

8.1 Introduction The application of the fundamental principles and computational methodologies described in previous chapters is at the heart of the understanding of the photophysical processes driving the operation of many light-driven technologies. One prototypical example of these technologies is based on the exploitation of solar light to produce clean energy. As a matter of fact, the Sun is supplying an unlimited amount of energy in the form of irradiation to the Earth surface, which can be converted into electricity by means of photovoltaics (PV) devices (1). In this viewpoint, PV technologies appear as one of the most attractive strategies to shift toward more sustainable sources of energy, which will potentially enable, or at least contribute to, the reduction of the pollution generated by the human activities, and the supply of energy for a continuously growing population (2). Nowadays, most of commercial solar cell technologies are built by the integration of amorphous Si (a-Si)-based materials as photoactive semiconductors (3), which indeed present several drawbacks such as high economic and environmental cost of production, large mechanical rigidity, or low device performances for low-intensity light sources. Alternatively, during the last decades, intensive research efforts have been devoted to the development of the so-called “third-generation solar cells,” characterized by lower costs, high flexibility, and integration of earth

Theoretical and Computational Photochemistry. https://doi.org/10.1016/B978-0-323-91738-4.00013-0 Copyright © 2023 Elsevier Inc. All rights reserved.

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abundant materials (4). Nonetheless, their commercial exploitation has been hampered by their relatively low device efficiencies in comparison with the ones achieved by crystalline (but high cost manufacturing) materials (GaAs, InP) (5). Currently, the most promising technologies of this generation are the dye-sensitized solar cells (DSSCs), organic photovoltaics (OPVs), and perovskite solar cells (PSCs). In this chapter, we will focus on the first type of solar cells, whereas the other two technologies will be discussed in detail in the subsequent two chapters of this book. Notably, DSSCs present the advantages of requiring simple preparation methods, having an excellent flexibility, good stability, and being exploitable for indoor applications since they can work under diffuse-light conditions (6,7). Although evidence of charge injection from an organic dye into a semiconductor in an electrochemical cell was reported at the late 1960s (8), only in 1991, O’Regan and Gr€atzel achieved the first practical use of dye sensitizers grafted onto mesoporous semiconductor films to generate an electric current (9), paving the way to massive research efforts and industrial interest in the DSSC technology (10). The main components and the basic device operation for a typical n-type DSSC (Gr€atzel’s solar cell) are presented in Fig. 8.1. The functioning mechanism involves the following steps: (1) the dye sensitizer absorbs the solar light and is promoted to a charge-separated electronically excited state (D*); (2) the photogenerated electron is injected into the conduction band (CB) of the semiconductor, where the dye sensitizers are grafted; (3) the dye recovers its neutral state (D) by transferring the photogenerated hole to the electrolyte redox mediator (4), which is then regenerated by the reduction prompted by the metallic counter electrode acting as catalyst, thus closing the circuit which is sealed by means of a transparent conductive oxide (TCO) layer, and finally generating the electric current (11,12). As schematized in the energy diagram of Fig. 8.1, the difference between the quasi-Fermi level (Ef) of the photoanode under illumination and the redox mediator potential (R+/R) will determine the potential generated by the cell (the so-called open-circuit voltage VOC). Then, the corresponding overall efficiency is calculated as follows: ¼

V OC ∗FF∗J SC , PI

(8.1)

where FF (fill factor) is a dimensionless parameter that reflects the internal and external cell resistances, Jsc is the photocurrent density at short circuit, and PI is the intensity of the incident light. In this context, Jsc is estimated as the integral over the solar spectrum of the monochromatic incident photon to current conversion efficiency (IPCE) at short circuit, which can be quantified by following this equation: IPCE ¼ LHE∗ finj ∗ fcoll ,

(8.2)

FIG. 8.1 Scheme of the device operation of a prototypical n-type DSSC presenting the diagram for the energetic levels of cell components. Continuous green (light gray color in print version)/orange (gray color in print version) arrows are used to indicate the desired flow of electrons and redox shuttle reactions, whereas dashed red (dark gray color in print version) lines depict the undesired recombination processes. The numbering with all the steps involved in the process of current generation appears on the right side of the image.

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217

TABLE 8.1 Solar cell device characteristics for the four most performant DSSC devices reported in the literature for each dye sensitizer family mentioned in the text. Family D-p-A organic

Dye ALEKA-1/LEG1 XY1b/Y123 ZL003 MS5/XY1b

Zn(II) Porphyrin

GY50 SM315 SM342/Y123

Ru(II) complex

Fe(II) complex

Anchoring Silyl/COOH CN-COOH COOH CN-COOH COOH COOH COOH

Redox [Co(phen)3]

CE

h

Year

Ref

3+/2+

GNP/Au

14.3

2015

(13)

2+/+

PEDOT

13.1

2018

(14)

Pt

13.6

2019

(15)

[Cu(tmby)2] [Co(bpy)3]

3+/2+

[Cu(tmby)2]

2+/+

PEDOT

13.5

2021

(16)

[Co(bpy)3]

3+/2+

Pt

12.8

2014

(17)

[Co(bpy)3]

3+/2+

GNP

13.0

2014

(18)

[Co(bpy)3]

3+/2+

GNP

12.7

2017

(19)

3+/2+

Pt

14.2

2020

(20)

SGT-021/SGT-149

COOH

[Co(bpy)3]

CYC-B11

COOH

Z960

Pt

11.5

2009

(21)

C106

COOH

EL02

Pt

12.1

2010

(22)

C101

COOH

I /I3

Pt

11.7

2011

(23)

N719

COOH

I /I3

PtCoFe

12.3

2017

(24)

C1

COOH

I /I3

PEDOT

0.9

2020

(25)

ARM13

COOH

el3(MgI2/TBAI)

PEDOT

1.4

2021

(26)

ARM130

COOH

el3(MgI2/TBAI)

Pt

1.8

2021

(27)

COOH

3+/2+

Pt

1.3

2021

(28)

FeCD

[Co(bpy)3]

For all devices reported here, mesoporous TiO2 was used as a photoelectrode. CE, counter electrode; FeCD, [Fe(cpbmi)(dtapbmi)]2+; GNP, graphene nanoplatelet; phen, phenanthroline; TBAI, tetrabutylammonium iodide salt; tmbpy, 4,40 ,6,60 -tetramethyl-2,20 -bipyridine.

where LHE is the light harvesting efficiency of the photoelectrode, finj is the quantum yield of electron injection, and fcoll is the electron collection efficiency at TCO. Overall, we can thus conclude that cell efficiency will be governed by the competition between the desired operation processes, involving generation, transport, and collection of charges (steps 1–4 indicated by green arrows (light gray color in print version) in Fig. 8.1), and the charge recombination losses (steps 5–6 marked with red dashed arrows (dark gray color in print version)). The first reported efficiency for DSSCs in 1991 amounted to 7.1% (9), and after three decades of active research in the field, the efficiency for these technologies has been pushed until reaching 14.3% (13) (see Table 8.1) under standard solar light irradiation (29) and above 34% when employing artificial light sources (16,30). Such an improvement in the device efficiencies has been realized by a careful optimization of the working conditions and the device components integrating the cell (31). The most common semiconductor materials employed as photoanodes in DSSCs are mesoporous films of TiO2 nanocrystals due to their enhanced surface contact area, where dyes are grafted, thus significantly boosting light harvesting. The most recent advances achieved in TiO2 film optimization concern the control of the characteristics (porosity, chemical composition) of the material to tune the interface with the electrolyte and the conductive glass, or to modify its intrinsic properties via doping or light scattering (32,33). Other types of nanomaterial morphologies employed in DSSC photoanodes include nanosheets, nanowires, nanorods, and nanofibers (34). Alternatively, other materials such as ZnO (35,36) and SnO2 (37,38) nanostructures have been used as photoanodes in DSSCs as well, but unfortunately, their lower chemical stability under illumination and their deeper CB edge energies, respectively, have somehow lowered their device efficiencies (no more than 7.5% (39) and 6.4% (40) for ZnO and SnO2 photoanodes, respectively). It is noteworthy to mention that in the classical Gr€atzel’s solar cell, the working electrode consists of an n-type semiconductor, where the photogenerated electrons are injected. However, the flow of electrons in the cell can be reversed by using the dye sensitizer to transfer holes into the valence band (VB) of a p-type semiconductor, which is acting as a photocathode (normally NiO) (41), forming the so-called p-type DSSC. Although this idea was first introduced in 1999 by Lindquist et al., the device efficiencies achieved by p-type devices were negligible (42). A promising avenue consists in combining a photoanode and a

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photocathode in a tandem device that enables to collect efficiently a larger amount of solar light, thus pushing the theoretical maximum performance, which can be achieved by a single-junction device up to a Shockley-Queisser limit of 43% (43). Nonetheless, the rapid recombination taking place at the photocathode resulted in rather limited overall efficiencies, being 4.1% the largest one reported to date for tandem DSSC devices (44). Regarding the materials employed for the transport of the photogenerated holes and electrons, the most common family is the one represented by liquid electrolytes made of organic, inorganic, or ionic solvents coupled with a redox mediator (45,46). The most widely employed redox couple is I /I3 due to its suitable potentials for many dyes, good solubility, and high conductivity. Despite this, nowadays the most efficient DSSCs are systemically fabricated by employing transition metal complexes (TMCs), such as Cu(II/I)- or Co(III/II)-based electrolytes, in view of their improved stability and their excellent electrochemical behavior (47,48). Nevertheless, for the commercial exploitation of DSSC technologies, it would be highly desirable to avoid the use of liquid components, by replacing traditional electrolytes with quasi-solid and solid hole transport materials (HTM) such as small organic molecules, polymers, and inorganic or metal complex materials, forming the so-called solid-state DSSCs (ssDSSCs) (49–51). On the other hand, Pt-based materials have dominated the role of counter electrodes in DSSC technologies during several decades due to their excellent conductivity and catalytic activity (52,53). However, during the last years, some increasing efforts have been devoted to use catalysts displaying a better energetic alignment with the typical TMCs used as redox couples, and based on Earth-abundant materials (54,55). Among them, carbon materials such as graphene or polymers (i.e. PEDOT: PSS) alone or combined in the form of hybrid junctions have shown the best device performances (see Table 8.1). Finally, as one can expect, the dye sensitizer is one of the principal ingredients of a DSSC, and its chemical versatility offers a huge space to boost the device efficiency by modulating the optical and redox properties (56). An highly efficient dye should possess, indeed, a wide and intense optical absorption in the Vis and nIR regions associated with a long-lived charge transfer (CT) excited state, possibly electronically coupled to the oxide CB states, and ground- and excited-state oxidation potentials, which properly match the redox potential of the mediator and the semiconductor CB, respectively, as shown in Fig. 8.1. We can mention here three main families of dye sensitizers due to their historical relevance and their higher power conversion efficiencies when employed in DSSCs: the push-pull organic sensitizers, porphyrin-based dyes, and TMC dyes (57,58). Push-pull organic dyes possess an electron-donating group (D) connected to an electron withdrawing moiety (A) via pconjugated bridges (also known as D-p-A architecture) (59). In addition to their high tunability and good absorption in the red portion of the solar spectrum, organic dyes also present the advantage of being metal-free materials, allowing for simple synthesis procedures, low production costs, and reduced environmental impact (20,60,61). Thanks to their long excitedstate lifetimes and their strong absorption in the Vis region (62,63), Zn(II) porphyrins combined with D-A groups have emerged as highly performant sensitizers, with record efficiencies of about 13% (18). Furthermore, their main disadvantage, represented by their tendency to aggregate at the liquid/solid semiconductor interface, can be effectively mitigated, as shown by the record reported efficiencies of 14.2% (20), by using them in combination with other types of dye (i.e., organic D-p-A ones) in co-sensitized devices. Undoubtedly, TMCs, with Ru(II) polypyridyl complexes holding the place of honor, are historically the most studied and employed sensitizers in DSSC applications. As a matter of fact, these dyes were the first ones to be integrated in DSSC devices in the pioneer work of Gr€atzel and O’Regan (9), and the ones displaying the largest efficiencies till the last decade (see Table 8.1). TMCs possess many desirable properties such as a favorable photoelectrochemical behavior, high stability of their oxidized states, and a wide absorption range from the Vis to the nIR regime. However, the main reasons behind the success of Ru(II)-based sensitizers are the long lifetime and the excellent directionality of their metal-ligand charge transfer (MLCT) states, as we will explain in more detail in the last three sections of this chapter (64–66). Unfortunately, these dyes present important drawbacks, which have significantly limited their commercial application, such as low extinction coefficients and, more importantly, the scarcity and potential toxicity of ruthenium (67). Thus, during the last years, the field of TMC sensitizers is moving toward the development and the use of compounds owing earth-abundant and ecologically friendly first row d-block metal centers (68), such as Cu (69,70) and Fe (71). Notably, [Cu(bpy)2]+ exhibits similar photophysics as [Ru(bpy)3]2+ complexes (72) (vide infra), but their lower absorption coefficients in the Vis region and their facile ligand redistribution have hampered their use as sensitizers (67). Despite these issues can be partially solved by means of ligand functionalization, the efficiencies achieved by Cu(I) complex sensitizers have not exceeded 4.7% (73). In case of Fe, even if displaying the same d6 electronic configuration as Ru, its photophysics is completely different (71), as we will further develop later, thus resulting in relatively low device efficiencies (Table 8.1). Nonetheless, due to their high technological relevance (iron is the fourth most common element in the Earth’s crust), there is a growing interest in exploiting its peculiar photophysics (74). The first attempt to employ Fe complex dyes as sensitizers was reported in the late 1990s by Ferrere and co-workers (75,76) who synthetized Fe(II) polypyridine complexes displaying a similar chemical structure as prototypical Ru(II) complexes used in DSSC devices (N3 and N719 in Fig. 8.2 left). However, as we will better explain in Section 8.4, the populated metal to ligand CT (MLCT) excited states of these dyes undergo an ultrafast

First-principles modeling of dye-sensitized solar cells Chapter

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FIG. 8.2 Chemical structures of the most performant Ru(II) (left) and Fe(II) (right) complex dye sensitizers according to Table 8.1.

deactivation into metal-centered (MC) states (77), thus impeding the electron injection into the electrode and their practical application in photovoltaics. Notably, the biggest breakthrough for the development of Fe sensitizers came only in 2013 with the use of N-heterocyclic carbene (NHC) ligands to replace the traditional polypyridyl metal-binding groups (see Fig. 8.2 right) (78). The strong s-donor characteristic of these ligands is able to effectively destabilize the low-lying MC states (78,79), thus impeding the ultrafast MLCT deactivation and allowing to reach excited-state lifetimes in the ps scale. For example, MCLT lifetimes of 16–18 ps were reported for the homoleptic Fe(II)-NHC complex (C1) (80–82) endowed with the carboxylic functionalities required to be grafted to the TiO2 electrode. A further improvement of the excited-state lifetimes can be reached via modification of the NHC ligands (83,84), by increasing, for instance, the ligand aromaticity (82), or by employing diazene central cores (85). Remarkably, nowadays ligand engineering has allowed Fe-carbene complexes to reach MLCT state lifetimes in the ns scale (86). Despite the overall improved lifetimes, efficient utilization of Fe dyes in DSSCs is still limited, with the highest device efficiencies amounting to 1.83% (ARM130 in Table 8.1) (27), being still about 6–7 times lower with respect to those of Ru (II)-based solar cells. However, this field is still at its infancy (note that in the last two years, the device efficiencies reported for Fe-DSSCs have been boosted by one order of magnitude), and there is still plenty of room to improve the interfacial charge separation, for instance, via the incorporation of electron-accepting moieties close to the anchoring groups, or further tuning the redox shuttles or the electrodes. As attested by the huge number of theoretical and computational studies published in the last decades, frequently carried out in close collaboration with experiments, first-principles calculations have been proved to be extremely useful to shed light on the optoelectronic properties of both the individual dyes and their interface with the electrodes and the electrolyte environment (87–92), effectively contributing to boost the device efficiencies (93,94). The rationalization of the main physical processes underlying the device operation via theoretical modeling has driven, indeed, the development of DSSC technologies (6). In this viewpoint, here we will discuss the application of the state-of-the-art first-principle calculations to the field of DSSCs, focusing our attention on the TMC dyes. We refer the reader to some recent reviews to get a deeper knowledge of the recent advances of DSSC technologies (6,7,10), and of the modeling of other dye families (91,92,95). We will start by a short overview of the methodologies and models, which have been employed in the DSSC field to simulate the properties of dye sensitizers and their interface with the TiO2 semiconductor. Afterward, we will illustrate how these methods can be suitably employed for describing the optoelectronic properties of TMC sensitizers. First, we will discuss the last advances in modeling Ru(II) dye sensitizers, focusing on the development of panchromatic dyes via the enhancement of their singlet-triplet state couplings. Afterward, we will show how modeling of the dye’s photophysics can be used to rationalize and optimize the MLCT state lifetimes in Fe(II) complexes. To conclude, we will illustrate how appropriate modeling of the electrode/dye/electrolyte interfaces can be used to get a better understanding of the different injection/recombination processes responsible of the photocurrent generation, and how these calculations can guide in the design of complexes, leading to improved interfacial charge separation characteristics.

8.2 Computational modeling of DSSCs: Methods, limitations, and practical strategies 8.2.1 Generalities As exemplified in Fig. 8.1, DSSCs are rather complex systems involving components and processes of different nature, and this makes their theoretical investigation an extremely challenging task. From a computational point of view, it is compulsory to develop realistic models and accurate methodologies to reliably describe the main properties of the cell components (dye sensitizers, semiconductors, and electrolytes) and the interactions between them. The fundamental information

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that one can get from atomistic simulations is (i) the dye’s ground- and excited-state equilibrium structures, ground-state oxidation potential (GSOP), optical absorption spectra, and excited-state oxidation potential (ESOP); (ii) the semiconductor band gap and conduction and valence band density of states; and (iii) the electrolyte/hole conductor redox properties. The models employed to reproduce TiO2 or, more in general, semiconductor surfaces can be periodic 2D slabs or finite cluster (nanoparticles) systems (96). The choice will depend on the targeted properties and on the appropriateness of the finite models in reliably reproducing the electronic structure of the solid material. For instance, TiO2 nanoparticles (97,98) have been shown to deliver electronic properties similar to those of periodic systems, opening the way to their systematic use, in a time-dependent density functional theory (TD-DFT) framework in combination with localized atomic orbital (LAO) basis sets and hybrid exchange and correlation functionals, for the calculation of the optical properties of the dye@TiO2 interfaces. This is, however, a peculiarity of the material, which yields size-independent electronic properties above a size of c. 1–2 nm. The situation is completely different, for example, for ZnO and for related ZnX materials (with X ¼ S, Se, and Te), as well as for WO3, which all exhibit strong quantum confinement effects (99–101). For the simulation of the dye-sensitized interface, the first step is to determine the adsorption mechanism of the dye onto the semiconductor, the nature and topology of the dye@semiconductor excited states, and the lining up of ground- and excited-state energy levels at the heterointerface. These properties, along with an estimate of the electronic coupling, constitute the fundamental parameters determining the electron injection and dye regeneration processes (102–106). As illustrated in Table 8.1, among the different types of anchoring functionalities, the ones employed in the most performant devices are carboxylic groups, which can be grafted to the semiconductor surface by following a mono- or bidentate fashion (107–109). The main drawback of cyanoacrylic and carboxylic groups is their poor stability in water environments that has, for example, limited their utilization in dye-sensitized photoelectrocatalytic cells for water splitting (110,111). On the contrary, phosphonic acid provides excellent anchoring stability, with an adsorption strength estimated to be approximately 80 times higher than that of carboxylic acid, and negligible desorption in the presence of water (11,112). Many alternative anchoring groups have been proposed and tested, such as hydroxyl (113), silanes (114), silatranes (115), and hydroxamate (116) groups. Dealing with solid/liquid interfaces, the inclusion of the solvent environment in the simulations is essential to reliably describe the photophysical behavior of the various cell components, especially if one wants to directly compare with experiments. The most common approach to take solvent effects into account at a low computational cost is to use implicit polarizable models, where the solvent is treated as continuum dielectric medium, being the polarizable continuum models (PCM) (117) and the COnductor-like Screening Model (COSMO) (118) the most common implementations available in commercial quantum chemistry and solid-state codes. Unfortunately, these approaches can only capture bulk solvation effects, while specific solute-solvent interactions are supposed to take place at the interface, one needs to explicitly include the solvent in the simulations. A realistic description of explicit solvent effects and of its dynamics requires sufficiently long ab-initio molecular dynamics (AIMD) (109,119,120) or classical (121–123) MD simulations; hybrid Quantum/Molecular Mechanics (QM/MM) schemes, where the most relevant parts of the system (usually the solute and the first solvent layers) are described at the QM level, whereas the rest (the outer solvent layers) is treated classically, have been also largely applied (124–126). Finally, concerning the dye/semiconductor/electrolyte system interdependence, one should consider many important phenomena taking place at the dye/semiconductor, semiconductor/electrolyte, dye/electrolyte, dye/dye, and dye/co-absorbent interfaces (95).

8.2.2 Electronic structure and optical properties of dyes in solution DFT (127) and TD-DFT (128,129) are the most popular computational tools employed for the ground- and excited-state properties of DSSC materials due to their excellent compromise between accuracy and computational cost. However, standard DFT functionals such as the ones based on the general gradient approximation (GGA) (i.e., PW91 (130), PBE (131), or BLYP (132)) usually fail in the description of the electronic structure and optical properties of dye sensitizers (133,134). Pure DFT methodologies, indeed, present problems to recover the correct 1/R asymptotic behavior of the potential energy, yielding large underestimations for semiconductor band gaps and excited states with a significant long-range CT characteristic, and in the case of molecules with spatially extended p systems (135–138). While this error dramatically affects organic push-pull dyes and their interfaces with the semiconductor (89,119,139,140), for TMCs, the CT problem is limited to some extent by the substantial overlap of metal and ligand states characterizing the starting and arriving orbitals in MLCT excitations typical of these complexes. The use of opportunely tuned hybrid functionals, incorporating a variable amount of nonlocal Hartree-Fock (HF) exchange, partially corrects the wrong asymptotic behavior (134,141–143). Other popular approaches are based on introducing an increasing fraction of HF exchange as the interelectronic separation increases; the long-range corrected (LC) functionals (144–147), CAM-B3LYP (148), and oB97XD (149) methods belong to this family of range-separated functionals. Finally, it is worth mentioning a recent and promising new family of DFT functionals, namely, the local hybrids (150–155), whose tunability seems to be favorable for the description

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of hybrid DSSC interfaces (140). In this approach, different amounts of exact exchange are used at each point in space, thus allowing much more flexibility with respect to global hybrid and long-range corrected functionals. Standard DFT methods also lack accuracy in treating long-range polarization effects that give rise to the van der Waals dispersion interactions, which, on the contrary, are properly taken into account by more expensive correlated ab-initio methodologies, such as second-order Møller-Plesset perturbation theory (MP2) (156), coupled cluster (CC) (157), or the random phase approximation (RPA) (158). However, various correction schemes have been proposed to properly recover the dispersion interactions at the DFT level at a moderate computational cost (159). These approaches are based on nonlocal density-based functionals, such as van der Waals (vdW-DF, vdW-DF2, vdW-DF3) (160) or Vydrov and Van Voorhis functionals (VV10) (161); semiclassical C6-based potentials, such as the Grimme’s corrections (D2, D3) (162), Thatchenko-Scheffer (TS) models (163), or Becke Johnson (BJ) damping functions (164); and effective one-electron potentials, such as the semilocal meta-GGA Minnesota set of functionals (M06, M06-2X) (165). Finally, another major drawback of TD-DFT methods is the well-known triplet (and multiplet) instability, which can be partially mitigated by resorting in the Tamm-Dancoff approximation (TDA) (166), although it still may result in largely underestimated triplet excitation energies (167). In the particular case of TMCs, where intersystem crossings (ISC) involving several multiplicities often take place during the excited-state dynamics (168), the accurate estimation of such energies becomes crucial. It is well known that pure exchange and correlation (xc) functionals tend to stabilize low-spin states, whereas the inclusion of HF exchange favors the high spin configurations (169–171). Therefore, the reparameterization of standard hybrid DFT functionals with a slightly reduced amounts of HF exchange, as for instance, the modified B3LYP functional with a 15% of exchange (the so-called B3LYP*) (172), is one of the most popular strategies employed to model TMCs, as Fe(II) and Co(II) complexes (173–176). In a similar way, the reparameterization of the short-range exact exchange in RSH functionals also provided accurate high and low spin state energy (EHS/LS) values (177). In the case of meta-GGA functionals, they have shown a good performance in the description of the optical properties of Fe complexes bearing strong ligand fields (178). Furthermore, the so-called “double hybrid” functionals (179) have also shown to accurately estimate the relative EHS/LS energies (180,181). Very recently, a cheaper computational alternative to these methods has been proposed by employing the Hubbard correction (DFT + U) (182) with a U-corrected density, which yielded to similar EHS/LS energies as the most accurate hybrid DFT approaches (183). Finally, it is worth mentioning that the performance of TD-DFT to properly describe a large number of low energy quasi-degenerated states having a strong multiconfigurational nature and distinct CT characteristics has revealed very system-dependent (184). In this context, more sophisticated computational approaches as, for instance, multiconfiguration pair DFT (MP-DFT) (185) or wavefunction methods such as CASPT2 (186), NEVPT2 (187), combined CASPT2/CC (188), or multireference CI (189), should be employed instead (187,190), especially in those complexes with a more marked metal-ligand bond covalent characteristic, which exhibit a huge dependence of the HF xc fraction of the DFT functional chosen (191,192). The presence of heavy atoms used as metal centers in TMCs usually results in the appearance of relativistic effects, affecting their electronic structure at different extent (187,193). The splitting of the dye’s energy levels and the population of certain forbidden triplet states originated by the coupling between the singlet and triplet states, spin-orbit coupling (SOC), can heavily impact their optoelectronic response, especially in the lowest energy absorption region of the spectra. The full resolution of the Dirac-Kohn-Sham (DKS) equations is not computationally affordable for medium-size systems as Ru(II) dyes, and due to this, many different approximations to these equations have been developed during the last years. Among them, the most common approach to treat relativistic effects is the 2-component zero-order regular approximation (ZORA) (194,195), which is able to disentangle the scalar relativistic (SR) effects from the corrections associated with the inclusion of the spin-orbit coupling (SOC) operator. To understand the influence of the SOC between the lowest energy triplet and singlet states for a given molecular system, one can resort to a simple mono-electronic SOC model, where the oscillator strength of the lowest singlet-triplet excitations (fST) can be expressed as follows: f ST ¼

hC T i f , ET ES s

(8.3)

where CS/CT are the singlet/triplet wavefunctions, HSOC is the spin-orbit coupling Hamiltonian, Es/ET are the singlet/triplet state energies, and fS is the singlet oscillator strength. The matrix element at the numerator ‹CSjHSOCjCT› represents the strength of the coupling between the spin-free singlet and triplet states, and it can be approximated by the amount of metal characteristic in the metal-based highest-occupied molecular orbital (HOMO) of the complex (196). In this regard, with the aim of illustrating the implementation of these techniques in the modeling of TMCs, Section 8.3 will provide a resume of the main theoretical studies dedicated to elucidate the influence of SOC in the properties of the Ru(II) complex dyes. Note that at this point, we will stick the discussion to the works done on Ru(II) and Os(II) dyes since relativistic effects in Fe(II) complexes are expected to be very low in view of the low mass of its metal center.

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8.2.3 Electronic structure and optical properties of semiconductor materials and dye-sensitized interfaces As discussed for the dyes, pure DFT functionals such as the ones based on the local density approximation (LDA) usually lead to an underestimation of the experimental gaps of semiconductors as TiO2 (197), WO3 (198), or NiO (199). This error can be corrected by adding a tuned fraction (10%–20%) of HF exact exchange in form of hybrid functionals (200–203). As a matter of fact, however, hybrid functionals become an expensive method when modeling systems with large dimensions within a plane wave basis set (204). In this regard, some computationally less expensive methods such as DFT + U (182) or DFT-1/2 (205) have emerged during the last years as affordable solutions to overcome the DFT band gap issue in semiconductors as TiO2. However, the need for an unbiased and accurate method enabling to fairly reproduce the electronic structure of both TiO2 and the dye sensitizer has turned the attention to some many-body perturbation theory (MBPT) methods such the ones based on the GW formalism (206), which has recently become a prominent tool to investigate the interfacial energetics in DSSCs (119,207,208). Concerning the dye@TiO2 excited-state properties, the large size of these systems clearly makes multiconfigurational correlated wavefunction methods, such as CASSCF, CC, or CI, computationally prohibitive, and the reported calculations are limited to the interface with small TiO2 clusters (209). As a result, hybrid DFT functionals in combination with cluster models remain the most common tool to access to the dye@TiO2 optical properties (89). Nonetheless, the rapid development of MBPT approaches based on the combination of GW and the resolution of the Bethe-Salpeter equation (210,211) (GW/BSE) has recently allowed their implementation in the study of the excited states of these types of interfaces (212,213). Still, the computational load needed to study realistic dye@TiO2 models (owing hundreds or thousands of atoms) at the DFT level is still considerable, thus opening the door to the search of less expensive approaches such as tight binding (TB) models. Among these techniques, the TB method, whose parametrization is based on the second-order expansion of the DFT energies with respect to the charge density fluctuations (the so-called SCC-DFTB method) (214), is the most employed one. Indeed, the parametrization of Ti atoms to treat both bulk and molecular systems in DFTB (215) has allowed the application of this approach to extended dye@TiO2 interfaces, by providing realistic structures via geometrical relaxations (216,217) and AIMD simulations (218–222), or by calculations of their excited-state properties via its time-dependent implementation (TD-DFTB) (223). The electronic structure of the dye@semiconductor interfaces can provide indications on the magnitude of the coupling between the states of the dye and the semiconductor, which determine the injection/recombination properties. The simplest method to estimate the injection rates is based on the Newns-Anderson model (224,225), which assumes that the coupling between the dye and semiconductor states is directly connected with the broadening of the projected density of states (PDOS) relative to the lowest-unoccupied molecular orbital (LUMO) of the sensitizer. An alternative still simple approach to estimate the recombination/injection ET reactions describes these phenomena as a nonadiabatic radiative process. Then, the recombination/injection rates can be calculated as a function of two factors: the squared electronic coupling matrix element between the donor (dye’s HOMO or LUMO) and the acceptor (semiconductor valence (VB) or conductance bands (CB)) and the Franck-Condon weighted density of states (FCDS) which defines the probability of reaching a nuclear configuration where the donor-acceptor electronic states have the same energy, which, in the case of semiconductors, reduces to a pure density of states r(E). Thus, by assuming a weak coupling and resorting to the Fermi’s golden rule, as proposed by Thoss and co-workers (226), these rates can be estimated by employing the formula: kinj ¼

2p X 2 V dk rðEk Þ: ħ k

(8.4)

Here, kinj represents the sum over the manifold of k acceptor states of interest, and the product jVdkj2r(Ek) defines the socalled probability distribution G(Ek). A more refined method to investigate the interfacial dye@semiconductor ultrafast charge dynamics is based on the simulation of the laser-induced electron migration processes, where the laser-driven electronic wave packets are treated by means of the time-dependent many-body configuration interaction (TDCI) method (227). Very interestingly, this approach has allowed to evaluate the influence of the many-body interactions (228) and to reach a real-time spatial representation (229) of the interfacial CT phenomena in dye@TiO2 systems. Most of the methods commented earlier are applied to cluster models with a relatively medium size. For that reason, with the aim of evaluating the interfacial dye@TiO2 electron transport phenomena in extended systems, one can resort to nonequilibrium Green function (NEGF) theory applied with the adequate boundary conditions (229,230). More reliable information about the kinetics of the electron transfer phenomena occurring at the dye@TiO2 interface can be obtained by means of quantum or nonadiabatic dynamics (231,232) based on the time-dependent propagation of the electronic wavefunction, as obtained from

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semiempirical Hamiltonians (233–235) or from the DFT orbitals (236–238), combined with mixed-classical (239), or full quantum dynamics (240) simulations. On the same foot, much more expensive real-time propagation TD-DFT has also been emerged as a useful technique to access to the photoinduced electron transfer processes when combined with mixed classical quantum dynamics (241,242).

8.2.4 Machine learning and semiempirical methods applied to DSSCs Finally, it is worthwhile to mention the recent attempts of integrating artificial intelligence (AI)- and machine learning (ML)-based approaches to the high-throughput screening of materials used as device components in DSSCs. This can be done by establishing reliable structure-property relationships obtained by the combination of accurate theoretical calculations and available experimental data. For instance, Ma et al. combined a database of 233 organic dyes with ML screening to discern eight potential organic dyes displaying PCEs larger than 9% among 10,000 possible candidates (243). Ramanujam and co-workers designed a new ML route capable of predicting novel high performant dye sensitizers, where 75% of them showed an improvement in PCE when compared to the reference dyes used to build the model (244). One step further, Cooper et al. applied high-throughput screening to identify new materials displaying panchromatic optical absorption by selecting 6 combinations of co-absorbers (among 9431 dye candidates), which exhibited performances in the same order of magnitude as Ru(II) complexes (245). On the same vain, these methodologies can also be employed to predict a given property that can be used as a descriptor of the device efficiency, thus allowing to identify which are the most important properties driving the cell performance. For the sake of illustration, Sutar et al. used ML to predict the best synthetic conditions to achieve ZnO electrodes with enhanced performances (246), while Venkatraman et al. applied similar methods to identify which dye sensitizers are prone to undergo a larger red shift when adsorbed on TiO2 films (247). All these results are a clear fingerprint of the huge potential of combining ML with large databases and high accurate calculations to assist the experimentalist to the design and optimization of highly efficient DSSC architectures.

8.3 Design rules for Ru(II) sensitizers: The role of spin-orbit coupling (SOC) Due to their unsurpassed success, most of the theoretical works have been dedicated to the study of Ru(II) dyes (mainly N3 ([Ru(dcbpyH2)2(NCS)2], with dcbpyH2 ¼ 4,40 -dicarboxy-2,20 -bipyridine) and its salt N719. The first computational work was carried out in 1997 by Restmo et al., who employed semiempirical calculations to access to the electronic structure of the best-performing N3 dye (248). Only in the early 2000s, the pioneering works on the excited-state properties of Ru(II) sensitizers (N3 and the related salt N719) based on TD-DFT calculations appeared in the literature (249,250). Standard hybrid functionals such as PBE0 and B3LYP gave accurate results in the description of the three main bands, centered at 2.6, 3.4, and 4.2 eV, dominating the absorption spectrum of the N3 dye (see Fig. 8.3 left). The inclusion of solvent effects, even only by an implicit solvation model, was proved to be mandatory to describe the optoelectronic properties of this class of Ru-based dyes, and the adopted methodology was able to reproduce the 0.2 eV red shift and the appearance of a low-energy absorption shoulder experimented by the N3 spectrum when substituting the dCNS ligands by Cld groups

FIG. 8.3 Comparison of the TD-DFT simulated (blue—dark gray color in print version) and experimental (red lines—gray color in print version) spectra for the cis-[Ru(4,40 -COO-2,20 -bpy)2(X)2]4 , X ¼ NCS (left), and Cl (right) dyes in water solution. The chemical structures for both dyes are represented inset. (Figure adapted with permission from reference De Angelis, F.; Fantacci, S.; Selloni, A.; Nazeeruddin, M. K. Time Dependent Density Functional Theory Study of the Absorption Spectrum of the [Ru(4,40 -COO–2,20 -bpy)2(X) 2]4- (X ¼ NCS, Cl) Dyes in Water Solution. Chem. Phys. Lett. 2005, 415, 115–120. https://doi.org/10.1016/j.cplett.2005.08.044. Copyright 2005 Elsevier.)

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(see Fig. 8.3 right) (251). Moreover, the experimental solvatochromism shown by the N3 sensitizer going from ethanol to water was accurately predicted and related to a decreased dipole moment in the excited state with respect to the ground state, which translates into a higher stabilization of the ground state in solvents of increasing polarity (249). The following studies on Ru-based complexes have confirmed the importance of taking into account the surrounding effects (193,252– 254). From the analysis of the TD-DFT calculations shown in Fig. 8.3, the two bands appearing in the Vis region were assigned to mixed RudNCS/Cl-to-bipyridine-p* transitions, whereas the band in the UV region was attributed to local bipyridine p ! p* transitions. Very interestingly, the electron withdrawing nature of the dNCS/-Cl moieties localizes the photogenerated holes around this groups, and due to this, the lowest absorption bands of these complexes can be considered as NCS/Cl ligand-to-bipyridine transitions, rather than typical MLCT states. This feature is at the origin of the efficient intramolecular charge separation and the correct flow of charge toward the ligands bearing the carboxylic anchoring groups, thus making the N3 the system of reference to reach performant DSSC devices. The computational strategy setup for N3 was extended to its salt N719 and to the amphiphilic heteroleptic [Ru(dcbpyH2) (tdbpy)(NCS)2] and N621 dye (dcbpyH2 ¼ 4,40 -dicarboxy-2,20 -bipyridine, tdbpy ¼ 4,40 -tridecyl-2,20 -bipyridine) to evaluate the effect of protonation and counterions on the electronic structure, and redox and optical properties of the investigated dyes (255). The good agreement between the simulated properties and the experimental data has allowed the use of DFT and TD-DFT calculations to screen new Ru-polypyridyl dyes before moving on to synthesis. Moreover, through computational modeling, structure-electronic/optical properties relationships have been established, thus providing design rules to extend the solar light absorption toward the nIR. As illustrated in Fig. 8.3, the absorption features of standard Ru(II) complexes such as N719 or N3 are reproduced by computing their singlet states at the TD-DFT level. These dyes display their lowest energy absorption at 450 and 530 nm for their dCNS and dCl ligand equivalents, respectively, which, in a certain manner, limits their capability to absorb red photons in the nIR region from solar light (255). With the objective of red shifting the absorption of Ru(II) complex, new dyes showing a panchromatic absorption in the nIR-Vis region appeared in the literature. In this respect, the first one of these compounds was the so-called black dye (BD), which was synthetized by employing one single terpyridine (tpy) and three dCNS ligands (see Fig. 8.4), and it showed a red-shifted absorption centered at 610 nm (256). Later on, Segawa and co-workers developed a new panchromatic dye (DX1) by substituting one of the dCNS ligand of BD by a phosphine coordinated group (see Fig. 8.4). As a result, this complex presents a lower energy absorption band up to c. 800 nm contrary to the previously reported Ru(II) dye bands, attributed to singlet-to-triplet transitions (257). It is well known SOC has a small impact in determining the absorption spectra of the investigated ruthenium complexes; indeed, SOC is generally considered an atomic property related to heaviest elements; nevertheless, as a general trend, the SOC induces in this class of compounds a slightly red-shifted absorption tail, corresponding to the contribution of singlet-to-triplet excitations, with a concomitant slight reduction of the more intense singlet-to-singlet transitions, giving rise to the absorption maximum region, in line with expectations. With the aim of favoring the direct singletto-triplet transitions, similar dyes to DX1 owing modified phosphine groups bearing weaker ligand fields (DX2 and DX3) (258) and (DX4-DX6) (259) were synthetized and characterized (Fig. 8.4). Due to their enhanced SOC interaction, the lowest absorption features of this class of TMC sensitizers were shifted up to 1000 nm (258,259). In view of the experimental evidence, the treatment of relativistic effects became a major issue for the modeling of panchromatic dyes (260). The first work tackling the impact of SOC in the optoelectronic response of Ru(II) dye sensitizers was conducted by Daul et al., which analyzed the nature of MLCT states of the prototypical [Ru(bpy)3]2+ complexes, and pointed to the mixing of their singlet and triplet states as the origin of the degeneracy the MLCT absorption bands (261). However, this degeneracy was further measured experimentally by quantifying the HOMO splitting of the [M(bpy)3]2+ (M ¼ Fe, Ru, Os) dyes, which, indeed, was found only significant for M ¼ Os, whereas M ¼ Ru, Fe dyes showed a low-energy splitting below 0.3 eV (262). This low influence of the relativistic effects on the [Ru(bpy)3]2+ electronic structure was confirmed by some of us and extrapolated to N3 by comparing the optoelectronic properties of both dyes calculated at the SR and SOC FIG. 8.4 Chemical structures of the panchromatic Ru(II) dye complexes based on tpy ligands.

