First-Principle Calculations of Solvated Electrons at Protic Solvent

Apr 2, 2009 - When the solvent is a protic solvent, the interaction is mainly the hydrogen bond (HB). In the present study, we focus on the wetting, i...
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J. Phys. Chem. C 2009, 113, 7236–7245

First-Principle Calculations of Solvated Electrons at Protic Solvent-TiO2 Interfaces with Oxygen Vacancies Takanori Koitaya,† Hisao Nakamura,* and Koichi Yamashita Department of Chemical System Engineering, Graduate School of Engineering, the UniVersity of Tokyo, Tokyo 113-8656, Japan ReceiVed: October 30, 2008; ReVised Manuscript ReceiVed: February 24, 2009

Heterogeneous electron transfer dynamics at molecule-metal or molecule-metal oxide interfaces are a central issue for many fields in surface science. Recent time-resolved two-photon photoemission studies observed electron solvation at protic molecule-metal or -metal oxide interfaces in photoinduced charge transfer processes. Although the solvated electron is expected as a functional state (for instance, highly catalytic active site, promoter of charge transfer at devices), details are not yet well understood because of the strong dependence of specific atomistic-level solvent-substrate interactions on interfaces, e.g., ordered hydrogen bonds or defects at surfaces. We focus on the interfaces of H2O/TiO2(110) and CH3OH/TiO2(110) and have applied ab initio DFT calculations for several structures with changing coverage and adsorption types and after inserting oxygen vacancies. The electronic structures of wet-electron states and hydrogen bonds at interfaces have been analyzed using the calculated results. I. Introduction Photochemistry on metal and semiconductor surfaces is one of the most active research fields in surface science, solid catalytic chemistry, and electrochemistry.1-7 The keys to photochemical processes are the properties of excited electronic states and their dynamic behavior. In isolated molecular systems and related homogeneous systems, the focused excited-state can easily be identified, at least conceptually, as one of the Born-Oppenheimer states associated with well-defined potential energy surfaces. Using a few (often only two) coupled potential energy surfaces, the electron transfer (ET) dynamics of an excited electron can be described.8-10 On the other hand, the electron excitation and transfer at surfaces (interfaces) are much more complicated because of the existence of the bulk substrate connected to the surface region. Although the initial excitation processes on surfaces can be classified into direct and indirect excitations,11-13 both types can consist of an electronic localization in the interface or adsorbed layers, with states that trigger photochemical reactions through energy conversion from electrons to nuclear motions. Furthermore, the de-excitation dynamics of the localized electron dominate the lifetime (resonance time) of the reactive state; hence, they control the reactivity of photochemical species and the efficiency of energy conversion from/to electron current.14 Because the substrate works as an electron reservoir, a main de-excitation channel is the ET from the interface to the bulk conduction band, where electronic states distribute continuously. Now, the ET is heterogeneous, and it is not straightforward to extend homogeneous ET theories, which are usually based on the picture of identical electronic states or a set of a few potential energy surfaces. Therefore, an alternative point of view, i.e., one-particle (electron) kinetics or one-particle dynamics, is more convenient to describe the ET than the use of Born-Oppenheimer states or potential energy surfaces, which require total electronic wave functions.12-17 * Corresponding author. E-mail: [email protected]. † Present address: The Institute for Solid State Physics, The University of Tokyo, Chiba 277-8581, Japan.

Recent developments in spectroscopic techniques such as twophoton photoemission (2PPE) and time-resolved 2PPE (TR2PPE) provide useful insights into the photoexcitation mechanism, the properties of the resonant intermediate state, and the trace of its dynamics; thus, the ET at the interface can be directly observed.18,19 Various inorganic systems covered by organic molecules or molecular films have been studied by 2PPE and TR-2PPE.20-31 These experimental studies show clear evidence of the formation of small polarons or electron solvation at the interface, which are caused by strong couplings between localized electron and nuclear degrees of freedom. Although the terms “small polaron” and “solvated electron” are usually treated as the same theoretical concept,32 we distinguish between the two terminologies here. The term small polaron is used to represent a quasiparticle formed by coupling between an electron and the internal vibrational modes of molecules in low-coverage molecular adsorbed systems or deformation of lattice of molecular crystals. On the other hand, we reserve the term solvated electron to represent a quasiparticle created in two steps: (i) formation of a “wet-electron” state by trapping on a localized site in the interface and (ii) further stabilization by the reorganization of surrounding solvent molecules. Using this terminology, the stabilization of the solvated electron will be decomposed into the two contributions “wetting” and “solvent response”.29,31,32 The wetting relates to trapping of an electron by direct chemical bonding interactions with the solvent. When the solvent is a protic solvent, the interaction is mainly the hydrogen bond (HB). In the present study, we focus on the wetting, i.e., the wetelectron state in the entire solvation, and simply refer to it as the solvated electron. Because the reorganization of the surrounding solvent is the step after wetting, our approach can be regarded as a first-order approximation for (photoinduced) solvation. Note that a solvated electron is localized on the surface but can be delocalized over the two-dimensional region parallel to the surface. Solvated electrons for H2O/Cu and H2O/Ru systems have been studied extensively by Wolf’s group.22,27,33 In water-metal

