Structure and Stability of TiO2-B Surfaces: A Density Functional Study

Andrea Vittadini,*,† Maurizio Casarin,† and Annabella Selloni‡

Istituto di Scienze e Tecnologie Molecolari del CNR (CNR-ISTM) and CR-INSTM “Village”, Dipartimento diScienze Chimiche, UniVersita di PadoVa, Via Marzolo 1, I-35131 PadoVa, Italy, and Chemistry Department,Princeton UniVersity, Princeton, New Jersey 08540

ReceiVed: July 30, 2009; ReVised Manuscript ReceiVed: September 28, 2009

We investigate the structure and energetics of low-index surfaces of the TiO2-B polymorph by means ofperiodic density functional theory calculations within the generalized gradient approximation. The bulk structurecontains two nonequivalent Ti ions, one of them exhibiting an octahedral coordination, while the other issquare-pyramidal. When exposed at the surface, these two types of ions display different relaxation schemes,which ultimately tend to make them more similar. On the basis of the computed surface energies and of theWulff construction, we predict for TiO2-B a pseudohexagonal prismatic equilibrium shape and an averagesurface energy practically identical to that of TiO2-anatase.

TiO2-B is a titanium dioxide form first synthesized byMarchand in 19801 and successively identified also in naturalsamples.2 It has been thus proposed to represent the fourthnatural polymorph of TiO2, in addition to the well-known rutile,anatase, and brookite phases. Its structure, which can bedescribed as composed of edge- and corner-linked pairs of edge-sharing octahedra, is in fact generated from the anatase structureby regular shear parallel to (103) of anatase, which is theobserved orientation of the interface between the two minerals.2

Both TiO2-anatase and TiO2-B are able to replace topotacticallyperovskite (CaTiO3).3 TiO2-B has been also proposed to be thebasic constituent of titania nanotubes (TNTs)4 and has beenidentified as the final product of TNT decomposition.5,6 Fur-thermore, its open structure, characterized by large and continu-ous channels, makes it well suited as a host for intercalation.7

This has motivated great interest in the use of TiO2-B for energystorage applications, particularly as electrode material forrechargeable Li ion batteries.8,9 Despite these interesting features,theoretical studies of TiO2-B have been so far scarce: only veryrecently density functional theory (DFT) calculations have beenperformed on bulk TiO2-B, to understand its structural andvibrational properties,10 and the intercalation of Li ions.11 Surfaceproperties have not been investigated yet even though they areessential for understanding the reactivity of this TiO2 polymorph.

In this paper we focus on the low-index surfaces of TiO2-B,examine their structures and stabilities by means of DFTcalculations, and also compare some of our findings withprevious work on the more common rutile, anatase, and brookitepolymorphs. We performed the calculations by using thePerdew-Burke-Ernzerhof12 (PBE) gradient-corrected exchange-correlation functional as well as the pseudopotential plane-wave(PP-PW) method as implemented in the PWSCF code of theQUANTUM-ESPRESSO package.13 We used Vanderbilt ultra-soft pseudopotentials14 including 2s and 2p shells for O and 3s,

3p, 3d, and 4s states for Ti. The plane-wave kinetic energy cutoffwas set to 25 and 200 Ry for the smooth part of the wavefunctions and for the augmented density, respectively. Thiscomputational setup is the same as that adopted in previouscalculations of TiO2 surfaces15,16 so that a meaningful compari-son of the properties of the various TiO2 polymorphs can bemade.

Structure and Stability of Bulk TiO2-B. TiO2-B is the leastdense of the four natural polymorphs of TiO2.3 It has amonoclinic structure with 8 (4) TiO2 units in the conventional(primitive) unit cell and two nonequivalent Ti sites. Weoptimized the lattice constants and the internal parameters ofthe TiO2-B bulk phase by using the BFGS algorithm and theprocedure proposed by Bernasconi et al.17 with A ) 50 Ry, E0

