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Charge and Energy Transfer Dynamics in Molecular Systems
Comprehensive resource offering knowledge on charge and energy transfer dynamics in molecular systems and nanostructures
Charge and Energy Transfer Dynamics in Molecular Systems provides a unified description of different charge and energy transfer phenomena in molecular systems with emphasis on the theory, bridging the regimes of coherent and dissipative dynamics and thus presenting classic rate theories as well as modern treatments of ultrafast phenomena. Starting from microscopic models, the common features of the different transfer processes are highlighted, along with applications ranging from vibrational energy flow in large polyatomic molecules, the motion of protons in solution, up to the concerted dynamics of electronic and nuclear degrees of freedom in molecules and molecular aggregates.
The newly revised and updated Fourth Edition contains a more detailed coverage of recent developments in density matrix theory, mixed quantum-classical methods for dynamics simulations, and a substantially expanded treatment of time-resolved spectroscopy.
The book is written in an easy-to-follow style, including detailed mathematical derivations, thus making even complex concepts understandable and applicable.
Charge and Energy Transfer Dynamics in Molecular Systems includes information on:
- Electronic and vibrational molecular states, covering molecular Schrödinger equation, Born—Oppenheimer separation and approximation, Hartree-Fock equations and other electronic structure methods
- Dynamics of isolated and open quantum systems, covering multidimensional wave packet dynamics, and different variants of density operator equations
- Interaction of molecular systems with radiation fields, covering linear and nonlinear optical response using the correlation function approach
- Intramolecular electronic transitions, covering optical transition and internal conversion processes
- Transfer processes of electrons, protons, and electronic excitation energy
Providing in-depth coverage of the subject, Charge and Energy Transfer Dynamics in Molecular Systems is an essential resource for anyone working on timely problems of energy and charge transfer in physics, chemistry and biophysics as well as for all engaged in nanoscience and organic electronics.
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Veröffentlichungsjahr: 2023
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Table of Contents
Cover
Title Page
Copyright
Preface to the Fourth Edition
Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
1 Introduction
2 Electronic and Vibrational Molecular States
2.1 Introduction
2.2 Molecular Schrödinger Equation
2.3 Born–Oppenheimer Separation
2.4 Electronic Structure Methods
2.5 Potential Energy Surfaces
2.6 Adiabatic versus Diabatic Representation of the Molecular Hamiltonian
2.7 Condensed‐phase Approaches
2.8 Supplement
References
Further Reading
Notes
3 Dynamics of Isolated and Open Quantum Systems
3.1 Introduction
3.2 Time‐dependent Schrödinger Equation
3.3 The Golden Rule of Quantum Mechanics
3.4 The Nonequilibrium Statistical Operator and the Density Matrix
3.5 The Reduced Density Operator and the Reduced Density Matrix
3.6 Quantum Master Equation
3.7 The Reservoir Correlation Function
3.8 Reduced Density Matrix in Energy Representation
3.9 Coordinate and Wigner Representation of the Reduced Density Matrix
3.10 The Path Integral Representation of the Density Matrix
3.11 Hierarchy Equations of Motion Approach
3.12 Coherent to Dissipative Dynamics of a Two‐level System
3.13 Trajectory‐based Methods
3.14 Generalized Rate Equations: The Liouville Space Approach
3.15 Supplement
References
Further Reading
Notes
4 Interaction of Molecular Systems with Radiation Fields
4.1 Introduction
4.2 Absorption of Light
4.3 Nonlinear Optical Response
4.4 Field Quantization and Spontaneous Emission of Light
References
Further Reading
Notes
5 Vibrational Dynamics: Energy Redistribution, Relaxation, and Dephasing
5.1 Introduction
5.2 Intramolecular Vibrational Energy Redistribution
5.3 Intermolecular Vibrational Energy Relaxation
5.4 Polyatomic Molecules in Solution
5.5 Quantum–Classical Approaches to Relaxation and Dephasing
References
Further Reading
Notes
6 Intramolecular Electronic Transitions
6.1 Introduction
6.2 The Optical Absorption Coefficient
6.3 Absorption Coefficient and Dipole–Dipole Correlation Function
6.4 The Emission Spectrum
6.5 Optical Preparation of an Excited Electronic State
6.6 Internal Conversion Dynamics
6.7 Supplement
References
Further Reading
Notes
7 Electron Transfer
7.1 Classification of Electron Transfer Reactions
7.2 Theoretical Models for Electron Transfer Systems
7.3 Regimes of Electron Transfer
7.4 Nonadiabatic Electron Transfer in a Donor–Acceptor Complex
7.5 Bridge‐Mediated Electron Transfer
7.6 Nonequilibrium Quantum Statistical Description of Electron Transfer
7.7 Heterogeneous Electron Transfer
7.8 Charge Transmission Through Single Molecules
7.9 Photoinduced Ultrafast Electron Transfer
7.10 Supplement
References
Further Reading
Notes
8 Proton Transfer
8.1 Introduction
8.2 Proton Transfer Hamiltonian
8.3 Adiabatic Proton Transfer
8.4 Nonadiabatic Proton Transfer
8.5 The Intermediate Regime: From Quantum to Quantum–Classical Hybrid Methods
8.6 Proton‐coupled Electron Transfer
References
Further Reading
Notes
9 Excitation Energy Transfer
9.1 Introduction
9.2 The Aggregate Hamiltonian
9.3 Exciton–Vibrational Interaction
9.4 Regimes of Excitation Energy Transfer
9.5 Transfer Dynamics in the Case of Weak Excitonic Coupling: Förster Theory
9.6 Transfer Dynamics in the Case of Strong Excitonic Coupling
9.7 Optical Properties of Aggregates
9.8 Excitation Energy Transfer Including Charge‐transfer States
9.9 Exciton–Exciton Annihilation
9.10 Supplement
References
Further Reading
Notes
Index
End User License Agreement
List of Tables
Chapter 9
Table 9.1 Classification of the Coulomb interaction matrix elements in Eq. (...
Table 9.2 Förster radii for typical biological donor–acceptor systems.
List of Illustrations
Chapter 1
Figure 1.1 The problem of the interaction between electrons and nuclei is tr...
Figure 1.2 The total system consisting of a relevant system () interactin...
Figure 1.3 Mixed quantum–classical description of condensed phase dynamics. ...