10.1021/jp809596q CCC: $40.75  2009 American Chemical Society Published on Web 04/02/2009

DFT Calculations on Protic Solvent-TiO Interfaces systems, solvation is governed by the degree of amorphous or crystalline ice thin film. They found that HB and its broken site, which is called a “dangling H atom”, play an important role in the formation of a wet-electron state as well as in the solvent response by dipolar interactions. Similar solvation can be considered for water-metal oxide interfaces, and these will be more significant than metal interfaces for technology and practical applications such as photocatalysis and hydrophilicity induced by light.23,25 Onda et al. performed TR-2PPE measurements for H2O molecules on a TiO2 surface and observed a solvated electron.25 They found a wet-electron state at 2.4 ( 0.1 eV above the Fermi level only when both OH and H2O were chemisorbed on the surface. In addition, the TR-2PPE experimental results for CH3OH/TiO2 by Li et al. showed the wet-electron state at energy levels similar those for H2O, but a solvated electron in CH3OH/TiO2 has a much longer lifetime than in H2O/TiO2, and the proton-coupled electron transfer (PCET) mechanism is suggested as the de-excitation processes.23 Water is always concerned in any surface phenomena because of its abundant existence in air, and methanol works as a sacrificial species when water is photodecomposed at a TiO2 surface; thus, the analysis of solvated electrons and HB (dangling H) for these two solvent molecules is very important to elucidate the efficient reactivity of TiO2 photocatalysis.34,35 Compared with metal surfaces, the electronic structure of the protic solvent-TiO2 interface is not as well understood. Furthermore, the existence of oxygen vacancies could change the chemisorption, e.g., from molecular adsorption to dissociative adsorption.36 We try to clarify the existence of solvated electrons on both H2O/TiO2 and CH3OH/TiO2 interfaces and analyze their solvent-dependent properties by ab initio calculations. In particular, we focus on the distribution and energy of the solvated electrons and their qualitative correlation with the HB strength at the interface. For each case, the concrete system (e.g., vacancies and coverage) is modeled systematically to perform first-principle calculations by using density functional theory (DFT), just as in the theoretical study by Zhao et al.37 The organization of this paper is as follows. In section II, the results of DFT calculations are presented for TiO2, H2O/ TiO2, and CH3OH/TiO2 surface systems. Details of the electronic structure and comparisons with previous work are also shown. In section III, we analyze a wet-electron state and HB by introducing a simple model and presenting the resulting DFT data. Relations between the location of an oxygen vacancy, coverage, and an acceptor site for solvated electrons will be discussed. Furthermore, we discuss the correlation between solvated electrons and HB, which could be a measurement of preferred acceptors of the solvation in the interface region. section IV presents a summary and conclusions. II. DFT Calculations of TiO2 Surfaces Covered by Protonic Solvent Molecules II-1. Setup. We adopted a slab model for all systems, TiO2, H2O/TiO2, and CH3OH/TiO2, for both stoichiometric surfaces and surfaces with oxygen vacancies, which we called defective models, and we calculated their electronic properties and optimized structures by using periodic DFT. The TiO2 surface is taken as the (110) crystallographic face, which is the major face in the rutile structure. From now on, we omit the notation (110) because all of our calculations are for the (110) surface structure. In Figure 1, clean and defective models of the TiO2 surface are presented. For the DFT calculations, we adopted the PBE as the XC functional. The basis set employed was the pseudoatomic orbital (PAO) at the double-ζ with polarization

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Figure 1. The structure of the TiO2(110) rutile surface. Large (gray) and small (red) spheres represent Ti and O atoms, respectively. An oxygen vacancy for a defective surface can be obtained by removing the bridging oxygen atoms labeled as Ob. The ligand titanium is labeled as Ti5c4+.