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levels of theory (263). Regarding their electronic properties, only a small destabilization of 0.1 eV was found for the HOMOs without implying a modification in the orbital delocalization upon including SOC. In the case of the absorption features, SOC inclusion induced only a small intensity decrease and a red shift in the lower energy region of the calculated spectrum (see Fig. 8.5A). Despite this, the small but sizable differences in the relative position of the singlet and triplet states of [Ru(bpy)3]2+ and N3 pointed to the possibility of enhancing the singlet-triplet coupling via ligand modification (263). In this regard, in view of the remarkable singlet-triplet interactions reported for the panchromatic dyes BD and DX1, which enabled to build high performant DSSC devices (11.4% for both dyes) (257,264), some of us proceeded to evaluate the impact of relativistic effects in the photovoltaic behavior of BD and DX1 (265). The larger HOMO energy splitting observed in DX1 than BD is attributed to their different electronic structures, where HOMO and HOMO 1 (of dxy and dxz characteristic in DX1) are switched in BD and displayed a reduced amount of metal characteristic. Due to this, the calculated spectrum of BD was not modified upon inclusion of SOC (Fig. 8.5B), whereas for DX1, it promoted a small shift absorption maximum (0.05 eV) and, more importantly, a broadening of the MLCT band prompted by this energy splitting (see Fig. 8.5C). As a result, the SOC-induced spectral broadening slightly enhances the light-harvesting efficiency, and it consequently contributes additional photocurrent (2.3 mA/cm2, 32% of increase) in DX1-sensitized DSSCs (Fig. 8.5D) (265). In view of the positive impact of singlet-triplet couplings in the device performance, further theoretical studies were carried out with the aim of boosting relativistic effects via chemical engineering of ligands. For instance, Kanno et al. showed that the substitution of the dCNS and dCl ligands in N3 and DX1 by iodine groups can be employed to increase the strength of SOC due to the heavier mass of this atom (266). On the same vain, Mishima et al. found very small modifications of the absorption features of DX1 by changing the substituents of the phosphine ligand and confirmed

FIG. 8.5 Simulated absorption spectra calculated at the SR (blue—dark gray color in print version) and SOC (red—light gray color in print version) levels of theory, together with their respective experimental spectra (black lines) for the (A) N3, (B) BD, and (C) DX1 dyes; and (D) maximum JSC (mA/ cm2, dotted lines) and IPCE curve (%, full lines) for DX1, as computed at the SR (blue—dark gray color in print version) and SOC (red—light gray color in print version) levels, against the ASTMG173 solar spectrum (black line). (Images adapted with permission from reference Ronca, E.; De Angelis, F.; Fantacci, S. Time-dependent density functional theory modeling of spin-orbit coupling in ruthenium and osmium solar cell sensitizers. J. Phys. Chem. C. 2014, 118, 17067–17078. https://doi.org/10.1021/jp500869r; Fantacci, S.; Ronca, E.; De Angelis, F. Impact of spin-orbit coupling on photocurrent generation in ruthenium dye-sensitized solar cells. J Phys Chem Lett 2014, 5, 375–380. https://doi.org/10.1021/jz402544r. Copyright 2014 American Chemical Society.)

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that relativistic effects are higher for these dyes with respect to BD due to the higher covalent characteristic of these ligands (255). Notably, TD-DFT SOC calculations showed that the large red-shifted absorption observed in DX3 is originated by the reduced energy difference between the two lowest energy triplet states, resulting in a higher coupling with the singlet states with respect to DX1 and DX2 (258). Finally, the introduction of a conjugated thiophene group coupled to the tpy ligand for DX4 and DX5 yielded to enhanced oscillator strengths for singlet-to-singlet and singlet-to-triplet transition, whereas the extent of singlet-triplet couplings remained unaltered (259). Overall, despite DX3 represents the upper limit for the red-shifted absorption in panchromatic Ru(II) complexes, relativistic effects can be significantly boosted by employing heavier atoms as metal centers as for instance Os (260,267,268). Despite their promising red-shifted absorption, the use of such as heavy atoms should be somehow limited due to the toxicity and scarcity issues commented before.

8.4

Modeling the photophysics of Fe(II) metal complexes: Tools and findings

As discussed before, the success of Ru-based complexes in DSSCs and other applications is essentially due to their longlived triplet 3MLCT states, up to microsecond timescales. As already mentioned, polypyridyl Fe(II) complexes, having their six valence electrons paired in the lowest energy metal-centered d orbitals, experience ultrafast (c. 50 fs) deactivation to low-lying metal-centered (MC) states, via the triplet 3MC and ultimately the quintuplet 5T2 states (269,270), thus impeding any efficient utilization of photoinduced electron transfer reactions. As exemplified in Fig. 8.6A, the main difference in the photophysics of Ru(II) vs Fe(II) complexes lies in the inversion of the MLCT and MC states energetics, which originates from the fundamental electronic structure features of the first-row metals (269). The absence of a radial nodal plane at a large distance from the nucleus for n ¼ 3 and l ¼ 2 (3d) wavefunctions causes a reduced screening of the nuclear charge and thus a “contraction” of the 3d orbitals (71). This, in turn, results in a decreased orbital overlap between the metal and the ligands and, thus, in a lowered ligand-field strength. This intrinsic weaker ligand-field splitting in Fe(II)-polypyridyl octahedral complexes compared to the Ru(II) analogous yields a stabilization of the MC states in the former, whereas the MLCT states, whose energy is related to the metal oxidation potential, remains essentially unchanged. Due to the distinct relaxation pathways that Fe(II) complexes can follow from the MLCT to the MC states, their photophysics is quite intricate and usually involves many possible cascade intersystem crossings (ISCs) (see Fig. 8.6B) (271). Many different strategies have been pursued to increase the lifetime of the Fe(II) complex MLCT states featuring the destabilization of the MC states by increasing the ligand strengths via chemical functionalization with strong s-donating (78,79,84), combined s- and p-donating (272,273), s-donating, and p-accepting groups (274,275); or the population of the 5MLCT state as obtained by further decreasing the ligand field to achieve a high-spin ground state (5MC) in highly strained complex structures (276,277). In this context, a computational setup able to track accurately the relative energies and relaxation decays of the MLCT and MC states via potential energy surface (PES) curves has been at heart of the recent developments of the Fe(II) complex for dye-sensitized technologies (278). Nonetheless, this is not a trivial task since one needs to track the different electronic, structural, and environmental effects governing the excited-state relaxation pathways. The simplest computational approach employed to this purpose is based on the computation of the equilibrium (adiabatic) energies for the MLCT

FIG. 8.6 (A) Schematic representation of the MLCT states decays of [Ru(bpy)3]2+ and [Fe(bpy)3]2+ complexes, where solid/dashed black arrows indicate light absorption/emission (labs/lem) mechanisms, and red curly arrows (gray color in print version) denote vibrational relaxation and nonadiabatic population processes. Note that MLCT and MC surfaces have been represented by continuous and dotted lines, respectively. (Image adapted with permission from reference Franc
es-Monerris, A.; Gros, P. C.; Assfeld, X.; Monari, A.; Pastore, M. Toward Luminescent Iron Complexes: Unravelling the Photophysics by Computing Potential Energy Surfaces. ChemPhotoChem. 2019, 3, 666–683. https://doi.org/10.1002/cptc.201900100. Copyright 2019 Wiley-VHC; (B) energy diagram representation of the distinct models for 1MLCT ! 5MC(5T) relaxation pathways proposed in the literature. Image adapted with permission from reference Nance, J.; Bowman, D. N.; Mukherjee, S.; Kelley, C. T.; Jakubikova, E. Insights into the Spin-State Transitions in [Fe(tpy)2]2+: Importance of the Terpyridine Rocking Motion. Inorg Chem 2015, 54, 11259–11268. https://doi.org/10.1021/acs.inorgchem.5b01747. Copyright 2015 American Chemical Society.)

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and MC states in their triplet and quintuplet geometries (273). Calculations of adiabatic energy levels, however, does not give information concerning the crossing points between the involved excited states and the associated energy barriers; this requires the characterization of the corresponding PESs to identify the different nonadiabatic events that lead decay pathways. A computationally affordable and conceptually simple approximation to PESs can be obtained by fitting the energies and the most relevant vibrational frequencies of the optimized excited states with harmonic functions (279,280). This technique was employed by Warnmark and co-workers to shed light on the origin of the long-lived 3MLCT states (528 ps) observed for an homoleptic Fe(II) complex-bearing bidentate NHC ligands. The authors attributed this exceptionally long lifetime to the presence of a relatively high energetic barrier (0.12 eV), which should be overcome to access to the conical intersection with the 3MC state, whose minimum lies lower in energy with respect to the 3MLCT minimum (see Fig. 8.7A) (281). Unfortunately, for certain cases, the PESs estimated by this method are not realistic, especially for those structures lying far from the Frank-Condon region or in those systems where the relaxation decays are not following a harmonic shape. Alternatively, PESs can be explicitly calculated by linearly interpolating one or several coordinates relevant for the photoprocess (282). This approach can deliver accurate enough energy profiles of the states of interest, rough (upper bound) estimate of crossing points, and reliable insights into the operative decay mechanisms and their specific competition. In the particular case of Fe complexes, the metal-ligand distances and ligand-metal-ligand angles represent the main geometrical parameters governing their excited-state energies (283) due to the antibonding characteristic of the orbitals populating the 3MC and 5MC states. The energy scans for these two coordinates have allowed to study the relaxation pathways of some Fe(II) complexes via the construction of their respective 2D PES curves, demonstrating that ISCs in [Fe(terpy)2]2+ cannot be described by using a single configuration coordinate (284). For getting more accurate PES curves including a larger number of molecular motions, one can implement certain interpolation algorithms to elucidate the energy paths between two given optimized structures, as for instance, the high and low spin geometries. In this regard, Jakubikova and co-workers employed a sparse grid interpolation algorithm to get an approximation of the energy paths between the ground and the 1,3MLCT and 3,5MC states of a terpyridine ligand complex [Fe(tpy)2]2+ ISCs. This work pointed to the relative rock motion of one of the ligands Y as the main geometrical coordinate dictating the crossing points between the studied states (see Fig. 8.7B) (272). However, when reconstructing PESs by interpolations between different minima, each point is not necessarily an energy minimum, and thus, the calculated pathway is not the lowest energy route. A more refined and computationally onerous approach consists in the calculation of minimum energy paths (MEPs), where any point represents a minimum in all directions perpendicular to the path (276). In this context, different computational schemes have been implemented. One of these techniques consists in considering the intrinsic reaction coordinate (IRC), which estimates the transition states (TSs) between two local minima (even with different multiplicities) by resorting, for instance, to analytic nuclear gradients (269,285). Some of us also employed this approach to elucidate the effect of the fac/mer isomerism on the photophysics of a bidentate pyridyl-carbene Fe(II) complex (286,287). We reported the full DFT PES landscape for both isomers by computing the MEPs from the 3MLCT to 3MC minima adopting the IRC algorithm (Fig. 8.8). Transient absorption spectroscopy (TAS) measurements suggested that the 3MLCT decay for both fac and mer isomers proceeds with a parallel

FIG. 8.7 (A) Scheme of the parabolic energy surfaces for the main states of the [Fe(btz)3]2+ is depicted on the left part of the image, as a function of its M-L distances; and (B) PESs for the 1A, 1,3MLCT, and 3,5MC for the [Fe(tpy)2]2+ complex as a function of the Rax and Y coordinates, while keeping fixed ˚ , as represented in the image on the right part. All energies are relative to the interpolated minimum of the 1A state. (Images the value of Req ¼ 2.01 A adapted with permission from references Nance, J.; Bowman, D. N.; Mukherjee, S.; Kelley, C. T.; Jakubikova, E. Insights into the Spin-State Transitions in [Fe(tpy)2]2+: Importance of the Terpyridine Rocking Motion. Inorg. Chem. 2015, 54, 11259–11268. https://doi.org/10.1021/acs.inorgchem.5b01747; € V.; Lomoth, R.; Persson, Cha´bera, P.; Kjaer, K. S.; Prakash, O.; Honarfar, A.; Liu, Y.; Fredin, L.A.; Harlang, T. C. B.; Lidin, S.; Uhlig, J.; Sundstrom, € P.; Warnmark, K. FeII Hexa N-Heterocyclic Carbene Complex with a 528 ps Metal-To-Ligand Charge-Transfer Excited-State Lifetime. J. Phys. Chem. Lett. 2018, 9, 459–463. https://doi.org/10.1021/acs.jpclett.7b02962. Copyright 2015 and 2018 American Chemical Society; for panels (A) and (B) respectively.)

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FIG. 8.8 Lowest-lying singlet, triplet, and quintet PESs of (A) fac-Fe-NHC and (B) mer-Fe-NHC. Arbitrary nuclear coordinate values are used to label the structures obtained along the decay path. CI, conical intersection; STC, singlet-triplet crossing; TQC, triplet-quintet crossing. (Image adapted with permission from reference Franc
es-Monerris, A.; Magra, K.; Darari, M.; Cebria´n, C.; Beley, M., Domenichini, E.; Haacke, S.; Pastore, M.; Assfeld, X.; Gros, P. C.; Monari, A. Synthesis and Computational Study of a Pyridylcarbene Fe(II) Complex: Unexpected Effects of fac/mer Isomerism in Metal-to-Ligand Triplet Potential Energy Surfaces. Inorg. Chem. 2018, 57, 10431–10441. https://doi.org/10.1021/acs.inorgchem.8b01695. Copyright 2018 American Chemical Society.)

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two-state mechanism: a fast component (2–3 ps and 3–4 for ps mer and fac, respectively) and a slow one (15–20 ps). In addition to this small difference in the fast component, the two isomers also showed different branching ratio, and hence, the probability of the two decay pathways. Indeed, while for the fac arrangement, the fast/slow branching ratio is 57%, and for the mer isomer, it accounts for 87%. In both isomers, based on the calculated PESs, the fast component was ascribed to the decay of the T2 state (T2/ S0 crossing points), whereas the slow component was assigned to the deactivation of the T1 state (T1/S0 crossing point). On the contrary, the differences in the branching ratio of the two components was interpreted based on the steeper MEP computed for the mer isomer (see Fig. 8.8B) with respect to that of the fac arrangement (see Fig. 8.8A). However, IRC algorithms still present some limitations due to the fact that optimizing TSs in the excited state can be rather problematic since in Fe(II) complexes, as in many other systems, the reaction coordinate is often complex and involves coupled variations in bond lengths, angles, and dihedrals, and the accuracy is highly dependent on the initial guess used as starting geometry. To overcome this problem, the nudged elastic band (NEB) method (288) optimizes a number of intermediate images along the reaction path between the reactant and the product. Each image finds the lowest energy possible while maintaining equal spacing to neighboring images by adding spring forces along the band between images and by projecting out the component of the force due to the potential perpendicular to the band (288). This methodology accounted for 0.2 eV the energy barrier between the 3MLCT and 3MC states in [Fe(tpy)2]2+ complexes bearing cyclometallating ligands with degenerate triplet states. Such a high-energy barrier may trap the system in the initially populated 3 MLCT state. Indeed, the authors suggested a 3MLCT Ð 3MC equilibrium process highly influenced by the deactivation of the 3MC state via its crossing with the ground state, requiring a barrier of 0.18 eV, also depending on the SOC magnitude between the 3MC and S0 states (273). Before closing this section, it is noteworthy to mention the attempts made to extract direct information on the excitedstate lifetime by theoretically investigating the excited relaxation pathways, through excited-state quantum dynamics simulations (74). In this direction, Papai and co-workers performed quantum wavepacket dynamics simulations on top of PESs built as solutions of a spin-vibronic Hamiltonian describing the nonadiabatic vibronic coupling (289). This work elucidated the excited-state relaxation process in a [Fe(bmip)2]2+ (bmip ¼ 2,6-bis(3-methyl-imidazole-1-ylidene-pyridine) complex: 1 MLCT ! 3MLCT relaxation occurs within 100 fs, followed by population of the 3MC states. The 3MLCT-3MC deactivation is slower than [Fe(bpy)3]2+ due to a high-excited-state barrier and modest SOC along two dominant FedL breathing modes modulating the FedN and FedC bond distances (290). The same approach showed that the substitution of the methyl moieties of this complex by tert-butyl groups promotes the stabilization of the 1MC states, thus enabling their population from the 1,3MLCT states close to the Frank-Condon geometry (173). The use of on-the-fly nonadiabatic dynamics is computationally prohibitive when dealing with TMCs, characterized by a high density of quasi-degenerate and strongly coupled states that could be populated. Moreover, one needs to propagate a large number of trajectories to achieve a statistically significant representation of the process. Recently, the Gonzalez’s group reported state-hopping dynamics simulations with a linear vibronic coupling model to understand the first steps of the excited-state relaxation in a Fe-NHC complex (291), and it evidenced the ultrafast population (50 fs) of its triplet states upon irradiation with a small component of MC characteristic, which leads to nonradiative recombination (292). Although no direct comparison with experiments is provided in this work, it nicely shows the potentiality of the nonadiabatic dynamics approach to shed light on the photophysics of TMCs.

8.5 Interfacial properties of Fe-NHC-sensitized TiO2 The first photovoltaic characterization of the Fe(II)-NHC complex C1 (Fig. 8.2) was reported by Gros end co-workers in 2015 with the first realization of iron-based DSSCs (80). Despite their relatively long lifetimes and suitable energetic alignments with both the TiO2 CB edge and the I /I3 oxidation potential, the homoleptic Fe(II)-NHC complex yielded extremely low photocurrent and photovoltage, resulting in weak power conversion efficiencies (0.13%) (80). It is important to highlight here that besides a sufficiently long 1,3MLCT lifetime, being mandatory for injecting electrons into the TiO2 CB, the measured overall cell efficiency finally depends on the specific properties of the dye-sensitized interface (adsorption configuration, electronic coupling, charge generation, charge recombination, etc.). On the contrary, a subsequent work by W€arnmark and co-workers (81) claimed a record injection efficiency, almost unitary, for the same C1 complex, attributing the low photovoltaic performances reported in Ref. [80] to a fast recombination of the injected electron with the oxidized dye. In view of this, after the first proof of the practical use of Fe sensitizers in solar energy technologies, the main strategies followed to boost their device efficiencies have followed two different directions: the chemical engineering of the dye structure (26–28,280,293,294) and the optimization of the electrolyte composition (25,26,295,296).

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Before analyzing the interfacial dye@TiO2 properties of Ru(II) vs Fe(II) complexes, we will briefly discuss the nature of their lowest energy absorption features. Regarding the Fe(II)-NHC complexes, the heteroleptic dye, ARM13, displays two absorptions bands in the Vis region centered at 461 nm and 385 nm, originated from MLCT transitions toward the anchoring ligand; and a local p ! p* transitions of the unanchored ligand, respectively. Unfortunately, due to the low donor characteristic of the Fe(II) metal center, holes start to delocalize around the unanchored ligand, thus making the charge separation less efficient due to its local characteristic and facilitating electron recombination. The inclusion of a second carboxylic group in the homoleptic dye C1 induces a small red shift of the absorption bands and an increase in the absorption intensity (see Fig. 8.9 left). Due to their symmetry, the hole/electron contributions of the two ligands to the MLCT state are equal (Fig. 8.9). Nonetheless, this symmetry will be broken upon the deprotonation of the ligand which is attached to the surface. Notably, for the deprotonated C1 (called C1(D) in the following), its lowest energy MLCT transition is pointing toward the protonated ligand, and therefore, in the case of being populated, this will yield to a CT moving in the opposite direction to the charge injection into the TiO2 surface. Interestingly, the insertion of p-conjugated bridges (i.e., a phenylene or thiophene moieties) between the carboxylic group and its correspondent NHC ligand in the heteroleptic dyes makes the MLCT states acquiring a local p-p characteristic around these moieties, thus detrimental for the charge separation due to the lower extent of electron delocalization around the carboxylic anchoring groups (16% vs 7%–8% for ARM13 and their p-conjugated anchoring ligand analogues) (26). Finally, the functionalization of ARM13 with an electron donor moiety (as in ARM130 or FeCD) resulted in a considerable enhancement of the absorption band harvesting, while the nature of the MLCT transition remained unaltered (Fig. 8.9). Overall, the different natures of the lowest energy MLCT bands of the Fe (II) vs Ru(II) dyes in terms of charge separation and directionality already provide some important hints to explain the superior performance reached by the latter complexes, but let us consider their interface with TiO2 and some possible strategies to mitigate these drawbacks. The first work dealing with the interface of the Ru(II) complex with TiO2 was reported in 2005 by Person and coworkers, which studied the absorption of N3 attached to a (TiO2)38 cluster at the TD-DFT level, and predicted an MLCT injection time which amounted to 10 fs (297). During the following years, many works appeared in the literature with the aim of discerning which are the most stable anchoring binding modes for that dye and their relation with the injection rates, and reported injection lifetimes which ranged from few to 100 fs depending on the anchoring mode (87,298,299). In one of these works, some of us reported a TD-DFT study of the N3 dye adsorbed on a (TiO2)82 cluster with three carboxylic moieties attached to the semiconductor (one bidentate and two monodentate binding modes) (89). The computed absorption spectrum in the lowest energy region was nicely reproducing the experimental one (see Fig. 8.10-a), with the most prominent transition (the one with the largest oscillator strength) corresponding to the S18 state (555 nm). By looking at the charge density difference plot between S0 and S18 depicted in Fig. 8.10B, it is possible to appreciate how, apart from the adequate CT directionality and separation, the excited electrons are largely delocalized along the TiO2 surface, which is a clear fingerprint for a direct ultrafast injection process. Focusing on the interfacial energetic alignment presented in Fig. 8.10C, the HOMO of the dye in the N3@TiO2 lies far from the CB/VB of TiO2, thus avoiding undesired recombination processes, whereas its LUMO lies about 0.3 eV above the CB edge of TiO2, assuring the correct thermodynamics of the electron injection, and its position in a densely populated region of TiO2 states yields to a strong coupling with the semiconductor (89). It is noteworthy to highlight that in this paragraph, we focused only on the electronic and absorption properties of the N3@TiO2 interface as a illustrative way to discern the main differences with respect to the Fe(II)

FIG. 8.9 Simulated absorption spectra (left) and natural transition orbitals (NTOs) (277) for the most prominent lowest energy states (right) for C1, ARM13, and ARM130, as calculated at the TD-B3LYP*/6-311G** level of theory with the CPCM to treat the methanol solvent environment. The absorption wavelengths and oscillator strengths for each state represented in the NTOs are also reported. The isovalue used to represent the isodensity plots was set to 0.02 a.u.

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FIG. 8.10 (A) Normalized simulated (blue—dark gray color in print version) and experimental (red—gray color in print version) absorption spectra of N719 on TiO2, with the inset representing its calculated density of singlet (black) and triplet (magenta—light gray color in print version) excited states; (B) S0 ! S18 charge density difference plot of the N719@TiO2 system, where blue (dark gray color in print version)/yellow colors (light light gray color in print version) represent the photogenerated electron/hole densities; c) energetic level alignment of the ground and excited states for the N719@TiO2 system (left) and calculated DOS of the interacting and isolated TiO2 cluster (right). (Image adapted with permission from reference Pastore, M.; Fantacci, S.; De Angelis, F. Modeling Excited States and Alignment of Energy Levels in Dye-Sensitized Solar Cells: Successes, Failures, and Challenges. J. Phys. Chem. C. 2013, 117, 3685–3700. https://doi.org/10.1021/jp3095227. Copyright 2013 American Chemical Society.)

dye@TiO2 systems, whereas many other works have been dedicated to the interfaces of another types of Ru(II) dyes with TiO2 (such as BD) (300–302) and resulted in similar results as the ones observed with N3. The first theoretical work investigating the impact of the cyclometallation of the pyridine ligands in the MLCT lifetimes and injection rates of the [Fe(tpy)2]2+ complexes was reported by Jakubikova and co-workers (303). They found that although the use of carbene ligands assists in the stabilization of the 1MLCT states, they can be detrimental to the injection due to the localization of the electron density around the carbene groups, instead of the carboxylic anchoring moieties, resulting in injection times in the order of 100 fs (303). This work represented the first evidence of the suitability of employing NHC ligand Fe(II) complexes for photovoltaic applications. In this regard, Fredin and co-workers studied the electronic structure of the heteroleptic NHC complex interface (ARM13@TiO2) showing that NHC complexes exhibit a suitable energy level alignment and LUMO couplings with respect to the TiO2 CB, yielding electron injection rates of about 100 fs, as estimated on the basis of a Newns-Anderson model (280). Nonetheless, these calculations were not able to fully rationalize the low device efficiencies first reported when employing this dye as sensitizer, and the main question related with the fast recombination occurring at the semiconductor interface was still unclear (304). In this context, some of us investigated the interface with TiO2 for the prototype homoleptic and heteroleptic NHC Fe(II) complex dyes (C1 and ARM13) (293). Fig. 8.11 presents the PDOS along with the molecular and semiconductor parts of the dye@TiO2 interface. All considered complexes remained tilted on the semiconductor surface, forming an angle with respect to the surface plane of approximately 45 degrees, which was consistent with the previous studies (280). The energetic alignment for the three complexes resembled the one schematized in Fig. 8.10C, with the dye HOMOs lying above the VB and the LUMO above the TiO2 CB edge. In the case of the HOMOs, although these levels are mainly delocalized over the molecular backbone, they also exhibit a considerable mixing with the TiO2 VB states (30%–50%), as evidenced by the tail of the TiO2 VB extending through the dye’s HOMOs PDOS, which can be originated by the close contact between the tilted dyes and the surface. This high HOMO coupling with both CB and VB states leads to high recombination rates (in the ns or ps timescale) and thus to an inefficient charge injection/recombination at the interface. On the contrary, the energetics and spatial

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FIG. 8.11 PDOS over the dyes (black/red colors—gray color in print version for C1/ARM13) and TiO2 surface (blue—dark gray color in print version), and main dye’s MOs isodensity plots for the C1@TiO2 and ARM13@TiO2 systems. The corresponding percentage of dye’s delocalization and energies in eV are reported for each MO plot. Vertical bar intensities represent the percentage of the dye’s contribution to the total system as calculated by Mulliken population analysis. (Image adapted with permission reference Pastore, M.; Duchanois, T.; Liu, L.; Monari, A.; Assfeld, X.; Haacke, S.; Gros, P. C. Interfacial Charge Separation and Photovoltaic Efficiency in Fe(II)-Carbene Sensitized Solar Cells. Phys. Chem. Chem. Phys. 2016, 18, 28069–28081. https:// doi.org/10.1039/c6cp05535d. Copyright 2016 Royal Society of Chemistry.)

distribution of the unoccupied molecular states differ as a function of the dye. In the case of C1, the dye’s LUMO is essentially not coupled with the TiO2 CB states (with 93% localized on the dye backbone), whereas the LUMO + 1 exhibits a strong coupling with the semiconductor (2% dye’s contribution). As a result, the primary charge transfer channel from the lowest MLCT state appears to point toward the opposite direction to the charge injection into the semiconductor. Similar PDOS features were also observed for ARM13, thus demonstrating that the use of heteroleptic dyes allows obtaining the correct CT directionality. The calculated injection times for the strongly coupled LUMO levels were in the few fs timescale, thus showing an ultrafast injection process in analogous way as the one reported in Ru(II) complex dyes (293). With the objective of evaluating the impact of the inclusion MgI2 additives in the electrolyte composition, which allowed to reach record efficiencies in Fe-NHC DSSCs (25), we followed the same computational approach used by some of us to model the effect of Li+ cations at the indoline D102@TiO2 interfaces (305), consisting of the addition of a Mg2+ cation lying on the top of the TiO2 surface (26). The comparison of the PDOS among the molecular backbone and the

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semiconductor corresponding to the dye-Mg@TiO2 and dye@TiO2 surfaces for the homoleptic C1 and heteroleptic ARM13 complexes is presented in Fig. 8.12. Notably, the Mg2+ cation strongly interacts with the dye’s anchoring group, ˚ ). Therefore, one may expect a more pronounced impact lying very close to oxygen atoms of the carboxylic moiety (2.3 A of the cation in those molecular levels localized on the anchoring NHC ligands. The main effect prompted by the presence of the cation in the PDOS features of the pristine surfaces is a negative energy shift, which is especially prominent for the unoccupied orbitals of both dyes. In the case of C1, the pristine surface PDOS presents the same characteristics as the ones described before (293) in terms of the nature of the non-/strongly coupled LUMO/LUMO + 1 levels. Nonetheless, since the

FIG. 8.12 Normalized PDOS over the dye (purple—light gray color in print version)/green—gray color in print version for C1/ARM13), TiO2 surface (orange—light gray color in print version), and Mg2+ cation (magenta—light light gray color in print version) for the C1@TiO2, C1@Mg-TiO2, ARM13@TiO2, and ARM13@Mg-TiO2 (from the top to the bottom) systems, as obtained from Mulliken population analysis. Vertical bars are used to represent the dye’s states conforming the PDOS. The isodensity plots (isovalue 0.02 a.u.) displayed in the onsets of the PDOS plots correspond to the two lowest unoccupied MOs of the dye. (Image adapted with permission from reference Reddy-Marri, A.; Marchini, E.; Diez-Cabanes, V.; Argazzi, R.; Pastore, M.; Caramori, S.; Gros, P. C. Record Power Conversion Efficiencies for Iron (II)-NHC-Sensitized DSSCs from Rational Molecular Engineering and Electrolyte Optimization. J. Mater. Chem. A. 2021, 9, 3540–3554. https://doi.org/10.1039/x0xx00000x. Copyright 2021 Royal Society of Chemistry.)

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unoccupied orbitals localized on the anchoring ligand are more affected by the electrostatic effect of the cation, this level becomes the LUMO when Mg2+ is adsorbed at the interface, thus contributing to the desired reversal of the interfacial charge separation direction. Regarding ARM13, the incorporation of the cation induces a shift of its LUMO levels, thus falling in the same energy region as the stabilized TiO2 CB edge. Since the CB edge possesses lower density of states, one may expect that the injection rates will decrease upon the shift prompted by the cation. Nonetheless, the larger coupling magnitudes between the LUMO and the CB edge states lead to an increase of about 20% in the injection rates with respect to the values reported with the pristine TiO2 surfaces, both for ARM13 and C1 (26). Similar stabilization effects induced by the presence of lithium (Li+) and tetrabutylammonium (TBA+) additives in the electrolyte composition were found in [Fe(bpy-dca)2(CNS)2]2+ complex dye-sensitized interfaces (233). Overall, it has been demonstrated the change of the nature of the dye LUMO in C1, the decrease in the injection driving forces, and the largest injection rates, derived from a more efficient coupling with the CB edge, were at the origin of the enhancement of the device performances (26). Recently, we tackled the modeling of the ARM130-based DSSCs, for which record power conversion efficiencies were recorded. We investigated the electronic structure and charge generation properties of the ARM130@TiO2 interface, comparing these results with the ones from the unsubstituted dye ARM13 (27). The similar anchoring geometries, injection, and recombination rates with respect to the reference ARM13@TiO2 system demonstrated that donor substitution is not significantly affecting the interfacial charge generation processes. As a result, the higher efficiencies achieved with ARM130 can be only ascribed to the enhancement of the light harvesting of the MLCT band, as shown in Fig. 8.9 left (27).

8.6

Conclusions

In this chapter, we have presented an extensive overview of the recent advances achieved in modeling DSSC devices, taking as reference two of the most representative and relevant TMC dye sensitizers used in these technologies, due to their rich photophysics: Ru(II) and Fe(II) complexes. After providing a small review of the device operation and the main components integrating the cells, we have focused our discussion on the state-of-the-start theoretical tools used nowadays to model these components and their interfaces, emphasizing the current challenges and limitations, and finishing by illustrating the main trends in the field aimed to couple these models with experimental data and artificial intelligence to carry out a data-driven device design. In a subsequent step, we have evidenced how these methodologies can be applied to tackle the main strategies to optimize TMC dye sensitizer-based cells: (i) the red shift and intensification of the lowest energy absorption for Ru(II) dyes via enhancement of the SOC effects; (ii) extension of the MLCT lifetimes in Fe(II) complexes by investigation of their minimum energy relaxation paths; and (iii) improvement of the CT occurring at the Fe(II) dye@TiO2 interface by analyzing the underlying injection/recombination processes and the nature of the dye MLCT bands. These examples clearly evidenced the active role played by the computational chemistry in the last progress reached in the DSSC field and corroborated the idea that the development of accurate and affordable methods to treat complex systems (i.e., solvent/electrolyte environments) and phenomena (i.e., interfacial ET processes or ISCs of the dye states), coupled with experimental measurements dedicated to the fundamental understanding of the physical processes taking place during the device operation, may have a central role in the development of highly efficient DSSC devices in future.

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Chapter 9

Solar cells: Organic photovoltaic solar cells Daniele Fazzi Universita` di Bologna, Dipartimento di Chimica "Giacomo Ciamician", via F. Selmi, Bologna, Italy

Chapter outline 9.1 Introduction 9.1.1 Organic photovoltaics 9.1.2 OPV materials 9.1.3 Models to describe charge generation in OSCs 9.2 Excitonic processes: Excited states at the donor/ acceptor interfaces 9.2.1 Electronic structure methods to describe the excited states at D/A OPV interfaces 9.2.2 Analytical tools to characterize the excited-state wavefunction 9.2.3 Examples of polymer/fullerene OPV interfaces

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9.3 Time-dependent processes: Excited-state dynamics in donor and donor/acceptor domains 9.3.1 Excited-state dynamics: Brief overview of nonadiabatic surface hopping and multiconfiguration time-dependent Hartree methods in the context of OPV 9.3.2 Excited-state dynamics of oligothiophenes as prototypes for P3HT 9.3.3 Examples of polythiophene-/fullerene-based interfaces 9.4 Conclusions References

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9.1 Introduction 9.1.1 Organic photovoltaics As we have seen in the previous chapter, the photovoltaic effect is the ability of materials to convert light (photon) into electrical current (voltage potential). When the active materials are organic p-electron-conjugated molecules or polymers, we refer to organic photovoltaics (OPV). OPV is a vibrant research area within the field of organic electronics (OE), and it is focused on understanding the generation of charges (holes and electrons, also called polarons) via light excitation of organic semiconductors (1). Photocurrent generation in organic materials such as phthalocyanines and acenes single crystals was first reported in the 1960s. Initial power conversion efficiency (PCE, electricity produced by the solar cell under solar irradiation) was very low (PCE < 0.1%) (2,3). Since the early 2000s, the PCE of organic solar cells (OSCs) has rapidly increased, rising from 3% in 2001 to record values of 18.2% in recent years (4). In analogy to the inorganic silicon-based solar cells, in which a p-/n-type junction is employed to create free charge carriers, in OPV, electron donor (D) and acceptor (A) materials are blended to create D/A domains and interfaces, namely, regions where holes and electrons can be efficiently generated, separated, and transported. Donors and acceptors are organic materials characterized by different ionization potentials (IPs) and electron affinities (EAs). In a simple picture, the free energy offset (DG) between the IPs and EAs at the D/A interface is the driving force for the charge generation processes (5). The photovoltaic effect in OPV D/A solar cells can be summarized into the following steps (Fig. 9.1): (i) light absorption and excited-state formation; (ii) excited-state transfer and relaxation at the D/A interfaces; (iii) excited-state separation and generation of holes and electrons on the D and A domains, respectively; and (iv) charge transport and collection. Within the context of condensed matter, excited states undergo the name of exciton (i.e. bound electron-hole pair); therefore, the OPV effect can be reformulated as: (i) exciton formation, (ii) exciton transfer and relaxation at the D/A interfaces, (iii) exciton separation and generation of holes and electrons, and (iv) charge transport and collection (1).

Theoretical and Computational Photochemistry. https://doi.org/10.1016/B978-0-323-91738-4.00008-7 Copyright © 2023 Elsevier Inc. All rights reserved.

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FIG. 9.1 Schematic representation of a D/A OPV interface. Lines sketch polymer chains (donor domain) and circles fullerene-like molecules (acceptor domain). Upon light absorption, an exciton (electron-hole pair) is formed. Electrons are represented as white circles, and holes as black circles. The exciton, generated in the light-absorbing (donor) domain, can diffuse toward the D/A interface to populate a charge-transfer (CT) state, with the hole localized on the D domain and the electron on the A domain. The Coulomb capture radius (rC) and the thermalization length (a) are reported recalling the classical theory (i.e., Onsager’s theory, see the next paragraph—Models to describe charge generation in OSCs) for charge separation. If a  rC, the electron and hole separate. Upon exciton separation, the hole and electron diffuse within the organic materials, leading to an electric current. (Reprinted (adapted) with permission from Clarke, T. M.; Durrant, J. R., Charge Photogeneration in Organic Solar Cells. Chem. Rev. 2010, 110 (11), 6736–6767. Copyright 2022 American Chemical Society.)

In this chapter, theoretical and computational approaches to describe the electronic processes in D/A OPV interfaces will be reviewed. Given the broadness of the field, the reader will be referred toward specific reviews and perspectives for further details and in-depth case studies.

