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Progress in Surface Science 84 (2009) 30–68

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Progress in Surface Science

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Review

Photoinduced electron dynamics at thechromophore–semiconductor interface: A time-domainab initio perspective

Oleg V. Prezhdo a,*, Walter R. Duncan a,b, Victor V. Prezhdo c

a Department of Chemistry, U. Washington, Seattle, WA 98195, USAb Department of Chemistry, Seattle U., Seattle, WA 98122, USAc Institute of Chemistry, Kochanowski U., Kielce, Poland

a r t i c l e i n f o

Commissioning Editor: X. Zhu

0079-6816/$ - see front matter � 2008 Elsevier Ltdoi:10.1016/j.progsurf.2008.10.005

* Corresponding author.E-mail address: [email protected] (O.

a b s t r a c t

The chromophore–semiconductor interface offers a classic exam-ple of an interaction between an organic molecular species andan inorganic bulk material. The interface provides the foundationfor a new, promising type of solar cell and presents a fundamen-tally important case study for several fields, including photo-, elec-tro- and analytical chemistries, molecular electronics, andphotography. Scientists employ different concepts and terminolo-gies to describe molecular and solid states of matter, and these dif-ferences make it difficult to describe the interface with a singlemodel. At the basic atomistic level of description, however, thischallenge can be largely overcome. Recent advances in non-adia-batic molecular dynamics and time-domain density functional the-ory have created a unique opportunity for simulating the ultrafast,photoinduced processes on a computer very similar to the way thatthey occur in nature. The progress report is a review of these state-of-the-art theoretical tools. It offers a comprehensive picture of avariety of electron transfer processes that occur at the interface.The topics of discussion include electron injection from the chro-mophore to the semiconductor, electron relaxation and delocaliza-tion inside the semiconductor, back-transfer of the electron to thechromophore and to the electrolyte, and regeneration of the neu-tral chromophore by the electrolyte. The ab initio time-domainmodeling is particularly valuable for understanding these dynamicfeatures of the ultrafast electron transfer processes, which cannotbe represented by a simple rate description. For example, the sim-ulations show that what appears as a single step, such as electron

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O.V. Prezhdo et al. / Progress in Surface Science 84 (2009) 30–68 31

injection, is in fact an average over many distinct elementary pro-cesses, and that very different vibrational modes drive electrontransfer, depending on the process, the system, and the experimen-tal conditions. The report focuses in particular on the electronicdonor–acceptor interaction, atomic motions, electron-vibrationalcoupling, surface termination, thermal effects, electron transfermechanisms and fluctuations from the average behavior.

� 2008 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.1. Time-domain density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.2. Non-adiabatic molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2.1. The classical-path and Ehrenfest approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.2.2. Fewest switching surface hopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.3. The electron transfer mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3. Interface structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1. Interface geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2. Electronic properties of the interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4. Injection from the chromophore to the semiconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.1. The role of atomic motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2. The distribution of initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.3. The electron injection mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4. Temperature effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.5. Electron transfer to hydrated Ti4þ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5. Electron dynamics inside the semiconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.1. Distribution of the injected electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.2. Relaxation to the conduction band edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.3. Delocalization into bulk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6. Back electron transfer processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.1. Atomic motions relevant for the back electron transfer processes . . . . . . . . . . . . . . . . . . . . . . . . . 596.2. Electron recombination with the chromophore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606.3. Electron loss to electrolyte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.4. Electron relaxation processes in the presence of both chromophore and electrolyte. . . . . . . . . . 626.5. Electron transfer from electrolyte to the chromophore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7. Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

1. Introduction

The photoinduced electron transfer (ET) processes that occur at the interface between organicchromophores and inorganic semiconductors are responsible for the initial charge separation in thedye-sensitized semiconductor solar cell, which is also known as the Grätzel cell [1]. Many experimen-talists [1–30] and theoreticians [31–50] have focused on Grätzel cells because they present a promis-ing alternative to the more costly traditional solar cells made of silicon. The ET at the molecular-bulkinterface is central to a number of other fields as well, including molecular electronics [51–54], photo-catalysis [55–57], photo-electrolysis [58], photography [3,59], lithography [60], and quantum reactioncontrol [61,62]. In addition, interfaces between solid-state materials and large molecules are encoun-tered in bio-analytical chemistry [63], bio-mechanics [64] and drug delivery [65]. Given this variety,we have chosen to study a chromophore interacting with a TiO2 surface, which presents a well-char-acterized, fundamental system for studying ET.

32 O.V. Prezhdo et al. / Progress in Surface Science 84 (2009) 30–68

The chromophore–semiconductor interface serves multiple functions in the Grätzel cell systems, asillustrated in Fig. 1. Absorption of a photon hm promotes an electron from the ground state to the ex-cited chromophore state. The photoexcited electron injects into the semiconductor, sinject. Once insidethe semiconductor surface, the electron simultaneously delocalizes into the bulk, sbulk, and relaxesdown in energy to the conduction band (CB) edge, srelax. The Coulombic attraction to the chromo-phore-cation, the presence of surface trap states, the interaction with the electrolyte, and other factorsmay keep the electron close to the chromophore, or even draw the delocalized electron back to thesurface. As a result, the electron can return to the chromophore ground state, schrom, or the electrolytecan approach the surface and capture the electron, selectr. The main function of electrolyte is to deliveranother electron to the chromophore from the counter-electrode and to regenerate the neutral chro-mophore ground state, sregen. These processes can occur in parallel, but they compete with each otherand ultimately determine the efficiency of the Grätzel cell. In a perfect cell, all photoexcited electronsinject into the semiconductor surface, delocalize into the semiconductor bulk, reach the counter-elec-trode (performing useful work along the way), and are delivered by the electrolyte back to the chro-mophore ground state. Relaxation of the electron inside the semiconductor CB results in voltagelosses. ET from the semiconductor directly back to the chromophore ground state or to the electrolytedecreases the current and photon-to-electron yield.

In spite of the enormous amount of effort that researchers have devoted to the chromophore–semi-conductor interface, many details of the interfacial ET dynamics remain unknown and are hard to elu-cidate. These difficulties trace back to the stark differences between the two interface components:molecules, which are studied by chemists, and solid-state materials, which are studied by physicists[66,67]. For example, molecules have discrete, localized electronic states, unique vibrational spectraand well-defined directional bonds. Solid-state materials, on the other hand, have continuous bandsof delocalized electronic and vibrational states, can be easily modified and doped, and contain manydefects that disrupt regular bonding patterns. Furthermore, the properties of a solid-state semicon-ductor surface strongly depend on its preparation and termination, the interaction with surrounding

Energy

Ground

Excited

hν

Chromophore

τchrom

τ inject

Electrolyte

Conduction Band

τbulk

Valence Band

Semiconductor

τrelax

τelectr

τregen

3.2 eV

Fig. 1. Schematic representation of the electronic energy levels in the chromophore–semiconductor system and the ultrafastphotoinduced processes that occur at the interface. The left-side corresponds to the semiconductor, while the right-siderepresents the chromophore and the electrolyte. The chromophore ground state resides in the semiconductor band-gap. As aresult, the system remains in the ground state in the absence of light. A photon promotes the ground state electron to thechromophore excited state, and the electron injects into the semiconductor surface. After that, it simultaneously delocalizesinto the bulk and relaxes down in energy. Upon relaxation, and especially if it is trapped near the surface, the electron cantransfer to the chromophore ground state or to the electrolyte, if the latter approaches the surface. The electrolyte is needed inorder to regenerate neutral chromophore for the next photovoltaic cycle. All of these processes occur on femto- and picosecondtimescales and have been modeled at the atomistic level by ab initio non-adiabatic molecular dynamics.


vapor, solvent, or other chemicals, pH, and temperature. The electronic structure of the bulk materialis also affected by its size; smaller crystals exhibit quantum confinement, which dramatically altersthe electronic and vibrational characteristics of the material.

Faced with these difficulties, we turn to ab initio time-domain atomistic simulation, which providesa unique perspective on the interfacial ET and most closely mimics the processes as they happen innature. The atomistic representation complements the simplified phenomenological models [34–37,68–71], which allow scientists to systematically investigate the influence of various interfacecharacteristics on the ET dynamics by varying model parameters. The atomistic simulation treatsthe interface in full, realistic detail and describes the evolving geometric and electronic structure ofthe chromophore, the surface, and the electrolyte in real time. The approach explicitly considers bind-ing between the chromophore, the semiconductor and the electrolyte, the chemical species terminat-ing the surface, bulk and surface defects, and the complex many-body electronic and electron-vibrational interactions. An ab initio treatment of the interface makes it possible to avoid fittingparameters and to build the theory starting from such fundamental laws of physics as the Coulomblaw. A minimal amount of parameterization occurs at the level of individual atoms or other basicbuilding blocks. The complex chromophore–semiconductor interface is then constructed with thesebasic blocks without further parameter tweaking. The time-domain representation, as opposed tothe energy domain, is particularly valuable with large systems and ultrafast processes, includingthe current case, in which individual energy levels are hard to resolve because of the very fine energyspacing and the time–energy Heisenberg uncertainty relationship. Using the techniques described be-low, the theorists can model and interpret the time-resolved laser experiments and follow the ET pro-cesses with an unprecedented level of detail.

The current progress report differs from the earlier review [41] and account [42] of the electrondynamics at the chromophore–semiconductor interface in several key ways. The review [41] focusedsolely on the photoinduced electron injection, which is the first of the six processes considered in thepresent report, and covered extensively the theoretical work performed by many other groups using avariety of computational techniques. The account [42] gave a brief description of the photoinducedinterfacial phenomena, barring any equations or extended discussion of the simulation results. Inthe current report we take the opportunity to discuss our own work at a relatively technical level,and to cover every aspect of the interface that we have learned about from the time-domain ab initiosimulations thus far [40,72–77].

The paper is constructed as follows: The theory section describes the two components of the time-domain atomistic simulation, involving coupled evolution of the electronic and vibrational subsys-tems, and defines the adiabatic and non-adiabatic (NA) ET mechanisms. The next section thenswitches to the specific processes that the electron undergoes at the interface after the initial photo-excitation. Following a discussion of the geometric and electronic structure of the interface, the focusmoves to the photoinduced electron injection from the chromophore to the semiconductor. Two spe-cific systems illustrate the role that atomic motions play in the injection process, the distribution ofthe initial conditions for the ET, the adiabatic and NA injection mechanisms, and thermal effects.We then describe the dynamics of the injected electrons inside the semiconductor, consider the dis-tribution of the injected electrons, and compare the electron relaxation to the edge of the semiconduc-tor CB with the delocalization into the semiconductor bulk. Finally, the back-ET processes arediscussed, including the electron recombination with the chromophore, the electron loss to the elec-trolyte, and the regeneration of the neutral chromophore by ET from the electrolyte. We conclude witha summary of the ET dynamics at the chromophore–semiconductor interface and some final remarks.

2. Theory

Time-domain atomistic simulation of the excitation dynamics is comprised of two closely inter-twined components. The first is an approach for describing the electronic and nuclear structure ofthe system and the coupling between the two subsystems. The second is an algorithm for propagatingthe coupled electron–nuclear evolution. By necessity, the electrons are treated quantum-mechanicallyby one of the established electronic structure approaches. Since the system under consideration here


involves a periodic solid-state material, its electronic structure is particularly well described by den-sity functional theory (DFT), which efficiently includes many-body electron correlation effects. Thenuclei, however, can be treated classically or semiclassically since they are much heavier and slowerthan electrons. There exists no generally accepted prescription for mixed quantum-classical or semi-classical dynamics, although a number of schemes have been proposed. Among the multitude of ap-proaches the two most common schemes – Ehrenfest and surface hopping (SH) – as well as thesimplest – the classical-path approximation – are described below.

2.1. Time-domain density functional theory

The three-dimensional single-particle electron density is the central quantity in DFT [78]. In theKohn–Sham (KS) representation [79], which ensures that the density corresponds to a multi-dimen-sional Ne-electron wave function, the single-particle density is represented by the sum of the densitiesof the occupied KS orbitals uaðx; tÞ

qðr; tÞ ¼XNe

a¼1

juaðr; tÞj2: ð1Þ

The evolution of the electron density is derived by applying the time-dependent (TD) variationalprinciple to the KS energy,

Efuag ¼XNe

a¼1

huajKjuai þXNe

a¼1

huajV juai þe2

2

Z Zqðr0; tÞqðr; tÞjr� r0j d3rd3r0 þ Excfqg: ð2Þ

The right-hand side of the above expression includes the kinetic energy, the electron–nuclearCoulomb attraction, the electron–electron Coulomb repulsion, and the exchange-correlation energyfunctional. The TD variational principle gives a set of equations for the evolution of the KS orbitals[78,80–82]

i�h@uaðr; tÞ

@t¼ Hðr;R; tÞuaðr; tÞ; a ¼ 1;2; . . . ;Ne; ð3Þ

The Hamiltonian H is time-dependent, both through the external potential created by the nuclearmotion, as well as through the electron density. The density dependence of the Hamiltonian couplesthe above single-particle equations of motion.

We solve the TD KS equations (3) by expending the TD KS orbitals uaðr; tÞ in the basis of adiabaticKS orbitals u

�kðr; RðtÞÞ

uaðr; tÞ ¼X

k

cakðtÞu

�kðr; RðtÞÞ; ð4Þ

that are obtained by solving the time-independent DFT equations for the current nuclear positionsusing well-developed codes, such as VASP [83,84]. Inserting this expansion into the TD KS equation(3) gives the evolution of the expansion coefficients

i�h@

@tca

j ðtÞ ¼X

k

cak ðtÞð�kdjk þ djkÞ; ð5Þ

where �k is the energy of the adiabatic KS orbital u�

k, and djk is the NA coupling between orbitals u�

j andu�

k. The NA coupling djk is calculated numerically [85] as the overlap of the adiabatic KS orbitals u�

j andu�

k at sequential molecular dynamics (MD) time steps Dt

djk ¼ �i�h u�

jj$Rju�

k

D E� dR

dt¼ �i�h u

�j@

@t

��u� k

� �ð6Þ

� � i�h2Dtðhu�

jðtÞju�

kðt þ DtÞÞi � hu�

jðt þ DtÞju�

kðtÞiÞ:

Having computed the NA coupling, we propagate Eq. (3), using one of the numerical algorithms de-signed for solving the TD Schrödinger equation [86].


The difference in the characteristic timescales of the electronic and nuclear motions allows us tovary independently the electronic and nuclear time steps, and speed up the calculation, while main-taining its accuracy. The TD KS equations (5) describing the fast electronic dynamics are propagatedusing 10�17–10�18 s steps. At the same time, the significantly slower nuclear motions are evolved with10�15 s steps. The timescale separation produces substantial computational savings, since the mosttime-consuming part of the calculation, associated with the update of the adiabatic KS orbitals, ener-gies and NA coupling, is performed much less frequently than the update of the electronic coefficients.

2.2. Non-adiabatic molecular dynamics

Once the approach for evolving the electrons has been established, the evolution of the nuclei, RðtÞin Eq. (3), needs to be defined. This so-called quantum back-reaction problem, which prescribes howthe quantum subsystem influences the classical subsystem, can be a matter of heated debate and can-not be resolved in a generally acceptable way [87–95]. In contrast, the dependence of the quantumsubsystem on the classical external potential (as described above) raises little controversy; the influ-ence of the classical subsystem on the quantum one is given by the parametric dependence of Eq. (3)on the classical coordinates RðtÞ. Here, we outline the three most common prescriptions for the quan-tum back-reaction.

2.2.1. The classical-path and Ehrenfest approximationsThe simplest solution to the quantum back-reaction problem is provided by the classical-path

approximation (CPA), which circumvents the problem by assuming that the nuclear evolution is inde-pendent of the electronic evolution and is a given a priori, while the electron dynamics, in contrast, dodepend on the nuclear dynamics [96]. For example, one can use the nuclear trajectory obtained for theground electronic state in order to describe the NA dynamics of the photoexcited electron [72,74]. Thisapproach provides a very efficient approximation, since a single nuclear trajectory can be used tomodel multiple processes. When considering the photoinduced dynamics at the chromophore–semi-conductor interface, a nuclear trajectory obtained for the ground electronic state can be used in theCPA to study electron injections at different energies, back-ET, electron relaxation and delocalizationinside the semiconductor. The CPA remains valid as long as the forces that are experienced by the nu-clei change little as the electronic density evolves. The force changes can be evaluated, for instance,relative to the thermal kinetic energy that is available to the nuclei. The CPA clearly breaks down ifan excitation of a stable molecule results in molecular dissociation or isomerization. On the otherhand, the CPA works well for the chromophore–semiconductor interface, since no bond breaking orother major geometric changes take place during any of the considered processes, and since the sys-tem comprises hundreds of electrons, such that the excitation of a single electron produces minorchanges in the overall electron density (Eq. (1)).