Figure 1.4 Scheme of molecular potential energy surfaces including the level...
Figure 1.5 Hydrogen bonding, which governs the
proton transfer
(
PT
) dynamics...
Figure 1.6
Excitation energy transfer
(
EET
), which occurs after optical prep...
Chapter 2
Figure 2.1 Potential energy curves for different adiabatic electronic st...
Figure 2.2 Orbital diagram for water calculated using Hartree–Fock theory (t...
Figure 2.3 Schematic view of a typical potential energy curve of a diatomic ...
Figure 2.4 Schematic view of a potential energy curve typical for isomerizat...
Figure 2.5 Schematic view of typical ground and excited state potential ener...
Figure 2.6 The displacement vectors for the three normal modes of water. The...
Figure 2.7 Harmonic oscillator potential together with the eigenfunctions fo...
Figure 2.8 Shifted harmonic oscillator potential surfaces for two electronic...
Figure 2.9 (a) Schematic plot of a two‐dimensional PES. The coordinate i...
Figure 2.10 Two‐dimensional Cartesian reaction plane for the hydrogen atom t...
Figure 2.11 Potential energy curves for the ground and the lowest excited st...
Figure 2.12 (a) Diabatic (dashed) and adiabatic (solid) potential energy cur...
Figure 2.13 (a) Two intersection diabatic harmonic PESs along the so‐called ...
Figure 2.14 (a) Dipole moment of O (left). (b) Macroscopic electrostatic ...
Figure 2.15 Molecule (doubly‐charged glyphosate) and cavity within a dielect...
Figure 2.16 Potential energy surface along a reaction coordinate describing ...
Chapter 3
Figure 3.1 (a) Schematic view of a typical situation encountered in condense...
Figure 3.2 Survival probability for a system with eigenstates (). The ene...
Figure 3.3 Coupling of the single‐state to the manifold of states as des...
Figure 3.4 Coupling of the manifold of initial states to the manifold of f...
Figure 3.5 Ohmic spectral density with cut‐off (dashed line), Eq. (3.301), ...
Figure 3.6 Transitions among five different quantum states of the relevant...
Figure 3.7 Visualization of different time‐sliced paths leading from to ...
Figure 3.8 Transition amplitude following from Eq. (3.414) for different (...
Figure 3.9 Dissipative dynamics in a coupled two‐level system as obtained fr...
Figure 3.10 Dissipative dynamics in a coupled two‐level system as obtained f...
Figure 3.11 Surface hopping (FSSH) versus exact description of wave packet r...
Figure 3.12 Graphical scheme for second‐order rate computation. The diagonal...
Figure 3.13 Graphical scheme for fourth‐order rate computation. The three di...
Figure 3.14 Three different pathways in the graphical scheme of the fourth‐o...
Chapter 4
Figure 4.1 Scheme of a three‐pulse experiment with the different pulses wi...
Figure 4.2 Scheme of a pump–probe experiment with the pump and probe pulse b...
Figure 4.3 Schematic view of different scenarios of two‐dimensional spectros...
Chapter 5
Figure 5.1 Pump–probe signal corresponding to the ion yield after excitati...
Figure 5.2 Pump–probe spectroscopy of the vibrational dynamics of a diatomic...
Figure 5.3 Mixing of zeroth‐order vibrational states due to a Fermi resonanc...
Figure 5.4 Preparation of bright (zeroth‐order) states (for example, by lase...
Figure 5.5 Hierarchical structure of IVR as described by the tier model. A s...
Figure 5.6 IVR in a polyatomic system. (a)–(c) The IVR process in a three‐di...
Figure 5.7 Photodissociation of a diatomic molecule () in a rare gas lattic...
Figure 5.8 Instantaneous normal‐mode density of states for an cluster at 1...
Figure 5.9 Spectral densities obtained using GLE‐based simulation according ...
Figure 5.10 Molecular dynamics simulations of HgI in ethanol solution. (a) C...
Figure 5.11 Schematic view of vibrational relaxation out of the harmonic osc...
Figure 5.12 Infrared pump–probe signal showing the vibrational relaxation of...
Figure 5.13 Multiquantum relaxation processes in a two‐level system. Downwar...
Figure 5.14 Solvent‐assisted vibrational energy cascading after excitation o...
Figure 5.15 (a) Fluctuations of the fundamental transition frequency of the ...
Figure 5.16 (a) IR absorption spectrum of N–HN and N–HO vibrations in an a...
Chapter 6
Figure 6.1 Ground and excited state PESs of a diatomic molecule versus bond...
Figure 6.2 Pump–probe spectroscopy of wave packet dynamics. (a) Wave packet...
Figure 6.3 Steady‐state absorption and fluorescence spectrum (a) of a peryle...
Figure 6.4 Internal conversion of the population of electronic levels with e...
Figure 6.5 Description of optical absorption as a curve‐crossing problem wit...
Figure 6.6 Stick spectrum of the absorption described by Eq. (6.31). The wei...
Figure 6.7 Absorption (solid line) and emission (dashed line) spectrum (in a...
Figure 6.8 From wave packet motion to the absorption spectrum of a Morse os...
Figure 6.9 Numerical results for the absorption spectrum of FNO obtained u...
Figure 6.10 Linear absorption spectrum for a curve crossing system (states
Figure 6.11 Linear absorption coefficient for an MBO model, Eqs. (6.57) and ...
Figure 6.12 Population of an excited state PES via an ultrashort laser pulse...
Figure 6.13 Experimental verification of the energy gap law, Eq. (6.119). Th...
Figure 6.14 Ultrafast internal conversion dynamics in a three electronic sta...
Chapter 7
Figure 7.1 ET from sodium to benzyl halide resulting in bond breaking and be...
Figure 7.2 Photoinduced ET from triethylamine (TEA) to IrPS ([Ir(ppy)(bpy)]
Figure 7.3 Chromophores of the photosynthetic bacterial reaction center (sid...
Figure 7.4 (a) ET in a schematically drawn DA complex. The initial and final...
Figure 7.5 ET reaction of an excess electron in a HOMO–LUMO scheme of a DA c...
Figure 7.6 Photoinduced ET (a) and hole transfer (b) reaction in a HOMO–LUMO...
Figure 7.7 Porphyrin (D)–fullerene (A) dyads with a phenyl (a) and an additi...