) 30 Ry, and σ ) 4 Ry to correct for finite basis-set effects,We sampled the Brillouin zone (BZ) of the conventionalmonoclinic unit cell with a 2 × 4 × 2 k-point mesh. Theconvergence threshold for the residual atomic forces was 0.03eV Å-1, while the pressure tolerance in the cell optimizationwas 50 bar. The optimized constants (a ) 12.297 Å, b ) 3.755Å, c ) 6.625 Å, � ) 107.0°) and the bond distances (see Figure1 and Table 1) are in very good agreement with the most recentXRD data10 [a ) 12.197(8) Å, b ) 3.7537(15) Å, c ) 6.6535(4)Å, � ) 107.16(8)°] and differ by ∼0.1% with respect toprojector-augmented planewave (PAW) DFT calculations byYahia et al.,10 while slightly larger differences are found withrespect to the PP-PW-DFT calculations of Panduwinata andGale.11

The framework structure of TiO2-B (Figure 1) is usuallydescribed as a stack of (011)-oriented sheets consisting of doublelayers of edge-sharing TiO2 octahedra, the sheets being con-nected through the corners of the facing octahedra. In a recentstudy,10 however, it was found that one of the Ti-O distancesexceeds the usual 2.20s2.25 Å range of single Ti-O bonds,and it was proposed that the environment of the correspondingTi atom is distorted square-pyramidal rather than octahedral.Our calculations (see Figure 1 and Table 1) confirm this finding:the optimized Ti*(1)...O(2) distance is 2.359 Å, so that the

* Address correspondence to this author. E-mail: andrea.vittadini@unipd.it.† CNR-ISTM, CR-INSTM, and Universita di Padova.‡ Princeton University.

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2009, 113, 18973–18977

structure of TiO2-B is indeed better described as made ofoctahedra (Ti ions) and square pyramids (Ti* ions). The differentstereochemistries of the Ti and Ti* ions can be rationalized byconsidering that a (hypothetical) Ti*(1)-O(2) bond would bringto the formation of 4-fold coordinated O ions (see Figure 1),which are not observed in any of the naturally occurring TiO2

phases.Previous DFT calculations on the relative stabilities of the

rutile, anatase, and brookite TiO2 polymorphs found anatase asthe most stable form, followed by brookite and finally byrutile,16,18,19 an ordering inversely correlated with the densityof the different phases. This ordering, however, is opposite tothat found in experiment, indicating the limitations of currentlyavailable DFT functionals in predicting the relative stability ofdifferent metal oxide structures.15,19 The present calculationsconfirm this trend: TiO2-B is found to be even more stable thananatase and thus the most stable TiO2 polymorph, its enthalpybeing 13.0, 7.3, and 2.6 kJ mol-1 lower than those of rutile,brookite, and anatase, respectively. Although no precise ther-modynamic data are available yet for TiO2-B, this finding isclearly in contrast with the available experimental evidence and

with the low abundance of TiO2-B relative to the other TiO2

polymorphs.20

Structure and Stability of TiO2-B Surfaces. There is a largebody of evidence indicating that, despite the above-mentionedlimitations in describing the phase stability of TiO2, current DFTapproaches provide an overall satisfactory description of itssurface properties, notably of the structure and energetics ofTiO2 stoichiometric surfaces.15,16,21 In the next paragraphs, wethus use DFT-PBE calculations to investigate the structure andstability of several stoichiometric low-index surfaces which wereselected on the basis of qualitative scrutiny, where the densityand the type of the exposed ions were considered.

The TiO2-B surfaces were represented with periodicallyrepeated slab models, using an ∼12 Å vacuum space to decouplethe surfaces. All the atoms were allowed to relax. Because eachslab exposes two equivalent surfaces, the surface energy γ wasevaluated by the formula:

where A is the area of the surface, n is the number of TiO2

stoichiometric units contained in the slab, Eslab is the total energyof the slab model, and Ebulk is the total energy of a single TiO2

stoichiometric unit as obtained from the calculations on bulkTiO2-B. The main results for all the investigated surfaces aresummarized in Table 2.

Structure of the (001) Surface. The (001) surface wasmodeled with a nonprimitive orthorhombic cell (size: 3.7546Å × 12.2974 Å × 31.3640 Å), while the BZ was sampled witha 4 × 2 × 1 grid.

