Theoretical Study of Photoinduced Electron-Transfer Processes in the

J. Phys. Chem. C 2010, 114, 18481–18493

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Theoretical Study of Photoinduced Electron-Transfer Processes in the Dye-Semiconductor System Alizarin-TiO2 Jingrui Li,*,†,‡ Ivan Kondov,⊥,† Haobin Wang,§ and Michael Thoss*,‡ Department of Chemistry, Technical UniVersity of Munich, Lichtenbergstrasse 4, D-85747 Garching, Germany, Institute for Theoretical Physics and Interdisciplinary Center for Molecular Materials, Friedrich-Alexander-UniVersität Erlangen-Nürnberg, Staudtstrasse 7/B2, D-91058 Erlangen, Germany, and Department of Chemistry and Biochemistry, MSC 3C, New Mexico State UniVersity, Las Cruces, New Mexico 88003 ReceiVed: May 12, 2010; ReVised Manuscript ReceiVed: August 17, 2010

Photoinduced electron-transfer processes in the dye-semiconductor system alizarin-TiO2 are studied. The study is based on a recently developed method, which uses first-principles electronic structure calculations to characterize the system and to parametrize a model Hamiltonian including electronic-vibrational coupling. On the basis of this modeling procedure, accurate quantum dynamical simulations are performed, employing the multilayer multiconfigurational time-dependent Hartree method. The results of the simulations show that the electron injection process in this system takes place on an ultrafast femtosecond time scale and is accompanied by significant electronic coherence effects. A detailed analysis reveals that the electron-transfer process proceeds via a two-step mechanism involving an intermediate state localized at the dye-substrate interface. I. Introduction Photoinduced electron-transfer (ET) reactions at dyesemiconductor interfaces represent an interesting example of surface ET processes. In particular, the process of electron injection from an electronically excited state of a dye molecule into a semiconductor substrate has been studied in great detail experimentally in recent years.1-26 This process presents a key step for photonic energy conversion in nanocrystalline solar cells.2,7,10,27,28 Employing femtosecond spectroscopy techniques, which allow the observation of ultrafast photoreactions in real time, it has been shown that electron injection processes at dye-semiconductor interfaces often take place on an ultrafast subpicosecond time scale.3,10,11,15-17,29 For example, an electron injection time as fast as 6 fs has been reported for the system containing alizarin adsorbed on TiO2 nanoparticles in timeresolved experiments.15 Other interesting aspects of ultrafast interfacial ET reactions include the nonequilibrium character of the process, the effect of electronic-nuclear coupling,11,16 and the influence of intermediate states localized at the chromophore-substrate interface19,20,25,26 The theoretical study of interfacial ET reactions requires a quantum mechanical description for the overall processes including a characterization of the electronic structure of the system as well as a simulation of the ET dynamics taking into account the coupling to the nuclear degrees of freedom. To this end, a variety of different methods have been developed and employed.30-61 The electronic structure of dye molecules adsorbed at semiconductor substrates, in particular titanium oxide, has been studied by employing cluster models of nanoparticles32,47,50,57,59,60 or the slab model with periodic bound* To whom correspondence should be addressed. † Technical University of Munich. ‡ Friedrich-Alexander-Universität Erlangen-Nürnberg. § New Mexico State University. ⊥ Present address: Steinbuch Center for Computing, Karlsruhe Institute of Technology, Herrmann-von-Helmholtz-Platz 1, 76344 EggensteinLeopoldshafen, Germany.

ary conditions to describe an extended surface.37-39,42,43,49,52,53 The dynamics of electron injection at dye-semiconductor interfaces has been studied by employing first-principles simulations37,39,42,43,47,49,51,53 as well as models based on a parametrized Hamiltonian.30,31,33-36,41,44,45,48,54-56,58 While the former class of methods typically use an approximate classical treatment of the nuclear dynamics, the model-based approaches often allow a full quantum dynamical treatment. Recently, we have proposed a first-principles-based approach62 to study the quantum dynamics of ET in dye-semiconductor systems. This method employs a representation of the Hamiltonian in localized donor and acceptor states. The donor and acceptor states as well as other parameters of the ET Hamiltonian are determined using a partitioning method based on electronic structure calculations. On the basis of this modeling, the quantum dynamics of the ET process is simulated using the multilayer multiconfigurational time-dependent Hartree (ML-MCTDH) method.63 This approach has been successfully applied to investigate photoinduced interfacial ET processes in different dye-semiconductor systems.62,64 In this work, we apply this methodology to study ET dynamics in the system alizarin-TiO2 (anatase). This system has been studied in detail experimentally by employing ultrafast laser spectroscopy.15,22,24 The electronic structure and spectra of alizarin-TiO2 (anatase) were studied by Duncan et al.46 Furthermore, theoretical simulations of ET dynamics in the related system alizarin-TiO2 (rutile) have been carried out by employing an ab initio nonadiabatic molecular dynamics approach.43,49 The remainder of this paper is organized as follows. In section II, we outline the theoretical approach, including the model Hamiltonian as well as the partitioning method employed to determine the electronic energies and the donor-acceptor coupling matrix elements. Furthermore, a brief account of the ML-MCTDH method used to simulate the ET dynamics is given. Section III presents the results of the electronic structure calculations and the dynamical simulations of the ET process.

10.1021/jp104335k  2010 American Chemical Society Published on Web 10/14/2010

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Figure 1. Physical model of a photoinduced ET process including the relevant electronic states labeled by Vg, Vdd, and {Vkk}. Also indicated is the photoexcitation of the vibrational wavepacket from the ground state Vg to the donor state Vdd at t ) 0, which initiates the ET process.

In particular, the mechanism of the ET process is analyzed. Section IV concludes with a summary.

partitioning scheme based on electronic structure, in particular, density functional theory (DFT) calculations for a complex of the dye molecule with a finite TiO2 cluster. The scheme for defining the localized donor and acceptor states |ψd〉 and |ψk〉 in the Hamiltonian is based on three steps, (i) a partitioning of the Hilbert space in a donor and acceptor group using a localized basis, (ii) a partitioning of the Hamiltonian according to the donor-acceptor separation, and (iii) a separate diagonalization of the donor and acceptor blocks of the partitioned Hamiltonian. In the present paper, we work within the mean-field singleelectron picture. Thus, we identify the effective Hamiltonian with the Fock (or Kohn-Sham) matrix and use the orbitals and orbital energies to represent the corresponding system states and the energies in the partitioning method. In the first step, the set of (Gaussian-type) atomic orbitals of the overall system {|φj〉} is divided into two groups, the donor group {|φjd〉}, which comprises the orbitals centered at the atoms of the dye molecule, and the acceptor group {|φja〉}, which includes the orbitals centered at the TiO2 cluster. Since it is advantageous to work with orthogonal orbitals,66,67 the set of atomic orbitals of the overall system is orthogonalized according to Lo¨wdin68,69

II. Theory and Methods To study ET in the dye-semiconductor system alizarin-TiO2, we have used a methodology developed recently, which combines a first-principles-based model Hamiltonian and accurate quantum dynamical simulations using the ML-MCTDH method. The methodology has been outlined in detail previously.62 Therefore, here, we give only a brief summary of the method and some details specific to the application considered in the present work. A. ET Hamiltonian. Within the first-principles-based model used here to study ET dynamics in dye-semiconductor systems, the Hamiltonian is represented in a basis of the following localized electronic states which are relevant for the photoreaction: the donor state of the ET process |ψd〉 (which, in the limit of vanishing coupling between the chromophore and semiconductor substrate, corresponds to the product of an electronically excited state of the chromophore and an empty conduction band of the semiconductor) and the (quasi-)continuum of acceptor states of the ET reaction |ψk〉 (corresponding in the zero-coupling limit to the product of the ground state of the chromophore cation and a conduction band state of the semiconductor substrate). Thus, the ET Hamiltonian reads