Figure 7.8 Bridge‐mediated ET between a donor and an acceptor level connecte...
Figure 7.9 (a–c) Possible HET reactions between a molecule represented in a ...
Figure 7.10 Ultrafast HET between an Ru(2,2′‐bipyridyl‐4,4′‐dicarboxylic aci...
Figure 7.11 Single‐molecule transistor including different charging states o...
Figure 7.12 One‐dimensional sketch of the pseudopotential , introduced in E...
Figure 7.13 PES of the DA complex according to Eq. (7.25) versus a single no...
Figure 7.14 Donor and acceptor PESs versus a single reaction coordinate. The...
Figure 7.15 Ultrafast ET in a system of two coupled PESs with donor, , and ...
Figure 7.16 The coupled PES of a DA complex versus a single reaction coordin...
Figure 7.17 Schematic representation of the different ET regions. The horizo...
Figure 7.18 Potential energy surfaces for a DA complex in harmonic approxima...
Figure 7.19 The normal region (a), the activationless case (b), and the inve...
Figure 7.20 ET rate versus driving force of the reaction for a DA complex sh...
Figure 7.21 PES for the case of independent vibrational coordinates of the d...
Figure 7.22 PESs for the ET in the case of a single high‐frequency intramole...
Figure 7.23 Bridge‐mediated ET using a molecular wire of
p
‐phenylenevinylene...
Figure 7.24 Bridge‐mediated ET between a donor and an acceptor level. (a) A ...
Figure 7.25 Bridge‐mediated ET between a donor and an acceptor level. (a) ET...
Figure 7.26 Length dependence of the overall ET rate at room temperature for...
Figure 7.27 HET between perylene attached by different bridge anchor groups ...
Figure 7.28 (a) IV characteristics of the single‐molecule transistor of Figu...
Figure 7.29 Charge transmission through a single molecule represented in a H...
Figure 7.30 Vibrational contributions in the IV characteristics of a single ...
Figure 7.31 Vibrational contributions to the charge transmission through a b...
Figure 7.32 Electronic ground‐state PES as well as donor and acceptor diabat...
Figure 7.33 Probability distribution of the vibrational coordinate, Eq. (7.2...
Figure 7.34 Population of the acceptor in a system of a coupled donor and ...
Figure 7.35 Photoinduced ET in a molecular triad (a). Upon excitation of the...
Chapter 8
Figure 8.1 (a) Single PT in malonaldehyde, which is one of the standard exam...
Figure 8.2 Protonated water networks in bacteriorhodopsin encompass differen...
Figure 8.3 Coherent oscillations in a hydrogen bond after ultrashort IR puls...
Figure 8.4 Two‐dimensional IR spectroscopy monitoring PT between water and h...
Figure 8.5 Infrared transient transmission change due to stimulated emission...
Figure 8.6 (a) Potential energy profile along a PT reaction coordinate, for ...
Figure 8.7 Infrared absorption spectra show clear signatures of hydrogen bon...
Figure 8.8 Empirical correlation between the N–H stretching frequency and th...
Figure 8.9 Schematic view of two‐dimensional PES for linear (a) and quadrati...
Figure 8.10 One‐dimensional potential energy curve along a PT coordinate w...
Figure 8.11 PES and eigenfunctions of (in‐plane) PT in 3,7‐dichlorotropolone...
Figure 8.12 Quantum–classical hybrid treatment of the hydride transfer rea...
Figure 8.13 Schematic view of the potential energy curve for PT in the adiab...
Figure 8.14 The probability for the proton to be in the reactant configurati...
Figure 8.15 Wave packet dynamics of a seven‐dimensional quantum model mimick...
Figure 8.16 Potential energy curves for nonadiabatic PCET according to Eq. (...
Chapter 9
Figure 9.1 Excitation energy transfer between an energy donor molecule D and...
Figure 9.2 Excitation energy transfer between an energy donor molecule D and...
Figure 9.3 Cylindrical J‐aggregate of an amphiphilic dye molecule. (a) Monom...
Figure 9.4 (a) Time scales of energy flow and charge separation in the photo...
Figure 9.5 Fluorescence resonance energy transfer (FRET) between a CdSe/ZnS ...
Figure 9.6 Schematic illustration of exciton motion in a chromophore complex...
Figure 9.7 Transition density (Eq. (9.18)) for the to transition of pe...
Figure 9.8 (a) Rylene diimide dyad with a
perylene diimide
(
PDI
) donor and a...
Figure 9.9 Schematic illustration of the presence of a singly excited state ...
Figure 9.10 (a) PESs according to the model of Eq. (9.89) for a dimer with o...
Figure 9.11 Schematic representation of different EET regimes. The strength ...
Figure 9.12 EET in a DA pair. (a) Both molecules are represented by electron...
Figure 9.13 EET in a donor (PDI) acceptor (TDI) pair (chemical structure – (...
Figure 9.14 Combined DOS for DA EET (). Solid lines: , Eq. 9.132; dashed l...
Figure 9.15 Transfer rates for a DA heterodimer ( cm) coupled to a reservo...
Figure 9.16 Dissipative dynamics in a regular chain‐like aggregate of seven ...
Figure 9.17 Dissipative exciton dynamics according to Eq. (9.167) for a line...
Figure 9.18 EET in the FMO complex (cf. Figure 9.4). (a) One‐exciton eigenen...
Figure 9.19 Dependence of the amplitudes for transitions between the ground ...
Figure 9.20 Molecular structure of the dye TDBC (5,5,6,6‐tetrachloro‐1,1‐...
Figure 9.21 (a) Distribution of site energy shifts relative to an experiment...
Figure 9.22 Schematic of the exact (a), one‐particle (b), and two‐particle (...
Figure 9.23 Oscillator strengths for transitions from the electronic and vib...
Figure 9.24 HOMO–LUMO scheme of EET based on a two‐electron exchange via cha...
Figure 9.25 PES of the DA complex undergoing 2ET‐assisted EET. The present s...
Figure 9.26 Exciton dynamics and dissociation in a model of a polythiophene–...
Figure 9.27 Scheme of exciton–exciton annihilation in a molecular trimer wit...
Figure 9.28 Time dependence of exciton density (fraction of excited molecule...