The Ehrenfest approximation provides a classic solution to the quantum back-reaction problem[97,98,89,40,82]. The classical variables are coupled to the expectation values of the correspondingquantum-mechanical operators in the spirit of the Ehrenfest theorem [99]. This form of quantum-clas-sical coupling gives an average, mean-field back-reaction, as expressed by the quantum force

Fq:m: ¼ �rRhWðr; tÞjHðr; RÞjWðr; tÞi; ð7Þ

which is computed as the gradient of the expectation energy of the quantum energy. Provided that thewave function evolves according to the TD Schrödinger equation, the gradient can be moved insidethe quantum averaging [96]. If the quantum particle evolves into a superposition of multiple states,the classical-particle experiences the Ehrenfest force (Eq. (7)), which is an average of the forces corre-sponding to these states. This is in contrast with the full quantum description, in which the wave-packet describing the heavy particle branches into multiple wave-packets correlated with differentelectronic states. These branched wave-packets experience different forces that are associated withthe relevant electronic state. The lack of the branching constitutes one of the drawbacks of the Ehren-fest and CPA schemes [100]. Their other major drawback is the lack of detailed balance, which makestransition upward in energy less likely than transitions downward by the Boltzmann factor and en-sures that the system reaches thermal equilibrium in the long-time limit [101,102].


When these limitations are kept in mind, both the CPA and the Ehrenfest methods can be used tostudy those interfacial ET processes that exchange small amounts of energy, such as the initial electroninjection and its delocalization into semiconductor bulk. Processes that involve significant energy ex-change between the electronic and vibrational degrees of freedom and include relaxation inside thesemiconductor CB, back-ET to the chromophore, electron loss to the electrolyte, and ET between elec-trolyte and the chromophore require more sophisticated treatments, such as the SH procedure de-scribed below.

2.2.2. Fewest switching surface hoppingFewest switches SH (FSSH) is a popular and well-tested approach to NA MD [103]. It is designed [100]

to work in condensed phase systems and to account for both branching [100] and detailed balance [101],which are missing in the Ehrenfest and CPA schemes. FSSH can be interpreted as a quantum masterequation, in which the transition rates are non-perturbative and evolve in time. The time dependenceof the FSSH transition probabilities reproduces the short-time behavior of quantum transitions, includ-ing the quantum Zeno [104] and anti-Zeno [105] effects. The dependence of the transition probabilitieson the vibrational coordinates creates electron-vibrational feedback, which works in both directions.

The probability of hopping from state j to state k during the time step Dt is given in FSSH by

gjkðt;DtÞ ¼ max 0;bkjDtajjðtÞ

� �; ð8Þ

where

akjðtÞ ¼ c�kðtÞcjðtÞ; and ð9Þ

bkj ¼ 2�h�1Imðakjhu

�kjHju

�jiÞ � 2ReðakjdkjÞ:

If the calculated gjk is negative, the hopping probability is set to zero; a hop from state j to state kcan occur only when the electronic occupation of state j decreases and the occupation of state k in-creases. This feature of the algorithm minimizes the number of hops and is responsible for the nameof the technique, ‘‘fewest switches”. In the adiabatic representation, Eq. (4), the imaginary term in Eq.(9) vanishes. The nuclear velocities are rescaled after a hop to conserve the total energy of electronsand nuclei in the standard formulation of FSSH [85,100]. If the kinetic energy available to the nucleialong the direction of the NA coupling is insufficient to accommodate an increase in the electronic en-ergy, the hop is rejected. The hop-rejection creates the detailed balance between upward and down-ward transitions [101].

In the simplified implementation of FSSH [76], which assumes that the energy exchanged betweenthe electronic and vibrational degrees of freedom during a hop is rapidly re-distributed among allvibrational modes, the velocity-rescaling and hop-rejection are replaced by multiplying the probabil-ity (Eq. (8)) for transitions upward in energy by the Boltzmann factor. This simplification leads to sig-nificant improvements in the computational efficiency of FSSH, since it permits use of the ground statenuclear trajectory to determine the TD potential that drives the electron dynamics, Eq. (5), just as inthe CPA. The FSSH algorithm has been employed to study the relaxation inside the semiconductor CB,the transfer of the electron back to the chromophore, the electron loss to the electrolyte and the ETfrom the electrolyte and the chromophore.

Our implementation of FSSH within TDDFT [106] and its simplified version [76] describe excitedelectronic states by promoting electrons between KS orbitals. The more sophisticated implementationof FSSH within TDDFT uses carefully chosen linear combinations of KS excitations [107] or linear re-sponse TDDFT excited states [108]. These approaches are generally more accurate and, at the sametime, are more computationally expensive. Refs. [109–112] discuss the calculation of the NA couplingfor such schemes.

2.3. The electron transfer mechanisms

ET processes can take place by one of the two alternative mechanisms, adiabatic and NA [10,113].These two mechanisms are characterized by distinct sets of properties and show a very different depen-dence on the details of the interface configuration. Therefore, it is critical, from both fundamental and


practical view points, to first establish the mechanism of a particular ET event. In the adiabatic mech-anism, the electronic coupling between the chromophore and the semiconductor is large, and ET occursby means of a concerted motion of nuclei that drive the system through a transition state. The electronremains in the same Born–Oppenheimer, adiabatic state that changes localization from the chromo-phore to the semiconductor along the ET reaction coordinate. Adiabatic ET can be efficient regardlessof whether the electron can transfer to only a single acceptor state, for instance from a chromophoreto a Ti4þ ion in solution, or to one of many TiO2 surface states [12,114–118]. Adiabatic ET shows Arrhe-nius temperature dependence and requires an active energy exchange between different vibrationalmodes. Since adiabatic ET is constrained to one electronic state, relatively small amounts of energycan be exchanged this way between electronic and vibrational degrees of freedom. Thus only electroninjection and delocalization into the bulk can occur adiabatically. Electron relaxation inside the TiO2 CB,back-transfer to the chromophore ground state, and ET involving the electrolyte are all NA processes inwhich a significant amount of energy flows between electronic and vibration degrees of freedom. Eventhe injection can proceed non-adiabatically, for instance in the systems involving long chromophore–semiconductor bridges.

The electronic donor–acceptor coupling decays exponentially with the distance between the donorand acceptor species. At some point, the coupling becomes too small to support the adiabatic mecha-nism, and the ET proceeds non-adiabatically via a quantum-mechanical transition from the chromo-phore state to one of the acceptor states in the TiO2 CB manifold. The NA transfer can be as fast asthe adiabatic transfer, provided that multiple electron acceptor states are available in the system.Therefore, efficient NA ET requires the chromophore excited state to be energetically located deep in-side the TiO2 CB. Any process that lowers the energy of the donor state, for instance the electron-vibra-tional or single-triplet relaxation in the chromophore, will bring the donor energy closer to the CB edgeand will lower the injection efficiency. At the same time, adiabatic ET can occur right at the CB edge andtherefore avoid the energy and voltage losses that are associated with the electron relaxation inside theTiO2 CB. On the other hand, weak chromophore–semiconductor coupling, which results in NA injection,reduces the rate of transfer of the injected electron back onto the chromophore and can thus improvethe overall current. The different dependence of the photovoltaic current and voltage on the chromo-phore–semiconductor coupling creates optimization possibilities for the solar cell performance.

The adiabatic transfer can be described by the Marcus transition state theory, while the NA ET istypically treated by a perturbation theory such as the Fermi Golden Rule [10,113]. Intermediate do-nor–acceptor couplings can entail a combination of both ET mechanisms. In order to distinguish be-tween the adiabatic and NA ET mechanisms, the following mathematical procedure is used duringthe simulation [40]. The portion of the excited electron that remains on the dye is determined by inte-grating the electron density of the photoexcited state over the region of the simulation cell that isoccupied by the dye,
Z
dyeqðr; tÞdr ¼

XNe

a¼1

Zdyejuaðr; tÞj

2dr ¼XNe

a¼1

Xi;j

ðcai Þ�ca

j

Zdye

u� �

i ðr;RðtÞÞu�

jðr;RðtÞÞdr: ð10Þ

The time derivative of this density

dR

dye qðr; tÞdr

dt¼XNe

a¼1

Xi;j

d½ðcai Þ�ca

j �dt

Zdye

u� �

i u�

jdrþ ½ðcai Þ�ca

j �dR

dye u� �

i u�

jdr

dt

( )ð11Þ

describes the progress of the ET process and can be separated into the adiabatic and NA contributions.The first term arises from changes in the occupations of the adiabatic states and, hence, represents NAET. The second term gives changes in the electron density for fixed occupations of adiabatic states andrepresents adiabatic ET.

3. Interface structure

The details of the chromophore–semiconductor interface depend on the chromophore, the semi-conductor; the relative position, orientation, and binding between the two species; the chromo-


phore–semiconductor bridge; the solvent; the electrolyte; and other factors. The general propertiescan be understood with a few representative examples. The TiO2 semiconductor has a long historyin photoelectrochemistry [119,120], and is chosen as a cheap and robust oxide, in contrast to the moreexpensive and delicate pure silicon which is commonly used in solar cells. It can be made highly por-ous in order to expose a large surface area for interaction with chromophores. The chromophores arechosen to actively harvest the visible light that penetrates through the ozone layer and reaches thesurface of the Earth. They should bind to TiO2, match the semiconductor energy levels, and be photo-chemically and thermally stable.

The properties of the individual constituents that form the interface are relatively well understood,but they are discussed differently in different scientific communities. Chromophore molecules arestudied by chemists, while solid-state semiconductors are investigated by physicists. This inconsis-tency means that a proper description of the interface as a whole, combining the periodic propertiesof the semiconductor with the local molecular structure, remains a challenge.

3.1. Interface geometry

The majority of chromophore–semiconductor ET studies are carried out with TiO2 nanoparticles,which exist as a mixture of anatase and rutile crystal forms and have a variety of surfaces [121–123] (Fig. 2). Rutile(110) is one of the most stable and best studied surfaces. It has a well-describedideal structure, but several different kinds of defects can complicate the depiction. These include stepedges, oxygen vacancies, line defects, and impurities. Variations in the structure can significantlyinfluence the chemistry of surface reactions. Under high vacuum, the surface will reconstruct in orderto saturate the dangling bonds as much as possible. Bare TiO2 is highly chemically reactive and willinteract with those substances that are present in solution and in the air. Molecular adsorption isproblematic even on ideal surfaces. In particular, the interaction of rutile(110) with water, which pro-vides a termination layer both in solution and in the air, is complex; it can involve both molecular anddissociative absorption, and there is debate about which is the dominant form [124,125]. Low surfacecoverage encourages dissociation, which is followed by molecular adsorption as the coverage level in-creases [123,126]. Defects favor dissociative binding, while hydrogen bonding to other physisorbedwater molecules stabilizes molecular adsorption. Rutile(100) has been subjected to less investigation,but it has several advantages when used in theoretical calculations. The interaction of water with ru-tile(100) is straightforward, and a general form of the stable surface termination can be easily estab-lished. In simulation, the (100) surface allows for a smaller cell. Rutile(001) has a relatively highenergy compared to (110) and (100) [123].

Many experimental studies use TiO2 colloids that are prepared by hydrolysis of TiCl4 in cold water[127]. Such colloids have high anatase content. Anatase surfaces are less well understood because theyare harder to investigate experimentally; large anatase crystals are more difficult to obtain than aretheir rutile counterparts because bulk rutile is more stable than bulk anatase. Below a critical size,however, anatase is the more preferable substance, because of its lower surface energy relative to ru-tile. Above the critical size, a mixture of anatase and rutile particles can be converted to pure rutile byheating. The great variety of TiO2 structures, surfaces, impurities and defects create significant chal-lenges to theoretical as well as experimental studies. One would like to presume that the results ob-tained for one surface will be characteristic of the entire class of interfacial ET reactions.

The chromophores that are used in the chromophore-sensitized TiO2 photovoltaic cell fall into twobroad categories (Fig. 2): purely organic conjugated molecules, such as alizarin [12–14], and transi-tion-metal/ligand complexes, such as Ru(dcbpy)2(NCS)2. The latter, also known as the N3 dye, hasemerged as a classic sensitizer for mesoporous solar cells [128]. The photoactive ligands in the tran-sition metal chromophores are also conjugated organic molecules. They are, however, smaller than thepurely organic chromophores. The chromophores bind to the TiO2 surface with oxygen-containingsubstituents such as hydroxy, carboxy, and phosphoric acid groups. Binding via the hydroxy group re-sults in a very strong chromophore–semiconductor coupling, since only a single oxygen atom sepa-rates the chromophores from the semiconductor. The coupling decreases with carboxy binding and,further, with binding through the phosphoric acid group. In many instances, chromophore stability,synthetic and surface conditions, and other factors require longer bridges. The additional bridging

Rutile (110)

Rutile (100)

Anatase (101)

- Titanium - Oxygen

N

N

N

N

Ru

NCS

NCS

COOH

HOOC

HOOC

COOH

Ru(dcbpy)2(NCS)2

O

O

HO

OH

Alizarin

Fig. 2. Representative semiconductor surfaces and molecular chromophores. The most commonly used TiO2 semiconductorexists in two crystalline forms: anatase and rutile. Surface terminating oxygens interact with chromophores and water. Twotypes of oxygens are seen. Those that protrude from the surface have unsaturated bonds and are chemically reactive. Thebridging oxygens located between surface titaniums have no dangling bonds and can interact with the adsorbed species viahydrogen bonding. The chromophores can be classified into two types: those containing a transition metal, e.g. Ru, that isligated with smaller organic species, and those that are larger purely organic molecules, such as alizarin. Transition metalchromophores are used more commonly in the Grätzel cell because of their excellent electrochemical stability and their morefavorable electron transfer properties. The great variety of organic chromophores can provide additional flexibility with thespectral energy range for light-harvesting.


groups, most typically ðCHÞn and ðCH2Þn, slow down ET both in the forward and in the backward direc-tions [46] and may be advantageous when back-ET from TiO2 to the chromophore becomes a practicalconcern. The hydroxy, carboxy, phosphoric acid and ðCHÞn groups preserve the chromophore–semi-conductor conjugation, since they simply extend the p-electron system of the chromophore. TheðCH2Þn bridges break the conjugation and significantly decrease the electron donor–acceptor coupling.Chromophores can also physisorb onto the semiconductor. Physisorption becomes the dominantmode of chromophore–semiconductor interaction in the absence of chemical and hydrogen bonding


or other specific interactions. Large conjugated molecules can lie flat on the surface and interact withthe semiconductor by van der Waals interactions. Chromophore molecules that provide multiple bind-ing groups, such as alizarin with its two hydroxyls and Ru(dcbpy)2(NCS)2 with multiple carboxyls(Fig. 2) can span more than one Ti atom and generate bidentate binding (see the bottom part ofFig. 3). In many cases, it is safe to assume that binding to the semiconductor does not alter the chro-mophore geometry. Sometimes, particularly with bidentate binding, the chromophore may becomedistorted in order to provide a better match with the surface geometry [44]. The redox mediator doesnot permanently bind to the semiconductor surface and shuttles between the surface and the counter-electrode. The reduced form of the electrolyte molecule, typically [129] I�2 , can be attracted towardsthe surface and has the opportunity to replace an OH� group and to create a transient chemical bond.

3.2. Electronic properties of the interface

The rates and yields of the ET reactions at the chromophore–semiconductor interface are deter-mined by the energies of the chromophore and semiconductor states, the electronic coupling betweenthese states, and the electron-vibrational interaction [113]. The gap between the valence and conduc-tion bands of bulk TiO2 is around 3.2 eV (Fig. 1). Specifically, the gaps are [130] 3.035 eV for rutile and3.420 eV for anatase TiO2. The work-function of TiO2 strongly depends on surface preparation. Thevacuum level of the bare, stoichiometric rutile(110) surface is around 5.6 eV above the bottom ofthe TiO2 CB [131]. The work-function of water covered rutile(110) at pH = 1 is much smaller. In thiscase, the CB starts 4.5 eV below the vacuum level [119], which is close to the redox potential of thenormal hydrogen electrode. The CB of TiO2 is created by the d-orbitals of the titanium atoms. TheTiO2 valence band (VB), in contrast, is formed by the valence electrons of the oxygen atoms. Surfaceeffects induce substantial changes in the electronic structure of the bulk [20]. Localized surface statesappear both within the bands and inside the gap [12,114–118,126]. The dangling bonds of unsaturatedsurface atoms introduce multiple states that lower the gap energy to a few 10ths of an electron-volt.