H ) Tnucl + |ψd〉Vdd〈ψd| +

|φ˜ n〉 )

where S denotes the overlap matrix of atomic orbitals, with matrix elements Sjj′ ) 〈φj |φj'〉. The new basis functions obtained (|φ˜ n〉) exhibit a minimal deviation from the original ones in a least-squares sense, and hence, their localization is preserved. In particular, the classification with donor (|φ˜ nd〉) and acceptor (|φ˜ na〉) orbitals is still valid. The new set of orthogonal basis functions is then used to partition the Fock (or Kohn-Sham) matrix from the converged SCF (or DFT) calculation into the two (donor and acceptor) subspaces. The Fock matrix in the orthogonal basis is given by

F˜ ) S-1/2FS-1/2

(2.3)

where F denotes the Fock matrix in the original atomic orbital basis. The Fock matrix can be arranged in the following donor-acceptor block structure

∑ |ψd〉Vkk〈ψk|

∑ (|ψd〉Vdk〈ψk| + |ψk〉Vkd〈ψd |)

(2.2)

j

F˜ )

k

+

∑ (S-1/2)jn|φj〉

(

F˜dd F˜da Fãd Fãa

)

(2.4)

(2.1)

k

Here, Vdd and Vkk denote the energies of the electronic donor and acceptor states, respectively. The off-diagonal matrix elements Vdk characterize the donor-acceptor coupling, and Tnucl denotes the kinetic energy of the nuclear degrees of freedom. The physical model of the ET reaction is schematically illustrated in Figure 1. B. Determination of Electronic Energies and DonorAcceptor Coupling Matrix Elements. The parameters of the ET Hamiltonian (eq 2.1) are determined from first-principles electronic structure calculations employing a partitioning method. The details of this method, which is motivated by the projectionoperator approach of resonant electron-molecule scattering,65 have been presented previously.62 The method employs a

Separate diagonalization of the two (donor and acceptor) blocks of the Fock matrix, F˜dd and Fãa, and the corresponding transformation of the off-diagonal parts, result in the following prediagonalized block structure F)

)

(

(

Fdd Fda Fad Faa

)

d1 0 · · · 0 d2 · · · l l · ·.

Fda

Fad

a1 0 · · · 0 a2 · · · l l · ·.

)

(2.5)

Photoinduced Electron-Transfer Processes Alizarin-TiO2 The corresponding donor and acceptor orbitals, which can j na〉, are given as the eigenvectors j nd〉 and |φ be denoted as |φ of F˜dd and Fãa, respectively. The diagonal blocks of the thusobtained Fock matrix contain the energies of the localized chromophore states and those of the semiconductor substrate states, respectively. The off-diagonal blocks contain the electronic coupling matrix elements between chromophore and substrate sites. Identifying the donor state |ψd〉 with j nd〉 (based, e.g., on the orbital energy or one of the states |φ the transition dipole moment to the ground state) and the j ka〉, the electronic energies acceptor states |ψk〉 with the states |φ of donor and acceptor states are given by the corresponding diagonal elements of the prediagonalized Fock matrix (dn and {ak}), while the donor-acceptor coupling matrix elej da)nk. ments are given by Vdk ) (F The partitioning method discussed above is not limited to dye-semiconductor systems containing a finite semiconductor cluster but can, in principle, also be applied to a dye molecule adsorbed on an extended surface. One possibility is to employ a slab model and electronic structure calculations with periodic boundary conditions. Alternatively, the effect of an infinite semiconductor substrate can also be described using surface Green’s function techniques.70 Within this method, the effect of the infinite substrate enters via the self-energy. In this paper, we have used a simpler approximate version of this method to mimic the effect of an extended surface. Thereby, a constant imaginary part is added to the atomic orbital energies (in the j ka〉) at the outer atoms of the TiO2 substrate. orthogonal basis |φ In the results presented in section III, a value of 1 eV for the imaginary part has been used. Employing this approach, the interaction of the donor state with the acceptor states is fully characterized by the continuous function

Γ(ε) ) 2π

∑ |Vdk|2δ(ε - εk)

(2.6)

k

which is sometimes also called the energy-dependent decay width of the donor state. It describes the density of states of the semiconductor substrate weighted by the donor-acceptor coupling strength. The details of this method and a practical approach to approximately calculate Γ(ε) are described in ref 62. C. Characterization of Nuclear Degrees of Freedom. To characterize the nuclear degrees of freedom, the partitioning procedure outlined above has to be performed for each nuclear geometry, thus resulting in diabatic potential energy surfaces Vii(Q) and coordinate-dependent donor-acceptor coupling matrix elements Vdk(Q). If many nuclear degrees of freedom are important, as in the system considered below, such a global characterization of the potential energy surfaces is not feasible. Therefore, in the present paper, we adopt the strategy used in our previous work, where a local low-order polynomial expansion of the diabatic potential matrix elements Vij(Q) around the equilibrium geometry of the neutral ground state is employed. The corresponding vibrational parameters are determined based on electronic structure calculations for the isolated chromophore, thereby neglecting the coupling to the semiconductor substrate and to the phonons of the semiconductor. Since the ET in the system considered in this paper occurs on an ultrafast time scale of about 10 fs, these approximations are expected to be appropriate. The approach has been described in detail elsewhere.55,59 Briefly, we perform a vibrational analysis of the isolated chromophore in the electronic ground state and employ the

J. Phys. Chem. C, Vol. 114, No. 43, 2010 18483 harmonic approximation for the corresponding potential energy surface

Vg(Q) ) εg +

1 2

∑ Ωl2Ql2

(2.7)

l

Here, Ql denotes the lth normal mode (with frequency Ωl) and εg is the ground-state equilibrium energy (mass-scaled coordinates and atomic units are used throughout this paper). The latter is obtained from an electronic structure calculation for the overall (dye-semiconductor) system. We assume the donoracceptor coupling matrix elements Vdk to be approximately independent of the nuclear geometry71 and expand the diabatic potential energy surfaces around the equilibrium geometry of the electronic ground state, Q0

∑ κljQl + ∑ γlkjQlQk

Vjj(Q) ) Vjj(Q0) +

l

(2.8)

lk

In the simplest approximation, only the linear terms of the expansion are taken into account; the frequencies are approximated by their ground-state values, and Dushinsky rotation72 of the normal modes is neglected. In this way, we obtain

∑ κljQl + 21 ∑ Ωl2Ql2

Vjj(Q) ) Vjj(Q0) +

l

(2.9)

l

This approximation has been used successfully to describe Franck-Condon and resonance Raman spectra.73 It is also used in the linear vibronic coupling model of conical intersections74 and in the Marcus theory of ET.75 Within the description of the nuclear degrees of freedom employed here, the parameters of the diabatic potential energy surfaces of the donor and acceptor states are obtained from the potential energy functions of the excited state of the neutral chromophore and the ground state of the cation of the chromophore, respectively. Accordingly, we have

Vdd(Q) ) εd +

∑ κldQl + 21 ∑ Ωl2Ql2

(2.10a)