Figure 9.29 Photon‐mediated EET in a DA complex. Both molecules are represen...
Figure 9.30 Transfer rates of photon‐mediated EET versus DA distance and for...
Figure 9.31 2ET‐assisted EET in a DA complex (identical molecules; use of th...
Guide
Cover Page
Title Page
Copyright
Preface to the Fourth Edition
Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
Table of Contents
Begin Reading
Index
Wiley End User License Agreement
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Charge and Energy Transfer Dynamics in Molecular Systems
Fourth Edition
Volkhard May and Oliver Kühn
Authors
Dr. Volkhard May
Humboldt-Universität zu Berlin
Institut für Physik, Newtonstraße 15
12489 Berlin
Germany
Prof. Dr. Oliver Kühn
Universität Rostock
Institut für Physik
Albert‐Einstein‐Str. 23‐24
18059 Rostock
Germany
Cover Image: Figure courtesy of Oliver Kühn
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Print ISBN: 978‐3‐527‐33978‐5
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Preface to the Fourth Edition
The third edition of this book has been published more than 10 years ago. During this time, the field of molecular charge‐ and energy‐transfer processes evolved tremendously. Therefore, we consider it timely to provide an update of our 2011 edition. Among the most important developments in quantum dissipative dynamics has been the further elaboration of nonperturbative and non‐Markovian approaches such as the hierarchy equation of motion (HEOM) method. Although already known at the time of the publication of the third edition, the past decade has witnessed not only numerous applications but also a broader development of methods being based on the hierarchy idea. For the common system‐reservoir models it provides a feasible numerical exact reference such that HEOM can be considered a game changer. The derivation of the HEOM and applications are covered in this fourth edition. In this context, the discussion of spectral density models has also been considerably enhanced. A second method that has matured to a versatile tool for studying various transfer processes is two‐dimensional spectroscopy. In the expanded Chapter 4, we provide an introduction into the formulation of two‐dimensional spectroscopy in terms of dipole correlation functions, knowing that a full account of this fascinating method is far beyond the scope of this book. Besides these two mentioned additions, we have included many minor modifications introducing up‐to‐date material in terms of both methodology and applications. We have also replaced some of the older examples from the literature by more recent ones.
Each chapter contains a section entitled “Further Reading”, which should serve as a starting point to explore the original literature. Additionally, at the end of each chapter, the reader will find a brief list of references pointing to the sources of the given examples and to the origins of those fundamental concepts that have been directly integrated into the text. As in previous editions, we emphasize that these lists are by no means exhaustive. It is not the purpose of this book to review all relevant literature on molecular charge and energy transfer dynamics.
Among the recent developments that are not covered in this book, we would like to mention the fields of Attosecond Physics and X‐ray Spectroscopy. While femtochemistry successfully explored nuclear dynamics over the past three decades, Attosecond Physics has the focus on the dynamics of electronic degrees of freedom on time scales where the nuclei are essentially frozen. In contrast to, for instance, Marcus theory of electron transfer where the electron motion is intimately connected to nuclear rearrangement, the driving forces of attosecond electron dynamics are different. They originate, for instance, in electron correlations or spin–orbit coupling. X‐ray science has a long tradition, but it is the availability of novel light sources offering, for instance, intense X‐ray flashes that enable time‐resolved studies of charge and energy transfer. In contrast to optical spectroscopy of valence transitions, core‐level excitation by X‐ray radiation is element specific, thus providing a complementary local view on electronic structure changes. The recent development in these two areas has already been impressive, but many exciting insights into the dynamics of charge‐ and energy‐transfer processes are yet to come. A number of concepts and methods introduced in this book, such as the correlation function description of transfer rates, can be adopted to serve these fields. However, in particular, phenomena due to the interaction of molecular systems with strong external fields require a different theoretical framework.
As with the previous editions, this book would not have been possible without the help and the many discussions with a number of students, postdocs, and colleagues. In particular, we would like to express our sincere thanks to A. A. Ahmed, O. S. Bokareva, S. I. Bokarev, F. Fennel, F. Gottwald, S. D. Ivanov, S. Karsten, X. Liu, S. Lochbrunner, Th. Plehn, P. A. Plötz, S. P. Polyutov, T. Pullerits, B. Röder, M. Schröter, J. Schulze, M. F. Shibl, J. Seibt, L. Wang, Y. Zelinskyy, Y. Zhang, and D. Ziemann.
The work on the manuscript of this fourth edition greatly benefited from the scientific atmosphere provided by the Rostock Collaborative Research Center Sfb 652 “Strong Correlations and Collective Effects in Radiation Fields” and Sfb 1477 “Light–Matter Interactions at Interfaces” and the Berlin Sfb 951 “Hybrid Inorganic/Organic Systems for Opto‐Electronics” funded by the German Research Foundation.
Volkhard May and Oliver KühnBerlin and Rostock, December 2022
Preface to the Third Edition
The continued interest in our book since its first publication in 2000 and its second edition in 2004 triggered the idea to prepare a third edition in order to account for more recent developments in the field of molecular charge‐ and energy‐transfer research.
Following the concept of the previous editions, we start by providing some general background on gas‐ and condensed‐phase interaction potentials and Hamiltonians, now including a discussion of quantum mechanics/molecular mechanics hybrid methods for the explicit treatment of condensed‐phase environments. The methodologically oriented Chapter 3 on the dynamics of quantum systems has been extended in several respects. Most notable is an exposition on the calculation of transfer rates within the Liouville space approach. Here, special emphasis is put on the fourth‐order rates that are crucial for an understanding of the more involved electron‐ and excitation energy‐transfer processes. Further, we give an account on the multiconfiguration time‐dependent Hartree method that in recent years has been proven to be a versatile tool for the numerically exact treatment of the quantum dynamics of thousands of degrees of freedom.
The introduction to basic theoretical concepts has been expanded by a new Chapter 4 devoted to some general aspects of the interaction between light and molecular systems. This comprises a derivation of the interaction Hamiltonian in dipole approximation, an introduction to field quantization for the description of emissions, and an outline of the basics of linear and nonlinear spectroscopy. In the second edition, laser control was covered in a separate chapter. For the present edition, we have incorporated a discussion of the topic into the text, which reflects the development of laser control into an almost routine tool for the investigation of molecular dynamics phenomena. The theoretical foundations and, in particular, optimal control theory are now part of Chapter 4, while the various applications are covered in Chapters 7–9.
