theoretical studies on anatase and less common tio 2 phases: bulk,...

Theoretical Studies on Anatase and Less Common TiO2 Phases: Bulk,Surfaces, and NanomaterialsFilippo De Angelis,† Cristiana Di Valentin,‡ Simona Fantacci,† Andrea Vittadini,§ and Annabella Selloni*,∥

†Computational Laboratory for Hybrid Organic Photovoltaics (CLHYO), Istituto CNR di Scienze e Tecnologie Molecolari, Via Elcedi Sotto 8, I-06123 Perugia, Italy‡Dipartimento di Scienza dei Materiali, Universita di Milano-Bicocca, I-20125 Milano, Italy§Istituto CNR per l’Energetica e le Interfasi (IENI), c/o Dipartimento di Scienze Chimiche, Universita’ di Padova, I-35131 Padova,Italy∥Department of Chemistry, Princeton University, Princeton, New Jersey 08544, United States

CONTENTS

1. Introduction A2. Anatase Bulk: Electronic Properties C

2.1. Remarks on Theoretical Methods C2.2. Band Structure C2.3. Polaronic States D

2.3.1. Photoinduced Excitons and IntrinsicPolarons D

2.3.2. Dopants Induced Polarons F2.3.3. Donors Induced Polarons (H and Li) F2.3.4. Surface Self-Trapped Polarons G

2.4. Intrinsic Defects States G2.4.1. Oxygen Vacancies G2.4.2. Titanium Interstitials H2.4.3. Defect-Induced Ferromagnetism H2.4.4. Oxygen Interstitials H

3. Anatase Surfaces: Energetics, Structure, andReactivity H3.1. Clean Surfaces H

3.1.1. Surface Energies H3.1.2. Surface Structure I3.1.3. Surface Defects K

3.2. Adsorption of Small Molecules L3.2.1. Water L3.2.2. Molecular Oxygen O3.2.3. Methanol and Formic Acid P3.2.4. Water, Methanol, and Formic Acid at

Step Edges on Anatase (101) P4. TIO2 Sensitization and Applications to Hybrid and

Organic Photovoltaics P4.1. Dye and Coadsorbent Effects on TiO2 P4.2. Insulating Metal Oxide Monolayers Q

4.3. Polymers’ Morphology on TiO2 R4.4. Quantum Dots on TiO2 S

5. Modeling TiO2 Nanoparticles S5.1. Shape, Size, and Phase Stability of TiO2

Nanocrystals U5.2. Molecular Dynamics of TiO2 Nanoparticles W5.3. Electronic Structure of TiO2 Nanocrystals Y

6. Less Common TiO2 Phases AE6.1. Three-Dimensional Systems AE

6.1.1. Point Defects and Doping AF6.2. Surfaces AG6.3. Two-Dimensional Systems: Nanolayers,

Nanosheets, and Films AH6.3.1. Nanosheets AH6.3.2. Adsorption AI6.3.3. Point Defects and Doping AI6.3.4. Supported Films AJ

6.4. One-Dimensional Systems: Nanotubes AJ7. Concluding Remarks ALAuthor Information AM

Corresponding Author AMAuthor Contributions AMNotes AMBiographies AM

Acknowledgments ANReferences AN

1. INTRODUCTION

Driven by growing concerns for environmental and energyissues, interest in semiconductor-based heterogeneous photo-catalysis has increased considerably over the last decades.Photocatalysis allows the use of sunlight for the destruction ofhighly toxic molecules and remediation of pollutants; for theselective, synthetically useful redox transformation of specificorganic compounds; for the production of hydrogen, and theconversion of solar energy to electric power.1−12 Due to itsabundance, nontoxicity, and high stability under a variety ofconditions, the most widely used material in heterogeneousphotocatalysis is titanium dioxide (TiO2).

3,5,6,13,14 Accordingly,

Special Issue: 2014 Titanium Dioxide Nanomaterials

Received: February 7, 2014

Review

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there has been a tremendous amount of studies on widelydiverse aspects of TiO2 (nano)materials, ranging from theirsynthesis and characterization to atomic scale experimental andtheoretical investigations of their fundamental physical andchemical properties.13,15−20 Even restricting to theoretical/computational studies, it would be hard to provide anexhaustive overview of all the available work on TiO2. Forthis reason, in this manuscript we almost completely ignore thework on rutile TiO2, for which recent reviews areavailable,13,14,17,20 and focus mainly on theoretical/computa-tional studies of anatase TiO2, and other phases relevant toTiO2 nanomaterials. As the majority of the available studies,most of the work described in this review is based on densityfunctional theory (DFT) calculations,21,22 which have generallyproven quite successful at both predicting TiO2 properties andexplaining experimental observations on TiO2 systems over thelast 20 years. DFT has also several difficulties22 andlimitations,23 however, and theoretical studies of TiO2 materialsbased on a variety of more advanced techniques are reviewed inthe following. To treat systems of large sizes, e.g., nanoparticlesand interfaces, atomistic simulations based on classicalinteratomic potentials are often performed. These studies arereviewed only partially, focusing on work particularly relevantto photocalysis and photovoltaics.Of the three natural polymorphs of TiO2, rutile, anatase, and

brookite, rutile is the thermodynamically most stable bulkphase, while anatase is very common and stable in nanoma-terials.24,25 Interestingly, for a long time it has been challengingfor first-principles electronic structure calculations to describecorrectly the relative stabilities of the rutile and anatase bulkphases,26−29 and only recently the inclusion of dispersioninteractions, not present in usual DFT calculations, has beenshown to be essential for reproducing the observed greaterstability of rutile with respect to anatase.30 Besides being stablein nanoparticles, the anatase phase shows also the highestphotocatalytic activity,31 making it the most interesting phasefor use in high surface area photocatalytic and photovoltaicdevices.4 A key role in all these applications is played by theelectronic properties. For instance, the positions of theconduction and valence band edges relative to the potentialsof relevant redox couples determine whether a photocatalyticreaction can occur or not; the band gap determines the opticalabsorption, which has an essential role in the performance ofphotocatalytic devices; the states near the valence andconduction band edges have a major influence on the electricalconductivity and chemical reactivity. For these reasons, we startthis review with a survey of the electronic properties of bulkanatase (section 2). We first review some of the issues andrecent progress in the study of the band structure of pureanatase, focusing on the description of the band gap. Anaccurate theoretical description of the band gap is indeedimportant because it would allow quantitative prediction of theenergies of trap states, defect, and impurity levels, as well as theinfluence of doping on optical absorption; all these quantitiesare crucial for the design of TiO2 materials with improvedproperties. Also in section 2, we discuss the character ofelectron and hole states in anatase, i.e. whether charge carriers(originating from photoexcitation, doping or intrinsic reducingdefects) are in delocalized band states or are coupled to thelattice polarization to form more or less localized polaronicstates. The polaronic picture is supported by variousexperimental observations, but there are also significantdifferences between the results of spectroscopic and transport

measurements that are not fully understood. Theoretically, theanswer to the above question depends critically on theelectronic structure method used for the calculations, and thequestion is not completely settled yet. Polaronic states havebeen obtained using “non-standard” DFT methods, but thedegree of localization of the polaronic state depends on manydetails and is difficult to determine.Surfaces have a prominent role in nanomaterials, where a

large fraction of atoms is at surface sites, and are also essentialfor understanding the material’s reactivity. In section 3 we thusreview studies on the structure and reactivity of extendedanatase surfaces, focusing on the (101), (001), and, to a lesserextent, (100) surfaces. These are the crystal faces that are mostfrequently exposed by anatase (nano)crystals, including naturalanatase samples, and are also the surfaces for which the largeramount of theoretical studies is available. Similarly, in reviewingstudies of the surface reactivity, we have chosen to focus ononly a few representative probe molecules, which are importantin photocatalysis and/or photovoltaics, notably water, oxygen,methanol, and formic acid. Of these, water is of particularinterest because of its central role not only in photocatalysis butalso in a variety of other fields, ranging from the synthesis ofTiO2 nanomaterials and biomaterials to geochemistry andenvironmental chemistry.32−34

Surface functionalization/modification is widely used toinduce new material’s properties that can be exploited fortechnological applications. In this review we focus on TiO2surfaces’ modifications relevant to hybrid photovoltaics anddye-sensitized solar cells (DSCs). In hybrid photovoltaicdevices a film of sintered TiO2 nanoparticles behaves aselectron acceptor and transporter, to convey the photoexcitedelectrons from an electron donor. In DSCs the TiO2 surface isfunctionalized (sensitized) by a dye (or in general a lightabsorbing material) that is adsorbed on the TiO2 surface. Manytheoretical investigations on the interaction of dyes with TiO2have focused mainly on the properties of the adsorbate. Insection 4 we review computational studies where the focus israther on the modification of the TiO2 properties (notably theconduction band energies) upon functionalization with dyes,oxide overlayers, polymers, and quantum dots. Reviewed workranges from studies exploring the correlation between the opencircuit voltage Voc, a fundamental parameter of DSC devices,and the dipole moment of adsorbed species or electrolyteadditives, to molecular dynamics studies addressing therelationship between the morphology of polymer-function-alized TiO2 and the improvement of the device efficiency.In many technological applications, TiO2 is present in the

form of nanoparticles/nanocrystals. In section 5, we present anoverview of computational studies of the properties of TiO2nanoparticles in relation to these applications, particularlyphotocatalysis and hybrid/organic solar cells. The shape, sizeand crystal phase markedly influence the stability of TiO2nanoparticles. Studies addressing these issues are reviewed insection 5.1. In particular, there have been several studies on thephase transitions between the anatase and rutile polymorphs,including the effects of aqueous environments with differentpHs. Molecular dynamics simulations based on force fieldpotentials have opened the possibility to model the sintering ofTiO2 nanoparticles; these studies are reviewed in section 5.2.The evolution of modern parallel computers and computationalalgorithms have further expanded the scope of first-principleselectronic structure simulations, allowing an accurate picture ofthe interplay between the structural and electronic factors

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underlying the properties of realistic TiO2 nanocrystals of fewnm size. We discuss these results with emphasis on the role ofelectron/hole traps in photoelectrochemical and photocatalyticprocesses. The role of surface passivation and sintering of TiO2nanocrystals on the electronic properties of model systems isalso discussed.Finally, section 6 presents an overview of less common TiO2

polymorphs, including brookite (the third natural form ofTiO2), TiO2(B) (also observed in nature but only in traceamounts), the so-called “high-pressure TiO2 phases”, and thelepidocrocite-like polymorph, which forms 2D sheets. Insection 6.1, we review studies on the bulk and surfaceproperties of these polymorphs, including defects, doping,and the adsorption of small molecules. Section 6.2 focuses on2D nanomaterials, notably nanosheets and supported films,while section 6.3 discusses 1D nanomaterials, with particularfocus on the mechanism of formation of nanotubes.

2. ANATASE BULK: ELECTRONIC PROPERTIES

2.1. Remarks on Theoretical Methods

As pointed out in the introduction, most of the calculationsdiscussed in this review are based on DFT.21 This approach hasgained a prominent position in the general scenario ofcomputational materials science thanks to its rather highaccuracy at a relatively low cost. In most cases, the local densityapproximation (LDA) or the generalized gradient approxima-tion (GGA)35,36 is used for the exchange correlation functional.Because of these approximations, however, DFT suffers of aresidual electron self-interaction and an improper description ofelectronic correlations,22,37 causing a number of problems, suchas the underestimation of band gap values and the over-delocalization of electrons in defects or impurity states, whichare commonly predicted to be shallower than they should be.Another major issue is the fact that, being a ground-statetheory, DFT is intrinsically not suitable to describe excitedsystems.Viable approaches for the correction of the electronic self-

interaction are DFT+U,38,39 where an on-site Hubbard Urepulsion is added on selected localized orbitals, and hybriddensity functional methods,40,41 which include a fraction ofexact Hartree−Fock exchange in the DFT exchange-correlationfunctional. These approaches generally provide an improveddescription of the one electron spectrum (e.g., the fundamentalgap and photoemission energies), but a limitation is that theydepend on the chosen value of U or the percentagecontribution of exact exchange in the case of hybrid functionals.The U term can be derived from first-principles,39 but it is oftenempirically set to the value that correctly reproduces theexperimental fundamental band gap. Unfortunately, thispractice can have severe consequences on the description ofother properties,42 for example of defects states in the gap, as itwill be discussed later in this review. As for hybrid functionals,recently a new generation of screened hybrid densityfunctionals (also named short-range hybrid functionals) hasbeen developed which neglect the long-range Hartree−Fock-like exchange,43 and better reproduce the band gaps and otherproperties of solid state materials.44−46 However, DFT+Umethods have a much lower computational cost in comparisonto hybrid functionals, and therefore are often preferred to thelatter in many studies of oxide materials.To calculate excitation energies (e.g., the optical spectrum),

the knowledge of the static ground-state density is not

sufficient. There are two main approaches to go beyond DFTin these cases,23 notably: (i) many-body perturbation theory(MBPT) in the GW approximation and the Bethe-Salpeterequation; (ii) time-dependent DFT (TD-DFT). The latterapproach is appealing for being simpler and computationallymore affordable than MBPT, but the typical approximationsthat are used for the exchange-correlation kernel of TD-DFTfail to reproduce the optical absorption spectra of extendedsystems, which are instead well described by solving the Bethe-Salpeter equation of MBPT.47 For solid materials, MBPTmethods are used to describe both the optical spectra and,more frequently, the single particle excitations by the GWapproximation, with increasing applications to large and morerealistic models.48 The situation is different for finite systems,e.g. molecules and clusters, for which TD-DFT methods arewidely used and work quite well in most cases.49,50

2.2. Band Structure

Anatase has a tetragonal lattice (P42/mnm) with four TiO2units per unit cell forming chains of slightly elongated TiO6octahedra. The Ti−O bonding is largely ionic with somecovalent contribution.51 The dielectric properties show asignificant anisotropy with static (optical) dielectric constantsof 22.7 (5.41) and 45.1 (5.82) for polarization parallel andperpendicular to the c axis, respectively.52 The large differencebetween static and optical dielectric constants is already anindication of the importance of lattice relaxation effects in thismaterial.The electronic structure of semiconducting oxides is

challenging to describe even for state-of-the-art computationalmethods. The two most critical quantities to be accuratelyreproduced are the band gap and the band edge positions withrespect to the vacuum reference level. For anatase, theminimum fundamental band gap is indirect, with the bottomof the conduction band (CB) at Γ and the top of the valenceband (VB) close to the X point, along the Δ (or Γ-X) direction.Two relevant gaps should be considered: the fundamental orelectronic band gap and the optical band gap. Experimentally,the fundamental band gap was determined to be 3.2 eV withelectrochemical measurements at room temperature (RT),31

while for the optical band gap a value of ∼3.4 eV was obtainedwith optical absorption measurements at 4 K.53

Theoretically, the band gap is generally determined from thedifference of the lowest unoccupied and highest occupiedKohn−Sham eigenvalues. As mentioned in the previoussection, this is significantly underestimated by DFT-LDA andDFT-GGA. On the contrary, hybrid functionals, with a typical20−25% contribution of exact exchange, overestimate theanatase band gap.54 A reduction of this contribution to 12−15%makes the Kohn−Sham gap match the experimentalfundamental gap of TiO2.

54 For the screened hybrid functionalHSE different values have been reported, ranging from 3.58 to3.89 eV.55,56 With the DFT+U method, very large andunphysical U values (U = 6 eV) for the on-site correction onTi 3d states are required to reproduce the experimental bandgap, whereas the use of the self-consistent linear responsederived U value (U = 3.23 eV) improves only slightly the bandgap given by GGA.56 Interestingly, the situation highlyimproves when an additional on-site correction on the O 2pstates is introduced, and in this case the computed band gapmatches the experimental one.56

For bulk stoichiometric anatase, given the limited size of thesystem, quasiparticle calculations with the GW approximation


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have recently become available (Figure 1a).57−59 BothG0W0

58,59 and self-consistent GW (scGW)57 calculations arereported in the literature, but they all appear to overestimatethe band gap. G0W0 results are dependent on the starting point,while scGW calculations are known to give too large band gaps,especially for polar materials.57 The causes for this error havebeen discussed in the literature.57 Better starting points for theG0W0 approach are provided by calculations based on hybridfunctionals (HSE)60 or the DFT+U.61

UV light absorption causes electrons to be excited in theconduction band with holes left in the valence band. Thus, tocorrectly reproduce the optical band gap associated with thisevent, the inclusion of two-particle interactions, i.e., theinteraction between the photoinduced CB electron and VBhole, is necessary. Bethe-Salpeter equations have been used tosimulate the experimental optical spectra of bulk anatase TiO2in several studies.58,59,62 In particular, so-called BSE@GWcurves match nicely the experimental spectral shape59,63 ofanatase for both xy and z polarizations, radically improving thepoor description obtained with an independent particleapproach (RPA; Figure 1b). It should be noted, however,that even these sophisticated BSE@GW calculations do nottake into account important effects such as electron−phononcoupling (indirect processes) and self-trapping phenomena,whereby exciton creation induces a local ionic relaxation whichtraps the exciton (see section 2.3).As mentioned above, the second critical issue is the correct

alignment of the anatase valence and conduction bands withrespect to the vacuum. Two quantities need to be accuratelycomputed: the workfunction or ionization potential (IP), andthe electron affinity (EA) of the stoichiometric material.Recently the IP of bulk anatase was estimated considering aQM/MM cluster model,64 and employing a hybrid meta-GGAmethod.65 The authors noticed that the difference betweentheir computed value of the IP (8.3 eV) and the commonlyaccepted band gap value for anatase (3.2 eV), corresponds to

an estimated EA of 5.1 eV, in good agreement with theexperimental result of 5.1 eV.66

2.3. Polaronic States

The question whether charge carriers in TiO2 behave as quasi-free electrons and holes or are coupled to the latticepolarization to form polarons has been the subject of long-lasting debates. The electron transport, which is a key aspect ofmost applications of TiO2 (e.g., photocatalysis, new generationphotovoltaics, etc.), depends strongly on the nature of theelectronic states. Charge carriers can originate from photo-excitation, reduction, or doping. We discuss these differentcases separately in the following sections.

2.3.1. Photoinduced Excitons and Intrinsic Polarons.Ultraviolet photoexcitation of anatase TiO2 causes the creationof excitons which can become self-trapped. The existence ofself-trapped excitons in bulk anatase is proven by the largeStoke-shift of the photoluminescence emission peak, centeredat 2.3 eV,67,68 and is also indicated by the Urbach taildependence of the absorption peak edge with temperature.53 Inorder to be able to describe the self-trapping process of excitonsand polarons within density functional model calculations, it isnecessary to introduce some correction to the self-interactionerror. As mentioned in the previous section, this can be done byreplacing part of the semi(local) exchange with the exactHartree−Fock exchange or by introducing a Hubbard U termwhich penalizes the delocalized solutions. For example, thetriplet exciton has been modeled in a periodically repeated bulkanatase supercell (96-atoms) with the hybrid functional B3LYP,first in the singlet (S0) ground state relaxed atomic structureand then, through internal coordinates optimization, in thetriplet (T1) relaxed (self-trapped) atomic structure.69 (As thetriplet state of lowest energy, the triplet exciton can bedetermined by standard DFT minimization techniques with theconstraint S = 1 for the total spin.) The reorganization energyassociated with the exciton self-trapping is estimated to beabout 0.6 eV, while the computed T1-S0 photoluminescence

Figure 1. (a) Electronic band structure of anatase bulk along the high symmetry directions of the first Brillouin zone. Black lines indicate the GGAcalculation, yellow dots indicate the values obtained after G0W0 corrections. (b) Imaginary part of the dielectric constant for anatase, in-planepolarization xy left and out-of-plane polarization z right, calculated by GGA random-phase approximation (RPA@PBE, dashed blue line) using G0W0on top of GGA (RPA@GW, dotted light green) and via Bethe-Salpeter equation (BSE@GW, black solid line). The experimental spectrum soliddotted black is also shown for comparison purposes (Cardona, M.; Harbeke, G. Phys. Rev. 1965, 137, A1467). Insets: BSE spectrum (BSE2, red solidline) calculated by including in the screening calculation the proper G0W0 electronic gap. Reproduced with permission from ref 59. Copyright 2010American Physical Society.


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emission peak is at 2.6 eV. Given the approximations of themodel and the approach used, the agreement with experimentcan be considered excellent (see Figure 2). The triplet excitonis highly localized on two nearest-neighbor Ti6c

3+ and O3c−

ions, although lattice relaxation effects extend to largerdistances.