There are two possible terminations for the (001) surface,hereafter indicated as “(001)-I” and “(001)-II” (see Figure 2).The presence of unstable titanyl groups makes the type-IItermination less favorable with respect to the type-I termination(1.11 vs 0.40 J m-2). If we exclude the lepidocrocite form, whichis intrinsically two-dimensional in nature, the (001)-I surfaceactually appears to be one of the most stable TiO2 surfaces, its

Figure 1. Equilibrium structure of bulk TiO2-B. The gray lines showthe boundaries of the conventional monoclinic cell. Equivalent atomsare marked with a prime or a double prime. Two cells are shown, witha polyhedral (top) and a ball-and-stick (bottom) representation. In thisand in the following figures, red and gray spheres correspond to O andTi atoms, respectively.

TABLE 1: Optimized Theoretical and Experimental BondDistances (Å) for Bulk TiO2-Ba

Bond (bond type) PP-PW-DFTb PAW-DFTc XRDc

Ti*(1)-O(1) (AS) 1.777 1.780 1.78(2)Ti*(1)....O(2) (AL) 2.359 2.364 2.29(9)Ti*(1)-O(2)′ (×2) (E/B) 1.960 1.965 1.960(3)Ti*(1)-O(4) (E/B) 2.039 2.037 2.04(3)Ti*(1)-O(3) (E/B) 1.920 1.926 1.87(6)Ti(2)-O(1)′ (A) 1.862 1.867 1.85(2)Ti(2)-O(3) (E) 2.175 2.144 2.10(4)Ti(2)-O(4)′ (×2) (E) 1.938 1.942 1.94(0)Ti(2)-O(2) (E) 2.157 2.152 2.11(3)Ti(2)-O(3)′ (A) 1.860 1.868 1.850(7)

a AS, AL, E, and B stand for “axial short”, “axial long”,“equatorial”, and “basal”, respectively. Labels are defined inFigure 1. b This work. c From ref 10.

TABLE 2: Computed Surface Energies γ for the Unrelaxed/Relaxed Low-Index Surfaces of TiO2 (B) and TheirFractional Abundance

γ (J m-2)

Surface Unrelaxed Relaxed Abundance (fraction)

(001)-I 0.54 0.40 0.45(001)-II 2.02 1.11 0(100) 1.65 0.76 0.19(010) 1.48 0.69 0(110) 1.25 0.61 0.36

γ ) [Eslab - nEbulk]/2A

Figure 2. Structure of the relaxed TiO2-B (001) surface with (a) type-Itermination and (b) type-II termination. (001)⊥ indicates the directionnormal to the (001) surface.

18974 J. Phys. Chem. C, Vol. 113, No. 44, 2009 Letters

formation energy being similar to those of the rutile (110) andanatase (101) surfaces.15,21

The (001)-I surface (see Figure 2a) exposes two types of5-fold (5c) coordinated cations, Ti(1) and Ti*(2), one 2-fold(2c) coordinated anion, O(1), and two 3-fold (3c) coordinatedanions, O(2) and O(3). Due to their different bulk environments,the two exposed 5c cations relax in opposite directions (seeTable 3): Ti(1) undergoes an inward relaxation, while Ti*(2)slightly relaxes outward by 0.01 Å. Similarly, the O(3) ions,which are basal ligands of Ti*(2), stay almost unperturbed,whereas the O(2) ions, equatorial ligands of Ti(1), strongly relaxoutward. Even though the surface O-3c ions usually favor ansp3-type pyramidal configuration, their relaxation appears to belargely driven by the tendency to compensate for the reducedcoordination of the surface cations. The changes in bond lengthconfirm the opposite behavior of Ti(1) and Ti*(2): whereas theTi(1)sO(4) bond is shortened (from 1.862 to 1.822 Å), alengthening occurs for Ti(2)sO(5) (from 1.777 to 1.789 Å).Overall, the stereochemistry differences between Ti(1) andTi*(2) cations are found to be reduced at the surface.