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The Role of Chlorine in the Formation Process of “CH3NH3PbI3-xClx” Perovskite. Adv. Funct. Mater. 2014, 24, 7102–7108. 126. Lee, M. M.; Teuscher, J.; Miyasaka, T.; Murakami, T. N.; Snaith, H. J. Efficient Hybrid Solar Cells Based on Meso-Superstructured Organometal Halide Perovskites. Science 2012, 338 (6107), 643–648. 127. Stone, K. H.; Gold-Parker, A.; Pool, V. L.; Unger, E. L.; Bowring, A. R.; McGehee, M. D.; Toney, M. F.; Tassone, C. J. Transformation From Crystalline Precursor to Perovskite in PbCl2-Derived MAPbI3. Nat. Commun. 2018, 9 (1), 3458.

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128. Colella, S.; Mosconi, E.; Pellegrino, G.; Alberti, A.; Guerra, V. L. P.; Masi, S.; Listorti, A.; Rizzo, A.; Condorelli, G. G.; De Angelis, F.; Gigli, G. Elusive Presence of Chloride in Mixed Halide Perovskite Solar Cells. J. Phys. Chem. Lett. 2014, 5 (20), 3532–3538. 129. Meggiolaro, D.; Mosconi, E.; Proppe, A. H.; Quintero-Bermudez, R.; Kelley, S. O.; Sargent, E. H.; De Angelis, F. Energy Level Tuning at the MAPbI3 Perovskite/Contact Interface Using Chemical Treatment. ACS Energy Lett. 2019, 4 (9), 2181–2184. 130. Emara, J.; Schnier, T.; Pourdavoud, N.; Riedl, T.; Meerholz, K.; Olthof, S. Impact of Film Stoichiometry on the Ionization Energy and Electronic Structure of CH3NH3PbI3 Perovskites. Adv. Mater. 2016, 28 (3), 553–559. 131. Yajima, T.; Hikita, Y.; Minohara, M.; Bell, C.; Mundy, J. A.; Kourkoutis, L. F.; Muller, D. A.; Kumigashira, H.; Oshima, M.; Hwang, H. Y. Controlling Band Alignments by Artificial Interface Dipoles at Perovskite Heterointerfaces. Nat. 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Saliba, M.; Orlandi, S.; Matsui, T.; Aghazada, S.; Cavazzini, M.; Correa-Baena, J.-P.; Gao, P.; Scopelliti, R.; Mosconi, E.; Dahmen, K.-H.; De Angelis, F.; Abate, A.; Hagfeldt, A.; Pozzi, G.; Graetzel, M.; Nazeeruddin, M. K. A Molecularly Engineered Hole-Transporting Material for Efficient Perovskite Solar Cells. Nat. Energy 2016, 1 (2). 141. Torres, A.; Rego, L. G. C. Surface Effects and Adsorption of Methoxy Anchors on Hybrid Lead Iodide Perovskites: Insights for Spiro-MeOTAD Attachment. J. Phys. Chem. C 2014, 118 (46), 26947–26954. 142. Yin, J.; Cortecchia, D.; Krishna, A.; Chen, S.; Mathews, N.; Grimsdale, A. C.; Soci, C. Interfacial Charge Transfer Anisotropy in Polycrystalline Lead Iodide Perovskite Films. J. Phys. Chem. Lett. 2015, 6 (8), 1396–1402. 143. Soler, J. M.; Artacho, E.; Gale, J. D.; Garcı´a, A.; Junquera, J.; Ordejo´n, P.; Sa´nchez-Portal, D. The SIESTA Method for Ab Initio Order-N Materials Simulation. J. Phys. Condens. Matter 2002, 14, 2745–2779. 144. 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Chapter 11

Thermally activated delayed fluorescence Leonardo Evaristo de Sousa and Piotr de Silva Department of Energy Conversion and Storage, Technical University of Denmark, Kongens Lyngby, Denmark

Chapter outline 11.1 11.2 11.3 11.4

11.1

Introduction Excited states calculations Condensed phase effects Role of charge transfer and local excited states

293 295 297 299

11.5 Vibronic effects and rate calculations 11.6 Synopsis References

302 306 307

Introduction

Thermally activated delayed fluorescence (TADF) is a photophysical phenomenon resulting in light emission from an electronically excited molecule after a thermal upconversion from a lower lying triplet state to an emissive singlet state. TADF was first experimentally observed in 1930 by S. Boudin (1), who studied the phosphorescence of the eosin Y compound. It took around 30 years to verify that the observed long-lived emission occurred at a different energy than phosphorescence and involved a thermal activation energy of around 10 kcal/mol (2). The phenomenon has been named E-type fluorescence to distinguish it from the P-type delayed fluorescence, where the latter resulted from triplet-triplet annihilation. Both types of delayed fluorescence are viable mechanisms for converting typically dark triplet excited states into potentially emissive singlet states; therefore, they are often considered triplet-harvesting mechanisms (3). Despite its potential to convert the energy deposited in molecular excited states to light, the applications of TADF have been very limited for a long time due to the low efficiency of the process. Nevertheless, the idea seemed attractive for improving the efficiency of organic lightemitting diodes (OLEDs), whose operation is based on electroluminescence, that is, converting electrically generated excited states to photons. The recent and ever-growing interest in TADF materials for OLEDs has started, thanks to the pioneering works of Adachi and collaborators (4–6). They laid the foundations for the molecular design of TADF emitters with internal quantum efficiencies reaching 100% and competitive external quantum efficiencies in OLED devices. Since then, extensive work by numerous research groups worldwide has resulted in developing new concepts for TADF-based materials and led to a fairly detailed understanding of what transpired to be a complicated photophysical mechanism. Thanks to this progress, TADF materials are not only becoming commercially viable for OLEDs but also finding applications in areas like lasing (7), photocatalysis (8), bioimaging, and sensing (9,10). Since TADF is a thermally activated process, its rate depends on the activation barrier. Assuming that the emitter has a closed-shell singlet ground state and internal conversion processes are fast enough, the upconversion happens between the lowest-lying triplet, T1, and the lowest-lying singlet, S1, states in a process dubbed reverse intersystem crossing (rISC). The activation energy needed for the upconversion corresponds to the singlet-triplet energy gap DEST ¼ E(S1) E(T1) and is usually positive. Therefore, the rISC rate has often been modeled with an empirical Arrhenius-like equation for the temperature dependence of reaction rates krISC ¼ Ae

DEST =kB T

,

(11.1)

where A is a prefactor, kB is the Boltzmann constant, and T is the temperature. The relative alignment of the key electronic states and the electronic transitions contributing to TADF are illustrated in Fig. 11.1 Based on Eq. (11.1), it became clear what the molecular design strategy should be to assure efficient rISC and, consequently, high delayed fluorescence rates. Namely, one should design emitters such that DEST is minimized and ideally approaching 0. To understand how it could be achieved, a simple two-state model (4) can be invoked. If we assume that both

Theoretical and Computational Photochemistry. https://doi.org/10.1016/B978-0-323-91738-4.00010-5 Copyright © 2023 Elsevier Inc. All rights reserved.

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FIG. 11.1 Diagram of the alignment of the excited states and photophysical processes responsible for TADF.

S1 and T1 states are one-electron excitations between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO), their energy splitting is twice the HOMO-LUMO exchange integral KHL: Z Z fH ðr 1 ÞfL ðr 1 ÞfH ðr 2 ÞfL ðr 2 Þ dr 1 dr 2 (11.2) DEST ¼ 2K HL ¼ 2 jr 1 r 2 j Therefore, to minimize DEST, the emitters should have a low-lying HOMO!LUMO transition where both orbitals are spatially separated as much as possible. The HOMO-LUMO separation has been practically realized in the donor (D)-acceptor (A) molecular architectures. The excitation from the high-lying HOMO of the donor to the low-lying LUMO of the acceptor generated a chargetransfer state characterized by a small singlet-triplet gap. The proposed systems were either exciplexes (11), where the small orbital overlap was achieved through the spatial separation of molecules, or chemically bonded D-A dyads, where the near-orthogonal dihedral angle between the D-A units assured an almost completely broken conjugation (12,13). Extensions of the D-A concept include connecting multiple donor units to an acceptor core to form D-AD triads or, more generally (D)n-A systems (12,13), but also D-A-based polymers (14), macrocycles (15), and dendrimers (16). Recently, a new class of emitters based on the multiple resonance effect (MRE) has shown much promise (17). In MRE molecules, the HOMO and LUMO have a characteristic interlacing pattern, where both orbitals are peaked at different but neighboring atoms. The HOMO and LUMO orbitals of exemplary D-A and MRE emitters are included in Fig. 11.2. The initial successes in the molecular design based on the HOMO-LUMO separation allowed the TADF field to grow rapidly and attract many researchers. This push helped the community realize that the simple design rule focused solely on minimizing DEST through maximizing the charge-transfer character of S1 and T1 states is flawed or limited at best. The realization came from the fact that while pure CT states would indeed make DEST small, the rISC and the fluorescence rates would be negatively impacted. First, rISC is a nonradiative electronic transition between states of different spin multiplicities; therefore, it is forbidden unless the states can mix, to some extent, through spin-orbit coupling (SOC). According to El-Sayed’s rule (18), such transition is only possible between states differing in the electronic character so that the total angular momentum is conserved after the spin flip. Therefore, an efficient rISC is not possible between pure CT singlet and triplet states, despite the vanishing energy gap. Additionally, the negligible overlap between HOMO and LUMO directly leads to a vanishing oscillator strength for the S1!S0 transition, which is determined by the transition dipole moment between the frontier orbitals. In this case, even if rISC was possible and sufficiently efficient, the delayed fluorescence would not, and the upconverted excitons would decay mostly nonradiatively, defeating the purpose for any light-emission applications. The limitations of the two-state CT model and the resulting molecular design strategies called for a more thorough understanding of the TADF mechanism. It became apparent that in addition to the singlet-triplet gap, other factors play a decisive role in the delayed fluorescence efficiency and in assuring high quantum yields. Since then, numerous reports have focused on the importance of the environment in the excited states’ energy alignment, the role of local excitations (LEs) in allowing appreciable rISC and fluorescence rates (19), and the coupling of the electronic states with molecular

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FIG. 11.2 HOMO (left) and LUMO (right) orbitals of a donor-acceptor, D-A (top) and multiple resonance effect, MRE (bottom) TADF emitters.

vibrations, which enable the effective mixing of the CT and LE states (20). These effects will be reviewed in the following sections of the chapter. From the beginning of the development of TADF materials, theory and computational modeling played an essential role in the mechanistic understanding of the process and the design of new materials. The initial studies mainly focused on direct DEST calculations and illustrating the extent of the electron-hole separation. However, since then, the theoretical studies have proved instrumental for developing the state-of-the-art material-level understanding of the TADF mechanism and are often the basis for the rational design of new materials. In this chapter, we will review the various fundamental aspects related to the computational and theoretical studies of TADF. We do not aim to cover the rich literature on the materials or device design, but the main focus is on highlighting the methodological challenges and developing a conceptual understanding of how different physical effects determine the overall efficiency of the process.

11.2

Excited states calculations

It should be clear that the quantitative and qualitative studies of electronic excited states are central to the development of TADF materials. Experimentally, such studies are predominantly done using optical spectroscopy techniques, both steadystate and time-resolved. Based on the spectra collected under various conditions (e.g., in different solvents or at different temperatures), the energies and characters of the excited states can be elucidated. At the same time, quantum-mechanical electronic structure calculations can directly reveal similar properties and can be either complementary to the experiment to gain deeper insights or replace the experiment in the context of virtual materials design. Considering that simulations are always based on some model and entail many approximations, it is necessary to compare the computational and experimental results and understand where the differences might be coming from. When discussing the electronic structure of TADF molecules, the relevant low-lying excited states are typically assigned a well-defined energy and character (CT, LE, mixed), which allows for determining, for example, the singlettriplet energy gap. It is already a conceptual approximation as it disregards the fact that molecules vibrate, and both the excitation energies and characters fluctuate during this motion. Experimentally, the state’s energy is understood either as the position of the peak or the onset of an associated band in the spectrum. Readout of these quantities can sometimes be hindered due to poorly resolved or overlapping bands. The situation is not much clearer for simulations. Even though one can get well-defined energies and characters for a specific molecular geometry, the differences in equilibrium structures in different excited states complicate the analysis. For example, DEST can be calculated from vertical excitation energies at the S0, S1, T1 equilibria, adiabatic gap between the S1 and T1 minima or the activation energy to the S1-T1 minimum energy crossing point. In general, the results will be very different, and it is not always clear which of them corresponds to the

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experimentally reported value. Additionally, the uncertainty related to the shortcomings of the computational method may be greater than the quantity of interest. Despite these reservations, atomistic simulations have proven to be a very valuable tool in the studies of TADF materials, and the community has gathered a significant experience about their capabilities and limitations. Electronic structure calculations to support and rationalize the experimental findings were already used in the 2012 seminal TADF paper by Uoyama et al. (4). They used the time-dependent density functional theory (TD-DFT, see Chapter 3) to calculate properties like DEST, emission wavelengths, Stokes shifts, reorganization energies, and natural transition orbitals (NTOs) of the proposed 4CzIPN emitter. Considering that the size of typical TADF molecules can exceed 100 atoms, TD-DFT is the method of choice for routine excited-state calculations, and it has been used in countless studies of TADF molecules. Unfortunately, the computational convenience of TD-DFT comes at a price. First, the results are highly dependent on the approximate exchange-correlation functional chosen and, in some cases, may be completely unreliable. The situation is particularly grave for CT excitations (21), that is, precisely those relevant for TADF, a well-known failure of TD-DFT calculations using local exchange functionals. In particular, the results are highly sensitive to the amount of the nonlocal exact exchange (EXX) used in hybrid functionals. The smaller the EXX admixture, the more pronounced CT character and lower energy of the CT state, effectively leading to an underestimation of DEST. Therefore, significant effort has been put into validating different functionals and DFT-based protocols to calculate DEST. Huang et al. attempted to devise a protocol for optimizing the optimal amount of EXX based on the predicted degree of excited-state charge separation (22). The optimal percentage varied for different emitters but typically exceeded 30%. In general, DEST prediction proved to be challenging with global hybrid functionals with a fixed admixture of EXX. Another approach for mixing local and nonlocal exchange is based on the range separation of the Coulomb potential, where the EXX admixture reaches 100% but only at large interelectronic separations. Such functionals are believed to be able to describe CT excitations, but usually at the expense of the description of LE states. For TADF compounds, off-the-shelf rangeseparated hybrid (RSH) functionals tend to overestimate emission energies and DEST (22,23). A widely implemented approach to TD-DFT calculations with RSH functionals involves the so-called optimal tuning of the range separation parameter. The idea is based on enforcing Koopman’s theorem, which states that the HOMO energy should be equal to the negative of the vertical ionization potential. The work by Penfold (24) and Sun et al. (25) demonstrated the high accuracy of optimally tuned RSHs for predicting the excited-state energetics of TADF molecules; therefore, it has become a common choice. Among other DFT excited-state methods, it is worth mentioning those based on the DSCF approach, where ground-state-like calculations are done with non-Aufbau electron configurations and circumvent the known issues of TD-DFT with CT states. In this context, Hait et al. (26) and Mewes (27) showed that restricted open-shell Kohn-Sham and maximum-overlap methods yield a good description of CT states of TADF emitters. The downside of DSCF methods is that they are state-specific, allowing for optimizing only one excited state of a predefined character at a time. Regardless of the computational efficiency and accuracy of TD-DFT, analysis of the simulation results and the underlying theory gives some additional insights into the interpretation of DEST. First, it is clear that the picture of S1 and T1 as pure CT states resulting from a transition between well-separated HOMO and LUMO is wrong. Often T1 is an LE state completely localized either on the donor or the acceptor, and the alleged CT states are, in fact, of a mixed CT/LE character with differing orbital contributions. The mixing of CT and LE diabatic states can be explained with a four-state model (28) discussed later in the chapter. Second, even in the case of a pure HOMO!LUMO electronic configuration of both spin states, the TD-DFT DEST is given by the expression:      DEST ¼ 2ðfH fL jfL fH Þ + 2 fH fL f a,b (11.3) xc ðr1 , r2 , oÞ fL fH , b where (fHfLj fLfH) is the exchange integral in Eq. (11.1) and f a, xc (r1, r2, o) is the cross-spin frequency-dependent exchange-correlation kernel. The frequency dependence is neglected in the universally applied adiabatic approximation; nevertheless, the electronic correlation can affect the DEST beyond the exchange effects. Considering differing electronic configurations of S1/T1 and the correlation contribution, the prevailing notion that DEST ¼ 2KHL in the TADF literature is clearly wrong and should not be used. While the electronic correlation effects might be small for typical D-A-based TADF molecules, it has been recently discovered that the excited states of MRE TADF emitters have a substantial contribution of double excitations (29). The presence of doubly excited configurations in the wavefunction means that the electronic correlation is significant. What is more, adiabatic TD-DFT is known to fail in this case due to the missing frequency dependence responsible for generating additional solutions beyond one-electron transitions. Consequently, TD-DFT has been shown to overestimate DEST by up to 0.5 eV, rendering the approach useless for such systems. Pershin et al. have demonstrated that correlated wavefunctionbased methods like CC2 can capture this effect for a series of MRE emitters (29). In fact, the correlation-driven stabilization

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of the singlet excited state may be so significant that it outweighs the exchange-driven stabilization of the triplet state. In this case, the S1 state drops below T1, and the resulting singlet-triplet gap is negative. The possibility for such correlationinduced state inversion was analytically proven by de Silva in 2019 and computationally demonstrated for the cyclazine molecule (30). The ST (singlet-triplet) relative energy inversion was also discovered independently by Ehrmaier et al. in heptazine (31), isoelectronic with cyclazine. Both molecules have an interlacing HOMO/LUMO pattern analogous to MRE emitters, but even more pronounced. Both pioneering reports on the ST inversion demonstrated the failure of adiabatic TD-DFT and the need to resort to correlated wavefunction methods like ADC, CCSD, and CIS(D) that include doubly excited configurations explicitly. Later it was shown that double-hybrid TD-DFT, which includes an admixture of many-body correlation energy, can also capture negative DEST (32). The success of MRE emitters and the possibility of the ST inversion, which results in a thermodynamically favored rISC, highlight the importance of rationally exploiting electron correlation in the TADF materials design. At the same time, the failure of TD-DFT calls for the development of new cost-effective methods for the modeling of photophysical properties of such emitters.

11.3

Condensed phase effects

The experimental studies of TADF usually are done either in solutions or in the solid state. TADF emitters can be dissolved in organic solvents of varying polarity, which largely affect emission wavelengths due to large solvatochromic shifts of CT states. The polarity of the solvent also has a pronounced effect on the electronic transition rates, including the rISC rate, which is critical for TADF. Depending on the type and concentration of chromophores as well as the solvent, the formation of molecular aggregates is possible, which affects the measured optical properties. When it comes to the solid-state materials, the active layers of OLEDs are typically amorphous films comprising emitters as dopants embedded in a host matrix made of a different organic material. The solid-state environment can impact the spectroscopic properties of dopants, similar to that of a solvent. However, additional effects related to the static disorder, inhibited molecular vibrations, and bimolecular processes like exciton-exciton annihilation can lead to additional complications in the interpretation of the experiments. Sometimes these condensed-phase effects can be ignored when building computational models, but often they are crucial for developing the understanding of the TADF-related mechanisms or for successful computational materials design. Therefore, it is necessary to understand how different environmental effects are coupled with the atomistic and electronic structure of TADF emitters and their influence on overall material efficiency. The major environmental factor occurring in both liquid and solid phases is the dielectric effect, that is, the polarization of the surrounding materials in response to the electronic transition of the emitter. The transition alters the strength of emitter-environment electrostatic interactions, and the states with larger changes in the dipole moment are stabilized more strongly. It means that the relative energetic alignment of CT and LE states can be dramatically influenced by the polarizability of the environment, in particular affecting the emission wavelengths and DEST. Since the alignment and mixing of CT and LE states have proved to be crucial for the TADF efficiency, the role of the environment has been extensively studied both experimentally and computationally. Macroscopically, the material’s polarizability is quantified by the dielectric constant e(o), which is a frequency-dependent quantity describing the response to the external time-dependent electric fields. In equilibrium situations, when the environment had enough time to adjust its electronic and nuclear degrees of freedom to the changed electronic state of the emitter, the static dielectric constant, e0, is the most relevant. However, electronic transitions are happening on time scales where only electrons have time to respond, and their response is better described by the high-frequency dielectric constant, often calculated as the square of the refractive index: ehf ¼ n2. Including environmental effects in the electronic structure calculations is challenging. The most computationally efficient and widespread approach is to use implicit solvation models (33), where the emitter is placed in a cavity surrounded by a polarizable continuum described by the dielectric constant. While including implicit solvation in the self-consistent field (SCF) calculations is relatively straightforward, combining it with response-based theories like TD-DFT is less so. Nevertheless, nonequilibrium implicit solvation methods have been developed in the linear response (34) and state-specific (35) formalisms. They use both static and high-frequency dielectric constants to capture the equilibrated solvation of the initial state and nonequilibrium state after the electronic transition. In fact, only the state-specific formulation can treat CT excitations properly, which becomes problematic when multiple states are of interest. Other viable approaches based on the macroscopic description of the environment include accounting for the dielectric screening at the level of the range separation parameter tuning in RSH functionals (36), combining equilibrium solvation with DSCF excited-state methods (27,37), or including the screening effects in a model Hamiltonian of an emitter (38). An explicit atomistic description of the environment is an alternative to the implicit solvation models. The main challenges here are computational efficiency due to the explicitly modeled system’s size and the configurational space sampling due to the inherent disorder and thermal fluctuations. Obtaining realistic configurations of TADF emitters in host matrices

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is possible via classical force field-based molecular dynamics (MD, see Chapter 2) or Monte Carlo (MC) simulations. The computational protocols are often designed to mimic the thin film preparation, for example, by vapor deposition (39,40) or solution processing (41). In any case, the result is an atomistically resolved morphology model, from which individual molecules or molecular aggregates can be extracted to provide input for quantum-mechanical simulations, which can now replace the rest of the material with a continuum model. The advantage of this approach for simulating optical properties is the available distribution of conformations accessible in the condensed phase at a finite temperature. It has been shown that the critical properties of TADF like emission energies, DEST, and transition rates are very sensitive to the conformation, especially to the dihedral angle between the D-A units (42,43). The intermolecular interactions in organic active layers may lead to different distributions of conformers, further modulated by molecular vibrations, so such material-level simulations can give important insights into the correlation between the material’s type, morphology, and electronic properties. Explicit atomistic representation of the environment can also be used to model its dielectric response and the resulting stabilization of electronic states of the embedded emitters. Full quantum-mechanical calculations of the emitterenvironment system are not practical; therefore, polarizable embedding schemes where the emitter and the rest of the system are described at different levels of theory have been developed to this end. Polarizable quantum mechanics/ molecular mechanics (QM/MM) methods (44,45) use classical force fields to describe the environment, and the polarization of the electronic degrees of freedom is modeled through induced multiple moments or fictitious charges bound to the nuclei via a harmonic potential. The interaction Hamiltonian between the QM and MM parts describes the electrostatic interactions between the electronic density of the emitter and both static and induced charges in the environment. The use of MD-sampled morphologies enables accounting not only for the conformational disorder and molecular vibrations but also for the heterogenous electrostatic environment felt by individual emitters. The combined MD and polarizable QM/MM simulations have shown that disorder leads to broad distributions of polaron energies in OLED emission layers and that its static and dynamic components are of comparable magnitude (46). Another study using microelectrostatic embedding demonstrated that excitation energies and DEST also have broad distributions and that vibrations dynamically modulate the LE/ CT character of electronic states, greatly impacting the photophysical rates (47). For some TADF emitters, the stronger stabilization of the singlet states due to the more pronounced CT character resulted in a negative DEST. It was later demonstrated that the polarization of the environment due to the electronic excitations can be formally treated as double excitation, bearing some similarity to the correlation-driven stabilization of singlets, where both effects can lead to the DEST inversion (30). A practical limitation of the QM/MM and similar schemes based on the sampling of MD trajectories is that the nuclear relaxation due to the electronic transition is not captured. Developing force fields for the excited-state potential energy surfaces is possible, but combining the sampling of the configurational space in both ground and excited states with electronic structure calculations is cumbersome and has not been commonly applied. The methods and studies of condensed phases discussed so far concerned mostly properties of individual TADF emitters embedded in an environment that can respond to the electronic transitions. Another important class of phenomena enabled by intermolecular interactions are charge and energy transport processes. In electroluminescence, the molecular excited states are formed through the recombination of electrons and holes, which first need to be transported from the electrodes to the recombination sites in a series of incoherent hops between interacting molecules. Similarly, the exciton may also hop between molecules before a radiative or nonradiative relaxation happens. There are two energy transfer mechanisms: (i) Dexter energy transfer, which requires a wavefunction overlap; therefore, it can happen only between neighboring molecules, and (ii) F€orster energy transfer (FRET), which requires only that the emission and absorption spectra of the accepting and donating molecules overlap to some extent. Because of its long-range character, FRET is usually the dominant process in terms of the impact on photophysical properties. The interactions between traveling excitons may be a source of efficiency loss, for example, in the triplet-triplet annihilation mechanism, where two triplet excitons combine to form only one singlet state. In other cases, efficient energy transport is desired, for example, to transfer the exciton from a host molecule to a TADF emitter or from a TADF sensitizer to a fluorescent emitter. The latter process is the basis of hyperfluorescent OLED materials (48), which can offer higher color purity than pure TADF-based active layers. Such transport processes are inherently connected to the condensed phase conditions, and because of the relatively large separation between FRET donors and acceptors, their atomistic simulations usually require some level of coarse graining. There are several approaches to computing FRET rate constants; the difficulty lies in modeling the vibronic effects, which can be accounted for through the static electronic structure or quantum dynamics calculations. The latter approach was used by Lyskov et al. (49) and Giret et al. (50) to study sensitizer-emitter pairs in hyperfluorescent materials. Since the FRET rate constant depends on the distance and orientation between the molecules, the morphology plays a critical role in the efficiency of the process. To study the connection between the morphology and energy transfer in hyperfluorescent OLEDs, Cho et al. used MD-generated morphologies combined with electronic structure calculations (51). The actual exciton

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dynamics can be directly simulated with the kinetic Monte Carlo (KMC) method, in which the molecules that can be excited are mapped on a lattice of sites. FRET rates, together with possibly other relevant rates, are calculated from first principles or otherwise assumed based on simpler models. Gottardi et al. used KMC simulations to evaluate the loss processes in hyperfluorescent OLEDs that lead to the efficiency roll-off at high currents (52). Recently, we have studied hyperfluorescence-based active layers using a combination of first-principles and KMC simulations and demonstrated that a direct triplet-to-singlet FRET is a viable triplet harvesting mechanism, alternative to rISC (53). The simulations have revealed that both mechanisms are operational and, depending on the material, either of them can be the dominant one.

11.4

Role of charge transfer and local excited states

In the previous sections, we have mentioned that LE states participate in the rISC mechanism and are critical for its efficiency, something that was not fully recognized in the early days of TADF materials. In the following, we will look closer at their role in the electronic structure of TADF molecules and in enabling appreciable rISC and fluorescence rates. As previously discussed, the standard two-state model of TADF consists of two adiabatic states, one singlet (S1) and one triplet (T1), both with pure charge transfer character. These states can be written as linear combinations of two spin-mixed configuration state functions jCT1> and jCT2> that differ by a spin flip of unpaired electrons (Fig. 11.3A). TADF molecules that can be well described by this simple model suffer, however, from two shortcomings: first, the fact that when the S1 state is CT in nature, the corresponding transition tends to display very low oscillator strength. This means that fluorescence rates in such a molecule will be low, resulting in reduced quantum efficiency (36). Second, due to the necessity of conservation of total angular momentum, the spin-orbit coupling between states of similar electronic character should approach zero, in accordance with El-Sayed’s rules (18). In this sense, efficient TADF molecules must depart from the characterization provided by the two-state model. Following this idea, a four-state model was then proposed (28). In this model, the building blocks are two sets of two opposite spin configuration state functions, one set composed of CT states (jCT1> and jCT2>) and one of LE states (jLE1> and jLE2>). Incidentally, the CT and LE labels usually refer to the dominant character of the adiabatic excited states, even though these states may in general have a mixed LE/CT character. Here, we treat them as diabatic representations (54–57). These diabatic states are states of well-defined electronic character, as opposed to adiabatic states, which diagonalize the Hamiltonian and thus have well-defined energies. These diabatic states are then used as FIG. 11.3 Energy level diagrams and electronic configurations illustrating (A) a two-state and (B) a four-state model of TADF. (Modified from de Silva, P.; Kim, C. A.; Zhu, T.; van Voorhis, T. Extracting Design Principles for Efficient Thermally Activated Delayed Fluorescence (TADF) From a Simple Four-State Model. Chem. Mater. 2019, 31 (17), 6995–7006. Copyright 2019 American Chemical Society.)

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a basis in terms of which the adiabatic states are written. These states are thus mixtures between CT and LE states, as illustrated in Fig. 11.3B. The mixed and somewhat different character of S1 and T1 states in TADF molecules has been confirmed through TD-DFT calculations with a tuned RSH functional (58). The states in the new basis are not eigenfunctions of the spin operator and the Hamiltonian is not diagonal, taking rather the following form: 0 1 K CT

t

0

KX

t

t

KX

DE

K LE

KX

t

K LE

DE

0

K b¼B H @ CT

KX

C A

(11.4)

In this expression, the energy of the CT states has been subtracted from the diagonal, making the DE term the energy difference between CT and LE states. The off-diagonal terms correspond to the different couplings: KCT and KLE are the couplings between the two CT and LE states, whereas t and KX couple CT and LE states of equal and opposite spin configuration, respectively. Under the four-state model, it is in principle possible to design the diabatic composition of the S1 and T1 states in such a way as to optimize three properties that are key to TADF efficiency. First, lowering the DEST to facilitate rISC and then increasing the spin-orbit coupling by keeping the diabatic character of the two states different and improving quantum efficiency by including LE contributions in the S1 state. Of course, even though it is clear that the four-state model is more general than the two-state model, it does not mean that more complex models involving a higher number of states cannot be possible. Therefore, the extension to a model with N diabatic states is straightforward. The number of states necessary to describe satisfactorily a given molecule should vary from system to system since these models assume that the diabatic basis set spans the space occupied by at least N adiabatic states. It is an interesting problem, thus, to verify for several TADF molecules what the minimum model able to describe them is. We have recently conducted such study and showed that the four-state model was indeed appropriate for a good variety of TADF molecules (59). The study made use of a diabatization procedure that decomposed several singlet and triplet adiabatic states into their diabatic components and selected those components that could best serve as basis set to reconstruct N adiabatic states. The quality of the model was measured by the lowest norm of the adiabatic states that could be reproduced by the new basis, which had to be at least equal to 0.8. This procedure was applied to several TADF molecules using optimized geometries at S0, S1, and T1 states. In all cases, the four-state model was shown to be sufficient when applied to S0 geometries, but for excited geometries, some molecules required a six- or even an eight-state model. For four molecules—MCZ-XT, PTZ-XT, XAC-CM, and ACRSA—the four-state model was appropriate for all three geometries considered (59). Considering these four molecules, we may look into the composition of the adiabatic states. In Fig. 11.4, the diabatic decomposition of the four states are shown for the four TADF molecules. We see that for all but one molecule, the first excited state is mainly a CT state. The ACRSA S1 state, however, is a mixture with a stronger contribution from a localized excitation in the acceptor unit (LEA). The first triplet state in these four molecules, on the other hand, has a major LE participation, showing how the simpler two-state model is clearly unsuitable for these molecules. The location of the LE contribution may also vary, as seen for instance, in the case of PTZ-XT, whose the LE state is localized in the donor (LED) in opposition to the other three molecules. With the four-state model in hand, it is possible to obtain the elements of the corresponding diabatic Hamiltonian of Eq. (11.4). These are presented in Fig. 11.5. There are several features worthy of discussion. First, we observe that two couplings (KCT and KX) practically vanish in all cases. This is expected as these couplings are associated with overlaps between either two CT states or between a CT and an LE state of opposite spin configuration, whose orbitals are localized in different parts of the molecule. Considering now the two remaining couplings, KLE and t, it can be seen that they can assume nonzero values. In the case of KLE, particularly high values can be obtained as the local excitations display large spatial overlap, as opposed to the previous case of KCT. Finally, the t coupling has a behavior that can be connected to degree of purity of the adiabatic states that was previously shown in Fig. 11.5. When the degree of mixture is low, the t couplings tend to zero. Deviations from zero, on the other hand, are seen the more mixed the characters of the states are. The last element of the diabatic Hamiltonian is DE, the energy difference between CT and LE states. This is usually a positive number as the S1 state is usually higher in energy than T1, and S1 state tends to be of CT character, whereas T1 is more often of the LE type. This is different for ACRSA, a molecule for which both states have a higher LE character, resulting in higher energy states having CT property. The diabatic Hamiltonian has a dependence on molecular geometry as its components change once the four-state model is computed for different conformations. We also know that conformational changes can affect the efficiency of TADF in a given molecule (43). By generating several molecular conformations and calculating the four-state diabatic Hamiltonian of

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FIG. 11.4 Diabatic decomposition into donor-to-acceptor charge transfer (CT), acceptor-to-donor charge transfer (rCT), localized excitation in the donor (LED), and localized excitation in the acceptor (LEA) states that compose the four-state models for ACRSA, MCZ-XT, PTZ-XT, and XAC-CM in ground-state optimized geometries.

FIG. 11.5 Elements of the diabatic Hamiltonian (in eV) calculated for four TADF molecules in their corresponding optimized ground-state geometry.

each, it is possible to obtain a connection between these elements and the four main parameters that control the photophysics of a TADF molecule: the optical gap, oscillator strength, DEST, and spin-orbit coupling. The first two parameters control the efficiency and wavelength of fluorescence, whereas the latter two control the efficiency of the rISC mechanism. Fitting the data obtained for several conformations of the NCFCZ molecule with a LASSO regression model, it was then possible to look for the diabatic Hamiltonians that could optimize some photophysical properties: fluorescence rate (kf), rISC (krISC) rate, and overall TADF efficiency, which means maximizing the product between kf and krISC. The results for these three optimization procedures are shown in Fig. 11.6A. To maximize fluorescence rates, it stands out that KCT values are increased indefinitely. This is equivalent to reducing the spatial separation of CT orbitals, which effectively turns CT states into LE states. In turn, to maximize rISC rates, the best Hamiltonian has a null KCT value, associated with a lower DEST. A nonzero t value suggests the existence of some amount of mixture between LE and CT

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FIG. 11.6 (A) Diabatic Hamiltonians that result from the maximization of fluorescence rates, rISC rates, and overall TADF performance. (B) Diabatic Hamiltonians that optimize TADF performance constrained by the color of the emitted light. All values in eV.

diabatic states, which is expected to increase spin-orbit couplings. Finally, for the total TADF rate optimization, the optimal Hamiltonian looks similar to the one for the rISC optimization, but with lower DE and KLE values. The differences between these two Hamiltonians reflect the necessary trade-offs for simultaneous optimization of rISC and fluorescence. Finally, if we constrain the color of the emitter and again optimize the Hamiltonian for the total TADF efficiency, we obtain the results of Fig. 11.6B, in which the most notable changes seen when one moves from blue to red emitters is that the energy difference between LE and CT states is reduced as well as the KLE coupling. Furthermore, it is noted that overall TADF efficiency decreases as well for lower energy emitters. The interested reader is referred to Ref. (59) for a more detailed analysis.