Electrons and holes that do not undergo radiativerecombination can follow different paths and thus separate.Transient photoconductance measurements indicate that therecombination rate in anatase is much lower than in rutile,70

probably as a consequence of the indirect minimum band gap.Photogenerated charges in TiO2 have been probed also byelectron paramagnetic resonance71,72 and by infrared spectros-copy73,74 and found to form localized polaronic states at low

temperature which can be self-trapped through latticereorganization. The application of polaron theory to transitionmetal oxides, including TiO2, has been discussed by Austin andMott75 in terms of effective mass, mobility and hopping. Theactivation barrier for electron transport via polaron hopping hasbeen estimated within the Marcus-Emin-Holstein-Austin-Motttheory using the DFT+U method and found to be ∼0.3 eValong both the [100] and [201] directions in bulk anatase76

(Figure 3).Electron polarons in anatase are usually identified as Ti3+

species where the electron is largely trapped at one Ti latticesite which is characterized by elongated Ti−O bonds. The sizeof the polaron cannot be easily established and is still underdebate. From a theoretical point of view, the polaronic natureof extra electrons in the perfect anatase lattice has beendescribed using either hybrid functionals69 or DFT+U.76,77 Thesize and degree of localization of the electron polaron can beaffected by the fraction of exact exchange or the value of U.HSE06 calculations with 25% of exact exchange did not findelectron self-trapping at regular Ti lattice sites in anatase,78,79 atodds with the rutile bulk case where Ti3+ species were identifiedwith the same functional.78,80 With the hybrid functionalB3LYP, instead, it is found that a fraction of about 0.76 electronis localized on a single Ti ion and the remaining 0.22 electron isshared by the four next-nearest Ti ions.69 As a result, ∼0.98 ofthe electron charge is confined within a sphere of about 6 Å inradius, even though the polaron lattice reorganization has alonger range. The corresponding electron polaron self-trappingenergy is estimated to be 230 meV with B3LYP,69 suggestingthat electron polarons in bulk anatase are rather weakly bound.This picture has been confirmed and reinforced by recentexperimental studies on anatase samples heavily doped with O-vacancies.81,82 Here the radius of the electron polaron stateswas estimated to be ∼20 Å and the corresponding energy 10 to100 meV below the bottom of the CB (Figure 4).Hole self-trapping has been obtained with different hybrid

functionals and with DFT+U,69,77−79 even though the value of

Figure 2. Schematic representation of the S0-T1 excitation, excitonself-trapping and emission energies for bulk anatase TiO2 as computedwith spin polarized B3LYP calculations on a 96-atoms supercell model.Experimental emission energy from refs 67 and 68.

Figure 3. (a) Schematic diagram of polaron e− transfer. In the initial state A, with structure qA, the electron is localized on the left Ti ion, while in thefinal state B, with structure qB, the electron is localized on the right Ti ion. At the transition state C, with structure qC, a thermal transfer regime, theelectron is shared between the two Ti ions. (b) General features of Marcus-Emin-Holstein-Austin-Mott theory for symmetric polaron transfer. Thepotential-energy surfaces of the initial state A and final state B are shown with equilibrium structures qA and qB. The initial state is Ti

3+−Ti4+ and thefinal state is Ti4+−Ti3+. The coincidence state, or transition state, between the two states is shown as qC. The reorganization energy corresponds tothe energy of the final state B at the geometric configuration qA. The diabatic activation energy is shown as ΔG*. The adiabatic energy curves areshown as dashed lines, with the electronic coupling matrix element VAB given as twice the energy difference between the two adiabatic states.Reproduced with permission from ref 76. Copyright 2007 American Physical Society.


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the self-trapping energy at a fully coordinated bulk anatase Osite depends significantly on the method used, as it varies from0.2 eV with HSE (estimated from the adiabatic transition level)to 0.74 eV with B3LYP (determined as a difference of totalenergies). Hole-associated acceptor states in anatase aregenerally found to be deeper in the gap than electron-associated donor states.55,69 Such traps have been experimen-tally proved to exist by EPR studies with 17O labeling.71 Thecomparison of measured hyperfine coupling constants withcomputed ones is fairly satisfactory.69

2.3.2. Dopants Induced Polarons. The formation ofelectron and hole polarons in anatase TiO2 can be also inducedby substitutional nonisovalent dopants (Figure 5). A number ofexperimental studies have analyzed the electronic properties ofn-type Nb-doped anatase, where niobium is substituted at a Ti

site, see, e.g., refs 83, 84. Transparent Ti1−xNbxO2 films with x≥ 0.01 have been found to exhibit metallic behavior.83 Theexistence of electron polarons in niobium, antimonium andfluorine doped anatase was probed by EPR measurements.85,86

Several computational studies have also been reported. Typicalsubstitutional dopants used to create electron polarons inanatase are Nb, Sb, V, Ta, and F,18,55,87−90 while Al, Sc, Ga, andIn are used to create hole polarons.55,91 Analogously to whatdescribed in the previous section, the character of the impuritystates depends critically on the density functional method usedfor the calculations. If (semi)local functionals are used,87,90 a V,Nb, or Ta impurity gives rise to a broad resonance in theconduction band, and therefore the extra electron coming fromthis impurity is in a delocalized CB state. On the contrary, GGA+U88 and B3LYP18 calculations indicate the formation of alocalized polaron. At odds with these findings, but in line withwhat reported above for intrinsic self-trapped polarons, theHSE functional predicts that the electron from a Nb impurity isin a completely delocalized state,55 a result that can moredirectly explain the metallic behavior of Nb-doped anataseobserved experimentally. Results similar to the HSE ones wereobtained also with another screened exchange (sX) hybridfunctional,92 whereas highly localized Ti3+ polarons were foundfor Nb-doped rutile.55,92

Hole polarons deriving from substitutional doping in bulkanatase have been systematically studied by Deak et al. bymeans of the HSE hybrid functional.55 They considered Al, Ga,In, Sc, and Y substitutional doping. For the Al case, theycomputed the photoluminescence emission peak from the holetrapping state at 1.4 eV, which compares well with theexperimental value of 1.7 eV. Al incorporation into the anatasestructure was also investigated with a semiempirical approach.91

Other computational studies examined Al-doping in the rutilephase.93,94

2.3.3. Donors Induced Polarons (H and Li). Hydrogen95

and lithium96,97 are particularly interesting donor dopants(Figure 5). Hydrogen atoms donate the valence electron andform O−H bonds with the lattice oxygens, as probed by various

Figure 4. (a) An as-grown anatase single crystal. (b) The BZ of anatase. (c),(d) constant energy maps at EF (T = 20 K, hν = 85 eV) of electron-doped anatase (001) in the kxky (c) and kxkz (d) planes, respectively. The blue lines outline the boundaries of the 3D BZs. (e) E vs k dispersion ofthe bottom of the conduction band for a sample with ne ≈ 3:5 1019 cm−3. (f) ARPES intensity measured at k = kF for a sample with ne ≈ 5 × 1018

cm3. The solid line is a Franck−Condon line shape. Voigt peaks of width E = 90 meV (fwhm) are separated by 108 meV, while intensities follow aPoisson distribution. (g−i) Cartoon of the polaron formation induced by the photoemission process, showing the solid in its ground state (g) andtwo possible final states (h and i) of ARPES. Reproduced with permission from ref 82. Copyright 2013 American Physical Society.

Figure 5. Ball and stick representation of pure and stoichiometricanatase TiO2 bulk structure. On the left top side, spin density plotassociated with Ti3+ centers deriving from one lattice oxygen vacancy.On the left bottom side, spin density plot associated with an interstitialTi3+ ion. On the right top side spin density plot of Ti3+ speciesderiving from n-type dopants (e.g., F). On the right bottom side spindensity plot of Ti3+ species from donor dopants (e.g., H). B3LYPcalculations on a 96-atoms supercell. From ref 18. Copyright 2009American Chemical Society.


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Infrared studies for mixed anatase/rutile (Degussa) and purerutile nanoparticles.73,95 In GGA+U and B3LYP calculationstwo types of solutions exist where the electron is eitherdelocalized or almost entirely transferred to the Ti ion boundto the newly formed hydroxyl species,98 while only a totallydelocalized solution is obtained with the HSE functional.55

Lithium atoms intercalate in the lattice interstices and alsodonate the extra electron to form lattice Ti3+ species.96 TheGGA+U picture of the Li-induced states is a strongly localizedpolaron at a Ti3+ site, associated with the appearance of a defectstate about 1 eV below the conduction band.99 As usual, localand semilocal DFT calculations describe the excess electroniccharge as distributed over all the Ti atoms in the supercell,giving a metallic system with partial occupation of the bottomof the conduction band (see references in ref 99). Core leveland valence photoemission measurements on LixTiO2

96

indicate an electronic charge transfer to the Ti 3d level ofabout 0.85 ± 0.10 as well as the formation of a localized defectstate about 1 eV below the conduction band, in very goodagreement with the GGA+U calculations.2.3.4. Surface Self-Trapped Polarons. Electrons and

holes that do not recombine may diffuse from the bulk to thesurface, where they can trigger redox transformations ofadsorbed species.100 The polaron self-trapping energy is largerat the surface, thus making the electron and hole diffusion fromthe bulk to the surface energetically favorable, according toB3LYP calculations on anatase (101) slab models.69 Note thatelectrostatic interactions between electrons (or holes), as wellas surface charging and band bending effects are not included inthese simplified models. Electrons are trapped at surface under-coordinated Ti sites with an energy gain of 0.6 eV, while holesare trapped at bridging (2-fold coordinated) Oxygen surfacesites with an even larger gain of 1.4 eV.69 Holes can be trappedalso by surface hydroxyls. These species are often invoked whendiscussing surface reactivity. Surface hydroxyls are consideredto be excellent hole traps since hydroxyl radicals have beenoften observed through EPR spectroscopy. Recently, it hasbeen possible to prove by means of a hybrid density functionalstudy that hydroxyl groups on the anatase (101) surface aretruly capable of hosting a hole.69 However, it was also shownthat this is only possible when they are not involved inhydrogen bonding with other hydroxyl species on the surfacebecause the interaction with a proton through hydrogenbonding inhibits the hole trapping by the hydroxyl, preferringother surface low coordinated oxygen species (bridgingoxygens), as trapping sites.Subsurface layer ions could also trap electrons or holes, as

found for the rutile (110) surface where the most stable Titrapping site is in the first subsurface layer (0.15 eV better thana Ti5c surface trap), according to a PBE+U study.101 It shouldbe kept in mind, however, that polarons will hop from one siteto the next at finite temperature and will thus visit many surfaceand subsurface sites in a relatively short time scale.76

2.4. Intrinsic Defects States

Intrinsic defects are extremely important since they determinethe electrical conductivity and optical properties of TiO2.Slightly reduced anatase samples, resulting from commonthermal treatments, are electrically conductive. Furtherreduction induces the characteristic blue color observed inmany samples. Understanding the electronic structure mod-ifications induced by these types of defects is thus crucial inorder to be able to tailor and control the material properties.

2.4.1. Oxygen Vacancies. The removal of a neutral oxygenatom from stoichiometric bulk anatase leaves two excesselectrons in the lattice which becomes reduced. This processcan be easily observed after thermal annealing treatments. Forthis reason TiO2 is known as a reducible semiconducting oxide.It is also generally recognized that partial reduction of TiO2 isassociated with the formation of Ti3+ species. For example, it isobserved that, following the creation of O vacancies, a smallpeak in the photoemission spectrum of anatase single crystalsappears at around 1 eV binding energy, which is assigned toTi3+ species.102,103 Very recently it was shown that anatase TiO2can be tuned from an insulator to a polaron gas to a weaklycorrelated metal as a function of the electron doping by UVphoton irradiation creating oxygen vacancies in the bulk of thematerial.82 In particular, the single polaron picture valid at lowdensity of defects is found to break down as charges added tothe conduction band progressively screen the electron−phononinteraction at the basis of the polaron self-trapping.The electronic model of an oxygen vacancy defect in anatase

has evolved significantly during the past decade. Before 2008, itwas usually accepted that oxygen vacancies in anatase do notproduce levels in the band gap but rather give rise todelocalized resonant states at the bottom of the conductionband.104 Following the pioneering work on rutile,105−108 severalstudies showed that if some self-interaction correction isintroduced in the calculations, either in terms of exact exchangein hybrid functionals or in terms of a Hubbard U correction inDFT+U, the oxygen vacancy states become localized on few Tilattice sites with an associated polaronic distortion98,109(Figure5). At the same time, defect levels appear in the band gap atabout 1 eV below the bottom of the conduction band,consistent with experiment.102,103 This general picture of theelectronic structure for oxygen deficient anatase has beenconfirmed by many studies.56,77−80,110 However, depending onthe computational method and setup, the details of theelectronic states associated with the VO defect are different. (i)In DFT+U studies there is a significant dependence of thedefect state energy levels in the band gap on the value of Uapplied on the Ti 3d states. Values of U between 3 and 4 eV areoften considered most appropriate since they give rise to defectstate level positions consistent with experiment (∼1 eV belowthe bottom of the conduction band), even though they do notproperly reproduce the band gap of anatase. In particular, the Uvalue determined by the self-consistent linear responseapproach,39 U ≈ 3.2−3.3 eV for Ti atoms in the anataselattice, fits well in this range of values. On the other hand, largerU values that better reproduce the band gap (U > 5 eV) lead tovery undesirable effects such as defect levels inside the valenceband. Hybrid functional calculations show also a dependence ofthe position of the defect states on the fraction of exactexchange in the functional. For hybrid functionals, however,values which provide a reasonable band gap (20−25% as inB3LYP or HSE), also give reasonable positions of the defectstates in the gap. (ii) The two excess electrons deriving fromthe removal of the oxygen atom may prefer to be spin paired(close-shell solution)78 or spin parallel (open-shell solu-tion).77,98,56,109 (iii) The two excess electrons can be trappedat different Ti ions: either on the two undercoordinated Tiions56,77,78 or on one undercoordinated Ti ion and on aneighboring 6-fold Ti ion,56,98 or even, as found in somestudies, only one electron is trapped at a Ti3+ site, while thesecond remains delocalized in a resonant conduction bandstate.56,98,109 From a more technical point of view, we note that


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it is often important to introduce a small local lattice distortionof the initial atomic configuration in order to converge to alocalized electronic state. Different choices of the initial latticedistortion generally lead to different localized solutions andmany different initial configurations should thus be sampled inorder to identify the most stable localized solution. A thoroughstudy of this type was carried out by Deskins et al.111 for an O-vacancy on the rutile TiO2(110) surface.This variety of possible solutions is an indication that at finite

temperature the electron polarons associated with an oxygendeficiency are likely to hop from one cationic site to the next,with some preference for the ones adjacent to the defect. Thispicture is also supported by an ab initio molecular dynamicsstudy on the rutile polymorph.112 The polaronic nature ofoxygen vacancy states is also crucial for understanding the dualcharacter of these defects: on the one hand their behavior isthat of deep traps, as probed by photoemission measurements,on the other hand they induce an n-type conductivity.56 In fact,the large energy gain associated with the polaronic relaxation orreorganization effectively drops the energy cost of adiabatic orthermodynamic electronic transitions (involved in the electrontransport) with respect to that of vertical electronic transitions(probed by the spectroscopic techniques).56

2.4.2. Titanium Interstitials. Reduced titania can berepresented by two chemical formulas, TiO2‑x and Ti1+xO2.The former corresponds to the situation where there is adeficiency of oxygen atoms (as discussed in the previousparagraphs), while the latter represents the situation wherethere is an excess of titanium atoms. The true nature of reducedanatase and rutile TiO2 has been debated for a long time,113

and it is possible that both type of defects, oxygen vacanciesand Ti interstitials, coexist in nonstoichiometric TiO2 indifferent concentrations, depending on the sample preparationand treatment. Experimental evidence of the existence and roleplayed by Ti interstitials have been reported in the literature,especially for rutile,114,115 but more recently also for anatase.116

The Ti interstitial species, which is highly symmetric in anoctahedral (Oh) coordination in bulk rutile,117,118 is a lowsymmetry species in anatase (C2v).

104,117,119 It is relevant tonote that it is actually such low symmetry and the largedistortion caused by the insertion of a Ti atom in the latticeinterstice that makes this defect a good electron trapping site(Figure 5), even at the GGA level of theory. The introductionof some exact exchange or of a Hubbard U parameter enhancesthe trapping ability, producing lower defect states in the gap.Differently from the case of oxygen vacancies, however, for a Tiinterstitial a state in the band gap is predicted also by PBEcalculations;110,117 a similar result is found also in the case ofrutile.115 Depending on the method used, there will be a largerportion of electron trapped at the interstitial Ti ion; thisfraction is 0.61 in PBE but becomes 0.86 in B3LYP (Tii

3+). Theother three electrons introduced by the addition of one Ti atom(for the overall neutrality) are donated to the lattice and theyare found to be distributed on the next shell of lattice Ti ions,forming other Ti3+ species.There are many analogies between the vertical and adiabatic

transition energy levels of the interstitial Ti species and those ofthe oxygen vacancies in anatase.56 Thermodynamic transitionlevels are much shallower than vertical transition levels, asobserved for oxygen vacancies. The barycenter of the transitionlevels is about the same for the oxygen vacancy transitions (+1/0 and +2/+1) and for the Ti interstitial transitions (from +1/0to +4/+3). Thus, the model of interstitial titanium species for

reduced anatase can also explain spectroscopic features andtransport properties in a way analogous to the oxygen vacancymodel.

2.4.3. Defect-Induced Ferromagnetism. Undoped ana-tase TiO2 thin films have been found to have room-temperatureferromagnetic behavior,120 and experimental evidence exist thatthe magnetic properties are related to the presence of oxygenvacancies.120 Similar results have been obtained also on singlecrystals of rutile which are supposed to be free from anyunwanted impurity. These findings triggered a lot of computa-tional work aimed at rationalizing the observed behavior.Nonmagnetic, ferromagnetic and antiferromagnetic solutionsfor the two extra electrons left by the removal of a neutraloxygen atom have been investigated and compared in a numberof studies, leading to contrasting solutions. According toLSDA,121 GGA,122 and GGA+U (U = 4 eV)123 studies, oxygenvacancies in anatase are not related to the so-called d0

ferromagnetism since they do not induce any appreciablemagnetic moment, even at a rather high concentration (TiO1.75in ref 123). On the contrary, GGA+U calculations with both U= 3.5 and 5.8 eV124 show that the extra electrons, in a 48-atomsanatase bulk supercell model (TiO∼1.95), convert two Ti4+ intotwo Ti3+ ions resulting in a local magnetic moment of about 1.0μB per Ti

3+ ion. The antiferromagnetic solution is found to belower in energy than the ferromagnetic one with both the Uvalues considered.124

In summary, there is no unifying and conclusive outcomefrom the existing studies, so that the origin of the magneticproperties of undoped TiO2 is still an open question.

2.4.4. Oxygen Interstitials. Only few studies of interstitialoxygen species are available. These show that an additionalneutral oxygen atom prefers to bind to a lattice oxygen atomforming an O−O bond,104,125 instead of being stabilized as acharged species in the middle of an interstice. Interestingly,oxygen interstitials are predicted to be good electron traps byGGA+U calculations.125 The extra electron occupies a σ* state,which leads to a significant elongation of the O−O bond from1.484 to 1.970 Å.

3. ANATASE SURFACES: ENERGETICS, STRUCTURE,AND REACTIVITY

Because of difficulties in growing sufficiently large singlecrystals, for many years most of the available experimentalinformation on the surface chemistry of anatase has been basedon studies of disperse samples. Over the past decade, however,the situation has changed significantly with the development ofvarious growth techniques, and currently large anatase samplesin the form of epitaxial thin films126 and bulk material82 areavailable. This has led to a considerable increase in the numberof experimental studies on anatase surfaces, including severalatomic scale surface science studies, which, in turn, havemotivated a large amount of theoretical studies.

3.1. Clean Surfaces

3.1.1. Surface Energies. Surface energies have a centralrole in the thermodynamic, structural, and chemical propertiesof TiO2 surfaces. Besides determining the relative stabilities ofdifferent structures and terminations and thus the crystal shape,surface energies are essential for understanding the phasestability of different TiO2 polymorphs, notably the phasetransformation from anatase to rutile in nanocrystalline TiO2.In addition, surface energies are related to the surface reactivity,as a low surface energy generally corresponds to a low


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reactivity. These reasons, coupled with the fact that surfaceenergies are difficult to measure experimentally, explain whyanatase surface energies have been the subject of numeroustheoretical studies. In the work reviewed in this section, surfaceenergies and corresponding equilibrium crystal shape arecalculated assuming perfectly clean, stoichiometric and defect-free bulk-terminated surfaces. As defects, steps, and adsorbatesare generally present on the surfaces of real materials, thecrystal shape calculated under the above assumptions should inprinciple be very different from the shapes observed inexperiment. Interestingly enough, however, this “ideal” anatasecrystal shape is rather common in colloidal particles as well asin natural anatase samples.The structure and stability of anatase surfaces were first

studied by Oliver et al.,127 who performed atomistic simulationsbased on classical interatomic potentials128 and compared thesurface energies and equilibrium shapes of anatase and rutileTiO2. For anatase, they found that the (101) and (001) surfacesdominate the morphology, as observed experimentally fornatural anatase crystals, with the (001) surface slightly morestable than the (101) one. The latter result does not agree withthe observed predominance of (101) facets in the morphologyof most anatase samples. Based on the computed surfaceenergies of Oliver et al.,127 Zhang and Banfield24 performed athermodynamic analysis of the phase stability of nanocrystallineanatase and rutile. Their study predicts that anatase becomesmore stable than rutile when the particle size is smaller than∼14 nm, consistent with the observation that nanomaterialsgrow preferentially in the anatase phase and transform to rutilewhen the particle size increases.24

First-principles calculations of the energetics of anatasesurfaces have been reported by several groups.26,129−136 Inthese studies, surfaces are usually modeled as slabs of finitethickness with periodic boundary conditions in the surfaceplane, and surface energies are determined from the differencebetween the total energy of the slab and the total energy of anequal number of TiO2 units in the bulk phase, divided by thetotal exposed area. Computed surface energies for a few lowindex bulk-terminated surfaces of anatase are presented inTable 1. By hybrid functional B3LYP calculations on unrelaxedor partially relaxed surface models, Beltran et al130 found thatthe relative energies of low Miller index surfaces follow thesequence (001) < (101) < (100) < (110). As shown by otherDFT studies,26,129,134 however, relaxation has a major effect onthe surface energetics, e.g. it can reduce the surface energy bymore than 50%. For the relaxed surfaces, the relative energiesfollow the sequence (101) < (100) < (001) < (110), which canbe related to the density of undercoordinated Ti surface atomson the different surfaces.26 The same sequence is obtained witha variety of DFT functionals, including LDA, GGA, anddifferent types of hybrid functionals.26,131,132,134−136

Once the surface energies are known, the equilibrium shapeof a macroscopic crystal can be determined via the Wulffconstruction. The computed Wulff shape for anatase is shownin Figure 6. As typically found for natural samples, it consists of

a truncated tetragonal bipyramid exposing majority (101) andminority (001) facets.26,129,131,132 The most stable (101) facetsdominate the crystal surface, e.g. constituting more than 94% ofthe exposed surface according to ref.26,129 Using DFT surfaceenergies for anatase and rutile, thermodynamic studies of theanatase vs rutile phase stability as a function of nanoparticle sizepredicted that at low temperature a phase transition occurs atan average anatase diameter of about 9.3−9.4 nm.132 Theequilibrium shape of anatase is further discussed in section 5.1,where the influence of different adsorbed species is considered.