The applications start with Chapter 5, where the discussion of vibrational dynamics has been expanded to include quantum‐classical approaches to the calculation of pure dephasing‐induced line broadening. Chapter 6, focusing on intramolecular electronic transitions, now contains a section on pump–probe spectroscopy and its role in interrogating molecular dynamics in the condensed phase. Major changes in Chapter 7 on electron transfer include the incorporation of heterogeneous electron transfer at surfaces as well as of single‐molecule electron transfer in the context of molecular electronics. The quantum dynamics treatment of proton‐transfer reactions has flourished recently due to the development of time‐dependent multiconfiguration approaches, as mentioned above; an example is discussed in Chapter 8. Finally, Chapter 9, on excitation energy (Frenkel exciton) transfer, has been substantially rewritten. Topics that have been added include Dexter transfer and two‐electron‐assisted as well as photon‐mediated exciton transfer.
The “Suggested Reading” section of the previous editions, which served to give a systematic starting point to explore the original literature, has been merged into the main text to become a list of “Further Reading” suggestions at the end of each chapter. As before, we would like to emphasize that these lists are by no means exhaustive; that is, it is not the purpose of this book to review all the relevant literature on the title subject.
While working on the manuscript of this third edition, we enjoyed the inspiring atmosphere of the Berlin Collaborative Research Center (Sfb450) “Analysis and Control of Ultrafast Photoinduced Reactions” and the Rostock Sfb652 “Strong Correlations and Collective Effects in Radiation Fields.”
Finally, we wish to thank E. Petrov (Bogolyubov Institute for Theoretical Physics, Kiev) and L. Wang (University of Science and Technology, Beijing) for reading parts of the new manuscript and K. Mishima (University of Tokyo) for drawing our attention to corrections required for the second edition.
Volkhard May and Oliver KühnBerlin and Rostock, October 2010
Preface to the Second Edition
The positive response to the First Edition of this text has encouraged us to prepare the present Revised and Enlarged Second Edition. All chapters have been expanded to include new examples and figures and also to cover more recent developments in the field. The reader of the First Edition will notice that many of the topics that were addressed in its “Concluding Remarks” section have now been integrated into the different chapters.
The introduction to dissipative quantum dynamics in Chapter 3 now gives a broader view on the subject. Particularly, we elaborated on the discussion of hybrid quantum‐classical techniques that promise to be able to incorporate microscopic information about the interaction of some quantum system with a classical bath beyond the weak coupling limit. In Chapter 4, we give a brief account on the state‐space approach to intramolecular vibrational energy and the models for treating the intermediate time scale dynamics, where the decay of the survival probability is nonexponential. Chapter 5 now compares different methodologies to compute the linear absorption spectrum of a molecule in a condensed‐phase environment. Furthermore, the basic aspects of nonlinear optical spectroscopy have been included to characterize a primary tool for the experimental investigation of molecular transfer processes. Bridge‐mediated electron transfer is now described in detail in Chapter 6 also including a number of new examples. Chapter 7 on proton transfer has been supplemented by a discussion of the tunneling splitting and its modification due to the strong coupling between the proton‐transfer coordinate and other intramolecular vibrational modes. Chapter 8 dealing with exciton dynamics has been considerably rearranged and includes now a discussion of two‐exciton states.
Finally, we have added a new Chapter 9, which introduces some of the fundamental concepts of laser field control of transfer processes. This is a rapidly developing field which is stimulated mostly by the possibility to generate ultrafast laser pulse of almost any shape and spectral content. Although there are only few studies on molecular transfer processes so far, this research field has an enormous potential not only for a more detailed investigation of the dynamics but also with respect to applications, for instance, in molecular‐based electronics.
Following the lines of the First Edition, we avoided to make extensive use of abbreviations. Nevertheless, the following abbreviations are occasionally used: DOF (degrees of freedom), ET (electron transfer), IVR (intramolecular vibrational redistribution), PES (potential energy surface), PT (proton transfer), QME (quantum master equation), RDM (reduced density matrix), RDO (reduced density operator), VER (vibrational energy relaxation), and XT (exciton transfer).
We have also expanded the “Suggested Reading” section which should give a systematic starting point to explore the original literature and also to become familiar with alternative views on the topics. Additionally, at the end of each chapter, the reader will find a brief list of references. Here, we included the information about the sources of the given examples and refer to the origin of those fundamental concepts and theoretical approaches that have been directly integrated into the text. We would like to emphasize, however, that these lists are by no means exhaustive. In fact, given the broad scope of this text, a complete list of references would have expanded the book's volume enormously, without necessarily serving its envisaged purpose.
It is our pleasure to express sincere thanks to the colleagues and students N. Boeijenga, B. Brüggemann, A. Kaiser, J. Manz, E. Petrov, and B. Schmidt, who read different parts of the manuscript and made various suggestions for an improvement. While working on the manuscript of this Second Edition, we enjoyed the inspiring atmosphere, many seminars, and colloquia held within the framework of the Berlin Collaborative Research Center (Sfb450) “Analysis and Control of Ultrafast Photoinduced Reactions.” This contributed essentially to our understanding of charge‐ and energy‐transfer phenomena in molecular systems. Finally, we would like to acknowledge the financial support from the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie (O. K.).
Volkhard May and Oliver KühnBerlin, September 2003
Preface to the First Edition
The investigation of the stationary and dynamical properties of molecular systems has a long history extending over the whole century. Considering the past decade only, one observes two tendencies: First, it became possible to study molecules on their natural scales, that is, with a spatial resolution of some Ångström () and on a time scale down to some femtoseconds (). And second, one is able to detect and manipulate the properties of single molecules. This progress comes along with a steadily growing number of theoretical and experimental efforts crossing the traditional borderlines between chemistry, biology, and physics. In particular, the study of molecular transfer processes involving the motion of electrons, protons, small molecules, and intramolecular excitation energy resulted in a deeper understanding of such diverse phenomena as the photoinduced dynamics in large molecules showing vibrational energy redistribution or conformational changes, the catalysis at surfaces, and the microscopic mechanisms of charge and energy transfer in biological systems. The latter are of considerable importance for unraveling the functionality of proteins and all related processes such as the primary steps of photosynthesis, the enzymatic activity, and the details of the repair mechanisms in DNA strands, to mention just a few examples. In a more general context, molecular electronics, that is, the storage and processing of information in molecular structures on a nanometer length scale, has also triggered enormous efforts. Finally, with the increasing sophistication of laser sources, first steps toward the control of chemical reaction dynamics have been taken.
