gertz likhtenshtein solar energy conversionthree important material classes of ceramics, polymers,...

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Gijsbertus de W

ithStructure, D

eformation, and

Integrity of Materials

1

This fi rst integrated approach to thermomechanics deals equally with the atomic scale, the mesoscale of microstructures and morphology, as well as the macroscopic level of actual components and workpieces for applications. With some 85 examples and 150 problems, it covers the three important material classes of ceramics, polymers, and metals in a didactic manner. The renowned author surveys mechanical material behavior at both the introductory and advanced level, providing read-ing incentive to both students as well as specialists in such disciplines as materials science, chemistry, physics, and mechanical engineering. Backed by fi ve appendices on symbols, abbreviations, data sheets, mate-rials properties, statistics, and a summary of contact mechanics.

Volume I: Fundamentals and ElasticityA. Overview � Constitutive BehaviourB. Basics � Mathematical Preliminaries � Kinematics � Kinetics � Thermodynamics � C, Q and S Mechanics � Structure and BondingC. Elasticity � Continuum Elasticity � Elasticity of Structures � Molecular Basis of Elasticity � Microstructural Aspects of Elasticity

Gijsbertus de With is full professor in materials science and head of the Laboratory of Materials and Interface Chemistry, Eindhoven University of Technology, the Netherlands. He graduated from Utrecht University and received his Ph.D in 1977 from the University of Twente on the structure and charge distribution of molecular crystals. His research inter-ests include the chemical and mechanical processing as well as the thermomechanical behaviour of materials. He is a member of the advisory board of the J. Eur. Ceram. Soc. and the recently founded Coating Science International Confer-ence.

Volume 1 of 2

Gertz Likhtenshtein

Solar Energy ConversionChemical Aspects

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Gertz Likhtenshtein

Solar Energy Conversion

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Gertz Likhtenshtein

Solar Energy Conversion

Chemical Aspects

The Author

Prof. Gertz LikhtenshteinDepartment of ChemistryBen Gurion UniversityP.O. Box 65384105 Beer-ShevaIsrael

All books published by Wiley-VCH are carefullyproduced. Nevertheless, authors, editors, and pub-lisher do not warrant the information contained inthese books, including this book, to be free of errors.Readers are advised to keep in mind that statements,data, illustrations, procedural details or other itemsmay inadvertently be inaccurate.

Library of Congress Card No.: applied for

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from theBritish Library.

Bibliographic information published bythe Deutsche NationalbibliothekThe Deutsche Nationalbibliothek lists this publica-tion in the Deutsche Nationalbibliografie; detailedbibliographic data are available on the Internet athttp://dnb.d-nb.de.

# 2012 Wiley-VCH Verlag & Co. KGaA,Boschstr. 12, 69469 Weinheim, Germany

All rights reserved (including those of translation intoother languages). No part of this book may be repro-duced in any form – by photoprinting, microfilm,or any other means – nor transmitted or translatedinto a machine language without written permissionfrom the publishers. Registered names, trademarks,etc. used in this book, even when not specificallymarked as such, are not to be considered unprotectedby law.

Composition Thomson Digital, Noida, IndiaPrinting and Binding Markono Print Media Pte Ltd,SingaporeCover Design Schulz Grafik-Design, Fußgönheim

Printed in SingaporePrinted on acid-free paper

Print ISBN: 978-3-527-32874-1

Contents

Preface IX

1 Electron Transfer Theories 11.1 Introduction 11.2 Theoretical Models 11.2.1 Basic Two States Models 11.2.1.1 Landau–Zener Model 11.2.1.2 Marcus Model 31.2.1.3 Electronic and Nuclear Quantum Mechanical Effects 51.2.2 Further Developments in the Marcus Model 71.2.2.1 Electron Coupling 71.2.2.2 Driving Force and Reorganization Energy 91.2.3 Zusman Model and its Development 171.2.4 Effect of Nonequilibrity on Driving Force and Reorganization 211.2.5 Long-Range Electron Transfer 241.2.6 Spin Effects on Charge Separation 281.2.7 Electron–Proton Transfer Coupling 291.2.8 Specificity of Electrochemical Electron Transfer 331.3 Concerted and Multielectron Processes 38

References 40

2 Principal Stages of Photosynthetic Light Energy Conversion 452.1 Introduction 452.2 Light-Harvesting Antennas 462.2.1 General 462.2.2 Bacterial Antenna Complex Proteins 472.2.2.1 The Structure of the Light-Harvesting Complex 472.2.2.2 Dynamic Processes in LHC 482.2.3 Photosystems I and II Harvesting Antennas 492.3 Reaction Center of Photosynthetic Bacteria 532.3.1 Introduction 532.3.2 Structure of RCPB 562.3.3 Kinetics and Mechanism of Electron Transfer in RCPB 58

V

2.3.4 Electron Transfer and Molecular Dynamics in RCPB 642.4 Reaction Centers of Photosystems I and II 682.4.1 Reaction Centers of PS I 692.4.2 Reaction Center of Photosystem II 722.5 Water Oxidation System 76

References 84

3 Photochemical Systems of the Light Energy Conversion 913.1 Introduction 913.2 Charge Separation in Donor–Acceptor Pairs 923.2.1 Introduction 923.2.2 Cyclic Tetrapyrroles 933.2.3 Miscellaneous Donor–Acceptor Systems 1013.2.4 Photophysical and Photochemical Processes in Dual