3.1.2. Surface Structure. 3.1.2.1. Anatase (101). It isknown from LEED experiments137 that clean anatase (101) hasthe same (1 × 1) periodicity of the bulk-terminated surface,consistent with the low formation energy of this surface (seeTable 1). Figure 7 shows the characteristic sawtooth profile of

Table 1. Surface Energies (in J/m2) of Selected Low Index Anatase Surfaces, As Obtained in Different DFT Studies

surface LDA, refs 26 and 129 PBE, refs 26 and 129 PW91, ref 131 PW91, ref 132 PBE, ref 134 PBE0, ref 136 PW1PW, ref 135

(101) 0.84 0.44 0.435 0.35 0.609 0.65 0.64(100) 0.96 0.53 0.533 0.39 0.712 0.79 0.81(001) 1.38 0.90 0.984 0.51 1.082 1.27 1.36(110) 1.09 1.024 0.81 1.34 1.38

Figure 6. Equilibrium shape of a TiO2 crystal in the anatase phase,according to the Wulff construction and the calculated surface energiesof refs 26 and 129. Reprinted with permission from ref 26. Copyright2001 American Physical Society.

Figure 7. Side view of the anatase (101) surface. Relevant structuralparameters are given in Table 2


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anatase (101) viewed along the [010] direction. The surfaceexposes both 5-fold (Ti5c) and 6-fold (Ti6c) coordinated Tiatoms, as well as 2-fold (O2c) and 3-fold (O3c) oxygens. Thesurface atoms undergo significant displacements from the idealbulk-like positions upon relaxation.26,131,132,134 This is partic-ularly evident for the surface O3c and Ti5c atoms, which relaxoutward and inward, respectively, thus causing a small ripple onthe sawtooth profile. Computed bond distances on the relaxedsurface and their deviations from the ideal values are reportedin Table 2. An important feature is that the Ti−O bondsformed by the bridging oxygens with their neighboring Tiatoms are ∼5−10% shorter than average. As a result, therelaxed surface is predicted to be quite rigid.Given that anatase (101) and rutile (110) are the most stable

surfaces of anatase and rutile, respectively, it may be interestingto ask which of the two surfaces has lower surface energy.Lazzeri et al.26,129 found the two surfaces to have nearly thesame energy at the LDA level, whereas rutile (110) has asignificantly (0.13 J/m2) lower surface energy using the PBEfunctional. Labat et al.133 computed the anatase (101) andrutile (110) surface energies using Hartree−Fock, DFT-LDA,DFT-PBE, B3LYP, and PBE0. They found the differences

between the two surfaces to be always small (≤0.1 J/m2) exceptwith B3LYP, which predicted the rutile (110) surface morestable by ∼0.15 J/m2. Using the hybrid PW1PW functional,Esch et al.135 studied the dependence of the computed surfaceenergies on the slab thickness. For well converged calculations,their results finally show that the surface energies of rutile(110) and anatase (101) are essentially identical.

3.1.2.2. Anatase (001). The bulk-terminated TiO2(001)exposes coordinatively unsaturated Ti5c and O2c atoms, as wellas fully coordinated O3c, see Figure 8a. In DFT calculations, thetwo bonds formed by each bridging O2c with its neighboringTi5c become strongly inequivalent upon relaxation,26 with bondlengths of ∼2.2 and 1.8 Å. The surface energy, however, doesnot change significantly and remains large in comparison to the(101) surface (see Table 1), suggesting that an energeticallymore favorable surface structure might exist. In fact,experimental studies on anatase (001) films grown epitaxiallyon SrTiO3 revealed that the surface is reconstructed, with a (1× 4) periodicity with respect to the bulk.138−140 To explain thisstructure, Herman et al.139 suggested a model with (103) and(−103) microfacets that was subsequently shown to beinconsistent with STM data.140 Liang et al. proposed an

Table 2. Bond Distances (Å) on the Relaxed (101) Surface and Their Deviations from the Unrelaxed Value (in Parentheses),from DFT-PBE Calculations26a

O1 O2 O3 O4 O5 O6

Ti1 1.83 (−8.6%) 1.98 (+2.0%) 2.07 (+3.3%) 1.78 (−8.4%)Ti2 1.84 (−5.0%) 2.01 (+0.3%) 1.94 (−0.3%) 2.10 (+5.1%) 2.11 (+8.7%)Ti3 2.04 (+5.2%) 1.98 (−1.2%) 1.94 (0.0%)

aAtom labels are as in Figure 7.

Figure 8. Ball and stick models of (a) unreconstructed anatase (001); (b) ADM model of reconstructed anatase (001)-(1 × 4), showing ridgesrunning along the [100] direction; (c) unreconstructed anatase (100) surface.


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“added-and-missing row” (AMR) model,140 but also in this casethe computed STM image141 turned out to be very differentfrom the experimental one. Based on DFT calculations, Lazzeriand Selloni141 proposed the “ad-molecule” (ADM) model, seeFigure 8b, for which the theoretical STM image agrees nicelywith the experiment.140,142 The ADM model is constructed byperiodically replacing [100] rows of bridging O2c atoms of theunreconstructed 1 × 1 surface with rows of TiO3 species. Thisresults in “ridges” on which the Ti atoms are 4-foldcoordinated. The computed surface energy of the 1 × 4ADM structure is 0.48 Jm−2, nearly half that of theunreconstructed surface, indicating that this reconstructionstabilizes the surface quite effectively. The primary mechanismof this stabilization is the relief of the surface stress.141 On thebulk-terminated surface, the large Ti5c−O2c−Ti5c bond angles(∼156°) and Ti5c−O2c bond lengths (1.96 Å) give rise to asignificant tensile surface stress. In the 1 × 4 ADM structure,the insertion of an extra TiO2 row causes a compression whichmakes the surface bonds shorter, 1.80−1.85 Å, and the bondangles smaller, ∼120°; as a result, the tensile stress is alsostrongly reduced.141

An alternative “ad-molecule” model was suggested byIgnatchenko et al.,143 in which water was added instead ofTiO2 to the original unreconstructed (0 0 1) surface in periodic(1 × n) rows. Different supercells with n original unit cells wereconstructed and their energy was minimized with and withoutwater. The surface energy of the hydrated periodic model of the(0 0 1) anatase surface was found to go through a minimum atn = 3, i.e. for a (1 × 3) periodicity, rather than the (1 × 4)periodicity observed in experiment. Based on these results, itwas suggested that water, and perhaps some other molecules,could play a similar stabilization role for the (0 0 1) anatasesurface as the TiO2 ad-molecule in the ADM model,141

depending on the environment.The ADM model of anatase TiO2(001)-(1 × 4) has been

recently questioned on the basis of microscopic andspectroscopic experiments indicating that this surface, whenoxidized, is essentially inert with respect to water adsorption.144

In the ADM model, instead, water dissociates on the ridges dueto the presence of reactive 4-fold Ti sites, see section 3.2.1.2.To explain their findings, the authors proposed a modifiedADM model where the Ti atoms on the ridges are 6-fold(rather than 4-fold) coordinated due to the presence of oxygenadatoms, and suggested that the reconstructed (001) surface isreactive only when reduced.144 Another modified ADM modelhas been recently proposed by Xia et al.145 on the basis of highresolution STM measurements. This model includes 3-foldcoordinated bridging oxygen atoms beneath some of the addedTiO3 rows of the ADM model.3.1.2.3. Anatase (100). Although not present in the Wulff

shape of anatase, the (100) surface is very frequent in thenanoparticles used for photocatalysis and has been recentlyreported to have a higher photocatalytic activity than both the(101) and (001) surfaces.146,147 The bulk-truncated (100)surface, Figure 8(c), shows flat regions separated by narrowchannels running along the [010] direction. The outermostlayer exposes Ti5c, O2c and O3c atoms, while the second layer atthe bottom of the channels exposes fully coordinated Ti and Oatoms. For this surface, the effects of relaxation are qualitativelysimilar to those found on the (101) surface. In particular, thesurface O3c and Ti5c atoms are predicted to relax outward andinward, respectively, resulting in a small (∼0.35 Å) corrugationof the first layer.26,131,132,148

According to the low surface energy predicted by DFTcalculations (Table 1), the relaxed bulk-truncated (100) surfaceshould be quite stable. On the sputtered and annealed (100)surface, however, LEED and STM experiments have detected a(1 × n) reconstruction characterized by bright ridges runningalong the [010] direction.149,150 This structure has beenqualitatively explained in terms of a (101)-microfacetedmodel,149 but so far no detailed computational study hasbeen reported.

3.1.3. Surface Defects. The most common defects onTiO2 surfaces are oxygen vacancies and steps, with theconcentration of surface oxygen vacancies being very sensitiveto the oxygen pressure, temperature and duration of theannealing used to prepare the surface. Under the typicalpreparation conditions used for surface science experiments, alarge (5−10%) concentration of surface oxygen vacancies ispresent on the most stable (110) surface of rutile. For a longtime it was assumed that the same would hold also for thesurfaces of anatase, but recent experimental and theoreticalwork has shown that this is actually not true. In this section wereview this work and further discuss step edges, which have alsoa strong influence on the surface reactivity.

3.1.3.1. Oxygen Vacancies and Ti Interstitials. Due to theirstrong influence on the reactivity, defects on TiO2 surfaces havebeen extensively investigated for many years.13 In particular, awea l th o f expe r imenta l 1 3 , 1 5 1− 1 5 5 and theore t i -cal156−159,101,105−108,112,113,160,161 studies have focused onsurface oxygen vacancies (VO’s) on rutile (110), for which aconcentration of the order 5−10% is typically observed insurface science experiments. For anatase (101), on the otherhand, the concentration of surface VO’s was much lower undersimilar preparation conditions.13,162,163 To rationalize thisdifference, it was suggested that the formation energy ofsurface O-vacancies may be larger on anatase than on rutilebecause the removal of a surface O2c results in two Ti5c cationson rutile (110), whereas one Ti5c and one 4-fold Ti (Ti4c) areformed on the anatase (101) surface.16 To obtain more detailedinsight, Cheng and Selloni performed DFT-GGA calculationsof the formation energies of O-vacancies at several surface andsubsurface sites of anatase (101), anatase (001)-(1 × 4), andrutile (110).119,164 Their results show a significant differencebetween the anatase and rutile surfaces, that is supported alsoby DFT+U calculations.119 On rutile (110) the formationenergy of surface O2c vacancies is lower than that of subsurfaceVO’s, in agreement with other DFT studies.159,165 On anatasesurfaces, instead, O-vacancies have lower formation energy inthe subsurface and bulk than at the very surface (see Figure9).119,164 The predominance of subsurface defects at the

Figure 9. Various surface and subsurface O vacancies (left) and theircorresponding formation energies (right) at the anatase (101) surface.The ball and stick model on the left shows only part of the six-layer-thick slab used for the calculations. Red and light blue spheresrepresent O and Ti atoms, respectively. Yellow spheres highlight theinvestigated oxygen vacancies sites.


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anatase (101) surface was confirmed by STM measurements ona freshly cleaved anatase TiO2(101) sample, which showed analmost perfect surface with very few subsurface impurities andadsorbates.166 Surface oxygen vacancies were not present in thissample but could be induced by electron bombardment, andfound to migrate to subsurface sites at temperature larger than200 K.167 From time-lapse STM images, the activation energiesfor subsurface migration were estimated to lie in a rather widerange, between 0.6 and 1.2 eV,167 due to inhomogeneouslydistributed subsurface defects in the reduced sample. Incomparison, DFT-GGA calculations of O-vacancy diffusionpredict the barrier for subsurface-to-surface migration to beabout 0.75 eV.119,167

The electronic structure of O-vacancies at anatase surfaces isnot fully understood. A number of DFT(GGA)+U studies withvarious U values have been reported,119,168,169 but the extent ofthe analysis and the quality of the models is still insufficient toallow clear conclusions to be drawn on important aspects suchas, e.g., the extent of the polaronic distortion and the preferredlocalization sites of the excess electrons.In ref 119, the binding sites and diffusion pathways for a Ti

interstitial (Tiint) at the anatase (101) surface were also studied.The most stable surface binding site is between two adjacentO2c atoms along the [010] direction, where the interstitial iscoordinated to four oxygens. In all the subsurface sites, Tiint iscoordinated to five oxygen atoms, as in the bulk, and the defectformation energy decreases steadily as Tiint moves away fromthe surface to the bulk. Altogether the interstitial is predicted tobe ∼1.2−1.3 eV more stable at a bulk site than at the surfaceand the highest barrier in the diffusion pathway from thesurface to a deep subsurface site is less than 0.5 eV.3.1.3.2. Steps. Steps are very common surface defects that

have a strong influence on the reactivity of metal oxides. Adetailed study of the structure and energetics of monatomicsteps on anatase (101) was reported by Gong et al.170 On thissurface steps occur along a few well-defined directions and giverise to characteristic trapezoidal islands (Figure 10). The

parallel sides of all trapezoids are oriented along the [010]direction, and the two nonparallel sides are directed along[−111] and [11−1]. Due to the lack of mirror plane symmetry,the two parallel sides of the trapezoidal islands are non-equivalent and one of the two trapezoidal island orientationsindicated in Figure 10 (right panel) is preferred. Step formationenergies were calculated for the edges of the islands sketched in

Figure 10. The structures of the steps were constructedmaintaining the TiO2 stoichiometry and for some stepsdifferent possible terminations were examined. To determinestep formation energies, vicinal surfaces were considered;details of the procedure are given in ref 170. From the results, itwas inferred that the preferred orientation of the islands is theone indicated by the full line in Figure 10. This assignment wassupported by the agreement between the theoretical andexperimental STM images for a step along the [11−1]direction.

3.2. Adsorption of Small Molecules

3.2.1. Water. Due its relevance to photocatalysis, watersplitting, and other important applications,171 the interaction ofwater with TiO2 surfaces has been the subject of countlessinvestigations over the last decades. Several reviews articles arealso available,3,7,8,13,172,173 including a recent review specificallyfocused on theoretical studies of titania-water interactions.174

Here we shall mainly consider work reported after that review(ca. 2009).

3.2.1.1. Anatase (101). The computed adsorption structureof water on undefected anatase (101) is shown in Figure11.174,175 Water adsorbs in molecular form with the oxygenforming a dative bond with an undercoordinated surface Ti5catom while the hydrogens form H-bonds with two bridging O2catoms. This structure, first predicted theoretically,175 issupported by various experimental observations, includingatomic-scale STM images.176,177

The influence of subsurface defects on water adsorption inthe dilute limit was investigated by Aschauer et al.116 Thecalculations predict a strong preference for water to adsorb inthe vicinity of the subsurface defects. The H2O adsorptionenergy at these sites is higher than on the stoichiometricsurface, in agreement with the experimental finding that thewater desorption temperature increases when the surfacereduction increases.116 Molecular adsorption, strongly favoredon the defect free surface, is preferred also in the presence of asubsurface oxygen vacancy, whereas dissociation becomesslightly favored in proximity of a Ti interstitial in the secondlayer. The barrier for a water molecule to dissociate is muchsmaller on the reduced surface (∼0.25 eV) than on thestoichiometric one (∼0.52 eV).More recently, Zhao et al. considered also the effect of

doping by studying the adsorption and dissociation of a watermolecule on p-type N-doped and n-type V-doped178 anatase(101). They found that water dissociation remains unfavorableon the V-doped surface, whereas it becomes favorable withrespect to molecular adsorption on the N-doped surface.As the character of water adsorption in the dilute limit has

been largely established, recently increasing attention has beendevoted to the properties of adsorbed water layers and theanatase/water interface. Raju et al.179 studied the adsorptionand dissociation of water at 300 K using ReaxFF reactive forcefield simulations. In these simulations, water dissociation wasobserved on the defect-free surface for water coverages of 0.75ML and higher, indicating that the dissociation is assisted bythe surrounding water molecules. The water dissociationpercentage (WDP) was of ∼10% at 1 ML coverage, andshowed only minor variations up to 3 ML water coverage, whilethe terminal WDP (TWDP) increased steadily up to ∼22% at 3ML coverage. These findings, however, do not agree with first-principles molecular dynamics (FPMD) studies (see below)and temperature-programmed desorption (TPD) experi-

Figure 10. Left: STM image of anatase TiO2 (101) showingpreferential orientations of monatomic steps; one isolated trapezoidalisland is highlighted by a red circle. Right: Schematic plot of possibleisland shapes and orientations on anatase TiO2 (101); five differenttypes of steps are identified and labeled as A−E. Adapted from ref 170.Copyright 2006 Nature Publishing Group.


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ments,176 in which no evidence for water dissociation onanatase (101) was obtained.The effect of a subsurface Ti interstitial on the structure and

reactivity of thin water layers on anatase (101) was investigatedby Tilocca and Selloni180 using DFT-GGA and DFT+U basedFPMD simulations at 160 K (the temperature at whichdesorption starts to occur in TPD experiments149). StandardDFT-GGA and the DFT+U method were found to predictsimilar energetic and dissociation barrier for an isolatedadsorbed water molecule, as well as very similar structuralfeatures for an adsorbed water monolayer on the reducedsurface, thus justifying the use of simple DFT-GGA for thesimulation of thicker water layers. Compared to the defect-freesurface, the subsurface defect was found to enhance the surfacereactivity and to lead to a more disordered structure of the firstwater layers adsorbed on the reduced surface. In particular, nowater dissociation was observed in rather long simulations for awater monolayer (ML), bilayer (BL), and trilayer (TL) on thedefect-free surface, whereas dissociated water was present onthe defected surface.FPMD simulations of the interface between anatase (101)

and liquid water have been reported by a few groups. Sumita etal.182 concluded that the structure of the interface ischaracterized by a first layer of water molecules adsorbedmolecularly at Ti5c sites and a second layer forming strong H-bonds with the O2c surface atoms. These two layers form astable bilayer at the interface due to the presence of short H-bonds between first and second layer water molecules (Figure12), as first pointed out and analyzed in detail by Mattioli etal.181 Cheng et al.183 analyzed the number of surface sitescoordinated to water molecules during a rather long (∼30 ps)FPMD simulation at 300 K. They found that on average about75% of the Ti5c sites and ∼85% of the surface O2c atoms werebonded to water molecules during the simulation.183 Theexistence of a stable water bilayer at the interface was recentlydiscussed also by Zhao et al.,184 who used static DFTcalculations to study adsorbed water up to 8 ML coverage incombination with force field simulations of the anatase/liquidwater interface.As a further advancement, recent theoretical studies have

addressed the oxidation of water on various TiO2 surfa-ces.185−187 For anatase, in particular, Li et al.187 determined themechanism and energetics of the oxygen evolution reaction(OER) on the (101), (001), and (101) surfaces in aqueousenvironment, described using a continuum solvation model.

Their results show that the OER is not sensitive to the localsurface structure, as very similar mechanisms are found on thedifferent surfaces. Instead the overpotential is stronglyinfluenced by the position of the valence band maximum. Forall surfaces, the rate limiting step is the first proton removalleading to the formation of an OH radical. A detailed analysis ofthe kinetics of this step has been presented recently.188 In thelatter study the liquid water environment is explicitly describedand the energy profiles of the proton-coupled electron transfer(PCET) are determined by means of hybrid functionalcalculations. The results suggest that the first PCET issequential, with the electron transfer (ET) following theproton transfer and occurring via an inner sphere process,

Figure 11. Adsorption structures of (a) H2O, (b) CH3OH, and (c) HCOOH on anatase TiO2(101) terraces. C atoms are in deep gray and H inwhite. The dashed lines indicate H-bonds between molecules and surface O2c.

Figure 12. (A) Structure and (B) charge density difference Δρ = ρ[2H2O/anatase] − (ρ[H2O](I) + ρ[H2O](II) + ρ [anatase]) for a pairof water molecules adsorbed on anatase (101). Red (blue) regions in(B) indicate an increase (decrease) of the charge density. The arrowhighlights the charge transfer from Ti5c to O2c atoms via the adsorbedwater molecules. From ref 181. Copyright 2008 American ChemicalSociety.