The ever‐growing precision of the experiments requires on the theoretical side to have microscopic models for simulating the measured data. For example, the interpretation of optical spectroscopies in a time region of some tenths of femtoseconds demands for an appropriate simulation of the molecular dynamics for the considered system. Or, understanding the characteristics of the current flowing through a single molecule in the context of scanning tunneling microscopy needs detailed knowledge of the electronic level structure of the molecule as well as the role of its vibrational degrees of freedom. These few examples already demonstrate that advanced theoretical concepts and numerical simulation techniques are required, which are the combination of methods known from general quantum mechanics, quantum chemistry, molecular reaction dynamics, solid‐state theory, nonlinear optics, and nonequilibrium statistical physics.
Such a broad approach is usually beyond the theoretical education of chemists and biologists. On the other hand, quantum chemistry and chemical reaction dynamics are quite often not on the curriculum of physics students. We believe that this discrepancy quite naturally does not facilitate communication between scientists having different backgrounds. Therefore, it is one of the main intentions of the present book to provide a common language for bridging this gap.
The book starts with an introduction and general overview of different concepts in Chapter 1. The essentials of theoretical chemical physics are then covered in Chapter 2. For chemistry students this will be mostly a repetition of quantum chemistry and in particular the theory of electronic and vibrational spectra. It is by no means a complete introduction into this subject but intended to provide some background mainly for physics students. The prerequisites from theoretical physics for the description of dynamical phenomena in molecular systems are presented in Chapter 3. Here, we give a detailed discussion of some general aspects of the dynamics in open and closed quantum systems, focusing on transfer processes in the condensed phase.
The combination of qualitative arguments, simple rate equations, and the powerful formalism of the reduced statistical operator constitutes the backbone of the second part of the book. We start in Chapter 4 with a discussion of intramolecular transfer of vibrational energy that takes place in a given adiabatic electronic state. Here, we cover the limits of isolated large polyatomic molecules, small molecules in a matrix environment, up to polyatomics in solution. In Chapter 5, we then turn to processes that involve a transition between different electronic states. Special emphasis is put on the discussion of optical absorption, which is considered to be a reference example for more involved electron‐vibrational transfer phenomena such as internal conversion, which is also presented in this chapter. Chapter 6 then outlines the theoretical frame of electron‐transfer reactions, focusing mainly on intramolecular processes. Here, we develop the well‐known Marcus theory of electron transfer, describe nuclear tunneling and superexchange electron transfer, and discuss the influence of polar solvents. In Chapter 7, it will be shown that, even though proton transfer has many unique aspects, it can be described by adapting various concepts from electron‐transfer theory. The intermolecular excitation energy transfer in molecular aggregates is considered in Chapter 8. In particular, the motion of Frenkel excitons coupled to vibrational modes of the aggregate will be discussed. In the limit of ordinary rate equations, this leads us to the well‐known Förster expression for the transfer rate in terms of emission and absorption characteristics of the donor and acceptor molecules, respectively.
By presenting a variety of theoretical models that exist for different types of transfer processes on a common formal background, we hope that the underlying fundamental concepts are becoming visible. This insight may prepare the reader to take up one of the many challenging problems provided by this fascinating field of research. Some personal reflections on the current and possible future developments are given in Chapter 9.
The idea for writing this book emerged from lectures given by the authors at the Humboldt University Berlin, the Free University Berlin, and at the Johannes Gutenberg University Mainz during the past decade. These courses have been addressed to theoretically and experimentally oriented undergraduate and graduate students of Molecular Physics, Theoretical Chemistry, Physical Chemistry, and Biophysics, being interested in the fast developing field of transfer phenomena. The book is self‐contained and includes detailed derivations of the most important results. However, the reader is expected to be familiar with basic quantum mechanics. Most of the chapters contain a supplementary part where more involved derivations as well as special topics are presented. At the end of the main text, we also give some comments on selected literature, which should complement the study of this book.
Of course, this book would not have been possible without the help, the critical comments, and the fruitful discussions with many students and colleagues. In this respect, it is a pleasure for us to thank I. Barvik, N. P. Ernsting, W. Gans, L. González, O. Linden, H. Naundorf, J. Manz, S. Mukamel, A. E. Orel, T. Pullerits, R. Scheller, and D. Schirrmeister. We are also grateful for continuous financial support that has been provided by the Deutsche Forschungsgemeinschaft, in particular through the Sonderforschungsbereich 450 “Analysis and Control of Ultrafast Photoinduced Reactions.”
Volkhard May and Oliver KühnBerlin, September 1999
1Introduction
The understanding of transfer phenomena in molecular systems calls for a unified theoretical treatment that should have its foundation in a microscopic definition of the constituent parts and their interactions. There are three important questions that need to be answered in this regard. First, what is the appropriate theoretical description of the molecular system? Second, what is the form of the dynamical equations that describe the transfer process? And third, how can the computed results be related to experimental observations?
In what follows the term “molecular system” will cover single molecules and simple molecular aggregates as well as larger arrangements of molecules such as supramolecular complexes. In particular, molecules embedded in different types of environments will be of interest. Here, the scope ranges from molecules in solution to biological macromolecules such as membrane‐bound protein complexes. The common link between these molecular systems is that they show transfer processes. By “transfer process,” we understand the flow of vibrational energy and the dynamics of electrons, protons, and electronic excitation energy.
From a general point of view, quantum mechanics gives the framework for all phenomena occurring in molecular systems. Given the broad scope of transfer processes to be discussed, it is clear that an exact quantum mechanical treatment is impossible if we go beyond the level of simple model systems. Therefore, it is a particular challenge for theory to develop versatile models that provide answers to the initially raised three questions.