Fig. 3. Simulation cells composed of rutile surfaces sensitized with representative transition metal and organic chromophores.The TiO2 surfaces are terminated with dissociated water molecules, as expected in solution.


Surface reconstruction and healing reopen the gap. The termination of the surface with dissociatedwater or other chemically bonded species brings the gap close to the bulk value. The energies of mostsurface states that exist in this case lie within the bands [116–118,126]. As an n-type semiconductor,TiO2 creates a negatively charged surface region by trapping electrons at the surface when it is in con-tact with air, liquid, or metal. In order to preserve electrical neutrality, a layer of positive chargesdevelops just within the semiconductor, causing a shift in the electrostatic potential and a bendingof band energies upward in the region near the surface. An accumulation of positive charge carriersat the surface moves the bands down. This is possible, for instance, in solutions with low pH [14].The accumulation of charge at the interface can also create a Schottky barrier to the electron injection.The surface charging effects are significantly decreased when the surface is chemically terminated.

The chromophore photoexcited states are similar in both organic and transition metal chromoph-ores (Figs. 2 and 3). They are formed by the p�-electron orbitals of the conjugated systems. The groundstate of the purely organic chromophores is also a p-state, but the ground state of the transition metalchromophores is localized on the n-orbitals that are occupied by the metal’s undivided electron pairs.The purely organic chromophores contain large p-conjugated systems that create small p-p* energygaps within the visible spectrum. The energy gap in the transition metal chromophores is determinedby the n–p* energy difference and can be tuned regardless of the p–p* splitting, which depends on theextent on the p-electron system. Therefore, the ligands that are attached to the transition metal arerelatively small. The p-conjugated system of the purely organic chromophores can be made smallerby oxygens, which lower the excited state energy as a result of the stronger electron affinity relativeto the carbons. The alizarin chromophore shown in Figs. 2 and 3 contains oxygens and is smaller thanother organic chromophores, such as perylene [4,5], that contain only carbons. Compact p-electronchromophores such as catechol are used to sensitize TiO2 as well. However, the visible spectral bandin such cases relies on strong chromophore–semiconductor coupling and involves direct transitionfrom the p-ground state into the TiO2 CB, bypassing the p*-excited state that is high in energy[45,132,133]. The relatively minor differences in the photoexcited states of the organic and transitionmetal chromophores result from the larger excited state delocalization in the purely organic chro-mophores and the positive charge on the transition-metal. Additional complications with the transi-tion metal chromophores arise from the strong spin-orbit coupling that the metal induces. The spin-orbit coupling provides a high probability of intersystem crossing into triplet states, which may lie be-low the semiconductor CB [9,29,30].

While the energy levels of the isolated chromophore and semiconductor species are typically wellknown from both experimental and theoretical work, the electronic coupling is not so well understoodand thus can be measured only indirectly or computed theoretically. In the weak coupling limit thatgenerates NA ET (see Section 2.3) the chromophore and semiconductor states are perturbed only a lit-tle. However, in the cases of strong chromophore–semiconductor interaction, the electronic states ofthe two species change significantly. They mix, and the photoexcited state becomes delocalized be-tween the chromophore and the semiconductor surface. The properties of the surface states may thenbe influenced by the chromophore, sometimes, for example, becoming strongly localized. The align-ment of the chromophore, semiconductor and electrolyte energy levels represented in Fig. 1 is repre-sentative of most systems. In some cases, e.g. alizarin, the photoexcited state can be located close tothe bottom of the TiO2 CB, complicating the electron injection dynamics and requiring strong donor–acceptor coupling. If the chromophore–semiconductor coupling is extremely strong, an absorbed pho-ton can promote an electron directly into the CB, bypassing the injection step [38,39,74].

Thermal atomic motion can have a strong effect on the geometric and electronic structure of theinterface [39,72,77]. The binding energy of water and electrolyte molecules to the surface may beweaker than kBT , and a structure that is stable at zero Kelvin can be substantially perturbed at a finitetemperature. Thermal fluctuations in the atomic coordinates influence the electronic energy levels,primarily those of the localized states of the chromophore and the electrolyte. The chromophoreand electrolyte state energies shift relative to the TiO2 CB, which contains delocalized states and re-mains relatively stationary with respect to atomic motions. The fluctuation of the molecular state en-ergy closer and farther away from the CB edge, as well as between regions of higher and lower CB statedensity, affects the ET reaction rates and mechanisms and creates an inhomogeneous distribution ofinitial conditions. The energy fluctuations occur over a large distribution of frequencies, with the time-


scales starting from 10s of femtoseconds and extending to the picosecond range. The faster vibrationsoriginate in the stretching and bending of chemical bonds. The slower ones result from large-scale mo-tions, such as the bending of the whole chromophore molecule with respect to the surface. The low-frequency motions modulate the relative positions of the energy levels, while the higher frequencymodes determine the electron-vibrational coupling that drives the photoexcited electron dynamics.

4. Injection from the chromophore to the semiconductor

Time-domain ab initio simulations illustrate many features of the electron injection process. Nei-ther static ab initio electronic structure calculations, nor reaction-rate theories, nor more advancedphenomenological treatments of ET can fully capture the processes as they occurs in nature. Thereal-time atomistic simulations, however, approximate the reality most closely. The simulations areable to model both the initial photoexcitation process and the ensuing non-equilibrium electron–pho-non dynamics. We focus on the processes that follow the photoexcitation.

In order to illustrate the photoinduced electron injection at the chromophore–semiconductor inter-face we consider two systems [72,73] (Fig. 3). The top system is part of a transition metal chromo-phore, such as Ru(dcbpy)2(NCS)2 (Fig. 2). The full chromophore requires large simulation cells andcannot yet be treated by ab initio NA MD. The model chromophore is composed of isonicotinic acid,which is part of the transition metal ligand, a model transition metal (Ag) that requires only two li-gands, and the second ligand ðCN�Þ. The alizarin–TiO2 system shown in the bottom of Fig. 3 has beenextensively studied experimentally [12–14]. The isonicotinic acid chromophore binds to TiO2 via thecarboxy group, which forms one chemical bond with a Ti atom. Alizarin is bonded chemically to thesemiconductor via its two hydroxy oxygens and creates a stronger electronic donor–acceptor cou-pling. As a quinone, alizarin can exist in several isomeric forms [134,133]. The structure shown inFig. 3 corresponds to the most stable binding conformation.

4.1. The role of atomic motions

Atomic motions of the chromophore and of the semiconductor surface play many key roles in thephotoexcited electron’s dynamics [39,72,77]. They create an inhomogeneous distribution of initialconditions for the electron injection and its subsequent evolution, and they drive both adiabaticand NA ET events by carrying the system over a transition state during adiabatic ET and creatingthe NA coupling (6). Atomic motions also influence the strength of the electronic donor–acceptor cou-pling and are responsible for the dissipation of the injected electron’s energy. Thermal fluctuations ofthe atoms can be quite significant, especially in the case of loosely bound species at the interface.Additional kinetic energy is provided by the interaction of the nuclei with the excited electron.

Fig. 4 presents the evolution of the excited electronic state energies in the isonicotinic acid and aliz-arin systems that are shown in Fig. 3. The bottom panel of Fig. 4 gives the localization of the photo-excited state on the chromophore in the alizarin system. The majority of the states in the top twopanels are bulk and surface states that represent the TiO2 CB (gray lines). Only the first excited stateof each chromophores falls within the shown energy range; it is indicated by the dark bold line. Theenergies of the photoexcited states are sensitive to the positions of the atoms in the chromophore–TiO2 system and oscillate with the amplitude of several 10ths of an electron-volt. The oscillation isfairly small relative to the excitation energy of several electron-volts, however, it has a significant ef-fect on the positioning of the excited state with respect to the TiO2 CB. The density of states (DOS) ofthe TiO2 CB increases with energy, such that the chromophore excited state can interact with substan-tially fewer semiconductor states near the oscillation minimum than near the oscillation maximum.

The energies of the CB states also vary depending on the atomic coordinates. However, this varia-tion is much less significant than is the fluctuation of the chromophore state energy. This lesser sig-nificance comes about because the CB states are delocalized, and the effect of local atomicfluctuations averages out. In contrast, the chromophore states are localized on only a few atomsand are therefore sensitive to the exact positions of these atoms. The photoexcited state in the isoni-cotinic acid system is positioned relatively deep inside the TiO2 CB. Only rarely does it fluctuate below

-5.0isonicotinic acid

-5.2

-5.4

-5.6

-5.8

-5.0

alizarin

500400300200100 0sf,emiT

Ener

gy, e

V

-5.2

-5.4

-5.6

-5.8

0

2.0

4.0

6.0

8.0

1

Loca

lizat

ion

alizarin

Fig. 4. Evolution of the chromophore excited state and TiO2 conduction band (CB) energies (top two panels) and of thephotoexcited (PE) state localization on the chromophore (bottom panel). Atomic motions induce significant fluctuations in theelectronic energies, particularly of the chromophore states. The energy of the isonicotinic acid chromophore (see top panel inFig. 3) fluctuates primarily inside the CB, while the alizarin excited state oscillates near the bottom of the CB. Every time thealizarin state dips below the CB edge, the PE state becomes well localized on the chromophore. While inside the CB, the PE stateis significantly delocalized onto the semiconductor. Here and in Figs. 6, 8, 11, 14, the energy origin is set to the vacuum level ofthe DFT calculation.


the CB edge. The situation is the opposite in the case of the alizarin state. It fluctuates around the CBedge, spending a significant fraction of time below the edge. This difference is reflected in the ET pro-cess, as described below.

The bottom panel of Fig. 4 shows the localization of the photoexcited state on the chromophore inthe alizarin–TiO2 system. The localization data correspond to the energy evolution shown in the mid-dle panel. The localization is also very sensitive to the atomic motion. If the alizarin excited state isinside the CB, it couples very strongly to the TiO2 states. The donor–acceptor coupling is large, because


only oxygen atoms in the hydroxy groups stand between the chromophore and the semiconductor(Fig. 3). As a result, the chromophore and semiconductor states mix, and the photoexcited state isdelocalized between the two subsystems. If the alizarin state is below the CB edge, it does not mixwith the semiconductor states, even though the electronic coupling remains strong. In such cases,the photoexcited state is localized on the chromophore.

Fig. 5 presents Fourier transforms (FT) of the fluctuations of the photoexcited state energy andlocalization shown in Fig. 4. The FTs reveal the nature of the atomic motions that induce these fluctu-ations. Fig. 5a deals with the energy fluctuations in the isonicotinic acid and alizarin systems. A singledistinct peak at 1600 cm�1 dominates the FT in the isonicotinic acid case. This frequency is associatedwith a carbon stretching motion, which plays the key role in this small molecule. Since the first excitedstate of the chromophore is a p*-state localized on the ring carbons, an oscillation of these carbonsmodulates the energy of the state. The shortening of the C–C distance increases the interaction be-tween the Pz electrons of the carbon atoms. As a result, the splitting between the p-bonding andp*-antibonding orbitals grows, and the energy of the p* excited state increases. On the other hand,elongation of the C–C distance diminishes the interaction between the Pz electrons of the carbonatoms, resulting in a smaller splitting and a lower excited state energy. In contrast, the FT of the photo-excited state energy fluctuation in the alizarin system shows a range of frequencies primarily in thelow energy region, 700 cm�1 and below. The difference from isonicotinic acid stems from two factors.First, alizarin itself is a significantly bigger molecule with a larger set of frequencies, most of which arein the lower energy range. Second, the donor–acceptor coupling is stronger with alizarin than withisonicotinic acid. This is because alizarin connects to the semiconductor via two bridges, each contain-ing only one oxygen atom, while isonicotinic acid binds to the semiconductor via a single bridge con-sisting of two atoms, oxygen and carbon. Due to the stronger electronic coupling, the photoexcited

200050011000500 0

40003000200010000

Am

plitu

de, a

rbitr

ary

units

004300420041

02x

Frequency, cm-1

a

b

Fig. 5. Fourier transforms (FT) of the fluctuations of (a) excited state energy and (b) localization that are shown in Fig. 4. Theenergy of the isonicotinic acid chromophore (see top panel of Fig. 3) fluctuates at the C–C stretch frequency of 1600 cm�1, asdepicted by the solid black line in part (a). Because alizarin is bigger than isonicotinic acid, its excited state energy fluctuatesover a range of lower frequency modes, as depicted by the dashed red line. The localization of the alizarin state oscillates at thesame frequencies as the energy. In addition, the localization shows minor dependence on the motion of the hydroxyl groupsterminating the TiO2 surface, as shown in the insert in part (b). The electronic energy shows no response in the OH frequencyrange.


state of alizarin delocalizes onto the semiconductor, which contains heavy Ti atoms and contributeslow frequency modes to the FT spectrum.

The FT of the excited state localization in the alizarin–TiO2 system shown in Fig. 5b is also domi-nated by the low frequency modes and is similar to the FT of the energy. There is a difference in thehigh frequency range corresponding to hydrogen motions. The fluctuation in the photoexcited stateenergy shows no contribution from hydrogens. It is not surprising, since this p-electron state is notlocalized on hydrogens at all. At the same time, the localization is sensitive to the high frequencyhydrogen motion, as emphasized by the insert in Fig. 5b. Further analysis reveals that the hydrogensthat influence the state localization are not the chromophore hydrogens, but rather those of the OHgroups that terminate the semiconductor surface (see Fig. 3). The strong dipole moments of the sur-face hydroxy groups contribute to the electronic chromophore–semiconductor coupling and affect thestate mixing. In general, the electronic wave function that determines the state localization is sensitiveto a wider range of factors than the electronic energy.

4.2. The distribution of initial conditions

The fluctuations in the atomic coordinates generate a distribution of the photoexcited state ener-gies and localizations and create an inhomogeneous ensemble of initial conditions for the photoin-duced ET processes [72,77]. This fact is well exemplified with the alizarin–TiO2 system in Fig. 6,which depicts the localization of the photoexcited state on the chromophore as a function of the stateenergy (circles) superimposed on the TiO2 DOS. Below the CB edge, the TiO2 DOS is low, and there isvery little mixing between the alizarin and semiconductor states. The localization of the photoexcitedstate on alizarin is therefore close to a value of one (filled circles). As the energy increases, progres-sively more semiconductor states mix with the chromophore state. Under these circumstances thelocalization decreases (empty circles), and a significant fraction of the photoexcited electron spreadsonto TiO2. In some instances, the photoexcited state is well localized on the chromophore. In othercases, over half of the photoexcited state density is on the semiconductor. The large variation in thelocalization data at the energies inside the CB is somewhat unexpected and indicates that even in aperfect system, with no defects and well-defined bulk and surface structures, the ET process is veryinhomogeneous and changes from sample to sample.

The localization of the photoexcited states that are inside the TiO2 CB depends on several factorseven in a perfect system. These include the presence of a surface state near the chromophore state,both energetically and spatially, and the strength of the electronic donor–acceptor coupling, which de-pends not only on the relative positioning of the chromophore and the semiconductor, but also on the

0

2.0

4.0

6.0

8.0

1

8.4-0.5-2.5-4.5-6.5-8.5-Ve,ygrenE

Loca

lizat

ion

TiO2 D

ensity of States

Fig. 6. Localization of the photoexcited (PE) state in the alizarin–TiO2 system as a function of energy (circles), plotted togetherwith the TiO2 density of states (solid red line). If the PE state is below the TiO2 conduction band, it is well localized in thechromophore (filled circles). If the PE is inside the band, it may or may not be significantly delocalized onto the semiconductor(empty circles), depending on the chromophore–semiconductor coupling and the energy match between the chromophore andsurface states.


geometry of the bridge and the surface terminating OH groups. The number of semiconductor statesthat can couple to the chromophore at a particular energy fluctuates over time. In a given sample thedensity of TiO2 surface states at the alizarin state energy may be substantially greater or less than theaverage. Even if the density of the acceptor states is the same, the spatial overlap between them andthe chromophore excited state varies substantially between the configurations, depending on the cur-rent geometry of the docking region and orientation of the surface OH groups. The chromophore ex-cited state strongly couples and mixes with the states of the semiconductor when the latter producesan instantaneous nuclear fluctuation that supports those surface states that are energetically and spa-tially close to the molecular state. Additional factors that create further inhomogeneity in the initialconditions, but have not yet been accounted for in the simulations, include the configuration of thesolvent, the fluctuations in the chromophore–semiconductor bonding, defects, and the presence ofother chromophores and injected charges. The atomistic simulations show that the chromophore–semiconductor interface cannot be viewed as a single homogeneous object, and that its propertiesvary a great deal, depending on multiple factors.