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which is facilitated by a shared hole state shared between thetwo oxygen ions involved in the transfer.3.2.1.2. Anatase (001). Vittadini et al.175 first reported that

water adsorption on unreconstructed anatase (001) is stronglyexothermic at low coverage and associated with a majorrestructuring of the surface: the water molecule dissociates,disrupting one of the bonds of the bridging oxygens, see Figure13. This result was confirmed by several other studies,131,189,190

and can be related to the large surface energy and tensile strainof the clean unreconstructed (001) surface,141,173 which makeadsorption extremely favorable. The structure of adsorbedwater is not as clear at higher coverage, however. At 1 ML, inparticular, different configurations have been proposed, such asa mixed molecular-dissociated monolayer where half of theH2O molecules are adsorbed dissociatively and form H-bondswith a “second layer” of intact molecules,131,175 or a fullydissociated monolayer with terminal/bridging OH groups at allTi5c/O2c sites of the clean surface.190

The structure of an adsorbed water monolayer on anatase(001) has been recently revisited by Selcuk et al.,191 whooptimized a very large number of structures in a 2 × 4 supercellin order to accurately determine the most stable configurationsfor water adsorbed on both the bulk terminated and (1 × 4)reconstructed (001) surfaces at different coverages up to 1 ML.Their results show some differences from previous studies,

which were based on smaller unit cells. For the bulk terminatedsurface at 1 ML coverage, Selcuk et al.191 obtained a mixeddissociated-molecular structure where only 25% of the watermolecules is dissociated, while the other molecules form H-bonds with the surface and between themselves (Figure 14).Interestingly, the surface structure formed by the dissociatedwater molecules is essentially identical to the model for thereconstructed (001) surface proposed by Ignatchenko et al.143

(see section 1.2.2), and is also similar to the ADM model inFigure 8, which may be the reason for its stability. Thecomputed adsorption energy is 0.98 eV per molecule for thestructure in Figure 14, against 2.38 eV at 0.25 ML coverage.Incidentally, we note that incomplete dissociation of the firstlayer of adsorbed water molecules was observed also in FPMDsimulations of the anatase (001)/liquid water interface,182 butthe authors attributed this effect to the use of a 3 × 3 supercellpreventing full dissociation.182

Figure 14 shows also the structure of one water monolayeron reconstructed anatase (001)-(1 × 4). This surface issignificantly less reactive than the unreconstructed one,consistent with the relative stabilities of the two surfaces.141

The site of highest reactivity for anatase (001)-1 × 4 is theridge, where water adsorbs dissociatively with a computedenergy of 1.54 eV, whereas water is only weakly adsorbed onthe terraces.191,192

Figure 13. Adsorption structures of (a) H2O, (b) CH3OH, on anatase TiO2(001)-1 × 1. C atoms are in deep gray and H in white.

Figure 14. Optimized structure of one monolayer of water adsorbed on the unreconstructed (left) and (1 × 4) reconstructed (right) anatase (001)surface.


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It is interesting at this point to ask which of the twostructures in Figure 14 is more stable. The computed freeenergy diagram for the anatase (001) surface in the presence ofwater vapor191 shows a close competition in the region of waterchemical potential relevant to liquid water. While thereconstructed surface appears to remain energetically morestable also in the presence of water, additional studies, e.g.,FPMD simulations in a full liquid environment, may be neededto completely sort out this issue.3.2.1.3. Anatase (100). Despite recent experimental

evidence that anatase (100) may be the low-index anatasesurface with highest photocatalytic activity,146 studies of the(100) surface and its interaction with water are still relativelyscarce. At low coverage, all available studies have consistentlypredicted that water adsorption in dissociative form is slightlymore stable than molecular adsorption,131,143,190,193−195 seeFigure 15. Zhao et al.195 determined also the water dissociationbarrier and found it to be ∼0.34 eV. Dissociated andundissociated configurations were found to have very similarenergies also at higher coverages. Arrouvel et al.131 found thatmixed structures of dissociated and undissociated molecules areslightly more stable than fully dissociated structures, whereas afully dissociated monolayer was slightly more stable accordingto Barnard et al.190 and Ignatchenko et al.143 In reactive forcefield simulations at 300 K,179 the fraction of first layer watermolecules that were dissociated was ∼15% and ∼33% at 1 and3 ML coverage, respectively, indicating that the dissociation iswater-assisted.3.2.2. Molecular Oxygen. Molecular oxygen plays a key

role in many TiO2-based photocatalytic processes; in particular,O2 adsorbed on TiO2 surfaces is known to act as an electronscavenger and is often used to suppress electron−holerecombination, which increases the lifetime of the excitedstate and thus the yield of the photocatalytic reaction. Electrontransfer from the surface to the O2 molecule is essential foroxygen adsorption. In fact, O2 does not adsorb onstoichiometric TiO2; excess electrons are required. As titaniasamples are very often reduced,13 excess electrons originatingfrom oxygen vacancies and titanium interstitials are typicallypresent in the material.Theoretical studies of O2 adsorption on reduced anatase have

mostly focused on the (101) surface. The structural, electronic,and vibrational properties of O2 adsorbed in superoxo (O2

−)and peroxo (O2

2−) forms were investigated by Mattioli etal.196,197 As shown in Figure 16, both species adsorb in a side-on configuration and exhibit O−O bond lengths of ∼1.33 and1.46 Å, respectively. The calculations further predicted peroxospecies to be more stable than superoxo ones whenever theFermi energy is above the middle of the gap, as it is always thecase in reduced TiO2. The greater stability of the peroxidespecies was confirmed by Aschauer et al., who studied O2

adsorption on a reduced surface with a subsurface oxygenvacancy198 or Ti interstitial.199 These authors also found that itis energetically favorable for O2 to adsorb in the vicinity of thedefect, e.g., the adsorption energy at the Ti5c site closest to VOis ∼0.5 eV larger than at a site ∼5 Å farther away from it.198

Upon adsorption, the extra charge associated with the defect istransferred to the O2 molecule, converting it to an O2

2− specieswith electronic states in the anatase band gap. More recently,however, First-Principles Molecular Dynamics (FPMD) simu-lations at T ≈ 220 K showed that the configuration with O2adsorbed atop a subsurface VO is only metastable.200 In thepresence of adsorbed O2 the oxygen vacancy prefers indeed tomigrate from subsurface to the surface, where it recombineswith the adsorbed O2 to form a bridging dimer species (O2)O.This process is strongly exothermic, with an energy gain ofabout 1.6 eV from the state with O2 adsorbed atop a subsurfaceVO to the bridging dimer (O2)O state. This theoreticalprediction has been verified experimentally by STM.200

The predicted stability of O22− species seems to contradict

the experimental observation of stable O2− species by

EPR.201,202 This difficulty has motivated recent hybridfunctional calculations of the charge transfer reaction between

Figure 15. Adsorption structures of (a) H2O, (b) CH3OH, and (c) HCOOH on anatase TiO2(100). C atoms are in deep gray and H in white.

Figure 16. Optimized geometries, desorption enthalpies, and O−Ostretching frequencies of (A) a physisorbed O2 molecule, (B) anadsorbed superoxo (O2

−) species, and (C) an adsorbed peroxo (O22−)

species on anatase (101). From ref 196. Copyright 2006 AmericanChemical Society.


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reduced anatase (101) and molecular O2 by Li et al.203 It wasfound that although the peroxo O2

2− species is energeticallymore stable than O2

−, there is a significant barrier, ∼0.3−0.4eV, that must be overcome to transfer an additional electron to*O2

− and transform it to a peroxide, whose formation is insteadbarrierless. The existence of this barrier can thus explain whyexperimentally superoxo species are often observed.3.2.3. Methanol and Formic Acid. Experimental inves-

tigations on the adsorption of organic molecules on anatasesingle crystal surfaces are scarce.204−206 Here we consider onlymethanol and formic acid, as an extensive review has recentlybeen given by Thomas and Syres.207 Besides being a simpleprototype for organic compounds as well as an importantmolecular probe for the investigation of surface properties,methanol is frequently used as a hole scavenger in photo-catalysis. For these reasons, the interaction of methanol withanatase surfaces has been the subject of several studies.Experimentally, X-ray photoelectron spectroscopy (XPS)measurements on anatase (101) did not show any evidenceof methanol dissociation, whereas features of methoxy groupswere detected by TPD on the same surface and attributed toCH3OH dissociation at step edges.176 Similarly, sum-frequencygeneration (SFG) experiments on a nanoparticulate anatasefilm indicate that molecular adsorption dominates at regularsurface sites, whereas chemisorption with methoxy formation isrelated to defects.208,209

In theoretical studies, trends in the adsorption structures ofmethanol on different anatase surfaces appear to be similar tothose of water and reflect the relative energies of these surfaces.Molecular adsorption is preferred on undefected anatase (101),see Figure 11, and the adsorption energy is only slightly smallerthan that of water.193,210 By contrast, dissociative adsorptionwas predicted to occur on both the (001)189 and (100)193

surfaces, Figures 13 and 15, respectively. The stability of themolecular adsorption state on the (101) surface has beenconfirmed by recent B3LYP hybrid functional calculations.211

The latter study further showed that methanol can trap holesonly when it is dissociatively adsorbed on the surface.Carboxylic acids and carboxylates are widely used anchoring

groups for the modification of TiO2 with functional molecules,notably dye sensitizers in dye sensitized solar cells.4,212 This hasmotivated several theoretical/computational studies of theadsorption properties of formic acid (HCCOH), which is thesimplest species containing a carboxylic group. On anatase(101), GGA calculation indicate that the most stable HCOOHadsorption structure is a molecular monodentate configurationwhere the carbonyl oxygen binds to a surface Ti5c while thehydroxyl hydrogen forms a H-bond with a bridgingO2c,

193,213,214 see Figure 11. This finding has been confirmedby B3LYP hybrid functional calculations,211 but the differencebetween the undissociated monodentate and dissociatedbridging bidentate forms is only 0.05 eV with B3LYP,211

against 0.24 eV with DFT-GGA.213 HCOOH adsorption in thebridging bidentate form was predicted to be favorable also onthe (100)193 (Figure 15) and unreconstructed (001)192

surfaces. For the reconstructed (001)-(1 × 4) surface,calculations based on the ADM model192 indicate that formicacid adsorbs preferentially on the ridges where it has abidentate chelating configuration. This assignment is supportednot only by the computed total energies but also by the goodagreement of the calculated STM image with experiment.142

3.2.4. Water, Methanol, and Formic Acid at StepEdges on Anatase (101). Gong and Selloni193 reported

extensive DFT calculations showing that the reactivity of stepedges is very similar to that of the extended surfaces which areexposed at their facets. Adsorption of water and methanol atsteps with (112) facets (denoted D-(112) in ref 193) isnondissociative even though stronger than at (101) terraces. Bycontrast, dissociative adsorption is preferred for water andmethanol at steps with (100) facets (denoted B-(100) in ref193). At both step D-(112) and B-(100) the adsorption energyof H2O is lower than that of CH3OH, even though the twomolecules have very similar adsorption configurations. This canbe attributed to the relatively higher acidity (weaker RO-Hbond strength) of CH3OH with respect to H2O. On the flatTiO2(101) surface, instead, H2O has a slightly higheradsorption energy than CH3OH (0.78 vs 0.73), due to thefact that H2O can form two H-bonds with O2c, while CH3OHcan form only one (see Figure 11). For HCOOH, whilemolecular adsorption is favored on (101) terraces (Figure 11),a dissociated bidentate adsorption geometry is favored at bothD-(112) and B-(100) step edges. This can be attributed to thefact that the HCOO moiety in bidentate configuration can formtwo O−Ti5c bonds with reactive Ti5c atoms at the edges. Thisalso suggests that the binding of HCOOH with stepped anataseTiO2(101) surfaces is more “robust” than that of H2O andCH3OH, consistent with various experimental evidence.

4. TIO2 SENSITIZATION AND APPLICATIONS TOHYBRID AND ORGANIC PHOTOVOLTAICS

Due to its abundance and stability against photocorrosion,TiO2 is an interesting material for solar energy conversion, yetit is not very efficient. One of the most serious drawbacks ofTiO2 is its large band gap, Eg∼ 3.2 eV, which results in theabsorption of only a small portion of the solar spectrum in theUV region. A widely used strategy for overcoming thisdifficulty, is to “sensitize” TiO2 by attaching suitable moleculesto its surface. A large body of theoretical/computationalinvestigations on sensitized TiO2 surfaces has been reported,especially with reference to Dye-sensitized solar cells(DSCs),4,12,215−217 in which a dye molecule is covalentlygrafted to TiO2. Many theoretical studies have examined theelectron injection step, which is the primary charge generationevent in DSCs, and the dye adsorption mode on TiO2.

218−268

Since various reviews dealing, in full or in part, withcomputational approaches to DSCs are available,249,260,269−273

including another contribution to this issue,274 in the presentreview we focus almost exclusively on the effect of theadsorbate on the underlying TiO2 electronic structure.

4.1. Dye and Coadsorbent Effects on TiO2

A crucial ingredient of the DSC efficiency is the open circuitvoltage, VOC, which is essentially given by the differencebetween the TiO2 CB edge and the redox potential of theelectrolyte. The energetic of the TiO2 CB in DSCs is known todepend on several factors, such as the local pH,215,275−278 theconcentration of potential determining ions (e.g., Li+),276,279,280

and also on the nature of the electrolyte solvent.279,281 Whilethe role of surface adsorbed molecules, including the dye, indetermining the TiO2 CB energetics is generally lessclear,282−289 an interesting correlation between the dipolemoment of coadsorbing species, mainly substituted benzoicacids, and VOC was observed by Ruhle et al.,

284 who pointed outa linear relation between the dye coverage (N), the dipole (μ)component normal to the surface (θ is the molecule tilting


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angle) and the potential shift (ΔV) at the interface affecting theTiO2 CB energy: ΔV = (εε0)/(Nμ cos θ)De Angelis et al.290 proposed that the high VOC measured for

DSCs sensitized by the prototypical N719 ruthenium dyeoriginates from the special dye adsorption mode, involving twoor three carboxylic anchoring groups residing on differentbipyridines, see Figure 17, while most other ruthenium dyeshave only two anchoring units. DFT calculations for the dyeadsorbed on a model TiO2 cluster revealed also a dipole-induced shift of the TiO2 CB, indicating that the sensitizer’sadsorption mode influences the position of the TiO2 CB.Kusama et al. reported a combined experimental and

theoretical study which showed a clear correlation betweenthe dipole moment of (liquid) electrolyte additives and themeasured VOC in DSC devices.288 These authors investigatedthe adsorption of nitrogen-containing heterocycles such aspyrazole, imidazole, 1,2,4-triazole, pyridine, pyrimidine, pyr-azine, and 4-t-butylpyridine (TBP) on TiO2 anatase (101),(100), and (001) surfaces, using periodic DFT calculations;optimized structures for the (101) surfaces are shown in Figure18. All the investigated structures displayed a negative (i.e.,upward) shift in the TiO2 Fermi level upon adsorption of N-containing heterocycles, that the authors correlated to theadsorbate dipole moment component normal to the TiO2surface plane.Mattioli et al.291 investigated the interaction of a oxo-Ti-

phtalocyanine (TiOPc) with the anatase TiO2 (101) surface.The interaction was found to take place mainly through the Ti-oxo phtalocyanine group, with creation of a Ti-oxo-Ti (surface)bond. The equilibrium geometries and isosurfaces of differenceelectron densities for such system are reported in Figure 19. Anelectron density rearrangement accompanies the oxo-Ti(surface) bond formation, which was also found to lead to anegative TiO2 CB shift (i.e., the TiO2 CB was pushed at higherenergies relative to the molecular HOMO). A much strongerinteraction was observed when the molecule was interactingwith the (101) anatase surface doped by a Ca2+ ion. In this casea pronounced charge density rearrangement involving also theTiOPc carbon atoms was found, leading to a substantial chargetransfer from the TiOPc molecule to the TiO2 surface.Although the “dipole effect” can explain a number of results

related to DSCs, the adsorbate/TiO2 interaction usually is morecomplicated than the simple electrostatic (EL) effect mediated

by the adsorbate dipole. When an adsorbate binds to asemiconductor surface, one should indeed consider also theeffect of the charge transfer (CT) between the dye and thesemiconductor which may accompany the adsorbate/semi-conductor physisorption or chemisorption (bond formation).Both the EL and CT terms can vary with the surface coverage,so it is important to check also this quantity for meaningfulcomparisons with experiment.Ronca et al.292 used a charge displacement (CD) analysis293

to investigate the adsorption of several prototypical organicdyes and coadsorbents on a TiO2 cluster model, quantifyingand rationalizing the separate effects of EL and CTcontributions to the TiO2 CB energetics. These authorsinvestigated a series of dyes with vastly varying dipole momentand electron donating properties considering both dissociativebridged bidentate (BB) and molecular monodentate (M),adsorption modes.292 Investigation of the charge displacementcurves showed significant charge in the interface region, of theorder of 0.3−0.4 electrons. Based on the CD analysis, theyproposed a simple interpretative model by expressing the totalTiO2 CB shift, ΔCBTOT, as the sum of the two main effectsstrictly related to the dye sensitizer:292 ΔCBTOT=ΔCBEL+ΔCBCT.

4.2. Insulating Metal Oxide Monolayers

An appealing strategy to reduce recombination losses in DSCsis to introduce an “insulating oxide” layer between the TiO2surface and the sensitizing dye. Typically, these oxide layers aregrown by atomic layer deposition (ALD), which is an idealtechnique to deposit close to monolayer films. Terranova andBowler investigated by periodic DFT calculations the interfacebetween amorphous Al2O3 (a-Al2O3) and the TiO2 anatase(101) surface, and further studied the sensitization of theensuing interface by a ruthenium dye.294 They considered twoa-Al2O3 thicknesses, and relaxed the structure of the resulting a-Al2O3/TiO2 interfaces, see Figure 20. The DOS of the bareTiO2 and of the two a-Al2O3/TiO2 interfaces are also reportedin Figure 20. Although in the thick coating, a net dipolemoment (0.67 eÅ) was present along the directionperpendicular to the interface, the authors considered it to betoo small to significantly affect the TiO2 energy levels. ThePDOS for the bare and a-Al2O3-coated TiO2 are indeed quitesimilar, even though some states appear close to the VBmaximum, possibly reflecting hybridization with Al2O3 states.

Figure 17. Adsorption geometries of two prototypical Ru(II)-complexes: (left) N719 with three anchoring points (one bidentate bridging and twomonodentate) and (right) C106 with two monodentate anchoring points.


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The calculated DOS and energy level alignment were similar tothe electronic structure for the same interface in the study byWang et al. .295 The calculated interaction energies of a formicacid probe were found to increase on the a-Al2O3 overlayerwith respect to TiO2, suggesting a stronger dye loading on theformer oxide. This trend was confirmed also for the N3 dyebinding to a-Al2O3, for which a largely increased binding energywas obtained compared to TiO2.4.3. Polymers’ Morphology on TiO2

As an alternative to molecular dyes, donor polymers can beused as light-harvesters in hybrid solar cells based on TiO2 orZnO. The effect of selective interactions at the interface of

polymer−TiO2 hybrid solar cells was investigated by Canesi etal. in a combined experimental and computational inves-tigation.296 These authors reported a step-change improvementin the solar cell device performance, which was enabled byengineering the hybrid interface by the insertion of anappropriate molecular interlayer between the polymer donorand the TiO2 semiconductor. By positron annihilationtechniques, it was experimentally observed that the presenceof a 4-mercaptopyridine (4-MP) interlayer led to better contactbetween the oxide and polymer phases and a closer polymerpacking at the interface. Canesi et al. performed classicalmolecular dynamics simulations based on a model potential forthe P3HT polymer/TiO2 interface with and without thepresence of a 2-mercaptopyridine (2-MP) and 4-MP interlayer.2-MP and 4-MP are both able to bind to TiO2 by virtue of thenitrogen lone pair and to possibly interact with the polymer bytheir S−H groups. In a subsequent study, Malloci et al.297

extended the study on the effect of interlayers on P3HTadhesion to TiO2 to 2-MP, PYR (pyridine), 4-MP, and TBP(tert-butylpyridine). A survey of the ensuing optimizedstructures is reported in Figure 21. Ordered monolayers werefound for PYR and 4-MP, a partially ordered monolayer forTBP, and a disordered one for 2MP. Interestingly, theseauthors also found a correlation between the dipole moment ofinterlayer molecules and the Voc of the corresponding solarcells, in line with what discussed above for DSCs.When considering a polymer/metaloxide hybrid, usually the

polymer does not form covalent bonds with the inorganicmaterial, and therefore covalent bonds do not contribute to theadhesion. This is clearly at variance with dye-sensitized solar

Figure 18. Geometry-optimized structures of N-containing hetero-cycles adsorbed on a TiO2 anatase (101) surface: (a) pyrazole, (b)imidazole, (c) 1,2,4-triazole, (d) pyridine, (e) pyrimidine, (f) pyrazine,and (g) TBP. From ref 288. Copyright 2008 American ChemicalSociety.

Figure 19. Equilibrium geometries and isosurfaces of differenceelectron densities of (A) a TiOPc molecule bonded to the (101)anatase surface and (B) a TiOPc molecule bonded to a p-dopedanatase surface. Electron density difference maps show the displace-ments of electronic charge induced by the molecule−surfaceinteraction. Red surfaces cover areas where the difference is positive,blue surfaces, where it is negative. From ref 291. Copyright 2009American Chemical Society.