11.5

Vibronic effects and rate calculations

In the preceding section, we have explained the mixing of CT and LE states from the diabatic representation perspective. An important conclusion is that the diabatic composition of the excited-state wave functions depends on the conformation of a molecule. This is direct evidence for the importance of vibronic effects and their role in boosting the TADF efficiency. Here, we further develop the understanding of vibrational effects in the TADF mechanism and discuss the methods for calculation of the relevant rates that properly account for fluctuations of the electronic character along the vibrational modes. When it comes to estimating the rates of (r)ISC, several methods have been employed using different levels of perturbation theory. The starting point is most often Fermi’s golden rule:   2  2p X Ei D  kISC ¼ e kT  Cf H SOC Ci  d Ef Ei (11.5) ħZ f

in which Ci and Cf refer to the combined nuclear and electronic molecular wave functions corresponding to the initial and final states of energy Ei and Ef, respectively. Z is the system’s partition function at temperature T, k is Boltzmann’s constant, ħ is Planck’s reduced constant, HSOC is the spin-orbit coupling operator, and the delta function ensures conservation of energy in the transition. Under the Born-Oppenheimer approximation, the nuclear (f) and electronic (c) wave functions can be separated, resulting in E 2    2 D 2p X Eij D   kISC ¼ e kT  cf H SOC ci   ffk jfij  d Efk Eij (11.6) ħZ f , j, k Here, the electronic spin-orbit coupling is taken as independent of nuclear coordinates, with the vibrational effects being restricted to the vibrational overlap terms, also known as Franck-Condon factors. Coupling the molecular system with a classically treated bath representing the molecular environment (solvent degrees of freedom), this expression takes the more familiar form (60):

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kISC ¼

D   2  ffi  cf H SOC ci  rffiffiffiffiffiffiffi p X ħZ

lkT f , j, k

e

Eij kT

D E2 ðEfk Eij +lÞ2   4lkT  ffk jfij  e

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(11.7)

in which l is the reorganization energy of the bath. From this point, since there is a very large number of Franck-Condon factors, a usual approach to simplify the calculations is to separate vibrational contributions into classical and nonclassical. The classical part comprises the low-frequency modes—those whose energy is much lower than the thermal energy kT— whose effect is accounted for by adding to the bath reorganization energy the low-frequency intramolecular reorganization energy. The nonclassical high-frequency modes, on the contrary, have their contributions bundled in a single effective vibrational mode. Furthermore, by accounting only for transfers taking place from the lowest vibrational level of the initial electronic state (j ¼ 0) and employing the harmonic approximation, one may resort to an analytical expression for the overlap factors. This procedure results in the famous Marcus-Levitch-Jortner (MLJ) formula: D   2  ffi 2  cf HSOC ci  rffiffiffiffiffiffiffi p X e S Sn ðEf 0 Ei0 +nħo+lÞ 4lkT kISC ¼ e (11.8) lkT n n! ħ

where S is the Huang-Rhys factor associated with the effective mode of frequency o and n indexes vibrational levels. This very popular expression has been the method of choice for estimating (r)ISC rates in several works (58,61,62). It is particularly convenient since the parameters it requires—such as excitation energies, Huang-Rhys factors, and electronic spinorbit couplings—are readily available in several quantum chemical packages. However, one important shortcoming of Eq. (11.8) is that the spin-orbit coupling is computed at a reference geometry, typically the optimized structure at the initial electronic state. This means, however, that this procedure fails to take into account vibrational contributions to the coupling term, which often results in underestimated rates, especially for the case of reverse intersystem crossing. From an experimental standpoint, the necessity of taking into consideration these effects in describing (r)ISC became clear after it was demonstrated that preventing rotations around the donor-acceptor bond could change the contributions of TADF and phosphorescence to the photoluminescence of a series of phenothiazine-dibenzothiophene-S,S-dioxide molecules (63). However, theoretical tools to address these effects had already been developed several years prior in order to reflect these effects into rate estimations. The first option to include vibrational effects in the rate calculations is by going beyond the Condon approximation. This can be achieved by expanding the coupling term in a Taylor series of the normal coordinates Qa. D    D    X ∂ Cf H SOC ðQa ÞCi   D  Qa Cf H SOC Ci  Cf H SOC ðQ0 ÞCi + ∂Qa a D  (11.9)   2 ∂ Cf H SOC ðQa ÞCi X 1 + Qa Qb + … 2 a, b ∂Qa ∂Qb

Here, Q0 refers to the coordinates at the equilibrium geometry. The zeroth-order term of the expansion corresponds to the Franck-Condon (FC) approximation that we had before, whereas the inclusion of linear terms is referred to as the HerzbergTeller (HT) approximation. Performing this expansion of the spin-orbit coupling corresponds to accounting for the vibronic spin-orbit mechanism of singlet-triplet mixing and has been put into practice in different applications (64–68). Inclusion of these expansion terms in the rate equation results in rather convoluted expressions even under the harmonic approximation. The main difficulty in this case is the computation of the derivative terms—electronic and vibrational, which have to be obtained numerically for each individual normal mode. Another improvement on rate calculations comes from going beyond the first-order perturbation theory. In this case, the perturbation to the original Hamiltonian is considered to consist of the sum of the nuclear kinetic energy and the spin-orbit operators, which amounts to treating intersystem crossing and internal conversion on equal footing. As such, the spin-orbit term can be written as (69) D       D D  Cf H SOC Cn Cn T N Ci X      Cf H SOC Ci  Cf H SOC ðQ0 ÞCi + E n Ef n D     (11.10)   X Cf T N Cn Cn HSOC Ci + En Ei n

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The terms involving summations correspond to the so-called spin-vibronic mechanism. Their actual implementation is not particularly straightforward, as can be seen in the original 1971 paper (69). More recent approaches use, instead of excited state, molecular and quantum dynamics to incorporate these effects (70–72). In spite of the similarities in naming, the spinvibronic effect is different from the previously discussed vibronic spin-orbit corrections. In the spin-vibronic mechanism, ISC between a singlet and a triplet state is mediated by a third state, usually another triplet state close in energy, to which the first one is nonadiabatically coupled. This idea stems from some theoretical considerations and experimental observations. As mentioned before, the spin-orbit coupling between states of the same electronic character is expected to vanish, which would prevent ISC between two pure CT states. In addition, it was shown that the states involved in the TADF process could be affected individually by environmental factors (73,74). This indicates that these states must have different electronic character, which brings about the idea of a localized excitation as a possible intermediate between two CT states of opposite spin (75). Recently, this spin-vibronic mechanism has been highlighted in several works as essential for the accurate estimation of ISC rates as its inclusion results in rISC rates that are sometimes orders of magnitude larger than those predicted by the standard first-order perturbation theory (20,75–78). It is worth mentioning that while the vibronic spin-orbit and the spin-vibronic mechanisms are of different nature, these two contributions cannot be told apart in practice and their relative importance can vary from system to system (79,80). As we have seen, the inclusion of more sophisticated approximations carries with itself significant computational overhead along with the necessity of specialized knowledge for its application, which prevents a more widespread use by researchers from other fields. As such, the development of methods that can provide a better trade-off between the accuracy, computational cost, and ease of applicability to different molecular systems is of crucial importance. To achieve these objectives, such methods would have to include non-Condon and spin-vibronic effects, while avoiding the explicit computation of all different couplings or the need to run costly dynamics simulations. Aiming to fill this gap, a new approach, based on the so-called nuclear ensemble method (NEM), was recently proposed (81). In the nuclear ensemble method, explicit computation of vibrational integrals is substituted by Monte Carlo integration. This is achieved by means of judicious sampling of hundreds of molecular geometries on which single-point calculations are run. Results from these various calculations are averaged together to produce rate estimates. NEM was originally used to simulate absorption and fluorescence spectra (82), serving as a useful alternative to the Franck-Condon/Herzberg-Teller methods that would fail in the case of flexible molecules that undergo significant geometrical changes upon excitation (83). For the particular application in the estimation of (r)ISC rates, the NEM-derived rate expression is (81) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 fQk gÞ+lÞ M D    2 ðDE2fi ð2lkT+s 1 2p 1 X  2Þ ð   kISC ¼ (11.11)  cf H SOC Qk ci  e ħ ð2lkT + s2 Þ M k¼1 In this expression, both the spin-orbit coupling and the singlet-triplet gap DEfi are written as a function of the set of normal coordinates {Qk} since they are computed at each of the ensemble’s geometries. In addition, s is a phenomenological broadening term typically made equal to the thermal energy kT. Finally, M corresponds to the number of geometries in the ensemble, which are sampled from the following distribution r for a set of 3N 6 harmonic oscillators at temperature T, N being the number of atoms in the molecule 0 112

3N Y6 mi oi mi oi Q2i ħoi @  A exp rðQk , T Þ ¼ tanh (11.12) ħoi ħ 2kT 2pħ sinh i¼1 kT

Here, mi, oi, and Qi correspond, respectively, to the reduced mass, frequency, and displacement in the direction of the mode indexed by i. Importantly, the sampling must be performed using the parameters from the initial electronic state of a given transition. This means that in the case of S1 ! T1 ISC, for example, the geometries in the ensemble are sampled from the first singlet state. The rate’s temperature dependence is also caught by the approach as temperature affects the shape of the distribution, in addition to being present explicitly in the rate formula. Even though the NEM rate expression holds some similarity with the MLJ formula, we previously saw (Eq. 11.8) there are some key differences, the most relevant of which being the fact that couplings and energy differences are calculated at several different molecular configurations. This ensures that non-Condon effects are taken into account, even though no vibrational terms had to be computed directly. The effects of the ensemble calculation of spin-orbit couplings and DEST have been shown to be most significant, explaining partially the reason behind the observed tendency for the MLJ formula to underestimate rISC rates. This is illustrated in Fig. 11.7, which shows the distributions of spin-orbit couplings and DEST for S1 and T1 ensembles calculated for two TADF molecules—PTZ-DBTO2 and 4CzIPN—to which the method was

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FIG. 11.7 Distribution of spin-orbit couplings and singlet-triplet gaps (DEST) obtained from the S1 and T1 nuclear ensembles of PTZ-DBTO2 (A–D) and 4CzIPN (E–H). The red dashed lines (gray in print version) mark the values taken from equilibrium geometries. The green dashed lines (light gray in print version) show the ensemble averages. (Modified from de Sousa, L. E.; de Silva, P. Unified Framework for Photophysical Rate Calculations in TADF Molecules. J. Chem. Theory Comput. 2021, 17 (9), 5816–5824. Copyright 2021 American Chemical Society.)

applied. These distributions are compared with their mean values and the values obtained from the corresponding optimized geometries. It can be seen that these two values matching each other is the exception, rather than the rule, again demonstrating the inadequateness of estimates that neglect vibrational effects. Since non-Condon effects are accounted for in the NEM, we may say that this method includes vibronic spin-orbit corrections much like the Herzberg-Teller approximation. How about the spin-vibronic mechanism? As we have seen, in this mechanism, the transition from triplet to singlet states happens by means of another triplet state, coupled nonadiabatically to the first. In the case of the PTZ-DBTO2 molecule, it was shown that there were two energetically closely triplet states T1 and T2 with different electronic character. The lowest one corresponded to a localized excitation in the donor fragment, whereas the second was a charge transfer state. When coupling between these two states was ignored, rISC rates were severely underestimated (20). Since in the NEM, we consider only direct transitions from one state to another, it would appear at first glance that the spin-vibronic mechanism could by no means be included in this approach. It pays, however, to take a look at the geometries that compose the triplet ensemble of this molecule and the electronic character of their lowest triplet state. Fig. 11.8 shows two examples of such geometries along with the natural transition orbitals (NTOs) FIG. 11.8 Highest occupied natural transition orbitals (HONTOs) and lowest unoccupied natural transition orbitals (LUNTOs) for the T1 excitation in two different geometries sampled from the PTZ-DBTO2 ensemble. Whereas the first configuration has a charge transfer character, the second one presents a localized excitation character. (Modified from de Sousa, L. E.; de Silva, P. Unified Framework for Photophysical Rate Calculations in TADF Molecules. J. Chem. Theory Comput. 2021, 17 (9), 5816– 5824. Copyright 2021 American Chemical Society.)

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for the T1. It can be seen that even though the two NTOs correspond to transitions to the lowest triplet state, the electronic character of such state is different for each configuration: one is a charge transfer state, while the other is a local excitation. Each contributes to the overall rISC rate, which means that the NEM incorporates transitions from both CT and LE triplet states. As mentioned before, the different vibronic contributions (first or second order) are in practice indistinguishable, so it is also not possible to disentangle how much each is captured by the NEM. Having ascertained the correspondence between the NEM and the relevant vibronic mechanisms, we turn to the actual performance of the method. The NEM was used for (r)ISC rate estimations in PTZ-DBTO2 and 4CzIPN. Energy and spinorbit couplings were calculated using the Tamm-Dancoff approximation (84) (TDA) with M062X functional and 6-31G(d, p) basis set. For PTZ-DBTO2, the S1 ! T1 ISC rate was predicted to be 1.6  108 s 1, which is about one order of magnitude larger than the experimental estimate of 1.0  107 s 1. The T ! S1 rISC rate, on the contrary, was calculated at 4.5  105 s 1, underestimating the experimental value by only a factor of 3 (14,85). These results compare favorably with estimations from quantum dynamics with a spin-vibronic Hamiltonian applied to the same system, which yielded an ISC rate of 5  105 s 1 and a rISC rate of 7  104 s 1 (20). This provides further weight to the claim that the NEM incorporates spin-vibronic effects in its sampling procedure. In the case of 4CzIPN, good agreement was seen between calculated (5.8  107 s 1) and experimental ISC rates (5.1  107 s 1). The estimated rISC rate (3.3  104 s 1), on the contrary, is two orders of magnitude lower than the reported experimental values (2.7  106 s 1) (86) pointing at the necessity of further improvements in the method. One immediate extension to the NEM is the estimation of rates for transitions involving higher lying electronic states. The original paper focused only on transfers between the first singlet and first triplet states. Even though, as we have seen, the ensemble is able to capture contributions from other diabatic states accessible via vibrations, this is no longer the case when it comes to higher lying electronic states. Fortunately, the extension of the NEM to other electronic states is straightforward, especially when considering transfers from S1 to higher triplet states. In this case, the same S1 ensemble can be used to provide rates for transfers to several triplet states, with only the need to calculate the appropriate reorganization energies. The situation is more complicated when it comes to transfer from higher energy states as they would formally require ensembles of their own, which, in turn, require geometry optimization and normal mode analysis in these higher excited states. Since in some cases, this may not be feasible due to the high computational cost of such operations, approximate estimates can still be produced using the already generated ensembles. This has allowed for more complex photophysical scenarios to be modeled, helping explain experimental results (87,88). Another important improvement to the NEM relates to how solvent effects are included in the calculations. In the original NEM paper, solvent effects to the excitation energies were modeled by means of a linear-response polarizable continuum model (PCM) (89). However, charge transfer states tend to suffer strong stabilization upon solvation, the more so the higher the solvent’s polarity. This effect produces a strong solvatochromic shift that is not well described by PCM. Considering the key role played by charge transfer states in TADF and the fact that experimental measurements are always conducted in an environment—whether it be in solution or solid state—capturing solvent effects properly becomes fundamental. In this sense, one of the first improvements to the NEM concerns the use of a perturbative approximation to the state-specific solvation model, which was shown to accurately reproduce experimental solvent shifts (90,91). Application of this method in conjunction with nonempirically tuned long-range corrected DFT functionals shows great improvement in the accuracy of emission spectrum simulations, which serves as a good indicator of more reliable (r)ISC rate estimates as well (92,93). As mentioned earlier, the NEM aims to combine accuracy, reasonable computational cost, and ease of use. We have discussed the theoretical aspects that justify its use and we have mentioned that from a computational point of view, the overall cost is diluted into the cost of several parallelizable TD-DFT single-point calculations. This makes the question of ease of use the only one left. Fortunately, the whole NEM has been packaged into the NEMO program (94), an application that interfaces with the QChem quantum chemistry software (95). Taking as input frequency calculations (conducted either with QChem or Gaussian software), NEMO generates the proper nuclear ensembles, has a built-in batch scheme for running calculations, and is able to simulate absorption and emission spectra in addition to calculating fluorescence, phosphorescence, and (r)ISC rates. This combination of features gives one the capability to paint a complete photophysical picture of any molecular system. In addition, a module for estimating exciton transport properties—F€orster radii and exciton diffusion lengths—is also available (96).

11.6

Synopsis

The objective of this chapter was to introduce the reader to the theory of the photophysical processes in TADF materials and the atomic-scale computational methods for their simulations. We took a somewhat historical perspective, starting from the

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initial understanding of TADF based on a simple model of two CT states with singlet and triplet spin multiplicities. The breakdown of the model to explain the role of LE states and molecular vibrations has led to the development of improved models and, consequently, to a much more complete understanding of TADF. At the same time, molecular modeling has been used extensively to support the experimental results and to expedite the TADF materials discovery. The computational studies have also come a long way from inaccurate and straightforward TD-DFT calculations of excited states to much more sophisticated material-level simulations, including condensed phase effects, nuclear excited-state dynamics, and photophysical rate calculations. In the review, we have focused on four aspects of theoretical studies of TADF materials, that is, (i) electronic structure calculations, including predicting DEST as the key parameter; (ii) accounting for condensed phase effects like strong stabilization of CT states by a polarizable environment or exciton dynamics particularly relevant to hyperfluorescent materials; (iii) the role of local excitations in enabling efficient rISC and fluorescence; and (iv) calculations of photophysical rates, which inherently rely on the proper treatment of molecular vibrations. This dissection into four separate problems is, to some extent, arbitrary and motivated by our work in these areas. From a broader perspective, all these effects are interrelated as vibrations enable the LE and CT states to mix. This mixing, in turn, modulates the energies of the excited states, which are further affected by the dynamic response of the environment. When it comes to the future, there is still plenty of room to improve our theoretical understanding of TADF materials and, consequently, to accelerate the materials discovery. The recent work on the MRE emitters and the discovery of the possibility of inverted DEST calls for new molecular designs and the development of robust and efficient excited-state methods that can properly capture double excitations. The multiscale simulations of materials will continue to reveal mesoscopic effects and ultimately enable us to better understand the material-level structure-property relationships like the relation between the molecular composition and the efficiency roll-off at high currents. Efficient and accurate calculations of photophysical rates will allow the computational materials discovery efforts to directly target quantum efficiencies of materials, instead of relying on simple parameters like DEST and oscillator strength computed at equilibrium geometries. We expect to see the development of complex computational frameworks combining all these elements, perhaps together with data-driven approaches like machine learning and artificial intelligence, to discover new TADF materials for a host of possible applications.

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A Quantum Dynamics Study of the Ultrafast Relaxation in a Prototypical Cu(I)–Phenanthroline. J. Phys. Chem. A 2014, 118 (42), 9861–9869. 72. Curchod, B. F. E.; Rauer, C.; Marquetand, P.; Gonza´lez, L.; Martı´nez, T. J. Communication: GAIMS—Generalized Ab Initio Multiple Spawning for Both Internal Conversion and Intersystem Crossing Processes. J. Chem. Phys. 2016, 144 (10), 101102. 73. Dias, F. B.; Bourdakos, K. N.; Jankus, V.; Moss, K. C.; Kamtekar, K. T.; Bhalla, V.; Santos, J.; Bryce, M. R.; Monkman, A. P. Triplet Harvesting With 100% Efficiency by Way of Thermally Activated Delayed Fluorescence in Charge Transfer OLED Emitters. Adv. Mater. 2013, 25 (27), 3707–3714. 74. Santos, P. L.; Ward, J. S.; Data, P.; Batsanov, A. S.; Bryce, M. R.; Dias, F. B.; Monkman, A. P. Engineering the Singlet–Triplet Energy Splitting in a TADF Molecule. J. Mater. Chem. C 2016, 4 (17), 3815–3824.

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75. Gibson, J.; Penfold, T. J. Nonadiabatic Coupling Reduces the Activation Energy in Thermally Activated Delayed Fluorescence. Phys. Chem. Chem. Phys. 2017, 19 (12), 8428–8434. 76. Chen, X.-K.; Zhang, S.-F.; Fan, J.-X.; Ren, A.-M. Nature of Highly Efficient Thermally Activated Delayed Fluorescence in Organic Light-Emitting Diode Emitters: Nonadiabatic Effect Between Excited States. J. Phys. Chem. C 2015, 119 (18), 9728–9733. 77. Chen, X.-K.; Kim, D.; Bredas, J.-L. Thermally Activated Delayed Fluorescence (TADF) Path Toward Efficient Electroluminescence in Purely Organic Materials: Molecular Level Insight. Acc. Chem. Res. 2018, 51 (9), 2215–2224. 78. Marian, C. M. Mechanism of the Triplet-to-Singlet Upconversion in the Assistant Dopant ACRXTN. J. Phys. Chem. C 2016, 120 (7), 3715–3721. 79. Penfold, T. J.; Gindensperger, E.; Daniel, C.; Marian, C. M. Spin-Vibronic Mechanism for Intersystem Crossing. Chem. Rev. 2018, 118 (15), 6975– 7025. 80. Marian, C. M. Spin-Orbit Coupling and Intersystem Crossing in Molecules. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2012, 2 (2), 187–203. 81. de Sousa, L. E.; de Silva, P. Unified Framework for Photophysical Rate Calculations in TADF Molecules. J. Chem. Theory Comput. 2021, 17 (9), 5816–5824. 82. Crespo-Otero, R.; Barbatti, M. Spectrum Simulation and Decomposition With Nuclear Ensemble: Formal Derivation and Application to Benzene, Furan and 2-Phenylfuran. Theor. Chem. Accounts 2012, 131 (6), 1237. 83. de Sousa, L. E.; Ribeiro, L. A.; de Almeida Fonseca, A. L.; da Silva Filho, D. A. Modeling the Emission Spectra of Organic Molecules: A Competition Between Franck–Condon and Nuclear Ensemble Methods. J. Phys. Chem. A 2016, 120 (27), 5380–5388. 84. Hirata, S.; Head-Gordon, M. Time-Dependent Density Functional Theory within the Tamm–Dancoff Approximation. Chem. Phys. Lett. 1999, 314 (3– 4), 291–299. 85. Penfold, T. J.; Dias, F. B.; Monkman, A. P. The Theory of Thermally Activated Delayed Fluorescence for Organic Light Emitting Diodes. Chem. Commun. 2018, 54 (32), 3926–3935. 86. Ishimatsu, R.; Matsunami, S.; Shizu, K.; Adachi, C.; Nakano, K.; Imato, T. Solvent Effect on Thermally Activated Delayed Fluorescence by 1,2,3,5-Tetrakis(Carbazol-9-Yl)-4,6-Dicyanobenzene. J. Phys. Chem. A 2013, 117 (27), 5607–5612. 87. Goya, T.; Zimmermann Crocomo, P.; Hosono, T.; Minakata, S.; Sousa, L. E.; Silva, P.; Data, P.; Takeda, Y. A New Entry to Purely Organic Thermally Activated Delayed Fluorescence Emitters Based on Pyrido[2,3-b]Pyrazine-Dihydrophenazasilines Donor-Acceptor Dyad. Asian J. Org. Chem. 2022, 11 (3). 88. Hosono, T.; Decarli, N. O.; Crocomo, P. Z.; Goya, T.; de Sousa, L. E.; Tohnai, N.; Minakata, S.; de Silva, P.; Data, P.; Takeda, Y. The Regioisomeric Effect on the Excited-State Fate Leading to Room-Temperature Phosphorescence or Thermally Activated Delayed Fluorescence in a Dibenzophenazine-Cored Donor–Acceptor–Donor System. J. 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Izumi, S.; Govindharaj, P.; Drewniak, A.; Crocomo, P. Z.; Minakata, S.; de Sousa, L. E.; de Silva, P.; Data, P.; Takeda, Y. Comparative Study of Thermally Activated Delayed Fluorescent Properties of Donor–Acceptor and Donor–Acceptor–Donor Architectures Based on Phenoxazine and Dibenzo[a,j]Phenazine. Beilstein J. Org. Chem. 2022, 18, 459–468. 94. NEMO. https://github.com/LeonardoESousa/NEMO. 95. Shao, Y.; Gan, Z.; Epifanovsky, E.; Gilbert, A. T. B.; Wormit, M.; Kussmann, J.; Lange, A. W.; Behn, A.; Deng, J.; Feng, X.; Ghosh, D.; Goldey, M.; Horn, P. R.; Jacobson, L. D.; Kaliman, I.; Khaliullin, R. Z.; Kus, T.; Landau, A.; Liu, J.; Proynov, E. I.; Rhee, Y. M.; Richard, R. M.; Rohrdanz, M. A.; Steele, R. P.; Sundstrom, E. J.; Woodcock, H. L.; Zimmerman, P. M.; Zuev, D.; Albrecht, B.; Alguire, E.; Austin, B.; Beran, G. J. O.; Bernard, Y. A.; Berquist, E.; Brandhorst, K.; Bravaya, K. B.; Brown, S. T.; Casanova, D.; Chang, C.-M.; Chen, Y.; Chien, S. H.; Closser, K. D.; Crittenden, D. 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F.; Yang, J.; Yeganeh, S.; Yost, S. R.; You, Z.-Q.; Zhang, I. Y.; Zhang, X.; Zhao, Y.; Brooks, B. R.; Chan, G. K. L.; Chipman, D. M.; Cramer, C. J.; Goddard, W. A.; Gordon, M. S.; Hehre, W. J.; Klamt, A.; Schaefer, H. F.; Schmidt, M. W.; Sherrill, C. D.; Truhlar, D. G.; Warshel, A.; Xu, X.; Aspuru-Guzik, A.; Baer, R.; Bell, A. T.; Besley, N. A.; Chai, J.-D.; Dreuw, A.; Dunietz, B. D.; Furlani, T. R.; Gwaltney, S. R.; Hsu, C.-P.; Jung, Y.; Kong, J.; Lambrecht, D. S.; Liang, W.; Ochsenfeld, C.; Rassolov, V. A.; Slipchenko, L. V.; Subotnik, J. E.; van Voorhis, T.; Herbert, J. M.; Krylov, A. I.; Gill, P. M. W.; Head-Gordon, M. Advances in Molecular Quantum Chemistry Contained in the Q-Chem 4 Program Package. Mol. Phys. 2015, 113 (2), 184–215. 96. de Sousa, L. E.; Bueno, F. T.; e Silva, G. M.; da Silva Filho, D. A.; de Oliveira Neto, P. H. Fast Predictions of Exciton Diffusion Length in Organic Materials. J. Mater. Chem. C 2019, 7 (14), 4066–4071.

Chapter 12

DNA photostability Lara Martı´nez-Ferna´ndeza and Antonio Franc
es-Monerrisb a

Departamento de Quı´mica, Facultad de Ciencias and Institute for Advanced Research in Chemistry (IADCHEM), Campus de Excelencia UAM-CSIC, Universidad Auto´noma de Madrid, Madrid, Spain, b Institut de Cie`ncia Molecular, Universitat de Vale`ncia, Valencia, Spain

Chapter outline 12.1 Photophysics of canonical nucleobases in the gas phase. 312 Photostability 12.1.1 Absorption properties in the gas phase 312 12.1.2 Photophysical paths for purine nucleobases 312 12.1.3 Excited-state dynamics of purine nucleobases 314 12.1.4 Photophysical paths for pyrimidine nucleobases 314 12.1.5 Excited-state dynamics of pyrimidine 315 nucleobases 12.2 Photophysics of canonical nucleobases in solution. Impact of the solvent effects into the photostability 316 12.2.1 Purine nucleobases 316 12.2.2 Pyrimidine nucleobases 317 12.3 Photophysics of modified nucleobases. Impact of the 319 substitution effects into the photostability 12.3.1 Addition of external groups into the pyrimidine/purine core 319

12.3.2 Substitution of internal groups into the pyrimidine/purine core 12.4 Photophysics of canonical nucleobases in DNA/RNA environments. Photostability mechanisms 12.4.1 Single monomers embedded in a DNA/RNA environment 12.4.2 DNA/RNA light absorption and excited-state delocalization 12.4.3 Watson-Crick base pairing and interstrand charge transfer states. A doorway to proton transfer and photostability 12.5 Final remarks and future perspectives Acknowledgments References

321

322 322 323

325 327 328 328

Nucleic acids are primordial life ingredients. They contain the nucleobase (NB) sequence that determines the primary structure of any protein and, therefore, constitute the set of instructions that regulate cell functioning and, ultimately, life on Earth as we know it. However, important discoveries in the last decades indicate that the current chemical structure of nucleic acids is the result of a long and subtle natural selection process matured since the very beginning of life emergence (1–4). DNA/RNA NB are aromatic systems able to absorb part of the UV wavelengths of Sunlight reaching the ground surface (l > 320 nm), whereas the more energetic radiation is blocked by the stratosphere/mesosphere. This part of radiation reached the earth more intensely during the prebiotic era due to the absence of the current ozone layer. The absorption of this UV light by NBs originates electronic excited states. Subsequently, the population of these excited states and the associated excess of energy and electron density redistribution can trigger photochemical reactions that can compromise the nucleic acids’ structural integrity and therefore their biological function (5), being among the most common photoinduced lesions the [2 + 2] cycloadditions between pyrimidine NBs (the so-called cyclobutane pyrimidine dimers or CPDs) (6–9). Therefore, it is reasonable to think that only the privileged structures able to fulfill nucleic acids’ biological functions while minimizing the consequences of the intense UV irradiation that reached the planet surface survived the natural chemical evolutionary pressure over ages (1–4). In the last two decades, a vast amount of experimental (10–12) and computational (11–15) studies have delineated the photophysical decay mechanisms of NBs. It is now well established that NBs’ excited states efficiently return to their electronic ground state (photostability), being photochemical reactions minor channels. Interestingly, small chemical modifications of the NB core can alter this preferred mechanism resulting in longer-living excited states or, even, populate more reactive triplet states (16). It is, then, of utmost importance to understand why the particular structure of NB confers this photostability and what could be related with their selection as building-blocks of DNA during the prebiotics. The characterization of NB behavior under light absorption is the result of a strong multidisciplinary effort, in which the development of sophisticated theoretical methodologies and high-resolution time-resolved experimental techniques have Theoretical and Computational Photochemistry. https://doi.org/10.1016/B978-0-323-91738-4.00001-4 Copyright © 2023 Elsevier Inc. All rights reserved.

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played a major role in revealing, at an electronic structure level, the sequence of events that take place in the DNA/RNA molecular photoresponses. In addition, the synergy between the paramount biological (and almost philosophical) importance of the NB photophysics and the huge amount of available data has converted these systems as usual molecules to benchmark and fine-tune new theoretical methodologies and experimental techniques. In this chapter, we shall summarize the photophysics of NB and how the preferred deactivation mechanisms can be altered by several factors such as solvent effects, chemical modifications, or the DNA double-strand surrounding.

12.1

Photophysics of canonical nucleobases in the gas phase. Photostability

This section describes the light absorption properties and the excited-state decay mechanisms of isolated DNA/RNA NBs in the gas phase. Note that while these systems can exist in different tautomers, only the biologically relevant canonical DNA structure is considered in this book chapter. The reader is referred to extensive reviews published elsewhere for results concerning other tautomers (14). The gas-phase vertical absorptions and the main theoretical grounds of the static photoreaction paths started to be studied in the early 2000s decade (17–24), whereas the first dynamic studies appeared in the second half of the same decade (25–28). However, these phenomena are often revisited with more sophisticated or new methodological implementations (29–37), to study additional features (35,38–41), with extended systems that capture more physics (42–44), or with benchmark purposes (45–51). The goal of the present section is to provide a homogenized overview of the current consensus on the DNA/RNA photophysics in the gas phase, highlighting also some of the latest advances in this field. An exhaustive revision is out of the scope of the present book chapter, for more details, the reader is referred to extensive reviews published elsewhere (11,13–15). The influence of solvent effects, NB chemical modifications, and the multichromophoric nature of nucleic acids will be covered in the next sections.

12.1.1 Absorption properties in the gas phase Canonical DNA/RNA NBs (Schemes 12.1 and 12.2) absorb UV light mainly through the population of the optically active states of 1(p,p*) nature, as summarized in Table 12.1. The vertical energy ordering of the state mostly accessed upon irradiation (i.e., bright state) depends on the NB. For uracil and thymine, the 1(p,p*) state is the second singlet excited state (S2), whereas the 1(n,p*) state lies at lower energies (S1) (13,15). This has important implications in the excited-state decay paths, since the role of the 1(n,p*) state in the return to the ground state from 1(p,p*) was intensely debated in the literature (13–15). In contrast, the bright 1(p,p*) state of cytosine is S1, being the 1(nO,p*) state above in energy. In adenine, two 1(p,p*) states, labeled La and Lb, and a 1(n,p*), fall in the UV absorption region; however, the largest absorption probability (oscillator strength, f) is ascribed to the 1(p,p* La) state. The vertical energy ordering of these three states is much more controversial and method-dependent, as discussed in detail in Ref. (15). In any case, there is consensus in the fact that, in the gas phase, the mentioned states of adenine are very close in energy. Guanine has three similar lowest-lying singlet excited states to those of adenine, nevertheless, both La and Lb 1(p,p*) states have similar absorption capacities. The energy ordering is much clearer in guanine as compared to adenine: all methods predict the 1(p,p* La) state as S1, whereas its Lb counterpart and the 1(n,p*) state lie at higher energies (13,15).

12.1.2 Photophysical paths for purine nucleobases The calculation of the relevant potential energy surfaces within the so-called static approach, i.e., the kinetic energy of the system is neglected, and all efforts are devoted to map the potential energy of the system defined by the set of nuclei and electrons, elucidates the main aspects of the photochemical decays with a rich variety of computational strategies at a moderate computational cost (13,63,64). In addition, it allows necessary benchmarks of theoretical methods and comparisons SCHEME 12.1 Purine (Pur) core and atom labeling. Canonical purine nucleobases Adenine (Ade) and Guanine (Gua). (Credit: The authors of this chapter.)

DNA photostability Chapter

12

313

SCHEME 12.2 Pyrimidine (Pyr) core and atom labeling. Canonical pyrimidine nucleobases Cytosine (Cyt) and Uracil (Ura) (Thymine: 5methylUracil). (Credit: The authors of this chapter.)

TABLE 12.1 Vertical absorption energies of the lowest-energy bright state computed with different correlated electronic structure methods at the corresponding Franck-Condon geometries. NB

State

TDDFT

CC

DFT/ MRCI

RASPT2

CASPT2

NEVPT2

MRCISD+Q

Exp

Uracil

1

5.17 (52)

5.34 (53)

5.33 (54)

5.20 (45)

5.02–5.23 (55–57)

5.39 (58)

5.76 (58)

5.1 (10)

Thymine

1

4.95 (52)

5.20

5.15 (54)

5.00 (45)

4.89–5.06 (55–57)

5.18 (58)

5.71 (58)

4.8 (10)

Cytosine

1

4.63 (52)

4.66 (53)

4.69 (54)

4.66 (45)

4.41–4.68 (55,56)

4.78 (58)

4.89 (58)

4.6 (10)

Adenine

1

4.97 (52)

5.25 (53) 5.23 (59)

5.15 (54)

5.18 (45)

5.30–5.35 (55,56,60,61)

5.22 (58)

–

5.2 (10)

Guanine

1

4.88 (52)

4.98 (53)

4.86 (59)

4.68 (45)

4.93 (55,62)

5.14 (58)

–

4.6 (10)

(p,p*) (p,p*) (p,p*) (p,p* La)

(p,p* La)

References provided within parenthesis. CASPT2, complete-active-space second-order perturbation theory; CC, coupled-cluster based methods; DFT/MRCI, density functional theory/multireference configurations interaction; MRCISD+Q, internally contracted multireference configuration interaction singles and doubles with Davidson correction; NEVPT2, n-electron valence state perturbation theory; RASPT2, restricted-active-space second-order perturbation theory; TDDFT, time-dependent density functional theory. The reader is referred to Ref. (15) for the comparison of more levels of theory. Table inspired from Giussani, A.; Segarra-Martı´, J.; Roca-Sanjua´n, D.; Mercha´n, M. Excitation of Nucleobases From a Computational Perspective I: Reaction Paths. Top. Curr. Chem. 2015, 355, 57–97.

with some experimental observables, such as excited-state lifetimes, by identifying potential energy penalties in the surfaces’ shapes. Guanine. Studies based on high-level multireference methods (4,62) indicate that the most efficient decay of the 1(p,p* La) state, populated after excitation at low energies, is the barrierless decay through the corresponding minimum energy path (MEP) toward the interstate crossing (conical intersection) (65,66) with the ground state (gs/p,p* La)CI. The absence of energy barriers correlates with one of the excited-state lifetime measured for guanine, in particular, the ultrafast (sub picosecond) lifetime of 148 fs (67). The decay is mediated by the out-of-plane distortion of the methanamine fragment (C2NH2), so in the following, we refer to this (gs/p,p* La)CI structure as C2-CI (Fig. 12.1A). The population of the lowest-lying 1 (n,p*) state and the trapping in the corresponding excited-state minimum would be responsible of longer lifetimes (67). From this minimum, either a direct access to the ground state through the (gs/n,p* La)CI or the population transfer to the 1 (p,p* La), which in turn decays to S0, are predicted (4,62). The (gs/n,p* La)CI is characterized by an out-of-plane distortion of the C]O fragment, being labeled as C6-CI (Fig. 12.1A). The triplet states can be accessed either along the 1(p,p* La) path or close to the C2-CI region (68). Adenine. Seminal works based on the CASPT2//CASSCF approach reported by Serrano-Andr
es, Mercha´n, and Borin (60,61) indicated that the gradient-driven MEP for the optically active 1(p,p* La) state leads to a barrierless crossing with the ground state through the ring puckering at C2 in conjunction with the out-of-plane distortion of H2, (gs/p,p* La)CI (C2-CI in Fig. 12.1A). This is the dominant decay path and has been ascribed to the ultrashort excited-state lifetime of 100 fs (67). The population of the other singlet excited states 1(p,p* Lb) and 1(n,p*), vertically lower in energy according

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III Applications

FIG. 12.1 Schematic description of the major channels along the deactivation mechanisms of (A) purine nucleobases (both bases share the main mechanisms features) and (B) pyrimidine nucleobases (cytosine solid, thymine/ uracil dashed) in the gas phase. Ade, adenine; CI, conical intersection; Cyt, cytosine; Gua, guanine; ISC, intersystem crossing; MIN, minimum; Thy, thymine. (Credit: The authors of the present book chapter.)

to CASPT2 determinations (although the energetic ordering is method-dependent (15)), can take place through state crossings along the mentioned 1(p,p* La) MEP or, less likely, by direct light absorption (13). The population of the 1 (n,p*) state is associated with longer excited-state lifetimes and even to the population of triplet states due to the presence of an energy barrier to access the crossing (C6-CI). As in the guanine case, this structure is characterized by the ring puckering at the C6 position and the out-of-plane disposition of the NH2 group (69), which mediates the decay to the ground state (15). The topology and the interpretation of the 1(n,p*) potential energy surface has been intensely debated in the literature dealing with the decay of photoexcited adenine (15).

12.1.3 Excited-state dynamics of purine nucleobases The static description depicted in the previous section provides the grounds of the decay mechanisms. The next natural research step is, thus, to simulate the excited state evolution over time to quantify the competition of the different available photophysical routes, their time scales, and quantum yields. Nonetheless, technical difficulties and the very high computational cost of these methodologies have led to a progressive sophistication of the simulations, particularly, but not limited to, the amount of electron correlation included in the QM scheme, which can greatly impact the simulation outcomes (70–73). Guanine. The nonadiabatic simulations departing from the bright 1(p,p* La) state show consensus in determining the direct and ultrafast decay to the ground state through the C2-CI puckered conical intersection. The participation of the 1(n,p*) state is therefore secondary (14). Adenine. In clear contrast to guanine, nonadiabatic excited-state dynamics simulations for adenine show a clear population transfer from the 1(p,p* La) state to 1(n,p*) (14), ascribed to the subpicosecond excited-state lifetime experimentally measured (67). The return to the ground state from 1(n,p*), predicted in the simulations from 440 to 1120 fs, is shorter than the experimental observations although both experimental and theoretical approaches are in reasonable agreement (14). Further simulations reported by Plasser et al. (74) with the ADC(2) method concluded that both C2-CI and C6-CI mediate the deactivations, although the latter structure is more relevant at higher energies.

12.1.4 Photophysical paths for pyrimidine nucleobases Cytosine. The potential energy surface of the cytosine bright 1(p,p*) state shows a shallow although well-defined minimum between the Franck-Condon (FC) region and the conical intersection with the ground state, connected with the latter structure through a small energy barrier (Fig. 12.1B) (24). The conical intersection that allows the degeneracy (gs/p,p*)CI is characterized by a twist of the C5-C6 bond (C5/C6-CI in Fig. 12.1B), as recently demonstrated via laser experiments and multiconfigurational CASPT2//CASSCF determinations (33). This particular topology could correlate with a slightly slower decay and the longer excited-state lifetimes measured for cytosine with respect to other NB (67) and the fluorescence dependence on the temperature (75).