(DZP) level. For core electrons, the Troullier-Martins normconserving pseudopotential with Kleinman-Bylander nonlocal projector was used. All calculations were performed with the SIESTA program package.38 To check the validity of DZP basis, in particular, for analysis of a wet electron state in H2O/TiO2, we prepared another basis termed as DZP++, which is DZP but optimized for H and O to represent an anion state by changing charge during the optimization procedure for PAOs. The resulting DZP++ basis is more diffuse PAO than the standard PAO of DZP. Most calculations and analysis were performed by DZP, and we will refer to the adopted basis set explicitly only when the calculation is compared with the results by DZP++. To check the validity of our setup, we first calculated the bulk rutile TiO2 with 7 × 7 × 10 k-points. The resulting set of optimized lattice parameters is a ) 4.73 Å, c ) 3.07 Å, and u ) 0.30, where u is an internal parameter defined in ref 39. These values are in good agreement with experimental values, 4.59 Å, 2.95 Å, and 0.31, respectively.39,40 The calculated parameters were used for all surface calculations in this work. The TiO2 surface was represented by a slab containing six layers plus a vacuum gap of 10 Å, with the three-dimensional periodic boundary conditions. We adopt the terminology “surface unit cell” as 2a × c (along the [11j0] and [001] directions, respectively), which is the minimum cell to construct clean TiO2. In the present study, we adopted a unit cell of the slab model, which is called a “surface cell” as follows. For the clean TiO2, H2O/TiO2, and stoichiometric CH3OH/TiO2 surfaces, a 2a × 2c structure, i.e., 1 × 2 surface unit cells, was employed as the surface cell. To represent a defective surface containing 25% bridging oxygen (Ob) vacancies and reduced CH3OH/TiO2(110) surfaces, a 2a × 4c structure was used as the surface cell, which contains 1 × 4 surface unit cells. The k-points were set to 5 × 6 × 1 and 5 × 3 × 1 for the former and latter surface cells, respectively. In the geometry optimization procedure, only the top two layers were relaxed. The molecules were adsorbed only on the relaxed side of the slab. It is well-known that some physical properties (e.g., the surface stress tensor) and the favored adsorption type (molecular or dissociative: see section II-2) are sensitive to the XC functional adopted in the DFT calculation,41-43 even if the XC functionals belong to the same-level categories, e.g., LDA or GGA. However, at least within the same-level XC functionals, thegeometricoptimizedstructureandthebandgap(HOMO-LUMO gap) are not so sensitive if the adsorption type is fixed. To confirm this, we also performed the same calculations using

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other GGA XC functionals. For instance, the use of revised PBE changed the lattice constants, projected density of states, and energy gap by only 2% (or less). Further, the difference between spin-polarized and unpolarized calculations was negligible. Therefore we performed all calculations and analysis in the present systems using only the PBE functional with the nonspin-polarized DFT. II-2. TiO2 Surface. The surface energy of the frozen stoichiometric surface can be defined by

Esurf,fixed )

Efixed - NTiO2ETiO2 2S

(1)

where Efixed is the total energy of the unit cell, in which each atom is fixed at the bulk position. NTiO2 is the number of TiO2 moieties in the surface cell, and ETiO2 is the energy of a TiO2 moiety in the bulk. The term S represents the surface area of the adopted slab. We found that the resulting frozen surface energy, Esurf,fixed, was 1.12 J/m2. For the relaxed surface, the surface energy can be evaluated by

Esurf )

Eopt - NTiO2ETiO2 - Esurf,fixedS S

(2)

where Eopt is the total energy of the geometrically optimized slab. The result was Esurf ) 0.767 J/m2. The obtained relaxed surface energy agrees well with the energies reported in previous studies (0.73 J/m2 in ref 44 and 0.81 J/m2 in ref 45). There are four Ob atoms in the surface cell (i.e., 1 × 4 surface unit cells). A reduced TiO2 surface can be modeled by removing one Ob; hence, it contains oxygen vacancies at 25% concentration. The energy required for the formation of an oxygen vacancy can be expressed as follows

Evac ) Ereduce + 1/2EO2 - Eclean

(3)

where Ereduce is the total energy of the reduced surface. Eclean represents the totals energy of the (clean) stoichiometric surface. EO2 is the energy of an isolated gas-phase oxygen molecule. The calculated Evac was 3.08 eV, which agrees well with the result of Oviedo et al. (3.07 eV).46 II-3. H2O/TiO2. Adsorption of H2O onto the TiO2 surface was calculated for the 0.5 and 1.0 ML coverage, which have one and two H2O molecules per one surface cell, respectively. At a coverage of 0.5 ML, the molecular state (MS) and the dissociated state (DS), which represent molecular and dissociative adsorption, respectively, were considered for the adsorbates. For the 1.0 ML case, three conformations were calculated; the dissociation rates were 0% (2MS), 50% (MS/DS), and 100% (2DS). Recall that these five systems represent the stoichiometric H2O/TiO2 surface. Several STM results show that water adsorbs dissociatively at the Ob vacancy, and a pair of hydroxyl groups (ObH) is created by the dissociative adsorption of an H2O molecule.45,47,48 In other words, the dissociative adsorption of H2O on the vacancy site can split into two single ObH species by borrowing a remaining Ob atom from TiO2. Thus, a basic structure of H2O/TiO2 with 25% Ob vacancy concentration can be represented by replacing one of the two Obs in a clean TiO2 cell with an ObH species. As a result, the size of 1 × 2 is sufficient for the surface cell to construct the defective models for H2O/TiO2 created by the 25%