It is interesting to note that the thinnest model for the (001)-Isurface (obtained by truncating the bottom surface of the slabshown in Figure 2a with a plane bisecting the Ti(5)sO(7) andTi(4)sO(9) bonds) is almost identical to the “pentacoordinatednanosheet” found in a recent study of unsupported TiO2-anatase(101) thin films and predicted to be the most stable TiO2

nanosheet after lepidocrocite.22 The only differences betweenthe “pentacoordinated nanosheet” and a regular TiO2-B (001)slab are a slight contraction of the lattice constants [11.887 ×3.677 Å to be compared with the 12.297 × 3.755 Å size of thesurface cell] and the higher symmetry exhibited by thenanosheet, where all first layer cations are equivalent.

As a final remark to this section, we note that the relaxationof the inner ions of the (001)-I slab in Figure 2 is negligible,which is once more indicative of the intrinsic stability of thissurface.

Structure of the (100) Surface. The (100) surface wasmodeled with an orthorhombic cell (size: 3.7546 Å × 6.625 Å× 31.000 Å), while the BZ was sampled with a 4 × 2 × 1grid. This surface exposes two 5c-Ti, Ti(1) and Ti*(2); two 2c-O, O(1) and O(2), and two 3c-O, O(3) and O(4) (see Figure 3).Of these ions, only Ti(1) and O(1) are truly undercoordinated.Interestingly, the first layers of this surface are isostructural witha single sheet of “step-2” titanate, whose layered structure wehave recently shown to be related to that of protonated-TiO2

nanotubes.23 This could provide a natural explanation to the fact

that the final product of the decomposition of TiO2 nanotubesis TiO2-B. Incidentally, step-2 titanates were also proposed asintermediates in the transformation of step-4 titanates to TiO2-B.24 Furthermore, given the close relationship between layeredtitanates and the TiO2-anatase (001) surface,23,25 it follows thatthe TiO2-B(100) surface can be also related to the anatase (001)one.

Analysis of the surface relaxations (see Table 4) shows thatthe ion displacements at the TiO2-B (100) surface are strongerand extend to deeper layers than those at the (001) surface. Thisis in line with the large difference in stability between the twosurfaces (see Table 2). Similarly to the (001)-I surface, the Ti(1)-5c cation strongly relaxes inward, whereas Ti*(2) relaxesoutward by a smaller amount. The surface O-2c ions, however,show a behavior very different from that on TiO2-B (001)-I.On TiO2-B (100), there is a large modification of theTi*(2)′sO(1)sTi(1) and Ti(1)sO(2)sTi*(2) angles, whichchange from 110.2° to 115.6° and from 168.2° to 142.8°,respectively, indicating a preference for an sp2-type hybridizationin both cases (in the bulk, the almost linear coordination of O(2),generally very unfavorable, is stabilized by packing and long-range effects). This suggests the presence of surface Ti-O πbonds, which is confirmed by the values of the Ti(1)sO(1) andTi*(2)sO(2) distances, 1.811 and 1.766 Å, respectively.

Structure of the (010) Surface. The (010) surface wasmodeled with a nonprimitive monoclinic cell (size: 12.2974 Å× 6.625 Å × 30.000 Å, γ ) 107.0°), while the BZ was sampledwith a 2 × 2 × 1 grid. The relaxed surface is shown in Figure4. Note that, because of the slab symmetry, two sets ofequivalent atoms are present in the slab, e.g. O(1) is equivalentto O(1)′.

This surface exposes a large variety of ions; taking intoaccount the symmetry, these are as follows: one 4c cation,

TABLE 3: Atomic Displacements (in Å, fromBulk-Terminated Positions) of the (001)-I Surfacea

Direction

Label [010] [100] (001)⊥

Ti(1) 0.00 -0.01 -0.04Ti*(2) 0.00 +0.06 +0.01Ti*(3) 0.00 +0.01 -0.02Ti(4) 0.00 +0.01 +0.02O(1) 0.00 +0.01 -0.01O(2) 0.00 -0.02 +0.24O(3) 0.00 -0.03 -0.03O(4) 0.00 +0.02 0.00O(5) 0.00 -0.01 -0.01O(6) 0.00 0.00 +0.01O(7) 0.00 0.00 0.00O(8) 0.00 0.00 -0.02

a Labels are shown in Figure 2a.