4.3. The electron injection mechanisms

The ET can occur by either the adiabatic or the NA mechanism (see Section 2.3). During adiabaticET, the electron remains in the same electronic state, while the state itself changes its nature. Nuclearmotion carries the system over a transition state. The donor state is located on one side of the transi-tion barrier, and the acceptor state is on the other. A small barrier relative to the nuclear kinetic energygives fast adiabatic ET. In contrast, NA processes involve electron jumps between different adiabaticstates. If the initial adiabatic state before the jump is localized on the electron donor, and the finalstate is on the acceptor, the NA jump produces ET. While adiabatic ET can be thought of as a purelyclassical-mechanical process, and may or may not involve quantum-mechanical tunneling throughthe barrier, NA ET is necessarily a quantum-mechanical event. The two mechanisms have very differ-ent properties, as discussed in Section 2.3. Both mechanisms can compete with each other and act inparallel during the electron injection.

Fig. 7 shows the averaged progress of the photoinduced electron injection in the isonicotinic acidand alizarin systems. The overall ET is separated into the adiabatic and NA contributions. The alizarincase is further separated into the ET from the initial states that are below and above the TiO2 CB edge(see Fig. 6). Each line is fit to an exponential function, except for the low energy alizarin case, which isdiscussed separately below. The ET in the isonicotinic acid occurs on a 5 fs timescale, in agreement withthe experimental data [18]. The ET coordinate starts around 0.6, indicating that nearly 60% of the elec-tron is already inside the TiO2 surface as soon as the photon is absorbed. This is because the chromo-phore–semiconductor coupling is strong, the chromophore molecule is small, and mixing of thechromophore and semiconductor electronic states results in a significant contribution of the semicon-ductor to all the adiabatic states of the system, including the photoexcited state. The transfer is primar-ily adiabatic, again due to the strong donor–acceptor coupling. A notable NA component is observed,though. If 55% of ET takes place by the photoexcitation, 30% is transferred adiabatically and 15%non-adiabatically. The adiabatic transfer is very fast, because the chromophore excited state is locatedin the region of high TiO2 DOS. The small oscillations in the total and adiabatic ET data in the isonicot-inic system relative to the fit line are similar to those observed by Willig and co-workers with perylene[68] and are due to coherent nuclear vibrations. The 20 fs oscillation period agrees with the 1600 cm�1

frequency that is most prominent in the oscillation of the photoexcited state energy (Fig. 5a).The high-energy alizarin data are very similar to those in the isonicotinic acid case. A little less

transfer occurs during the photoexcitation, because alizarin is a larger molecule, and mixing withthe semiconductor states affects alizarin states to a lesser extent. The adiabatic mechanism dominatesover the NA mechanism even more, because the chromophore–semiconductor coupling is stronger.The coupling is facilitated by two single-atom oxygen bridges, rather than one two-atom carbon-oxy-gen chain, as in the isonicotinic acid (Fig. 3). The low-energy alizarin data are qualitatively differentfrom the rest of the plots. The initial state is almost entirely localized on the chromophore, and theET coordinate starts close to zero. The adiabatic ET component in this case evolves non-exponentiallyat the early times. This occurs because adiabatic ET can take place only if the photoexcited state enters

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Fig. 7. Electron transfer (ET) progress in the isonicotinic acid and alizarin systems. The bottom panels represent ET in thealizarin system averaged separately over the high and low energy initial conditions that correspond to the filled and emptycircles in Fig. 6. Each panel shows the overall ET and the contributions from the adiabatic and non-adiabatic (NA) mechanisms.The initial value of the ET coordinate indicates the fraction of the photoexcited state that is localized on the chromophore. Theisonicotinic acid system exhibits a larger relative contribution from the NA mechanism, because of the weaker donor–acceptorcoupling that results from the longer chromophore–semiconductor bridge (Fig. 3). The alizarin high energy data resemblesthose for isonicotinic acid. The alizarin low energy data shows an initial Gaussian component associated with the time requiredfor the photoexcited state to enter the TiO2 conduction band.


the TiO2 CB. Thus, in the very early stages the NA ET dominates over the adiabatic ET. The first deriv-ative of the adiabatic ET coordinate with respect to time is zero at time zero, which is not possible withexponential time-dependence. Both adiabatic and NA transfer components are slower at the lowerenergies. Because the TiO2 DOS decrease with decreasing energy (Fig. 6) a longer wait is required tofind a transition state for adiabatic ET. Similarly, smaller DOS at lower energies provides fewer semi-conductor states to couple to, also leading to slower NA ET. The ET in the alizarin–TiO2 system aver-aged over all of the initial conditions is essentially exponential, because the fraction of the initialconditions that are below the CB-edge is relatively small, less that 50% (see Figs. 4 and 6). It is quiteremarkable [73] that photoexcitation below the CB can lead to fast and efficient electron injection!

The study of the photoinduced electron injection in the isonicotinic system at a low temperatureshowed that adiabatic and NA mechanisms carry similar contributions to the ET process. The in-creased role of the NA ET at a low temperature is observed, because the adiabatic ET rate exponentiallydecreases with temperature, while the NA rate is relatively temperature independent. In the systems,in which the chromophore is separated from the semiconductor by longer bridges, in particular thosethat break the direct electronic conjugation between the two subsystems, the ET occurs primarily bythe NA mechanism, and the application of the Fermi Golden Rule is entirely appropriate [10,113]. Theeffect of temperature on the adiabatic and NA ET rates is discussed next.

4.4. Temperature effects

The efficiency of Grätzel cells is remarkably temperature independent [135]. At the same time, theadiabatic ET rate should depend on temperature exponentially [10,113]. The NA ET rate depends on


the nuclear velocities that determine the NA coupling, Eq. (6). Therefore, the NA rate should depend onthe nuclear kinetic energy and temperature as well, but to a lesser extent than the adiabatic rate. Ther-mal atomic motions play multiple additional roles in the ET dynamics, as discussed above. They createan inhomogeneous distribution of initial conditions; modulate state energies, localizations, and do-nor–acceptor couplings; and transform excess electronic energy into heat. The experimental tempera-ture insensitivity of the ET dynamics creates an apparent contradiction to the overall importance ofthermal effects.

Having performed [77] a series of simulations on the alizarin–TiO2 system at temperatures rangingfrom 50 to 350 K, here we contrast the lowest and highest temperatures. Zero temperature calcula-tions show that the alizarin photoexcited state is located at the TiO2 CB edge [132,133]. Therefore,thermal nuclear motions that drive the photoexcited state in and out the CB (Fig. 4) should play a par-ticularly important role relative to other chromophores, such as isonicotinic acid, whose photoexcitedstate is deeper inside the TiO2 CB. The alizarin example should, therefore, show stronger temperaturedependence of the electron injection process than an average chromophore.

The kinetic energy that is present in the system at a finite temperature generates atomic fluctua-tions around the equilibrium geometry. The inserts in Fig. 8 illustrate the amplitude of the thermalatomic motions. Comparing typical snapshots from the ab initio trajectories at 50 and 350 K, we ob-serve that the TiO2 structure changes very little at the increased temperature, while the alizarin mol-ecule and the hydrogens that terminate the interface move significant distances at 350 K. Thesechanges in the chromophore geometry affect the photoexcited state energy. Out-of-plane bending dis-tortions perturb p-conjugation and raise the energy. The chromophore excited state enters the semi-conductor CB, generating favorable conditions for the ET.

The distribution of the photoexcited state energies created by the thermally induced fluctuations ofthe system geometry is plotted in Fig. 8 with a dashed line. The distribution is created by multiple nu-clear vibrations and can be well represented by a Gaussian. It is significantly broader at the elevatedtemperature. The figure also shows the thermally averaged DOS of the TiO2 CB (solid line). The DOSfeatures become broadened and smeared out at the higher temperature, even though the semiconduc-tor atoms move very little. Small fluctuations in the TiO2 coordinates generate disorder in the crystal

50 K

350 K

-5.8 -5.6 -5.4 -5.2 -5 -4.8Energy, eV

Fig. 8. Distribution of the photoexcited state energy (dashed lines) and the TiO2 density of states (DOS) at 50 and 350 K. Thestate energy distribution and DOS are broadened at the elevated temperature. The insert illustrates the extend of the thermallyinduced fluctuations in the molecular geometries.


structure and lower the CB edge. Both effects, the broadening of the excited state energy distributionand the lowering of the CB edge, are favorable for the electron injection. At higher temperatures, thephotoexcited electron is able to interact with a higher density of TiO2 states and injects more easily.Thermal atomic motions also affect the chromophore–semiconductor electronic coupling and, there-fore, the extent of delocalization of the photoexcited state onto the semiconductor. Although the ex-cited state can be localized on the chromophore at any energy, generally, at higher energies a largerfraction of the state is delocalized on the semiconductor. Therefore, increased temperatures in thealizarin–TiO2 system raise the contribution of the electron injection component that occurs directlyduring photon absorption.

The photoinduced electron injection from alizarin into TiO2 occurs through a combination of adia-batic and NA mechanisms (see Section 2.3). We know from the arguments above that an increase intemperature should lead to faster transfer, but how much faster? By fitting the adiabatic ET component(Fig. 7) for each temperature to an exponential function, we can plot the logarithm of the adiabatic rateas a function of inverse temperature, the top panel of Fig. 9. With some scatter, the data follow Arrhe-nius behavior, indicating that adiabatic injection can be modeled using the traditional expression

Fig. 9.accessitheory,

kad ¼ m expð�DG=kBTÞ; ð12Þ

where DG is the Gibbs free energy of activation, and m is the frequency of nuclear motion over the bar-rier, T is temperature and kB is the Boltzmann constant [113]. Linear regression of the data givesDG ¼ 4:0� 10�2 eV, which corresponds to kBT at 50 K. Because the calculated energy of activation isso small, the adiabatic ET in the alizarin–TiO2 system should depend on temperature only atT < 100 K. This helps explain the small temperature dependence of the experimental ET rate.

The frequency prefactor m that is determined by the Arrhenius fit equals 2:8� 1014 s�1. This num-ber exceeds all of the vibrational frequencies associated with the heavy atoms in the alizarin–TiO2 sys-

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Average Density of States

Nonadiabatic rate

Adiabatic electron transfer (ET) rate as a function of temperature, and non-adiabatic (NA) ET rate as a function of theble TiO2 density of states (DOS). The adiabatic rate shows an Arrhenius dependence on temperature, as in the Marcuswhile the NA rate depends on the DOS roughly linearly, as in the Fermi golden rule.


tem; the heavy atom frequencies range from 1013 to 1014 s�1. This fact implies that the adiabatic rateconstant (12) represents a sum of individual rate constants for several reaction pathways. MultipleTiO2 surface states couple to the chromophore state and contribute to the overall ET process. The spe-cific TiO2 state that accepts the electron at a given nuclear configuration depends on the direction ofthe atomic motion on the multi-dimensional potential energy surface.

NA ET occurs by a quantum transition from the chromophore state onto one of the states from theCB continuum. In this case, the ET rate can be described by the Fermi Golden Rule [10,113]

kNA ¼2p�hjV j2qðEchromÞ; ð13Þ

where V is the donor–acceptor coupling, and qðEchromÞ is the semiconductor DOS at the energy of thechromophore donor state Echrom. Increased temperatures will push the energy of the chromophore statehigher into the CB, where the DOS is larger, leading to a faster transfer. As demonstrated in Fig. 8, theTiO2 DOS varies significantly over the distribution of the photoexcited state energies. The average DOSentering Eq. (13) was computed for each temperature by multiplying the distribution by the DOS andintegrating over the energy. As expected, this average accessible DOS increased with temperature.

The plot of the NA ET rates vs. the average accessible TiO2 DOS is shown in the bottom panel ofFig. 9. Assuming a constant value of the chromophore–semiconductor coupling, the plot should be lin-ear according to the Fermi Golden Rule (13). It is not, but it does show a general trend of faster transferwith increasing DOS. The deviations from linearity arise from the assumption of constant chromo-phore–semiconductor coupling V in Eq. (13). The coupling is clearly not constant and varies with bothenergy and time.

The above results are general for organic chromophores and transition metal complexes (Figs. 2 and3). Similar to an organic chromophore, the photoexcited state of a transition metal complex is a p*-orbi-tal. In both cases, the electron donor–acceptor coupling involves interaction between d-orbitals of theTi atoms and the p*-orbital. The thermal atomic motions are primarily those of the chromophore andligand, and show little contribution from the semiconductor. Therefore, the results should not be sen-sitive to a specific choice of the TiO2 surface nor to the type of semiconductor in general. Extending theconclusions that were obtained for the alizarin–TiO2 system to other chromophore–semiconductorinterfaces, it is necessary to note that the temperature effects should be even less pronounced for anaverage interface. The alizarin excited state lies close to the edge of the TiO2 CB. This condition createsa minor barrier to the adiabatic ET and puts a fraction of the PE-state distribution in the low TiO2 DOSregion. In a typical system, the photoexcited state is well inside the CB, in the region where the acceptorDOS is already high, and where an adiabatic transition state can be found by moving not only uphill butalso downhill in energy. This configuration requires very little thermal energy to initiate the electroninjection, as observed in the experiments [4,8,11,135] and predicted by the analytic theories[8,10,34–37]. While the ET is promoted by atomic vibrational motions, and therefore the ET rates dodepend on temperature, the dependence is very minor relative to other experimental factors, such asvariations in the TiO2 surfaces, chromophore–semiconductor binding, defects, and solvent, etc.

4.5. Electron transfer to hydrated Ti4þ

The initial step of electron injection from the photoexcited chromophore to the semiconductor CBinvolves localized surface states [12,114–118]. Typically, the TiO2 acceptor states in the alizarin–TiO2

system are localized within the first three surface layers of the semiconductor [74]. The isonicotinicacid system shows an even stronger localization of the acceptor state [40,72]; the Ti atom that isbound to the acid can contribute over 20%. Moreover, the calculated electronic spectrum of alizarinthat is attached to a solvated Ti4þ ion [132] matches the experimental spectrum of alizarin at theTiO2 surface [12]. Catechol is closely related to alizarin, and, as with alizarin, its spectrum when at-tached to TiO2 has similar features to its spectrum when attached to Ti4+ in solution – except in thearea masked by bulk TiO2 absorption [136]. In addition to the static spectra, the local nature of thechromophore–semiconductor interaction motivates a close comparison between the ET dynamics inthe surface [74] and the solution [75] systems. If the photon absorption is very similar in the twocases, will the ET that follows the absorption also be similar?


In the alizarin/TiO2 system (Fig. 3) the electron is on average 30% delocalized onto the semiconduc-tor after the photoexcitation (Fig. 7). The ET continues non-radiatively with a time constant of 7.9 fsand levels off at close to 90%, due to the finite dimensions of the simulation cell. The adiabatic mech-anism controls the dynamics and is not only faster, but it also reaches a much higher value than the NAcomponent. The adiabatic transfer is very efficient in this case, because alizarin is directly bound toTiO2, which creates a strong interaction between alizarin’s conjugated p-electron system and theTiO2 CB. The energies of the strongly coupled donor and acceptor states cross as a result of atomic mo-tion. Quantum jumps that are responsible for the NA transfer take place on a slower timescale.

The ET dynamics of alizarin attached to the hydrated titanium ion shown in the top panel of Fig. 10are different in several key ways. 85% of the photoexcited state is, on average, localized on the chro-mophore, compared to 70% in the surface system. The extent of ET onto Ti4þ is minor and leaves nearly50% of the electron on the chromophore. The transfer occurs on the 32 fs timescale, four times slowerthan for bulk TiO2. The adiabatic transfer rate is dramatically slowed down from 7.1 fs with bulk TiO2

(Fig. 7) to 64 fs in solution (Fig. 10). The NA ET is also slower. However, the difference in NA ET time-scales is only a factor of 2, compared to the factor of 9 for the adiabatic ET. The NA mechanism is moreefficient than the adiabatic mechanism and therefore dictates the overall ET in the Ti4þ system.