∑ κlaQl + 21 ∑ Ωl2Ql2

(2.10b)

l

Vkk(Q) ) εk +

l

l

l

where εd and εk denote the energy of the donor and acceptor states (at the equilibrium geometry of the ground state), respectively, which are obtained from an electronic structure calculation of the overall system. The electronic-vibrational coupling constants κld and κla are obtained from the gradients of the excited state of the neutral chromophore (corresponding to the donor state) and the ground state of the cation of the chromophore (corresponding to the acceptor states) at the equilibrium geometry of the ground state of the neutral chromophore. The details of the electronic structure calculations as well as the specific parameters for the system

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considered here (alizarin at titanium oxide) are described below. In the results presented below, we will also consider for analysis the purely electronic dynamics that is obtained with the nuclear degrees of freedom frozen in their equilibrium positions. The corresponding approximate electronic Hamiltonian is given by

Helec ) |ψd〉εd〈ψd | +

∑ |ψk〉εk〈ψk|

+

k

∑ (|ψd〉Vdk〈ψk| + |ψk〉Vkd〈ψd|)

(2.11)

k

D. Observables of Interest and Dynamical Method. Several observables are of interest to analyze heterogeneous ET reactions in dye-semiconductor systems. In the present work, we focus on the electron-injection dynamics, which is most directly reflected by the time-dependent population of the donor state

|Ψ(t)〉 )

J

p

∑ ∑ ... ∑ Aj j ...j (t) ∏ |φjκ (t)〉 j1

Thereby, we have assumed that the system is initially prepared by an ultrashort laser pulse in the donor state |ψd〉. In principle, the photoexcitation may also result in a direct population of acceptor states.49 An analysis of this mechanism would require the inclusion of the laser pulse in the simulation, which is beyond the scope of the current study and will be the subject of future work. The initial state of the nuclear degrees of freedom is specified by the Boltzmann operator exp(-βHnucl,g) of the nuclear Hamiltonian in the electronic ground state

Hnucl,g ) Tnucl +

1 2

∑ Ωl2Ql2

(2.13)

l

with the kinetic energy of the nuclei given by Tnucl ) ∑l Pl2/2. To simulate the quantum dynamics of this system, we use the multilayer (ML) formulation63 of the multiconfiguration time-dependent Hartree (MCTDH) method76-79 in combination with an importance sampling scheme80 to describe the thermal initial conditions (all calculations in this paper are performed at a temperature of 300 K). The method as well as applications to different reactions in the condensed phase have been described in detail previously.44,63,81 Here, we only briefly introduce the general idea and give some details specific to the application in this work. The ML-MCTDH method63 is a rigorous variational approach to study quantum dynamics in systems with many degrees of freedom. It extends the previously proposed MCTDH method76-79 to treating significantly larger systems. In the original (singlelayer) MCTDH method, the overall wave function is expanded in terms of time-dependent configurations

j2

12

jp

p

κ)1

κ

(2.14)

Each configuration is a Hartree product (for systems with distinguishable particles) of the “single-particle” functions (SPFs) |φjκκ(t)〉, where p denotes the total number of single particle (SP) degrees of freedom and κ is the index of a particular SP group. In this single-layer MCTDH method, the SPFs are represented by the full configuration interaction (CI) expansion of the time-independent basis functions. This limits the application of the MCTDH method to a few tens of degrees of freedom.76-79 In the ML-MCTDH theory, the full CI construction of the SPFs is replaced by a recursive, layered expansion of the timedependent wave function. Following eq 2.14, this recursive expansion reads

|Ψ(t)〉 )

Pd(t) ) Tr[exp(-βHnucl,g)|ψd〉〈ψd | exp(iHt)|ψd〉〈ψd | exp(-iHt)] Tr[exp(-βHnucl,g)] (2.12)

∑ AJ(t)|ΦJ(t)〉 ≡

∑∑ ∑ ...

j1

|φj(κ) (t)〉 ) κ

j2

jp

∑ ∑ ... ∑ B i1

|V(κ,q) iq (t)〉 )

p

Aj1j2...jp(t)

i2

(κ) jκ (t)〉

Q(κ)

κ,jκ i1i2...iQ(κ)(t)

iQ(κ)

R2

(2.15a)

κ)1

RM(κ,q)

∏ |V

(κ,q) iq (t)〉

(2.15b)

q)1

∑ ∑ ... ∑ C R1

∏ |φ

M(κ,q) κ,q,iq R1R2...RM(κ,q)(t)

∏ |ξ

κ,q,γ Rγ (t)〉

(2.15c)

γ)1

l q where Aj1j2...jp(t), Biκ,j1i2κ...iQ(κ)(t), CRκ,q,i (t), . . . are the expansion 1R2...RM(κ,q) coefficients for the first, second, third, . . . layers, respectively; (κ,q) κ,q,γ |φ(κ) jκ (t)〉, |Viq (t)〉, |ξRγ (t)〉, . . . are the SPFs for the first, second, third, . . . layers, respectively. The notations beyond the first layer are as follows. In eq 2.15b, Q(κ) is the number of (level 2) SP groups for the second layer that belong to the κth (level 1) SP group in the first layer, that is, there are a total of p Q(κ) second-layer SP groups. Continuing along the mul∑κ)1 tilayer hierarchy, M(κ,q) in eq 2.15c is the number of (level 3) SP groups for the third layer that belong to the qth (level 2) SP group of the second layer and the κth (level 1) SP group of the first layer, resulting in a total of ∑pκ)1 ∑Q(κ) q)1 M(κ,q) third-layer SP groups. Such a recursive expansion can be carried out to an arbitrary number of layers. To terminate the multilayer hierarchy at a particular level, the SPFs in the deepest layer are expanded in terms of time-independent configurations. For example, in the four-layer version of the ML-MCTDH theory, the fourth layer is expanded in the time-independent basis functions/configurations, each of which may still contain several Cartesian degrees of freedom. Applying the Dirac-Frenkel variational principle

〈δΨ(t)|i

∂ ˆ |Ψ(t)〉 ) 0 -H ∂t

(2.16)

with the functional form in eq 2.15, the equations of motion can be obtained as

Photoinduced Electron-Transfer Processes Alizarin-TiO2

˙ (t)〉L1 coefficients ) H ˆ (t)|Ψ(t)〉 i|Ψ

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(2.17a)

(κ) ˆ 〉(κ)(t)|φ i|φ˙ κ(t)〉L2 coefficients ) [1 - Pˆ(κ)(t)][Fˆ (κ)(t)]-1〈H _ (t)〉 (2.17b)