Figure 3. Structure of the relaxed TiO2-B (100) surface. The (100)⊥symbol indicates the direction normal to the (100) surface.

TABLE 4: Atomic Displacements (in Å, fromBulk-Terminated Positions) of the (100) Surfacea

Direction

Label [010] [001] (100)⊥

Ti(1) 0.00 +0.16 -0.33Ti*(2) 0.00 +0.09 +0.22Ti*(3) 0.00 +0.10 -0.20Ti(4) 0.00 -0.09 +0.10O(1) 0.00 -0.29 -0.12O(2) 0.00 +0.02 +0.35O(3) 0.00 -0.13 -0.24O(4) 0.00 +0.09 +0.10O(5) 0.00 -0.16 -0.21O(6) 0.00 +0.04 +0.15O(7) 0.00 -0.04 -0.25O(8) 0.00 +0.06 -0.03

a Labels are shown in Figure 3.

Letters J. Phys. Chem. C, Vol. 113, No. 44, 2009 18975

Ti*(1); two 5c cations, Ti(2), Ti*(4); one 6c cation, Ti(3); three2c anions, O(1), O(3), and O(8); and three 3c anions, O(2), O(4),and O(6). These ions are arranged to approximately form twolayers, where Ti*(1), Ti(2), O(1), O(2), and O(3) belong to theouter one. We can see clear structural analogies between TiO2-B(010) and the anatase TiO2(100) surface. On TiO2-B (010), the4c-Ti*(1) and 5c-Ti(2) ions undergo displacements of similaramounts, which is justified by the fact that they are both trulyundercoordinated ions. Furthermore, as on the anatase (100)surface, the 3c O(2) and O(4) ions undergo a large outwardrelaxation (see Table 5), in line with the strong preference fora pyramidal configuration of 3c-O atoms. By contrast, both the2c O(1) and O(3) ions, which show weak horizontal displace-ments, strongly relax along [001] so as to strengthen theTi*(1)-O(1) and Ti(2)-O(3) bonds (from 2.039 to 1.864 Åand from 2.157 to 1.947 Å, respectively). This compensatesfor the undercoordination of the corresponding cations. Simi-larly, also the “backbonds” of the undercoordinated Ti*(1) andTi(2) cations become shorter: the Ti*(1)-O(5) and Ti(2)-O(7)distances are reduced from 1.960 to 1.829 Å and from 1.938 to1.793 Å, respectively. Considering that the Ti*(1)-O(4)′distance, which corresponds to the axial bond in the bulkstructure, is substantially lengthened (from 1.777 to 1.837 Å),it appears that the surface once again relaxes making thecoordination environment of the two types of exposed cationsmuch more similar than they are in the bulk.

On a larger scale, we find that the TiO2-B (010) relaxationyields a sort of coupling of the ion layers, so that there is atightening of the bonds between the first and the second layer,whereas the interactions between the second and the third layerare weakened.

Structure of the (110) Surface. The (110) surface wasmodeled with a monoclinic cell (size: 6.625 Å × 6.429 Å ×

30.000 Å, γ ) 106.24°), while the BZ was sampled with a 2 ×2 × 1 grid. The structure of the relaxed surface is shown inFigure 5, whereas the atomic relaxations are reported in Table6.

The unrelaxed surface exposes the following ions: one 4ccation, Ti*(1); two 5c cations, Ti(2) and Ti*(4); one 6c cation,Ti(3); four 2c anions, O(1), O(2), O(3), O(6); and two 3c anions,O(4), O(5). The truly undercoordinated ions are Ti*(1), Ti(2),O(1), and O(2). The relaxations of the first two, the pyramidal-type Ti*(1) and octahedral-type Ti(2) cations, are qualitativelysimilar and lead to strong outward relaxations of the O(3) andO(4) anions. Moreover, both undercoordinated cations show astrengthening of the “backbonds” [Ti*(1)-O(7) from 1.960 to1.807 Å and Ti(2)-O(8) from 1.938 to 1.780 Å] and of thebonds with the undercoordinated oxygens [Ti*(1)-O(1) from2.039 to 1.838 Å and Ti(2)-O(2) from 2.157 to 1.928 Å].