Flourophore–Nitroxide Molecules (FNO) 1063.2.4.1 System 1 1073.2.4.2 Systems 2 1093.3 Electron Flow through Proteins 1133.3.1 Factors Affecting Light Energy Conversion in Dual

Fluorophore–Nitroxide Molecules in a Protein 1163.3.2 Photoinduced Interlayer Electron Transfer in Lipid Films 118

References 121

4 Redox Processes on Surface of Semiconductors and Metals 1274.1 Redox Processes on Semiconductors 1274.1.1 Introduction 1274.1.2 Interfacial Electron Transfer Dynamics in Sensitized TiO2 1274.1.3 Electron Transfer in Miscellaneous Semiconductors 1304.1.3.1 Single-Molecule Interfacial Electron Transfer in

Donor–Bridge–Nanoparticle Acceptor Complexes 1304.1.4 Redox Processes on Carbon Materials 1334.2 Redox Processes on Metal Surfaces 1364.3 Electron Transfer in Miscellaneous Systems 144

References 147

5 Dye-Sensitized Solar Cells I 1515.1 General Information on Solar Cells 1515.2 Dye-Sensitized Solar Cells 1535.2.1 General 1535.2.2 Primary Grätzel DSSC 1555.3 DSSC Components 1585.3.1 Sensitizers 1585.3.1.1 Ruthenium Complexes 1585.3.1.2 Metalloporphyrins 1595.3.1.3 Organic Dyes 161

VI Contents

5.3.1.4 Semiconductor Sensitizes 1675.3.2 Photoanode 1685.3.3 Injection and Recombination 1715.3.4 Charge Carrier Systems 1755.3.5 Cathode 1825.3.6 Solid-State DSSC 185

References 189

6 Dye-Sensitized Solar Cells II 1996.1 Optical Fiber DSSC 1996.2 Tandem DSSC 2026.3 Quantum Dot Solar Cells 2086.4 Polymers in Solar Cells 2116.5 Fabrication of Solar Cell Components 2196.6 Fullerene-Based Solar Cells 223

References 229

7 Photocatalytic Reduction and Oxidation of Water 2357.1 Introduction 2357.2 Photocatalytic Dihydrogen Production 2367.2.1 Photocatalytic H2 Evolution over TiO2 2367.2.2 Miscellaneous Semiconductor Photocatalysts for H2 Evolution 2377.2.3 Photocatalytic H2 Evolution from Water Based on Platinum and

Palladium Complexes 2407.3 Water Splitting into O2 and H2 2417.3.1 Thermodynamics and Feasable Mechanism of the Water Splitting 2417.3.2 Mn Clusters as Water Oxidizing Photocatalysts 2427.3.2.1 Structure and Catalytic Activity of Cubane Manganese Clusters 2427.3.2.2 Catalytic Activity and Mechanism of WOS in Manganese Clusters 2447.3.3 Heterogeneous Catalysts for WOS 2487.3.3.1 General 2487.3.3.2 Photocatalysts Based on Titanium Oxides 2507.3.3.3 Miscellaneous Semiconductors for the WOS Catalysis 251

References 256

Conclusions 261References 263

Index 265

Contents VII

Preface

World energy consumption is about 4.7 1020 J and is expected to grow at the rate of2% each year for the next 25 years. Since the emergence of the apparition of theimpending energy crises, various avenues are being explored to replace fossil fuelswith renewable energy from solar power. In the past decades, the development ofadvanced molecular materials and nanotechnology, solar cells, and dye-sensitizedsolar cells, first of all, has initiated a new set of ideas that can dramatically improveenergy conversion efficiency and reduce prices of alternative energy sources. Never-theless, there are many fundamental problems to be solved in this area. Commercialcompetition of the new materials with existing fossil energy sources remains one ofthe most challenging problems for mankind.

This book embraces all principal aspects of structure and physicochemical actionmechanisms of dye-sensitized collar cells (DSSCs) and photochemical systems oflight energy conversion and related areas. A large body of literature exists on thissubject and many scientists have made important contributions to this the field. TheInternet program SkiFinder shows 44979 references for “dye sensitized” and 8493references for ‘‘dye-sensitized collar cells.’’ It is impossible in the space allowed inthis book to give a representative set of references. The author apologizes to those hehas not been able to include. More than 1000 references are given in the book, whichshould provide a key to essential relevant literature.

Chapter 1 of the monograph is a brief outline of the contemporary theories ofelectron transfer in donor–acceptor pairs and between a dye and a semiconductor.Principal stages of light energy conversion in biological photosystems, in which theNature demonsrates excellent examples for solving problems of conversion of lightenergy to energy of chemical compounds, are described in Chapter 2. The lightenergy conversion in donor–acceptor pairs in solution and on templates is thesubject of Chapter 3. Chapter 4 describes redox processes on the surface of semi-conductors and metals. Chapters 1–4 form the theoretical and experimental back-ground for the central Chapters 5 and 6. In Chapter 5, a general survey is made offundamentals of the primary Gertzel dye-sensitized solar cell and its rapid develop-ment. Advantages in design of new type of dye-sensitized solar cells such as opticalfiber, tandem, and solid-state DSSC and fabrication of its components are reviewedin Chapter 6. Chapter 7 gathers information on recent progress made in photo-catalytic reduction and oxidation of water.

IX

The monograph is intended for scientists and engineers working on dye-sensi-tized collar cells and other molecular systems of light energy conversion and relatedareas such as photochemistry and photosynthesis and its chemical mimicking. Thebook can be used as a subsidiary manual for instruction for graduate and under-graduate students of university chemistry, physics, and biophysics departments.