Chapter 2 addresses the first question discussing the steps that lead us from the formally exact to some approximate molecular Hamilton operator. Given a molecule in gas phase (vacuum) as shown in Figure 1.1a, the Born–Oppenheimer separation of nuclear and electronic motions can be performed. Here, the molecular wave function is split up into an electronic and a nuclear part, a procedure that is justified by the large mass difference between both types of particles. This results in a Schrödinger equation for the electronic wave function alone, for given fixed positions of the nuclei. Calculating the electronic energy spectrum for different positions of the nuclei, one obtains potential energy surfaces that govern the motion of the nuclei. These potential energy surfaces are at the heart of our understanding of stationary molecular spectra and molecular dynamics. If nuclear and electronic motion are adiabatically separable, that is if the coupling between different electronic states is negligible, one can carry out the Born–Oppenheimer approximation. Under certain conditions, however, the so‐called nonadiabatic transitions between different electronic states as a consequence of the nuclear motions take place.
Figure 1.1 The problem of the interaction between electrons and nuclei is transformed to some tractable level by employing the Born–Oppenheimer separation of their motions. (a) Three‐atomic molecule (O) with the electron density shown for the equilibrium distance (left) as well as for a stretched bond (right). The electron density adjusts instantaneously to the configuration of the nuclei. As a result, a potential energy curve is formed determining the dynamics of the bond distance coordinate. (b) If the molecule is taken from the gas into the condensed phase, its stationary and dynamic properties have to take into account the interaction with the surrounding molecules. This may give rise, for instance, to a change in equilibrium geometry and electron density (figure courtesy of Ashour Ahmed).
If we move from the gas to the condensed phase as shown in Figure 1.1b, the effect of the molecule–environment interaction has to be taken into account. The simplest way to do this is to add an additional external potential to the molecular Hamiltonian. Often, the environment can be described as a macroscopic dielectric, and its influence can be judged from its dielectric properties.
Having discussed the stationary molecular properties, we turn in Chapter 3 to the second question related to molecular dynamics. Here, the reader will become familiar with the concepts ranging from incoherent to coherent transfer events. The connection between these limits is provided by the relevant time scales; of particular importance is the relation between intramolecular relaxation and intermolecular transfer times. In view of experimental advances in ultrafast spectroscopy, our treatment reflects the historical evolution of knowledge about molecular dynamics from simple transfer rates to quantum mechanical wave packet dynamics.
An important ingredient for the theoretical modeling is the concept of an open molecular system interacting with its environment (reservoir) by collision processes or via other means of energy exchange. A schematic illustration of this situation is given in Figure 1.2. The relevant system may represent any type of molecule, but it may also comprise selected so‐called active degrees of freedom of a particular molecule.
Figure 1.2 The total system consisting of a relevant system () interacting with a reservoir () is completely described by the quantum‐statistical operator . By means of a reduction procedure, one can focus on the relevant system using the reduced statistical operator . Effects of the – interaction are still accounted for. In addition, the system may be influenced by external fields (wiggly line).
The most general description of the total system, plus , is given by the quantum‐statistical operator , as indicated in the left‐hand part of Figure 1.2. This operator is based on the concept of a mixed quantum state formed by and its macroscopic environment. However, the operator contains much more information than will ever be needed, for instance, to simulate a particular experiment. Indeed, it is the relevant system whose properties we are interested in. Making use of a reduction procedure, we obtain a reduced statistical operator that contains the information on the dynamics of only but including the influence of the environment (right‐hand part of Figure 1.2). When deriving equations of motion for the reduced statistical operator, the so‐called quantum master equations, a number of approximations have to be invoked. Most fundamental in this respect will be the assumption of a weak interaction between the system and the reservoir , which in practice requires a proper separation into relevant and environmental coordinates for the molecular system at hand. Under certain conditions, however, a numerical exact description of the dynamics of the relevant system becomes possible. If there is no interaction at all, the quantum master equation is equivalent to the time‐dependent Schrödinger equation. This is the regime of coherent dynamics. If the interaction is not negligible, however, the system dynamics gradually changes with increasing coupling strength from a partially coherent one to an incoherent one. The incoherent motion of a quantum system is commonly described using ordinary rate equations that are based on the Golden Rule rate expression of quantum mechanics.
The concept of the statistical operator provides a quantum‐statistical description of and. However, in many situations it is sufficient to describe by means of classical mechanics. Then, can be characterized by a wave function , and the dynamics of the environmental degrees of freedom is governed by Newton's equations. Often, the dynamics is split up in such a way that the classical particles move in the mean field of the quantum particle. This situation is visualized in Figure 1.3.
The overwhelming amount of data on transfer processes in molecular systems is obtained by spectroscopic techniques working in the infrared, the visible to ultraviolet, and, more recently, also in the X‐ray region. We will discuss the third question related to experimental observation mostly in the context of spectroscopy, with focus on the infrared to ultraviolet domain. As a means of preparation, Chapter 4 gives a brief account on the general theoretical concepts of the interaction of molecular systems with the electromagnetic radiation field. Further, a formulation of linear and nonlinear spectroscopy in terms of correlation functions will be introduced.
Figure 1.3 Mixed quantum–classical description of condensed phase dynamics. The classical particles move in the mean field generated by the quantum particle described by the wave function .
The general concepts presented in Chapters 2–4 are then applied to describe different transfer phenomena. In principle, transfer processes can be classified according to the type of transferred particle. In addition, one can distinguish between intra‐ and intermolecular particle transfer. The common frame is provided by the molecular Schrödinger equation together with the Born–Oppenheimer separation of electronic and nuclear motions as mentioned above.
The coupled nuclear dynamics in polyatomic molecules that might be immersed in some condensed phase environment is treated in Chapter 5. We show how an initially prepared vibrational state decays while its excitation energy is distributed over all possible environmental modes, as illustrated in the left‐hand part of Figure 1.4. For small polyatomic molecules, the energy flow out of the initial state is called intramolecular vibrational energy redistribution. For condensed phase situations, the dissipation of energy into the environment is called vibrational energy relaxation. In both cases, the transferred objects are the quanta of vibrational energy.
The preparation of the initial state can be due to an optical transition between two electronic states as a consequence of the interaction between the molecular system and an external electromagnetic field (cf. Figure 1.4). In Chapter 6, we discuss the processes of photon absorption and emission sketched in Figure 1.4. It will be shown that the coupled electron‐vibrational dynamics responsible for the absorption line shape can be described by a combined density of states that is the Fourier transform of some correlation function. This theoretical result will turn out to be quite general. In particular, we show that different types of transfer processes can be accommodated in such a framework. For example, the internal conversion dynamics of nonadiabatically coupled electronic states (right‐hand part of Figure 1.4) can, in the incoherent limit, be described by a combined density of states.