i|V˙ (κ,q)(t)〉L3 coefficients ) (κ,q) (κ,q) (κ,q) (t)][ρˆ L2 (t)]-1〈Hˆ 〉L2 (t)|V_(κ,q)(t)〉 (2.17c) [1 - PˆL2 i|ξ˙ (κ,q,γ)(t)〉L4 coefficients ) (κ,q,γ) (κ,q,γ) (κ,q,γ) (t)][ρˆ L3 (t)]-1〈Hˆ 〉L3 (t)|ξ_(κ,q,γ)(t)〉 (2.17d) [1 - PˆL3 l where the mean-field operators, reduced densities, and projection operators are defined in ref 63. For clarity, we refer to the top layer as the level 1 (L1) SP space, the second layer as the level 2 (L2) SP space, and so forth. The time derivatives in eq 2.17, denoted by an overhead dot on the left-hand side of each equation, are meant to be carried out only with respect to the expansion coefficients of a particular layer that appeared in that equation. For example, the time derivative in eq 2.17a acts only on the L1 expansion coefficient Aj1j2...jp(t); the time derivative in eq 2.17b is only on the L2 expansion coefficient Biκ,j1i2κ...iQ(κ)(t); and so forth. Note that for the N-layer version, there are (N + 1) levels of expansion coefficients because the SPFs in the deepest layer need to be expanded in time-independent basis functions/configurations. In this sense, the conventional wavepacket propagation method is a “zero-layer” MCTDH approach. The inclusion of several dynamically optimized layers in the ML-MCTDH method provides more flexibility in the variational functional, which significantly advances the capabilities of performing wavepacket propagations in complex systems. This has been demonstrated by several applications to quantum dynamics in the condensed phase including many degrees of freedom.44,62,63,80-84 In the calculation considered below, up to two dynamical layers are employed. III. Results and Discussions A. Characterization of the System. The methodology outlined in section II has been applied to study electron transfer in the dye-semiconductor system alizarin-TiO2. To investigate the quantum dynamics of electron injection from alizarin to the TiO2 substrate, we have considered complexes consisting of alizarin and anatase TiO2 cluster substrates of different sizes (two examples are depicted in Figure 2). All geometric parameters of the TiO2 clusters were taken from the X-ray structure of bulk anatase.85 In order to avoid artificial effects due to dangling bonds, the clusters were saturated by adding hydrogen atoms and hydroxyl groups to the boundary oxygen and titanium atoms, respectively, in a way that neutral closedshell clusters with high coordination of all titanium and oxygen atoms were obtained. Similar cluster models for the (101) anatase surfaces have been used in refs 86 and 87. Specifically, we have considered complexes with seven different TiO2 clusters, which comprise between 1 and 4 (101) layers of anatase with 10 TiO2 units per layer or between 1 and 3 (101) layers with 18 TiO2 units per layer. The results obtained for the ET dynamics are qualitatively similar for all cluster sizes considered (see below). Therefore, only the results obtained for the largest complex alizarin-(TiO2)54(H2O)58 (Figure 2b) will be discussed in detail.

Figure 2. Complexes of alizarin with titanium oxide clusters of different sizes, (a) (TiO2)10(H2O)18 and (b) (TiO2)54(H2O)58.

The following protocol was used to obtain the structure of the alizarin-(TiO2)54(H2O)58 complex: (i) The geometry of the alizarin anion C14H6O42- attached to a (TiO2)10(H2O)18H22+ substrate (corresponding to one (101) anatase layer) was optimized. Thereby, all nuclear degrees of freedom of the TiO2 substrate were kept frozen, and a 1,2-bidentate adsorption mode was used. This mode is energetically the most favorable, as indicated by test calculations. Furthermore, the two protons detached from alizarin were attached to two surface oxygen atoms adjacent to the reaction center. (ii) Using the thus-obtained geometry and relative orientation of the alizarin-TiO2 complex, alizarin was attached to the larger TiO2 substrates. The geometry optimization and the characterization of nuclear degrees of freedom of isolated alizarin were performed with TURBOMOLE88 using DFT with the B3LYP functional and the TZV(P) basis set. The geometry optimization of alizarin-(TiO2)10(H2O)18 was carried out with DFT using the Gaussian 03 package89 with the B3LYP functional and the 3-21G basis set. The single-point electronic structure calculations

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Figure 3. Energy-level scheme of the investigated complex alizarin-(TiO2)54(H2O)58. From left to right: energy levels of the isolated chromophore alizarin, energy levels of the donor orbitals (obtained from the partitioning procedure), which are localized in the adsorbate, energy levels of the overall complex, energy levels of the acceptor orbitals (obtained from the partitioning procedure), which are localized in the semiconductor substrate, and energy levels of the pure titanium oxide cluster (TiO2)54(H2O)60. The selected donor state |ψd〉 as well as the correlations among some energy levels relevant for the ET reaction are indicated.

for all complexes were performed using the TURBOMOLE package with the B3LYP functional and the SV(P) basis set. B. Energy-Level Scheme, Donor-Acceptor Separation, and Coupling Matrix Elements. An important aspect of interfacial ET reactions is the energy-level scheme, in particular, the location of the energy levels of the dye molecule relative to those of the semiconductor substrate. Figure 3 shows the energies of the molecular orbitals of the overall alizarin(TiO2)54(H2O)58 complex as well as those of the donor and acceptor orbitals obtained by the partitioning procedure described in section II.B. Also shown, in comparison, are the energy levels of the isolated chromophore alizarin and those of the isolated (TiO2)54(H2O)60 cluster. The energies of the orbitals of the isolated (TiO2)54(H2O)60 cluster exhibit a dense level structure with a valence and a conduction band separated by a band gap. The calculated value for this band gap is 2.4 eV, which is smaller than the calculated (4.0 eV90) and experimental (3.4 eV91) values of anatase TiO2 nanoparticles. This underestimation of the band gap is presumably due to the added hydrogen atoms and hydroxyl groups, which are used to saturate the cluster. A detailed analysis (data not shown) reveals that the lowest unoccupied orbitals have predominant contributions from the oxygen atoms of saturation groups. As has been discussed previously in a study of similar clusters,86 this deficiency of the saturated cluster model is not expected to have a significant influence on the electronic levels involved in the interfacial reactions. Defining the “true” lower edge of the conduction band by the energy of the lowest unoccupied (3d) orbital of the titanium atoms, a value of about 3.3 eV is obtained for the “band gap” in (TiO2)54(H2O)60, which is in good agreement with to the experimental value for bulk anatase. The energy-level scheme of the overall system (middle panel of Figure 3) shows that the adsorption of the chromophore alizarin on the titanium oxide cluster changes the level structure noticeably. Two energy levels are introduced in the lower part of the band gap through the adsorption of the alizarin molecule. Analysis of the orbitals corresponding to these two levels shows that they can be related to the HOMO and the (HOMO-1) of the isolated alizarin molecule. Thus, the two highest occupied levels of alizarin remain located in the band gap during the adsorption and thus retain their discrete structure in the complex.

Li et al. On the other hand, the lowest unoccupied levels of alizarin are located energetically in the conduction band of TiO2. As a consequence, in the complex, these levels are dissolved in the dense manifold of conduction band levels. The partitioning procedure results in orbitals localized in the alizarin chromophore and the (TiO2)54(H2O)58 substrate, which are depicted in Figure 3 as donor and acceptor levels, respectively. The energy-level scheme of the acceptor orbitals in Figure 3 shows a structure very similar to that of the isolated (TiO2)54(H2O)60 cluster. As discussed above for the isolated TiO2 cluster, the acceptor levels in the upper part of the band gap are predominantly localized at the oxygen atoms of the saturation groups. An example for such an orbital is shown in Figure 4a. The electronic coupling between these levels in the upper part of the band gap and the donor levels of alizarin is negligible in the complex, and thus, these states do not participate in the ET reaction. The energy-level scheme of the orbitals localized at the chromophore shows two energy levels in the band gap which correspond to the HOMO and (HOMO-1) of the overall system and are closely related to the HOMO and (HOMO-1) of the isolated chromophore. For instance, the overlap between the HOMO of the complex and the projected donor orbital that is associated with the HOMO of alizarin is larger than 0.80, thus demonstrating the close relation. The donor state of the ET reaction is chosen as the localized orbital that corresponds to the LUMO of an isolated alizarin. Time-dependent (TD) DFT calculations isolated alizarin have shown that the HOMO-LUMO excitation contributes more than 95% to the S0 f S1 excitation. Due to the coupling to the dense manifold of conductance band states of TiO2, the donor level does not have a predominant overlap to a single level of the complex. The local character of two frontier orbitals localized at the chromophore and the resemblance between them and the corresponding orbitals (HOMO and LUMO) of an isolated alizarin are illustrated in Figure 5. The acceptor states included in the dynamical calculation of the ET process comprise all orbitals localized in the TiO2 substrate that are associated with unoccupied orbitals in the conduction band of the isolated titanium oxide cluster. As in the isolated TiO2 cluster, the acceptor orbitals are dominated by the 3dxy, 3dyz, and 3dxz orbitals of titanium atoms due to the 3d level splitting caused by the octahedral coordination. Figure 4b depicts a projected acceptor orbital which is delocalized in the substrate part of the complex. An important parameter for the electron injection dynamics is the location of the donor level relative to the conductance band minimum. In our model, the donor level is located about 0.7 eV above the conduction band minimum, where the latter is defined by the lowest unoccupied 3d orbital of the titanium atoms, as discussed above. This location is higher than that reported by Duncan et al., where the donor level has an energy close to the conduction band minimum.43,46,49,51 This difference is presumably due to the different adsorption motifs and forms of titanium oxide (rutile vs anatase) considered and may also be caused by the different functionals and basis sets used. It should also be noted that the donor level, as defined in our method, is not a molecular orbital, that is, an eigenstate of the overall complex. The coupling to the acceptor states will result in a lowering of the energy of the corresponding resonance in the overall complex. Another key factor for heterogeneous ET reactions is the strength and distribution of donor-acceptor coupling. Figure 6 shows the modulus of the donor-acceptor coupling matrix elements Vdk for the system investigated. It is seen that the first-