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Although correlated methods predict that the dark 1(nO,p*) and 1(nN,p*) states lie vertically above the aforementioned bright 1(p,p*) state (15), the involvement of the former in photophysical deactivation routes has been debated in the literature since the adiabatic minimum of these states could be accessible from the 1(p,p*) surface. For instance, Blancafort and Robb (76) proposed a decay mechanism mediated by a three-state conical intersection (nO,p*/p,p*/gs)CI, evidencing the strong 1(p,p*) and 1(nO,p*) mixing in this region. Kistler and Matsika (77) characterized three additional three-state conical intersections, connecting them with the mentioned (gs/p,p*)CI structure. Meanwhile, the role of the triplet states have been also studied in detail (15), most likely accounting for the longer-lived excited-state transients detected in the experiments (67). Early studies (68,78) suggested 1(p,p*) to 3(p,p*) population transfers; however, later dynamic studies found mixed 1(p,p*)/1(n,p*) character of the excited state in the operative ISC mechanisms. Fig. 12.1B indicates ISC from the 1(n,p*) state. Uracil/thymine. The potential energy surfaces for these two NBs are very similar since the only difference between these two molecules is the presence of a methyl group at 5 position, which does not participate directly in the electronic excitation, although it can affect the dynamics (see next section) (14). For these systems, the decay mechanism (Fig. 12.1B) is sensitive to the electronic structure method and the nature of the S1 and S2 states. Highly correlated multiconfigurational methods such as CASPT2 predict the 1(p,p*) state as S2 (13). The MEP of the initially accessed 1(p,p*) state connects with the ethene-like conical intersection structure with the ground state (gs/p,p*)CI, also characterized by a C5-C6 bond twist, through a downward, barrierless path (C5/C6-CI in Fig. 12.1B) (13). This photostability channel repopulates the ground state by converting the excited state into vibrational energy in a subpicosecond timescale, as observed in transientabsorption experiments (67). Within this interpretation, the longer excited-state lifetimes would be ascribed to internal conversions from the 1(n,p*) and 3(p,p*) to the ground state. On the other hand, other studies provide alternative interpretations, in which the ultrafast excited-state component is ascribed to the decay from 1(p,p*) into the 1(n,p*) state and the longer-lived lifetime to the decay of latter to the ground state (see, for instance, Ref. (79)). The existence of a minimum on the 1(p,p*) that may trap the system in the S2 state has been also reported (80), although recent excited-state dynamics with strongly correlated methods discard this possibility (47,50,81).

12.1.5 Excited-state dynamics of pyrimidine nucleobases Cytosine. Dynamic studies indicate that the 1(p,p*) state decays to the ground state mostly through a conical intersection structure (gs/p,p*)CI characterized by the C6-puckering (C5/C6-CI) (14). The key role of this distortion in mediating the excited-state deactivation is supported by experimental recordings on 5-substituted cytosine derivatives, whose excitedstate lifetime shows dependence on the substitution pattern (33,82). Besides this photophysical pathway in the singlet manifold, Mai et al. (83) observed an ultrafast singlet-to-triplet ISC from 1(n,p*) to 3(n,p*) in surface-hopping arbitrary-coupling (SHARC) dynamics including the possibility of intersystem crossing (ISC) (Fig. 12.1B). The authors propose the triplet-state trapping as the explanation for the very long excited-state component detected experimentally (84). Improta, Santoro and co-workers also studied the photodynamics of higher 1(p,p*) states populated upon deep UV light absorption through quantum dynamics, showing ultrafast decays to the lowest-lying singlet excited states (30). Thymine/uracil. Both uracil and thymine have comparable dynamics, although it seems that the methyl group at C5 position in thymine may slightly slow down its relaxation dynamics (14). The initially accessed 1(p,p*) state either repopulates the ground state through (gs/p,p*)CI crossings or undergo S2/S1 internal conversion with the 1(n,p*) state through the corresponding (n,p*/p,p*)CI (15). The involvement of the 1(p,p*) state minimum in the relaxation mechanisms of thymine/uracil was also a matter of intense debate in the literature, and different mechanisms have been reported depending on the amount of electron correlation included in the simulations (14,15). However, recent excited-state dynamic simulations for uracil with the multistate CASPT2 method reported by Matsika and co-workers (50) indicate that there is no trapping on S2. Recent second-order algebraic-diagrammatic-construction (ADC(2)) simulations (42) and quantum dynamics based on linear vibronic coupling models (36) are in reasonable agreement. Instead, the 1(p,p*) population is transferred to the 1(n,p*) state in an ultrafast manner. As stated by the authors, the simulations were not relaxed to the ground state due to high computational requirements, and it is thus expected that machine learning (85,86) implementations should assist much longer simulation time scales at this level of theory. Nevertheless, on the grounds of multistate hybrid TDDFT surface hopping dynamics, Furche and co-workers (46) report that the 1(n,p*) decay to the singlet ground state takes place in 14 ps, whereas the mixed-reference spin-flip TDDFT nonadiabatic dynamics performed by Park et al. (32) indicate that the decay takes place in 6.1 ps. It shall be noted that previous surface-hopping CASSCF dynamics on uracil (87) and thymine (88) including triplet states support the competitive, ultrafast population of triplet states. For uracil, after 1 ps of simulation, the triplet-state population is 20%–25% while the return to S0 is only 10% (87).

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Summary: l

l

The spectroscopic state is the S1 1(p,p*) for guanine and cytosine, whereas for uracil/thymine and adenine there is discrepancy between different electronic structure methods. Potential energy surfaces and dynamics:  Barrierless pathways connect the bright states from the Franck-Condon region toward deactivation funnels to the ground state, disrupting the ring planarity at the C2 (purines) and C5-C6 (pyrimidines) positions.  Some involvement of dark 1(n,p*) and triplet 3(p,p*) states is not discarded, resulting in longer living lifetimes due to the presence of barriers preceding the corresponding funnel C6-CI (purines) and C2-CI (pyrimidines) and intersystem crossing processes.

12.2 Photophysics of canonical nucleobases in solution. Impact of the solvent effects into the photostability Throughout this section, we will analyze how solvent affects the absorption spectra in terms of energies and intensities plus the changes observed in the global shape of their potential energy surfaces and/or excited state dynamics. We will focus in some relevant examples of the bibliography, being a complete review out of the scope of the present chapter (15). In general, three different approaches have been used for the solvent treatment by the “DNA” community, namely, implicit (either via polarizable continuum model [PCM] or conductor-like screening model [COSMO]), quantum mechanics/molecular mechanics (QM/MM) schemes or via the inclusion of solvent molecules within the QM part. We refer to more specialized papers discussing the advantages/drawbacks of each model (89–91), just mentioning here the one used for each specific case.

12.2.1 Purine nucleobases 12.2.1.1 Absorption spectra Guanine. A QM(CASPT2/6-31G*)/MM(TIP3P) study (92) revealed that when guanine is in water solution, both p,p* states, 1(p,p* La) and 1(p,p* Lb), are stabilized (red-shifted) in 0.16 and 0.33 eV, respectively. The dark 1(n,p*) state is, instead, blue-shifted in +0.2 eV. These shifts produce a state ordering inversion between S2 1(n,p*) and S3 1(p,p* Lb) in water. Interestingly, water affects the oscillator strength of both pp* states but in different directions, 1(p,p* La) becoming less intense, whereas 1(p,p* Lb) slightly increases its oscillator strength and becomes the brightest state. This is in nice agreement with the experimental absorption spectrum in which the second band due to the 1(p,p* Lb) state is more intense than the band arising due to 1(p,p* La). Similar results were found when considering explicit solvent molecules at the TDDFT level of theory (93). Calculations with heavily hydrated clusters also reported coherent results (94). Adenine. The trend described above for guanine is conserved in adenine when comparing gas phase and water absorption spectra: the 1(p,p* La) and 1(p,p* Lb) states are red-shifted, and the dark state, 1(n,p*), is strongly blue-shifted according to both explicit TDDFT and QM(CASPT2)/MM(Monte Carlo quantum mechanics) calculations (95,96). Beyond water, one of these studies also considered the effect of acetonitrile, via PCM calculations (95). CASSCF (0.02 eV) and TDDFT ( 2AP > thienoG and (C) Efficient triplet population through several competing mechanisms: thiobases and azabases. (Credit: The authors of the present book chapter.)

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transition 1(p,p*) ! 1(n,p*), and thereby, larger energy barriers are associated to this pathway. An important conclusion is that a single water molecule at that position is equivalent to having bulk water. Although alternative mechanisms as proton transfer or ISC were explored, they were found to be less relevant (111).

12.3.1.3 Oxo incorporation (dC]O) 2-Oxopurine (2oxoP). The absorption spectra and potential energy surfaces of 2oxoP were characterized with calculations using three different levels of theory namely MS-CASPT2, ADC(2) and TD-M062X (112). The 1(p,p*) excited state is stabilized at the FC region, i.e., red-shifted, compared with the bare purine base (113) and 2AP (112). Despite this similarity in the effect of amino or oxo inclusion at the C2 position in the absorption spectra, the effect along the potential energy surfaces is slightly different. The MEP and optimization predict a quite planar region of the potential energy surface from the FC region toward a 1(p,p*) minimum. However, contrary to 2AP, 2oxoP does not show any transition state preventing the decay through the C6-CI, even though this CI is higher in energy compared with the minimum. So, both systems share the presence of a “stable” 1(p,p*) minimum, with some possibilities for fluorescence and/or ISC to the triplets, but this minimum should be loner living for 2AP compared to 2oxoP. 6-Oxopurine (6-oxoP or Hypoxanthine 5 HPX). A combined study sketched the excited state relaxation process of HPX by fluorescence upconversion experiments and MRMP2/SA-CASSCF(18,13)/6-31G+(d) calculations (114). The experimental observations, fluorescence lifetimes 100 Ps) Deactivations. J. Phys. Chem. Lett. 2018, 9, 2373–2379. 206. Ibele, L. M.; Sanchez-Murcia, P. A.; Mai, S.; Nogueira, J. J.; Gonza´lez, L. Excimer Intermediates En Route to Long-Lived Charge-Transfer States in Single-Stranded Adenine DNA as Revealed by Nonadiabatic Dynamics. J. Phys. Chem. Lett. 2020, 11 (18), 7483–7488. 207. Li, J.-H.; Zuehlsdorff, T. J.; Payne, M. C.; Hine, N. D. M. Photophysics and Photochemistry of DNA Molecules: Electronic Excited States Leading to Thymine Dimerization. J. Phys. Chem. C 2018, 122 (22), 11633–11640. 208. Jakhlal, J.; Denhez, C.; Coantic-Castex, S.; Martinez, A.; Harakat, D.; Douki, T.; Guillaume, D.; Clivio, P. SN-and NS-Puckered Sugar Conformers Are Precursors of the (6–4) Photoproduct in Thymine Dinucleotide. Org. Biomol. Chem. 2022, 20 (11), 2300–2307. 209. Olaso-Gonzalez, G.; Roca-Sanjuan, D.; Serrano-Andres, L.; Merchan, M. Toward the Understanding of DNA Fluorescence: The Singlet Excimer of Cytosine. J. Chem. Phys. 2006, 125, 231102. 210. Zuluaga, C.; Spata, V. A.; Matsika, S. Benchmarking Quantum Mechanical Methods for the Description of Charge-Transfer States in p-Stacked Nucleobases. J. Chem. Theory Comput. 2020, 17 (1), 376–387. 211. Doorley, G. W.; McGovern, D. A.; George, M. W.; Towrie, M.; Parker, A. W.; Kelly, J. M.; Quinn, S. J. Picosecond Transient Infrared Study of the Ultrafast Deactivation Processes of Electronically Excited B-DNA and Z-DNA Forms of [Poly(DG-DC)]2. Angew. Chem. Int. Ed. 2009, 48, 123–127. 212. Gobbo, J. P.; Sauri, V.; Roca-Sanjuan, D.; Serrano-Andres, L.; Merchan, M.; Borin, A. C. On the Deactivation Mechanisms of Adenine-Thymine Base Pair. J. Phys. Chem. B 2012, 116 (13), 4089–4097. 213. Chen, J.; Thazhathveetil, A. K.; Lewis, F. D.; Kohler, B. Ultrafast Excited-State Dynamics in Hexaethyleneglycol-Linked DNA Homoduplexes Made of A
T Base Pairs. J. Am. Chem. Soc. 2013, 135 (28), 10290–10293. 214. Jouybari, M. Y.; Green, J. A.; Improta, R.; Santoro, F. The Ultrafast Quantum Dynamics of Photoexcited Adenine–Thymine Basepair Investigated With a Fragment-Based Diabatization and a Linear Vibronic Coupling Model. J. Phys. Chem. A 2021, 125 (40), 8912–8924. 215. Sobolewski, A. L.; Domcke, W. Ab Initio Studies on the Photophysics of the Guanine-Cytosine Base Pair. Phys. Chem. Chem. Phys. 2004, 6 (10), 2763–2771. 216. L€ owdin, P.-O. Proton Tunneling in DNA and Its Biological Implications. Rev. Mod. Phys. 1963, 35 (3), 724. 217. L€ owdin, P.-O. Quantum Genetics and the Aperiodic Solid: Some Aspects on the Biological Problems of Heredity, Mutations, Aging, and Tumors in View of the Quantum Theory of the DNA Molecule. In Advances in Quantum Chemistry; L€owdin, P.-O., Ed.; Vol. 2; Academic Press, 1966; pp. 213–360. 218. Sauri, V.; Gobbo, J. P.; Serrano-Perez, J. J.; Lundberg, M.; Coto, P. B.; Serrano-Andres, L.; Borin, A. C.; Lindh, R.; Merchan, M.; Roca-Sanjuan, D. Proton/Hydrogen Transfer Mechanisms in the Guanine-Cytosine Base Pair: Photostability and Tautomerism. J. Chem. Theory Comput. 2013, 9 (1), 481–496. 219. Schwalb, N. K.; Temps, F. Ultrafast Electronic Relaxation in Guanosine Is Promoted by Hydrogen Bonding With Cytidine. J. Am. Chem. Soc. 2007, 129 (30). 9272–+. 220. Guallar, V.; Douhal, A.; Moreno, M.; Lluch, J. M. DNA Mutations Induced by Proton and Charge Transfer in the Low-Lying Excited Singlet Electronic States of the DNA Base Pairs: A Theoretical Insight. J. Phys. Chem. A 1999, 103 (31), 6251–6256.

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221. Hammes-Schiffer, S.; Stuchebrukhov, A. A. Theory of Coupled Electron and Proton Transfer Reactions. Chem. Rev. 2010, 110 (12), 6939–6960. 222. Hammes-Schiffer, S. Proton-Coupled Electron Transfer: Classification Scheme and Guide to Theoretical Methods. Energy Environ. Sci. 2012, 5 (7), 7696–7703. 223. Hammes-Schiffer, S. Proton-Coupled Electron Transfer: Moving Together and Charging Forward. J. Am. Chem. Soc. 2015, 137 (28), 8860–8871. 224. Franc
es-Monerris, A.; Gattuso, H.; Roca-Sanjua´n, D.; Tun˜o´n, I.; Marazzi, M.; Dumont, E.; Monari, A. Dynamics of the Excited-State Hydrogen Transfer in a (DG)
(DC) Homopolymer: Intrinsic Photostability of DNA. Chem. Sci. 2018, 9 (41), 7902–7911. 225. Groenhof, G.; Schaefer, L. V.; Boggio-Pasqua, M.; Goette, M.; Grubmueller, H.; Robb, M. A. Ultrafast Deactivation of an Excited Cytosine-Guanine Base Pair in DNA. J. Am. Chem. Soc. 2007, 129 (21), 6812–6819. 226. Green, J. A.; Yaghoubi Jouybari, M.; Asha, H.; Santoro, F.; Improta, R. Fragment Diabatization Linear Vibronic Coupling Model for Quantum Dynamics of Multichromophoric Systems: Population of the Charge-Transfer State in the Photoexcited Guanine–Cytosine Pair. J. Chem. Theory Comput. 2021, 17 (8), 4660–4674. 227. Franc
es-Monerris, A.; Segarra-Martı´, J.; Mercha´n, M.; Roca-Sanjua´n, D. Theoretical Study on the Excited-State p-Stacking Versus Intermolecular Hydrogen-Transfer Processes in the Guanine–Cytosine/Cytosine Trimer. Theor. Chem. Accounts 2016, 135 (2), 1–15. 228. Alexandrova, A. N.; Tully, J. C.; Granucci, G. Photochemistry of DNA Fragments Via Semiclassical Nonadiabatic Dynamics. J. Phys. Chem. B 2010, 114 (37), 12116–12128. 229. Markwick, P. R. L.; Doltsinis, N. L. Ultrafast Repair of Irradiated DNA: Nonadiabatic Ab Initio Simulations of the Guanine-Cytosine Photocycle. J. Chem. Phys. 2007, 126 (17), 175102. 230. Martinez-Fernandez, L.; Improta, R. Photoactivated Proton Coupled Electron Transfer in DNA: Insights From Quantum Mechanical Calculations. Faraday Discuss. 2018, 207, 199–216.

Chapter 13

Fluorescent proteins M.G. Khrenovaa,b and A.P. Savitskya,b a

A.N. Bach Institute of Biochemistry, Research Centre of Biotechnology of the Russian Academy of Sciences, Moscow, Russia, b Department of Chemistry,

Lomonosov Moscow State University, Moscow, Russia

Chapter outline 13.1 13.2 13.3 13.4

13.1

Introduction Modeling of absorption spectra F€ orster resonance energy transfer Photochemical reactions

337 338 341 343

13.5 Concluding remarks Acknowledgment References

345 346 346

Introduction

Fluorescent proteins are already recognized tools for bioimaging (1,2), superresolution spectroscopy (3,4), and analytical applications (5–7). By now, the whole visible window and neighboring near UV and IR parts of the electromagnetic spectrum are covered by known fluorescent proteins (8). The major group of fluorescent proteins has the secondary structure of 11-sheets b-barrel with the chromophore formed autocatalytically from three amino acids during the maturation. These proteins were found in marine organisms such as jellyfish, Anthozoa, and many others (9–15). Numerous amino acid substitutions were suggested to improve and diversify photochemical and photophysical properties due to variations of both chromophore structure and its surrounding. The chromophore group of GFP (green fluorescent protein), that was the first discovered member, is composed of phenyl and imidazolidinone parts bridged with the methylene group (Fig. 13.1). Fluorescent proteins with the b-barrel structures may consist of chromophores with benzyl, imidazole, and indole groups instead of a phenyl moiety, possibly extending the conjugated p-system due to variations of the molecular fragments attached to the imidazolidinone ring. Another group of fluorescent proteins was obtained from the LOV (light, oxygen, voltage) proteins with the flavin chromophore group (Fig. 13.1). Substitution of the cysteine residue to alanine, C426A, introduced to the LOV protein abolished the light-induced chemical reaction leading to the formation of an unstable covalent bond between the chromophore and the side chain of the cysteine C426 (16). Later, other amino acid substitutions were introduced to enhance the photostability of these proteins and the fluorescence quantum yield, with the most popular member called iLOV (16). Flavin-based fluorescent proteins have advantages compared with GFP-like proteins. For instance, their smaller size and oxygenindependent fluorescence are the advantages of these markers. Still, their absorption and emission bands are far from the biotransparency window that limits their utilization in tissues. All flavin-based fluorescent proteins studied by now have similar absorption and emission band maxima and there is still a question of whether it is possible to redshift spectral features. Fluorescent proteins with the absorption and emission band maxima shifted to the near IR region are characterized by linear tetrapyrrole (bilin) chromophores (Fig. 13.1) (17–21). Bilins have the largest conjugated p-system among the considered fluorescent protein chromophores that explains the smallest energy gap between the ground and the excited electronic states. Typically, these fluorescent proteins have a biliverdin molecule covalently bound to a cysteine residue of the protein and are called phytochromes. In this chapter, we mainly focus on modeling methods that can be utilized for the calculation of the photophysical and photochemical properties of fluorescent proteins. In this respect, the GFP-like proteins undergo diverse processes that can be studied using different methods. Therefore, in the following subsections, we will illustrate the quantum chemistry methods, molecular dynamics, and combined quantum mechanic/molecular mechanic (QM/MM) methods on the examples related to the GFP-like proteins. Theoretical and Computational Photochemistry. https://doi.org/10.1016/B978-0-323-91738-4.00015-4 Copyright © 2023 Elsevier Inc. All rights reserved.

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FIG. 13.1 Structures of the flavin-based fluorescent proteins, GFP-like proteins, and biliverdin-binding domain of phytochromes and their corresponding chromophores. The protein structure is shown in the cartoon representation and the chromophores with the van der Waals spheres. Derivatives of the GFP-like protein chromophore are depicted with R1 and R2 substituents. The R1 variations are due to different amino acid residues forming the chromophore (tyrosine, phenylalanine, histidine, or tryptophane) while the R2 part might include the extension of the conjugated p-system if additional oxidation steps occur during the maturation or after photoconversion. Here and in the next figures, green (dark gray color in print version), red (gray color in print version), blue (light gray color in print version), orange (light light gray color in print version), and white colors refer to the carbon, oxygen, nitrogen, phosphorus, and hydrogen atoms, respectively.

13.2

Modeling of absorption spectra

GFP-like proteins with the pHBDI (4-(4-hydroxybenzylidene)-1,2-dimethyl-1H-imidazol-5(4H)-one, shown in Fig. 13.1) chromophore or its analogs with the more extended conjugated p-systems in different protonation states absorb in the visible part of the spectrum and emits even in the near IR. Crude estimates of absorption and emission band maxima wavelengths can be done, if one knows the chromophore structure, i.e., the size of the conjugated p-system and the protonation state (Fig. 13.1). However, precise estimates require explicit calculations of the energy differences between the ground and excited electronic states or the usage of the QSPR-type (quantitative structure-property relationship) relations. The most widely used approach to estimate the absorption and emission properties of these chromophores is the timedependent density functional theory (TDDFT) that allows obtaining fast calculations of the electronic transition energies. Various benchmark studies have been already done for the pHBDI and their analogs with the extended p-system for both, isolated molecules (22–25) and within the fluorescent protein surrounding (26,27). The accuracy of the results strongly depends on the selected DFT functional (23,28–30), even for the gas phase systems. In proteins, chromophore polarization by the protein environment as well as deviation from the planarity further complicates the accurate energy difference estimates. Utilization of different DFT functionals to the same molecular systems can give estimates of the excitation energy differing by about 0.5 eV. Generally, benchmark studies cover three types of functionals: GGA, hybrid, and Coulombattenuated functionals. From the theoretical side, the latter two groups are supposed to be of higher quality as those combine semilocal DFT exchange and exact Hartree-Fock exchange. For the anionic pHBDI chromophore, the best estimates with respect to the experimental excitation energy value are obtained with GGA functionals as the BLYP (31,32) and PBE (33,34) ones, whereas for neutral and cationic pHBDI the best performance was found for the hybrid functionals (25). Authors state that occasional quantitative agreement for certain protonated states obtained with the GGA and hybrid functionals is most likely fortuitous, and utilization of the same functionals to other chromophores does not guarantee small errors in the excitation energy estimates. The alternative is the utilization of multiconfigurational perturbation theories, such as second-order complete active space perturbation theory (CASPT2) (27,35,36), second-order extended multiconfiguration quasi-degenerate perturbation theory (XMCQDPT2) (22,37), etc., that can derive to more accurate estimates of the excitation energies (35,37–39). This group of methods is basically based on the wave functions obtained at the complete active space self-consisted field (CASSCF) (40) level that requires manual construction of the active space and selection of molecular orbitals.

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The exception is the scaled opposite spin configuration interaction singles with perturbative doubles method (SOS-CIS(D)) (41) that is able to accurately predict vertical excitation energies for some systems and does not require manual inspection of molecular orbitals during calculations. However, despite their accuracy (42–44), all perturbation theory-based methods are time consuming. Therefore, on the one hand, we have fast and “black box” TDDFT methods with accuracy far beyond 0.1 eV and, on the other hand, time-consuming methods with manual expertise requirements from the multiconfigurational perturbation theory family with an accuracy of approximately 0.1 eV (45–51). This accuracy might be enough if one has to attribute experimentally observed absorption or emission bands of chromophores with quite different spectra. However, these methods can hardly be utilized if dealing with a set of similar systems, for example, different fluorescent proteins with the same GFP-like chromophore. The excitation band maximum of GFP is 488 nm (52), and the 0.1 eV accuracy corresponds to the 20 nm uncertainty on the wavelength scale in this region. Thus, straightforward calculations of the S0 (singlet ground state)-S1 (first singlet excited state) excitation energies cannot discriminate different species of the same colour. In contrast, the yellow fluorescent protein (YFP) (53), that is a GFP variant with an excitation energy shifted to 514 nm due to the p-stacking interaction of the chromophore with the tyrosine residue, could be distinguishable from the GFP by comparing computed absorption or emission energies. The state of things is worse for the red fluorescent proteins (RFPs) as the 0.1 eV uncertainty corresponds to the 40 nm in the red part of the spectrum, for example, 575–605 nm. Therefore, one cannot be confident when calculating S0-S1 transition energies of the RFPs with the same chromophore groups and different protein surroundings. This mentioned problem of the precise estimates of transition energies is due to the following. The methods mentioned above estimate the absolute energies of the ground and excited states and the transition energy is defined as their energy difference. This means that the selected method should nicely describe both states and that the error of the transition energy accumulates errors coming from both states unless the error cancelation occurs. Still, there is a need to estimate excitation energies with high accuracy. A prospective alternative to the direct calculations of energy differences between electronic states is the search for QSPR models that utilize the features of chromophores that are mainly responsible for the excitation energy values. Now, in this respect, there are two directions. The first is based on the geometry criterion—so called bond-length alternation (BLA)—that is calculated as a difference of two bond lengths on the methylene bridge of the chromophore (Fig. 13.2). To some extent, BLA is a measure of the equilibrium between two resonance states that determines the energy gap between the ground and excited states. The BLA index was successfully utilized for absorption estimates based on the crystal structures of the fluorescent proteins with the GFP-type chromophore (54). Still, utilization of the BLA index to RFPs with elongated chromophores was not successful (29). The prospective and physically grounded alternative to the BLA index was suggested by M. Drobizhev and coauthors (55). It was demonstrated that the electric field inside the fluorescent protein affects the excitation energy value. The internal electric fields in fluorescent proteins are strong; therefore, a quadratic Stark effect should be observed, that is the electronic transition energy shifts in the presence of the external electric field. The explanation is the following. It is assumed that the dipole moment of the molecule in the ground state, mS0, is different from the dipole moment in the excited state, mS1. Then, the dipole moment variation upon excitation (or, shortly, dipole moment variation), DMV ¼ mS1 mS0, is nonzero. From the computational point of view, this quantity can be estimated as the product of the charge, transferred upon excitation, and the distance between the barycenters of the negative and positive transferred charges (Fig. 13.3). The internal Stark effect due to the internal electric field can produce an excitation energy shift relative to the case of the absence of an electric field. The internal electric field is originated from the amino acid residues of the b-barrel close to the chromophore and its direction is fixed with respect to the chromophore; therefore, it will be the same regardless of the orientation of the fluorescent protein in solution. According to the quadratic Stark effect, the following equation can be written as: ES 0

S1 ,prot

¼ ES0

S1 ,vac

+ A
ðDMV vac
DMVÞ + B
jDMV j2

(13.1)

where the ES0 S1, prot is the excitation energy in protein, ES0 S1, vac is the excitation energy in vacuum, DMV is the vector of dipole moment variation upon excitation in protein, and DMVvac is dipole moment variation upon excitation in vacuum.

FIG. 13.2 Resonance forms of the tyrosine-based GFP-like chromophore. The bond-length alternation (BLA) index is the difference between the bond lengths of two bridging CdC bonds, rP, and rI, as depicted.

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FIG. 13.3 Electron density difference upon the S0–S1 excitation for the GFP-type chromophore. The pink (light gray color in print version) and cyan (gray color in print version) wireframe isosurfaces correspond to the increase and decrease of electron density upon electronic excitation, respectively; the contour values are 0.0015 a.u. The arrow identifies the negative charge transfer direction; it starts at the barycenter of the negative electron density difference (qCT) and ends at the barycenter of the positive electron density difference (q+CT).

The estimate of DMV can be obtained experimentally from one- and two-photon absorption cross-section measurements (55–58) and from Stark spectroscopy (54,59). This parabolic dependence was found experimentally for a series of fluorescent proteins from the mFruits family (55) and later for the proteins with the GFP-type chromophore (58). The theoretical estimate of the dipole moment variation upon excitation was first reported in Ref. (60) for a series of gasphase systems composed of the pHBDI chromophore in the anionic form stabilized by the hydrogen bonds between water molecules and the phenolic oxygen of pHBDI. Variations of excitation energies were due to the introduction of p-stacking partners to the model systems, and in this case, were substituted benzenes. Later, the study was extended to different chromophores and p-stacking partners, namely DsRed- and mOrange-type chromophores with the extended p-system and a chromophore that was obtained from the tryptophane instead of tyrosine in the chromophore group. p-stacking partners were both substituted benzenes and imidazols (28) that allowed obtaining both bathochromic and hypsochromic shifts relative to the bare chromophore. These calculations demonstrated that if there is no strong electric field, the linear Stark effect is observed, that is the excitation energy linearly depends on the calculated DMV (Fig. 13.4A). The DMV calculations were performed both at the CASSCF and TDDFT levels and demonstrated that CAM-B3LYP and oB97X-D3 functional succeeded in the evaluation of the DMV values. It was an important evidence that electron density change upon excitation is reproduced correctly within this protocol, although the transition energies are highly overestimated. Later, this approach was utilized for chromophores embedded in their respective fluorescent proteins. It was shown that the key to the cyan-to-yellow tuning in GFP-like proteins is polarization (36). To do this, a set of 5 fluorescent proteins with the same GFP-type chromophores were studied: the most red-shifted variant phiYFP (61), intermediate EYFP, Dronpa (62), EGFP, and the most blue-shifted mTFP0.7 (63) were selected. For these proteins, the absorption band maxima vary

FIG. 13.4 Dependencies between the vertical S0–S1 transition energies and DMV calculated at the TDDFT/CAM-B3LYP/cc-pvdz level: (A) molecular clusters containing a mOrange-type chromophore for which its vertical excitation energies are calculated at the XMCQDPT2/SA2-CASSCF(12/12)/ cc-pvdz level (28); (B) set of the models of the entire fluorescent proteins with the DsRed-type chromophores and experimental energies corresponding to the absorption bands maxima of the corresponding proteins (29).

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between 453 and 525 nm, covering the range of 0.4 eV on the energetic scale. Geometry optimization for the selected proteins was performed at the ONIOM QM(PBE0/6-31G(d))/MM(AMBER99) level and different protocols were selected to consider the influence of the electric field from the protein on the chromophore (36). The calculated excitation energies are found to be sensitive to the way of considering the chromophore surrounding. Utilization of nonpolarizable QM/MM models results in too limited variations of excitation energies. Considering only polarization of the ground state results in a large excitation energy difference between the blue- and red-shifted species. The best estimates of the transition energies can be obtained with the state specific accounting for polarization (36). Also, it is shown that the TDDFT method is worse than CASPT2 for some considered systems (36). The dipole moment variation upon excitation calculated at the TDDFT level and experimental excitation energies demonstrates quadratic dependence. This is in line with the strong internal electron field in fluorescent proteins predicted in experimental studies (58,59) and bridges experimentally observed macroscopic values with the calculated microscopic parameters. The extensive study of dependencies between calculated DMV values and experimental absorption band maxima was obtained for a set of red fluorescent proteins with the DsRed-type chromophore (Fig. 13.4B). Those were selected from different ancestors and the degree of similarity between the mRojo (mRojoA, mRojo-THSL, mRojo-VFAV, mRojo-TFAL, mRojo-VYGV, mRojo-VYGL, and mRojo-TYGV) (64) and RDSmCherry (RDSmCherry0.2 and RDSmCherry0.5) (65) families was large (90%). However, the similarity between eqFP670 (66), mKeima (67), and the mRojo family did not exceed 60%, according to the muscle protein sequence alignment analysis (68). Equilibrium geometry configurations for these systems were obtained at the QM(PBE0-D3/cc-pvdz)/MM(AMBER) level. A set of 7 hybrid functionals including CAM-B3LYP (69), LC-BLYP (70), M06-2X (71), BHHLYP (72), PBE0 (73), B1LYP (74), and B97M-D3BJ (75,76) were utilized for the TDDFT calculations at the QM/MM equilibrium geometries. Experimental excitation energies, corresponding to the absorption band maxima, demonstrate a quadratic dependence from the calculated DMV (29). However, the mean absolute errors (MAE) differ more than twice depending on the function used. The best candidates for the DMV estimates are CAM-B3LYP, M06-2X, and BHHLYP, whereas B1LYP and PBE0 are worse for the excitation energy estimates. Importantly, MAE for the best correlations is around 0.01 eV that is much smaller than the accuracy of the computational method for the direct estimates of the excitation energy. Importantly, analysis of the quadratic dependence obtained in the study allows extracting parameters with a physical meaning. For instance, the free term from Eq. (13.1) has the meaning of excitation energy in vacuum. According to the proposed relations, this value equals to 2.10  0.02 eV that agrees with the excitation energy calculated for the same chromophore in vacuum. Also, the DMVvac obtained for these models is 2.78  0.41 a.u. that is similar to the values obtained experimentally by Drobizhev et al. (55). To sum up, precise estimates of electronic excitation energies are difficult and can hardly be solved by the computationally fast TDDFT method. The multiconfigurational perturbation theory-based methods are much more expensive and precise but still can hardly exceed the accuracy of 0.1 eV. An alternative is the utilization of QSPR relations. The prospective descriptor is the dipole moment variation upon excitation that accounts for both dipole moments of the chromophore in the ground and excited states. Moreover, a physically based background is behind its utilization, that is the quadratic Stark effect, that should be observed for a chromophore in a strong electric field. Utilization of such QSPR-type relations allows one to obtain excitation energy estimates with an accuracy not exceeding 0.1 eV.

13.3

€rster resonance energy transfer Fo

F€orster resonance energy transfer (FRET) is a radiationless process that happens between a donor and an acceptor molecule if the fluorescence spectrum of the donor overlaps with the excitation spectrum of the acceptor (77). The efficiency of the energy transfer can be written as: E¼

R60 r 6 + R60

(13.2)

where r is the distance between the donor and acceptor and R0 is the F€orster radius. The latter is the distance between a donor and an acceptor that provides a 50% efficiency of energy transfer. It depends on the overlap integral that is an intrinsic feature of the donor-acceptor pair and is determined by the absorption band shape of an acceptor, emission band shape of a donor, extinction coefficient of an acceptor, and fluorescence quantum yield of a donor. Another key quantity is the orientation factor, k2, that is determined by the relative orientation of the donor and acceptor. It is evaluated as: k2 ¼ ð cos ’DA

3 cos ’AR cos ’DR Þ2

(13.3)

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FIG. 13.5 Example of the FRET pair from TWITCH-2B sensor composed of the triptophane-based mCerulean3 as an energy donor and cpVenus as an energy acceptor (78). Proteins are shown in cartoon representation and the chromophore in van der Waals spheres. Transition dipole moments of donor emission and acceptor absorption are marked as TDMD and TDMA, respectively. RDA is the distance between the donor and acceptor.

where ’DA, ’AR, and ’DR are angles between the transition dipole moments of the donor and acceptor (TDMD and TDMA, respectively), transition dipole moment of the acceptor and a vector connecting the donor and the acceptor (RDA), and transition dipole moment of the donor and the vector connecting the donor and the acceptor, respectively (TDMD, TDMA, and RDA are illustrated on the FRET pair composed of mCerulean3 and cpVenus in Fig. 13.5). The orientation factor varies between 0 and 4 depending on the relative orientation of TDMD, TDMA, and RDA. Proper orientation of an acceptor relative to a donor, and the corresponding increase of the k2, can enhance FRET efficiency that is proportional to the k2 according to the conventional formulae (77), whereas the low values of k2 abolish energy transfer. In the case of free donors and acceptors in solution k2¼2/3 is assumed. When constructing FRET systems, the knowledge of the transition dipole moment is crucial (78–81). Computational studies were performed to compute transition dipole moments for different types of chromophores within fluorescent proteins (30,82,83). First, computational studies were performed for the gas phase systems (82) with a set of 14 chromophores in both neutral and anionic states mimicking chromophores found in fluorescent proteins. Those covered the GFP-type chromophore and its extended analogs as well as chromophores are composed of amino acid residues that are different from tyrosine (tryptophan, histidine, amino-tryptophan, and phenylalanine) in the central part of the chromophore sequence. All transition dipole moments were calculated at the TDDFT(B3LYP/6-31 + G*) level. Later, simulations in molecular clusters, composed of a chromophore and nearby amino acids and water molecules or QM/MM models, were performed for the GFP, TagRFP, and KFP fluorescent proteins at the CIS level (83). Also, from the experimental side, interactions of fluorescent protein crystals with the polarized light were studied by complex spectroscopic techniques. Mathematical models were further applied to evaluate the transition dipole moment for a set of fluorescent protein chromophores (84). Those were the GFP-type chromophore in neutral and anionic states in diverse proteins such as the EGFP and the green form of mEos4b; variants with the extended p-system, mCherry and mTurquoise2 with the tryptophan-based chromophore (84). The computational and experimental studies are consistent and derive the following conclusion. Generally, the transition dipole moment is directed along the chromophore and, depending on the particular chromophore composition, its orientation is defined by the center of the ring formed by the side chain of the second amino acid residue (tyrosine and phenylalanine) of the triad and either of two carbon atoms of the imidazolidinone ring (Fig. 13.6A). For the chromophore containing a tryptophan fragment, the transition dipole moment is directed between the center of the five-membered ring of the tryptophan part and a nitrogen atom of the imidazolidinone ring (Fig. 13.6B) The knowledge of the transition dipole moment calculated for the individual components of FRET pairs can be utilized to estimate possible FRET efficiencies of these systems composed of fluorescent proteins.

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FIG. 13.6 Transition dipole moments in the chromophores of fluorescent proteins within the b-barrel structure are shown with violet vectors. Data is both from experimental (84) and computational (82,83) studies. (A) Chromophores derived from the tyrosine or phenylalanine with R1 equal to OH or O and H, respectively. Possible extension of the conjugated p-system of the chromophore is due to the R2 substituent. The triangle depicts variants of transition dipole moment depending on the particular chromophore. (B) Transition dipole moment for the tryptophane-based chromophore.