The external field interaction, on the other hand, provides the means for preparing nonequilibrium initial states that can act as a donor in a photoinduced electron‐transfer reaction, which is discussed in Chapter 7. The concerted electron‐vibrational dynamics accompanying electron‐transfer reactions can often be modeled in the so‐called curve‐crossing picture of two coupled potential energy surfaces representing two electronic states along a reaction coordinate (right‐hand part of Figure 1.4). Generalizations of this picture to larger molecular systems and to the case where the molecule is in contact with metal electrodes and a voltage is applied will also be discussed.
Figure 1.4 Scheme of molecular potential energy surfaces including the levels of the quantized motion of some reaction coordinate. After optical preparation of an electronically and vibrationally excited initial state (absorption), different transfer processes can occur. If the electronic state is not changed, but there is a coupling to some manifold of vibrational states, intramolecular vibrational energy redistribution (IVR) or vibrational energy relaxation (VER) can be observed. If there is some coupling to another electronic state, intramolecular internal conversion (IC), or electron transfer (ET) takes place. At the same time, one has VER as indicated by the wiggly lines. In addition, the system may return to the ground state by emitting a photon.
In contrast, the proton or hydrogen atom transfer investigated in Chapter 8 usually does not involve electronic transitions. In Figure 1.5, we have sketched a typical situation for intramolecular proton transfer that is realized as an isomerization reaction in the adiabatic electronic ground state. Since the proton has a rather small mass, tunneling processes may play an important role for proton transfer. The small mass ratio between the proton and the other heavy atoms provides the background for the introduction of a second Born–Oppenheimer separation. This will enable us to adapt most of the concepts of electron‐transfer theory to the case of proton transfer.
Figure 1.5 Hydrogen bonding, which governs the proton transfer (PT) dynamics, often leads to a double minimum potential along a reaction coordinate. The interaction between the proton and some environment may cause vibrational relaxation (wiggly lines).
Figure 1.6Excitation energy transfer (EET), which occurs after optical preparation of an electronically and vibrationally excited initial state (donor, left). The Coulomb interaction is responsible for deexcitation of the donor and excitation of the acceptor (right). The nuclear dynamics may be subject to relaxation processes (wiggly lines). Often, two independent nuclear (reaction) coordinates are used for the donor and the acceptor molecules.
In Chapter 9, we discuss excitation energy transfer or the so‐called exciton transfer in molecular aggregates as another example of coupled electron‐vibrational motion. In Figure 1.6, the mechanism of excitation energy transfer in the limit of localized excitations is shown. The donor (left) is initially excited, for example, by an external field. As a consequence of the Coulomb interaction, excitation energy is transferred between the excited molecule and some acceptor molecule (right). Often, donors and acceptors retain their chemical identity upon aggregation and, therefore, are usually described by different sets of nuclear (reaction) coordinates. In the incoherent limit, the rate of the process can be expressed in terms of an overlap integral between donor emission and acceptor absorption spectra. If the Coulomb interaction between different molecules becomes large enough, then excitation energy transfer has to be discussed by introducing quantum mechanical superposition states of all excited molecules, the so‐called Frenkel excitons. Their introduction gives a new view on excitation energy transfer via the motion of spatially delocalized states. A rigorous nonequilibrium quantum‐statistical model can describe both the incoherent and the coherent limits.
2Electronic and Vibrational Molecular States
This chapter provides the background material for the subsequent development of a microscopic description of charge‐ and energy‐transfer processes in molecular systems. After introducing the molecular Hamilton operator, we discuss the Born–Oppenheimer separation of electronic and nuclear motions as the key to the solution of the molecular Schrödinger equation. Next, the Hartree–Fock method, which is a simple yet very successful approach to the solution of the ground state electronic structure problem, is explained. It enables us to obtain, for instance the potential energy surface for nuclear motions. To prepare for the treatment of condensed‐phase situations, we further introduce the dielectric continuum model as a means for incorporating static solvent polarization effects into the electronic structure calculations.
The topology of the potential energy surface can be explored by calculating the first and second derivatives with respect to the nuclear coordinates. Of particular interest are the stationary points on a potential energy surface that may correspond to stable conformations of the molecule. In the vicinity of a local minimum it is often possible to analyze nuclear motions in terms of small amplitude normal mode vibrations. If one wants to model chemical reaction dynamics, however, the shape of the potential energy surface away from the stationary points is required as an input. We present two different approaches in this respect: the minimum energy reaction path and the Cartesian reaction surface model. Particularly, the latter will provide the microscopic justification for the generic Hamiltonians used later on to simulate molecular systems embedded in some environment. Finally, we discuss the diabatic and the adiabatic representations of the molecular Hamiltonian.
2.1 Introduction
The development of quantum theory in the 1920s was to a considerable extent triggered by the desire to understand the properties of atoms and molecules. It was soon appreciated that the Schrödinger equation together with the probabilistic interpretation of its solutions provides a powerful tool for tackling a variety of questions in physics and chemistry. The mathematical description of the hydrogen atom's spectral lines could be given and developed to a textbook example of the success of quantum mechanics. Stepping into the molecular realm, one faces a complicated many‐body problem involving the interactions of all electrons and all nuclei of the considered molecule. Its solution can be approached using the fact that nuclei and electrons have quite different masses, allowing their motion to be adiabatically separated. This concept was first introduced by Born and Oppenheimer in 1927. Within the Born–Oppenheimer adiabatic approximation, the simplest molecule, the hydrogen molecule ion, , can be treated.
From the electronic point of view, the appearance of one more electron, for instance in , necessitates the incorporation of the repulsive electronic interaction. Moreover, since one deals with two identical electrons, care has to be taken that the wave function has the proper symmetry with respect to an exchange of any two particle labels. In a straightforward way this is accomplished by the self‐consistent field method according to Hartree, Fock, and Slater. Despite its deficiencies Hartree–Fock theory has played an enormous role in the process of exploring the electronic structure of molecules. It still serves as the basis for many of the more advanced approaches used nowadays. In terms of practical applications to large systems, Density Functional Theory has emerged as the method of choice during the past decades.