vacancy concentration of TiO2. Two conformations of defective H2O/TiO2 surfaces were calculated. One contains only one set of ObH groups (denoted as 0.5 ML ObH), and the other has ObH plus 1.0 ML H2O adsorbed molecularly on Ti5c4+ (denoted as 0.5 ML ObH + 2MS). Calculated H2O/TiO2 systems are illustrated in Figure 2. One H atom of H2O points to the Ob and makes the adsorbate-substrate interaction, which causes dissociative adsorption. This attractive interaction can be treated as a kind of HB, and we also adopt this (extended) definition of HB.31,37 In the stoichiometric surface, two hydroxyl groups are formed by dissociation of a water on the surface. One is a terminal hydroxyl, which is labeled as OHt, and placed on the Ti5c4+ site. The other hydroxyl can be labeled as ObH and placed in the bridging oxygen row. The terminal hydroxyl, OHt, can be formed only on the stoichiometric surface. ObH still participates in the HB with the O atom of dissociated H2O. Another H atom points to the O atom of the neighbor H2O molecule. At the adsorption states of 1.0 ML coverage, the intermolecular O-H distance is around 2 Å (see Figure 2), which is comparable with the radial distribution function of liquid water (1.9 Å).49 Thus, the networks of HBs between water molecules along the (001) direction are formed. On the other hand, the intermolecular O-H distance at 0.5 ML is so long (>5 Å) that each molecule is nearly isolated on the TiO2 surface. Whether or not the H2O molecule can dissociate on a stoichiometric TiO2 surface has been a long-running controversy. Experimental studies suggest that a water molecule adsorbs molecularly on the stoichiometric TiO2 surface.36 However, results of previous theoretical studies are inconsistent with each other. Harris et al. performed DFT calculations of H2O/TiO2 surfaces and concluded that molecular adsorption is more stable than dissociative adsorption.41 In contrast, Lindan et al. showed that the dissociation was favored in all coverages.42 One of the main reasons for this difference in the theoretical results is the strong dependence of the XC functional in DFT;50 Harris et al. used the PW91 functional, but Lindan et al. employed the revised PBE. In addition, the basis set superposition error (BSSE) correction is an important issue in calculating the adsorption energy when the calculation is carried out using atomic orbitals.51 To correct the BSSE, the counterpoise (CP) method, which uses orbital-only ghost atoms to give the basis set the same accuracy, is usually applied.52 For instance, previous calculations on a pentacene on Au(001) surface have showed that adsorption energies corrected by the CP method are in very good agreement with BSSE-free plane wave results.53 Thus, we employed the CP method to deal with the BSSE for our DFT calculations. Adsorption energies of stoichiometric H2O/TiO2 surfaces can be evaluated by using the following equation

Eads ) Eopt + nEadsorbate - EH2O/TiO2

(4)

where Eopt is the total energy of the clean surface and n is the number of H2O molecules on the surface cell. Eadsorbate and EH2O/ TiO2 are the energy of the isolated molecule and the BSSEcorrected energy of the H2O/TiO2 system, respectively. With defective H2O/TiO2 surfaces, Eopt is substituted by Ereduce and EH2O/TiO2, which is obtained as the energy of the 1 × 2 surface cell multiplied by 2 to match the 1 × 4 size of the surface cell for the reduced system. The use of different unit cells brings additive error to adsorption energy; hence, we did not analyze the BSSE correction for the defective models. Calculated adsorption energies are given in Table 1. MS is more stable than DS for all coverage systems, and the results

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Figure 2. The optimized H2O adsorption structures (left panel, a-f) for each surface model, i.e., coverage, adsorption type, and defect models, are changed systematically. The labels MS and DS relate to the adsorption type, which are molecular and dissociative adsorption, respectively. The label ObH is a hydroxyl group on an Ob vacancy (see the text). The right panels of parts a-f are the distribution of the solvated electrons for the structure given in the left panel. The energy EΓ denotes the band energy of the relating state at the Γ point. The distribution of the solvated electrons (wet-electron state) is represented as a gray mesh clouds. Small (gray) spheres and dots (red) represent Ti and O atoms in the substrate. Oxygen atoms of water molecules are illustrated by large balls (light blue), and hydrogen atoms are represented as medium size balls (dark blue). The HBs are illustrated by dotted lines, where the thick and thin dots denote strong and weak HBs, respectively. EHB are onsets in kJ/mol. The HB lengths are also shown in parentheses in angstroms.

TABLE 1: Adsorption Energies (Eads) and Energies of Solvated Electron (Esol) for H2O/TiO2 Systemsa surface structure 0.5 0.5 1.0 1.0 1.0 0.5 0.5

ML ML ML ML ML ML ML

MS DS 2MS MS/DS 2DS ObH + 2MS ObH

Eads (eV/molecule)

Esol (eV)

0.83 0.77 0.95 0.93 0.73 1.16 1.16

4.98 5.25 4.89 5.02 4.86 3.14 4.28

a The six structures of the surfaces, which contain different coverages, adsorption type, and oxygen vacancies, are considered. The last two structures are defective models, and the BSSE correction is not applied. The notation ObH represents the hydroxyl group adsorbed on the Ob vacancy site (see the text).