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cells, as discussed previously. Nevertheless, strong electrostaticinteractions may occur between the ions of the surface and thepartially charged atoms in the polymers due to the ionicity ofthe metal oxide.298 This is, for instance, the case of poly(3-hexylthiophene) (P3HT), for which large atomic partial charges(up to 0.15e, where e is the electronic charge) are found. Asecond important issue to take into account is the effect of themorphology of the nanostructured substrate films convention-ally used in hybrid devices. Since polymer/metal oxideadhesion may affect the overall efficiency of polymer-basedsolar cells, investigating the morphology of this interface isimportant for further optimization of the related solar celldevices. Melis et al. investigated P3HT adhesion to flat andnanostructured TiO2 by means of classical molecular dynamicsbased on a model potential.298 The surface morphology of thetitania film, the local charge of the surface, or the presence ofdefects in the lattice can in principle strongly affect thepolymer/TiO2 interaction. Melis et al. found that when P3HTinteracts with TiO2 the adhesion energy is dominated byelectrostatic contributions. In addition, the nanomorphologydoes not necessarily increase the polymer adhesion with respectto the planar case, due to the strain introduced in the polymerby the curvature radius.4.4. Quantum Dots on TiO2

As a last example of TiO2 functionalization/sensitization forapplication to hybrid photovoltaics, we discuss the case ofquantum dots (QD)/TiO2 interactions. QDs are easily tunable

to absorb light at any wavelength, by exploiting quantum sizeeffects,299 have high extinction coefficients and large intrinsicdipole moments, and are photochemically robust. Long andPrezhdo258 investigated the interfacial properties of a PbSe QDadsorbed on a (110) rutile TiO2 surface. They found thatthermal fluctuations, as evaluated by molecular dynamicssimulations, may impact the system geometry, with the largestscale motion associated with QD displacements in the plane ofthe TiO2 surface. Figure 22 shows the electron densities of theinterfacial donor and acceptor states. The density of theelectron donor is delocalized over the whole QD, while theacceptor state is spread nearly uniformly across the TiO2 slab. Arather strong QD/TiO2 coupling was found even in theabsence of a QD/surface linker molecule, producing the mixingof the donor and acceptor energy levels shown in Figure 22. Assuch, a substantial QD → TiO2 charge-transfer is expected tooccur at the interface, similar to what discussed above formolecular dyes and for polymer donors.Patrick and Giustino300 simulated the interface between a 12

Å ultrathin stibnite Sb2S3 film and the (101) TiO2 anatasesurface, see Figure 23, as representative of the quantum-dotsensitizer systems employed in solid-state solar cells.301 Thestibnite LUMO was found to lie above the TiO2 CB edge,suggesting energetically favorable electron injection, and it wasshown to extend into the TiO2 substrate through the couplingbetween the S 3p and Ti 3d orbitals at the interface, therebyproviding a direct pathway for carrier injection. As such, thesituation is rather similar to the PbSe-sensitized TiO2 discussedabove. A key aspect shared by all the reviewed prototypicalsystems is the presence of strong electronic coupling, obtainedthrough the formation of covalent and/or charge-transferinteraction between the electron donor and the TiO2 acceptor,as an essential prerequisite for efficient solar energy conversion.

5. MODELING TIO2 NANOPARTICLESTiO2 nanoparticles or nanocrystals (here we use the two termsas synonyms) lie at the heart of many technologicalapplications, notably photocatalysis and hybrid/organic solarcells. An advantage of TiO2 nanoparticles over conventionalsemiconductors is that their phase, size, and morphology can bewidely tuned and tailored for specific target applications, thusproviding an efficient strategy for engineering more efficientphotoelectrochemical devices.The main property of TiO2 nanoparticles (and in general of

nanostructured semiconductors) is their high surface to volumeratio. Since most photoelectrochemical reactions mediated byTiO2 are initiated by surface adsorption of a chemical specieson TiO2, the high surface exposure ensures an overall highdensity of reactive centers. This feature is specifically exploitedin dye-sensitized solar cells (DSCs), in which the core of thedevice is represented by a film of sintered TiO2 nanoparticles,so that the exposed surface is magnified by ca. a factor 1000compared to a flat TiO2 surface. In turn, this allows a high dyeloading, which ensures complete light absorption with films ofjust 10 μm thickness (using conventional dyes, characterized bya molar extinction coefficient of ∼10−20.000 M−1 cm−1). Atthe same time, the high number of reactive sites introduced by,e.g., the nanoparticles edges, ensures a high reactivity of thesintered films in photocatalytic applications.At ambient pressures and temperatures, the rutile TiO2 phase

is more thermodynamically stable than the anatase phase,28

while the anatase phase is preferred for nanoparticles ofdimension ∼20 nm or smaller.302 The size range of the anatase

Figure 20. Top: front view of the thin and thick overlayers relaxedonto the (101) anatase substrate. Bottom: PDOS of the two oxides forthe three systems (coating 0, 3, and 9 Å). The highest occupied levelsare indicated by the vertical dashed lines. From ref 294. Copyright2012 American Chemical Society.


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to rutile phase transition for hydrothermal samples washowever shown to be sensitive to a variety of factors, such astemperature, pH, presence of impurities, reaction conditions,etc. In particular, the pH value of the sol−gel was shown tocrucially control, among other factors, the phase, size and shapeof the synthesized nanoparticles.303 A detailed understanding ofthe thermodynamic phase stability of TiO2 nanocrystals and the

factors affecting the anatase to rutile transformation is thus veryimportant for the control of nanoparticle nucleation andgrowth processes, in relation to target technological applica-tions.Despite the clear advantages in the use of sintered TiO2

nanoparticles, a significant drawback is represented by the slowelectron mobility in such disordered mesoscopic materials. Fornanostructured TiO2 films commonly employed in, e.g., DSCs,effective diffusion coefficient values in the range 10−8 to 10−4

cm2 s−1 have been measured, i.e. orders of magnitude smallerthan those observed for TiO2 single crystals.304,305 Thisobservation clearly suggests a high concentration of electron-trapping sites in the semiconductor film which slow down theelectron transport.306,307It has been suggested that inmesoporous TiO2 films made of sintered nanoparticles, theconduction band (CB) has a low energy tail of localized statesbelow the energy characterizing the onset of fully delocalizedconduction band states (also termed mobility edge). Kavan etal.308 and many others since then309−311 proposed the existenceof deep, surface trap states in the band gap, below the mostsignificant portion of the exponential tail of the DOS. Althoughthe effect of trapping states on electron transport inmesoporous nanocrystalline TiO2 has been extensivelyinvestigated,312−314 it is still unclear whether these statesoriginate from defects in the bulk and surface regions, from thegrain boundaries of the particles, from Coulomb trapping due

Figure 21. Ternary systems formed by the 2-MP, PYR, 4-MP, and TBP interlayers for P3HT adhesion on TiO2. For clarity, all hydrogen atoms havebeen omitted and the carbon atoms of P3HT are marked in white to distinguish them from those of the interlayer. From ref 297. Copyright 2013American Chemical Society.

Figure 22. Charge densities of (a) donor and (b) acceptor states at thePbSe QD/TiO2 interface. From ref 258. Copyright 2011 AmericanChemical Society.


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to interactions of electrons with the cations of the electrolyte,or from a combination of all these factors. Also, TiO2

nanocrystals of different shapes and sizes can have differenttypes of defects and trap states, along with a different value ofthe fundamental band gap or distribution of unoccupied states.Understanding the subtle interplay between the nature of trapstates in TiO2 nanocrystals and the size/shape/surfacefunctionalization of these materials is thus essential for furtheroptimization of their technological applications. Similarly, clearinsight into the parameters ruling the sintering process and thepossible consequences of sintering on the electronic propertiesof TiO2 nanoparticles is also crucial, since most technologicalapplications require sintering of the TiO2 nanoparticles andsintering of the titania nanoparticles takes place in typical drysynthetic approaches requiring thermal annealing.315

In this section, we present an overview of computersimulations aimed at predicting, rationalizing and under-standing the properties of TiO2 nanoparticles in relation tothe technological issues mentioned above. Due to thecomplexity and large dimensions of the investigated systems,most computational investigations have been based onatomistic molecular dynamics based on a model potential, oron thermodynamic model approaches, possibly augmented byor with parameters derived from DFT calculations on reducedperiodic surface slab models. Also, semiempirical tight-bindingDFT calculations have been employed both for structuraloptimization and for gaining some insight into the electronicproperties of the TiO2 nanoparticles. While various DFTstudies have been performed on clusters made by to ∼10−50TiO2 units, only recently first-principles DFT approaches havebeen employed to study realistic TiO2 nanoparticles of ∼3 nmsize, made of several hundred TiO2 units.

316,317 Recently, thestudy of small TiO2 nanocrystals has been approached also withpost-DFT methods based on perturbation theory, i.e. GW,

yielding improved results for the energy level alignment atheterointerfaces compared to DFT.237,318

5.1. Shape, Size, and Phase Stability of TiO2 Nanocrystals

As pointed out in section 3.1.1, the equilibrium shape of acrystal of an arbitrary material is given by the standard Wulffconstruction,319 which requires knowledge of the energies ofthe various exposed surfaces. The Wulff shape of anatase wasfirst derived by Lazzeri et al.26 for the case of clean dry surfaces(see Figure 6). Results for hydrated and hydrogenated surfaceswere later reported by Arrouvel et al.131 and Barnard et al.,320

who performed calculations also for rutile. In agreement withexperimental observations for naturally occurring anatase, theWulff construction shows that in anatase crystals the only twosurfaces exposed to the vacuum are the (101) and (001)surfaces, while for rutile the most stable surfaces are the (110),(101), and (100).To evaluate the thermodynamic stability of TiO2 anatase and

rutile nanoparticles as a function of the particle’s size andsurface functionalization, Barnard and Zapol320 estimated theGibbs free energy of formation of a nanocrystal of a material x,Go

x, in terms of the surface energy γxi for each surface i,weighted by the factors f i, such that Σi f i = 1:

∑ρ

γ= Δ + −G GM

e q f(1 )[ ]xo

f xo

x ii xi

(1)

where ΔfG0x is the standard free energy of formation of the

bulk (macroscopic) material, M is the molar mass, ρx is thedensity and e is the volume dilation induced by the surfacetension. The latter term cannot be ignored at the nanoscale andmay be approximated using the Laplace−Young equation321 forthe effective pressure, which is defined in terms of the surfacetension σ and is approximated by summing over the (weighted)surface tensions of the crystallographic surfaces present on thenanocrystal. The surface to volume ratio q and the weighting

Figure 23. Left: Atomistic model of the TiO2/Sb2S3 interface. The colored atoms represent the periodic repeat unit, and the view is along the TiO2[010] direction. Inset: Schematic representation of the stibnite-sensitized solar cell. Right: Isodensity plot of the Kohn−Sham LUMO state of Sb2S3at the TiO2 /Sb2S3 interface. The charge density is plotted in a plane through the Ti−S bond. The coupling between the S 3p states of the sensitizerand the Ti 3d states of the substrate, which provides a pathway for electron injection, is highlighted. Reprinted with permission from ref 300.Copyright 2011 John Wiley and Sons.


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factors f i must be calculated explicitly for each shape and thefacet therein. The effects of edges and corners were omitted inref 320, since they were expected to be small for relatively largenanocrystals. As discussed in the following, these effects havebeen explicitly evaluated by Hummer et al.322 and found to beimportant for particles size below ∼3 nm. According to eq 1,one can predict the Gibbs free energy of formation of ananocrystal by employing the experimental ΔfG

0x and ρx, while

explicitly calculating the surface energy and tension (γi and σi)for each surface, and the weighting factors f i. The surfacetension can be calculated by applying a two-dimensionaldilation to the slab in the plane of the surface and calculatingthe resulting free energy change. The results obtained by DFTcalculations of γi and σi for each surface have been employed byBarnard and Zapol to construct phase diagrams for the facetednanocrystals as a function of the nanocrystals size. Their resultspredict that the phase transition from anatase to rutile takesplaces at ∼12 300 TiO2 units for the clean surface,corresponding to anatase particles of diameter of ∼9.3 nm.For the partially hydrogenated and fully hydrogenated surfacesthe intersection points were found to occur at ∼10 800 and∼196 900 TiO2 units, respectively, corresponding to anatasenanocrystals with average diameters of ∼8.9 and ∼23.1 nm,respectively. This dramatic change indicates that the termi-nation of under-coordinated surface sites plays a major role indetermining the stability of nanoscale TiO2. Full hydrogenationof the nanocrystal surfaces is predicted to promote stability ofthe anatase phase, whereas hydrogenation of only the bridgingoxygens (partial hydrogenation) promotes stability of the rutilephase. These results are in agreement with the observedstabilization of anatase nanoparticles in acidic media.303

Barnard and Curtiss further investigated how the surfaceacid−base chemistry may influence the shape of the nanocryst-als and the anatase-to-rutile transition size.323,324 Theyconsidered the surfaces to be water-terminated and simulatedvariations in the surface pH by varying the ratio of hydrogenand oxygen atoms on the surface. The final shapes predicted foreach type of surface chemistry are shown in Figure 24. Anatase

nanocrystals with hydrogen-poor surfaces showed welldeveloped facets (100) and (010) surfaces, which appear asthe “belt” around the center of the nanoparticles. The effect ofchanging the surface chemistry upon the shape of the anataseand rutile nanoparticles is readily apparent. When hydrogen isdominant on the surface (or there is a greater fraction ofhydrogen present in the adsorbates), there is little change in theshape of the nanocrystals with respect to the (neutral) waterterminated nanoparticles; however, when oxygen is dominanton the surface, the nanoparticles of both polymorphs becomeelongated.The dependence of the anatase-to-rutile phase transition on

the surface chemistry is illustrated in Figure 25. This also showsthe possibility for phase transitions to be induced by a changein the absorbed groups on the surfaces. For instance, one canconsider the vertical guideline at 45 000 TiO2 units. Beginningwith hydrogen-rich surfaces (solid lines), anatase is thermody-namically stable. As the surface chemistry is neutralized to

Figure 24. Morphology predicted for anatase (top) with (a) hydrogenated surfaces (b) with hydrogen rich surface adsorbates, (c) hydrated surfaces,(d) hydrogen-poor adsorbates, and (e) oxygenated surfaces, and rutile (bottom) with (f) hydrogenated surfaces, (g) with hydrogen rich surfaceadsorbates, (h) hydrated surfaces, (i) hydrogen-poor adsorbates, and (j) oxygenated surfaces. From refs 323 and 324. Copyright 2005 AmericanChemical Society.

Figure 25. Free energy as a function of the number of TiO2 units foranatase and rutile with hydrogen-rich, hydrated, and hydrogen poorsurfaces. Vertical guidelines assist in the comparison of different stablephases at 45 000 and 75 000 TiO2 units during deprotonation (movingfrom the bottom upward). From refs 323 and 324. Copyright 2005American Chemical Society.


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correspond with hydrated surface, the anatase phase remainsstable. However, further deprotonation induces a phasetransition to rutile, since with hydrogen-poor surface chemistry(dotted lines), rutile is thermodynamically preferred at a size of45 000 TiO2 units. Then, one may consider the same procedurefollowing the guideline at 75 000 TiO2 units. Beginning withhydrogen-rich surfaces, anatase is again thermodynamicallystable. As the surface undergoes deprotonation, the anatasephase transforms to rutile, since for hydrated surfaces, rutile isthermodynamically preferred at a size of 75 000 TiO2 units.This phase transition for nanoparticles containing 75 000 TiO2

units occurs at an earlier stage of deprotonation thannanoparticles containing 45 000 TiO2 units. These resultsindicate not only that the opportunity exists for surfacechemistry induced phase transitions to occur, but that thesetypes of phase transitions are also size dependent.

5.2. Molecular Dynamics of TiO2 Nanoparticles

The main difference between the studies considered in thissection and those based on a thermodynamic approach in theprevious section is that in the latter an accurate electronicstructure method, i.e., DFT, is used to calculate the surfaceenergies in various conditions, which are then combined with anonatomistic thermodynamic approach. In molecular dynamicssimulations the structure of the nanoparticles retains itsatomistic description, but the interatomic potential is describedby a simple model force field. In these simulations, systems ofseveral thousand atoms are typically considered, which preventsthe use of electronic structure tools.All the MD simulations reviewed below employed the

Matsui-Akaogi force field.128 In this force field, the potential isexpressed as a combination of a repulsive exponential term, anattractive van der Waals term and an electrostatic contribution,for which partial charges q of +2.196 and −1.098 are assignedto titanium and oxygen, respectively. The main point ofstrength of this force field is its simplicity and the accuracy inreproducing the structural features and relative energies ofvarious TiO2 polymorphs against experimental data.One of the earliest molecular dynamics simulation of TiO2

nanoparticles was reported by Collins et al.,325 who investigatedan initially spherical rutile TiO2 nanocrystal made of 1245atoms, i.e., with a radius of 1.4 nm. The authors investigated thesystem in the 1000−3000 K temperature range, for simulationtimes up to 600 ps. The temperature conditions are thosetypical of the so-called “chloride” synthesis of TiO2 nano-particles. Interestingly, the initially spherical cluster structureevolved in time to a faceted arrangement, with microfacetscorresponding to the (100), (110) and (101) planes ofcrystalline rutile. Furthermore, the rutile phase was retainedthroughout the simulation, up to the melting point.Naicker et al.326 performed molecular dynamics simulations

of TiO2 nanoparticles in the anatase, brookite, and rutilephases, analyzing their structural features and surface energies.These authors investigated spherical nanoparticles of 2 to 6 nmsize for 3 ns time, exploring 300−2000 K temperature range.Although anatase was predicted as the lowest energy phase forlarge surface areas, no phase transformation was observedduring the simulation in the explored time/temperature regime.An interesting result of this study was the identification of four-and five-coordinated titanium ions at the nanoparticles surface,which showed shorter Ti−O bond lengths compared to six-coordinated atoms. These surface sites are responsible of the

peculiar reactive/electronic properties of nanoparticles, asdiscussed in section 5.3.The sintering of two spherical rutile nanoparticles was

simulated by molecular dynamics by Collins et al.,327 whoexplored the collision of ∼3 nm size particles initially separatedby ca. 4 nm in the 1200−2000 K temperature range. Thedynamics simulation was conducted for up to 1 ns time. Thenanoclusters were initially oriented so that the (100) directionof the underlying rutile phase was parallel to the direction ofmotion. The resulting collision could therefore be described asoccurring between the (100) faces of the nanoclusters, thoughthis picture is complicated by the roughness of the faces.Snapshots of a typical nanocluster collision at 1200 K areshown in Figure 26.

The initially separated nanoparticles are mutually attractedby long-range intermolecular forces. Contact between thesurfaces takes place after approximately 25−30 ps. The collisionis mediated by surface roughness, and there is visual evidence ofsurface distortion when the surfaces are close. The collisiontakes place without sign of fracture or rebounding and theunderlying phase remains rutile, although the formerobservation might be due to the initial preferential nanoparticleorientation. Following surface contact (third snapshot), there isan initial rapid reduction in configuration energy, which wasassociated with formation and widening of the “neck” betweenthe two nanoparticles and the subsequent relaxation of theunderlying lattice, particularly of the former surface layers,which occurs as a consequence of the nanoparticles contact.Along with the configuration energy decrease, a correspondingtemperature increase was observed, which is due to energyconservation.Koparde and Cummings328 reported molecular dynamics

simulations of TiO2 nanoparticles sintering, consideringspherical anatase and rutile particles of 3 and 4 nm size,investigating various possible initial orientations. MD trajecto-ries were propagated for 1 ns time, at temperature in the range573−1473 K. We notice that such MD times are still short incomparison to the characteristic sintering time of ca. 12 μsobserved for 3 nm TiO2 sintering at 1473 K.328 Neverthelessthe investigated time span is likely sufficient to study theimportant initial stages of sintering, also considering that aplateau in the potential energy was reached after ca. 0.5 nssimulation. The early stages of these MD simulations resemble

Figure 26. Time sequence of colliding TiO2 nanoparticles at (from leftto right): (a) 0.3 ps, (b) 25.5 ps, (c) 27.0 ps, and (d) 288.3 ps. Whiteand dark gray large spheres are oxygen, black and light gray smallspheres are titanium. Reprinted with permission from ref 327.Copyright 1997 Royal Society of Chemistry.