Gertz Likhtenshtein

X Preface

1Electron Transfer Theories

1.1Introduction

Electron transfer (ET) is one of themost ubiquitous and fundamental phenomena inchemistry, physics, and biology [1–34].Nonradiative and radiative ETare found to be akey elementary step in many important processes involving isolated molecules andsupermolecules, ions and excess electrons in solution, condensed phase, surfacesand interfaces, electrochemical systems and biology, and in solar cells, in particular.

As a light microscopic particle, an electron easily tunnels through a potentialbarrier. Therefore, the process is governed by the general tunneling law formulatedbyGamov [35]. The principal theoretical cornerstone for condensed phase ETwas laidby Franck and Libby (1949–1952) who asserted that the Franck–Condon principle isapplicable not only to the vertical radiative processes but also to nonradiativehorizontal electron transfer. The next decisive step in the field was taken by Marcusand his colleagues [2, 17, 36] and Hash [37]. These authors articulated the need forreadjustment of the coordination shells of reactants in self-exchange reactions and ofthe surrounding solvent to the electron transfer. They also showed that the electronicinteraction of the reactants gives rise to the splitting at the intersection of the potentialsurfaces, which leads to a decrease in the energy barrier.

1.2Theoretical Models

1.2.1Basic Two States Models

1.2.1.1 Landau–Zener ModelThe nonadiabatic electron transfer between donor (D) and acceptor (A) centers istreated by the Fermis golden rule (FGR) [38]

Solar Energy Conversion. Chemical Aspects, First Edition. Gertz Likhtenshtein.� 2012 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2012 by Wiley-VCH Verlag GmbH & Co. KGaA.

j1

kET ¼ 2pV2FCh

ð1:1Þ

where FC is the Franck–Condon factor related to the probability of reaching the termscrossing area for account of nuclear motion and V is an electronic coupling term(resonance integral) depending on the overlap of electronic wave functions in initialand final states of the process (see Figure 1.1).

At the transition of a system from one state to another, with a certain value ofthe coordinate Qtr, the energy of the initial (i) and final (f) states of energy termsis the same and the law of energy conservation permits the term–term transition

Figure 1.1 Variation in the energy of the systemalong the reaction coordinate: (a) diabatic terms ofthe reactant (i) and products (f); (b) adiabatic terms of the ground state (f) and excited state (f). V isthe resonance integral [9].

free energy inverted normal

reaction

coordinatesΔG0

ΔG0

ΔG*ΔG*

Δel

Δel

R

P

P

Figure 1.2 Energy versus reaction coordinatesfor the reactants and the products for normaland inverted reactions, and the inversion curve.The electron in the initial state requires apositive excitation energy Del for the normalreaction and a negative excitation energy - Delfor the inverted reaction (which could be directlyemitted as light). There is a positive energy

barrierDF� in both cases between the reactantsand the products that requires thermalactivation for the reaction to occur. This energybarrier as well as the energy for a direct electronexcitation vanishes for the inversion curve andthen the electron transfer becomes ultrafast.Reproduced from Ref. [61].

2j 1 Electron Transfer Theories

(Figure 1.2). Generally, the rate constant of the transition in the crossing area isdependent on the height of the energetic barrier (activation energy, Ea), thefrequency of reaching of the crossing area (y), and the transition coefficient (k):

ktr ¼ kn exp �Eað Þ ð1:2ÞThe transition coefficient k is related to the probability of the transition in the

crossing area (P) and is described by the Landau–Zener equation [39, 40]:

k ¼ 2Pð1þPÞ ð1:3Þ

where

P ¼ 1�exp �4p2V2

hvðSi�Sf Þ� �

ð1:4Þ

V is the electronic coupling factor (the resonance integral), v is the velocity ofnuclear motion, and Si and Sf are the slopes of the initial and final terms in the Qtrregion. If the exponent of the exponential function is small, then

P ¼ 4p2V2

hv ðSi�Sf Þ ð1:5Þ

and the process is nonadiabatic. Thus, the smaller the magnitude of the resonanceintegral V, the smaller is the probability of nonadiabatic transfer. The lower thevelocity of nuclear motion and smaller the difference in the curvature of the terms,the smaller is the probability of nonadiabatic transfer. At P¼ 1, the process isadiabatic and treated by classical Arrhenius or Eyring equations.

The theory predicts a key role by electronic interaction, which is quantitativelycharacterized by the value of resonance integralV in forming energetic barrier. If thisvalue is sufficiently high, the terms are split with a decreasing activation barrier andthe process occurs adiabatically. In another nonadiabatic extreme, where the inter-action in the region of the coordinateQtr is close to zero, the terms practically do notsplit, and the probability of transition i ! f is very low.

1.2.1.2 Marcus ModelAccording to the Marcus model [2, 3, 5, 17, 36], the distortion of the reactants,products, and solvent from their equilibrium configuration is described by identicalparabolas, shifted relative to each other according to the driving force of the value ofthe process, standard Gibbs free energy DG0 (Figure 1.2). Within the adiabaticregime (strong electronic coupling, the resonance integral V> 200 cm�1), in theframe of the Eyring theory of the transition state, the value of the electron transfer rateconstant is

kET ¼ hvkBT� �

exp� ðlþDG0Þ2

4lkBT

" #ð1:6Þ

1.2 Theoretical Models j3

where l is the reorganization energy defined as energy for the vertical electrontransfer without replacement of the nuclear frame. Equation 1.6 predicts the logkET�DG0 relationships depending on the relative magnitudes of l and DG0(Figure 1.3): (1) l > DG0, when log k increases if DG0 decreases (normal Marcusregion); (2) l¼DG0, the reaction becomes barrierless; and (3) l < DG0, when log kdecreases with increasing driving force.