J. Phys. Chem. C, Vol. 114, No. 43, 2010 18487 for other chromophores at TiO2.62,64 For an extended substrate, the donor-acceptor coupling can be characterized by the energydependent decay width of the donor state, Γ(ε), as defined in eq 2.6. The Γ(ε) function, which is obtained based on the discrete Vdk data as discussed in section II, is also shown in Figure 6. The position of the peak of Γ(ε) is in good agreement with the positions of the two acceptor states with the largest coupling to the donor state. C. Vibrational Degrees of Freedom and ElectronicVibrational Coupling. The vibrational frequencies Ωl and the corresponding electronic-vibrational coupling constants κld/a in the donor and acceptor states have been determined for all 72 vibrational modes of alizarin, as described in section II. All parameters are given in Table 1. For the lth normal mode, the electronic-vibrational coupling constants are related to reorganization energies via

λld )

λla

)

(κld)2 2Ωl2 (κla)2 2Ωl2

(3.1a)

(3.1b)

which are associated with transitions from the electronic ground state to the excited state and the cation of alizarin, respectively. The reorganization energy for the ET process for the lth normal mode, which corresponds to the transition from the excited state of the chromophore to the cation, is given by

λlET )

Figure 4. Examples of localized orbitals associated with unoccupied orbitals of the isolated titanium oxide cluster. Shown are (a) an “acceptor” orbital with energy in the upper part of the band gap (this orbital is predominantly localized at the saturation groups) and (b) a localized acceptor orbital associated with an unoccupied orbital in the conduction band of the isolated titanium oxide cluster.

principles-based model results in a distribution of donor-acceptor coupling matrix elements Vdk that exhibits a rather complicated structure. This finding agrees with results obtained previously

(κld - κla)2 2Ωl2

(3.2)

The corresponding total reorganization energies are given by . The calculated frequencies and the sum Λd/a/ET ) ∑l λd/a/ET l reorganization energies for the donor state and the ET transition are depicted in Figure 7. It is seen that the electronic-vibrational coupling is distributed over a rather large number of vibrational modes. The overall reorganization energies obtained are Λd ) 0.301 eV, Λa ) 0.240 eV, and ΛET ) 0.156 eV for the donor state, the acceptor state, and the ET transition, respectively. These values for the reorganization energies suggest a moderate electronic-vibrational coupling with respect to excitation but rather weak coupling to the ET process itself. For the simulation of the vibronic dynamics in the alizarin-TiO2 system, 37 of the total 72 vibrational normal modes were selected based on the ET-related electronic-vibrational coupling strength. The thus-selected modes incorporate more than 98% of the three total reorganization energies and are therefore expected to give a proper representation of the electronic-nuclear coupling in alizarin. D. ET Dynamics. The ET dynamics in the alizarin-TiO2 system has been simulated based on the first-principles models outlined in the previous section. Figure 8a shows the result of the simulation for the population of the donor state after photoexcitation (solid line). The initial decay of the population of the donor state reveals an ultrafast injection of the electron from the donor state localized at the chromophore into the quasicontinuum of acceptor states localized in the TiO2 substrate on a time scale of a few femtoseconds. The result also exhibits pronounced oscillations on different time scales. A comparison

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Figure 5. Selected localized adsorbate orbitals of the investigated complex obtained from the partitioning procedure (compared with the corresponding molecular orbitals of the isolated chromophore), (a) the localized orbital of the complex which is associated with the HOMO of isolated alizarin, (b) the HOMO of isolated alizarin, (c) the localized orbital of the complex which is associated with the LUMO of isolated alizarin (the donor orbital in the ET reaction), and (d) the LUMO of isolated alizarin.

The comparison between purely electronic and vibronic calculations also shows that the coupling to the vibrational modes results in a somewhat slower decay of the population of the donor state, in particular, in the short-time dynamics (shown by the inset of Figure 8a). This is due to the fact that during the dynamics, the nuclear wavepacket enters regions of phase space with an effectively smaller donor-acceptor coupling, an effect that has been discussed in detail previously.55 However, overall, the effect of electronic-vibrational coupling on the ET dynamics is rather small in the system investigated. This is a results of the ultrafast time scale of the ET process and the relatively small reorganization energy (0.155 eV). Figure 6. Modulus of the donor-acceptor coupling matrix elements Vdk (discrete lines) and the Γ(ε) function (continuous line). The red vertical line indicates the energy of the donor state εd.

with results of a purely electronic calculation (dashed line), where the nuclear degrees of freedom are frozen at their equilibrium geometry, reveals that the oscillations are due to electronic motion, that is, reflect electronic coherences. A further analysis of these oscillations is given below.

Due to the coherent oscillatory character of the dynamics, the electron injection dynamics cannot be characterized by a single rate constant. However, the overall time scale of about 5-10 fs found in the simulations agrees well with the experimental result of 6 fs15 for a colloidal solution of TiO2 nanoparticles and with previous mixed quantum-classical calculations.39,43,49 Very recent experimental results for alizarin adsorbed at nanostructured TiO2 films show a slower injection time.24



TABLE 1: Vibrational Parameters of the Selected Normal Modes of Alizarin Included in the Dynamical Simulation of The ET Processa mode no.

Ω

κd/Ω

κa/Ω

mode no.

Ω

κd/Ω

κa/Ω

10 12 15 18 21 23 25 27 29 32 34 39 41 43 44 45 46 47 48

329.0 395.2 428.2 477.1 586.5 629.4 674.9 702.5 770.6 846.6 911.5 1030.5 1072.5 1178.4 1192.9 1210.1 1223.3 1247.7 1291.2

282.5 271.3 179.4 217.1 -243.9 -226.9 231.8 -174.2 88.7 -381.4 261.8 393.4 -292.2 -84.1 -487.2 319.6 -416.5 643.8 568.4

83.9 398.8 149.5 29.3 -60.3 -378.8 138.0 -224.9 150.5 -280.0 246.0 427.1 -224.5 -170.3 -167.8 41.2 -538.0 519.1 120.3

49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 71 72

1305.2 1322.7 1348.7 1358.2 1378.2 1438.3 1486.7 1491.7 1511.2 1520.4 1611.3 1625.6 1629.2 1635.4 1674.2 1718.5 3305.4 3710.1

698.5 -265.8 -168.3 641.9 177.3 623.9 -450.3 120.9 -695.0 213.6 216.2 -780.8 407.7 90.4 580.7 -55.6 593.6 -163.4

199.2 -622.5 -120.7 -16.5 400.9 446.5 -859.1 -81.1 -389.9 301.5 93.3 -831.6 44.2 136.5 -486.5 427.6 613.3 -297.2

a Listed are the ground-state harmonic vibrational frequencies and the electronic-vibrational coupling constants in the excited state of alizarin and in the ground state of the alizarin cation. All data are given in cm-1.