Just as the similarities in the absorption spectra for the solution and surface systems are due to thesimilarities in the photoexcited states [12,136,132], the differences in the ET dynamics mainly result

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Fig. 10. Photoinduced electron transfer (ET) in alizarin bound to a ligated Ti4þ ion, as shown in the top panel insert. The bottompanel shows the evolution of the energies of the photoexcited (PE) state (bold line) and the Ti4þ states that are analogs of theTiO2 conduction band (CB). The insert of the bottom panel presents the Fourier transform of the PE energy. Since there are only afew acceptor states (compare with Fig. 4), the ET is slower than in alizarin bound to the TiO2 surface, and it is dominated by theNA mechanism. Higher frequency modes play a more important role in the present case (compare with the dotted red line inpart (a) of Fig. 5), because the system is lighter and contains only one Ti atom. The extent of the ET is minor, since both donorand acceptor states are delocalized over the whole system.


from the contrast between the electron acceptor states [74,75]. There are significantly fewer acceptorstates in the hydrated Ti4þ ion; only five d-orbitals in the Ti atom can potentially accept the photoex-cited electron. Of these, only the lowest two orbitals cross with the p-state of alizarin (bottom panel ofFig. 10). Thus, the adiabatic rate is dramatically slowed down. The NA ET is also slower, since there arefewer NA acceptor states, and they are farther in energy from the photoexcited state. The insert in thebottom panel of Fig. 10 shows the FT of the photoexcited state energy for the Ti4þ system. The FT isnotably different from that for the surface system (Fig. 5a). It shows significantly more higher fre-quency modes, because the system is smaller and lighter. The oscillations with frequencies at and be-low 700 cm�1 are induced by bending and torsional motions. The small peaks seen up to 1600 cm�1

are characteristic of C–C and C@O stretches. The electronic structure analysis [132] indicates thatthe differences in the photoinduced dynamics can be detected spectroscopically, and that the reportedpredictions regarding the photoexcitation dynamics in the alizarin/Ti4þ system can be verifiedexperimentally.

5. Electron dynamics inside the semiconductor

The injected electron can take several paths once it is inside the semiconductor (Fig. 1). Initially, theelectron transfers from the chromophore to a localized surface state [12,114–116,118], and the injec-tion typically occurs over a range of energies somewhere in the middle of the TiO2 CB. Thus, the sur-face electron can simultaneously relax down in energy to the bottom of the CB and move in spaceaway from the surface into the semiconductor bulk. This section discusses the distribution [77] ofthe electrons when they are first injected, and then the relaxation [76] and delocalization processes[74].

5.1. Distribution of the injected electrons

Thermal atomic motions generate a distribution of initial conditions for the ET process (Figs. 6 and8). As a result, the injected electrons occupy multiple surface states over a broad energy range, and theenergetic distribution of these states can be detected experimentally [5,137]. Since the ET occurs fasterthan relaxation, inside either the chromophore or the semiconductor, the distribution of the injectedelectrons is determined by the photoexcited state distribution and the ET process. The ET is essentiallyisoenergetic; large NA jumps in energy have low probability, and crossing a barrier during adiabatic ETrequires little activation energy (see Section 4.4). The photoexcited states that are below the CB edgeconstitute an exception [73]; they have to be lifted inside the CB by a fluctuation in order for the ET totake place. Therefore, the distribution of the injected electrons in the alizarin–TiO2 system mirrors thephotoexcited state distribution at high energies and is skewed at the lower energies, where the TiO2

DOS is smaller (Fig. 11a).The distributions of the injected electron energy were measured in the perylene–TiO2 system using

two photon photoemission [5]. Perylene is another purely organic molecule. It is bigger than alizarinand contains no oxygens in the primary conjugated system. In contrast to alizarin, perylene has aphotoexcited state that is approximately 0.5 eV above the TiO2 CB edge. At this energy, the TiO2

DOS is relatively constant and sufficiently high. Therefore, the distribution of the injected electronsis not skewed, as with alizarin (Fig. 11a) and it should perfectly mirror the distribution of photoexcitedstates. The measured spread of the injected electron energies in the perylene–TiO2 system is at least1 eV, which is by a factor of 2.5 larger than the distribution that was calculated for the alizarin system(Fig. 11a). Similar experiments performed with catechol also gave injected energy distributions on theorder of 1 eV [5,137]. Catechol is yet another purely organic molecule, and is smaller than alizarin. Theproperties of alizarin can be viewed as an average of those of catechol and perylene. Catechol’s excitedstate is very high in energy [132,133], however, and it does not participate in the electron injectionprocess. Instead, the strong coupling between catechol and TiO2 allows ET to occur directly duringphoton absorption, and the injected electrons occupy states near the CB edge [136]. In the alizarin sys-tem the direct ET process accounts for about 30% of the overall ET (see Section 4.3 and Fig. 7). The factthat the measured distributions are broader than the simulated one can be rationalized. The width of

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Fig. 11. Relaxation of the injected electron inside the TiO2 conduction band (CB). (a) Distribution of the injected electronenergies in the alizarin–TiO2 system (Fig. 3) at an ambient temperature. The distribution is skewed because it is constrained bythe TiO2 CB edge energy at�5.4 eV (see part (b)). (b) TiO2 density of states (DOS) as a function of energy. The arrows denote fourdifferent injection energies. The insert shows the probability distribution for a given amount of energy to be exchanged during asingle electron–phonon scattering event. (c) Population of the CB edge state as a function of time for the four injection events(indicated by the arrows in part (a)). The edge population reaches its Boltzmann equilibrium value within 200 fs, regardless ofthe electron injection energy.


the distribution in the alizarin–TiO2 system are diminished, since the distribution is skewed. Also, theexperimental data should definitely exhibit a larger degree of inhomogeneous broadening. Our simu-lation cell has only one perfect surface, with a single chromophore bound to the surface in one of themany possible conformations. The real system has a mixture of surfaces and chromophore–TiO2 bond-ing patterns, and most likely involves surface defects such as atom vacancies, step edges, and impu-rities. This inhomogeneity is responsible for a similar relationship between experimental andsimulated data for the photoexcited state. The experimental width of the excitation spectrum of aliz-


arin that was bound to TiO2 measured at half maximum is about 0.7 eV [12]. The calculated distribu-tion of the excited state energies is a little over half of that, 0.4 eV (Fig. 8) at 350 K.

5.2. Relaxation to the conduction band edge

The photoexcited electron that is still on the chromophore has a hard time relaxing in energy, be-cause the ground molecular state is much lower in energy, while triplet states have a different spinsymmetry and couple to the photoexcited singlet through spin-orbit coupling. Spin-orbit couplingis weak in purely organic chromophores and is enhanced by the presence of the transition metal[9,29,30]. The situation changes drastically once the electron is injected. The continuum of CB statesfacilitates rapid relaxation of the electron energy, which is released in the form of heat by excitingsemiconductor vibrations. The relaxation occurs by the NA mechanism, since it involves quantumtransitions between different electronic states [76].

The relaxation inside the TiO2 CB is detailed in parts (b) and (c) of Fig. 11. Part (b) of the figureshows the DOS of the TiO2 CB and uses arrows to indicate the four different electron injection energiesconsidered in the relaxation study. Arrows 1 through 3 model injection into three maxima in the TiO2

DOS, while arrow 4 represents the CB edge. The insert in Fig. 11b gives the probability distribution forthe quantum transition as a function of the energy difference between the initial and final states. Thedata shows that most transitions occur by small energy changes. This finding is consistent with thefacts that the relaxation occurs through the continuum of CB states with no gaps between the states,and that the vibrations promoting the relaxation involve heavy Ti atoms and therefore have low fre-quencies. The smallest amount of energy that can be accepted by a vibration is determined by thevibrational quantum of energy, which is small for low frequency modes. Our classical treatment ofthe nuclei is able to represent this effect properly. Although most transitions exchange small amountsof energies, large energy jumps do make a notable contribution to the energy relaxation. This obser-vation is reflected in the negative energy tail of the hop probability. As much as half an electron-volt ofenergy can be lost in a single hop, indicating the importance of multi-phonon processes. Still, themajority of the hops induce only small energy changes, as emphasized by the exponential decay ofthe probability distribution. The energy flow between the electronic and vibrational degrees of free-dom of the semiconductor is not uni-directional. Occasionally, vibrations can deposit energy intothe electronic degrees of freedom. The likelihood of a gain in the electron energy is, however, signif-icantly smaller than the likelihood of an energy loss, especially for large energy hops. This is consistentwith the detailed balance, which stipulates that in equilibrium, transitions upward in energy do takeplace, but with a probability that is smaller than the probability for the same transition down in en-ergy by the Boltzmann factor [101].

Fig. 11c presents the time-dependent population of the edge of the TiO2 CB, as represented in oursimulation cell (Fig. 3). Four different injection energies are considered, corresponding to part (b) ofthe figure. The data indicate that the movement of charge to the band edge happens at approximatelythe same rate, regardless of the electron injection energy. The differences among cases 1 through 3 areminor and fall within the statistical uncertainty of the simulation. Case 1, with the highest injectionenergy shows faster relaxation, because the TiO2 DOS is larger at higher energies, and multiple relax-ation pathways are available to the electron. Since there are a number of states close in energy at thebottom of the band, the relaxation creates a quasi-equilibrium set of populations of these states. Thequasi-equilibrium value of the CB edge population oscillates around 0.6 in the simulation. However,this value carries no physical meaning and depends on the size of the simulation cell. The relaxationtime is physically significant. Our simulation shows that relaxation inside the TiO2 CB occurs within200 fs, which is an order of magnitude slower than the injection (Fig. 7) and is similar in timescaleto the electron delocalization into the bulk, which is considered next. The intraband relaxation inthe presence of both chromophore and electrolyte is discussed later in the paper (see Section 6.4).

5.3. Delocalization into bulk

Parallel to the energy relaxation, the injected electron leaves the semiconductor surface and es-capes into the bulk. Initially, the injected electron occupies a fairly localized surface state. The simu-

Fig. 12. Evolution of the five largest adiabatic state occupations in the alizarin–TiO2 system (Fig. 3). The highest occupationdecreases as the electron hops from the photoexcited (PE) state onto other states via the NA mechanism. The 7.9 fs componentof the double-exponential fit corresponds to the initial electron injection from the PE state to a surface state. Delocalization ofthe injected electron from the surface state to TiO2 bulk is reflected in the 104 fs component of the fit.


lations show that this state resides within the first few layers of the semiconductor [73,74], and a sin-gle Ti atom can contribute up to 20% to such states [40,72]. The perturbations of the semiconductorthat create localized surface states can involve the chromophore itself, which both distorts the surfacegeometry and carries a positive electrostatic charge after the photoinduced injection, the solvent, andthe electrolyte. Surface defects can create particularly well-defined local states. Infrared measure-ments [138,139] show that the Fermi level of defective surfaces is quite close to the CB minimum.Occasionally, deep traps lie below the CB edge by much more that kBT. In either case, and especiallyin the latter situation, the photoexcited electron trapped near the surface has a chance to interact withboth the chromophore and the electrolyte, and to be annihilated by recombination with the positivecharge without performing any useful work. Therefore, electron delocalization into semiconductorbulk is critical for many applications.

Our simulations [74] indicate that the electron leaves the surface region and enters the semiconduc-tor bulk on a 100 fs timescale. This is in agreement with the available experimental data [140]. The ETfrom the surface to the bulk occurs in a sequence of NA electronic transitions starting at the surfacestate. Fig. 12 shows the time evolution of the five largest occupations of the adiabatic states in the aliz-arin–TiO2 system (Fig. 3). Since the electron injection in this case is primarily adiabatic, the NA transi-tions describe some other process. The decrease in the occupation of the photoexcited state in Fig. 12cannot be described by a single exponential decay and is instead fit by a double exponential. The faster7.9 fs exponential component has a minor amplitude and originates from the NA contribution to theelectron injection (compare with the 13 fs timescale for the NA ET in Fig. 7). The 13 fs NA ET componentis slower than the 7.9 fs fitting parameter, since not all NA transitions contribute to ET. The longer104 fs component of the fit describes the NA dynamics that follows the ultrafast electron injection.In agreement with the experimental timescale [140], it describes the spreading of the electron popu-lation from surface states into bulk states. None of the other four occupations shown in Fig. 12 reachany significant magnitude, indicating that there exist no special bulk states that draw the electron fromthe surface, as is expected. Generally, the delocalization process is not uniform in space, but ratherexhibits preferential directions, which depend on the symmetry of the photoexcited state. The symme-try of the chromophore state determines the symmetry of the initially occupied surface state, which inturn defines the direction of electron propagation into the semiconductor bulk [38].

6. Back electron transfer processes

In the ideal situation, the photoexcited electron is injected into the semiconductor, leaves the sur-face, travels to the counter-electrode, and then is brought via the electrolyte to the chromophore, hav-ing performed useful work. Often, however, the electron relaxes to the bottom of the TiO2 CB, becomestrapped at the surface and is eventually lost without performing useful work, by back-ET to the chro-


mophore ground state or to the electrolyte. As discussed in the previous section, the relaxation to thebottom of the CB occurs on a similar timescale as the delocalization into bulk. Trap states created bydefects in the semiconductor and interactions with the chromophore or electrolyte keep the relaxedelectron at the surface. Even if the electron manages to escape into the bulk, there remains a highprobability that the electron will return to the surface, since the surface area of the TiO2 is maximizedin order to attach as many chromophores to the semiconductor as possible. For instance, the semicon-ductor may be prepared as a collection of TiO2 nanoparticles that have been sintered together. In thatcase, the injected electron is never far from the surface and can easily return to it. The position of theelectron relative to the surface determines whether it can recombine with the positive charge remain-ing on the chromophore after the injection or transfer onto the electrolyte mediator. The mediator ispresent near the surface, since it brings electrons from the counter-electrode in order to regeneratethe neutral chromophore. While the electron injection process has been investigated experimentallyin great detail, the back-ET process has been studied to a lesser extent. The observed back-ET time-scales range from subpicosecond to milliseconds [141–144,136,145,16,12–15]. The faster rates areseen when the diffusion into the bulk is limited, and the electron remains at the surface. The slowertimescales are associated with electron diffusion into the bulk and back to the surface.

In order to investigate the back-ET processes, we performed a series of studies using a number ofsimulation cells [76], including one that contained the semiconductor, the chromophore, and the elec-trolyte (Fig. 13). Constructing a simple atomistic model for the electrolyte is not an easy task. The stan-dard Grätzel cell uses the I�=I�3 redox pair. One can argue that the key steps involve I��2 according to thefollowing chemical scheme [129]

I�3 � I2 þ I�;K1 ð14ÞI2 þ e� I��2 ;K2

2I��2 !kdisp

I�3 þ I�

I��2 þ e!kET 2I�:

The I��2 radical-anion can both donate an electron to the positively charged chromophore (reversereaction K2) and accept an electron from the semiconductor (reaction kET). We included it in the sim-ulation cell (Fig. 13). In order to preserve the overall electro-neutrality, the I��2 replaces an OH� groupat the surface. The I��2 is too large to fit into the space vacated by OH�, but it interacts most stronglywith the positively charged surface hydrogens and hovers close to the surface. The I��2 � TiO2 interac-tion is sufficiently strong to overcome thermal kinetic energy and maintain surface binding at roomtemperature. Even though both iodine atoms in the electrolyte molecule are negatively charged,one of the atoms makes a closer approach to the surface. The diatomic molecule becomes polarizedby the surface interaction and is tilted with respect to the surface (Fig. 13). The I2 neutral speciescan also accept an electron from the semiconductor (see reaction K2) however this species does notform a strong bond with the semiconductor surface and dissociates away from the surface at roomtemperature [76]. We have yet to consider the more complex ET processes involving electrolyte dif-fusion to and from the interface at the ab initio time-domain level. In the combined system, I��2 inter-acts both with the TiO2 surface and alizarin (Fig. 13). The I-I bond is 3.23 Å, which is similar to thebond length in the I��2 that is attached to the bare surface and substantially longer than in neutralI2. The latter fact is not surprising, because the extra electron occupies an anti-bonding orbital. Theinteraction between I��2 and alizarin occurs through the upper iodine atom, which is attracted bythe positively charged chromophore hydrogens.