The basic Marcus equation is valid in following conditions:

1) All reactive nuclear modes, that is, local nuclear modes, solvent inertial polar-ization modes, and some other kinds of collective modes, are purely classical.The electronic transition in the ET process is via the minimum energy at thecrossing of the initial and final state potential surfaces.

2) The potential surfaces are essentially diabatic surfaces with insignificant split-ting at the crossing and of parabolic shape. The latter reflects harmonicmolecular motion with equilibrium nuclear coordinate displacement and alinear environmental medium response.

3) The vibrational frequencies and the normalmodes are the same in the initial andfinal states.

The Marcus theory also predicts the Br€onsted slope magnitude in the normalMarcus region:

aB ¼ dDG#

dDG0¼ 1

21þ DG0

l

� �ð1:7Þ

The processes driving force (DG0) can be measured experimentally or calculatedtheoretically. For example,when solvation after the process of producingphotoinitiatedcharge pairing is rapid,DG0 can be approximately estimated by the following equation:

DG0 ¼ ED=Dþ �ðEA=Aþ þED*Þ�e2

eðrDþ þ rA�Þ ð1:8Þ

Figure 1.3 Variation in the logarithm of the rate constant of electron transfer with the driving forcefor the reaction after Marcus [50].


where ED=Dþ and EAþ =A are the standard redox potential of the donor and acceptor,respectively, ED� is the energy of the donor exited state, rDþ and rA� are the radii of thedonor and acceptor, respectively, and e is the medium dielectric constant.

The values of l can be roughly estimated within the framework of a simplifiedmodel suggesting electrostatic interactions of oxidized donor (Dþ ) and reducedacceptor (A�) of radii rDþ and rA� separated by the distance RDA with media ofdielectric constant e0 and refraction index n:

l ¼ e2

21n2� 1e0

� �1

rDþþ 1

rA�� 2RDA

� �ð1:9Þ

1.2.1.3 Electronic and Nuclear Quantum Mechanical EffectsThe nonadiabatic electron transfer between donor (D) and acceptor (A) centers istreated by the FGR (Equation 1.1). The theory of nonadiabatic electron transfer wasdeveloped by Levich, Dogonadze, andKuznetsov [41–43]. These authors, utilizing theLandau–Zener theory for the intersection area crossing suggesting harmonic one-dimensional potential surface, proposed a formula for nonadiabatic ET energy:

kET ¼ 2pV2

hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4plkBTp exp �ðlþDG0Þ

2

4lkBT

" #ð1:10Þ

Therefore, the maximum rate of ET at l¼DG0 is given by

kET ðmaxÞ ¼ 2pV2

hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4plkBTp ð1:11Þ

Involvement of intramolecular high-frequency vibrational modes in electrontransfer was considered [44–49]. For example, when the high-frequency mode (hn)is in the low-temperature limit and solvent dynamic behavior can be treated classi-cally [1], the rate constant for nonadiabatic ET in the case of parabolic terms is givenby

kET ¼

Xj

2pFjV2

hlkBTexp �ðjhvþ lsþDG0Þ

2

4lkBT

" #ð1:12Þ

where j is the number of high-frequency modes, Fj¼ e�S/j!, S¼ lv/hy, and lv and lsdenote the reorganization inside the molecule and solvent, respectively.

In the case of thermal excitation of the local molecular and medium high-frequency modes, theories mentioned before predicted the classical Marcus relationin the normal Marcus region. While in the inverted region, significant deviation onthe parabolic energy gap dependence is expected (Figure 1.4). The inverted Marcusregion cannot be experimentally observed if the stabilization of the first electrontransfer product for the accounting of the high-frequency vibrational mode occursfaster than the equilibrium of the solvent polarization with the momentary chargedistribution can be established. Another source of the deviation is the nonparabolicshape of the activation barrier [1].


A nonthermal electron transfer assisted by an intramolecular high-frequencyvibrational mode has been theoretically investigated [18]. An analytical expressionfor the nonthermal transition probability in the framework of the stochastic pointtransition approach has been derived. For the strong electron transfer, the decayof the product state can vastly enhance the nonthermal transition probability inthe whole range of parameters except for the areas where the probability isalready close to unity. If the initial ion state is formed either by forward electrontransfer or by photoexcitation, it may be visualized as a wave packet placed on theion free energy term above the ion and the ground-state terms intersection (seeFigure 6.1).

TheMarcus inverted region cannot be observed experimentally when term-to-termtransition in the crossing region is not a limiting step of the process as a whole(Figure 1.3) [50]. When ET reaction is very fast in the region of maximum rate, theprocess can be controlled by diffusion and, therefore, is not dependent on l, V2, andDG0. The integral encounter theory (IET) has been extended to the reactions limitedby diffusion along the reaction coordinate to the level crossing points where eitherthermal or hot electron transfer occurs [18]. IETdescribed the bimolecular ionizationof the instantaneously excited electron donor D� followed by the hot geminatebackward transfer that precedes the ion pair equilibration

2000 2000 6000

10–3

10–4

10–5

10–6

10–7

10–810000 14000 18000

Sc × 2.0

ω c = 1500 cm–1 λ = 2000 cm–1

T = 300 K

1.0

0.5

0.1

–ΔG cm–1

Figure 1.4 The energy gap dependence of thenuclear Franck–Condon factor, whichincorporates the role of the high-frequencyintramolecular modes. Sc¼D/2 is thedimensionless electron–vibration coupling,

given in terms that reduce replacement (D)between the minimum of the nuclear potentialsurfaces of the initial and final electronicstates [1]. Reproduced with permission.