of the present work show the same tendency as the results of Harris et al.41 The difference in energy between the two 0.5 ML conformations is quite small. Moreover, 2MS and MS/DS at 1.0 ML coverage are almost degenerate. Therefore, there is a possibility that some H2O molecules adsorb dissociatively on the (110) surface without oxygen vacancies in the present conformations. In contrast, the 2DS system is less stable and will be difficult to realize. The adsorption energy of 1.0 ML conformations except 2DS is larger than those of 0.5 ML, because the intermolecular HBs are formed at 1.0 ML. Note

that further detailed calculations and more careful analysis (e.g., systematic improvement of the XC functional, use of postHartree-Fock theories, dependence of coverage rate, including entropy effects) will be required to analyze “molecular vs dissociative adsorption” of H2O for the stoichiometric TiO2 surface. This is not a purpose of the present study, and we do not carry out further analysis for this problem. Furthermore, for later use, we performed the same calculations by using DZP++ and checked the basis dependence of the adsorption energy. We found that the difference of the energies by DZP and DZP++ is only within 1.5%, and it is negligible in the present purpose. We simply adopt all structures of both adsorption types for the analysis of the solvation in the next section. II-4. CH3OH/TiO2. Calculations for the methanol adsorption were carried out with the same procedures, i.e., the 1 × 2 and 1 × 4 sizes were taken as the surface cells for the stoichiometric and defective models, respectively. The adsorption structures of stoichiometric CH3OH/TiO2 can be represented by using the five models just as for H2O/TiO2, as shown in section II-3. We use the same notations, MS, DS, 2MS, MS/DS, and 2DS, for these five models. For the defective model, the first CH3OH adsorbs at the Ob vacancy site and splits into a methoxy group and a hydroxyl group. To represent defective CH3OH/TiO2, one Ob in the 1 × 4 cell is substituted by the methoxy, and one H

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Koitaya et al. of repulsive interactions between the close positions of the methyl groups. Our results show good agreement with the previous work of Bates et al.,55 and we concluded that our DFT calculations and models of the adsorption structure are valid to analyze the wet and solvated electron states for the present systems.

Figure 3. The distribution of the solvated electron (wet-electron state) on the CH3OH/TiO2 models. (A) Top view and (B) side view of MS/ DS structure. Part C shows defective surface models. Largest spheres (yellow) represent carbon atoms. Other symbols for atoms are the same as those in Figure 2 as well as the labels MS, DS, etc. The methoxy groups derived from dissociative adsorption at the oxygen vacancies are illustrated at the corner of the C atom in part C. The distribution of the solvated electrons is represented as dark mesh clouds.

TABLE 2: Adsorption Energies (Eads) and Energies of Solvated Electron (Esol) for CH3OH/TiO2 Systemsa surface structure 0.5 0.5 1.0 1.0 1.0 0.5 0.5

ML ML ML ML ML ML ML

MS DS 2MS MS/DS 2DS methoxy + 2MS methoxy

Eads (eV/molecule)

Esol (eV)

0.87 0.92 0.56 0.64 0.53 1.10 1.67

5.47 5.43 5.17 4.92 4.90 3.18 3.58

a The six structures of the surfaces, which contain different coverages, adsorption types, and oxygen vacancies, are considered. The last two structures are defective models, and the BSSE corrections are not applied. The label “methoxy” represents the adsorption of the methoxy group on the Ob vacancy site (see the text).

atom is added on an adjacent Ob. This model relates to the 25% oxygen vacancy in the CH3OH/TiO2 system. We adopted two models as our examples of defective surface models and performed calculations. In the first, a CH3OH molecule is adsorbed only at an Ob vacancy (0.25 ML methoxy). In the other, methanol molecules adhere to the vacancy site and Ti5c4+ sites except the neighbors to the Ob vacancy (0.25 ML methoxy + 2M/D). The coverage of molecular and/or dissociative adsoption in these defective models are determined by refereeing the recent temperature-programmed desorption (TPD) experiment,54 although the detail structures are different due to the different density of the oxygen vacancies for each study. In Figure 3, the conformation is illustrated. To estimate energetics, the BSSE corrections were applied to stoichiometric CH3OH/TiO2 surfaces. The resulting adsorption energies are listed in Table 2. In contrast to H2O/TiO2, the DS structure is more stable than the MS by 0.08 eV, and the most stable structure is partially dissociated MS/DS at the 1.0 ML coverage. Therefore, the dissociation of CH3OH on TiO2 is more feasible than that of H2O. The adsorption energies of the 1.0 ML coverage are smaller than the energies of 0.5 ML because