13.4

Photochemical reactions

All GFP-like fluorescent proteins are characterized by distinct absorption and emission spectra. Some of them can undergo light-induced photochemical reactions like cis-trans isomerization (85–87) of the chromophore that takes place through an S1–S0 conical intersection and is completed in the ground state. Also, irreversible photoconversion may take place (88–90) which usually corresponds to the elongation of the conjugated p-system of the chromophore. These chemical reactions partly happen on the excited state potential energy surface and so, they are difficult to be modeled. Although recent experimental studies using the serial crystallography technique revealed some crucial steps of these transformations (91,92), still there is a need for a detailed study of their mechanism. The fluorescence state of the GFP-like chromophore corresponds to its cis conformation and anionic protonation state (Fig. 13.7C). For some proteins, light irradiation may result in photoisomerization to the trans form that abolishes fluorescence (Fig. 13.7A). The representative GFP-type proteins undergoing this photoisomerization are Dronpa (93), SAASoti (90,94), its monomeric variant mSAASoti (95) and IrisFP (96). Chromophores of other photoswitchable fluorescent proteins exist in the trans form, being, in fact, chromoproteins. These proteins can be photoactivated and isomerized to the cis form (Fig. 13.7B) as KFP1 (97), asFP595 (98,99), and Padron (100). Both of these processes are reversible, and the thermal relaxation leads to the most stable form (Fig. 13.7). Photoswitching mechanism and back reaction by thermal isomerization are discussed in computational papers (101–103). These processes involve large transformations inside the binding pocket and are difficult to be studied. Moreover, photoisomerization description requires the utilization of multiconfigurational methods as it happens on excited state energy surfaces. Ground state thermal isomerization was studied in ref. (101) for the chromoproteins asFP595 and KFP that can be activated by light irradiation and isomerized to the fluorescent cis form. To do this, QM/MM molecular dynamics simulations FIG. 13.7 Reversible photoswitching mechanisms. (A) Photoswitching of the fluorescent protein from on-state (with cis anionic form) to off-state (with trans neutral form) and back thermal (ground state) isomerization. (B) Photoactivation of the fluorescent protein from off-state to on-state and back thermal isomerization. (C) Structures of the GFP chromophore in the cis anionic and trans neutral forms. The CdC]CdN dihedral angle that characterizes the cis and trans forms is highlighted.

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FIG. 13.8 Potential of mean force (PMF) of the ground-state isomerization of the chromophore in the asFP595 and KFP proteins. The reaction coordinate is a sum of two dihedrals depicted in the inset.

were performed. The quantum subsystem was treated at the DFT level, selecting the BLYP functional with empirical dispersion corrections and the QZV2P basis set with Goedecker-Teter-Hutter pseudopotentials. This basis set is a combination of Gaussian functions and plane waves that speeds up calculations. QM/MM molecular dynamics simulations with the umbrella sampling approach require an explicit choice of the reaction coordinate, selecting in this case the half sum of two dihedrals (Fig. 13.8). Two main questions were addressed in the study which is the influence of the A143G point mutation on the dark state recovery rate and which protonated state of the phenyl part of the chromophore promotes isomerization. A set of QM/MM MD trajectories with the harmonic potential centered at different reaction coordinate values were calculated to reconstruct the ground-state isomerization profile (Fig. 13.8). These calculations demonstrate that the thermal cis-trans isomerization of the anionic chromophore occurs slower in the KFP, an A143G variant of the asFP595, than in the asFP595 that is consistent with the experimental observations of the 7 times speedup of this reaction. Also, a 20 kcal/mol increase of the energy barrier in the case of the neutral chromophore isomerization in the asFP595 makes this process impossible. Another computational study of the thermal isomerization mechanism is related to the Dronpa and its mutants (102) and utilizes a QM/MM string method (103) followed by molecular dynamic simulations. The authors discriminate between different reaction mechanisms that may take place during isomerization. One of them is a one-bond flip that corresponds to the one-bond isomerization around the imidazole double bond. The second is the hula-twist mechanism, which is a concerted rotation of the adjacent central bonds of the chromophore. For gas-phase systems, usually the one-bond flip mechanism is preferable. However, due to the steric hindrances of the protein environment, the hula-twist mechanism could also take place when considering the chromophore inside the protein. The energy barriers in proteins were found to be much smaller for the hula-twist mechanism that supports this type of mechanism. Functional mode analysis demonstrates the importance of the residue at the 157 position that explains the slowdown of isomerization upon G157 amino acid substitution to bulkier valine and leucine residues. As aforementioned, the utilization of QM/MM MD umbrella sampling techniques shed light on the energy profile of thermal isomerization; however, these calculations are time consuming. Another approach that can qualitatively discriminate fluorescent proteins regarding their fast or slow isomerization in the ground state is the search for the neighboring amino acid residues that affect this process. It was done for the protein mSAASoti and its mutants (95). Classical MD runs of 100–200 ns were performed to study different conformations of amino acid residues close to the chromophore. It was shown that the conformational flexibility of the neighboring F177 residue might be the marker of the ground-state isomerization rate (Fig. 13.9). For slowly isomerizing proteins (such as the mSAASoti), the C-Ca-Cb-Cg dihedral of the F177 is mostly distributed around the 60°, whereas for proteins with fast isomerization (such as the C175A-mutated mSAASoti), this distribution is centered at around 160–170°. This cannot be explained by simple steric hindrance due to the changes in the size of the 175th amino acid residue. More likely, the general behavior of the protein-folding changes leads to a different conformational distribution of F177. Its more distant location facilitates isomerization, whereas if it comes closer, isomerization slows down. Thus, steric hindrance seems to be the main reason.

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FIG. 13.9 Conformations of the F177 side chain in fast isomerizing mSAASoti (cyan) and slow isomerizing mSAASoti C175A (magenta). (A) The C-Ca-Cb-Cg dihedral of the F177 distributions. (B) Alignment of different F177 side-chain conformations.

We also show an example of a computational study of the chromophore cis-trans photoisomerization on the excited state energy surface (39). QM/MM calculations at the CASPT2 level for the QM subsystem were utilized to study the S1 potential energy landscape along the isomerization coordinate. Minima, transition states, and conical intersections are located. It is shown that for Dronpa, rsFastLime, and rsKame, the stability of the conical intersection correlates with the photoswitching speed: slower switching is observed for the species with higher energies of the lowest lying conical intersection relative to the minimum on the S1. For the fastest species, rsFastLime, this energy is 3.7 kcal/mol, and for the slowest rsKame, it increases to 7.9 kcal/mol. However, the energy landscape for two other similar proteins, bsDronpa and Padron0.9, significantly differs from that of Dronpa, rsFastLime, and rsKame, making the previous conclusions less obvious. The isomerization process occurs with large structural reorganization in the protein-containing pocket; therefore, potential energy scans might be not enough for proper conclusions, and molecular dynamic-based approaches would shed light on the process, similarly to the already discussed studies focused on the ground-state isomerization. This was done for the reversible photoswitchable protein Dronpa at the QM(SA(2)-CASSCF(6,6)/3-21G)/MM(AMBER) molecular dynamic level (104). A set of 12 molecular dynamic trajectories were calculated and the photoisomerization mechanism was explicitly demonstrated. In particular, it was observed and quantified measuring the torsions involved in the isomerization in the QM/MM surface hoping that in the first singlet excited state, the hula-twist mechanism is preferable compared with the single-bond flip.

13.5

Concluding remarks

In this chapter, we covered the main photophysical and photochemical events that may happen in fluorescent proteins, namely with their chromophore groups. They absorb and emit light in different parts of the spectrum, depending on the chromophore structure. Being different, all of them have a conjugated p-system responsible for the photophysical properties. Therefore, all fluorescent proteins are similar from the computational side, being a protein macromolecule with the chromophore group buried inside. Now, there are already well-established QM/MM protocols that allow obtaining of model systems composed of the entire protein and the chromophore solvated by water molecules that are consistent with the available experimental structural X-ray and NMR data. Calculations of electronic excitation energy differences can be obtained at the quantitative to qualitative levels depending on the computational protocol and aim. For fast estimates, TDDFT approaches can be utilized. Depending on the selected functional, the differences between the computed and experimental values as well as values calculated with different functionals may vary by 0.5 eV or even more. Still, these calculations can be used to get qualitative information and knowledge on the electron density redistribution upon excitation. Multiconfigurational perturbation theory approaches are much more time consuming and usually require manual inspection of molecular orbitals constituting the active space. Ground-state reactions are usually studied within the QM(DFT)/MM method. If the transformations occur locally, it is enough to perform potential energy scans along the reaction coordinate. To allow reorganization of the protein surrounding, playing a key role in the photochemical properties of the chromophore, QM/MM molecular dynamic simulations are required. This is definitely true, for example, for the chromophore isomerization that involves conformational changes of the neighboring residues. The most difficult task is the study of photochemical reactions in excited states. This type

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of process requires much more computational effort as quantum chemical methods appropriate for the excited states should be utilized and, at the same time, considering the protein environment either within the QM/MM approach or as large QM clusters. Another direction of the computational studies is the search for correlations between the calculated microscopic parameters and observed macroscopic features. One can find a microscopic property that is mainly responsible for the macroscopic property of interest that can be calculated easier or with higher accuracy instead of direct calculation of the required quantity. For example, one can obtain the relation between the dipole moment variation upon excitation and the experimental excitation energy with high accuracy even if the calculations are performed at the TDDFT level. This is simpler than the direct calculations of the excitation energies using more accurate multiconfigurational perturbation theory. Analysis of the dynamic behavior of the protein through molecular dynamic simulations can shed light on the photoswitching rate, as the conformational flexibility of the protein affects both photochemical and thermal isomerization rates. The progress in computational facilities and the development of programming codes for both classical and quantum calculations extend the field of possible applications in the fluorescent protein field. Now, we are on the way from the explanation of the experimental observation to the rational design of novel systems with desired properties.

Acknowledgment M.G.K. acknowledges financial support from the Russian Science Foundation, grant number 22-13-00012.

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82. Ansbacher, T.; Srivastava, H. K.; Stein, T.; Baer, R.; Merkx, M.; Shurki, A. Calculation of Transition Dipole Moment in Fluorescent Proteins— Towards Efficient Energy Transfer. Phys. Chem. Chem. Phys. 2012, 14 (12), 4109. 83. Khrenova, M.; Topol, I.; Collins, J.; Nemukhin, A. Estimating Orientation Factors in the FRET Theory of Fluorescent Proteins: The TagRFP-KFP Pair and Beyond. Biophys. J. 2015, 108 (1), 126–132. 84. Mysˇkova´, J.; Rybakova, O.; Brynda, J.; Khoroshyy, P.; Bondar, A.; Lazar, J. Directionality of Light Absorption and Emission in Representative Fluorescent Proteins. Proc. Natl. Acad. Sci. 2020, 117 (51), 32395–32401. 85. Lukyanov, K. A.; Fradkov, A. F.; Gurskaya, N. G.; Matz, M. V.; Labas, Y. A.; Savitsky, A. P.; Markelov, M. L.; Zaraisky, A. G.; Zhao, X.; Fang, Y.; Tan, W.; Lukyanov, S. A. Natural Animal Coloration Can Be Determined by a Nonfluorescent Green Fluorescent Protein Homolog. J. Biol. Chem. 2000, 275 (34), 25879–25882. 86. Hoi, H.; Shaner, N. 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Chapter 14

Chemi- and bioluminescence: A practical tutorial on computational chemiluminescence Daniel Roca-Sanjua´n Instituto de Ciencia Molecular, Universitat de Vale`ncia, Vale`ncia, Spain

Chapter outline 14.1 Introduction 14.2 Design of the methodology 14.3 Identification of the molecule responsible for chemiexcitation 14.3.1 Walsh correlation diagrams 14.3.2 Reaction paths for the chemiexcitation of small models 14.3.3 “Activator”-“chemiluminophore” configuration 14.4 Reaction paths of the isolated system 14.4.1 Formation of the chemiluminophore

14.1

351 353 355 355 357 357 358 358

14.4.2 Chemiexcitation 14.4.3 Light emission 14.4.4 Identification of relevant parameters in challenging systems 14.5 Solvent effects 14.6 Dynamical aspects 14.7 A perspective on future research directions Acknowledgments References

359 359 360 360 362 363 363 363

Introduction

Chemiluminescence (CL) stands for the abbreviation of chemical luminescence and refers to the phenomenon in which a chemical reaction ends with the emission of radiation, normally UV/Vis light (1,2). In this sense, it can be considered the “opposite” of photochemistry, which is probably a concept more familiar to the reader. In photochemistry, light gives rise to a chemical transformation. Both processes of light-chemistry conversions, photochemistry and CL, involve excited electronic states and therefore belong to the field of the chemistry of the excited electronic states. CL is nowadays integrated into many technologies useful for scientific research and the welfare of society. It is used in DNA sequencing (3), immunoassays developed for biochemical and clinical analyte quantitative determination (4–7), enzyme-linked immunosorbent assay (ELISA) (8) or Western blot (9) detecting organic chemiluminescent products of enzymes, detection of traces of impurities (10) or target compounds in combustion processes (11) in biosensors, and glow sticks used for entertainment, recreation, or emergency services, among other uses. Motivated by both curiosity and also the applications of CL, many scientists have dedicated intense research efforts toward the understanding of the underlying mechanisms and efficiency improvement of the conversion of chemical energy into light (12). This task has been, however, complex mainly due to the involvement of excited electronic states and highenergy intermediates (HEI) in the phenomenon. HEIs are highly unstable (with short lifetimes) and reactive, which make their detection difficult by conventional analytic chemistry techniques. Electron-spin resonance (ESR) techniques combined with freezing procedures can immobilize the HEI and achieve their detection, although its complexity is high and it is not always successful (13). The development of sophisticated spectroscopy time-resolved techniques in the fs time-scale during the last decade has also allowed capturing short-life absorption and emission signals that can be associated with the said excited states (14). However, signal interpretation and its association with the excited states and intermediates involved in the mechanism are not trivial. In this context, computational chemistry arises as a powerful tool to monitor at the molecular scale the chemical transformations occurring in CL and allowing identification of the main actors. Furthermore, the rapidness of chemical changes occurring in CL is an advantage rather than a problem in computational Theoretical and Computational Photochemistry. https://doi.org/10.1016/B978-0-323-91738-4.00007-5 Copyright © 2023 Elsevier Inc. All rights reserved.

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FIG. 14.1 Scheme of a photochemical transformation via a nonadiabatic process (left) and the chemiluminescence phenomenon (right).

chemistry. Nevertheless, as shall be seen here, some cautions must be taken when choosing the methodological approach to address the study of CL. Fig. 14.1 illustrates the main features that characterize CL and (nonadiabatic) photochemistry from a computational viewpoint via a scheme based on the potential energy curves (PEC) of the ground and excited electronic states (1,15). Note that the plotting shows only a single nuclear position coordinate as representative of the process; nevertheless, the potential energy is a function of all the internal degrees of freedom and therefore the real picture would be a potential energy hypersurface (PEH). As can be seen, in photochemistry, a photon of light brings the molecule “vertically” (in the sense of the Franck-Condon principle) to the excited state, normally of singlet spin multiplicity as the ground state. Once there, the molecule relaxes vibrationally toward the regions of lower potential energy following the laws of physics. At some point, the molecule reaches a crossing between the excited and ground states where it can hop to the lower energy state in an ultrafast manner. Such crossing between states of the same spin multiplicity is the so-called conical intersection (CI). Evolution along the ground-state PEH increasing the values of the photochemical coordinate brings the system to the equilibrium structure of a new compound (photoproduct). Overall, in the described process, light energy induces a chemical transformation via a nonadiabatic (CI) process. On the other hand, in CL, the molecule in its singlet ground state surmounts the barrier related to the transition state (TS) of a thermal chemical transformation. In the neighborhood of the TS, there is a crossing with the excited state (CI if it is a singlet) and a hop to that excited state occurs. Once there, the system evolves toward a minimum in that state, from where a photon will be emitted via a radiative decay to the ground state. Notice that the description presented is a simple picture and variants of this scheme are common. For example, in photochemistry, excitation can occur not only at the lowest-lying singlet excited state (S1) but also at higher-energy excited states (Sn). Nonradiative decay is then happening in an ultrafast manner via several CIs between the excited states until reaching S1, according to Kasha’s rules (16). On the other hand, along the relaxation of the initially populated excited singlet states, crossings with triplet states, the so-called singlet-triplet crossing (STC), might occur. They allow population transfer to the triplet states if the coupling between the orbital and spin angular momenta (spin-orbit coupling, SOC) is nonnegligible. In CL, it is also common to find both CIs and STCs nearby the TS region, allowing both singlet and triplet chemiexcitation. It is worth anticipating at this point that this kind of hypersurface singularities (CI/STC) has strong electron correlation, which is due to the fact that at least two electronic configurations are relevant for their description. For example, Fig. 14.1 shows the biradicaloid and zwitterionic configurations that coexist in the photoisomerization of ethylene and the four biradicaloid configurations strongly interacting at the chemiexcitation region of the 1,2-dioxetane decomposition reaction. It is also worth mentioning that the topology of the PEH is a characteristic of the system. For example, in the mechanism of CL, we can find situations with a single CI/STC crossing or with two CI/STC crossings separated by a diradical minimum on the PEH.

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FIG. 14.2 Bottom-up strategy to address CL studies with computational chemistry (see text for details). CASPT2, complete-active space second-order perturbation theory; DFT, density functional theory; QM, quantum mechanics; MM, molecular mechanics; ISM, implicit solvent model; MD, molecular dynamics; AINAMD, ab initio nonadiabatic molecular dynamics.

In this chapter, the reader will find a practical tutorial on the step-by-step approach to address research on CL by using the tools of computational chemistry.a Firstly, relevant methodological aspects shall be briefly described, including comments on the choice of appropriate quantum chemistry methods and how to determine solvent effects and dynamical aspects. Secondly, hints will be given to facilitate the identification of chemical groups and key factors that favor efficient chemical excitation. Thirdly, the methodological strategy to obtain the CL mechanism for the isolated molecule will be detailed. Next, the chapter will show steps to analyze solvent effects and what should be done to get quantitative properties, allowing predictions of the efficiency and direct comparisons with experiments. Finally, the author’s perspective on future research directions can be found at the end.

14.2

Design of the methodology

When trying to define a convenient methodological strategy, it is important to follow a bottom-up approach, focusing first on the determination of the most important (or “first-order”) features of the CL mechanism and subsequently adding additional “second-order” contributions. This approach ensures an accurate modeling, reaching conditions and values directly comparable with experimental data or reproducing real conditions (Fig. 14.2). At the bottom, we can find high-level computations of the chemistry-to-light transformation in small molecular models (representative of the CL system). At this step, benchmarking of faster lower-level methods, useful for the next step involving the real CL molecule, can be done. Analyses of solvent effects (or interactions with the real environment) can be introduced at the next level. Finally, the determination of time-dependent or dynamical properties (rates, yields) can be evaluated in the last step. This is of course a general and ideal approach. The degree in which the user should follow it will depend on the problem at hand, the objective pursued, or how challenging the CL system is. Details on the methodological aspects to be considered in each of the three steps proposed are given in the following lines and are also illustrated in Fig. 14.2: – First step: Electronic-structure properties of the intermediate species. Diradical or biradicaloid configurations occurring at the crossing region in the chemiexcitation process (Fig. 14.1) are quite challenging since they are described by several electronic configurations of two unpaired electrons with singlet and triplet spin multiplicities (12). Such a. Complementary recommended readings are Vacher et al. (12), where the reader will find a complete revision of the knowledge on the field of CL with both theoretical and experimental insights, and Roca-Sanjua´n et al. (15) with an approach more focused on the PEC diagrams that characterize distinct CL mechanisms.

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structures correspond to CIs and STCs, which distribute the population among the distinct degenerated states. To properly characterize this type of structure, the safest approach is to use high-level post-Hartree-Fock wave function methods, in particular, multi-configurational quantum chemistry, which allows describing, with enough flexibility, multiple electronic configurations. Two of the most popular and practical methods belonging to this family are complete-active-space self-consistent field (CASSCF) and complete-active-space second-order perturbation theory (CASPT2) (17–20). The MOLCAS software (21) accounts for an efficient implementation of these methods, although other software packages could also be used such as MOLPRO (22), BAGEL (23), or TERACHEM (24), among others. MOLPRO has an effective optimizer of state crossings (CI/STC) and BAGEL and TERACHEM contain highly efficient algorithms for parallel processing. In particular, CASPT2 can get high accuracy in systems of relatively small size (for instance, luminol or firefly luciferin for determining static properties or smaller size systems for obtaining dynamical properties) (20). It shall be our reference for the electronic-structure characterization of the HEI and excited states. Recent improvements in the CASPT2 method (extended dynamically weighted) (25) can be used, if needed for challenging characterizations of CIs or STCs. The density matrix renormalization group (DMRG) technique (26) allows also extending the active orbital space although their application still must be better optimized for practical applications. Those approaches are, however, not efficient (or sometimes directly prohibitive) for big systems (macromolecular environments, such as solvation, protein, or biomembranes) or when attempting to obtain dynamical properties. In these situations, the user might be interested in the computationally less costly lower-level density functional theory (DFT) method that is implemented in a vast variety of codes, including Gaussian (27), Amsterdam Density Functional (ADF) (28), Q-Chem (29), TURBOMOLE (30), GAMESS (31), or ORCA (32,33), among many others. They allow the characterization of adiabatic chemical transformations of CL (involving a single PEH) but not for the nonadiabatic chemiexcitation process (with two or more interacting PEH). Time-dependent DFT (TDDFT) can allow finding excited-state equilibrium structures if double or higher-order excitations are not involved. Describing the interaction between np* and pp* excited states can be also difficult by using this method. In any case, the choice of the low-level method must be done carefully. Benchmarking with high-level methods is always needed. Open-shell DFT must be used for treating singlet ground-state diradical or biradicaloid configurations. Otherwise, an upper solution is obtained, which does not represent the minimum (and thus most probable) energy path. The open-shell solution can be reached if we force spin symmetry breaking by mixing singlet and triplet densities and/or applying time-dependent techniques to reach the correct diradical wave function. Further details are given in previous works (34,35). – Second step: Considering environmental effects. Determination of electronic-structure properties and mechanisms of the isolated CL molecule is important because they can be carried out with high-level multi-configurational quantum chemistry. Moreover, they ensure a correct description and selection of much faster methods that can reproduce the main properties. However, to reach a more realistic description, interactions with the nearby surroundings must be considered. Normally, we are interested in describing the process taking place in the solution conditions. Two common approaches are useful here: (1) Hybrid quantum mechanics/implicit solvent model (QM/ISM) strategy. The CL molecule, and (optionally but highly recommended) the strongly interacting solvent molecules from the first solvation shell, are included in the quantum chemical description. Then, implicit modeling (commonly, polarizable continuum model, PCM) (36) is used for the remaining solvent. Notice that not including any solvent molecule at the QM level avoids considering solvent participation in the chemical transformations. (2) Hybrid QM/MM methods (see examples in Refs. (12, 37)). The molecular structure directly involved in the CL mechanism is treated by QM methods while the rest of the system is treated by appropriate force fields at the MM level (solvent molecules and structural moieties of the CL molecule that do not affect qualitatively the mechanism). – Third step: Evaluation of time-dependent (sub- and supra-nanosecond) properties. Dynamic simulations are relevant to obtain lifetimes, rates, and yields of the distinct elementary steps occurring in the CL process. Two types of time scales must be distinguished, sub- and supra-nanosecond. Considering the current knowledge on CL, the chemiexcitation process takes place via CI and STC, which occur in the fs/ps lifetimes. STCs, although spin-forbidden, take place due to the non-negligible spin-orbit coupling (SOC), found in the CL systems, especially the ones based on the peroxide bond breaking. This is due to the interaction of singlet and triplet configurations with unpaired electrons located in perpendicular oxygen lone pair (n) and p* anti-bonding orbitals, allowing spin inversion (El-Sayed rules) (38). In this context, ab initio nonadiabatic molecular dynamics (AINAMD) is a convenient computational technique with practical implementation in the coupled programs SHARC (39) and MOLCAS (for CASSCF/CASPT2). SHARC interface with ORCA (32) can be useful for DFT and recent interfaces with COBRAMM (40) are promising for

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AINAMD with QM/MM in macromolecular environments. For CL elementary processes that take place at much longer times, such as reaction processes associated with TS (like the OdO bond breaking) or even bimolecular reactive associations (like the formation of the CL molecule by the interaction between the precursor molecule and molecular oxygen), biased dynamics based on free energy perturbation, such as “umbrella sampling” (41) or string method (42), or enhanced sampling approaches derived from metadynamics (43) are available. Notice here that thermal fluctuations of the environment are important for an accurate determination of the rates associated with the distinct slow steps of CL and must be considered at the QM/MM level. The coupled programs AMBER (or fDynamo) (44) and Gaussian16 (45) have an optimal implementation to carry out umbrella sampling. GROMACS (46), with PLUMED (47), is an optimal combination for enhanced sampling metadynamics.

14.3

Identification of the molecule responsible for chemiexcitation

Chemical excitation is a relatively rare property in organic and inorganic chemistry and therefore is not expected to happen in every molecule. An initial important part of the research on CL is to identify the molecular structure and geometrical coordinate that allows hopping from the ground to the excited state (this structure is called chemiluminophore, CLP) and the chemical functionalities that favor the process (so-called activator, ACT). For instance, Fig. 14.3 shows three structures that are involved in the CL of the firefly bio-organism (Fig. 14.3A–C) and two structures involved in the CL of luminol (Fig. 14.3D and E). Which one is responsible for chemiexcitation? To get a response, a helpful strategy is to appeal first to the Walsh correlation diagrams of orbitals and electronic states assisted by symmetry analyses according to the Woodward-Hoffmann rules (48,49) and next search for special chemical groups (ACT) that are well known to enhance the process (50). In the following sections, we shall give hints on how to use the Walsh correlation diagrams and identify key chemical functionalities.

14.3.1 Walsh correlation diagrams Walsh correlation diagrams of molecular orbitals and states correspond to a qualitative energy representation of orbitals and states versus a distortion coordinate, used to make predictions about the energetic evolution of each orbital and state along the geometrical change. The orbitals of interest are the chemically relevant ones, that is, the frontier orbitals (highestoccupied molecular orbital (HOMO), lowest-occupied molecular orbital (LUMO), and nearby orbitals from Hartree-Fock calculations) whose excitations can give rise to nearby electronic states. Symmetry classification of orbitals and states to a symmetry group which is preserved upon applying the geometrical distortion facilitates subsequent prediction of the evolution of the system on the ground or excited state with the use of Woodward-Hoffmann rules. Vacher et al. (12) give a detailed description of the use of Walsh diagrams and Woodward-Hoffman rules for the [2+2] cycloelimination of 1,2-dioxetane, which shall help the reader to understand why in Fig. 14.3 the firefly structures bearing such a ring (Fig. 14.3B and C) are potential CLPs. Here, we shall briefly illustrate their application for the ring opening of a small model of luminol, 1,2-dioxin (see Fig. 14.3E).

FIG. 14.3 Chemical structures of the CL mechanism of firefly (A–C) and luminol (D and E). The 1,2-dioxetane ring is highlighted in (B) and (C) and the 1,2-dioxin ring in (E).

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FIG. 14.4 Walsh molecular orbital (A) and electronic state (B) correlation diagrams for the ring opening or 1,2-dioxin and related isoelectronic systems. The notation (A) or (S) for the orbitals stands for asymmetric or symmetric, respectively, and denotes the symmetry property of each orbital with respect to the C2 axis. Dash and solid lines (right) denote diabatic and adiabatic representations, respectively. State symmetry labels are obtained by multiplying the symmetry of the orbitals with unpaired electrons according to the following rules: SS¼AA¼S, SA¼AS¼A; doubly occupied orbitals contribute with S. Adapted from Giussani, A.; Farahani, P.; Martı´nez-Mun˜oz, D.; Lundberg, M.; Lindh, R.; Roca-Sanjua´n, D. Molecular basis of the chemiluminescence mechanism of luminol. Chem. Eur. J. 2019, 25 (20), with permission by the John Wiley & Sons, Inc. Copyright © 2019.

Fig. 14.4A shows the Walsh diagrams for 1,2-dioxin and related systems using the cleavage of the dXdXd bond, where X stands for O, S, NH, and N , as distortion coordinate. Note that the process preserves a C2 symmetry axis; therefore, it is convenient to classify the orbitals and states as symmetric (S) or asymmetric (A) with respect to their symmetry operations of the C2 group. In the dOdOd bond-breaking process leading from the cyclic peroxide (reactant) to the dialdehyde structure (product), the bonding sS and anti-bonding s*A orbitals of the reactant located on that bond correlate with the two oxygen lone pairs (nS and nA, respectively) of the product, while the pS orbital describing the two dC]Cd double bonds in the reactant corresponds to a p*S orbital in the product (Fig. 14.4A). These correlations consequently describe a change of the bonding/anti-bonding nature, which translates into the presence of crossings or avoided crossings between the corresponding electronic states along the reaction coordinate. As can be seen in the Walsh correlation diagrams of states (Fig. 14.4B), in the reactant, the ground state describes a closed-shell structure in which the sS and pS orbitals are double occupied, (sS)2(pS)2 state; the lowest-lying (first) excited state corresponds to a single electronic promotion from pS to s*A, (sS)2(pSs*A) state; and at much higher energies, an excited state appears which is the result of a double excitation from pS to s*A, (sS)2(s*A)2 state. In the product, the ground state is a closed-shell structure in which the nS and nA are doubly occupied, (nS)2(nA)2 state; the first excited state corresponds to a single electronic promotion from nA to p*S, (nS)2(nAp*S) state; and a high-energy excited state can be found resulting from a double nA to p*S excitation, (nS)2(p*S)2 state. Along the transformation, the ground state of the reactant correlates with the doubly excited state of the product and the doubly excited state of the reactant correlates with the ground state of the product. Since the states have the same symmetry (S), an avoided crossing between them can be estimated at the region of the TS separating the two ground state minima. Meanwhile, the singly excited state of the reactant correlate with that of the product. This state has A symmetry and therefore can cross with both the ground and doubly excited states. The presence of crossings, which connect the original ground state with the excited manifold of the product, indicates that this system and coordinate distortion are capable of originating chemiexcitation upon ring opening. Note in the analysis done that the peroxide bond is not the only functionality in 1,2-dioxin that provides this molecule with chemiexcitation properties. The adjacent dC]Cd double bonds of the 6-membered ring are as important as the dOdOd bond. For instance, the reader can verify that removing those double bonds or placing only one double bond in the C(3) and C(4) atoms of the 1,2-dioxin ring will change the scenario. The original p-conjugated configuration of luminol (Fig. 14.3D) is therefore inappropriate for chemiexcitation. However, this still does not fully explain why the dNdNd bond present in Fig. 14.3D is exchanged by the dOdOd bond to give the structure in Fig. 14.3E. Is not possible any other re-organization of the molecular structure without the need of releasing N2? For instance, the systems displayed in Fig. 14.4 analogous to 1,2-dioxin with dNHdNHd or dN dN d instead of dOdOd are isoelectronic and the Walsh diagrams give rise to a similar picture as for 1,2-dioxin; they could be in principle potential CLPs. To find now the answer, we need to go one step further and improve the description of the energy profiles by computing the PEH reaction paths related to the ring opening, as shown in the next sub-section.

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TABLE 14.1 Data (in kcal/mol) computed by Giussani et al. (51) related to the activation energy barriers (DE#), energy STC gaps between S1 and S0 (ECI gap) and T1 and S0 (Egap ) at the TS, and energy differences between products and reactants (DE) for the ring-opening reaction of 1,2-dioxin (dOdOd) and analogous compounds with dNHdNHd and dN2dN2d bonds. DE#

ECI gap

ESTC gap

dNHdNHd

36.7

23.3

3.3

12.2

dN dN d

27.0

9.1

3.4

30.7

dOdOd

11.1

6.4

2.9

59.7

DE

14.3.2 Reaction paths for the chemiexcitation of small models Giussani et al. (51) characterized the chemiexcitation reaction paths for 1,2-dioxin and the systems with dNHdNHd and dN dN d using the CASPT2 method. Table 14.1 compiles the activation energy for the bond breaking (DE#), the STC S0/S1 (ECI gap) and S0/T1 (Egap ) energy gaps at the TS and the energy difference between products and reactants (DE). By CI STC comparing Egap and Egap of the studied models, dN dN d has an almost S0-S1-T1 degeneracy at the TS, while only S0 and T1 are significantly close in dNHdNHd. Regarding the activation energy (DE#), the value for dN dN d and dNHdNHd is more than twice and three times higher, respectively, than that for dOdOd. For DE, dN dN d and dNHdNHd release half and one-fifth, respectively, the energy released by dOdOd bond cleavage. All these data lead to the conclusion that 1,2-dioxin is the most favorable (efficient) situation for chemiexcitation kinetically (low activation barrier) and thermodynamically (large product stabilization) and in terms of the energy gap between the ground and excited states (low gap).

14.3.3 “Activator”-“chemiluminophore” configuration By the inspection of the relevant orbitals participating in the chemiexcitation process, it can be anticipated, without doing any calculation, that a total or partial electron addition into the s*A orbital of the dOdOd bond will weaken the bond, favoring the cleavage and decreasing the energy barrier for chemiexcitation. Indeed, this has been observed both theoretically and experimentally for efficient CL systems. The corresponding mechanism is the so-called chemically initiated electron exchange luminescence (CIEEL) or charge transfer-induced luminescence (CTIL), proposed by Schuster (52) and Takano et al. (53), respectively, which can be either intramolecular (Fig. 14.5A), when the donation arises from a fragment of the molecule, or intermolecular (Fig. 14.5B), when it happens from another molecule. The donor moiety (acting as the ACT) is characterized by a low ionization potential, and the charge donation requires p-conjugation, in the

FIG. 14.5 Reaction mechanisms for (A) the intramolecular and (B) intermolecular chemically initiated electron exchange luminescence (CIEEL) and charge transfer-induced luminescence (CTIL).

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intramolecular process, and p-stacking, in the intermolecular case (54). It is also common for instance in bioluminescence the presence of a chemical group that can interact with the environment and switch on or off the donation from the electron reservoir (50). Thus, in firefly dioxetanone, deprotonation of the phenol moiety lowers the ionization potential of the HOMO delocalized mainly over the benzothiazole fragment (see Fig. 14.3), which promotes the charge transfer and decreases the energy barrier for the decomposition of the dioxetanone ring (see the “Bioluminescence” contents of this book). In the dioxin-based derivative of luminol, the amino group facilitates the charge donation from the p-cloud and the whole p-conjugated system constitutes the ACT. Therefore, we can finally respond to the question stated in the introduction of Section 14.3 that the structure of luminol of Fig. 14.3 responsible for chemiexcitation must be the one at the bottom (E).

14.4

Reaction paths of the isolated system

Once we have identified the relevant features that are optimal for producing chemiexcitation, we can begin to determine: firstly, the reaction path that transforms the initial compound into the CLP; secondly, the mechanism that brings the molecule from the ground to the excited state in the real molecule; and thirdly, the route that gives rise to the species that emits light.

14.4.1 Formation of the chemiluminophore As mentioned in Section 14.3, the CLP might not be present in the starting molecule or, if present, it might be in an inactive state. In such a scenario, the starting structure is a precursor with much higher stability than the CPL, which is convenient for CL experiments/applications or biological functions. Then, the first step in the determination of the mechanism is to characterize the reactivity that brings the precursor to the molecular structure with optimal properties for chemiexcitation (CPL). In luminol, the starting point is 5-amino-2,3-dihydrophthalazine-1,4-dione (LH2) (Fig. 14.3D), which does not have chemiexcitation properties. The first step toward the generation of the chemiluminophore (that we know at this point that it should have the 1,2-dioxin group; see Section 14.3) is the double deprotonation of the cyclic nitrogen atoms. This deprotonation is achieved experimentally in a DMSO solution by adding a strong base. Next, computation of the energy profiles corresponding to the approach of an oxygen molecule (3O2) to the deprotonated species (L 2) indicates that they can easily react forming a bicyclic peroxide (EP 2, see Fig. 14.6). This happens as follows. Firstly, a reaction complex is formed in the triplet manifold (3[L…O2] 2min), which is lower in energy than the singlet one. Such a complex is characterized by partial negative charge donation from L 2 to 3O2. Nearby this region, an STC structure energetically accessible is present ([L …O2 ]STC). An electron is fully transferred at this point. Computation of the spin-orbit coupling (SOC) points to an allowed ISC process. From [L …O2 ]STC, a downhill reaction path on the singlet manifold is found toward EP 2. The energetics of the release of nitrogen show a favorable process toward the generation of CP 2, which contains 1,2-dioxin and can be considered therefore the CLP.

FIG. 14.6 Energy profiles for the triplet (T) and singlet (S) manifolds (dashed red (dashed gray in print versions) and solid blue (solid gray in print ˚ ) are shown. STC, versions) lines, respectively) for the L22 + 3O2 ! CP22 + 1N2 reaction process. The values for the OdO bond lengths (in A singlet-triplet crossing; TS, transition state. Adapted from Giussani, A.; Farahani, P.; Martı´nez-Mun˜oz, D.; Lundberg, M.; Lindh, R.; Roca-Sanjua´n, D. Molecular basis of the chemiluminescence mechanism of luminol. Chem. Eur. J. 2019, 25 (20), with permission by the John Wiley & Sons, Inc. Copyright © 2019.

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14.4.2 Chemiexcitation From the analyses carried out in a previous step on model systems (1,2-dioxin; Sections 14.3.1 and 14.3.2), we already have relevant information to analyze the chemiexcitation mechanism in the real molecule (CP22). In those sections, the chemically relevant orbitals were already identified and also reaction profiles for the coordinate-driving chemiexcitation were determined in 1,2-dioxin. At this point, the computational chemist must further determine the changes in the reaction paths that appear when moving from the small model to the real molecule. In luminol, when doing so, we observe two regions of crossing rather than one. Moreover, an intermediate space in which the ground and excited states are significantly close in energy is found, pointing to a large region of chemiexcitation (see Fig. 14.7). The reason for such a difference between 1,2-dioxin and CP22 is that the aniline group (acting as the ACT) stabilizes the ps* state in the latter by electron donation, which favors chemiexcitation by the CTIL/CIEEL mechanism.

14.4.3 Light emission Once the molecule is promoted to the excited electronic states via the chemiexcitation step, we must characterize the evolution there toward the emissive excited state and its minimum geometry on the PEH from where the formed species will be trapped in enough time until decaying by radiative emission. For luminol, reaction path computations from the late lowenergy CI (Fig. 14.7, 2ndCI) bring the molecule to the minimum of the dark np* state. The emission will not take place from here because the 2ndCI is only 0.05 eV higher in energy than the np* minimum and nonradiative decay toward the ground state by this crossing is much faster than radiative decay from the np* minimum. On the contrary, energy profiles computed from the high-energy CI (HECI) displayed in Fig. 14.7 follow a slightly different reaction coordinate which not only involves OO stretching but also bending of the CCC angles of the dioxin ring. The molecule evolves toward a proton transfer CI (PTCI) that mixes the np* state with a (pp*)PT state that has more efficient radiative decay. The proton transfer gives rise to the formation of the imino form of the 3-aminophthalate dianion on the excited (pp*)PT state (3AP22 im *). The evolution also implies a back electron transfer typical of the CTIL/CIEEL mechanism. Reaction path on the (pp*)PT state from PTCI reaches its minimum geometry, with an emission energy of 541 nm, in reasonable agreement with the experimental signals obtained for the CL of luminol in DMSO (485–510 nm) (55–58). The ground-state structure of the amino form of 3-aminophthalate dianion (3AP22 am; see structure in Fig. 14.7) is the final product of the overall CL process.