D� þA !w12w21½Dþ � � �

w32 l w23DA

A��!�w Dþ þA�

and its subsequent thermal recombination tunneling is strong. It was demonstratedthat the fraction of ion pairs that avoids the hot recombination is much smaller thantheir initial number when the electron tunneling is strong. The kinetics of recom-bination/dissociation of photogenerated radical pairs (RPs) was described with ageneralized model (GM), which combines exponential models (EMs) and contactmodels (CMs) of cage effect dynamics [31]. Kinetics of nonthermal electron transfercontrolled by the dynamical solvent effect was discussed in Ref. [11]. Recombinationof ion pairs created by photoexcitation of viologen complexes is studied by a theoryaccounting for diffusion along the reaction coordinate to the crossing points of theelectronic terms. The kinetics of recombination convoluted with the instrumentresponse function were shown to differ qualitatively from the simplest exponentialdecay in both the normal and the inverted Marcus regions. The deviations of theexponentiality are minimal only in the case of activationless recombination and arereduced evenmore by taking into consideration a single quantummode assisting theelectron transfer

1.2.2Further Developments in the Marcus Model

1.2.2.1 Electron CouplingVariational transition-state theory was used to compute the rate of nonadiabaticelectron transfer for a model of two sets of shifted harmonic oscillators [51]. Therelationship to the standard generalized Langevin equation model of electrontransfer was established and provided a framework for the application of variationaltransition-state theory in simulation of electron transfer in amicroscopic (nonlinear)bath. A self-consistent interpretation based on a hybrid theoretical analysis thatincludes ab initio quantum calculations of electronic couplings, molecular dynamicssimulations of molecular geometries, and Poisson–Boltzmann computations ofreorganization energies was offered [52]. The analysis allowed to estimate thefollowing parameters of systems under investigation: (1) reorganization energies,(2) electronic couplings, (3) access to multiple conformations differing both inreorganization energy and in electronic coupling, and (4) donor–acceptor couplingdependence on tunneling energy, associated with destructively interfering electronand hole-mediated coupling pathways. Fundamental arguments and detailed com-putations show that the influence of donor spin state on long-range electronicinteractions is relatively weak.

The capability of multilevel Redfield theory to describe ultrafast photoinducedelectron transfer reactions and the self-consistent hybrid method was investigat-ed [53]. Adopting a standardmodel of photoinduced electron transfer in a condensedphase environment, the authors considered electron transfer reactions in the normaland inverted regimes, as well as for different values of the electron transferparameters, such as reorganization energy, electronic coupling, and temperature.


A semiclassical theory of electron transfer reactions in Condon approximation andbeyond was developed in [54]. The effect of the modulation of the electronic wavefunctions by configurational fluctuations of the molecular environment on thekinetic parameters of electron transfer reactions was discussed. A new formula forthe transition probability of nonadiabatic electron transfer reactions was obtainedand regular method for the calculation of non-Condon corrections was suggested.Quantum Kramers-like theory of the electron transfer rate from weak-to-strongelectronic coupling regions using Zhu–Nakamura nonadiabatic transition formulaswas developed to treat the coupled electronic and nuclear quantum tunnelingprobability [55]. The quantum Kramers theory to electron transfer rate constantswas generalized. The application in the strongly condensed phasemanifested that theapproach correctly bridges the gap between the nonadiabatic (Fermis golden rule)and adiabatic (Kramers theory) limits in a unified way, and leads to good agreementwith the quantum path integral data at low temperature.

In work [56], electron transfer coupling elements were extracted from constraineddensity functional theory (CDFT). This method made use of the CDFTenergies andthe Kohn–Sham wave functions for the diabatic states. A method of calculation oftransfer integrals between molecular sites, which exploits few quantities derivedfrom density functional theory electronic structure computations and does notrequire the knowledge of the exact transition state coordinate, was conceived andimplemented [57]. The method used a complete multielectron scheme, thus includ-ing electronic relaxation effects. The computed electronic couplings can then becombined with estimations of the reorganization energy to evaluate electron transferrates. On the basis of the generalized nonadiabatic transition-state theory [58], theauthors of the work [59, 60] presented a new formula for electron transfer rate, whichcan cover the whole range from adiabatic to nonadiabatic regime in the absence ofsolvent dynamics control. The rate was expressed as a product of the Marcus theoryand a coefficient that represents the effects of nonadiabatic transition at the crossingseam surface. The numerical comparisons were performed with differentapproaches and the present approach showed an agreement with the quantummechanical numerical solutions from weak to strong electronic coupling.

A nonadiabatic theory for electron transfer and application to ultrafast catalyticreactions has been discussed in Ref. [61]. The author proposed a general formalismthat not only extends those used for the standard theory of electron transfer but alsobecomes equivalent to it far from the inversion point. In the vicinity of the inversionpoint when the energy barrier for ET is small, the electronic frequencies become ofthe order of the phonon frequencies and the process of electron tunneling isnonadiabatic because it is strongly coupled to the phonons. It was found that whenthe model parameters are fine-tuned, ET between donor and acceptor becomesreversible and this system is a coherent electron–phonon oscillator (CEPO). Theacceptor that does not capture the electronmay play the role of a catalyst (Figure 1.5).Thus, when the catalyst is fine-tuned with the donor in order to form a CEPO, it maytrigger an irreversible and ultrafast electron transfer (UFET) at low temperaturebetween the donor and an extra acceptor. Such a trimer system may be regulated bysmall perturbations and behaves as a molecular transistor.