III. Analysis of Solvated Electrons III-1. Models of Solvated Electrons and Hydrogen-Bond Strength. In a protic solvent, there are dangling H atoms. They do not participate in the HBs directly, but can be seeds of other weak HBs. These dangling H atoms are electropositive and provide preferred sites for excess electrons.28,31,56 In other words, the weak HB or its break (termination) relate to the existence of dangling H atoms. On the contrary, the existence of strong HBs makes the function of dangling H atoms weak. Therefore, the HBs control solvated electrons in a protic solvent. In heterogeneous systems constructed from a solid and a protic solvent, a similar solvation process can be considered. The key is the relation between the HBs at the interface and the distribution of solvated electrons in the surface region. To carry out a practical analysis for the solvated electron in heterogeneous systems of a protic solvent and a solid surface, defining a set of measurements for a solvated electron and for strength of HBs could be useful. First we adopt the energy of the solvated electron, denoted as Esol, which was introduced by Zhao et al.37 The term Esol represents the excess energy measured from the Fermi level of an electron trapped in the HBs, i.e., the self-energy of the wetelectron state after photoexcitation. When the molecular orbital relaxation by electronic excitation is omitted, the lowest unoccupied Kohn-Sham orbital in the subset of the unoccupied Kohn-Sham orbitals that satisfy the condition that they are localized around surface H atoms is a good first approximation for an initial wet-electron state (i.e., the target state of phototransition). We refer it as φLHKS. Then the value of Esol can be estimated by the following consistent method. First, we search the edge energy, E1, which is defined as follows

∫EE dE DH(E) ) 1 1

F

(5)

where DH(E) is the projected density of states (PDOS) on the surface H atoms and EF is the Fermi energy. Then Esol is expressed as the average energy weighted by the PDOS:37

Esol )

∫EE dE DH(E)E 1

F

(6)

Recall that we do not consider the reorganization of the surrounding solvent (response of solvent) as described in section I; thus, strictly speaking, Esol is the upper limit of the exact self-energy of the solvated electron, which will be obtained by subtracting the orbital-relaxation energy and reorganization energy because of the solvent response. The excitation energies estimated by the orbital energy of φLHKS from the Fermi level and Esol are distinct properties, although the relating PDOS should mainly come from φLHKS. Only when the orbital φLHKS is completely localized on the H atom and the PDOS of φLHKS is a delta function are the two properties equivalent. One also should not confuse Esol with the “solvation energy,” which is the stabilized energy and directly relates to the difference of the free energy caused by all effects of solvent.57 From the

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definition, the difference between Esol and the orbital energy of φLHKS can be treated as the lowest-order approximation of the solvation energy. Next, we introduce the HB energy, EHB, as the measure of the strength of the HB. It is difficult to calculate EHB directly by means of DFT. We have introduced several approximations to evaluate EHB. First, we decomposed EHB using the KitauraMorokuma scheme58,59

EHB ) Ees + Eex + Epl + Ect + mix

(7)

where Ees, Eex, Epl, and Ect represent the electrostatic, exchange repulsion, polarization, and charge transfer, respectively. The “mix” is a higher-order term. Then we assume that the first two terms of eq 7 dominate the value of EHB for the HBs in the surfaces, and the last three terms are not so sensitive to the type of HBs (in bulk, on the surface, etc.) Second, we expressed EHB using only the Ees term by introducing the approximation that the value of |Ees|/|Eex| is nearly constant. This is reasonable when the strength of the HB is moderate.60 If the electrostatic effect can be expressed as site-site interactions between the positive H atom and the negative O atom, EHB can be expressed as follows

EHB ≈ R

| | q˜Hq˜O rH-O

(8)

where q˜ is the effective valence charge on each atomic site, and rH-O is the length of the relating HB. To obtain a value of R, we used the adsorption energies of the MS and 2MS of H2O/ TiO2 systems obtained in section II. The adsorption energies are decomposed into three terms: the intermolecular HB, the molecular-substrate HB, and the interaction between the adsorbate O atom and the Ti5c4+ ion. Assuming the third term, the O-Ti interaction energy, is independent of the coverage, the coefficient R is determined uniquely. The introduced approximations for evaluating EHB are quite rough; thus, the EHB obtained is not a quantitative value. However, only the relative strength between various types of HBs is important for the present purpose, and hence, the use of eq 8 is sufficient to provide qualitative information about EHB for the present model. III-2. Solvated Electrons in H2O/TiO2. Because water molecules adsorbed on TiO2 can form several HBs with the adjacent water or the substrate Ob atom, the H2O/TiO2 system is a good example for analysis for the existence of solvated electrons as well as for checking our theoretical model. The calculated HB energies and distribution of solvated electron are shown in Figure 2. The HBs between the molecule and substrate are relatively strong, and they lead to the localization of solvated electrons over the H atoms pointing to the adjacent water. These values of EHB are comparable to the usual HB strength of liquid water (23.3 kJ/mol).61 Therefore, eq 8 is suitable for the present purpose. In the 2MS conformation, intermolecular HBs are formed, but these energies are very small, whereas the pairs of H2O molecules, which form strong HBs, are created in MS/DS and 2DS. With the model of the defective H2O/TiO2 surface, the ObH, which is the fragment of dissociated H2O and trapped on the Ob vacancy, does not participate in the HBs. This “free ObH” can work as a preferred acceptor site for a solvated electron (see Figure 2). Recall that this ObH is not the terminal OH. The distribution of the solvated electrons in the 0.5 ML ObH + 2MS conformation is in good agreement with the results