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those of ref 327 and showed the nanoparticles contact to takeplace within 20−25 ps, followed by formation of a neck, withassociated decrease of the configuration energy. Interestingly,the neck diameter was found to increase with temperature for 3nm anatase nanoparticles, while it was almost independent oftemperature in the case of 3 nm rutile nanoparticles. Also, nophase change was observed during the MD simulation. Tostudy the effect of crystallographic orientation in the sinteringof nanoparticles, the same authors328 rotated one of the twonanoparticles to ensure different crystallographic orientationsalong the axis the particles approach each other. MDsimulations carried out for 3 nm anatase particles showedthat neck formation occurred almost at the same time in all ofthe cases except the 180° rotation case, for which the particlesmoved away from each other. These simulations also indicatedthat a 90° orientation produced maximum interpenetration,suggesting that this initial configuration may be most favorable.This suggests that orientation of the nanoparticles is extremelyimportant in the process of sintering, which the authorsassociated with the relevant role played by dipole−dipoleinterparticle interactions.Koparde and Cummings329 later extended their MD study to

the sintering of 3 nm anatase nanoparticles with rutile and/oramorphous nanoparticles of the same size. The MD simulationswere conducted at initial temperatures of 973 and 1473 K for10 ns. Based on the calculation of the potential energy versussurface area of anatase and rutile nanoparticles at 1473 K, theseauthors predicted an anatase to rutile crossover at 1.65 nm,consistent with the results of Naicker et al.326 This crossoversize is much smaller than that predicted for faceted nanocryst-als. An interesting observation was that upon sintering of oneanatase and one rutile nanoparticles, the sintered nanoparticlesevolved toward the rutile phase, as inferred from the timeevolution of the simulated X-ray diffraction patterns from the1473 K simulation. Similar transformations to rutile areobserved in simulations involving amorphous + rutile nano-particles and those with anatase + amorphous + rutilenanoparticles. Thus, the authors concluded that the finalagglomerate will be in the rutile phase whenever one of the

sintering nanoparticles is in the rutile phase. In case of theanatase + amorphous simulation, the simulated X-raydiffraction patterns revealed that phase transformation led tobrookite, the third polymorph of titania. Notably, at initialtemperature of 973 K no phase transformation was observedfor any combination of particles over the simulated time scales,although the particles did undergo sintering. Altogether, theseresults indicate that enhanced ionic mobility in nanoparticlesclose to their melting points plays an important role in assistingphase transformations.Alimohammadi and Fichthorn330 reported classical MD

simulations on faceted anatase nanocrystals. These authorsconsidered charge-neutral anatase nanocrystals with variationsof the truncated tetragonal bipyramidal Wulff shape, aspredicted in refs 27 and 320 which contained both (101) and(001) facets. In addition to Wulff shapes, they considered alsoasymmetric nanocrystals, which mimicked possible off-Wulffshapes possibly occurring during crystal growth. Theseasymmetric nanoparticles had permanent dipole moments,while the symmetric nanocrystals did not. 50 different initialconfigurations were simulated for the symmetric nanoparticlesand 40 for the asymmetric nanoparticles. The total simulationtimes in the two-particle runs ranged between 1.0 and 5.0 ns,which, as previously mentioned, is too short to study the entiresintering process, but is long enough to observe the approach,coalescence, and initial restructuring of the nanocrystals aftercoalescence. A survey of the structural evolution representativeof the sintering dynamics of two symmetric nanoparticles ispresented in Figure 27.Figure 27a shows the initial configuration, where the center-

of-mass separation is ∼8.5 nm. After 100 ps (Figure 27b), theparticles have already adopted the relative orientation that theyassume upon aggregation. Aggregation begins when an (001)surface of one particle contacts the edge between two (101)surfaces of another particle (Figure 27c). The sintering ismediated by oxygen atoms on the edge between two (101)facets, which initially contacts one of the under-coordinatedtitanium atoms on the (001) surface of another particle, toform a “hinge” that joins the two nanoparticles at/near their

Figure 27. Snapshots of the aggregation of two large (3774 atoms) symmetric nanocrystals. These figures are taken (a) at the beginning of thesimulation, (b) after 100 ps, (c) after 160 ps, and (d) after 1.0 ns. Oxygen atoms are shown in red (dark) and titanium atoms are shown in white(light). From ref 330. Copyright 2009 American Chemical Society.


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edges, leading to the final arrangement, Figure 27d. In a few ofthe considered cases, interaction occurred by sintering via two(101) surfaces. The authors pointed at electrostatic inter-actions, generated by surface under-coordinated atoms at thenanoparticles edges, as being responsible of the observeddirectional preference for aggregation, while long-rangeinteractions were considered negligible. Further simulationsperformed for the asymmetric nanoparticles revealed a similaraggregation mechanism as found for the symmetric nano-particles, leading the authors to exclude a major role of dipole−dipole interparticle interactions in driving the sintering process,at variance with what reported in ref 328.

5.3. Electronic Structure of TiO2 Nanocrystals

While the thermodynamic approaches presented in section 5.1introduce electronic degrees of freedom into nonatomisticsimulations through the DFT-calculated surface and tensionenergies, and the MD results of section 5.2 have an atomisticresolution but are based on a model potential, it is only by anelectronic structure, possibly first-principles, simulation of TiO2nanocrystals that an accurate picture of the interplay betweenthe structural and electronic factors underlying the nanocrystalsproperties can be derived. As previously mentioned, the mainissue with electronic structure calculations is their unfavorablescaling (typically N3 or worse) with the number of electrons inthe system, which substantially limits the size amenable to first-principles simulations. Ab initio molecular dynamics are alsolimited by the time span which can be simulated, typically ofthe order of 10−20 ps. A static description of relatively large

clusters is thus nowadays possible, while to our knowledge noab initio MD simulation has been performed to extend theresults of classical MD discussed in section 5.2. Also, onlyrecently the study of sintered large nanocrystals has beenapproached by (semiempirical) electronic structure methods.Due to the large interest in TiO2 nanoparticles and

nanocrystals several semiempirical and molecular orbital studieswere reported starting in the early 1990s, see, e.g., refs331−333. Here, we restrict to the most recent studies, withemphasis on those which seem more relevant to technologicalapplications. Numerous studies on TiO2 nanocrystals have beenreported by Persson, Lunell, and co-workers. These authorsemployed general bonding principles to predict the structure ofindividual anatase nanocrystals.334 Among the selected criteria,they considered that the semi-ionic character of the bonding inTiO2 requires high coordination, balanced charge (i.e., no orvanishingly small dipole moment), and charge neutralitydistribution to be accomplished simultaneously, withoutresorting to artificial termination by embedding or saturation.Specifically, clusters with all oxygen atoms coordinated to atleast two titanium atoms, and all titanium atoms coordinated toat least four oxygen atoms, were considered. After generatingbare (TiO2) clusters with sizes of ca. 2 nm (n up to 68) basedon these criteria, Persson, Lunell and co-workers335 investigatedtheir structural and electronic properties by DFT calculations,Figure 28a. Despite these structures might not correspond tothe global minima for a given n, they represent suitable modelsof extended TiO2 surfaces that have been used, e.g., to study

Figure 28. Left: Geometries of (TiO2)n cluster models, n = 16, 28, 38, 46, 60, and 68. The starting structure (left), the PW/VSZ optimized geometry(middle), and side and top view of the B3LYP/VDZ optimized geometry (right). Right: DOS plots including the top of the valence band, thefundamental bandgap, and the bottom of the conduction band regions for the (TiO2)n nanocrystals with n = 16, 28, 38, 46, 60, and 68. Results fromB3LYP/VDZ//B3LYP/VDZ calculations. From ref 335. Copyright 2000 American Chemical Society.


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dye-sensitized interfaces.226 The electronic structures ofoptimized 1−2 nm nanoparticles, showed well developedband structures with essentially no electronic bandgap defectstates, Figure 28b. In all cases, the anatase crystal form waslargely intact with at most a few Ti−O bonds broken and a fewdefect sites formed.Another interesting aspect concerns the convergence of the

electronic properties with increasing cluster size.335 Thecalculated band edges were found to vary by less than 1 eVas a function of size, independent of the computational method.At the same time, the density of states (DOS) plots displaygradually emerging quasi-continuous valence and conductionbands with no apparent defect states in the band gap.According to the B3LYP/VDZ calculations, the calculatedband gap decreases monotonically from 5 to 4.6 eV when the

size is increased from n=16 to n=60, which roughlycorresponds to a doubling of the nanocrystal size. The lowestcalculated TDDFT excitations follow the trend in theHOMO−LUMO energy gaps but are consistently lower byca. 1 eV.An extensive search for global minimum structures of small

(n ≤ 15) TiO2 clusters was reported by Hamad et al.,336 whoemployed a combination of simulated annealing and MonteCarlo simulations, together with genetic algorithm techniques,with the energy calculated by means of an interatomic potential.A subsequent recalculation of selected structures by DFTallowed the authors to evaluate the accuracy of the modelpotential, which was found to provide results of increasedsimilarity to DFT as n increased. The selected minimumstructures did not retain the anatase structure and they showed

Figure 29. Top: Energy minimized model TiO2 nanoparticles. (A) 1 nm anatase, (B) 2 nm anatase, (C) 3 nm anatase, (D) 1 nm rutile, (E) 2 nmrutile, (F) 3 nm rutile. (Blue) titanium; (red) oxygen. Retention of the apical, three-coordinated Ti atoms on the 1 nm anatase particle was necessaryto preserve electrical neutrality. Bottom: Total DFT calculated surface energies of model 1, 2, and 3 nm anatase and rutile particles. Orangerepresents the contribution from the sum of the particle’s constituent crystallographic surfaces. Yellow represents the remainder, which is attributedto edges and defects. From ref 322. Copyright 2009 American Chemical Society.


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a preference for a central octahedron surrounded by a layer of4- and 5-fold coordinated Ti atoms.Barnard, Zapol, and co-workers337 employed DFT and

semiempirical calculations to investigate the structural andelectronic properties of a series of stoichiometric (TiO2)nnanocrystals, with n up to 455, characterized by the typicalbipyramidal shape. The employed semiempirical tight-bindingmethod delivered calculated surface energies of comparablequality to DFT results, allowing the authors to study largernanostructures. For the smaller n=35 nanocrystal, a sizablestructural relaxation compared to bulk anatase was obtained byDFT, which involved a contraction of all the bond lengths,including those in the center of the nanoparticle. The changesin the surface structure of the nanoparticle were dominated bythe outward relaxation of the O atoms, and the inwardrelaxation of the Ti atoms, which created a more rippledsurface. Not surprisingly, the relaxation energy and theassociated structural and electronic changes were found todecrease with increasing n.As pointed out by Qu et al.,338 a common feature of the

optimized nanoparticle structures discussed above,334−336

including larger stoichiometric bipyramidal nanocrystals,337 isthat they have at least two defective TiO groups at theirsurface. Because this may represent an artifact and becauseholes tend to localize at TiO defects, the use of these clustersfor studies of the photoactivity of TiO2 may be limited. Qu etal.338 thus proposed a new set of (TiO2)n structures, with n =10−16, that did not contain terminal TiO defects. It wasfound that the clusters tended to form sphere- or rodlikecompact structures, exhibiting some odd−even oscillations inthe band gap, with no convergence in the considered n range.Iacomino et al.339 performed DFT calculations on a

stoichiometric anatase (TiO2)29 cluster to investigate thedependence of the structural properties on the surfacechemistry. The cluster showed four dangling oxygen atomson the (001) surfaces. Adsorption of both atomic hydrogen andof dissociated water molecules on the nanocluster surfaceprevented a pronounced reconstruction of the surface, thusensuring a better crystalline organization of the atoms withrespect to the bare system. The calculated formation energiesalso showed surface hydration to lead to the most stablenanocrystal, in agreement with the experimental finding thatthe truncated bipyramidal morphology is typical of amoderately acidic environment. A band gap opening wasobserved for both the bare and hydrated nanocrystal withrespect to the bulk, while hydrogen coverage or oxygendesorption from the bare nanocrystal was found to introduceoccupied electronic states below the conduction levels.

Interestingly, for the bare and hydrated nanocrystal, theLUMO was located on the central portion of the nanocrystal,similar to what was found later for larger systems.317

Hummer et al.322 used DFT calculations combined with insitu X-ray diffraction to analyze the factors contributing to theanatase to rutile phase conversion. They investigatedstoichiometric anatase and rutile (TiO2)n nanoparticles with nup to 272 and 209, respectively, and compared the surfaceenergies calculated for the whole nanoparticles to thosecalculated using their surface slab models, see Figure 29.These calculations revealed a significant difference between thetotal calculated particle surface energies and the summedenergies of the constituent faces, which for small anataseparticles amounted to 63% of the total surface energy. Thedifferences were attributed to the contributions of thenanoparticle edges and corners, which led to an inversion ofthe relative stabilities of anatase and rutile when the surfacecontributions dominate, as is the case with nanoparticles.Including both surface and bulk terms, the transition fromnanoanatase to nanorutile was predicted at particles size slightlylarger than 3 nm, while without their inclusion a much slowerconvergence to phase equilibrium is predicted, in line with, e.g.,ref 320.An important contribution to the atomic level understanding

of the working mechanisms of TiO2 nanoparticles in photo-catalysis was given by Li and Liu,316 who performed extensiveDFT calculations on a series of TiO2 anatase nanoparticles withdifferent sizes and shapes, and quantitatively correlated theparticle size and shape with the photocatalytic activity of theoxygen evolution reaction. The authors focused their attentionon the rate-determining first proton removal step in the oxygenevolution reaction on anatase particles, while simulating thesurrounding water environment by a continuum solvationmodel. They considered 12 different (TiO2)n nanoparticles, ofdifferent size (n = 58−449) and different (flat or sharp)truncated-bipyramidal shape, with the under-coordinatedsurface atoms saturated by dissociated water molecules, seeFigure 30.Their results show that the HOMOs of the nanoparticles are

generally at lower energy compared to those of the extendedsurfaces, while the LUMOs are at similar energy for reasonablylarge particles. Importantly, the results of Li and Liu316 revealthat for nanoparticles larger than 2 nm, both the band gap andthe HOMO/LUMO energies converge rapidly toward those ofthe extended (101) and (001) surfaces, implying that thequantum size effect may only be significant in very small TiO2anatase particles, e.g., <2 nm. Moreover, while regular trendsare present for particles with similar shape, e.g., a band gap

Figure 30. Anatase decahedral nanoparticles models: (a) illustration of the sharp (left) and the flat (right) particles; (b) the geometrical parametersutilized in Wulff construction. The shape is determined by the ratio S = d[001]/d[101]. The calculated equilibrium S value in aqueous solution is 1.07,implying considerably flatter particles than those predicted in vacuo (S = 1.73), confirming that the shape of anatase nanoparticles could be sensitiveto the synthetic conditions. From ref 316. Coyright 2011 American Chemical Society.


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decrease with increasing size, the change in the band structurewith the nanoparticles shape is much less obvious. Thecalculated carrier effective masses showed the same trend asthe band gap and generally decreased by increasing the particlesize, indicating that the larger TiO2 particle would possess ahigher carrier mobility.By using oxygen evolving reaction as the probing reaction, Li

and Liu further revealed that the free energy change of the rate-determining step is sensitive to the particle shape andconcluded that the sharp nanoparticles possessed higherphotoactivity for the following reasons:

(i) The HOMO and LUMO are spatially separated in thesharp crystals, which is not the case in the flat crystals, seerepresentative examples in Figure 31. The HOMO−LUMOspatial separation is beneficial to the electron−hole pairseparation at the early stage of photoexcitation process, whichreduces the chance for the immediate recombination.(ii) The sharp crystals are dominated by {101} facets, and

these can promote the photoreduction reaction, in turnexpediting the consumption of photoelectrons and thus furtherreducing the chance of electron−hole recombination.

Figure 31. 3D isosurface contour plots of the frontier (HOMO and LUMO) orbitals of the four nanoparticles. (a) (TiO2)449, −2.25; (b) (TiO2)429,−1.82; (c) (TiO2)387, −1.36; and (d) (TiO2)315, −0.90. From ref 316. Copyright 2011 American Chemical Society.

Figure 32. Left: Converged geometry of reduced TiO2 cluster in continuum water, and the MO of excess electron is shown as the red and blueisocontour surfaces. Right: Converged geometry of TiO2 cluster with surface absorbed Li+ cation and also an excess electron. From ref 340.Copyright 2012 American Chemical Society.


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(iii) The OH production is the rate-determining step, andthe OH surface adsorption energy can be systematicallyimproved in the sharp nanoparticles, where more active siteare present.Overall, the authors concluded that very small oxide

nanoparticles do not possess the highest photoactivity, whichis realized in rather larger (∼15−20 nm) particles. This is quitedifferent from the conventional heterogeneous catalysis onmetals, where the higher activity is often observed on smallnanoparticles that contain a high concentration of low-coordinated surface defected sites.As it can be inferred from the studies above, the localization

of electron and holes in TiO2 nanocrystals plays an importantrole in photocatalytic and photoelectrochemical processes.Zhang et al.340 investigated electron transport and trapping insolvated rutile TiO2 nanoparticles, as models of the processesruling charge transport in dye-sensitized solar cells. Theyconsidered rutile clusters of increasing dimension, the largest ofTi61H116O180 stoichiometry, fully solvated by water molecules,and additionally included the effect of surrounding solvation bya polarizable continuum model. Using a modified B3LYPhybrid DFT functional, they investigated the electronicstructure of the rutile cluster and the localization of an excesselectron with and without a Li+ ion binding or intercalating tothe nanoparticle. Lithium ions are fundamental additives ofDSCs electrolytes which play a very important role in thecharge generation and collection processes. Isodensity plots ofthe excess electron molecular orbital in the bare and Li-boundnanocluster are reported in Figure 32. While the excess electronis predicted to be located at the center of the nanoparticle inthe absence of lithium ions, when Li+ is bound to a surfaceoxygen atoms the excess electron is mainly localized at thetitanium d-orbital closest to the Li+ cation, both in vacuum andin continuum water solvent. Similarly, for bulk-intercalated Li+

the excess electron is localized at the nearest Ti d-orbital,forming a Ti3+ site. Moreover, when Li+ is at or close to thesurface, the corresponding electron state is a relatively shallowtrap in continuum water. Zhang et al.340 further investigated theconduction band shift due to addition of Li+ ions, which is animportant property affecting the dynamics of electron injectionfrom excited dye molecules to TiO2 nanoparticle. An energydownshift of ∼0.07 eV was found for addition of one Li+, whichis comparable to other reports on dye-sensitized interfaces.341

An interesting aspect of the work by Zhang et al. is thesimulation of the lithium ion mediated electron hoppingbetween adjacent Ti sites. The authors explicitly calculated theenergy barriers associated with such hopping events, and foundvalues in the range of 0.12−0.25 eV, closely matchingexperimental values in the range of 0.10−0.27 eV.342

As a final example of electronic structure investigations onTiO2 nanoparticles, Nunzi et al.317 studied the nature ofelectronic trap states in realistic models of individual andsintered anatase TiO2 nanocrystals of ∼3 nm. After verifyingthe impossibility to generate truncated bipyramidal nanocrystalswhich were perfectly crystalline and stoichiometric, theyselected two models, (TiO2)411-H16 (1) and (TiO2)367(2), inwhich either all the under-coordinated dangling oxygen atomson the (001) surfaces were saturated by hydrogen atoms (1), orselected atoms from the (101) surfaces were removed to keepthe cluster neutral and stoichiometric (2). The latter strategy issimilar to that originally developed by Persson et al.334 forsmaller nanoclusters. Upon geometry optimization, the largeststructural distortions of 1 and 2 with respect to the bulk

crystalline structures occurred at the tetracoordinated Ti4+ sites(Ti4c) sites, which rearranged to a distorted tetragonalconfiguration. The two relaxed models showed similar bandgap and electronic Density of States (DOS). In particular, theunoccupied states of lowest energy for 1, of titanium t2gcharacter, are localized within the central part of the NC,mainly at the intersection of the (100) and (101) surfaces, seeFigure 33. We note the different LUMO spatial position with

respect to the nanocrystals by Li and Liu,316 Figure 31, whichwere calculated at a similar level of theory (DFT-GGA), butwere saturated differently; and the similar spatial position (i.e.,the central nanocrystal part) but different localization found forthe water-saturated rutile nanoparticle by Zhang et al.,340 Figure32, calculated using a hybrid functional. These differencessuggest that both saturation and the level of DFT descriptionmay influence the location of surface states in TiO2nanocrystals. In particular, the expected effect of the hybridDFT functional (or of a DFT+U correction) is to lead to morelocalized states.By increasing the energy, the unoccupied states are

progressively more delocalized, with the lowest energy state

Figure 33. Top: Isodensity plot for the LUMO (left) and a higherenergy CB states, at ca. 0.3−0.4 eV above the LUMO (right).Bottom:Contour plot of the space/energy (eV) diagram for the DOS ofunoccupied states of 1, scanned along the length of the NC. ThreeDOS areas are identified, corresponding to the central, intermediateand top/bottom NC regions. The right panel shows a pictorialrepresentation of such space/energy diagram, with colors shifting fromred to green to blue indicating localization in the related coloredregion of states of increasing energy. Redrawn from data in ref 317.


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completely delocalized over the NC structure being found at∼0.3−0.4 eV above the LUMO (Figure 33). To provide aquantitative picture of the unoccupied state energy localizationwithin the considered TiO2 NCs, Nunzi et al. reported acontour plot of the space/energy diagram for system 1, Figure33. In line with the qualitative analysis presented above, thisdiagram shows a substantial contribution to the low-energyportion of the DOS from states which are mainly localized onthe central part of the nanocrystal. By increasing the energy, thestates become progressively more delocalized over the entirenanocrystal structure, with the top/bottom (001) facetsmaximally contributing to the high energy portion of theDOS. Similar plots were obtained for 2. The localized surface

states constituting the bottom of the DOS may constitutetrapping sites for electron transport, and may further representrecombination sites between injected electrons and oxidizedspecies in the electrolyte. Surface saturation with watermolecules was found to rise to the energy of both theHOMO and LUMO by ∼0.5 eV, in line with the decrease ofthe work function experimentally observed upon hydration ofTiO2 surfaces,177 and with the results of ref 340. Surface-adsorbed H2O molecules also raised the energy and reducedthe number of localized states at the bottom of the unoccupiedDOS.Nunzi et al.317 further investigated whether the boundaries

between sintered nanoparticles can introduce electronic trap

Figure 34. Optimized geometries for two interacting nanocrystals (model 2) with the 101/101, 101/001, 001/001, and 100/001 interfaces, alongwith the corresponding DOS (curves of different colors) compared to that of the isolated model 2 (red curves) calculated at the DFTB level oftheory (σ = 0.18 eV). Redrawn from data in ref 317.