Two weakly coupled molecular units of donor and catalyst generate a CEPO. Thissystem is weakly coupled to a third unit, the acceptor (Figure 1.5a). An electroninitially on the donor generates an oscillation of the electronic level of the CEPO. Ifthe bare electronic level of a thirdmolecular unit (acceptor) is included in the intervalof variation, as soon as resonance between the CEPO and the acceptor is reached, ETis triggered irreversibly to the acceptor (Figure 1.5b). The authors suggested thatbecause of their ability to produce UFET, the concept of CEPOs could be an essentialparadigm for understanding the physics of the complexmachinery of living systems.

Perturbation molecular orbital (PMO) theory was used to estimate the electronicmatrix element in the semiclassical expression for the rate of nonadiabatic electrontransfer at ion–molecular collisions [62]. It was shown that the electron transferefficiency comes from the calculated ETrate divided by the maximum calculated ETrate and by dividing the observed reaction rate by the collision rate, calculated by thePMO treatment of ion–molecular collision rates.

1.2.2.2 Driving Force and Reorganization EnergySeveral works were devoted to models for medium reorganization and donor–acceptor coupling [63–78]. The density functional theory based on ab initiomoleculardynamics method combines electronic structure calculation and statistical mechan-ics and was used for first-principles computation of redox free energies at one-electron energy [66]. The authors showed that this is implemented in the frameworkof the Marcus theory of electron transfer, exploiting the separation in verticalionization and reorganization contributions inherent in Marcus theory. Directcalculation of electron transfer parameters through constrained density functionaltheory was a subject of the work by Wu and Van Voorhis [67]. It was shown thatconstrained density functional theory can be used to access diabatic potential energysurfaces in theMarcus theory of electron transfer, thus providing ameans to directlycalculate the driving force and the inner sphere reorganization energy. The influence

Figure 1.5 Principle of ET with a coherent electron–phonon oscillator [61].


of static and dynamic torsional disorder on the kinetics of charge transfer (CT) indonor–bridge–acceptor (D-B-A) systems has been investigated theoretically using asimple tight binding model [68]. Modeling of CT beyond the Condon approximationrevealed two types of non-Condon (NC) effects. It was found that if trot is much lessthan the characteristic time, tCT, of CT in the absence of disorder, the NC effect isstatic and can be characterized by rate constant for the charge arrival on the acceptor.For larger trot, the NC effects become purely kinetic and the process of CT in thetunneling regime exhibits timescale invariance, the corresponding decay curvesbecome dispersive, and the rate constant turns out to be time dependent. In the limitof very slow dynamic fluctuations, the NC effects in kinetics of CTwere found to bevery similar to the effects revealed for bridges with the static torsional disorder. Theauthors argued that experimental data reported in the literature for several D-B-Asystems must be attributed to the multistep hopping mechanism of charge motionrather than to the mechanism of single-step tunneling. Survival probability as afunction of time for D-B-A systems is shown in Figure 1.6.

The theory developed by Fletcher in Refs [71, 72] took into account the fact thatcharge fluctuations contribute to the activation of electron transfer, besides dielectricfluctuations. It was found that highly polar environments are able to catalyze the ratesof thermally activated electron transfer processes because under certain well-definedconditions, they are able to stabilize the transient charges that develop on transitionstates. Plots of rate constant for electron transfer versus driving force are shown inFigure 1.7, which is drawn on the assumption that electron transfer is nonadiabaticand proceeds according to Diracs time-dependent perturbation theory. On the

0.0

987

6

5

4

3

2

0.1

1

0.4 0.8 1.2

Time (ps)

Sur

viva

l pro

babi

lity

Figure 1.6 Survival probability as a function oftime for D-B-A systems containing three (solidline), four (dotted line), and five (dashed line)subunits. All curves were calculated for the

D-B-A system with the energy gap De betweenthe donor and the equienergetic bridge equal to1.2 eV. The valueof the charge transfer integralVwas taken to be equal to 0.3 eV [68].


theory, the relative permittivity of the environment exerts a powerful influence on thereaction rate in the highly exergonic region (the inverted region) and in the highlyendergonic region (the superverted region).

According to authors, nonadiabatic electron transfer is expected to be observedwhenever there is small orbital overlap (weak coupling) between donor and acceptorstates, so that overall electron transfer rates are slow compared to the mediadynamics. For strongly exergonic electron transfer reactions that are activated bycharge fluctuations in the environment, the activation energy was determined by theintersection point of thermodynamic potentials (Gibbs energies) of the reactants andproducts. The following equations for Greactants and Gproducts, which are the totalGibbs energies of the reactants and products (including their ionic atmospheres),respectively, were suggested:

Greactants ¼ 12Q21

14pe0

� �1

eð0Þ� �

1aDþ 1

aA� 2d

� �ð1:13Þ

Gproducts ¼ 12Q22

14pe0

� �1

eð¥Þþ f1 ½eð0Þ�eð¥Þ��

1aDþ 1

aA� 2d

� �ð1:14Þ

Q1 and Q2 are the charge fluctuations that build up on them, e(0) is the relativepermittivity of the environment in the low-frequency limit (static dielectric constant),e(¥) is the relative permittivity of the environment in the high-frequency limit (e� 2),aA is the radius of the acceptor in the transition state (including its ionic atmosphere),aD is the radius of the electron donor in the transition state (including its ionic

Figure 1.7 The rate constant for electron transfer (kET) as a function of the driving force (�DG0)and reorganization energy (l) on the Fletcher theory [71, 72]. Note the powerful catalytic effect ofpolar solvents (such as water) on strongly exergonic reactions.


atmosphere), and f1 is a constant (0< f1< 1) that quantifies the extent of polarscreening by the environment, d is the distance between the electron donor andacceptor. Figure 1.8 shows the Gibbs energy for electron transfer through anintermediate.