Figure 4. The solvated electron density as a function of the HB energy, EHB.

of Zhao et al.37 The solvated electron of 0.5 ML ObH is localized on each hydroxyl group. To clarify the relation between the HBs, the distribution or density of states (DOS) of the solvated electron, which belongs to the orbital of each H atom, is plotted as a function of its HB energy in Figure 4. A clear correlation can be found in Figure 4. A solvated electron populates the spatial region where the HB network is weak. On the other hand, the strong network excludes the distribution of a solvated electron. As a result, one can find that the distribution of a solvated electron is distinctly separated into the two regions (i.e., allowed and forbidden regions) as a function of the HB strength. A quite similar relation with the above distribution and HB strength is found in the solvated electron in homogeneous (protic) solution,62,63 and it indicates that solvated electrons do exist on an interface between a protic solvent and the TiO2 surface. Moreover, the calculated HB energies seem to give a better index to analyze correlation with distribution of a solvated electron than does the HB length as follows. In the MS/DS conformation (see Figure 2d), the highest density of the solvated electrons are found on the H atom, which has the largest HB length (2.81 Å) and smallest EHB (4.4 kJ/mol); i.e., both HB length and strength can give a reasonable correlation. They correctly predict that the H atom participating in the weakest HB has the most distribution. On the contrary, the other two HBs contained in the MD/DS conformations have very similar HB length (1.93 and 1.94 Å). The length of one HB is only 1.0% longer than that of the other. However, the density of the wet electron states relating to the shorter (i.e., tight) HB is almost twice as large as the density corresponding to the other HB. On the other hand, the difference of the two HB energies is about 50%. These are clear examples that the HB strength is a better index to analyze the correlation. Table 1 gives Esol values of H2O/TiO2 surfaces, and Figure 5 shows the PDOS of the adsorbed water and H atoms. Because the values of Esol are close to the Γ point energies of the relating band state (initial wet-electron state) in Figure 2, Esol can be regarded as the well-defined self-energy of the wet-electron state. In Figure 5, there are wide peaks from 1 to 4 eV for the PDOS originating from H2O for dissociative adsorptions (DS, MS/DS, and 2DS). This is reasonable because the bond between the O atom of the part of the dissociated molecule and Ti5c4+, with average lengths of 1.94 Å for the DS conformation, is shorter than the bond for molecular adsorbed H2O (2.39 Å on average). In addition, hybridization occurs between the atomic orbital relating to the O atom in H2O and the conduction band of the TiO2 substrate. The values of Esol correspond to the first peaks of PDOS of H atoms. The first peaks of 0.5 ML are sharp, while the shapes of 1.0 ML are slightly flat, because of the subtle

7242 J. Phys. Chem. C, Vol. 113, No. 17, 2009

Figure 5. The projected density of states for H2O (dotted line) and H atoms (solid line) in the various H2O/TiO2 surface models, denoted as a-g. The Fermi energy is set to 0. The labels MS, DS, and so on are the same as those used in Figure 2.

delocalization of solvated electrons along the [001] direction (see Figure 2). Here we briefly consider the validity of the DZP basis for the present analysis. The DZP basis is adequate to describe the ground-state of TiO2 and H2O/TiO2 systems in the present purpose as discussed in sections II-2 and II-3. However a wet electron state is a kind of (excited) anion state, and the use of a more diffuse basis will be better to describe it. Hence, we carried out the same calculations and analysis with using DZP++ to check the validity of our analysis by DZP. The structure of the PDOS of H atoms relating to wet electron states is very similar to the case of DZP, although the peak position of PDOS shifted to the lower energy value. The shifted values for all of the MS, 2MS, and MS/DS conformations are within 0.6 eV. The orbital energy of φHKS obtained by DZP++ also became about 0.7 eV lower than the one by DZP. The difference of their electron densities caused by the diffuse basis does not give serious effects to the present analysis, because the characters of spatial distribution such as an orientation or anisotropy are almost same. The resulting energy Esol shifted down about 0.7 eV, which is similar with the case of φHKS. Recall that these energy-shifts by changing the basis sets are common for the all of the adopted conformations, and the shifted values for the peak positions of PDOS, energies of φHKS, and Esol are similar (0.6-0.7 eV). Next we checked the correlation between the HB energies and the solvated electron densities by DZP++. It is well-known that Mulliken analysis often provides unphysical effective charge when diffuse basis set is adopted. Therefore, we adopted Lo¨wdin analysis to defineEHB by using eq 8. Note that Mulliken and Lo¨wdin charges agree well for DZP, and the Lo¨wdin charge by DZP++ and DZP are close to each other. Due to the long tail of basis for H, the distribution of the solvated electron by DZP++ is more delocalized, and hence, the differences of solvated electron densities on the two kinds of H atoms, which have weak and strong HB, are reduced. However, the tendency