FIG. 14.7 Mechanism of chemiexcitation and light emission of luminol, as concluded on the basis of CASPT2 data and analyses by Giussani et al. (51). Chemiexcitation giving rise to nonradiative decay is shown by means of dark arrows, while that resulting in light emission is indicated by white arrows. The chemical structures of relevant points on the potential energy hypersurfaces are shown together with the charge of the highlighted fragments (see text). CI, conical intersection; HEIC, high-energy CI; PT, proton transfer; PTCI, proton transfer conical intersection.

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FIG. 14.8 Luminol derivatives with methyl or phenyl substitutions at ortho (o-) and/or para (p-) position with respect to the amino group.

14.4.4 Identification of relevant parameters in challenging systems Knowledge acquired by characterizing the electronic-structure transformations that occur in the elementary steps of the CL mechanism can be subsequently utilized to identify magnitudes that can be quickly computed and be helpful in estimating efficiency changes upon introducing distinct substitutions in the parent molecule (59,60). Such analyses can be done if the researcher is interested in screening several molecules toward the search for more efficient CL systems. As an illustrative example, we shall revise how this was addressed for luminol derivatives with methyl or phenyl substitutions at ortho (o-) and/or para (p-) position with respect to the amino group by Mikroulis et al. (60) (see Fig. 14.8). By the inspection of the analyses done in Section 14.4.3, the reader should agree that important parameters that a priori can affect the chemiexcitation yield are the following: DE‡

Energy barriers between CP22 and the TS of the dOdOd bond-breaking reaction. This property has been previously shown to distinguish between highly efficient and less-efficient CL systems, with catalyzed and noncatalyzed mechanisms, respectively

DEp!s∗

Energy gap between the excited and ground states at the TS, which is related to the degree of accessibility of the excited state

Dr

Change of the charge of the carbonyl groups from the at the CP22 to the TS point. This shows any charge-transfer contribution that could activate the dOdOd bond breaking (resembling the CTIL or CIEEL mechanisms)

DE3AP∗-CP

Energy difference between the minimum of the (pp⁎)PT state and the equilibrium structure in the ground state of CP22 (see Fig. 14.7). It represents the energy released in such a process. Note that in the work by Mikroulis et al. (60) the (pp⁎)PT state 22 considered corresponds to the 3AP22 am * form and not to the 3APim * form because in water solution there is no proton transfer 22 (see Section 14.5). However, the same general trends are obtained for both 3AP22 im * and 3APam * regarding the present parameter

The study by Mikroulis et al. (60) concluded that DE‡ data were too similar in the studied luminol derivatives to obtain significant correlations with the experimental CL yields. Semi-quantitative explorations of the CIs, highly challenging in these systems, neither helped. Meanwhile, higher deviations were obtained for Dr and DE3AP∗-CP, which allowed the following rationale. p-substitution donates a negative charge to the cyclic peroxide moiety, thus assisting the cleavage of the dOdOd bond. This was correlated with the experimental observation that p-methylation greatly enhances the chemiluminescence yield, whereas phenyl substitution has a less marked effect. Substitution at the ortho position contributes less, with o-phenyl yielding somewhat better results, probably due to the higher release of energy required to generate the emissive species from the peroxide intermediate.

14.5

Solvent effects

Computations and analyses of Section 14.4 give already a good understanding of the internal CL properties. Nevertheless, the molecule is normally surrounded by an environment (in solution with solvents that can be polar or apolar, protic or nonprotic; embedded in biological macromolecules like proteins in bioluminescence, or in other condensed phases). Then, the theoretician should analyze the external effects that these conditions impose in the intrinsic reactivity. We shall focus here on CL in the solution. For the effects of the macromolecular embedding in proteins, the reader is referred to the “Bioluminescence” parts of this book. In CL in solution, we can distinguish two situations, nonreactive interactions that only tune the energy profiles obtained for the isolated system and reactive interactions with active participation of the solvent in the mechanism. In the first case, the QM/IMS method with all solvent molecules treated as a field with a certain dielectric constant or the QM/MM method treating all solvent molecules at the MM level are reasonable strategies to determine the energetic modifications of the reaction profiles. Yue et al. (61) used this approach to study the CL of luminol in the water environment. They

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FIG. 14.9 Chemical structures for the amino (3AP22 am) (A) and imino (3AP22 im ) (B) emitter forms of luminol and that corresponding to a functionalized luminol with an encapsulating (CAP) group (3APCAP22 im ) (C). “target” is a functionalization to drive the molecule toward specific parts of the cell.

demonstrated, by using QM, that the precursor of the CLP in these conditions is not the dianionic species, L22, but the monoanionic species with the deprotonation of the cyclic N atom closer to the noncyclic amino group (hereafter LH2). They also supported the key role of CP22 for chemiexcitation. For the second type of situation (reactive solvent), the study of CL in solution demands to include the solvent molecules strongly interacting with the CL compound in the quantum-chemistry description of the QM/IMS or the QM/MM approach. We shall see this part in detail by means of an example of the active participation of the solvent in producing the excited-state species responsible for CL of luminol. As described in Section 14.4.3, the excited-state reaction path computations to reach the emissive structure of the isolated luminol molecule after chemiexcitation on the HECI show a proton transfer from the amino group to the nearby carboxylate group. Thus, light emission takes place from the imino species (3APim22⁎, Fig. 14.9) rather than from the amino form (3APam22⁎, Fig. 14.9) (51). As mentioned above, this computed emission energy of 541 nm agrees with the experimental signal measured in DMSO and other aprotic solvents (485–510 nm). In water, the experimental value is significantly blue shifted (410–431 nm) (55,56,58,62–64). The QM/IMS methodology without considering solvent molecules at the QM level does not predict the shift. To give an explanation for the origin of such a big difference, the computational chemist can proceed as follows: (1) To carry out molecular dynamic simulations with classical mechanics (with force fields) on the ground-state 3APam22 molecule in a box of DMSO solvent molecules and other molecular dynamics simulations in a box of water molecules. Upon equilibrating the system and allowing enough time to have a representative ensemble of the conformation space, the first shell solvent configuration must then be identified. (2) To compute the excited-state reaction path connecting 3APam22 and 3APim22, considering also the chosen solvent molecules in step 1 at the QM level (TDDFT or CASPT2) and using IMS for the rest of the solvent. (3) To compute the emission spectra for the most stable species in both solvents. (a) A first estimation can be obtained by computing the vertical emission energy (VEE) and oscillator strength (f) at the excited-state minimum geometry. This should be roughly related to the band maximum. (b) For further improvements, a nuclear ensemble approach (NEA) can be used in which a Wigner distribution of excited geometries that represent the quantum vibrations around the minimum is computed. VEE and f are here computed for each geometry and then values are convoluted to generate the band shape. See Borrego-Sa´nchez et al. (65) for further details on the use of this strategy and find the Multispec program for running such computations in the GitHub repositories https://github.com/qcexval/multispec under an LGPL-2.1 License. By following the 3-step approach, Borrego-Sa´nchez et al. (66) found that while the excited-state proton transfer producing 3APim22 is favorable in DMSO, 3APam22 is more stable in water than 3APim22. The reason was the establishment of strong hydrogen bonds between the O(2) atom of the carboxylate group and solvent water molecules, not allowing the H(3) transfer from atom N(1) to O(2) (see Fig. 14.9). The emission spectrum simulated in water conditions is displayed in Fig. 14.10, showing now a reasonable agreement with the experimental value for water. The authors also proposed possible modifications to tune the relative stability of 3APim22 and 3APam22. Thus, by adding an encapsulation group to luminol, the imino form (3APCAPim22⁎, see Fig. 14.9) was found to be stabilized in water maintaining the emission properties of luminol in DMSO (higher emission intensity and longer wavelength emission; Fig. 14.10). These two properties are interesting for the development of variants for photodynamic therapy (PDT) for the treatment of cancer using CL rather than a laser of light. In PDT, light is used to activate a molecule, the so-called photosensitizer that, when irradiated, induces damage in the cells. The higher emission intensity of 3APCAPim22⁎ as compared to 3APam22⁎ in water (physiological conditions) implies more efficiency. Meanwhile, longer wavelengths are related to more penetration of light in the tissues.

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FIG. 14.10 Chemiluminescence intensities (e) determined by Borrego-Sa´nchez et al. (66) for the luminol-emissive species appearing in distinct conditions, the amino form of luminol in water (3APam22*), the imino form in DMSO (3APim22*) and the imino form of luminol with the encapsulation group in water (3APCAPim22*). Computations were done with the CASPT2 method using an approach combining the conformational sampling of the solvent and a quantum nuclear ensemble approach (NEA) of the chromophore excited-state vibrations. Shadow areas represent the statistical error of the approach. Notice that even though the data was obtained in the reference article for the stimulated emission, it is a good approximation to the spontaneous emission spectrum. Reprinted from Borrego-Sa´nchez, A.; Giussani, A.; Rubio, M.; Roca-Sanjua´n, D. On the chemiluminescence emission of luminol: Protic and aprotic solvents and encapsulation to improve the properties in aqueous solution. Phys. Chem. Chem. Phys. 2020, 22 (47), 27617–27625, with permission by the Royal Society of Chemistry Copyright © 2020.

14.6

Dynamical aspects

As mentioned above, the last step at the top of the bottom-up methodological approach proposed in Section 14.2 for a complete computational study of CL corresponds to obtain a kinetic model of the whole process. For this aim, rates of the chemical transformations occurring in the elementary steps of the process and competitive paths have to be computed by appropriate dynamic simulations to analyze the efficiency. With the same spirit as in the previous sections, we shall illustrate this step with a few chosen examples, focusing first on parts of the CL mechanism that are slower and next on ultrafast steps. Note, however, that dynamic simulations are significantly more computationally demanding and still highly challenging. Consequently, they are less abundant in the literature for the field of CL, and therefore we shall only give a brief overview. The diffusion and encounter energetics between 3O2 and firefly luciferin were determined by Navizet and co-workers (67) by using biased MD simulations, in particular, umbrella sampling, in the luciferase protein. Even though this process refers to bioluminescence, it can be related to the first step of luminol CL in which 3O2 approaches L22 and therefore it is worth discussing it here in line with the previous sections. Briefly, in such umbrella sampling computations, the free-energy profile was obtained for the process of bringing together the 3O2 and the precursor of the chemiluminescent molecule. The results obtained by Navizet and co-workers (67) indicate that the approach is barrier-free until reaching a short distance corresponding to the reaction complex. At this point, classical MD cannot be used and QM must be applied to further describe the subsequent reactivity. Similar conclusions were found by Yue and Liu (68) in another example of exploration of dynamical aspects with umbrella sampling. In this case, it refers to the inter-molecularly catalyzed CL, describing the encounter between a dioxetane molecule, 1,2-dioxetanedione, and an activator, rubrene. Regarding the characterization of the faster dynamical processes occurring in CL, that is the chemiexcitation step, these are important to describe the dynamics occurring during the chemiexcitation process and the subsequent evolution of the system on the excited-state manifolds toward the formation of the light-emitter species. Chemiexcitation involves several electronic states of singlet and triplet spin multiplicities with almost equal energies and therefore, with high hopping probability between the states. One relevant step forward in this type of modeling was achieved by Lindh and co-workers (69). They studied the dynamics of the 1,2-dioxetane molecule and its alkylated derivatives, using an approximate strategy in which only singlet states were modeled and the state hopping probabilities were treated with high accuracy by means of computing the nonadiabatic couplings (NACs). For luminol (our chosen molecule in this chapter), Liu and co-workers (70)

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have recently succeeded in obtaining the first dynamical information by using AINAMD with the CASSCF method and a faster, although less accurate, modeling of the hopping events than that provided by NACs, in particular the Zhu-Nakamura approach (71). Triplets were also included in the simulations. The CASSCF method is not able to model the excited-state proton transfer to generate 3AP22⁎ am . This process is highly affected by dynamical correlation and CASPT2 would be needed (51). Despite this, the authors found interesting results, in particular, an important chemiexcitation yield and triplet population.

14.7

A perspective on future research directions

Computational CL is nowadays a well-established tool for the research on this phenomenon of luminescence arising from a chemical transformation. Determinations of PEHs and the key features on them (minima, TS, CIs, STCs, reaction paths, f, SOC, r) are common in the field, although implying a much higher complexity as compared to adiabatic thermochemistry modeling. This is due to the participation of open-shell electronic configurations (at the TS and excited states), strong static electron correlation (at the CIs, STCs), and high effects of the dynamic correlation (at CIs, np*/pp* excited state interaction). The need for high-level methods (multi-configurational quantum chemistry) for at least some parts of the mechanism becomes a challenge especially for large molecules or for dynamic simulations with methods including both static and dynamic electron correlation, which require computational resources beyond the current practical limits. More research is still required to further increase the ratio “accuracy”/“cost” and allow creating quantitative kinetic models of the chemical transformations present in the CL phenomenon to get accurate quantum yields. This will be relevant for the design of more efficient CL agents in the field of biomedicine that are currently gaining high interest, such as PDT with CL (rather than a laser of UV light) for the treatment of tumors in internal tissues, especially when not easily accessible by surgery like in the brain (60,66,72,73). Chemiexcitation has been also demonstrated by in vivo experiments to be responsible for the generation of dimeric DNA lesions and the appearance of melanoma in mice in dark conditions (74–76) and might be at the chemical origin of some chronic diseases such as Parkinson’s or Alzheimer’s (77). The chemical mechanism occurring in the environment of nucleic acids surely implies several competitive chemical transformations between reactive oxygen species and biological constituents of the cell which demands complete and accurate kinetic models to evaluate in which conditions chemiexcitation is threatening human health.

Acknowledgments Funding support has been received by the Spanish “Ministerio de Ciencia e Innovacio´n/Fondos Europeos de Desarrollo Regional (MICINN/ FEDER)” within the projects CTQ2017-87054-C2-2-P, PID2021-127199NB-I00 and the Unit of Excellence Marı´a de Maeztu programme (CEX2019-000919-M).

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64. Giokas, D. L.; Christodouleas, D. C.; Vlachou, I.; Vlessidis, A. G.; Calokerinos, A. C. Development of a Generic Assay for the Determination of Total Trihydroxybenzoate Derivatives Based on Gold-Luminol Chemiluminescence. Anal. Chim. Acta 2013, 764, 70–77. 65. Borrego-Sa´nchez, A.; Zemmouche, M.; Carmona-Garcia´, J.; Frances-Monerris, A.; Mulet, P.; Navizet, I.; Roca-Sanjua´n, D. Multiconfigurational Quantum Chemistry Determinations of Absorption Cross Sections (s) in the Gas Phase and Molar Extinction Coefficients (E) In Aqueous Solution and Air-Water Interface. J. Chem. Theory Comput. 2021, 17 (6), 3571–3582. 66. Borrego-Sa´nchez, A.; Giussani, A.; Rubio, M.; Roca-Sanjua´n, D. On the Chemiluminescence Emission of Luminol: Protic and Aprotic Solvents and Encapsulation to Improve the Properties in Aqueous Solution. Phys. Chem. Chem. Phys. 2020, 22 (47), 27617–27625. 67. Berraud-Pache, R.; Lindh, R.; Navizet, I. QM/MM Study of the Formation of the Dioxetanone Ring in Fireflies Through a Superoxide Ion. J. Phys. Chem. B 2018, 122 (20), 5173–5182. 68. Yue, L.; Liu, Y. J. Three S0/S1Conical Intersections Control Electron-Transfer-Catalyzed Chemiluminescence of 1,2-Dioxetanedione. J. Chem. Theory Comput. 2021, 17 (6), 3483–3494. 69. Vacher, M.; Farahani, P.; Valentini, A.; Frutos, L. M.; Karlsson, H. O.; Galva´n, I.; Lindh, R. How Do Methyl Groups Enhance the Triplet Chemiexcitation Yield of Dioxetane? J. Phys. Chem. Lett. 2017, 8 (16), 3790–3794. 70. Yue, L.; Liu, Y. J. Two Conical Intersections Control Luminol Chemiluminescence. J. Chem. Theory Comput. 2019, 15 (3), 1798–1805. 71. Zhu, C.; Teranishi, Y.; Nakamura, H. Nonadiabatic Transitions Due to Curve Crossings: Complete Solutions of the Landau-Zener-Stueckelberg Problems and Their Applications. Advances in Chemical Physics, Prigogine, S. A. R., Ed, 117; Wiley, 2007; pp. 127–233. 72. Berg, K.; Theodossiou, T. A. Lumiblast; 2020. http://www.lumiblast.eu(Accessed 8 August 2020). 73. Pantelia, A.; Daskalaki, I.; Cuquerella, M. C.; Rotas, G.; Miranda, M. A.; Vougioukalakis, G. C. Synthesis and Chemiluminescent Properties of Amino-Acylated Luminol Derivatives Bearing Phosphonium Cations. Molecules 2019, 24 (21), 3957. 74. Premi, S.; Wallisch, S.; Mano, C. M.; Weiner, A. B.; Bacchiocchi, A.; Wakamatsu, K.; Bechara, E. J. H.; Halaban, R.; Douki, T.; Brash, D. E. Chemiexcitation of Melanin Derivatives Induces DNA Photoproducts Long After UV Exposure. Science 2015, 347 (6224), 842–847. 75. Premi, S.; Brash, D. E. Chemical Excitation of Electrons: A Dark Path to Melanoma. In DNA Repair; 44; Elsevier BV, 2016; pp. 169–177. 76. Premi, S.; Han, L.; Mehta, S.; Knight, J.; Zhao, D.; Palmatier, M. A.; Kornacker, K.; Brash, D. E. Genomic Sites Hypersensitive to Ultraviolet Radiation. Proc. Natl. Acad. Sci. U. S. A. 2019, 116 (48), 24196–24205. 77. Brash, D. E.; Goncalves, L. C. P.; Bechara, E. J. H. Chemiexcitation and Its Implications for Disease. Trends Mol. Med. 2018, 24 (6), 527–541.

Chapter 15

Chemi- and bioluminescence: Bioluminescence Isabelle Navizet Professor at Laboratoire Mod
elisation et Simulation Multi Echelle, Universit
e Gustave Eiffel, France

Chapter outline 15.1 Introduction 15.2 Bioluminescence, a reaction scheme of a chemiluminescent system catalyzed by a protein: Challenges for theoretical and computational researchers 15.2.1 Overview of a bioluminescent process 15.2.2 Generation of HEI 15.2.3 Decomposition of HEI to the light emitter 15.2.4 Emission of light 15.3 Tools and choices of the theoretical chemist: Divide to conquer 15.3.1 Performing calculation of a small chemiluminescent model in vacuum 15.3.2 Performing calculation of a chemiluminescent model in the solvent 15.3.3 Performing calculation of a bioluminescent model in the protein 15.3.4 Modeling spectral shape 15.4 Modeling formation of HEI: Case of firefly bioluminescent system 15.4.1 From D-luciferin substrate to D-luciferyl adenylate intermediate 15.4.2 Approach of dioxygen to the D-luciferyl adenylate intermediate

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15.4.3 Deciphering between reaction schemes for the reaction of dioxygen with the D-luciferyl adenylate intermediate 15.4.4 Formation of the dioxetanone ring: Addition-elimination mechanism? 15.5 Modeling decomposition of HEI leading to the light emitter in firefly 15.5.1 Failure of small models 15.5.2 Model in vacuum 15.5.3 Model in proteins 15.6 Modeling light emission 15.6.1 Challenges in modeling and experiments 15.6.2 Nature of the light emitter of firefly: The oxyluciferin 15.6.3 Use of analogs of firefly oxyluciferin 15.6.4 Influence of the protein on the emitted light color 15.6.5 Example of one mutation in luciferase 15.6.6 Different colors in different luciferases 15.6.7 Modeling emission spectra for substrate analogs 15.7 Conclusion References

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In this chapter, we will focus on the theoretical and computational approaches to study bioluminescent systems. Bioluminescence is the chemiluminescence phenomenon that produces light emission by a living organism. The best-known example of bioluminescence is that of fireflies, emitting flashes of light to attract their mates. Moreover, there is plenty of species, such as land animals (firefly, railroad worm, and click beetle), aquatic animals (jellyfish, protozoa, sponges, marine bacteria, dinoflagellates, crustaceans, mollusks, and shark), and vegetal (mushrooms) that are also known to be able to emit light (1). New biotechnologies using bioluminescent systems extracted from the living world and engineered for specific detection have made a lot of progress in the last half-century. For instance, bioluminescence is used as a biosensor to image biological processes in vivo. Readers interested in multiple applications can look at the reviews of Kaskova et al. (1), Love and Prescher (2), and Syed and Anderson (3). One of the challenges is to find biosensors that are able to emit in the 560–900 nm region of the light spectrum, a wavelength range that should preferably be used for probing tissues. Indeed, the oxy- and deoxyhemoglobin absorb light below 660 nm and water above 900 nm. Engineering new sensors by chemically modifying bioluminescent systems is a tool to provide sensors with desired emission energy. Theoretical and Computational Photochemistry. https://doi.org/10.1016/B978-0-323-91738-4.00011-7 Copyright © 2023 Elsevier Inc. All rights reserved.

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Even if bioluminescent phenomena have long been known, the studies at the molecular level of such complicated systems were possible only in the late 20th century-early 21st century. Progresses were made in the experimental technics on the extraction of proteins involved in this phenomenon, determination of crystal structures of proteins by X-ray, and spectroscopic measurements. It should be remarked that the chemistry Nobel Prize was decerned in 2008 to Shimomura, Chalfie, and Tsien for the discovery and development of the green fluorescent protein (GFP) of the Aequorea Victoria jellyfish, a protein that is activated by a bioluminescent reaction. In addition, progresses in the modeling methodology have been able to increase the memory and speed of the computers. The chemistry Nobel Prize was decerned in 2013 for the development of multiscale models for complex chemical systems by Karplus, Levitt, and Warshel. These theoretical methods are nowadays used for studying bioluminescent systems. In this chapter, we will present first the reaction scheme of bioluminescent systems. Then, we will expose the strategy and theoretical tools used for simulating bioluminescent processes. Finally, some examples of simulations on the firefly bioluminescent system are presented.

15.2 Bioluminescence, a reaction scheme of a chemiluminescent system catalyzed by a protein: Challenges for theoretical and computational researchers 15.2.1 Overview of a bioluminescent process In bioluminescent and chemiluminescent reactions, a light emitter is obtained in its excited state. The bioluminescent reaction differs from fluorescent experiments in the manner the system reaches its excited state. In fluorescent experiments or photoactivated phenomena, the system initially in its electronic ground state reaches the excited state by absorbing a photon. In a bioluminescent system, the excited state is reached by a series of thermal steps (see Fig. 15.1). For instance, it follows the electronic ground state potential energy surface, leading to an ultimate reaction intermediate whose decomposition path has the particularity to reach a crossing between the ground and first excited states. This particularity allows the system to thermally populate the excited state. Indeed, in bioluminescent systems, the system reaches a reaction step showing a low energetic barrier with a transition state easy to reach and with one (or more) conical intersection close to the transition state. The reaction intermediate formed before populating the excited state is then called high-energy intermediate (HEI) (4). Decomposition of the HEI leads to an excited state without involving a radiative phenomenon because the ground and first excited states cross throw at least one conical intersection. Like in fluorescent experiments, the final light emission comes from the radiative deexcitation of the product from an excited state with the same spin multiplicity as the ground state. Other radiative and/or non-radiative processes can also occur. Fig. 15.1 shows schematic potential energy curves along the reaction coordinate of a bioluminescent system involving one conical intersection (CI). As for chemiluminescent reactions, bioluminescent reactions can be divided into three main steps: first the generation of HEI by thermal reaction steps, decomposition of HEI that produces an excited light emitter, and physical electronic phenomena of light emission. These steps are detailed in the next paragraphs.

15.2.2 Generation of HEI In all bioluminescent systems, the chemiluminescent reaction occurs in the cavity of a protein. The substrate is called luciferin and the protein luciferase. The origin of the name comes from the Latin lucifer, “light-bearer”. Demonstration

FIG. 15.1 Schematic potential energy curve of a bioluminescent reaction. The reactants undergo some thermal reactions on the ground state (GS) until reaching a high-energy intermediate (HEI) that decomposes throw of a conical intersection (CI) near the transition state of the GS, allowing the excited state (ES) population. Light is emitted when the system radiatively relaxes from the light emitter minimum in the excited state to the ground state.

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of the presence of a substrate and a protein was first made by French scientist Raphael Dubois (1849–1929). The protein catalyzes the oxidation of luciferin leading to the light emitter, called oxyluciferin. Some luciferases like the ones in the bioluminescent systems of Aequorea jellyfish and Chaetopterus tubeworm have been given the name photoprotein by Shimomura and Johnson in 1966 (see the new edition of the book from Shimomura and Johnson (5) first published in 1966). Contrary to other luciferases, these bioluminescent systems showed a total light emission proportional to the amount of protein involved. In other bioluminescent systems, luciferase is acting as an enzyme leading to a total light emission independent of the amount of protein, i.e., the protein acts as a catalyst. Actually, the reaction scheme in photoproteins and luciferases is very similar. The main difference comes from the kinetics of the reaction steps. In photoproteins, the formation of a stable luciferin-photoprotein complex requires the addition of another cofactor (Ca2+ in aequorin) for the generation of the light emitter. This halosteric effect explains why photoproteins produce a flash of light when luciferin and calcium are added, rather than a prolonged glow like in other luciferin-luciferase systems. Inside the enzyme (luciferase or photoprotein), luciferin is oxidized to form the HEI. HEI is a peroxide, a derivative of 1,2-dioxetanone (see Fig. 15.2) or a noncyclic peroxide (see Fig. 15.3). The formation of HEI from luciferin can be induced by a reaction of luciferin with other cofactors. The cofactor is adenosine triphosphate (ATP) in the firefly (see Fig. 15.4) and nicotinamide adenine dinucleotide (NADH) in bacterial bioluminescence. Residues of the protein cavity can also be involved in the reaction. Oxidation is done by dioxygen in most of the cases and H2O2 in some cases of acorn worm (6) and Diplocardia longa earthworm (7). The dioxygen ground state is a triplet state, the ground state of luciferin is a singlet state, and the reaction leads to an HEI in its singlet state. Therefore, the reaction is a spin-forbidden process. Why and how does it occur efficiently? Theoretical investigations can give some answers.

15.2.3 Decomposition of HEI to the light emitter Decomposition of HEI leads to the light emitter in the singlet excited state. This decomposition involves at least one conical intersection between the ground and the first singlet excited state. Bioluminescent paths are specific to each bioluminescent system. It should be remarked that decomposition of dioxetanone moieties in bioluminescent systems does not follow the same paths as decomposition of simple chemiluminescent dioxetanones (4). The efficiency of the decomposition of HEI is the key to the bioluminescent process. Its proper description and the understanding of elements that favor it (nature of the HEI and influence of the protein) are therefore essential. Which are the paths followed by the chemical system? How many conical intersections can be found? What is the rate of the production of light emitter? What is the nature of the light emitter? Answers to these questions can be clarified by simulation studies.

15.2.4 Emission of light In vivo, the same luciferin is shared by different species, but the proteins can be different. Even proteins with high percent identity can lead to different colors (8). In vitro, one mutation in the protein can shift the color emitted by the bioluminescent system. For example, the S286N mutation in the Japanese Genji boraru luciferase leads to red emission while the wild type emits yellow-green (9). Changes in salinity, pH, and temperature lead to changes in the emission spectra for the same luciferin-luciferase system. At an electronic level, the energies of electronic transitions (and therefore the emitted color) depend on the molecular conformations and influence of the electric field of the solvent or protein environment. Actually, experimental emission spectra do not show one sharp emission line but rather a more complex emission band corresponding to many conformations and possibly many chemical forms of the light emitter (10). The chemical nature of the light emitter is difficult to be determined experimentally because the light emitter is not isolatable in its excited state inside the protein. Fast reactions like protonation/deprotonation can occur in the excited state as well as in the ground state and depend on the first shell of molecules (solvent and/or protein’s residues) around the molecule. Therefore, experimental isolation of the product of the reaction does not warrant finding the chemical nature of the light emitter. Simulations can help in deciphering the chemical nature of the light emitter and its protonation state. Simulations on bioluminescent systems give hints to understand the influence of key protein residues or environmental factors on the color of the emitted light. By applying tools similar to the ones used to simulate absorption spectra, emission spectra can be simulated by taking into account the protein environment and the flexibility of the system by considering a sampling of reached conformations.

15.3

Tools and choices of the theoretical chemist: Divide to conquer

The questions that can be addressed by computational studies on bioluminescent systems are multiple and the level of theory used to answer them will depend on the target, as well as available computer and human resources. Hypotheses

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FIG. 15.2 Commonly proposed reactions of known luciferins through a 1,2-dioxetanone high-energy intermediate (HEI). The oxidation process takes place in the molecular part circled in green in the luciferin. All structures are shown in their protonated neutral states. The protonation state of different species is in most cases an unsolved question. The first steps leading to HEI are not explicitly drawn being represented by two arrows. (Reprinted (adapted) with permission from Vacher, M.; Fdez. Galva´n, I.; Ding, B.-W.; Schramm, S.; Berraud-Pache, R.; Naumov, P.; Ferr
e, N.; Liu, Y.-J.; Navizet, I.; RocaSanjua´n, D.; Baader, W. J.; Lindh, R. Chemi- and Bioluminescence of Cyclic Peroxides. Chem. Rev. 2018, 118 (15), 6927–6974. Copyright (2018) American Chemical Society.)

will have to be made before the simulation and validated by comparison with experiments. In the following, we will give some approaches to be followed and choices to be made to study a bioluminescent system. In the next paragraphs, we will give concrete examples of theoretical studies published in recent years.

15.3.1 Performing calculation of a small chemiluminescent model in vacuum During the formation of HEI, one of the main steps is the oxidation of luciferin involving the triplet dioxygen molecule. Its study implies to properly describe intersystem crossings and a multistate description of the system. During the

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FIG. 15.3 Commonly proposed reactions of known luciferins through a nondioxetanone high-energy intermediate (HEI). The oxidation process takes place in the molecular part circled in green in luciferin. All structures are shown in their protonated neutral states. The protonation state of different species is in most cases an unsolved question. The first steps leading to HEI are not explicitly drawn being represented by two arrows. (Reprinted (adapted) with permission from Vacher, M.; Fdez. Galva´n, I.; Ding, B.-W.; Schramm, S.; Berraud-Pache, R.; Naumov, P.; Ferr
e, N.; Liu, Y.-J.; Navizet, I.; Roca-Sanjua´n, D.; Baader, W. J.; Lindh, R. Chemi- and Bioluminescence of Cyclic Peroxides. Chem. Rev. 2018, 118 (15), 6927–6974. Copyright (2018) American Chemical Society.)

FIG. 15.4 Reaction scheme of firefly’s bioluminescent reaction. The firefly D-luciferin reacts inside the protein cavity with ATP leading to a reaction intermediate D-luciferyl adenylate that is oxidized to a high-energy intermediate (HEI), the firefly dioxetanone. Decomposition of HEI leads to the oxyluciferin form product in its first singlet excited state. (Reprinted (adapted) with permission from Vacher, M.; Fdez. Galva´n, I.; Ding, B.-W.; Schramm, S.; Berraud-Pache, R.; Naumov, P.; Ferr
e, N.; Liu, Y.-J.; Navizet, I.; Roca-Sanjua´n, D.; Baader, W. J.; Lindh, R. Chemi- and Bioluminescence of Cyclic Peroxides. Chem. Rev. 2018, 118 (15), 6927–6974. Copyright (2018) American Chemical Society.)

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decomposition of HEI, conical intersections are involved in the description of the path from the ground state to the excited state. Multiconfigurational methods are the methods of choice for their study but limit the size of the system and basic functions that can be involved in the calculations. The complete active space self-consistent field (CASSCF) method requires choosing the number of electrons and orbitals relevant to the process (so-called active space). To study bondbreaking/-forming processes, at least the bonding and anti-bonding orbitals of that bond have to be included in the active space. Dynamic electronic correlation can be added for energy corrections (e.g., by complete active space second-order perturbation theory (CASPT2)) on top of CASSCF, which only relies on static electronic correlation. However, when studying electronic transitions and/or paths where no conical intersections are involved, density functional theory (DFT) and time-dependent DFT (TD-DFT) calculations can be used. The first studies are usually done on small chemiluminescent model systems in vacuum, taking only part of the molecule involved in the bioluminescent reaction. Then, this small model can be extended to more realistic models and at some point it is possible to consider the solvent or protein environment. The preliminary study of a chemiluminescent reaction of a small model in vacuum will help the researcher to choose the most proper parameters for the calculation such as the level of theory, basis sets, actives spaces, etc. Later, the results in vacuum will be compared with the ones of more realistic models to understand the influence of the presence of the protein or moieties that have not been considered first.

15.3.2 Performing calculation of a chemiluminescent model in the solvent Once the path of small models has been described, more realistic models should be investigated. At this point, the protonation state of species should be considered. A hypothesis should be made depending on experimental findings. Protonation and deprotonation of molecules can be different depending on the solvent used or the protein active site. Regarding the solvent, it can be considered explicitly or via an implicit model like the polarizable continuum model (PCM) (see for a review ref (11)). In some cases, bioluminescent luciferin and the corresponding reaction intermediates can have a quite large size. For them, less computationally demanding methods than CASSCF should be used for the description of electronic states, like TD-DFT. However, TD-DFT cannot be applied when state crossings are present in the path, as in bioluminescent processes and so, quantum mechanics/molecular mechanics (QM/MM) or two levels of quantum mechanics (QM/QM) methods can be used to properly describe the relevant part of the system (usually the small part already studied in vacuum) at a higher level, describing the rest of the system at a lower level.

15.3.3 Performing calculation of a bioluminescent model in the protein 15.3.3.1 Completing the protein Protein structures of the protein data bank (PDB) are determined by X-ray crystallography, and usually, their structures are not complete. The missing atoms and amino acids have to be added to build the proper system for study. Usually, the missing residues (referring to amino acids or biomolecule units) belong to external loops, which are not located in the protein cavity where the bioluminescent reaction occurs. Different programs are available to model the missing residues as DisGro (12), I-TASSER (13), or Modeler (14), among others. The Leap software of the Amber suite of program (15) is also able to build missing residues when the length of the loop is not too long. In the cases where the crystallographic structure is not available, artificial intelligence can be applied, for instance using the software Alphafold (16,17), to predict 3D structures of the protein. Protonation states of protein residues depend on the sample pH, being hence important to first determine their local pKa, possibly using programs like MolProbity (18), PROpKa (19), or H++ (20). Histidine residues have to be checked manually as their protonation state depends on their hydrogen bonds with other residues.

15.3.3.2 Docking the ligand in the protein Another difficulty coming with modeling bioluminescent processes is to have the “correct” position of the reactant in the protein active site. If the crystal structure has been determined with non-reactive analogs, manual modification of analogs to the target reactant can be made. Another possibility is the docking of the reactant in the protein cavity (for example using Autodock (21)), which provides diverse poses, i.e., protein/substrate conformations. The poses are evaluated by a score function that gives an estimation of the potential energy of interactions between the substrate and the residues of the cavity. Poses with similar conformations and substrate/protein interactions are also grouped in clusters. The researcher has to wisely choose one or more poses for further investigation. Previous experimental results of identifying residues that are important in the interaction with the substrate (for example when their mutation leads to no interaction) can be used

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to guide the researcher. The researcher can also take into account the available structures of other proteins that bind the same type of substrate (22). The researcher then chooses the most stable pose (taking into account the value of the score function or one of the calculated potential energy), or the average structure of the most abundant cluster, as recommended by Chang et al. (23). To complete the model, water molecules and ions to get a neutralized system are added. As the constructed model should be relaxed by molecular dynamics, the researcher needs to have an appropriate force field. Classical force fields are available for proteins, water molecules, and ions. For the substrate, classical force fields have to be built.

15.3.3.3 Getting the force field parameters for the substrate General force fields for organic molecules are available like the GAFF (generalized amber force field) force field compatible with amber force fields for proteins. These force fields may be sufficient to describe the interactions in the substrate when the purpose is to obtain a starting structure of the reactant in the ground state. However, it is not sufficient when dealing with excited states or very flexible molecules. For these cases, theoreticians are putting in place strategies for force fields parametrization, which consider the conformation changes of the molecule or its attributes in the excited state. Iterative processes like those in Refs. (24, 25) are good strategies.

15.3.3.4 Relaxing the structure The model comprising the protein/substrate should be relaxed to release constraints from its construction. Minimization and classical molecular dynamics (MD) are usually performed to release the steric clashes. Then, conformations for further calculations are extracted from the MD trajectories.

15.3.3.5 QM/MM calculations QM/MM methods allow taking into account the surrounding mechanical embedding and electrostatic embedding due to the constraint of the protein cavity. The researcher has to decide which part of the system is to be described at the high level, i.e., the QM level, and which part at a low level, i.e., with a classical force field. Contrary to the green fluorescence protein where the light emitter or chromophore is covalently bound to the protein, the substrate in bioluminescent systems interacts non-covalently with the protein. Therefore, a simple choice of setting the frontier between the QM and MM regions is to describe the substrate at the QM level and the rest of the system (protein, water, ions, and other cofactors) at the MM level. However, it is sometimes necessary to cut the covalent bonds between the part described at the QM level (or high layer) and the one described at the MM level (or low layer). This occurs for example when the level of theory for the QM level is very computationally demanding, and the substrate is too big to be described entirely at this level (only a part of the substrate is then included in the QM part/high layer). On the contrary, when residues or part of residues of the protein have to be included in the QM part, to consider their electronic contribution in the calculated propriety, then the selected QM level for this relatively big system is usually a noncomputational demanding one (such as DFT). In the case of covalent bonds between high and low layers, technics like the link-atom model (26,27), where atoms (usually hydrogen) are added to saturate the dangling bonds of the QM part, are used. For simulating reactivity, the difficulty remains in taking into account the flexibility of the protein during the reaction path. For simulating spectra, the vertical energy transitions and the corresponding oscillator strengths are calculated at the QM/MM level.

15.3.4 Modeling spectral shape Modeling emission spectra of the light emitter is a way to rationalize the color of bioluminescent systems. At a first approximation, the system oxyluciferin/luciferase can be optimized at the QM/MM level in its first singlet excited state. The difference of energy between the ground and first excited states at the obtained geometry can be compared to the energy corresponding to the wavelength of the maximum intensity of the emission spectra. This is quite useful for the first estimation of emission color shifts between similar systems. In order to have more accurate information on the shape of the spectra, classical MD simulations with a force field adjusted to mimic the behavior of the excited state should be performed in order to sample the conformations. The calculation of the vertical energy of a few hundred snapshots (upon reaching the statistical convergence) taken from the MD dynamics is a proper strategy to obtain the emission spectra by Gaussian convolution.