Figure 7. Reorganization energies of the intramolecular modes of alizarin associated with (a) the transition from the ground to the electronically excited state and (b) the ET transition.

The results in Figure 8a also show that the population of the donor state does not decay to zero for longer times. This is a result of the finite TiO2 substrate used in the calculation. Reflection of the wavepacket at the boundaries of the substrate results in recurrences in the population dynamics, for example, at 50 fs. The dependence of the ET dynamics on the size of the TiO2 substrate is illustrated in Figure 9. Figure 9a shows that an increase of the number of layers of the substrate from two

Figure 8. Population dynamics of the donor state after photoexcitation. Shown are results obtained for (a) the finite (TiO2)54 substrate and (b) the model of both results with vibronic coupling (solid lines) and without vibronic coupling (dashed lines) are depicted.

to four (each layer with 10 TiO2 units) results in an increase of the recurrence period (due to the reflection at the boundaries) by about a factor of 2. On the other hand, an increase of the lateral size of the TiO2 clusters results in a smaller intensity of the recurrences but has negligible effect on the recurrence period, as illustrated in Figure 9b. The results in Figure 9 also

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donor state h intermediate state(s) f acceptor states (3.3) The fact that the population of the donor state (in the extended system) decays for long time, that is, the oscillations do not persist, shows that these intermediate states cannot be truly bound surface states with energies in the band gap but rather surface resonances with energies in the conductance band. To analyze the oscillations in more detail, we consider for simplicity the purely electronic dynamics. The two-step ET mechanism can be seen more explicitly if the Hamiltonian is unitarily transformed to the form

Helec ) |ψd〉εd〈ψd | + |ψm〉εm〈ψm | + + |ψd〉Vdm〈ψm | + |ψm〉Vmd〈ψd | +

∑ |ψ˜ k′〉ε˜ k′〈ψ˜ k′| k′

∑ (|ψm〉Vmk'〈ψ˜ k′| + |ψ˜ k′〉Vkm′〈ψm|) k′

(3.4) Here, we have introduced the intermediate state

|ψm〉 )

V

∑ Vdmdk |ψk〉

(3.5)

k

with energy Figure 9. Dependence of ET dynamics on the size of the TiO2 cluster used to model the substrate. Shown are results for the vibronic dynamics obtained with the TiO2 clusters containing (a) 4 (solid line) and 2 (dashed line) layers with 10 TiO2 units per layer and (b) 3 layers with 18 (solid line) and 10 (dashed line) TiO2 units per layer.

εm )

V2dk

∑ V2 k

εk

(3.6)

dm

and coupling matrix element to the donor state demonstrate that the influence of the finite size of the TiO2 cluster is negligible on the time scale of the ultrafast ET process in alizarin. As discussed in section II, electron injection at an extended surface can be mimicked using absorbing boundary conditions by including an imaginary part in the orbital energies at the boundary of the substrate. As a result, the recurrences in the dynamics as well as the incompleteness of electron injection are quenched. The corresponding results, depicted in Figure 8b, agree well for short times with the results obtained for finite TiO2 substrates (Figure 8a) but decay to zero for longer times. E. Electronic Coherence Effects and Analysis of Electron Transfer Mechanism. The electron injection dynamics depicted in Figure 8 exhibits pronounced oscillations on a time scale of about 6 fs. These oscillations are also present in the calculations for the extended system (Figure 8b) and are therefore not a finite-size effect. The presence of the same oscillations in the results of purely electronic calculations (dashed lines in Figure 8) as well as the ultrafast time scale of 6 fs shows that these oscillations are not related to vibrational dynamics but rather to electronic motion, that is, they can be classified as electronic coherence effects. This finding suggests that the electron injection process proceeds with a two-step mechanism, that is, there exists a single (or a few) intermediate state(s) localized at the surface through which the electron in the originally populated donor state decays to the conduction band of the substrate according to the scheme

Vdm ) (

∑ V2dk)1/2

(3.7)

k

˜ k′〉 constitute the “secondary” acceptor states. The states |ψ Equations 3.6 and 3.7 show that the intermediate state |ψm〉 carries all coupling to the donor state and is in turn coupled to the secondary acceptor states. The intermediate state |ψm〉 for the alizarin-TiO2 system studied here is depicted in Figure 10. It is localized at the two titanium atoms that bind to the alizarin and is dominated by 3d orbitals of titanium. As to be expected from the distribution of their orbitals (cf. Figures 5c and 10), there is rather strong interaction between the intermediate state and the donor state (Vdm ) 0.336 eV). Furthermore, the energy of the intermediate state, εm ) -0.513 eV, is in good agreement with the maximum of the decay-width function of the donor state, Γ(ε) (cf. Figure 6) and is not far away from the donor level (εd ) -0.699 eV). The transformed Hamiltonian (eq 3.4) allows one to analyze the two-step mechanism of the ET process. The part of the Hamiltonian without the secondary acceptor states is given by the two-level system of the donor and the intermediate state d-m Helec ) |ψd〉εd〈ψd | + |ψm〉εm〈ψm | + |ψd〉Vdm〈ψm | + |ψm〉Vmd〈ψd | (3.8)



and describes Rabi oscillations of the electronic population between the donor and the intermediate state

Pd-m (t) ) 1 d 2 4Vdm

2

2 4Vdm + (εd - εm)2

sin

2 + (εd - εm)2t √4Vdm

(3.9)

2

with the Rabi-period

Td-m )

2π 2 + (εd - εm)2 √4Vdm

(3.10)

The value of the Rabi-period, Td-m ≈ 6 fs, agrees well with the oscillation of the donor-state population found for the overall system in Figure 8. Due to the coupling of the intermediate state to the secondary acceptor states, Vmk′, these oscillations are damped. Figure 11 shows an analysis of the population dynamics employing the intermediate state representation introduced above. After the initial excitation of the donor state, the population oscillates between the donor state and the intermediate state and decays from the intermediate state to the secondary acceptor states in the TiO2 substrate. The increase of the population of the secondary acceptor states in the substrate is almost monotonous. Thus, the overall ET mechanism can be considered as a coherent two-step procedure in accordance with the scheme given by eq 3.3. These findings are in accordance with previous theoretical studies and experimental results. Theoretical simulations37,42 for other chromophores adsorbed at TiO2 have shown that the injected electron is initially localized on titanium sites at the surface similar to those respresented by the intermediate state

Figure 10. Orbital that represents the intermediate state |ψm〉.