Koitaya et al. of the correlation between the HB strength and the distribution, i.e., “weak HB-large distribution” and “strong HB-small distribution” can be also found in the result of DZP++. The distribution of a solvated electron is separated by the strength of HBs around 10-15 kcal/mol, as the case of DZP. By the above results from DZP++ calculations, we conclude that our calculations by DZP are valid for our theoretical model and analysis of wet electron states. Although the absolute values of Esol are improved systematically by introducing diffuse basis, it does not affect our analysis as far as the absolute value of the energy is not required. It should be noted that the energy of the solvated electrons strongly depends on whether or not surface Ob vacancies exist. Solvated electrons of stoichiometric H2O/TiO2 surfaces are found to have very high energy levels (∼5 eV for DZP, ∼4.5 eV for DZP++). This result is consistent with the fact that no peak is observed in the 2PPE measurement of stoichiometric H2O/TiO2.25 There is little difference of Esol among the five stoichiometric conformations, suggesting that the extent of delocalization over water chains along [001] is small. On the other hand, the self-energies of solvated electrons on defective H2O/TiO2 are much lower, especially with 0.25 ML ObH + 2MS, which has a calculated Esol equal to 3.14 eV by DZP calculation (2.75 eV for DZP++); this indicates that a free ObH group plays an important role in forming solvated electrons. Although the value for the related initial wet-electron state is slightly higher than the experimental one (2.4 eV), it is acceptable for the present qualitative analysis, because excitation energies obtained by DFT (LDA and GGA level XC functionals) often include errors as large as 1 eV. Two reasons why solvated electrons on defective models can be formed by lower incident photon energy can be considered. First, when surface oxygen vacancies are created (i.e., the TiO2 surface is reduced), the Fermi energy shifts above that for the stoichiometric surface (0.86 eV shifted in the present calculations). Because Esol is determined by integration from the EF, the Esol of the defective models decreases relatively compared with that of stoichiometric surfaces. Physically, this is the effect of the surface property relating to the initial electronic excitation rather than the electronic structure of the wet-electron state. This is also clear from estimates of the (first-order) solvation energy (i.e., the difference between Esol and the related Kohn-Sham orbital energy). The difference between the two solvation energies is smaller than the difference between the two Esol values. Second, in the 0.25 ML ObH + 2MS conformation, the solvated electron is delocalized over three H atoms by free ObH. By comparison of the conformations of 0.25 ML ObH and 0.25 ML ObH + 2MS, the contribution by delocalization is estimated to be about 1 eV. The sum of these two energy shifts, i.e., surface effect (reduced) and delocalization over ObH moieties, is 1.86 eV. This is consistent with the difference in Esol between the 0.25 ML ObH + 2MS conformation and the stoichiometric surfaces, which takes the value from 1.72 to 2.11 eV. III-3. Solvated Electrons in CH3OH/TiO2. A CH3OH molecule has two kinds of H atoms: hydroxyl and methyl H atoms, denoted as Hh and Hm, respectively. The Hh atom is more electropositive than the Hm atom and forms a favorable site for a solvated electron, unless it participates in a strong HB. In the present stoichiometric CH3OH/TiO2 models, all Hh atoms of the methanol adsorbed on Ti5c4+ form HBs with the nearest Ob or the O atom of the dissociated methanol. Thus, there is no obvious acceptor site for solvated electrons on CH3OH/TiO2 surfaces, and the Hm atoms may participate in solvation as in homogeneous liquid methanol.25,64,65 In Figure 3, representative

DFT Calculations on Protic Solvent-TiO Interfaces

Figure 6. The projected density of states for CH3OH (dotted line) and H atoms for CH3OH/TiO2 surface models denoted as a-g. The Fermi energy is set to 0. The notation of the labels MS, DS, and so on are the same as those used in Figure 2.

distributions of solvated electrons on the stoichiometric (MS/ DS) and the defective model (0.25 ML methoxy + 2M/D) are shown. On the stoichiometric surface, solvated electrons are at both sides of the adsorbed CH3OH (see Figure 5B). The electron density is mainly localized around the Hm atoms, although the Hh atoms support delocalization of a solvated electron. This enables them to be delocalized along [001] rows, and the resulting tendency is consistent with the correlation between HB strength and the distribution of solvated electrons described in the previous subsection. In the model of the defective surface, there are free ObH groups on the surface because of the dissociative adsorption at the Ob vacancies. As seen in section II, they are the most preferred sites for solvated electrons. Therefore, the solvated electrons reside mostly on the free ObH, and they are more localized than those on stoichiometric surfaces (see Figure 3C). The PDOS of the CH3OH molecules and surface H atoms are shown in Figure 6. Just as with the H2O/TiO2 surfaces, dissociative adsorption leads to hybridization between the orbital of the O atom and the conduction band of the TiO2. Broad peaks of the PDOS of the CH3OH molecule appear at 1-4 eV above the Fermi energy. In the stoichiometric surfaces, the peaks of the PDOS relating to the H atoms are at high energy levels above the Fermi level (