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states at the bottom of the unoccupied states manifold. Theyconstructed models of sintered nanoparticles by attaching twoTiO2 NCs at their available surfaces. The structures of two NCsof type 2 with 101/101, 101/001, 001/001, and 100/001interfaces, optimized by semiempirical tight-binding DFT, areshown in Figure 34. The computed tight-binding DFT DOS forthe sintered configurations confirms that the shape of the DOStail is not largely affected by the boundaries betweennanoparticles. Although these calculations did not considerthe role of defects, e.g. oxygen vacancies, and for sinterednanocrystals did not include the disorder which is expected tocharacterize the mesoporous TiO2 film formation, these resultspointed at the presence of inherent trap states even in perfectlycrystalline TiO2 nanocrystals, due to the unavoidable presenceof 4-fold coordinated surface Ti(IV) ions.

6. LESS COMMON TIO2 PHASES

In addition to the rutile and anatase forms, more than a dozenTiO2 polymorphs are known. Brookite is the third natural formof TiO2, whereas only traces of the “B” phase are present innature. Many of the other polymorphs (Figure 35) are unstableunder ordinary conditions but can be present in particularconditions and possibly in nanoscale TiO2 samples, either asfree-standing or as supported systems. The synthesis andinvestigation of these less common TiO2 phases have beenlargely motivated by the quest for TiO2-based systems withband gaps narrower than those of rutile and anatase, and

therefore capable to harvest the sun light more efficiently thanthe latter. Although all the less-common TiO2 polymorphs canbe obtained following “soft chemistry” routes, some of them,viz. hollandite, columbite, baddeleyite, pyrite, fluorite, ramsdel-lite, cotunnite, and the so-called OI phase, can also be obtainedby high-pressure conversion of rutile or anatase. Therefore,they are usually referred to as “high-pressure TiO2 phases”.Unlike these high-pressure phases, the lepidocrocite-likepolymorph (section 6.3) has a strong tendency to form single2D sheets. In all the less common TiO2 polymorphs, thecoordination numbers range from 5 to 9 for titanium atoms andfrom 2 to 4 for oxygen atoms, where the highest values areusually present only in high pressure phases.

6.1. Three-Dimensional Systems

Brookite has a large (eight formula units) unit cell and ischaracterized by a distorted octahedral coordination around thecations. Without considering some early studies based on forcefields,343,344 the first theoretical investigation of brookite datesback to 1995.345 In that work, Mo and Ching determined thestructural parameters and the electronic bands by LDAcalculations, and compared them to those of rutile and anatase.Brookite was predicted to have a lower bulk modulus and alarger band gap with respect the other two natural forms.Subsequent hybrid functional calculations, however, found aband gap of 3.11 eV, which is lower than that of anatase.54

TiO2(B), the so-called fourth natural polymorph, has acentrosymmetric monoclinic cell containing eight stoichiomet-

Figure 35. Unit cells of less common TiO2 polymorphs: (a) brookite, I41/amd; (b) B, C/2m; (c) baddeleyte, P21/c; (d) fluorite, Fm3m; (e) spinel(λ-MnO2), Fd-3m; (f) hollandite, I4/m; (g) columbite (a-PbO2), Pbcn; (h) OI, Pbca; (i) pyrite, Pa-3; (j) ramsdellite, Pbnm; and (k) lepidocrocite-like, Cmcm. Red spheres are O atoms, and gray spheres are Ti atoms.


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ric units, and is characterized by the presence of layers linked by2-fold coordinated oxygens. Recent first-principle calculations(using both PBE and B3LYP),346 have revealed that half of theTi ions are not in a 6-fold, but rather in a 5-fold coordinatedenvironment (see Figure 36), which makes them chemically

nonequivalent. In tune with its open structure, the computedbulk modulus of TiO2(B) is lower than those of rutile andanatase, whereas the electronic bandgap (GGA, 2.68 eV;B3LYP, 4.28 eV) is predicted to be indirect and higher withrespect to the other low-pressure TiO2 forms. On the otherhand, the theoretical IR spectrum is very similar to that ofanatase, which makes these phases difficult to distinguish withvibrational spectroscopy.Somewhat surprisingly, studies on the brookite and B phases

were preceded by investigations on the more exotic high-pressure TiO2 forms. Dewhurst and Lowther347 first exploredthe mechanical properties of some of these phases by LDAcalculations. They found the sequence fluorite > baddeleyite(MI) ≈ columbite (II) > rutile > anatase for the bulk moduli, inline with the high-pressure nature of the first four phases.Different and conflicting results were reported in subsequentLDA calculations: Milman348 found a low bulk modulus forcolumbite, whereas a low B0 value was computed by Sasaki forbaddeleyite.349 Dubrovinskaia et al.350 studied the stability ofthe various polymorphs as a function of pressure by linearizedmuffin tin LDA calculations. They found that the hypotheticalpyrite structure is never stable, while rutile → columbite →baddeleyite transitions occur by raising the pressure. By furtherincreasing the pressure, an orthorhombic structure (the so-called OI phase) becomes stable, and finally this converts to thecotunnite(OII) structure. Whereas the latter was alreadyknown, OI was a new structure, which was studiedexperimentally and theoretically for the first time.The above results were largely confirmed by Harrison and

co-workers,28 who examined the effects of pressure for a largenumber of polymorphs using both GGA and Hartree−Fockcalculations. They predicted the sequence anatase < rutile <columbite < baddeleyite < fluorite < cotunnite for the bulkmoduli (the OI phase was not considered). Furthermore, byexamining enthalpy curves, they found that the anatase →columbite transition occurs at lower pressure (3.5 GPa) withrespect to the rutile→ columbite one (21 GPa), while thecolumbite → baddeleyite → cotunnite transitions occur at 31and 63 GPa, respectively. No stability range was found for thepyrite and fluorite cubic phases. However, these polymorphs

could have interesting solar energy applications, due to theirnarrow band gaps. Using augmented plane wave (APW) LDAcalculations, Mattesini et al.351 found indeed an indirect(direct) bandgap of 1.04 eV (1.79 eV) for the fluoritepolymorph, which is considerably lower than the band gapcomputed with the same approach for rutile, 1.88 eV (direct).The results are less promising for pyrite, for which thecomputed indirect and direct band gaps are 1.44 and 1.81 eV,respectively. These findings were confirmed by Kuo et al, whoexamined the rutile, anatase, columbite, baddeleyite, cotunnite,pyrite, and fluorite phases with plane-wave LDA calculations.352

6.1.1. Point Defects and Doping. Only a fewinvestigations have been devoted to the theoretical modelingof impurities and point defects in the less-common TiO2phases. Pan et al.353 performed GGA-DFT calculations toinvestigate the formation of common defects, such as Tiinterstitials, Ti vacancies, and O vacancies in brookite andcompared the results to analogous calculations for rutile andanatase. They found that the formation of the most stabledefects, i.e., Ti interstitials and O vacancies, is slightly morefavorable in brookite, suggesting that this polymorph may havea higher photocatalytic activity.Because of its use in rechargeable lithium ion batteries,

lithium is perhaps the most studied impurity in TiO2. Lithiationof less-common TiO2 phases was studied in a number ofpapers. By studying Li intercalation in rutile, Koudriachova355

discovered a new hexagonal phase, occurring for x > 0.75,which may have a role in electrode degradation. The sameauthor investigated also a ramsdellite-structured LiTiO2phase,356 and made a comparative study of Li intercalation inthe rutile, anatase, brookite, hollandite, ramsdellite, and spinel(Li0.5TiO2) phases.357 In the latter work, it was found thatintercalation properties are dominated by the strength of thecoupling between the electronic and the structural degrees offreedom. In an earlier comparative study of the rutile,ramsdellite, orthorhombic (imma) and spinel-type LiTiO2phases based on periodic Hartree−Fock calculations, Mack-rodt358 predicted an orthorhombic > spinel > ramsdellite >rutile stability order. Spinel-type structures have been studiedalso by Wagemaker using GGA-DFT calculations and thecluster expansion method.359 Whereas for x < 0.5 the Li ionsoccupy tetrahedral 8a sites, for 0.5 < x < 1 the Li0.5TiO2 phasewas found to coexist with a LiTiO2 phase where Li ions are atoctahedral 16c sites. The diffusion barrier for Li in spinel-typeLixTiO2 was recently estimated to be 0.68 eV by Liu et al.360

using the DFT+U approach.Most of the work on TiO2 lithiation has focused on the B

phase. Due to the large channels in its structure, the B phase isindeed ideally suited as a host for interstitial doping. After earlyextended-Huckel calculations by Nuspl et al.,361 several GGA-DFT studies on LixTiO2(B) models have been reported, withrather conflicting results. The energetics of diluted Li atomswas studied Panduwinata and Gale,362 who predicted that Liprefers to bind in a 5-fold coordinated site close to theoctahedral layer (A2 site, see Figure 37), while the mostfavorable diffusion path is along the open channels parallel tothe b axis, and involves A2→ C→A2 jumps (C sites are locatedat the middle of the b-axis channels, and at the center of thesquare planar arrangement of O atoms, see Figure 37). Areversed stability order for the A2 and C sites was found byArrouvel et al.,363 who predicted diffusion to occur along the bdirection through C → C jumps. In a subsequent work, whereDFT calculations were combined with powder neutron

Figure 36. Local coordination around the Ti ions in TiO2(B). The“Ti1” ion is in a 5-fold coordination. Reproduced with permissionfrom ref 346. Copyright 2009 American Institute of Physics.


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diffraction, Armstrong et al.364 identified three structures, viz.Li0.25TiO2, Li0.5TiO2, and LixTiO2 (x = 0.80.9). In the low-concentration case, Li was confirmed to prefer C sites. Forintermediate concentrations, the C site is unfavorable, and A1sites are occupied. Finally, at the highest Li concentration, A1and A2 sites turn out to be equally occupied. Different resultswere found by Dalton et al. using DFT calculations inconjunction with Monte Carlo simulations.354 As in ref 364, thesite occupation changes with x, but in a different way: for x <0.5, Li ions are predicted to occupy A1 sites on different planesalong the c-axis direction. At x = 0.75, an inversion of thestability order occurs, that causes A2 and C sites to bepreferred, so that, at x = 1.0, all the A2 and C sites areultimately occupied.Lithium incorporation through the TiO2(B) (001) surface

was investigated by Koudriachova.357 The (001) surface is themost stable one for TiO2(B) (see section 6.2), and is alsoexposed by nanowires, which have a cylindrical shape elongatedalong the b direction. Li ions are predicted to find a low-energydiffusion path bringing them from the A1 surface sites to bulk-type A2 sites (assumed to be most stable). This mechanismimplies that each Li has its own independent diffusion channel,which explains the good pseudocapacitive behavior of TiO2(B)nanowires with respect to Li ions. Site occupations wererecently studied with DFT+U calculations by Dylla et al.365 For3D systems the predicted stability sequence is A2 > C > A1 inthe dilute limit, whereas an alternating A1/A2 occupation isfavored at higher x. A stability sequence C > A2 > A1 wasinstead predicted for nanosheets, where the stability inversionof the A2 and C sites is related to a reduction of the Li+-Li+

interactions, due to lattice relaxations.6.2. Surfaces

The (010) and (001) surfaces of the brookite polymorph werefirst investigated with molecular dynamics simulations,366 whichrevealed that on both surfaces Ti ions in tetrahedralcoordination are present. In a subsequent, more detailed,DFT study,367 numerous surfaces were investigated, specifically(001), (111), (121), (011), (101), (210), (010), and (100),and the relationships with the surfaces of rutile and anatasewere discussed. On the basis of the stability of all the examinedsurfaces, the equilibrium shape of a brookite crystal wasdetermined through the Wulff construction (see Figure 38),from which the authors concluded that the average surfaceenergy of brookite is higher than that of anatase. The (210)surface, which is the most exposed one for brookite, is made ofthe same building blocks of the anatase (101) surface.Comparative investigations of the two surfaces indicate thatthe differences in the arrangement of the structural units

introduce specific reactions sites, which make the brookite(210) surface significantly more reactive than anatase (101).368

Classical molecular dynamics studies on the adsorption ofseveral proteins found a weaker adsorption on brookite than onanatase.369 For CO2, the interaction with brookite (210) wasfound similar to that with anatase (101).370 The surface→ CO2electron transfer is negligible, indicating that brookite may notbe a suitable photocatalyst for CO2 reduction. On the otherhand, the brookite (210) surface is an interesting support forgold nanoparticles in CO oxidation catalysis.371 In fact, brookitefavors a higher dispersion of gold clusters, which in turn have apeculiar ability to switch from 2D to 3D configurations. Thisreduces the coordination number of Au atoms, making themsuitable for strong adsorption and activation of CO and O2molecules.The structure and stability of the surfaces of the B

polymorph were first investigated by Vittadini et al. usingDFT-GGA calculations.372 Due to the presence of 2-fold Oanions and 5-fold Ti cations already in the bulk phase, thestructure of the TiO2−B surfaces is not straightforward todescribe. The (001) surface is by far the most stable, inagreement with experimental results of TiO2 heteroepitaxy,showing that the TiO2(B) phase grows under the form of(001)-oriented flat islands.373,374 The (001) surface exhibitstwo kinds of 5-fold Ti ions, one of which is not under-coordinated with respect to the bulk structure, while no trueundercoordinated O ion is exposed. This makes the surfaceparticularly stable. On the (100) surface, instead, there are oneundercoordinated Ti-5c ion and two O-2c ions, of which onlyone is genuinely undercoordinated. A similar situation is foundalso on the (110) surface.The equilibrium shape of TiO2(B)was computed under both

dry and wet conditions.372,375 As shown in Figure 39, the shapeis more elongated in the latter case. On the (001) surface, waterstabilizes the type-II termination, which is metastable in a dryenvironment, by converting the oxo ions into hydroxyls. Wateradsorbs dissociatively on the (100) surface,375,376 whereas thenature of the interaction with water is less definite for the othersurfaces, where a coverage-dependent behavior or a mixedmolecular/dissociative adsorption is generally found.375 Thehigh reactivity of the (100) surface toward water dissociationhas stimulated a number of studies aimed at clarifying the fateof prototypical molecules interacting with this surface.Methanol was found to dissociate on both the clean and thehydroxylated surface.377 Most interestingly, terminal OHgroups present on the hydroxylated (100) surface are able toextract protons from methanol. Adsorption structures and

Figure 37. Interstitial sites for Li interstitials in TiO2(B). Reprintedwith permission from ref 354. Copyright 2012 American ChemicalSociety. Figure 38. Equilibrium crystal shape of the brookite polymorphs as

obtained through the Wulff construction. Reproduced with permissionfrom ref 367. Copyright 2007 American Physical Society.


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energies for CO on the (100) hydroxylated surface were foundto be similar to those computed for other TiO2 surfaces.378

Surprisingly, while a CO → surface charge donation waspredicted to occur, the authors claimed that the interactionmainly involves the 5π* state.The interaction of formaldehyde with the clean and partially

hydroxylated (100) surfaces was studied by Liu et al.379 Threemain adsorption configurations were identified, see Figure 40.Two of these are dioxomethylene-like structures, where a trueundercoordinated O (a) and an intrinsic O-2c ion (b) isinvolved, respectively. The third case is a configuration wherethe adsorbate is singly coordinated to Ti-5c ion (c). The lowreactivity of the type-I termination of the (001) surface375 wasconfirmed in a comparative study of the adsorption of NH3 onthe (001) and (100) surfaces.380 Stimulated by the observationthat the TiO2(B) phase is selectively formed when the synthesisis carried out using ethylene glycol (EG) as solvent, Xiang etal.381 studied the adsorption of EG on the high-energy (010)surface.372 They found that EG interacts strongly with TiO2−B(010), forming a dissociatively adsorbed bidentate structure.As a consequence, the stability of the TiO2−B(010) surface isdrastically lowered, which explains the stabilization of the Bpolymorph in EG.The low-index surfaces of the columbite phase have been

investigated by Kuo et al. with LDA calculations.382 The surfaceenergy decreases in the order (100) > (001) > (110) > (010).This trend has been explained on the basis of the nature anddensity of undercoordinated Ti ion on the different surfaces. Inparticular, the (010) surface has the lowest surface energy, dueto the exposure of highly symmetrical TiO4 species.

6.3. Two-Dimensional Systems: Nanolayers, Nanosheets,and Films

In most of the current literature, “nanolayer” (NL), “nano-sheet” (NS), and film are used as interchangeable terms. In thisreview we use the term film to indicate supported layers, whilethe term nanosheet is used to indicate nanolayers of intrinsichigh thermodynamic stability. While nanosheets can be in somecases synthesized by a bottom-up approach, mostly assupported films, they are usually obtained by exfoliation oflayered titanates. Similarly to the more famous graphene, TiO2nanosheets are not only interesting for fundamental reasons,but also because they are ideally suited as building blocks forthe fabrication of nanoarchitectures with a wide range ofapplications.383

6.3.1. Nanosheets. We have already mentioned that thelepidocrocite-like phase tends to form highly stable nanosheets.Lepidocrocite-like NSs are formed by a core of four atomiclayers of alternating 4-fold coordinated O atoms and 6-foldcoordinated Ti atoms. This core layer is terminated at bothsurfaces by an atomic layer of 2-fold coordinated oxygens. Asshown in the following, lepidocrocite-like NSs are structurallyrelated to anatase (001) nanolayers. The first theoreticalinvestigation on lepidocrocite-TiO2 was performed by Sato etal. using DFT-GGA calculations.384 A stoichiometric unit of anisolated NS was found to be only 0.4 eV less stable than theanatase polymorph, whereas an hypothetical 3D stack wasfound to be 0.053 eV higher in energy, and thus unstable. Thepossible role of dispersion forces in stabilizing lepidocrocite-likestacks was later investigated by Forrer and Vittadini.385 Thestacking energy was found to be ∼9 kJ/mol, which isconsiderably lower than in typical layered oxides such asV2O5. This explains why lepidocrocite-TiO2 stacks are noteasily obtained as V2O5 ones are. The GGA study of Sato etal.384 predicted also a band gap of 3.15 eV, to be compared toband gap values of 2.28 eV for rutile and 2.67 eV for anataseobtained with the same approach. The wider band gap of thelepidocrocite-like phase should be attributed to the lowdimensionality of the system. No dispersion was observed inthe band structure along the stacking direction, while very smalldifferences occurred between the DOS of single- and stackedlepidcrocite NSs (see Figure 41), in agreement with thecomputed energetics. Recently, GW calculations have shownthe excitonic nature of optical transitions in the lepidocrocite-like phase, with a small blue shift when going from single NSsto NS stacks.386

While investigating the properties of Pt-supported TiO2nanolayers, Orzali et al.387 realized that 2 ML anatase (001)systems are unstable and spontaneously interconvert tolepidocrocite NS by a barrierless mechanism which involves

Figure 39. Equilibrium crystal shapes of the B polymorphs as obtainedthrough the Wulff construction: (a) dry conditions; (b) wetconditions. Reproduced with permission from ref 375. Copyright2010 The Royal Society of Chemistry.

Figure 40. Possible adsorption configurations for CH2O on TiO2-B(100). Reproduced with permission from ref 379. Copyright 2013 the PCCPOwner Societies.


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the relative sliding of the upper and lower half of the NS, seeFigure 42. Interestingly, similar transformations were observed

in simulations starting from anatase-derived nanoribbon modelsof various size.388 Motivated by the close relationship betweenthe anatase and lepidocrocite nanolayers, Vittadini and Casarinsystematically investigated the energetics of anatase nanolayersof increasing thickness oriented along the main low-indexdirections.389 They found that for 2D nanosystems the moststable polymorph is not anatase, as commonly assumed fornanoscale TiO2, but lepidocrocite-like TiO2. Furthermore, thelepidocrocite/anatase interconversion is not the only sponta-neous transformation occurring for TiO2 nanolayers. Whilethere is a general and expected tendency toward higher stabilitywith increasing thickness, several irregularities are present. Inparticular, 4 ML-thick layers oriented along (101) and (100)show an anomalous stability. For 2 ML (001) layers relaxationleads to a completely different structure where all the Ti ionsare 5-fold coordinated (see Figure 43). This structure, whichwas originally called “pentacoordinated nanosheet” in ref 173,was later identified as a TiO2−B (001) 2 ML system.374

In the above investigation, Vittadini and Casarin389 did notconsider 1 ML (101) layers, whose unrelaxed structure consistsof noninterconnected clusters. Evarestov et al.390 showedhowever that, when optimized, a (101) layer gives rise to adenser and remarkably stable film of hexagonal symmetry,consisting of one layer of 6-fold coordinated Ti ionssandwiched between two layers of 3-fold coordinated oxygens(see Figure 44). As pointed out by Evarestov et al., this NS canbe considered as originating from the fluorite (111) surface.390

6.3.2. Adsorption. Adsorption of water on lepidocrocite-like NSs was studied by Casarin et al.391 Here, we examine onlythe results of symmetric adsorption at both NS surfaces, sinceasymmetric adsorption leads to the formation of nanotubes,which will be examined in section 6.4. For 1/3 ML and 1/2 MLwater coverage, only molecular adsorption occurs when thelepidocrocite-type structure is maintained. Increasing thecoverage has the effect of strengthening intermolecularinteractions. By forcing water dissociation, thermodynamicallyfavored stepped tri- and dititanates are formed, whicheventually form ABA-type stacks.The adsorption and diffusion of gold atoms on 2 ML TiO2−

B NSs, used as models of films grown on Pt(110), were studiedin ref 373. Gold atoms prefer to adsorb on top of the oxygens,but binding energies are weak (0.30 eV) and diffusion barriersvery low (0.050.06 eV). By contrast, binding energies anddiffusion barriers are large, 2.96 and 2.4 eV, respectively, on the“kagome” reduced phase, which explains why experimentallyclusters are more easily formed on reduced films.