Figure 11. Analysis of the mechanism of the ET process based on the scheme given by eq 3.3. Shown are the population dynamics of the donor state (thin solid line), the intermediate state (thick dashed line), and the sum of the population of the secondary acceptor states (thick solid line). All results are obtained by employing a finite TiO2 substrate without including the electronic-vibrational coupling.

|ψm〉 (cf. Figure 10). Furthermore, experimental studies for dye-semiconductor systems with a slower injection time scale, in particular, chromophores adsorbed at ZnO, indicate a stepwise mechanism with an intermediate state that has been attributed to an interface-bound charge-separated pair state or exciplex state.19,20,25,26 Our results show that intermediate states can also be important in systems with very short electron injection times. In this case, the first step of the ET process may have significantly coherent character. IV. Concluding Remarks In this paper, we have studied the dynamics of photoinduced heterogeneous ET processes in the dye-semiconductor system alizarin-TiO2. The study was based on a recently developed method, which uses first-principles electronic structure calculations to characterize the system and to parametrize a model Hamiltonian including electronic-vibrational coupling. Within this model, the quantum dynamics of the ET process was simulated using the multilayer multiconfigurational time-dependent Hartree method. The results of the simulations reveal that the electron injection in the investigated system takes place on an ultrafast time scale of about 10 fs, which is in good agreement with experimental results.15 The results also show that the electron injection dynamics is accompanied by significant electronic coherence effects and thus cannot be characterized by a single rate constant. A detailed analysis shows that the ET process in the alizarin-TiO2 can be described by a two-step mechanism, which involves an intermediate state localized at the dye-semiconductor interface. The strong coupling between the donor and intermediate states results in coherent electronic motion, which is damped due to the interaction with the substrate. We have also performed an analysis of the electronicvibrational coupling in the ET process. The quantum dynamical simulations show that the coupling to the vibrational modes of the chromophore results in a somewhat slower injection dynamics. However, due to the ultrafast time scale of the ET process and the relatively small reorganization energy, the overall effect of electronic-vibrational coupling in the alizarin-TiO2 system is rather small. In the present application, we have assumed that the photoexcitation by an ultrashort laser pulse can be described by an instantaneous transition from the electronic ground state to the donor state. As shown previously,80 the dynamical methodology

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also allows one to include the laser pulse explicitly in the simulation of the ET process. The study of the influence of the laser pulse on the ET reactions, as well as effects due to Dushinsky rotation of the normal modes and anharmonicities of the potential energy surfaces, will be the subjects of future work. Acknowledgment. We thank Josef Wachtveitl and Oscar Rubio Pons for helpful discussions. The generous allocation of computing time by the Leibniz Rechenzentrum (LRZ), Munich, is gratefully acknowledged. This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) through the DFGCluster of Excellence Munich-Centre for Advanced Photonics and a research grant (M.T.), the BMBF (M.T.), the Fonds der Chemischen Industrie (M.T.), and the National Science Foundation (NSF) CAREER award CHE-0348956 (H.W.). References and Notes (1) Moser, J.-E.; Gra¨tzel, M. Chem. Phys. 1993, 176, 493. (2) Hagfeldt, A.; Gra¨tzel, M. Chem. ReV. 1995, 95, 49. (3) Rehm, J. M.; McLendon, G. L.; Nagasawa, Y.; Yoshihara, K.; Moser, J.-E.; Gra¨tzel, M. J. Phys. Chem. 1996, 100, 9577. (4) Burfeindt, B.; Hannappel, T.; Storck, W.; Willig, F. J. Phys. Chem. 1996, 100, 16463. (5) Martini, I. B.; Hodak, J.; Hartland, G. V.; Kamat, P. V. J. Chem. Phys. 1997, 107, 8064. (6) Ghosh, H. N.; Asbury, J. B.; Weng, Y.; Lian, T. J. Phys. Chem. B 1998, 102, 10208. (7) Wachtveitl, J.; Huber, R.; Spo¨rlein, S.; Moser, J.-E.; Gra¨tzel, M. Int. J. Photoenergy 1999, 1, 153. (8) Huber, R.; Spo¨rlein, S.; Moser, J.-E.; Gra¨tzel, M.; Wachtveitl, J. J. Phys. Chem. B 2000, 104, 8995. (9) Willig, F.; Zimmermann, C.; Ramakrishna, S.; Storck, W. Electrochim. Acta 2000, 45, 4565. (10) Asbury, J.; Hao, E.; Wang, Y.; Ghosh, H. N.; Lian, T. J. Phys. Chem. B 2001, 105, 4545. (11) Zimmermann, C.; Willig, F.; Ramakrishna, S.; Burfeindt, B.; Pettinger, B.; Eichberger, R.; Storck, W. J. Phys. Chem. B 2001, 105, 9245. (12) Ramakrishna, G.; Ghosh, H. N. J. Phys. Chem. A 2002, 106, 2545. (13) Kallioinen, J.; Benko¨, G.; Sundstro¨m, V.; Korppi-Tommola, J. E. I.; Yartsev, A. P. J. Phys. Chem. B 2002, 106, 4396. (14) Walters, K. A.; Gaal, D. A.; Hupp, J. T. J. Phys. Chem. B 2002, 106, 5139. (15) Huber, R.; Moser, J.-E.; Gra¨tzel, M.; Wachtveitl, J. J. Phys. Chem. B 2002, 106, 6494. (16) Schnadt, J.; Bru¨hwiler, P. A.; Patthey, L.; O’Shea, J. N.; So¨dergren, S.; Odelius, M.; Ahuja, R.; Karis, O.; Ba¨ssler, M.; Persson, P.; Siegbahn, H.; Lunell, S.; Mårtensson, N. Nature 2002, 418, 620. (17) Huber, R.; Moser, J.-E.; Gra¨tzel, M.; Wachtveitl, J. Chem. Phys. 2002, 285, 39. (18) Takeshita, K.; Sasaki, Y.; Kobashi, M.; Tanaka, Y.; Maeda, S.; Yamakata, A.; Ishibashi, T.; Onishi, H. J. Phys. Chem. B 2003, 107, 4156. (19) Furube, A.; Katoh, R.; Hara, K.; Murata, S.; Arakawa, H.; Tachiya, M. J. Phys. Chem. B 2003, 107, 4162. (20) Furube, A.; Katoh, R.; Yoshihara, T.; Hara, K.; Murata, S.; Arakawa, H.; Tachiya, M. J. Phys. Chem. B 2004, 108, 12583. (21) Anderson, N. A.; Lian, T. Annu. ReV. Phys. Chem. 2005, 56, 491. (22) Matylitsky, V. V.; Lenz, M. O.; Wachtveitl, J. J. Phys. Chem. B 2006, 110, 8372. (23) Ernstorfer, R.; Gundlach, L.; Felber, S.; Storck, W.; Eichberger, R.; Willig, F. J. Phys. Chem. B 2006, 110, 25383. (24) Dworak, L.; Matylitsky, V. V.; Wachtveitl, J. ChemPhysChem 2009, 10, 384. (25) Stockwell, D.; Yang, Y.; Huang, J.; Anfuso, C.; Huang, Z.; Lian, T. J. Phys. Chem. C 2010, 114, 6560. (26) Neˇmec, H.; Rochford, J.; Taratula, O.; Galoppini, E.; Kuzˇel, P.; Polka, T.; Yartsev, A. P.; Sundstro¨m, V. Phys. ReV. Lett. 2010, 104, 197401. (27) Hagfeldt, A.; Gra¨tzel, M. Acc. Chem. Res. 2000, 33, 269. (28) Gra¨tzel, M. Nature 2001, 414, 338. (29) Benko¨, G.; Kallioinen, J.; Korppi-Tommola, J. E. I.; Yartsev, A. P.; Sundstro¨m, V. J. Am. Chem. Soc. 2002, 124, 489. (30) Ramakrishna, S.; Willig, F. J. Phys. Chem. B 2000, 104, 68. (31) Petersson, Å.; Ratner, M. A.; Karlsson, H. O. J. Phys. Chem. B 2000, 104, 8498. (32) Persson, P.; Bergstro¨m, R.; Lunell, S. J. Phys. Chem. B 2000, 104, 10348. (33) Ramakrishna, S.; Willig, F.; May, V. Phys. ReV. B 2000, 62, 16330.