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Modeling formation of HEI: Case of firefly bioluminescent system

15.4.1 From D-luciferin substrate to D-luciferyl adenylate intermediate In the luciferase cavity, the firefly D-luciferin reacts first with the ATP in the presence of magnesium ions leading to an intermediate, D-luciferyl adenylate, covalently bonded to AMP (see Fig. 15.4). This first step, named adenylation reaction, has not yet been specifically studied for the firefly bioluminescent system. The reasons are that ATP hydrolysis inside the protein involved highly charged molecules that would polarize the surrounding protein residues, involving a very large QM system to be taken. Moreover, no structures with luciferin and ATP analogs have been crystallized yet and this step has not been identified as a rate-limiting step of the reaction. However, some investigations have been done on the reaction of the D-luciferyl adenylate intermediate with molecular oxygen to generate HEI. We will explain here the methodology used and the choices made to explore how the dioxygen approaches the intermediate and what is the mechanism for the formation of the dioxetanone ring, as reported in the article by Berraud-Pache et al. (28).

15.4.2 Approach of dioxygen to the D-luciferyl adenylate intermediate The approach of dioxygen to the substrate (D-luciferyl adenylate intermediate) in the cavity of the protein does not involve any bond breaking; therefore, this can be studied at the classical level. Classical MD and umbrella sampling show that dioxygen can move freely inside the protein without an energy barrier. Umbrella sampling consists of performing a series of MD simulations with, in this case, a harmonic constraint on the distance between the dioxygen and the substrate. The distance is progressively decreased forcing the two molecules to get closer. The weighted histogram analysis method (WHAM) removes the bias introduced by the umbrella potential to generate a free energy profile along the approaching path (29,30).

15.4.3 Deciphering between reaction schemes for the reaction of dioxygen with the D-luciferyl adenylate intermediate The first reaction step is deprotonation at the alpha-carbon of the carbonyl group of the D-luciferyl adenylate intermediate. Two hypotheses lead to three proposed mechanisms (see Fig. 15.5). In mechanisms (a) and (b) exposed in Fig. 15.5, a nearby histidine deprotonates the D-luciferyl adenylate (Fig. 15.5 1ab), leading to a carbanion substrate (Figure 15.5 2ab). Then, in case (a), a reaction between the carbanion and triplet dioxygen leads to a heterolytic formation of C-O bond (Fig. 15.5 reaction 3a to 5ab). In mechanism (b), a single electron is transferred from the negatively charged carbanion of the substrate to molecular oxygen, leading to the creation of a radical superoxide anion and a radical substrate (Fig. 15.5 4b). The homolytic bond formation of C–O follows (Fig. 15.5 reaction 4b to 5ab). In mechanism (c), hydrogen transfer occurs directly from D-luciferyl adenylate to molecular oxygen, rising to a singlet ground-state hydroperoxide molecule that forms back a homolytic bond with the resulting radical substrate (Fig. 15.5 reaction 1c to 3c to 5c). Starting with a model of deprotonated intermediate like compound 2ab of hypothesis (a) and (b), QM/MM calculations have shown a spontaneous electron transfer from the deprotonated intermediate to dioxygen when the molecular oxygen is inside the protein cavity, in favor of the pathway (b). The system initially in its triplet ground state undergoes an ISC as the two lower singlet and two lower triplet states are degenerated until the formation of the first C–O bond. A proton transfer from the protonated histidine residue to the formed superoxide anion can be observed leading to a hydroperoxide anion (mechanism (4b’) in Fig. 15.15), in favor of a mixed path between mechanisms (b) and (c) (28). In the study reported in Ref. (28), the complexity of the reaction has been overcome by dividing the different steps and choosing an adapted simulation for each of them. MS-CASPT2//SA-CASSCF/MM calculations have been performed to describe bond formation and breaking. Due to the size of the substrate, only a small part of the substrate and dioxygen was described at the QM level, considering the rest of the system at the MM level. Orbitals of active spaces have been carefully chosen to balance the constraint of the calculation time and trying not to leave apart important orbitals involved in the concerned bond modifications.

15.4.4 Formation of the dioxetanone ring: Addition-elimination mechanism? The last step of the formation of the dioxetanone intermediate is the closure of the ring with the formation of a C–O bond on the carbonyl carbon and leaving the AMP (Fig. 15.5 step 5 to 6). In the study of Berraud-Pache et al. (28), where only part of

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FIG. 15.5 Suggested pathways for the reaction of triplet molecular oxygen and D-luciferyl adenylate intermediate of firefly leading to singlet dioxetanone. (A) Deprotonation mechanism from histidine in the cavity (1ab to 2ab) and direct reaction with molecular oxygen (3a to 5ab). (B) Deprotonation of the intermediate from histidine in the cavity (1ab to 2ab) and formation of a superoxide anion (3b to 4b) before creating a homolytic bond (4b to 5ab). (C) Hydrogen of the intermediate captured by molecular oxygen (1c to 3c) followed by the homolytic formation of the C–O bond (3c to 5c). (B’) Reprotonation of the superoxide anion (4b’ to 3c) before its reaction with the radical (3c to 5c). Closure of the ring leads to firefly dioxetanone (6). The figure was created based on the results of the reference (28).

the molecule is described at the QM level (to be able to use a high-level QM method), step 5 to 6 involving the formation of the C–O bond could not be studied, unless artificially first breaking the bond between the carbonyl carbon and AMP group. Yu and Liu (31) have performed a QM/MM study of the last steps of mechanism (b): reaction of dioxygen on the deprotonated D-luciferin adenylate, starting after the ISC region, i.e., from 4b to 6 in Fig. 15.5. The aim of this last study is to locate the transition and intermediate states along the formation of the dioxetanone ring (31).

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FIG. 15.6 QM/MM partitioning used in the studies of Berraud-Pache et al. and Yu and Liu. In red is the part of the system described at the QM level and in blue the two bonds that constitute the frontier between the part treated at the QM level and the part treated at the MM level in the study of Berraud-Pache (28). The frontier is treated with the link-atom model. In the study of Yu and Liu (31), the whole represented system is treated at the QM level.

Yu and Liu have searched transition states and optimized reaction intermediates on the singlet ground state of the system (see Fig. 15.6 for the atoms taken into account in this study) in DMSO at the UMO6-2X/6-31G(d,p) level. Two reaction intermediates and three transition states have been identified. Then, in order to take into account the protein environment, the authors have used two-layered ONIOM method (32) with electronic embedding at the unrestricted DFT (UM06-2X/6311G(d,p)) level with broken-symmetry technology and spin projection method (33) for the QM level to optimize the transition states and the intermediates inside the protein, letting the side chain of six residues of the cavity relax. Fig. 15.7 shows the relative energy profile of the stationary points found by Yu and Liu. The calculation of the triplet state energy at different structures optimized at the ground state path confirmed the results from Berraud-Pache et al.: the ISC occurs before the formation of the C–O bond. In addition, Yu and Liu showed that the formation of the dioxetanone ring does not necessarily require leaving of the AMP group. However, the study of Yu and Liu still shows some limitations. Indeed, at the studied level of theory, TS3 has the characteristic of a transition state showing an imaginary frequency, but its QM/MM GS energy is lower than the one of the reaction intermediate Int2 in contradiction of what would be expected for a transition state between Int2 and P (see Fig. 15.7). Therefore, it is difficult to decipher from this latter study if the mechanism (from 5ab to 6 of Fig. 15.5) undergoes an addition-elimination mechanism or a synchronous creation-breaking of bonds in the protein. A higher level of theory should be applied for a better understanding.

FIG. 15.7 Optimized ground-state (GS) stationary points for the oxidation of the radical intermediate 4b (see Fig. 15.5) by superoxide anion O2 in the protein, depicting their energies in GS and triplet state (T1). RC is the starting structure. TS1, TS2, and TS3 are transition states optimized at the QM/ MM level starting from structures of the transition states found along the intrinsic reaction coordinate (IRC) in dimethyl sulfoxide (DMSO) on the ground state. Int1 and Int2 are reaction intermediates found in the GS profile after QM/MM optimization. P is the QM/MM optimized product of the system. (Based on Yu, M.; Liu, Y. A QM/MM Study on the Initiation Reaction of Firefly Bioluminescence—Enzymatic Oxidation of Luciferin. Molecules 2021, 26 (14), 4222.)

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Modeling decomposition of HEI leading to the light emitter in firefly

15.5.1 Failure of small models HEI of firefly’s bioluminescent system is firefly’s dioxetanone (see Fig. 15.4). HEI decomposes into carbon dioxide and oxyluciferin, the light emitter in its first singlet excited state. HEI decomposition of small models like 1,2-dioxetane and 1,2-dioxetanone leads to a highly populated triplet state, inducing a low efficiency of light emission (4). The firefly HEI decomposition is an example of reactivity where the study of a small model chemiluminescent system in vacuum is not sufficient to understand the reactivity of the bigger system. Contrary to the small models, the firefly dioxetanone moiety is bonded to an extended p-conjugated system, very rich in electrons. The p-conjugated moiety plays the role of activator for the reaction (34) and its presence explains the high experimental luminescence efficiency of 41% of the firefly bioluminescent system (35). Moreover, the p-conjugated moiety has a phenol group that can be protonated or deprotonated. HEI can in principle exist in its neutral or anionic form.

15.5.2 Model in vacuum The chemiluminescent reaction of the anionic and neutral form of firefly HEI was studied in vacuum (36,37). A DFT intrinsic reaction coordinate (IRC) minimum energy path was calculated, starting from the optimized transition state of decomposition of HEI. This results in a set of geometries along the maximum instantaneous acceleration path on the electronic ground state potential energy surface from the transition state toward the two local minima. TD-DFT and CASPT2// CASSCF calculations of the six lowest excited states were calculated along the path, taking into account solvation by polar or non-polar solvents with the polarizable continuum model (PCM). Depending on the protonated state of HEI, the decomposition path shows different schemes. Yue et al. (37) have calculated a high-energy barrier to reach the first singlet excited state for the decomposition of the neutral form, favoring an intersystem crossing (ISC) and the population of the triplet state, as it is the case for the 1,2-dioxetane and 1,2-dioxetanone systems. The existence of two transition states with a region where the excited singlet and triplet states are almost degenerated along the path (known as an entropic trap) favors the triplet state population. For the anionic form, the barrier is lower with only one transition state. Moreover, the reaction path shows at least two conical intersections (see Fig. 15.8). Analyzing the reaction path and the electronic density along it leads to the conclusion of a partial charge transfer mechanism during the dissociation, described as a charge transfer-induced

FIG. 15.8 Calculated CASPT2//CAM-B3LYP potential energy curves of the ground state GS and first excited single state S1 along the dissociation path of the deprotonated firefly’s dioxetanone in benzene solvent (PCM). The potential energy curve is colored in the graph depending on the character of the transition: green ball portion of the GS corresponds to partial electronic charge transfer (CT) from oxyluciferin to carbon dioxide; magenta triangles in the GS correspond to back-charge transfer (BCT), i.e., partial electronic charge transfer from carbon dioxide to oxyluciferin moiety. (Reprinted (adapted) with permission from Yue, L.; Liu, Y.-J.; Fang, W.-H. Mechanistic Insight into the Chemiluminescent Decomposition of Firefly Dioxetanone. J. Am. Chem. Soc. 2012, 134 (28), 11632–11639. Copyright (2012) American Chemical Society.)

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luminescence (CTIL) mechanism (38). These results were not in line with the proposition of Pinto da Silva et al. (39), promoting the neutral form to be involved in the reaction, with the argument that the phenol moiety can only be deprotonated in a polar environment.

15.5.3 Model in proteins Taking into account the protein’s local electrostatic field, Zhou et al. showed later that the energy barrier of the anionic form decomposition is lowered by the local electrostatic field created in the firefly Photinus pyralis luciferase’s cavity, but it is raised in the case of the neutral form (40). It should be noted that this study was done with only one conformation of the protein and still did not take into account the dynamic behavior of the protein. Studying the HEI decomposition inside the protein cavity from a dynamic point of view is still challenging. No crystallographic structure exists of the HEI inside the protein because it is unstable. Modeling it could be an alternative but the choice of the starting protein model structure (which global conformation of the protein? which protonation state of the residues? which conformation of the residues in the cavity?…) might be determinant. Moreover, the studies of the reaction path in vacuum and with implicit solvation are already very challenging concerning the QM level of the theory to be used. Conical intersections involving the hop from the electronic ground state to the excited state cannot be correctly described by TD-DFT, while a multi-configurational approach such as the CASSCF method can be used for this purpose, within the active space size limits. Development of methods to increase the active space like the density matrix renormalization group (DMRG) (41) will allow making such studies more feasible. The development of machine learning, as used for the analysis of the ab initio molecular dynamics simulation on the decomposition of dioxetane (42), might give some perspectives in the field.

15.6

Modeling light emission

15.6.1 Challenges in modeling and experiments The in vitro experimental bioluminescent spectra of a bioluminescent system result from the time-integrated total photon flux spectra from a reaction solution of luciferin and luciferase with cofactors. Firefly bioluminescent spectra are dependent on protein (different proteins from different species or engineered proteins lead to different color emissions), pH (43,44), temperature (45,46), and concentration of ions in the solution (47). The experiment is not easy to perform, and for this reason, usually, the absorption and fluorescence spectra of the light emitter (either in solvent solution or inside the protein) are recorded. Comparison of bioluminescence and fluorescence spectra (recorded after completion of the bioluminescent process) shows that the two do not necessary overlap. The excited chemical forms involved in the fluorescence experiment are not the same as the one obtained by the bioluminescent reaction. The possible discrepancy between the light emitter in the case of fluorescence experiment and bioluminescent is explained in Fig. 15.9. Theoretical simulations of the emission spectra help with a better understanding of the experimental data and interpretation of the discrepancies between bioluminescence and fluorescence spectra. One important question is to understand the factor that influences the spectral shape and color of the light emitted in bioluminescent systems. Understanding these factors leads to handling of the possible applications of these systems as sensors. Experimental measurements of emitted light are usually summarized by the intensity and the wavelength of the maximum emission spectra. This wavelength gives an indication of the color range of the emitted light, corresponding, in a simple approximation, to the wavelength calculated for the electronic transition from S1 to GS at the local minimum of the excited state. An estimation of the intensity of the emitted light is given from the calculation of the transition oscillator strength (f). To get a more complete picture of the absorption and emission spectra of the light emitter, more information than the electronic transition at a single point is mandatory. The line shape of the experimental emission spectra takes into account different sources of the broadening (this is also applicable to absorption spectra): - The chemical nature of the light emitter: Is the obtained spectrum a superposition of spectra from different species/ forms? - The contribution of more than one excited state. For emission, Kasha’s rule states that usually, for fluorescence and phosphorescence, photon emission occurs in an appreciable yield only from the lowest excited state of a given multiplicity (48). The emission spectrum is reasonably obtained by the calculation of transitions from the first excited state. For the absorption spectrum, more than one excited state contributes to the spectra (calculation of the transition in the range of the observed light should be made).

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FIG. 15.9 Different scenarios for the relative position of the bioluminescent state (BS) (obtained from HEI decomposition, black arrows) and the fluorescent state (FS) (green arrows). (A) There is no barrier; bioluminescence and fluorescence spectra match. (B) The barrier is low; the spectra would be a mix of BS and FS emissions. (C) The barrier between the bioluminescent state (BS) and the fluorescent state (FS) is high; the two experiments lead to different emission spectra.

- Inhomogeneous broadening due to the fluctuation of the solvation sphere and the resulting variation of the local electric field: Then, the statistical distribution of the electronic transition energies should be taken into account. An example of inhomogeneous broadening will be discussed in the subpart concerning modified substrates. - Homogeneous broadening due to the existence of a continuous set of vibrational sublevels in each electronic state: In very well-resolved experimental spectra, a vibrational structure can be observed. Modeling of the emission spectra obtained for a particular chemical form of the emitter is possible even though the experimental emission spectra are a mix of contributions of different chemical forms in equilibrium. Spectra simulation gives key information to decipher which chemical forms are predominant for the contribution of the experimental spectra. Indeed, the nature of the light emitter is still a debate, as during the reaction, protonation/deprotonation can occur in the excited state before light emission. A better understanding of the chemical nature of the light emitter is key to propose a modification of the protein or substrate to tune the emitted light color. Another question to be addressed is the estimation and simulation of the emission quantum yield. This question is more complicated to evaluate than the modeling of fluorescence spectra. The probability of radiative emission (i.e., electronic transition accompanied by photon emission), the reaction yield of the reaction scheme leading to the light emitter, and the yield of all possible relaxation channels (i.e., all nonradiative pathways that prevent the emission) participate in the total quantum yield. Assessing all these pathways in models including the protein environment is challenging in the study of bioluminescent systems.

15.6.2 Nature of the light emitter of firefly: The oxyluciferin Looking close to the nature of the light emitter proposed in Figs. 15.2 and 15.3, the presence of phenol groups, amino groups, and keto moiety implies possibilities of protonation/deprotonation of the acid-base group and keto-enol tautomerization. This can occur in the excited state (excited state proton transfer, ESPT), vacuum, solvent solution, or cavity of the protein. For firefly oxyluciferin, six forms of the light emitter are possible (Fig. 15.10). The advantage of the simulation over the experiments is that calculations can be done on one chosen chemical form. Experimentally, isolation of one form from the

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FIG. 15.10 The six forms of firefly oxyluciferin considering the acid/base protonation/deprotonation and keto/enol tautomerization reactions. (Reprinted (adapted) with permission from Garcı´a-Iriepa, C.; Gosset, P.; Berraud-Pache, R.; Zemmouche, M.; Dorkenoo, K. D.; Didier, P.; L
eonard, J.; Ferr
e, N. Simulation and Analysis of the Spectroscopic Properties of Oxyluciferin and Its Analogues in Water. 2018, 13. Copyright (2018) American Chemical Society.)

others is more difficult, even impossible when forms are in equilibrium. The control of pH can put acid/base systems in one of the predominant species but cannot prevent keto-enol tautomerization. The chemical nature of the light emitter and if the emitted light comes from one or more chemical forms of oxyluciferin is still under debate (10). To add impetus to the question, the electronic transition energies are calculated for the forms optimized in their first excited state in vacuum (49–51) or in water solution (24,52). Neutral form emissions are blue-shifted compared to deprotonated ones and can be excluded from the list of potential light emitters. Phenol-enolate-OxyLH form was also first excluded because of its relatively low oscillator strength and the probable phenolate nature of the dioxetanone intermediate. However, these last arguments have to be revised in light of the calculation of the pKa of phenol/phenolate and enol/enolate in the first excited state or ground state (40).

15.6.3 Use of analogs of firefly oxyluciferin To prevent tautomerization, analogs of the firefly light emitter, for which this tautomerization is blocked, have been synthetized (Fig. 15.11). The absorption and fluorescence spectra of these analogs in the solvent or protein have been recorded. As they are synthetic analogs of the firefly light emitter, the oxyluciferin, that is the product once the bioluminescent reaction has occurred, hence no bioluminescence experiments are made on the systems but fluorescence experiments can be performed, assuming that the fluorescence spectra are comparable with the bioluminescence ones. Simulations of the absorption and fluorescence spectra for the analogs and the corresponding natural forms give confidence in the transferability of the results obtained for the analogs to the expected contribution of the corresponding forms in the experimental spectra. For more information, the reader can see the works of Garcia-Iriepa et al. in solution and in protein (24,25,52).

15.6.4 Influence of the protein on the emitted light color Computational studies allow rationalizing the experimental emission color tuning depending on the experimental conditions (solvent, pH, protein, etc.). The environment can contribute to the change of emission color in two ways: changing the population ratio between the emitting forms (stabilizing one form against another and changing the relative contributions to the emission spectra) and/or changing the emission spectra of each form that contributes to the final spectra. Polarization of the environment stabilizes or destabilizes the ground state or the excited state of different species. Differences in stabilization between ground and excited states are especially important when the transition from the excited state to the ground state presents a large charge transfer (CT) character. Interactions that stabilize the ground state and/or destabilize the excited state lead to a larger energy gap and hence, a blue shift of the emission wavelength. Interactions that destabilize the ground state and/or stabilize the excited state lead to a smaller energy of the electronic transition and hence, a red shift of the emission wavelength. Specific interactions between the light emitter and solvent or protein surrounding, like hydrogenbond interactions, play the same role as polarization, i.e., modification of the overall emitted light color.

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FIG. 15.11 The six forms of natural firefly oxyluciferin and the corresponding synthetic analogs blocking the protonation/deprotonation states and keto/ enol tautomerization. (Reprinted (adapted) with permission from Garcı´a-Iriepa, C.; Gosset, P.; Berraud-Pache, R.; Zemmouche, M.; Dorkenoo, K. D.; Didier, P.; L
eonard, J.; Ferr
e, N. Simulation and Analysis of the Spectroscopic Properties of Oxyluciferin and Its Analogues in Water. 2018, 13. Copyright (2018) American Chemical Society.)

15.6.5 Example of one mutation in luciferase We discuss here an example of how QM/MM approaches can explain color modulation by a mutation in the protein. Simulation of emission of the oxyluciferin phenolate-keto-OxyLH form in a model of Luciola cruciata luciferase could explain the experimental red shift observed in the (S286N) mutant luciferase. Nakatsu et al. (9) crystallized and determined the structure of the wild-type Luciola cruciata luciferase and its red S286N mutant in 2006. They suggested that modulation of the emitted light depends on the open or closed shape of the cavity. According to the authors, the cavity structure of the S286N mutant remains open during the bioluminescent process, allowing the light emitter to relax before emitting red light. The native protein would close, preventing oxyluciferin from relaxing, leading to green light. To test this hypothesis, QM/MM studies of the phenolate-keto-OxyLH a form of oxyluciferin inside the cavity were performed (34). The process followed was to include oxyluciferin in the cavity, perform short classical molecular dynamics, and extract snapshots. Oxyluciferin was chosen to be described in the high layer and the rest of the protein in the low layer. QM/MM optimization setup consists of oxyluciferin described at CASSCF(14,16)/631G* in its first excited state and the rest of the system described at the MM level (AMBERParm99 force field). In addition, ˚ from oxyluciferin and side chains of six residues of the cavity were allowed to relax along water molecules at less than 5 A the simulations. Vertical electronic transition was calculated at the CASPT2//AMBERParm99 level of an optimized structure. The results showed that the hydrogen bond network involving the residues, water molecules, and the oxyluciferin’s benzothiazole moiety was responsible for the color modulation of the emitted light rather than the cavity size or conformation. The mutation S286N, present at the door of the cavity, changes the hydrogen bonding network and the polarity of the microenvironment around the phenolate moiety of oxyluciferin, destabilizing the HOMO orbitals and leading to a red shift of the emitted light. The influence of the presence of water molecules or hydrogen bonding on the color modulation was later shown experimentally by spectroscopic results on a single monohydrated molecule of oxyluciferin in the gas phase, showing the red-shifted emission due to the presence of the water molecule (53).

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15.6.6 Different colors in different luciferases Modulation of the emission color exists in vivo. For example, the Amydetes vivianii firefly (GBAv) emits green-blue light, shifted compared to the Japanese Luciola cruciata, a yellow-green bioluminescent system (YGLc). The glow-worm Phrixothrix hirtus emits red. Crystallographic structures and MD simulations followed by QM/MM calculations have highlighted the impact of luciferases on the bioluminescence color (54). The protocol used followed these steps: (1) docking oxyluciferin and AMP in the cavity of the proteins, (2) performing 10 ns classical MD simulations, and (3) calculating the vertical transition by the TD-DFT/MM approach. For GBAv, the conformation obtained by X-ray with the open C-terminal domain is not the conformation allowing bioluminescence. Thus, the folding of this domain (closing the cavity) is a necessary prerequisite for obtaining the emission. Moreover, the superposition of the resulting GBAv model and YGLc shows a displacement of the substrate in GBAv, exposing the emitting molecule to other amino acids than in YGLc and affecting the color emission. QM/ MM simulations explained the blue shift of the emission wavelength in Amydetes vivianii by a change of environment inside the cavity.

15.6.7 Modeling emission spectra for substrate analogs As bioluminescent systems are used as biosensors, the aim of the research is to propose sensors emitting on the red electromagnetic range. Both modification of the protein by mutations and chemical modification of the luciferin have been already explored to tune the emission color. We present here simulation of the emission spectra of firefly luciferin synthetic analogs studied in references (55, 56). Fig. 15.12 shows the experimental spectra of bioluminescence luciferin and three analogs and the simulated spectra of emission of their respective light emitters. Experimental spectra correspond to the bioluminescence (the light emitter, the product of the bioluminescence reaction, and the wavelength corresponding to the maximum intensity of the emitted light are represented in Fig. 15.12), and the simulated spectra correspond to the simulated emission from the light emitters inside the protein cavity.

FIG. 15.12 (A) Emission spectra of oxyluciferin (Natural_Oxy) and its analogs simulated with QM/MM methods considering 100 statistical MD snapshots (HWHM of 0.20 eV). (B) Experimental bioluminescence spectra. The red and blue shifts are represented by curved arrows. The structure of the natural firefly light emitter, Natural_Oxy, and the synthetic analogs (chemical modification highlighted in blue) are depicted. The values in nanometers (nm) in the frame correspond to the experimental bioluminescence. (Reproduced from M. Zemmouche, M.; Garcı´a-Iriepa, C.; Navizet, I. Light Emission Colour Modulation Study of Oxyluciferin Synthetic Analogues via QM and QM/MM Approaches. Phys. Chem. Chem. Phys. 2020, 22 (1), 82–91. https://doi.org/10.1039/C9CP04687A with permission from the PCCP Owner Societies.)

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In the following, we describe how to simulate the emission spectra for one analog (hereafter named oxyluciferin) taking into account the nuclear dynamics of the system, i.e., the inhomogeneous broadening due to the fluctuation of the structure and the environment. The homogeneous broadening due to the existence of a continuous set of vibrational sublevels in each electronic state is partly taken into account at the classical level (the force field allows bonds and angles to vibrate). Attempts to look more closely at the homogeneous broadening can be seen in reference (52). A 20 ns MD simulation of oxyluciferin (in its excited state) inside the water-solvated protein cavity and in the presence of AMP is performed. The parameters for the force field of oxyluciferin should correspond to the excited state. The first set of parameters with the reference bonds, angles, torsions, and charges obtained by optimization of oxyluciferin in the first excited state in vacuum is used. A MD simulation is performed with this first set of parameters and a snapshot is extracted and optimized at the QM/MM level to set new parameters. This process can be made iteratively until properties converge. In practice, one iteration leads to the convergence of emission spectra shape (24). A final 20 ns MD is performed, from which 100 snapshots are extracted. For each snapshot, the energy and oscillator strength of the transition from the first excited state to the ground state are computed at the QM/MM level. Homogeneous broadening is not calculated in this case and is added phenomenologically. The 100 energies are convoluted with a Gaussian function using a half-width of 0.2 eV. It should be noted here that in the study of Ref. (24) the emission spectra were simulated considering the oscillator strength with the Gaussian convolution to get the height of the band. Care should be taken to compare the simulated spectra with the experimental ones, having in mind that oscillator strengths, f, are proportional to the Einstein coefficient A12 (57), and the experimental spectra are recorded against the wavelengths and the correspondence between the spectra against wavelength and frequencies is not trivial (58).

15.7

Conclusion

In this chapter, we have presented how computational studies can help with the understanding of bioluminescent systems. Simulations have the advantage to study a concrete system that is difficult to isolate experimentally, testing the hypothesis. There are still limitations due to the fact that bioluminescent systems are relatively big. Modeling of the color modulation of the emitted light relies on the starting crystallographic structures and the sampling of different conformations of the protein, hence relying on the experimental data. Protein mutations can cause huge changes in protein conformation, and this should be taken into account in the modeling of the phenomena. Experimental-theoretical collaboration remains the driving force of scientific progress in the field of bioluminescence.

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e, N.; Liu, Y.-J.; Navizet, I.; Roca-Sanjua´n, D.; Baader, W. J.; Lindh, R. Chemi- and Bioluminescence of Cyclic Peroxides. Chem. Rev. 2018, 118 (15), 6927–6974. 5. Shimomura, O.; Johnson, F. H. Partial Purification and Properties of the Chaetopterus Luminescence System. In Bioluminescence in Progress; Princeton University Press, 2015; pp. 495–522. 6. Kanakubo, A.; Isobe, M. Isolation of Brominated Quinones Showing Chemiluminescence Activity from Luminous Acorn Worm, Ptychodera Flava. Bioorg. Med. Chem. 2005, 13 (8), 2741–2747. 7. Rudie, N. G.; Mulkerrin, M. G.; Wampler, J. E. Earthworm Bioluminescence: Characterization of High Specific Activity Diplocardia Longa Luciferase and the Reaction It Catalyzes. Biochemistry 1981, 20 (2), 344–350. 8. Carrasco-Lo´pez, C.; Lui, N. M.; Schramm, S.; Naumov, P. The Elusive Relationship between Structure and Colour Emission in Beetle Luciferases. Nat. Rev. Chem. 2021, 5 (1), 4–20. 9. Nakatsu, T.; Ichiyama, S.; Hiratake, J.; Saldanha, A.; Kobashi, N.; Sakata, K.; Kato, H. Structural Basis for the Spectral Difference in Luciferase Bioluminescence. Nature 2006, 440 (7082), 372–376. 10. Al-Handawi, M. B.; Polavaram, S.; Kurlevskaya, A.; Commins, P.; Schramm, S.; Carrasco-Lo´pez, C.; Lui, N. M.; Solntsev, K. M.; Laptenok, S. P.; Navizet, I.; Naumov, P. Spectrochemistry of Firefly Bioluminescence. Chem. Rev. 2022. 11. Tomasi, J.; Mennucci, B.; Cammi, R. Quantum Mechanical Continuum Solvation Models. Chem. Rev. 2005, 105 (8), 2999–3094. 12. Tang, K.; Zhang, J.; Liang, J. Fast Protein Loop Sampling and Structure Prediction Using Distance-Guided Sequential Chain-Growth Monte Carlo Method. PLoS Comput. Biol. 2014, 10 (4), e1003539. 13. Roy, A.; Kucukural, A.; Zhang, Y. I-TASSER: A Unified Platform for Automated Protein Structure and Function Prediction. Nat. Protoc. 2010, 5 (4), 725–738.

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Chapter 16

Photocatalysis Xin-Ping Wu, Ming-Yu Yang, Zi-Jian Zhou, Zhao-Xue Luan, Lin Zhao, and Yi-Chun Chu Key Laboratory for Advanced Materials and Joint International Research Laboratory for Precision Chemistry and Molecular Engineering, Feringa Nobel Prize Scientist Joint Research Center, Centre for Computational Chemistry and Research Institute of Industrial Catalysis, School of Chemistry and Molecular Engineering, East China University of Science and Technology, Shanghai, People’s Republic of China

Chapter outline 16.1 Introduction and historical overview 16.2 Fundamental mechanism of heterogeneous photocatalysis 16.2.1 Light absorption and photoexcitation 16.2.2 Charge migration and recombination 16.2.3 Photoredox reactions 16.3 Brief overview of computational methodologies 16.3.1 Kohn-Sham density functional theory (KS-DFT) 16.3.2 Multireference and multiconfigurational methods 16.3.3 Combined quantum mechanical and molecular mechanical (QM/MM) methods

16.1

387 388 388 389 389 390 390 391

16.4 Computational studies 16.4.1 TiO2 16.4.2 ZnO 16.4.3 MoS2 16.4.4 UiO-66 16.4.5 PCN-601 16.4.6 g-C3N4 16.5 Outlook Acknowledgments References

392 392 395 397 399 401 402 404 405 405

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Introduction and historical overview

IUPAC defined a catalyst as “a substance that increases the rate of a reaction without modifying the overall standard Gibbs energy change in the reaction,” (1) and the corresponding process is called catalysis. As a particular type of catalysis, photocatalysis refers to the process carried out under the action of light energy. In such a (photocatalytic) process, a chemical reaction is driven by a catalyst (the so-called photocatalyst) that is activated by light and has excited electrons and holes. The earliest mention of the term photocatalysis (photokatalyse) dates back to the textbook on photochemistry (photochemie) written by Plotnikov in 1910 (2). Subsequently, two studies on the photolysis of organic substances in the presence of uranyl (UO+2 ) salts also adopted the term photocatalysis (3,4). Almost two decades later, in 1929, Keidel studied the effect of rutile TiO2 on the light-induced aging of paints and fabrics (5). In 1938, Goodeve and Kitchener investigated TiO2 as a photosensitizer for dye bleaching with oxygen and, for the first time, introduced a few fundamental and central concepts in heterogeneous photocatalysis (6). Nevertheless, in the early years, photocatalysis was discovered to present, in some sense, a negative effect on the rate of chemical reactions. A breakthrough in photocatalysis was made by Honda and Fujishima in 1972 (7); they discovered that electrochemical photolysis of water can be achieved using an electrochemical cell constructed with a TiO2 electrode and a platinum electrode under light irradiation; it is widely accepted that this work led the modern developments in photocatalysis, and since then, significant research efforts have been devoted to finding potential photocatalysts (8–13). Among different types of photocatalysts, metal oxides are probably the most widely studied photocatalysts. TiO2 and ZnO stand out among a bunch of oxide photocatalysts for their superior photocatalytic activities for various reactions such as water splitting (14,15) and CO2 reduction (16,17). MoS2 is an emerging sulfide photocatalyst; this material shows layerdependent properties, and two-dimensional (2D) MoS2 with single or few layers shows great promise for photocatalysis (18,19). UiO-66 and PCN-601 are two representative metal-organic framework (MOF) photocatalysts with appealing features such as broad tunability and excellent stability (20–22). In recent years, graphitic carbon nitride (g-C3N4), which is composed of earth-abundant elements, has drawn considerable attention as a metal-free photocatalyst due to its fascinating electronic structure, high stability, and easy fabrication process (12,23,24). Theoretical and Computational Photochemistry. https://doi.org/10.1016/B978-0-323-91738-4.00009-9 Copyright © 2023 Elsevier Inc. All rights reserved.

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There are many other appealing heterogeneous photocatalysts as well as huge amounts of homogeneous photocatalysts that have been reported in the literature (25–27). Nevertheless, in this chapter, we mainly focus on the heterogeneous photocatalysts mentioned above, i.e., TiO2, ZnO, MoS2, UiO-66, PCN-601, and g-C3N4. The selected photocatalysts are discussed mainly from the computational point of view. In addition to the review of computational investigations, we also present the fundamental mechanism of heterogeneous photocatalysis and give an overview of the computational methodologies.

16.2

Fundamental mechanism of heterogeneous photocatalysis

Although photocatalytic mechanisms of different heterogeneous photocatalysts may differ in many details, all heterogeneous photocatalysts generally share the same fundamental mechanism which is composed of three basic processes: (i) light absorption and photoexcitation, (ii) charge migration and recombination, and (iii) photoredox reactions (Fig. 16.1) (8,28). Hence, these basic processes in heterogeneous photocatalysis are reviewed first, such that readers who are not familiar with the topic could have a general idea.

16.2.1 Light absorption and photoexcitation The initial stage of photocatalysis is photoexcitation, in which a photocatalyst produces excited electrons and holes by absorption of light (more precisely, photons). This process occurs only when the energy carried by the photon exactly matches the energy difference between two electronic levels. Then, the energy of the photon is likely to pump the electron at the lower level to the higher level, generating an excited electron at the higher level and leaving a hole at the lower level. The probability of photon absorption in an electronic transition can be expressed by a quantity, i.e., the oscillator strength (of the transition), which is closely related to the transition dipole moment. Note that multiphoton absorption, in which two or more photons are absorbed simultaneously, can occur as well; however, this is a rare event in photocatalysis under sunlight irradiation, and high optical intensities are generally required to increase the probability of such event (29–31). Photocatalysis driven by natural sunlight, which is a renewable energy source, is appealing. Around 40 percent of solar energy reaching the surface of the Earth is in the visible range which spans from approximately 400 to 700 nm. Therefore, in

FIG. 16.1 The basic processes in heterogeneous photocatalysis. CB and VB refer to the conduction band and the valence band, respectively.

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the photocatalysis community, it is often desirable to develop visible-light-response photocatalysts, and the (optical) band gap of a photocatalyst is of great concern (8,32–34). In this context, a narrow band gap is desirable. However, photocatalysts with very narrow band gaps could be inefficient in driving photocatalytic reactions (e.g., water splitting); this will be further discussed in Section 16.2.3.

16.2.2 Charge migration and recombination Photogenerated charge carriers (electrons and holes) are usually mobile. Migration of charge carriers is often essential; for example, charge carriers generated inside the bulk of a photocatalyst have to migrate to the catalyst surface to drive photocatalytic reactions. In practice, only a small amount of charge carriers can reach the photocatalytic active sites successfully due to the common charge recombination process, by which electrons and holes are annihilated (35,36). Charge recombination can either be radiative or nonradiative; the former decay channel is accompanied by photon emission, whereas in the latter decay channel, energy is released through phonon emission (28,37–41). Frequent charge recombination results in frustrating quantum efficiencies of most reported photocatalysts (42–45); the term quantum efficiency (more precisely, internal quantum efficiency) is defined as a ratio of the actual number of charge carriers reacted to the number of absorbed photons (46). Therefore, many photocatalytic studies have attempted to find out effective ways to suppress charge recombination (9,47–49). Controlled migration of electrons and holes into different regions of a photocatalyst is a promising approach as it causes spatial separation of charge carriers (48,50,51).

16.2.3 Photoredox reactions Photoredox reactions generally occur on the catalyst surface and are driven by charge carriers; more specifically, photoreduction reactions are driven by electrons, whereas holes can drive photooxidation reactions (8,50,52,53). It should be noted that only the charge carriers with sufficiently long lifetimes have opportunities to drive photoredox reactions before charge recombination (54,55). In addition to lifetime, the energies of the levels that the charge carriers reside in are crucial as well. In principle, to drive the photoreduction, the energy of the conduction band (CB) minimum (CBM) should be higher than the energy level of the corresponding reaction, whereas the photooxidation reaction can be driven when the energy of the valence band (VB) maximum (VBM) is lower than the energy level of that reaction (Fig. 16.2) (8,50). For example, water splitting, which is one of the most widely studied photocatalytic reactions, has two half-reactions, i.e., the hydrogen evolution reaction (HER): 2H+(aq) + 2e $ H2(g), and the oxygen evolution reaction (OER): H2O $ 2H+(aq) + ½O2(g) + 2e . To achieve overall water splitting, the CBM and the VBM of a photocatalyst should straddle the HER and the OER potentials. Note that the HER and the OER have the standard electrode potentials of 0 and 1.23 V, respectively (56), indicating that the energy difference of the HER and the OER levels is 1.23 eV and the minimum band gap requirement for overall water splitting is 1.23 eV. In practice, a minimum over-potential of 0.25 eV is required to ensure reasonable efficiency (57). Hence, photocatalysts with very narrow band gaps (