The four panels in Fig. 13 show the densities of the states that are relevant for both electron injec-tion and back-ET processes. The top panel depicts the photoexcited state of alizarin for the case inwhich it is localized primarily on the molecule. Even in this case, where there is little mixing betweenthe chromophore and the semiconductor states, the photoexcited state is shifted towards the end ofthe molecule that is directed towards the semiconductor. This is true even for isolated alizarin[12,132,133], and is stipulated by the presence of the electronegative hydroxy groups that attractthe electron density. The second panel of Fig. 13 gives the ground state of alizarin. It is spread evenlyover the whole molecule. A typical surface state that accepts the photoexcited electron and, under

Fig. 13. Densities of the alizarin ground state, alizarin excited state, a typical TiO2 surface state, and the I��2 electrolyte state inthe simulation combining alizarin, TiO2, and the electrolyte. The alizarin ground state is localized on the part of the moleculethat is close to the semiconductor, while the alizarin excited state is localized away from the semiconductor. The TiO2 state hasa large contribution from a single Ti atom.


favorable conditions, mixes with the photoexcited state is shown in the third panel. One can clearlysee the localized nature of the surface state, which has a significant contribution from a single Ti atom.One should note that the Ti atom that contributes most to the surface state shown in Fig. 13 is not theone that is bonded to the alizarin molecule, which has a much smaller contribution. In the exampleconsidered in the figure, the d-orbital of the Ti atom bound to alizarin contributes to the photoexcited


state (see the top panel). The bottom panel of Fig. 13 gives the density of the singly occupied I��2 orbital,which can both harvest an electron from the semiconductor and deliver an electron to the chromo-phore from the counter-electrode. The energies of these orbitals are shown in the top panel ofFig. 14 and are discussed in the following section, which considers the evolution of the electronic stateenergies and identifies the vibrational modes that couple to the electronic states and induce the ETprocesses.

CB edge - Ground State

CB edge - I2-

I2- - Ground State

0

0.5

1

1.5

0

0.5

1

1.5

0

2

4

0

2

4

0 500 1000 1500 2000Frequency, cm -1

Excited State - Ground State

Ampl

itude

, arb

itrar

y un

its

Fig. 14. Evolution of the electronic state energies in the alizarin/TiO2=I��2 system shown in Fig. 13, and Fourier transforms of therelevant energy gaps (compare with Figs. 4 and 5). The gap between the ground and excited states of alizarin couples both to theC–C stretching mode and to the lower frequency motions. The gap between the TiO2 conduction band (CB) edge and the alizaringround state corresponds to the back electron transfer (ET) process and couples better to higher frequency modes than does thephotoexcited (PE) state corresponding to the forward ET (see dotted red line in Fig. 5). This advantage occurs because the PEstate is delocalized onto the semiconductor, while the ground state is localized on alizarin. The gap between the CB edge and I��2 ,which reflects the electron loss to electrolyte, oscillates at frequencies around 100 cm�1, which corresponds to the I��2 stretchingmode. Finally, the gap between the I��2 state and alizarin ground state, which represents a regeneration of the neutralchromophore, couples to both the I��2 and alizarin modes; the coupling to the I��2 vibration is much more pronounced, however.


6.1. Atomic motions relevant for the back electron transfer processes

The evolution of the electronic energies of the chromophore, semiconductor, and electrolyte spe-cies comprising the simulation cell that is shown in Fig. 13 are presented in the top panel ofFig. 14. This panel can be compared with the middle panel of Fig. 4, which is generated from a simu-lation cell without electrolyte and which focuses on the evolution of the alizarin photoexcited stateand the TiO2 CB over a shorter period of time. The TiO2 CB states fluctuate less than the other states,since the CB states are delocalized over multiple TiO2 atoms and are not as sensitive to local atomicmotions. The position of the alizarin excited state is slightly shifted relative to the CB edge in the pres-ence of the electrolyte. However, the alizarin excitation energy remains the same, because the groundstate is also pushed upward in energy relative to the CB. By bringing the energies of the adsorbed spe-cies closer to the CB, this cooperativity effect will speed up the electron injection from alizarin to TiO2,the back-ET from TiO2 to alizarin, and the ET from TiO2 to the redox mediator. The energy level of theelectrolyte is positioned between the chromophore ground state and the TiO2 CB edge, as required bythe Grätzel cell design (Fig. 1). In the system with both alizarin and I��2 , the I��2 state moves slightly clo-ser to the CB, similar to the energy levels of alizarin [76]. Compared to the results of the zero temper-ature calculations [132], the higher temperature calculations produce thermal fluctuations of theatoms that increase the alizarin and I��2 state energies relative to the TiO2 CB. The energies are at theirlowest in the optimized geometry and tend to be increased by thermal distortions of the optimizedgeometry.

The amplitudes and frequencies of the atomic motions that drive various back-ET processes dependstrongly on the process itself. The four FT panels shown in Fig. 14 characterizes the following ETevents, proceeding from top to bottom: the relaxation of the photoexcited electron directly back tothe chromophore ground state, the transfer or the electron from the CB edge to the chromophoreground state, the electron loss from the CB edge to the electrolyte, and the delivery of the electronfrom the electrolyte to the chromophore ground state. The non-radiative intramolecular relaxationof the electron is facilitated by higher frequency motions than is the electron injection from the chro-mophore excited state to the TiO2 CB (Fig. 5a) or the back-ET from the TiO2 CB to the chromophoreground state. This difference comes about because the alizarin state energies fluctuate more thanthe CB. As a result, the energy gap between the excited and ground chromophore states fluctuatesmore than the energy differences between either of the chromophore states and a CB state. The elec-tron loss to the electrolyte is driven by very low frequency modes. The only significant peaks are below100 cm�1, and are due to the I-I stretch and I��2 bends relative to the surface. The largest peak at85 cm�1 corresponds to the 400 fs oscillation in the I��2 energy, which is prominent in the trajectoryshown in the top panel of Fig. 14; it is the I��2 stretching frequency.

Comparing the back-ET from the TiO2 CB edge to the chromophore ground state with the electronloss to the electrolyte, one notes that the back-ET to the electrolyte is promoted by much slowermodes. Therefore, one may expect that the NA coupling between the TiO2 and I��2 states should beweaker than between TiO2 and alizarin. On the other hand, if one compares the amplitudes of theFT, rather than the frequencies, those corresponding to the latter process are much higher. The energyof the I��2 state fluctuates much more than the energy of any other state, as can be clearly seen in thetop panel of Fig. 14. The NA coupling (6) depends on the nuclear velocity and therefore on both thefrequency and the amplitude of the atomic motion. Assuming a simple harmonic vibration, withthe coordinate changing as RðtÞ ¼ R0 cosðxtÞ; the nuclear velocity, dR=dt ¼ R0x sinðxtÞ, is propor-tional to the product of the amplitude R0 and the frequency x. As follows from the data shown inthe ‘CB edge – ground state’ and ‘CB edge – I��2 ’ panels in Fig. 14, the amplitude–frequency productand, therefore, the NA coupling are larger for the back-ET to the chromophore ground state.

In considering the ET between the electrolyte mediator and the chromophore, one wonderswhether the low frequency I��2 modes or the high frequency alizarin vibrations are the more important.Interestingly, the amplitudes of the I��2 modes are significantly higher than the amplitudes of the aliz-arin modes. Even the difference in the frequencies of these two types of motions cannot offset the dif-ference in the amplitude. Hence, the amplitude–frequency product and the NA coupling for the ETfrom the mediator to the chromophore is primarily created by the mediator. The electronic state local-ization argument that explains why chromophore states fluctuate much more than the TiO2 CB states


applies here as well. The I��2 state is localized over only two atoms, and it is much more sensitive to theinteratomic distances than the alizarin states that spread over more than a dozen atoms (Fig. 13). Lowfrequency motions are generally more important than the high frequency modes. The lower frequencymodes induce larger scale changes in the geometric structure and are therefore able to affect the ener-gies of the electronic states to a much larger extent. The inserts in Fig. 8 present another example ofsuch a mode that involves the bending of alizarin with respect to the TiO2 surface. This substantialmovement of the chromophore has the potential to significantly change the nature of the alizarin/TiO2 and alizarin/I��2 interactions, again illustrating the need to include dynamics in the characteriza-tion of the interfacial ET processes.

6.2. Electron recombination with the chromophore

Due to the high-surface area of chromophore-sensitized TiO2, an electron delocalized in bulk TiO2

has a high probability of finding a surface. Once at the surface, the electron will interact with the chro-mophores that are still positively charged after the injection. Surface structural defects, such as miss-ing atoms, edges, and unsaturated bonds, support trap states with energies below the CB edge. If theelectron finds a trap, it will remain at the surface for a long time, increasing the probability of itsrecombination with the chromophore or the electrolyte. The recombination of the electron with thechromophore takes place via a NA electronic transition in which a large amount of energy is ex-changed between the electronic and vibrational degrees of freedom.

Fig. 15a presents the evolution of the ET coordinate for the electron recombination process in thealizarin–TiO2 system, which is represented by the simulation cell shown in Fig. 3. Assuming that theelectron is trapped inside the first five surface layers, the calculations show [76] that the back-transferto the ground state of alizarin is two orders of magnitude slower than the initial electron injection [74]and occurs on a picosecond timescale. The transition is most efficiently facilitated by vibrationalmodes of intermediate frequencies (Fig. 14) which couple to the electronic states and provide a goodmatch between the electronic and vibrational energy quanta that are exchanged during the transition.Although the higher-frequency modes involving hydrogen atoms would have provided a better energymatch with the electronic gap, they do not take part in the electron return to the chromophore, sincethe chromophore and semiconductor electronic states that are engaged in the transition are localizedon the heavy atoms.

The simulation results [76] are in agreement with the experimental studies [12,14], which reportedmultiphasic recombination dynamics within a wide range of timescales from 400 fs to ns. The morerecent paper [14] characterizes the back-ET process in the alizarin–TiO2 system over a broad rangeof solvent pH. The pH change moved the CB edge by 0.42 eV, and at higher pH the excited electronswere forced to inject into surface trap states. The back-ET showed multiphasic kinetics that dependedon the acidity of the solution. In all pH values ranging from 2 to 9, the authors observed a picosecondback-ET component.

Similar timescales were observed with other chromophores. The back-ET transfer in theTiðcatecholÞ2�3 complex in solution (Fig. 10) took place within 200 fs. The corresponding process inthe catechol–TiO2 nanoparticle system showed multi-component kinetics with the fast 400 fs compo-nent being attributed to the electrons trapped near the surface. The back-ET dynamics occurred fasterin catechol than in alizarin because catechol is a smaller molecule and its ground state is closer to, andmore strongly coupled with, the semiconductor. The time-resolved experiments performed with theruthenium-polypyridyl complexes strongly coupled to TiO2 nanoparticles [16], indicated that 30% ofthe injected electrons recombined with a time constant of less than 2 ps. The fastest back-ET compo-nent observed by the same group with quinizarin-sensitized TiO2 was 600 fs, indicating that thecharge separation is longer lived in the ruthenium-based systems than in organic chromophore-sen-sitized TiO2 nanoparticles with similar dye-semiconductor interactions [145,143]. This is to be ex-pected, since the ground state of the transition metal complex is localized on the metal, where it isfurther away from, and more weakly coupled to, the surface. A comparison between the back-ETdynamics involving ruthenium-bipyridyl and porphyrin sensitizer dyes leads to the conclusion thatthe multiexponential nature of the recombination kinetics is not associated with properties of the sen-sitizer dye, but rather with heterogeneities and trap states in the TiO2 film [142].

0

2.0

4.0

6.0

0.8

Popu

latio

n

0

2.0

4.0

6.0

8.0

3210sp,emiT

I2-

b

cAlizarin

Alizarin

I2-

0

0.2

0.4

0.6

0.8

1

TiO2 AlizarinTiO2 I2

-e-

e- a

CB edge

Fig. 15. Electron transfer (ET) processes in the alizarin/TiO2=I��2 system (see Figs. 13 and 14). (a) Electron loss to electrolyte andalizarin ground state, considered separately. Provided that the I��2 molecule can approach the TiO2 surface near the electron, itaccepts the electron from the semiconductor very rapidly. At a slower rate, the electron is lost to the alizarin ground state. (b)Simultaneous electron loss to electrolyte and alizarin. Both the TiO2 conduction band edge and the I��2 states are populated onlytransiently in this case. Ultimately, the electron relaxes all the way down in energy to the alizarin ground state. (c) Regenerationof the neutral chromophore by ET from I��2 to alizarin. The ET dynamics in this case show a complicated non-exponentialbehavior, because the ET rate depends strongly on the orientation and the distance between the species. I��2 is only weaklybound to the surface and moves significantly around alizarin.


The back-ET process shows little temperature dependence [146]. Since the gap between the bottomof the TiO2 CB and the chromophore ground state is large, adiabatic ET requires crossing a huge barrierin the inverted Marcus region and is essentially impossible. The atomistic simulation confirmed thatthe back-ET occurs by means of the NA mechanism [76]. According to the Fermi Golden Rule (13) therate of NA transfer is proportional to the square of the coupling and therefore to the square of the nu-clear velocity and the first power of kinetic energy and temperature. The density of acceptor states inEq. (13) is independent of temperature, since the acceptor state is just the chromophore ground state.For instance, varying the temperature between 250 and 350 K will change the back-ET rate by only350/250 = 1.4, which constitutes a minor effect relative to the changes in the rate that are inducedby variations in TiO2 surfaces, chromophore–semiconductor binding, surface defects, solvent, and soforth.

6.3. Electron loss to electrolyte

The loss of the injected electron to the electrolyte mediator can be efficient compared with theback-ET to the chromophore, as demonstrated in Fig. 15a, which relates the two processes. The sim-ulation assumes that the electrolyte molecule I��2 has been able to diffuse close to the semiconductor


surface and replace one of the hydroxy groups, thereby attaching itself to the surface (Fig. 13). The rateof the electron transfer to the electrolyte is faster in this situation than the rate of back-ET to thechromophore ground state, because the electrolyte energy level is closer to the TiO2 CB and the elec-tron-vibrational coupling is stronger, as reflected in the large amplitude fluctuation of the electrolyteenergy level (Fig. 14).

The full-scale dynamics of the ET from TiO2 to the electrolyte mediator are harder to determinethan the chromophore–TiO2 ET, both experimentally and theoretically. A complete model of the elec-trolyte should involve a number of redox species, Eq. (14), and a solvent, and consider not only the ETprocess per se, but also electrolyte diffusion towards and away from the semiconductor surface. TheET rates involving the mediator should be very sensitive to the details of the diffusion process, such asthe fluctuation of the distance to the semiconductor, and the time spent in its vicinity. The structure ofthe solvent surrounding the mediator and the surface can play a critical role as well. Further compli-cations may arise because the mediator that has approached the semiconductor is also in close prox-imity to the chromophore.

6.4. Electron relaxation processes in the presence of both chromophore and electrolyte

In the presence of both the chromophore and the electrolyte, the electron injected into the TiO2 CBhas an opportunity to transfer to either species. If the chromophore and electrolyte molecules are lo-cated far from each other, the two transfer processes are independent. However, if the molecules areclose by, as would be expected, given that the negatively charged electrolyte is attracted to the chro-mophore-cation, the transfer processes will be coupled. The simulation cell shown in Fig. 13 allowedus to address this situation. The electron relaxation dynamics in the combined system involve multi-ple components [76]. The fastest component still corresponds to the relaxation inside the CB. Simul-taneously with the relaxation inside the band, but more slowly, the electron starts transferring to I��2and, in parallel, to alizarin. Finally, the electron leaves I��2 and accumulates in the chromophore groundstate. The maximum values of the transient occupations of the CB edge and I��2 are quite small, and thealizarin ground state is populated slightly faster than in the absence of I��2 . A large fraction of the trans-fer to the alizarin ground state derives directly from the states in the TiO2 CB, bypassing the mediator.