Li et al. (34) Ramakrishna, S.; Willig, F.; May, V. J. Chem. Phys. 2001, 115, 2743. (35) Schreiber, M.; Kondov, I.; Kleinekatho¨fer, U. J. Lumin. 2001, 9495, 471. (36) Ramakrishna, S.; Willig, F.; May, V. Chem. Phys. Lett. 2002, 351, 242. (37) Stier, W. M.; Prezhdo, O. V. J. Phys. Chem. B 2002, 106, 8047. (38) Persson, P.; Lunell, S.; Ojamaë, L. Chem. Phys. Lett. 2002, 364, 469. (39) Stier, W. M.; Prezhdo, O. V. Israel J. Chem. 2002, 42, 213. (40) Persson, P.; Bergstro¨m, R.; Ojamaë, L.; Lunell, S. AdV. Quantum Chem. 2002, 41, 203. (41) Ramakrishna, S.; Willig, F.; May, V.; Knorr, A. J. Phys. Chem. B 2003, 107, 607. (42) Rego, L. G. C.; Batista, V. S. J. Am. Chem. Soc. 2003, 125, 7989. (43) Stier, W. M.; Duncan, W. R.; Prezhdo, O. V. AdV. Mater. 2004, 16, 240. (44) Thoss, M.; Kondov, I.; Wang, H. Chem. Phys. 2004, 304, 169. (45) Wang, L.; May, V. J. Chem. Phys. 2004, 121, 8039. (46) Duncan, W. R.; Prezhdo, O. V. J. Phys. Chem. B 2005, 109, 365. (47) Rego, L. G. C.; Abuabara, S. G.; Batista, V. S. J. Chem. Phys. 2005, 122, 154709. (48) Wang, L.; Ernstorfer, R.; Willig, F.; May, V. J. Phys. Chem. B 2005, 109, 9589. (49) Duncan, W. R.; Stier, W. M.; Prezhdo, O. V. J. Am. Chem. Soc. 2005, 127, 7941. (50) Persson, P.; Lundqvist, M. J. J. Phys. Chem. B 2005, 109, 11918. (51) Duncan, W. R.; Prezhdo, O. V. J. Phys. Chem. B 2005, 109, 17998. (52) Nilsing, M.; Persson, P.; Ojamaë, L. Chem. Phys. Lett. 2005, 415, 375. (53) Abuabara, S. G.; Rego, L. G. C.; Batista, V. S. J. Am. Chem. Soc. 2005, 127, 18234. (54) Wang, L.; Willig, F.; May, V. J. Chem. Phys. 2006, 124, 014712. (55) Kondov, I.; Thoss, M.; Wang, H. J. Phys. Chem. A 2006, 110, 1364. (56) Sebastian, K. L.; Tachiya, M. J. Chem. Phys. 2006, 124, 064713. (57) Persson, P.; Lundqvist, M. J.; Ernstorfer, R.; Goddard, W. A., III; Willig, F. J. Chem. Theor. Comput. 2006, 2, 441. (58) Liang, K. K.; Lin, C.-K.; Chang, H.-C.; Hayashi, M.; Lin, S. H. J. Chem. Phys. 2006, 125, 154706. (59) Kondov, I.; Wang, H.; Thoss, M. Int. J. Quantum Chem. 2006, 106, 1291. (60) Lundqvist, M. J.; Nilsing, M.; Persson, P.; Lunell, S. Int. J. Quantum Chem. 2006, 106, 3214. (61) Duncan, W. R.; Prezhdo, O. V. Annu. ReV. Phys. Chem. 2007, 58, 143. (62) Kondov, I.; Cˇ´ızˇek, M.; Benesch, C.; Wang, H.; Thoss, M. J. Phys. Chem. C 2007, 111, 11970. (63) Wang, H.; Thoss, M. J. Chem. Phys. 2003, 119, 1289. (64) Li, J.; Nilsing, M.; Kondov, I.; Wang, H.; Persson, P.; Lunell, S.; Thoss, M. J. Phys. Chem. C 2008, 112, 12326. (65) Domcke, W. Phys. Rep. 1991, 208, 97. (66) Kurnikov, I. V.; Beratan, D. N. J. Chem. Phys. 1996, 105, 9561. (67) Galperin, M.; Toledo, S.; Nitzan, A. J. Chem. Phys. 2002, 117, 10817. (68) Lo¨wdin, P.-O. J. Chem. Phys. 1950, 18, 365. (69) Mayer, I. Int. J. Quantum Chem. 2002, 90, 63. (70) Xue, Y.; Ratner, M. A. Phys. ReV. B 2003, 68, 115406. (71) This corresponds to the Condon approximation, which is often employed in the ET systems and is expected to be valid for the relatively rigid systems considered in this paper. (72) Dushinsky, F. Acta Physicochim. URSS 1937, 7, 551. (73) Myers, A. B. Chem. ReV. 1996, 96, 911. (74) Conical Intersections: Electronic Structure, Dynamics and Spectroscopy; Domcke, W., Yarkony, D. R., Ko¨ppel, H., Eds. World Scientific: Singapore, 2004. (75) Marcus and, R. A.; Sutin, N. Biochim. Biophys. Acta 1985, 811, 265. (76) Meyer, H.-D.; Manthe, U.; Cederbaum, L. S. Chem. Phys. Lett. 1990, 165, 73. (77) Manthe, U.; Meyer, H.-D.; Cederbaum, L. S. J. Chem. Phys. 1992, 97, 3199. (78) Beck, M. H.; Ja¨ckle, A.; Worth, G. A.; Meyer, H.-D. Phys. Rep. 2000, 324, 1. (79) Meyer, H.-D.; Worth, G. A. Theor. Chem. Acc. 2003, 109, 251. (80) Wang, H.; Thoss, M. J. Chem. Phys. 2006, 124, 034114. (81) Thoss, M.; Wang, H. Chem. Phys. 2006, 322, 210. (82) Wang, H.; Thoss, M. J. Phys. Chem. A 2003, 107, 2126. (83) Thoss, M.; Domcke, W.; Wang, H. Chem. Phys. 2004, 296, 217. (84) Wang, H.; Thoss, M. Chem. Phys. Lett. 2004, 389, 43. (85) Burdett, J. K.; Hughbanks, T.; Miller, G. J.; Richardson, J. W., Jr.; Smith, J. V. J. Am. Chem. Soc. 1987, 109, 3639. (86) Bredow, T.; Jug, K. J. Phys. Chem. 1995, 99, 285.

Photoinduced Electron-Transfer Processes Alizarin-TiO2 (87) Redfern, P. C.; Zapol, P.; Curtiss, L. A.; Rajh, T.; Thurnauer, M. C. J. Phys. Chem. B 2003, 107, 11419. (88) Ahlrichs, R.; Ba¨r, M.; Ha¨ser, M.; Horn, H.; Ko¨lmel, C. Chem. Phys. Lett. 1989, 162, 165. (89) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich,

J. Phys. Chem. C, Vol. 114, No. 43, 2010 18493 S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03, revision E.01; Gaussian, Inc.: Wallingford, CT, 2004. (90) Nilsing, M.; Lunell, S.; Persson, P.; Ojamaë, L. Surf. Sci. 2005, 582, 49. (91) Kormann, C.; Bahnemann, D. W.; Hoffmann, M. R. J. Phys. Chem. 1988, 92, 5196.

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