6.3.3. Point Defects and Doping. Experimentallylepidocrocite-like NSs are obtained by delamination of alkalititanates, so they can easily incorporate metal dopants. Co-doped lepidocrocite NSs were studied by Osada et al. withLSDA calculations in conjunction with X-ray adsorption finestructure spectra and magnetic circular dichroism measure-ments.392 Their calculations show that substitutional Co ionsgive rise to a ferromagnetic electronic structure, characterizedby electron transfer to the Co 3d states. Titanium vacancies inlepidocrocite NSs were investigated by Ohwada et al. bytransmission electron microscopy (TEM) and DFT calcula-tions.393 These authors found that Ti vacancies are likely to giverise to larger defects, where two neighboring bridging oxygensare also missing. The energetics of oxygen vacancies wasstudied by Vittadini et al.394 Their computed formation energyof a neutral vacancy is in the range 5.05.5 eV, which is

Figure 41. Density of states of lepidocrocite-like single and stackednanosheets. Reproduced with permission from ref 384. Copyright2003 American Chemical Society.

Figure 42. Total energy curve, showing the barrierless conversion of a2 ML anatase nanolayer to lepidocrocite-like nanosheet. Replottedfrom the original data of ref 173.

Figure 43. Total energy curve, showing the activated conversion of a 4ML TiO2-anatase (101) nanolayer to a 2 ML TiO2-B (001) nanosheet.Replotted from the original data of ref 173.


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slightly larger than the values found for the rutile and anatasebulk phases.6.3.4. Supported Films. The first theoretical study of a

supported-TiO2 film, by Jennison et al.,395 concerned aPt(111)-supported film of TiO composition, which was takenas a model for TiOx-encapsulated Pt clusters observed onTiO2(110). One of the main results of that investigation wasthat metal−supported reduced films of TiO composition preferto arrange with Ti atoms at the metal/film interface.395 Zigzagfeatures experimentally observed by STM were attributed tothe O-layer strain, which induces an abrupt change from hcp tofcc arrangements of Ti atoms. More reduced TiOx phases weresubsequently obtained and theoretically investigated in a seriesof papers by Granozzi, Fortunelli and co-workers396−403 (seeref 404 for a recent review). In contrast to reduced films,oxygen atoms are present at the interface of metal−supportedTiO2 stoichiometric films. Studies of stoichiometric 2 ML-thickTiO2 films grown on (1 × 2)-Pt(110)387,388 show that theexperimental structure can be reproduced by assuming aslightly distorted lepidocrocite sheet. Experimentally, a (14 ×4) reconstruction is observed, which is dominated by darkstripes. Calculations show that these stripes are related to thepresence of an almost perfect coincidence between 14 substratecells and 13 lepidocrocite overlayer units along thelepidocrocite short b-direction. This leads to differences inthe local coordination of oxygens at the interface, whichgradually change from on-top to bridging along the b direction.Oxygens occupying bridge positions have a lower height withrespect to O atoms at on-top sites, and this difference leads tothe corrugation shown by the STM images. Similarlepidocrocite-type films on Pt(111) were found to becharacterized by a low surface-overlayer interaction, which isin agreement with the absence of features in the STM images ofthe film.405 A third type of supported lepidocrocite film wasobserved by Atrei et al.406 on the Ag(100) surface. In this case,a (5 × 1) reconstruction occurs along the long a-direction ofthe lepidocrocite-like film. However, a coincidence between thesubstrate and the overlayer along the short lepidocrocite b-axiswas found to be unlikely by comparing the energy required forthe film deformation with the adhesion energy. Thus, it wasconcluded that the film is composed by large patches of

unstrained lepidocrocite TiO2, interrupted by small patcheswhere the oxide layer matches the periodicity of the substrate.Due to their multilayer nature, TiO2(B) films grown on a Pt

substrate were simply modeled as unsupported TiO2(B)films.373 Hexagonal, fluorite-type films were grown onCu(100) by Atrei et al.407 DFT calculations indicate that thefilm strain due to the c(2 × 6) coincidence is more thancompensated by the interaction with the substrate. Further-more, analysis of the DFT results shows that the TiO2 filmloses its insulator character and the electron states close to theFermi level are dominated by the contribution from the Ti 3dorbitals.

6.4. One-Dimensional Systems: Nanotubes

The first theoretical investigation on TiO2 nanotubes wascarried out with DFTB, a DFT tight binding method, byEnyashin and Seifert.408 Nanotubes were constructed eitherfrom bilayers with the lepidocrocite-like structure (i) or frommonolayers of hexagonal structure (ii), which were defined asanatase-type, but were actually (111)-oriented monolayers offluorite-type TiO2, since monolayers of (101)-oriented anatase-TiO2 have been shown to convert spontaneously to (111)-oriented fluorite-TiO12.

390 For (ii), both single-wall nanotubesand nanorolls were examined (see Figure 45a,b), while for (i)single-wall structures made of the (n,0) and (0,n) type wereconsidered (see Figure 45c,d). The lepidocrocite-like nano-tubes, and in particular those of the (n,0) type, were found tobe unfavorable, due to their thickness. By contrast, nanotubesmade of the hexagonal monolayers were computed to be stableagainst nanostrips of the same size, suggesting that they couldrepresent viable models for real TiO2 nanotubes. As far as theelectronic structure is concerned, all the lepidocrocite nano-tubes were found to have a ∼4.5 eV indirect band gap, whereasfor those made of the hexagonal monolayer the gap is directand slightly smaller (4.2 eV).As previously pointed out, lepidocrocite-like and (001)-

oriented anatase bilayers are closely related, as the former isobtained from the latter through a nonactivated transformation,whereas dissociative adsorption of water inverts the process391

(see below). Given the large difference between the a latticeconstants of anatase and lepidocrocite, it was suggested thatasymmetric water adsorption could give rise to a spontaneouscurvature of the film, with the consequent formation of a

Figure 44. Structure (left) and PBE0 density of states (right) of 1 ML fluorite-type nanosheets. Reproduced with permission from ref 390. Copyright2011 Elsevier Science.


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nanotube. This possibility was proven using ribbon models, i.e.,models where the periodic boundary conditions are suppressedalong two directions. By extrapolating the results from suchmodels, nanotubes with an average diameter of 27 Å carryinghydroxyls only on the external surface were predicted toform.391 Interestingly, spontaneous partial dehydration of theinternal surface was observed to take place in DFTBsimulations of the structure of single wall NTs made oftrititanic acids by Enyashin and Ivanovskii.409 The same authorsextended their study to NTs formed by another family ofprotonated TiO2 films, i.e., metatitanic acid.410

Recently, the implementation of rotohelical symmetryconditions in the CRYSTAL code has finally offered thepossibility to carry out accurate density functional calculationson realistic nanotube models. Using this approach, Bandura etal.412 studied single wall (SW) nanotubes based on an anatase(101) ML (note that a TiO2 ML is made of three atomiclayers). With the same approach, Szieberth et al. studied SWnanotubes made of two lepidocrocite MLs.413 Their resultsconfirm that the (0,n) chirality is favored over the (n,0) onebecause of the lower strain. Furthermore, for diameters up to25 Å, a substantial reconstruction is observed both for the (0,n)and for the (n,0) nanotubes. The hyperpolarizabilities of theabove-described NTs were recently studied with the coupledperturbed Kohn−Sham approach.414

The strain properties of lepidocrocite NTs change drasticallywhen the thickness increases from 2 to 3 ML.415 If the thirdTiO2 ML is added on top of the film, a more anatase-likesurface is formed, while the structure of the bottom surface stillresembles the lepidocrocite structure. Furthermore, the O-bridges, which determine the direction of the surface stress, aremutually perpendicular on the top and bottom surfaces.Because of these structural differences, the bottom surface issubject to a large stress along <10>, while the top layer issubject to a lower strain along <01>. This effect, which issimilar to that predicted for asymmetrically hydroxylatedlepidocrocite bilayers,391 does not show up when the nanolayer

is studied with conventional periodic boundary conditions.389

As a consequence of the above-described structural strains onthe top and bottom surfaces, (n,0) nanotubes made of 3 ML ofanatase (001) have a negative strain (see Figure 46), i.e., theirformation is thermodynamically favored.

In another study, Ferrari and co-workers re-examined thecase of dititanate nanotubes.413 Several geometries were takeninto consideration i.e., DT1, where the NT derives from theplain rollup of a stepped H2Ti2O5 NS; DT2A, where water isdesorbed from the internal surface, while the external one issimilar to that of DT1; DT2B, which differs from DT2Abecause the external OH groups interact via H-bonds; DT2C,where the structure is turned to an asymmetrically hydroxylatedanatase/lepidocrocite one, similar to the one examined in ref391. Calculations indicate that asymmetric DT2X NTs(particularly those of DT2A type) have a lower strain, whiletheir strain vs diameter curve has a minimum, i.e., theyspontaneously tend to assume a curved shape. However, DT1NTs are thermodynamically more stable, and this is true alsowhen they are compared to lepidocrocite NTs. As for theelectronic properties, the band gap of dititanate NTs arepredicted to be larger (by ∼1 eV) with respect to lepidocrocite/anatase NTs.NTs based on 1-ML hexagonal fluorite (111) sheets have

been studied by Evarestov and co-workers.390,412 These authorsperformed a comparative study of SW NTs obtained from 1-ML fluorite with NTs obtained from 2-ML anatase (101) films,which are the thinnest sheets maintaining the rectangularstructure of anatase (101).389 Although the 1 ML fluorite sheetsare more stable than the 2 ML anatase sheets by 0.25 eV/TiO2unit, the NTs obtained from the latter are generally morestable, and have lower strain energies, at least for diameterssmaller than ∼25 Å. On the other hand, NTs made of thickeranatase sheets are found to be unstable. Concerning theelectronic structure, the band gaps of anatase-type NTs arelarger (up to 0.5 eV) with respect to the fluorite-type ones.Fluorite-based double-wall (DW) NTs (NTin@NTout)with

(n,n) “armchair” and (n,0) “zigzag” structures were alsostudied.416 The stability of the DW NTs was found to dependon the difference between the NT radii (ΔRNT), which fixes theinterwall distance, and on the diameters of the internal (Din)and external (Dout) NTs. The optimal configurations are (6,6)@(12,12) and (10,0)@(20,0), which correspond to ΔRNTvalues in the 4.55.0 Å range. Systems with smaller valuesfor ΔRNT or Din are unstable, whereas systems with larger

Figure 45. Structures of some TiO2 nanotubes: (a) a (20,0) NT and(b) a (16,16) NT, both obtained from fluorite-type nanosheets; (c) a(0,28) NT and (d) a (28,0) NT, both obtained from lepidocrocite-likenanosheets. Adapted with permission from ref 411. Copyright 2011The Royal Chemical Society.

Figure 46. Strain energy (Es) of TiO2 anatase(001) 3 ML nanotubesversus the nanotube diameter D [Å]. Reproduced with permissionfrom ref 415. Copyright 2011 American Chemical Society.


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values for ΔRNT or Dout correspond to quasi-independent pairsof NTs. Furthermore, zigzag configurations, and in particularthose with an inverse-stacking structures are most stable. In theelectronic structure, the strong interwall interactions cause thebandgaps of DW NTs to be considerably smaller than those ofSW NTs.In comparison to nanotubes, very little work has been done

on other 1D forms of less-common TiO2 polymorphs, such asnanoribbons (NRs). He et al.415 reported GGA calculations ofthe energetics and electronic properties of hexagonal fluorite-type NRs. Both armchair- (AR) and zigzag-terminated (ZZ)NRs were considered, and in the ZZ case both O- and Ti-terminated structures were included. As expected, Ti-terminated NRs are unstable, and reconstruct to an O-terminated superstructure where short and long O-bridges arepresent. Among O-terminated NRs, ZZ structures, particularlythose with an even number of Ti cations, are most stable (seeFigure 47). Band gaps for fluorite-type NRs, estimated from

GGA+U calculations, range from 2.93 to 4.04 eV, and are direct(indirect) for AR (ZZ) configurations. Furthermore, band gapsare higher for even NRs. Reduced ZZ NRs were studied, and Oions close to the edges were found to be most easily removed.Complete removal of one line of Ti ions brings to theformation of ferromagnetic NR.

7. CONCLUDING REMARKSIntense experimental and theoretical research efforts have led toconsiderable progress in the understanding of the bulk andsurface properties of anatase TiO2 crystals and nanocrystalsover the past decade. Nevertheless, there remain areas ofconsiderable debate, even at the very fundamental level, as wellas areas that are still largely unexplored. Starting from the bulk,the description of the bulk electronic structure and opticalproperties of anatase is still unsettled since even advancedtheoretical techniques often tend to overestimate thefundamental band gap (e.g., hybrid functionals and GWmethods) or underestimate the optical one (e.g., the BSEmethod). As pointed out in the Introduction, knowingaccurately the band gap is important because this determinesthe optical absorption, which has an essential role in theperformance of photocatalytic devices. An accurate determi-

nation of the band gap would also provide the basis forquantitatively predicting the energies of trap states, defect andimpurity levels, which are crucial for the design of TiO2materials with improved properties. Therefore, it is importantto clearly identify which are the aspects that are not taken intoaccount with satisfactory accuracy by the current theoreticalapproaches.Also not fully understood are the electron−phonon coupling

and self-trapping phenomena. The characteristics of theelectron and hole polaronic states (i.e., energy levels, radii,and effective masses), and how these change going, e.g., fromphotoexcited TiO2 to doped or reduced TiO2, are still underdebate. In the current theoretical studies a major problem isthat the size and degree of localization of the electron polarondepends critically on the fraction of exact exchange in hybridfunctional calculations or the value of U in DFT+U studies.The situation is less critical for the hole states, which aregenerally found to be strongly localized. In any case, additionalwork would be desirable, also from the experimental side, inorder to better understand the characteristics of electron andhole states as well as the connection between transport andspectroscopic measurements. This is especially important forpolaronic states at surfaces, because the energies of these statescan affect the reducing and oxidizing powers of electrons andholes in photocatalysis.69,417

There has been a lot of interest in assessing the relativephotocatalytic activities of different anatase surfaces in the lastseveral years. The expectation that the (001) surface could beextremely reactive stimulated intense experimental effortsaimed at synthesizing particles exposing a substantial amountof (001) facets.418,419 Recent reports, however, have been morecontradictory,146,420,421 pointing to the need for morecomprehensive fundamental investigations where both thesurface structure, including surface and subsurface defects, aswell as the interface with water (or, possibly, aqueous solution),are correctly taken into account. In this context, an importantfundamental problem is the description of the level alignmentbetween the TiO2 band edges and the relevant energy levels,e.g., the HOMO and the LUMO, of the adsorbate, notablywater, see, e.g., ref 278 and references therein. Also in this case,approaches beyond standard local and semilocal DFT,particularly hybrid functionals, have proven essential for acorrect description.188,278,417

Overall, the main issue to be addressed in the computationalcharacterization of TiO2-adsorbate interfaces, as used, e.g., inhybrid photovoltaics, is the ability to couple the accuracy oftypical quantum chemical methods for ground and excited stateproperties, with the large dimensions and complexity of theinvestigated systems. Most hybrid DFT methods seem toaccommodate such challenging accuracy issue, but at the cost ofa higher computational overhead. Furthermore, the inclusion ofsolvation effects is also important and is usually accounted in asimple and effective way by means of continuum solvationmodels. These approaches, however, are missing some of therelevant specific solute/solvent information, especially when thesolvent is also a reactant, as it is the case in the water splittingreaction. In these cases ab initio molecular dynamics can offeran important simulation tool but its application is limited inboth space and time scales. It seems likely that the future,grand-challenges in the simulations of TiO2 nanomaterials andtheir interfaces will be dominated by multiscale approaches,combining various levels of accuracy on different space/timescales. In this respect, the use of DFT-based tight-binging

Figure 47. Cleavage energies for fluorite-type nanoribbons witharmchair and zigzag configuration as a function of the NR width.Numbers are the lines of Ti ions present in the structure. Reproducedwith permission from ref 415. Copyright 2010 American ChemicalSociety.


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schemes may offer a reasonable alternative to conventionalDFT algorithms in terms of price/performance and encourag-ing results have been obtained in the specific simulation ofTiO2 structures.Finally, much needs still to be done for better characterizing

and understanding less common TiO2 phases. The availabletheoretical studies on these phases have focused almostexclusively on the bulk electronic properties, whereas theirsurface properties remain largely unexplored. Low-dimensionalforms like lepidocrocite-like TiO2, on the other hand, have beenmostly investigated for their intrinsic structural and electronicaspects, but little insight has been obtained on their catalyticactivity. Further fundamental insights are clearly desirable, bothfor the intrinsic scientific interest of these materials, and fortheir potential to lead to new or improved technologicalapplications.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

Author Contributions

The manuscript was written through contributions of allauthors. All authors have given approval to the final version ofthe manuscript. All authors contributed equally to thismanuscript.

Notes

The authors declare no competing financial interest.

Biographies

Filippo De Angelis is senior research scientist and deputy director atthe CNR Institute of Molecular Sciences and Technology, in Perugia,Italy. He is the founder and coleader of the Computational Laboratoryfor Hybrid/Organic Photovoltaics in Perugia. He earned a B.S. inChemistry 1996 and a Ph.D. in Theoretical Inorganic Chemistry in1999, both from the University of Perugia, Italy. He is an expert in thedevelopment and application of quantum chemical methods to thestudy of the structural, electronic, and optical properties of complexsystems including transition metals. His main research interest are inthe field of dye-sensitized and perovskite solar cells, employing first-principles computational methods to predict and interpret theproperties of new and existing materials. First-principles simulationsare also employed to investigate the active heterointerfacesconstituting hybrid/organic photovoltaic devices. He is the 2007recipient of the Raffaello Nasini Gold Medal of the InorganicChemistry Division of the Italian Chemical Society.

Cristiana Di Valentin received her Master’s degree in Chemistry from

the University of Pavia in 1997, and her Ph.D. degree jointly from the

University of Pavia and the Technische Universitat Munchen in 2000.

She was appointed as Assistant Professor at the University of Milano

Bicocca in 2002, and as Associate Professor of General and Inorganic

Chemistry in 2012. She has been a visiting scientist at Technische

Universitat Munchen, Princeton University, Universitat de Barcelona,

and Ecole Nationale Superieure de Paris. Her research interests range

from ab initio computational study of reaction mechanisms in organic

chemistry and homogeneous catalysis to heterogeneous catalysis,

photocatalysis, doped and defective semiconducting oxides, chemically

modified graphene, and carbon based materials for fuel cells.

Simona Fantacci is research scientist at the CNR Institute of Molecular

Sciences and Technology, in Perugia, Italy. She is cofounder of the

Computational Laboratory for Hybrid/Organic Photovoltaics

(CLHYO) in Perugia. She received a Ph.D. in Theoretical Inorganic

Chemistry from the University of Perugia in 1999 and was research

associate at Princeton University (USA) in 2002−2003. Her research

activity concerns the theoretical investigation of excited state

properties of transition metal complexes by means of DFT and

TDDFT methods, with focus on the linear and nonlinear optical

properties of polypiridyl complexes of Ru(II) and Ir(III) employed in

the fields of dye-sensitized solar cells, OLED devices and NLO

materials. Recently, she has worked on the modeling of the

photophysical properties and degradation processes of materials

relevant to the Cultural Heritage.


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mailto:[email protected]




Andrea Vittadini received his “laurea” degree (cum laude) inChemistry from the University of Padua in 1983. After a period oftraining as a Research Fellow of the Italian National Research Council(CNR), in 1988 he joined the CNR Istituto di Chimica e TecnologieInorganiche e dei Materiali Avanzati as a Research Scientist. In 1998he was appointed Senior Research Scientist, and moved to theUniversity of Padova unit of the CNR Institute of Molecular Sciencesand Technologies (now a part of the Institute for Energetics andInterphases). His expertise concerns the modeling of complexchemical systems by both quantum chemical and solid-state theoreticaltechniques. His current research interests include metal oxide surfaces,nanostructures metal oxides, functionalization of metal and metaloxide surfaces, and molecular self-assembly.

Annabella Selloni graduated in physics at the University “La Sapienza”(Roma, Italy), and received her Ph.D. degree from the Swiss FederalInstitute of Technology (Lausanne, Switzerland) in 1979. After apostdoc at the IBM- T.J. Watson research center in Yorktown Heights,she held positions at the University “La Sapienza”, at the InternationalSchool for Advanced Studies (Trieste, Italy), and at the University ofGeneva (Switzerland). In 1999 she joined the Department ofChemistry of Princeton University, where in 2008 she became theDavid B. Jones Professor of Chemistry. She has coauthored ∼250publications, mostly in the fields of surface physics and chemistry. Hercurrent research interests are mainly focused on metal oxide materials,surfaces and interfaces, photocatalysis, and photovoltaics.

ACKNOWLEDGMENTS

We are pleased to thank Ulrike Diebold, Francesca Nunzi, andGianfranco Pacchioni for many helpful discussions. Weacknowledge support from the following agencies: FP7 projectESCORT, for F.D.A. and S.F.; FIRB Project RBAP115AYN ofItalian MIUR, and LISA Project LI01p_VISFOTOCAT ofCINECA, for C.D.V.; PRIN-2010BNZ3F2, project DES-

CARTES, of the Italian Ministry of the University andResearch, for A.V.; DoE-BES, Division of Chemical Sciences,Geosciences and Biosciences under Award DE-FG02-12ER16286, for A.S.

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theoretical studies on anatase and less common tio 2 phases: bulk,...

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