An additional complication occurs in the back-ET process in the system that combines the chromo-phore and the electrolyte, because the I��2 orbital that is above the alizarin ground state is already occu-pied by an electron. Once the photoexcited electron leaves the chromophore, the I��2 electron cantransfer to the cation, and then the injected electron can transfer onto I��2 . The outcome is identicalto that of the single-step back-ET from the semiconductor to the chromophore. Hence, the presenceof the extra electron already occupying the electrolyte state increases the rate of overall back-ET tothe chromophore. Using the results obtained for the ET from the electrolyte to the chromophoreground state, which are discussed in the following section, we conclude that the photoexcited electronreaches the electrolyte state faster than the electrolyte electron transfers to the chromophore state.Therefore, the extra electron present in the I��2 state should not have a big effect on the overall se-quence of electron relaxation events.

6.5. Electron transfer from electrolyte to the chromophore

The positive charge that is created on the Grätzel cell chromophores by the photoinduced electroninjection are removed by the electrolyte mediator that brings electrons from the counter-electrode.The chromophore-cation and the mediator carry opposite charges and are attracted to each other.Since the chromophore is chemically attached to the semiconductor surface, the electrolyte–chromo-phore interaction occurs at the surface, as represented by the simulation cell shown in Fig. 13. Fig. 15cconsiders the dynamics of the electron loss process that involves the I��2 electrolyte molecule. An elec-tron placed in the I��2 state loses its energy and transfers to the alizarin ground state relatively slowly. Itmay seem surprising that electrolyte–chromophore ET is so slow, since the two states are closer in en-ergy than are, for instance, the TiO2 CB edge and alizarin ground state, between which the ET proceedsis faster (compare parts (a) and (c) of Fig. 15). The reason for this diminished speed is that the electrondonor–acceptor coupling is substantially weaker between the spatially separated chromophore and


electrolyte molecules than between the chemically bonded chromophore and semiconductor. Thenon-exponential change of the occupations of the alizarin and I��2 states seen in Fig. 15c indicates thatthe ET strongly depends on the chromophore–electrolyte distance. The step in the state occupations isdue to the change in the relative positions of the donor and acceptor species, caused by the slow dif-fusion of I��2 and the wagging motion of alizarin relative to the surface (Fig. 8). Additional averagingover the NA MD trajectories should smooth out the step. The electronic coupling between the alizarinand I��2 states proceeds not only through the space between the species, but also through the TiO2 sur-face bonds. In similar a vein, solvent molecules can provide yet another pathway for the electrolyte–chromophore electron exchange.

7. Summary and outlook

The photoinduced interfacial ET processes that have been investigated by the ab initio NA MD tech-niques are summarized in Fig. 16. The timescales shown in the figure derive from the simulationsperformed with the system comprised of the TiO2 semiconductor, alizarin chromophore and I��2 elec-trolyte mediator. Further, the rates of the ET processes have been obtained assuming that the electro-lyte molecule has diffused into a close proximity to the surface and that the injected electron has notyet delocalized beyond the first few surface layers. The simulation cells contained no surface defects,the surface was perfectly terminated with dissociated water molecules, and the chromophore wasbonded to the surface in the way that is most energetically favorable. The timescales shown in the fig-ure will vary for other systems, however, the overall kinetic scheme will remain the same in themajority of cases.

The interfacial electron dynamics start when the chromophore molecule absorbs a photon. Thiscreates an imbalance with a vacant chromophore ground state and an unstable chromophore excitedstate. The excited state rapidly moves into the semiconductor CB, creating a hot electron. The latterrelaxes to the bottom of the CB by coupling to phonons and simultaneously delocalizes into the bulk.Comparing the relevant timescales, one concludes that, in the absence of traps, the majority of the in-jected electrons move away from the surface. However, if the electron is trapped near the surface, or ifa large surface area leads it to return there repeatedly, it can go back to the chromophore and fill in theground state vacancy. The simulations indicate that if an electrolyte mediator approaches the ionizedchromophore, it efficiently transfers the electron that comes from the counter-electrode onto thechromophore, thereby regenerating the neutral chromophore and completing the photovoltaic cycle.At the same time, if electrolyte molecules are allowed to chemisorb onto the surface, the photovoltaicdevice will be very inefficient, since the electrons that are near the surface will be rapidly lost to theelectrolyte, even more rapidly than they return to the chromophore ground state.

Bulk e

TiO2

e-diffusion

Cold e

Hot e Chromophore Excited State

Chromophore Ground State

10 fs

100 fs

1ps

hν

0.25 ps

1ps Electrolyte

electrolyte diffusion

Solution

{ 100 fs

Fig. 16. Overview of the photoinduced electron dynamics at the chromophore–semiconductor interface initiated bychromophore photoexcitation. The timescales are derived from the alizarin–TiO2 simulation (Figs. 3 and 13) and vary withthe given system.


All interfacial ET processes considered in Fig. 16 proceed by the NA mechanism, involve quantum-mechanical transitions or tunneling between electronic states and can be described by the Fermi Gold-en Rule (13). The electron injection provides the only possible exception; it can occur by means of theadiabatic mechanism, in particular if the chromophore is bound to the semiconductor by a shortbridge and the chromophore–semiconductor coupling is very strong. Chromophores perturb the ex-tended electronic system of the semiconductor locally. If the chromophore–semiconductor couplingis strong, the combined system has to be considered explicitly in an atomistic calculation, but a rela-tively small representation of the semiconductor is sufficient. Most of the experimental data charac-terizing the effect that the bulk has on the chromophore can be modeled and understood withrelatively small-scale calculations that include moderate-sized portions of the semiconductor. Thisconclusion holds for both ground and low-energy excited chromophore states.

The variety of semiconductor surfaces, surface defects, chromophore types, chromophore–semi-conductor bridge lengths and binding patterns creates a broad distribution of interfacial ET events.Even with the perfect surfaces that we have considered in our time-domain ab initio simulation thusfar, atomic fluctuations significantly affect ET times and mechanisms. The type of surface termination,the presence of solvent, and other factors further broaden the ET event distribution, generating a richand complex picture of the photoinduced electron dynamics at the chromophore–semiconductorinterface. There exist subsets of faster and slower ET events in every system. Still, the average behavioris quite independent of the system details and can be understood using a small number of theoreticalparameters, such as the electron donor and acceptor energies, the donor–acceptor coupling, and thedensity of acceptor states.

Vibrational motions of the chromophore, the semiconductor, and the electrolyte atoms influencethe dynamics of ET at the interface in many different ways. Thermal disorder in the atomic coordinatesgenerates inhomogeneous distributions of electronic states. The state energies and the electrondonor–acceptor coupling vary substantially with atomic motion. While zero temperature quantum-chemical calculations produce an initial picture of the electron dynamics, explicit time-resolved mod-eling may notably alter the static description. The coupling of the electronic states to vibrationalmodes is responsible for not only the initial ET but also the subsequent electron relaxation. It also pro-motes electron loss back to the chromophore and to the electrolyte. The neutral chromophore is rec-reated by ET from the electrolyte mediator, and this process proceeds due to the NA couplinggenerated by vibrational motions. Generally, [147–151] high frequency stretching and bending modesprovide the largest NA coupling. Motion along low frequency degrees of freedom, including torsionaland intermolecular modes, changes the relative positions and orientations of the chromophore, semi-conductor and electrolyte, alters the energy levels, and mixes the electronic states.

On the practical side, the ab initio time-domain atomistic simulations have provided a number ofvaluable insights into the interface properties. The simulations showed that efficient electron injectionis possible right at the edge of the TiO2 CB. Compared to the traditional setups in which the chromo-phore excited state is positioned deep inside the CB, this scenario provides potential savings in solarcell voltage, since the extra energy required to inject higher into the CB that is subsequently lost toheat by relaxation inside the band is no longer needed. The electronic states of the chromophoreand electrolyte can now be shifted down in energy relative to the TiO2 bands (Fig. 1) and the voltagedetermined by the energy gap between the TiO2 CB edge and the energy level of the electrolyte can beincreased. Note that a different electrolyte mediator will be required is this case.

Another practical consideration comes from the understanding that the injected electron under-goes two simultaneous processes: delocalization into the bulk and relaxation to the bottom of theTiO2 CB and down into trap states. The timescales of these competing events are very similar, implyingthat if surface trap states do exist, they often can be populated after the injection. As a result, the elec-tron remains close to the ionized chromophore and electrolyte and can be rapidly lost to either spe-cies, generating substantial losses in the photovoltaic current and efficiency. Thus, trap states shouldbe annealed and avoided at all costs.

For optimal cell performance, the electrolyte should be kept away from close contact with the sur-face when it regenerates the neutral chromophore. If brought in touch with the surface, electrolytemolecules can strongly interact with it and even bond. Then, instead of delivering an electron fromthe counter-electrode to the chromophore, the electrolyte will rapidly uptake the photoinjected elec-


tron, creating a massive loss of efficiency. The problem is to some extent avoided in the existing solarcell designs, since the electrolyte density decreases near the semiconductor [128].

Of the two types of chromophores, those composed of a transition metal and small organic ligandsand those involving only large organic molecules (Fig. 2) the former type are superior for a number ofreasons, several of which become apparent from the time-domain atomistic simulation. Transitionmetal chromophores are preferable on the synthetic side, since they are significantly more stable pho-tochemically. There is little difference in the chemical stability of transition metal chromophores thatexist in the reduced and oxidized forms. At the same time, organic cations generated by the photoin-duced electron injection are much more reactive than their neutral counterparts. In addition to thiswell-known argument, transition metal chromophores provide further advantages. Their photoexcitedstates are located on the ligand and, therefore, are directed towards the semiconductor surface, sincethe binding occurs via the ligand. On the other hand, their ground states are localized on the metal andare situated farther away from the surface than are the excited states. The situation is often reversedwith purely organic chromophores, e.g. alizarin in Fig. 13. Here there exist no generic mechanism thatwould localize excited states closer to the surface than the ground states. Thus, the electronic struc-ture of the transition metal complexes favors forward electron injection and disfavors backward elec-tron transfer. This factor should increase solar cell currents and photon-to-electron conversion yields.Further, ET typically occurs faster if it is promoted by higher frequency vibrational modes. Once again,this factor plays a stronger role with the transition metal chromophores than with the organic mole-cules. Transition metal ligands are smaller than the molecules of purely organic chromophores. There-fore, the ligand vibrations have higher frequencies and accelerate the electron injection relative to thebig organic molecules. The opposite is true of the electron loss back to the chromophore ground state.Transitions metals are heavy elements and vibrate at low frequencies. Thus, the back-ET is sloweddown by the lower frequency modes of the transition metal chromophores compared to the modesof the organic chromophores.

The electron dynamics that occur at the chromophore–semiconductor interface provide an excel-lent case for studying the fundamental issues that arise when molecules interact with the solid-state.Molecular and bulk materials are opposites in nearly every respect. The structure of a molecule is un-iquely defined, while solids show multiple structures even when they have the same chemical com-position. Surfaces, edges, defects and dopants further complicate the solid-state structure. Moleculeshave a finite number of discrete electronic and vibrational states, while periodic materials form con-tinuous energy bands. The fraction of high frequency vibrational modes is much higher in moleculesthan in inorganic semiconductors such as TiO2, and the electron-vibrational coupling and electron cor-relation effects are much stronger in finite systems. Designs that combine organic and inorganic com-ponents have become increasingly common in recent years, as molecular and solid-state domainshave converged, with molecules being assembled into ever more complicated mesoscopic structures,[152] and periodic systems becoming miniaturized on the nanoscale [153]. The insights learned fromthe time-domain atomistic simulations of the ET phenomena that are detailed above apply to othermolecule-bulk interfaces as well.

The details of these interfacial ET dynamics have become available through recent theoretical ad-vances in DFT and NA MD. The time-domain ab initio description of the interface provides a clearunderstanding of the roles of molecular and atomic structure, binding, defects, chemical functionali-zation, solvent and electrolyte, through-bond and through-space electronic interactions, vibrationalmotions, electron-vibrational coupling, and many other factors, which are sensitive to system detailsand can be tuned with practical goals in mind. Generating this appealing picture is a computationallydemanding process, however, and it remains a state-of-the-art technology. As a result, only relativelysmall systems and ideal situations have been studied. The main computational difficulty resides in theelectronic structure calculations, which require computation of energies, forces and NA couplings, andmust be repeated multiple times for every step of nuclear dynamics. Semiempirical [49] or quantum-mechanical/molecular-mechanical [154] approaches can provide great computational savings, how-ever, they should be tested carefully to provide accurate energies, forces and wave functions for allatoms forming the interface. Classical nuclear dynamics requires little computational effort, but cre-ates additional challenges associated with the proper choice of the quantum-classical equations ofmotion and the need for semiclassical corrections for the nuclei. These conceptual challenges consti-


tute an active field of research [87–95,151]. The implementations of DFT and NA MD used in this workare made efficient by computer parallelizations. The plane-wave DFT code runs on multiple processes.Ensembles of NA MD trajectories required for statistical averaging are generated simultaneously onmultiple computer nodes. In spite of the computational and conceptual complexity, this exciting fieldis rapidly progressing and is beginning to allow researchers to expand to larger systems, longer time-scales, more examples, and finer details of the interfacial ET.

Acknowledgements

The authors are indebted to many experimentalist and theoretician colleagues for fruitful and illu-minating discussions, particularly to Tim Lian, Frank Willig, Arthur Nozik, Josef Wachtveitl, Victor Ba-tista, Haobin Wang, Michael Thoss, and Volkhard May. Bruce Duncan’s editing comments are highlyappreciated. The research was supported by NSF Awards CHE-0094012 and 0701517, DOE AwardsDE-FG02-05ER15755, and ACS PRF Awards 150393 and 41436-AC6.

References

[1] B. Oregan, M. Grätzel, Nature 353 (1991) 737.[2] R. Memming, Prof. Surf. Sci. 17 (1984) 7.[3] D. Liu, G.L. Hug, P.V. Kamat, J. Phys. Chem. 99 (1995) 16768.[4] B. Burfeindt, T. Hannappel, W. Storck, F. Willig, J. Phys. Chem. 100 (1996) 16463.[5] L. Gundlach, R. Ernstorfer, F. Willig, Prog. Surf. Sci. 82 (2007) 355.[6] H.J. Snaith, M. Grätzel, Phys. Rev. Lett. 98 (2007) 177402.[7] F. De Angelis, S. Fantacci, A. Selloni, M. Grätzel, M.K. Nazeeruddin, Nano. Lett. 7 (2007) 3189.[8] J.B. Asbury, E.C. Hao, Y.Q. Wang, H.N. Ghosh, T.Q. Lian, J. Phys. Chem. B 105 (2001) 4545.[9] J.B. Asbury, N.A. Anderson, E. Hao, X. Ai, T. Lian, J. Phys. Chem. B 107 (2003) 7376.

[10] N.A. Anderson, T.Q. Lian, Ann. Rev. Phys. Chem. 56 (2005) 491.[11] D.F. Watson, G.J. Meyer, Ann. Rev. Phys. Chem. 56 (2005) 119.[12] R. Huber, S. Spoerlein, J.E. Moser, M. Grätzel, J. Wachtveitl, J. Phys. Chem. B 104 (2000) 8995.[13] R. Huber, J.E. Moser, M. Grätzel, J. Wachtveitl, J. Phys. Chem. B 106 (2002) 6494.[14] V.V. Matylitsky, M.O. Lenz, J. Wachtveitl, J. Phys. Chem. B 110 (2006) 8372.[15] D.S. Zhang, J.A. Downing, F.J. Knorr, J.L. McHale, J. Phys. Chem. B 110 (2006) 21890.[16] G. Ramakrishna, D.A. Jose, D.K. Kumar, A. Das, D.K. Palit, H.N. Ghosh, J. Phys. Chem. B. 109 (2005) 15445.[17] S. Verma, P. Kar, A. Das, D.K. Palit, H.N. Ghosh, J. Phys. Chem. C 112 (2008) 2918.[18] J. Schnadt, P.A. Brühwiler, L. Patthey, J.N. O’Shea, S. Södergren, M. Odelius, R. Ahuja, O. Karis, M. Bässler, P. Persson, H.