Authors of paper [73] focused on themicroscopic theory of intramolecular electrontransfer rate. They examinedwhether or not and/or underwhat conditions thewidelyused Marcus-type equations are applicable to displaced–distorted (D-D) and dis-placed–distorted–rotated (D-D-R) harmonic oscillator (HO) cases. For this purpose,the cumulant expansion (CE) method was applied to derive the ETrate constants forthese cases. In the CE method, the analytical condition was derived upon which theMarcus-type equation of theGaussian formwas obtained for theD-DHOcase. In theframe of theory, the following equation for the ET rate constant was derived:

Wb! a ¼ jJabj2

h2

ffiffiffiffiffiffiffiffiffiffiffiph2

lkBT

sexp � ½hvabþhVabð0Þi�lþ l�

2

4lkBT

!ð1:15Þ

where DGab¼ hvab þ hVab(0)i� l and hvab¼Ea�Eb.The quantity hvab þ hVab(0)i has the following physical meaning. The quantity

hVab(0)i is the vibrational energy acquired in the final state through vertical or FCtransition from the initial state, averaged over the initial vibrational states undercondition of vibrational thermal equilibrium in the initial potential energy surface.

It was found that the reorganization energy and the free energy change for theD-DHO depend on the temperature. As a consequence, the preexponential factor of theET rate shows a temperature dependence different from the usual Arrhenius

Figure 1.8 Superimposed Gibbs energyprofiles in the vicinity of the electron trap T2.Trapping is thermodynamically reversible, sothe electron can return to T0 radiatively (R) via T1or nonradiatively (NR) via the inverted region.Both routes are kinetically hindered by the

extreme narrowness of the Gibbs energyparabola, however. This narrowness isconferred by the extremely nonpolarenvironment surrounding T2. Trapping state T3is the final acceptor [72].


behavior. The dependence of the Franck–Condonweighted density 1 on the degree ofdistortion for a model one-mode D-D HO system is presented in Figure 1.9.

The temperature dependence of the ETrate at different degrees of mixing for twomodes whose frequencies are 100 and 30 cm�1 is shown in Figure 1.10.

The influence of spatial charge redistribution modeled by a change in the dipolemoment of the reagent that experiences excitation on the dynamics of ultrafast

Figure 1.9 The dependent of the Franck–Condon weighted density 1 (FC DOS) (y-axis)on the degree of distortion for a model one-modeD-DHO system described in the text. Thedegrees of distortion are 0.9 in (a), 0.75 in (b),and 0.6 in (c). The peaks are the EGL calculated

with the exact TCF method. The thick dashedlines denote the energy gap law (EGL) of the ETrate EGL calculated with the CE method usingthe same parameters. The thin smooth lines arethe EGL calculated with the conventionalMarcus theory [73].


Figure 1.10 The temperature dependence ofthe ET rate at different degrees ofmixing for twomodes whose frequencies are 100 and 30 cm�1

in the lower level, without distortion. Theabscissa is the inverse temperature, labeled in1000/T. On the top, T (K) is also shown. (a) TheET rate on logarithmic scale versus the inverse

temperature, for four different angles of rotation(labeled in the legend in radian). (b) The ET ratemultiplied by the square root of thetemperature, on logarithmic scale, versus theinverse temperature. (c) The ET rate multipliedby the temperature, on logarithmic scale, versusthe inverse temperature [73].


photoinduced electron transfer was studied [22]. A two-center model based on thegeometry of real molecules was suggested. Themodel described photoexcitation andsubsequent electron transfer in a donor–acceptor pair. The rate of electron transferwas shown to depend substantially on the dipole moment of the donor at thephotoexcitation stage and the direction of subsequent electron transfer. The authorsof the work [74] have shown that the polarization fluctuation and the energy gapformulations of the reaction coordinate follow naturally from the Marcus theory ofouter electron transfer. The Marcus formula modification or extension led to aquadratic dependence of the free energies of the reactant and product intermediateson the respective reaction coordinates. Both reaction coordinates are linearly relatedto the Lagrangianmultiplierm in theMarcus theory of outer sphere electron transfer,so thatm also plays the role of a natural reaction coordinate.Whenm ¼ 0; F�ðm ¼ 0Þis the equilibrium free energy of the reactant intermediate X� at the bottomof its well,and when m ¼ −1; Fðm ¼ −1Þ is the corresponding equilibrium free energy of theproduct intermediate X. Atm ¼ −1 the free energy of reorganization of the solventfrom its equilibrium configuration at the bottom of the reactant. A theory of electrontransfer with torsionally induced non-Condon (NC) effects was developed by Jangand Newton [69]. The starting point of the theory was a generalized spin-bosonHamiltonian, where an additional torsional oscillator bilinearly coupled to other bathmodes causes a sinusoidal non-Condon modulation. Closed form time-dependentnonadiabatic rate expressions for both sudden and relaxed initial conditions, whichare applicable for general spectral densities and energetic condition, were derived.Under the assumption that the torsional motion is not correlated with the polaronicshift of the bath, simple stationary limit rate expression was obtained. Modelcalculation of ET illustrated the effects of torsional quantization and gating on thedriving force and temperature dependence of the electron transfer rate. The Born–Oppenheimer (BO) formulation of polar solvation is developed and implemented atthe semiempirical (PM3) CI level, yielding estimation of ET coupling elements (V0)for intramolecular ET in several families of radical ion systems [70]. The treatmentyielded a self-consistent characterization of kinetic parameters in a two-dimensionalsolvent framework that includes an exchange coordinate. The dependence of V0 oninertial solvent contributions and on donor–acceptor separation was discussed (seeFigure 1.11).