Siegbahn, S. Lunell, N. Martensson, Nature 418 (2002) 620.[19] R.D. McConnell, Renew. Sustain. Energy Rev. 6 (2002) 273.[20] B.A. Gregg, Coord. Chem. Rev. 248 (2004) 1215.[21] R. Argazzi, N.Y.M. Iha, H. Zabri, F. Odobel, C.A. Bignozzi, Coord. Chem. Rev. 248 (2004) 1299.[22] A. Furube, R. Katoh, K. Hara, T. Sato, S. Murata, H. Arakawa, M. Tachiya, J. Phys. Chem. B 109 (2005) 16406.[23] P. Piotrowiak, E. Galoppini, Q. Wei, G.J. Meyer, R. Wiewior, J. Am. Chem. Soc. 125 (2003) 5278.[24] J. Rochford, E. Galoppini, Langmuir 24 (2008) 5366.[25] A. Morandeira, I. Lopez-Duarte, M.V. Martinez-Diaz, B. O’Regan, C. Shuttle, N.A. Haji-Zainulabidin, T. Torres, E. Palomares,

J.R. Durrant, J. Am. Chem. Soc. 129 (2007) 9250.[26] A. Morandeira, G. Boschloo, A. Hagfeldt, L. Hammarstrom, J. Phys. Chem. B 109 (2005) 19403.[27] A. Morandeira, G. Boschloo, A. Hagfeldt, L. Hammarstrom, J. Phys. Chem. C 112 (2008) 9530.[28] S. Eu, S. Hayashi, T. Urneyama, Y. Matano, Y. Araki, H. Imahori, J. Phys. Chem. C 112 (2008) 4396.[29] J.K. McCusker, Acc. Chem. Res. 39 (2003) 876.[30] W. Henry, C.G. Coates, C. Brady, K.L. Ronayne, P. Matousek, M. Towrie, S.W. Botchway, A.W. Parker, J.G. Vos, W.R. Browne,

J.J. McGarvey, J. Phys. Chem. A 112 (2008) 4537.[31] M.K. Nazeeruddin, F.D. Angelis, S. Fantacci, A. Selloni, G. Viscardi, P. Liska, S. Ito, B. Takeru, M.G. Grätzel, J. Am. Chem. Soc.

127 (2005) 16835.[32] F. De Angelis, S. Fantacci, A. Selloni, M.K. Nazeeruddin, M. Grätzel, J. Am. Chem. Soc. 129 (2007) 14156.[33] I. Paci, J.C. Johnson, X.D. Chen, G. Rana, D. Popovic, D.E. David, A.J. Nozik, M.A. Ratner, J. Michl, J. Am. Chem. Soc. 128 (2006)

16546.[34] S. Ramakrishna, F. Willig, V. May, J. Chem. Phys. 115 (2001) 2743.[35] L.X. Wang, F. Willig, V. May, J. Chem. Phys. 126 (2007) 134110.[36] H. Wang, M. Thoss, J. Phys. Chem. A 107 (2003) 2126.[37] I. Kondov, M. Cizek, C. Benesch, H.B. Wang, M. Thoss, J. Phys. Chem. C 111 (2007) 11970.[38] L.G.C. Rego, V.S. Batista, J. Am. Chem. Soc. 125 (2003) 7989.[39] S.G. Abuabara, L.G.C. Rego, V.S. Batista, J. Am. Chem. Soc. 127 (2005) 18234.[40] W. Stier, O.V. Prezhdo, J. Phys. Chem. B 106 (2002) 8047.[41] W.R. Duncan, O.V. Prezhdo, Ann. Rev. Phys. Chem. 58 (2007) 143.[42] O.V. Prezhdo, W.R. Duncan, V.V. Prezhdo, Acc. Chem. Res. 41 (2008) 339.[43] P. Persson, S. Lunell, L. Ojamäe, Int. J. Quant. Chem. 89 (2002) 172.


[44] M. Odelius, P. Persson, S. Lunell, Surf. Sci. 529 (2003) 47.[45] P.C. Redfern, P. Zapol, L.A. Curtiss, T. Rajh, M.C. Thurnauer, J. Phys. Chem. B 107 (2003) 11419.[46] R. Katoh, A. Furube, A.V. Barzykin, H. Arakawa, M. Tachiya, Coord. Chem. Rev. 248 (2004) 1195.[47] C. Creutz, B.S. Brunschwig, N. Sutin, J. Phys. Chem. B 110 (2006) 25181.[48] B. Bruggemann, J.A. Organero, T. Pascher, T. Pullerits, A. Yartsev, Phys. Rev. Lett. 97 (2006) 208301.[49] P. Zapol, L.A. Curtiss, J. Comp. Theor. Nanosci. 4 (2007) 222.[50] A.M. Wodtke, D. Matsiev, D.J. Auerbach, Prog. Surf. Sci. 83 (2008) 167.[51] A. Nitzan, Ann. Rev. Phys. Chem. 52 (2001) 681.[52] A. Nitzan, M.A. Ratner, Science 300 (2003) 1384.[53] X.Y. Zhu, J. Phys. Chem. B 108 (2004) 8778.[54] C.D. Lindstrom, X.Y. Zhu, Chem. Rev. 106 (2006) 4281.[55] W. Zhao, W.H. Ma, C.C. Chen, J.C. Zhao, Z.G. Shuai, J. Am. Chem. Soc. 126 (2004) 4782.[56] T. Hirakawa, J.K. Whitesell, M.A. Fox, J. Phys. Chem. B 108 (2004) 10213.[57] S.O. Obare, T. Ito, G.J. Meyer, J. Am. Chem. Soc. 128 (2006) 712.[58] N.S. Lewis, J. Electroanal. Chem. 508 (2001) 1.[59] I.R. Gould, J.R. Lenhard, A.A. Muenter, S.A. Godleski, S. Farid, J. Am. Chem. Soc. 122 (2000) 11934.[60] R.D. Sytnik, V.V. Prezhdo, V.Y. Matveenko, G.M. Legoshin, Glass Sci. Technol. 74 (2001) 161.[61] W. Hess, A. Joly, K. Beck, M. Henyk, P. Sushko, P. Trevisanutto, A. Shluger, J. Phys. Chem. B 109 (2005) 19563.[62] S. Katano, Y. Kim, M. Hori, M. Trenary, M. Kawai, Science 316 (2007) 1883.[63] J. Wang, Acc. Chem. Res. 35 (2002) 811.[64] Y.V. Pereverzev, O.V. Prezhdo, M. Forero, E.V. Sokurenko, W.E. Thomas, Biophys. J. 89 (2005) 1446.[65] D.S. Kilin, K. Tsemekhman, O.V. Prezhdo, E.I. Zenkevich, C.v. Borczyskowski, J. Photochem. Photobiol. A 190 (2007) 342.[66] R. Hoffmann, Solids and Surfaces: A Chemist’s View of Bonding in Extended Structures, VCH Publishers, New York, 1988.[67] G. Malliaras, R. Friend, Physics Today 5 (2005) 53.[68] S. Ramakrishna, F. Willig, V. May, A. Knorr, J. Phys. Chem. B 107 (2003) 607.[69] L.X. Wang, F. Willig, V. May, J. Chem. Phys. 124 (2006) 014712.[70] M. Thoss, I. Kondov, H. Wang, Chem. Phys. 304 (2004) 169.[71] I. Kondov, M. Thoss, H. Wang, J. Phys. Chem. A 110 (2006) 1364.[72] W. Stier, O.V. Prezhdo, Isr. J. Chem. 42 (2003) 213.[73] W. Stier, W.R. Duncan, O.V. Prezhdo, Adv. Mater. 16 (2004) 240.[74] W.R. Duncan, W.M. Stier, O.V. Prezhdo, J. Am. Chem. Soc. 127 (2005) 7941.[75] W.R. Duncan, O.V. Prezhdo, J. Phys. Chem. B 109 (2005) 17998.[76] W.R. Duncan, C.F. Craig, O.V. Prezhdo, J. Am. Chem. Soc. 129 (2007) 8528.[77] W.R. Duncan, O.V. Prezhdo, J. Am. Chem. Soc. 130 (2008) 9756.[78] M.A.L. Marques, E.K.U. Gross, Ann. Rev. Phys. Chem. 55 (2004) 427.[79] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) 1133.[80] E. Runge, E.K.U. Gross, Phys. Rev. Lett. 52 (1984) 997.[81] R. Baer, D. Neuhauser, J. Chem. Phys. 121 (2004) 9803.[82] X.S. Li, J.C. Tully, H.B. Schlegel, M.J. Frisch, J. Chem. Phys. 123 (2005) 084106.[83] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558.[84] G. Kresse, J. Furthmüller, Comput. Mater. Sci. 6 (1996) 15.[85] S. Hammes-Schiffer, J.C. Tully, J. Chem. Phys. 101 (1994) 4657.[86] C. Leforestier, R.H. Bisseling, C. Cerjan, M.D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.D.

Meyer, N. Lipkin, O. Roncero, R. Kosloff, J. Comp. Phys. 94 (1991) 59.[87] I.V. Aleksandrov, Z. Naturforsch. 36A (1981) 902.[88] A. Anderson, Phys. Rev. Lett. 74 (1995) 621.[89] O.V. Prezhdo, V.V. Kisil, Phys. Rev. A 56 (1997) 162.[90] O.V. Prezhdo, P.J. Rossky, J. Chem. Phys. 107 (1997) 825.[91] O.V. Prezhdo, P.J. Rossky, J. Chem. Phys. 107 (1997) 5863.[92] R. Kapral, G. Ciccotti, J. Chem. Phys. 110 (1999) 8919.[93] J. Caro, L.L. Salcedo, Phys. Rev. A 60 (1999) 842.[94] O.V. Prezhdo, C. Brooksby, Phys. Rev. Lett. 86 (2001) 3215.[95] O.V. Prezhdo, J. Chem. Phys. 124 (2006) 201104.[96] J.C. Tully, in: B.J. Berne, G. Ciccotti, D.F. Coker (Eds.), Classical and Quantum Dynamics in Condensed Phase Simulations,

World Scientific, Singapore, 1998, pp. 489–514.[97] M.H. Mittelman, Phys. Rev. 122 (1961) 449.[98] J.B. Delos, W.R. Thorson, S.K. Knudson, Phys. Rev. A 6 (1972) 709.[99] P. Ehrenfest, Z. Phys. 45 (1927) 455.

[100] J.C. Tully, J. Chem. Phys. 93 (1990) 1061.[101] P.V. Parandekar, J.C. Tully, J. Chem. Phys. 122 (2005) 094102.[102] P.V. Parandekar, J.C. Tully, J. Chem. Theory Comput. 2 (2006) 229.[103] K. Drukker, J. Comp. Phys. 153 (1999) 225.[104] O.V. Prezhdo, P.J. Rossky, Phys. Rev. Lett. 81 (1998) 5294.[105] O.V. Prezhdo, Phys. Rev. Lett. 85 (2000) 4413.[106] C.F. Craig, W.R. Duncan, O.V. Prezhdo, Phys. Rev. Lett. 95 (2005) 163001.[107] N.L. Doltsinis, D. Marx, Phys. Rev. Lett. 88 (2002) 166402.[108] E. Tapavicza, I. Tavernelli, U. Rothlisberger, Phys. Rev. Lett. 98 (2007) 023001.[109] V. Chernyak, S. Mukamel, J. Chem. Phys. 112 (2000) 3572.[110] R. Baer, Chem. Phys. Lett. 364 (2002) 75.[111] S. Billeter, A. Curioni, J. Chem. Phys. 122 (2005) 034105.


[112] N.L. Doltsinis, D.S. Kosov, J. Chem. Phys. 122 (2005) 144101.[113] R. Memming, Semiconductor Electrochemistry, Wiley-VCH, Weinheim, 2001.[114] A.K. See, M. Thayer, R.A. Bartynski, Phys. Rev. B 47 (1993) 13722.[115] D. Ino, K. Watanabe, N. Takagi, Y. Matsumoto, J. Phys. Chem. B 109 (2005) 18018.[116] K. Onda, B. Li, J. Zhao, K.D. Jordan, J. Yang, H. Petek, Science 308 (2005) 1154.[117] B. Li, J. Zhao, K. Onda, K.D. Jordan, J.L. Yang, H. Petek, Science 311 (2006) 1436.[118] J. Zhao, B. Li, K.D. Jordan, J. Yang, H. Petek, Phys. Rev. B 73 (2006) 195309.[119] A.L. Linsebigler, G. Lu, J.T. Yates, Chem. Rev. 95 (1995) 735.[120] T.L. Thompson, J.T. Yates, Chem. Rev. 106 (2006) 4428.[121] V. Henrich, Prog. Surf. Sci. 50 (1995) 77.[122] D. Bonnell, Prog. Surf. Sci. 57 (1998) 187.[123] U. Diebold, Surf. Sci. Rep. 48 (2003) 53.[124] V. Shapovalov, Y. Wang, T.N. Truong, Chem. Phys. Lett. 375 (2003) 321.[125] Z. Zhang, P. Fenter, L. Cheng, N.C. Sturchio, M.J. Bedzyk, M. Predota, A. Bandura, J.D. Kubicki, S.N. Lvov, P.T. Cummings, A.A.

Chialvo, M.K. Ridley, P. Benezeth, L. Anovitz, D.A. Palmer, M.L. Machesky, D.J. Wesolowski, Langmuir 20 (2004) 4954.[126] J. Zhao, B. Li, K. Onda, M. Feng, H. Petek, Chem. Rev. 106 (2006) 4402.[127] J. Moser, M. Grätzel, J. Am. Chem. Soc. 105 (1983) 6547.[128] A. Hagfeldt, M. Grätzel, Acc. Chem. Res. 33 (2000) 269.[129] A.C. Fisher, L.M. Peter, E.A. Ponomarev, A.B. Walter, K.G.U. Wijayantha, J. Phys. Chem. B 104 (2000) 949.[130] H. Tang, F. Lévy, H. Berger, P.E. Schmid, Phys. Rev. B 52 (1995) 7771.[131] K. Onda, B. Li, H. Petek, Phys. Rev. B 70 (2004) 045415.[132] W.R. Duncan, O.V. Prezhdo, J. Phys. Chem. B 109 (2005) 365.[133] I. Kondov, M. Thoss, H. Wang, Int. J. Quant. Chem. 106 (2006) 1291.[134] V.V. Prezhdo, E.V. Ovsiankina, O.V. Prezhdo, J. Mol. Struct. 522 (2000) 71.[135] M. Grätzel, J. Photochem. Photobiol. C: Photochem. Rev. 4 (2003) 145.[136] Y. Wang, K. Hang, N.A. Anderson, T. Lian, J. Phys. Chem. B 107 (2003) 9434.[137] L. Gundlach, R. Ernstorfer, F. Willig, Phys. Rev. B 74 (2006) 035324.[138] D.S. Warren, A.J. McQuillan, J. Phys. Chem. B 108 (2004) 19373.[139] D.A. Panayotov, J.T. Yates, Chem. Phys. Lett. 436 (2007) 204.[140] S. Ramakrishna, F. Willig, A. Knorr, Surf. Sci. 558 (2004) 159.[141] T. Hannappel, B. Burfeindt, W. Storck, F. Willig, J. Phys. Chem. B 101 (1997) 6799.[142] Y. Trachibana, S.A. Haque, I.P. Mercer, J.R. Durrant, D.R. Klug, J. Phys. Chem. B 104 (2000) 1198.[143] G. Ramakrishna, H.N. Ghosh, A.K. Singh, D.K. Palit, J.P. Mittal, J. Phys. Chem. B. 105 (2001) 12786.[144] E. Hao, N.A. Anderson, J.B. Asbury, T. Lian, J. Phys. Chem. B 106 (2002) 10191.[145] G. Ramakrishna, A.K. Singh, D.K. Palit, H.N. Ghosh, J. Phys. Chem. B. 108 (2004) 4775.[146] J.E. Moser, M. Grätzel, Chem. Phys. 176 (1993) 493.[147] O.V. Prezhdo, P.J. Rossky, J. Phys. Chem. 100 (1996) 17094.[148] A.A. Mosyak, O.V. Prezhdo, P.J. Rossky, J. Mol. Struct. 485–486 (1999) 545.[149] B.F. Habenicht, C.F. Craig, O.V. Prezhdo, Phys. Rev. Lett. 96 (2006) 187401.[150] B.F. Habenicht, H. Kamisaka, K. Yamashita, O.V. Prezhdo, Nano. Lett. 7 (2007) 3769.[151] B.F. Habenicht, O.V. Prezhdo, Phys. Rev. Lett. 100 (2008) 197402.[152] L. Saiz, M.L. Klein, Acc. Chem. Res. 35 (2002) 482.[153] O.V. Prezhdo, Chem. Phys. Lett. 460 (2008) 1.[154] T. Vreven, K. Byun, I. Komaromi, S. Dapprich, J. Montgomery, K. Morokuma, M. Frisch, J. Chem. Theory Comput. 2 (2006)

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