In the work [76], it was demonstrated that constrained density functional theoryallowed to compute the three key parameters entering the rate constant expression:the driving force (DG0), the reorganization energy (l), and the electronic couplingHDA. The results confirm the intrinsic exponential behavior of the electroniccoupling with the distance separating D and A or, within the pathway paradigm,between two bridging atoms along the pathway. Concerning the through spacedecay factors, the CDFT results suggested that a systematic parameterization of thevarious kinds of weak interactions encountered in biomolecules should be under-taken in order to refine the global through space decay factors. Such a work hasbeen initiated in this paper. The hydrogen bond term has also been adjusted. Besidesthe refinement of the distance component, the authors underlined the appearance ofan angular dependence and the correlation factor between R and w.


Taking into account the volume of reagents, the theory gives the followingEquation 1.16 [77]:

l ¼ e2

21n2� 1e0

� �1

rDþþ 1

rA�� 2RDA

þ r3Dþ þ r3A��

2R4DA

� �ð1:16Þ

Further development of theory of reorganization energy consisted in taking intoconsideration the properties of medium and manner in which it interfaces with thesolute [65]. These properties must include both size and shape of the solute andsolvent molecules, distribution of electron density in reagents and products, and thefrequency domain appropriate to medium reorganization. When the symmetry ofdonor and acceptor is equivalent, reorganization energy can be generalized as

l ¼ CDgeff ðe2Þ 1reff �1

RDA

� �ð1:17Þ

where C¼ 0.5 is a coefficient, Dgeff is the effective charge, and reff is the effectiveradius of charge separated centers. More general theory of the reorganization energytakes the difference between energies of the reactant state and product state,UR andUP, with the same nuclear coordinates q, as the reaction coordinate [78]:

DeðqÞ ¼ UPðqÞ�URðqÞ ð1:18ÞIn this theory, the reorganization energy is related to the equilibriummean square

fluctuation of the reaction coordinate as

L ¼ 12ðb < De�hDeiÞ2 ð1:19Þ

Figure 1.11 ET dynamics in a low temperature case for different reorganization energies (seedetails in Ref. [87, 88]).


The atoms in the systems are divided into four groups: donor (D) and acceptor(A) sites of a reaction complex (as in protein), nonredox site atoms, and wateratoms as the environment. The following calculation determined eachcomponents contribution to De and, therefore, to the reorganization energy. Inthe case of thermal excitation of the local molecular and medium high-frequencymodes, before theories mentioned predicted the classical Marcus relation in thenormal Marcus region. While in the inverted region, significant deviation in theparabolic energy gap dependence is expected. The inverted Marcus region cannotbe experimentally observed if the stabilization of the first electron transfer productfor the accounting of the high-frequency vibrational mode occurs faster than theequilibrium of the solvent polarization with the momentary charge distributioncan be established. Another source of the deviation is the nonparabolic shape ofthe activation barrier.

The effect of solvent fluctuations on the rate of electron transfer reactions wasconsidered using linear expansion theory and a second-order cumulant expan-sion [79]. An expression was obtained for the rate constant in terms of the dielectricresponse function of the solvent andwas proved to be valid not only for approximatelyharmonic systems such as solids but also for strongly molecularly anharmonicsystems such as polar solvents.

Microscopic generalizations of the Marcus nonequilibrium free energy surfacesfor the reactant and the product, constructed as functions of the charging parameter,were presented [55]. Their relation to surfaces constructed as functions of the energygap is also established. The Marcus relation was derived in a way that clearly showsthat it is a good approximation in thenormal region evenwhen the solvent response issignificantly nonlinear. The hybrid molecular continuum model for polar solvation,combining the dielectric continuum approximation for treating fast electronic(inertialess) polarization effects, was considered [32]. The slow (inertial) polarizationcomponent, including orientational and translational solventmodes, was treated by acombination of the dielectric continuum approximation and a molecular dynamicssimulation, respectively. This approach yielded an ensemble of equilibrium solventconfigurations adjusted to the electric field created by a charged or strongly polarsolute. Both equilibrium and nonequilibrium solvation effects were studied bymeans of this model, and their inertial and inertialess contributions were separated.Three types of charge transfer reactions were analyzed. It was shown that thestandard density linear response approach yields high accuracy for each particularreaction, but proves to be significantly in error when reorganization energies ofdifferent reactions were compared.

1.2.3Zusman Model and its Development

The Zusman equation (ZE) has been widely used to study the effect of solventdynamics on electron transfer reactions [80–90]. In this equation, dynamics of theelectronic degrees of freedom is coupled to a collective nuclear coordinate. Appli-cation of this equation is limited by the classical treatment of the nuclear degrees of


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