Equilibrium Shape of TiO2-B. Figure 6 shows the Wulff26

shape of a TiO2-B crystal, calculated using the surface energiesreported in Table 2. This is a flat pseudohexagonal prism, wherethe bases are {001} facets, while the sides are {100} and {110}/{11j0} facets. Nanocrystals similar to this computed shape have

Figure 4. Structure of the relaxed TiO2-B (010) surface.

TABLE 5: Atomic Displacements (in Å, fromBulk-Terminated Positions) of the (010) Surfacea

Direction

Label [001] [100] (010)

Ti*(1) 0.00 -0.02 -0.11Ti(2) -0.05 -0.02 -0.15Ti(3) -0.02 +0.02 +0.17Ti*(4) 0.00 +0.02 +0.08O(1) -0.16 +0.06 +0.02O(2) 0.00 +0.04 +0.23O(3) -0.18 -0.06 -0.04O(4) +0.06 -0.05 +0.24O(5)′ +0.18 -0.10 -0.02O(6) -0.03 -0.01 +0.09O(7) -0.14 -0.06 -0.08O(8) -0.04 +0.01 +0.08

a Labels are shown in Figure 4.

Figure 5. Structure of the relaxed TiO2-B (110) surface.

TABLE 6: Atomic Displacements (in Å, fromBulk-Terminated Positions) of the (110) Surfacea

Direction

Label [001] [1-10] (110)

Ti*(1) 0.00 -0.06 -0.06Ti(2) -0.02 -0.14 -0.13Ti(3) -0.06 +0.05 +0.23Ti*(4) -0.06 -0.03 +0.08O(1) -0.09 +0.17 +0.04O(2) +0.03 +0.14 +0.11O(3) +0.08 +0.02 +0.30O(4) -0.05 +0.07 +0.32O(5) +0.02 +0.04 +0.02O(6) -0.05 0.00 -0.11O(7) +0.20 +0.19 -0.04O(8) -0.14 +0.06 -0.07

a Labels are shown in Figure 5.

Figure 6. Equilibrium shape of a macroscopic TiO2-B crystal, usingthe Wulff construction and the surface energies of Table 2.

18976 J. Phys. Chem. C, Vol. 113, No. 44, 2009 Letters

been actually observed for TiO2-B grown on Pt(111) substrates.27

From the areas of the prism facets we can compute the fractionalabundance of the various exposed surfaces (see Table 2). Byweighting the surface energies of the different facets by theirfractional abundance we can determine the average surfaceenergy, for which we find ⟨γ⟩ ) 0.54 J m-2, a value almostidentical to that previously found for anatase (using the samecomputational setup).15 Considering the exceptionally lowenergy of the TiO2-B(001) surface and the fact that the TiO2-Bpolymorph can be easily obtained in the form of nanosheets,we can understand why on the nanoscale the B polymorph cancompete with anatase.

In summary, we have studied the structure and stability ofthe main low-index surfaces of the TiO2-B polymorph by meansof periodic DFT calculations in the generalized gradientapproximation. Contrary to the experiment, the bulk phase iscomputed to be the most stable among all the TiO2 polymorphs,which confirms the limitations of currently available gradient-corrected functionals. The structure of the TiO2-B polymorphcontains two nonequivalent Ti ions, one of them exhibiting anoctahedral coordination, while the other is square-pyramidal.When exposed at the surface, these two types of ions displaydifferent relaxation patterns, which eventually make them moresimilar. On the basis of the computed surface energies and ofthe Wulff construction, we predict that the TiO2-B polymorphhas a pseudohexagonal-prismatic equilibrium shape and anaverage surface energy practically identical to that of the anatasepolymorph.

Acknowledgment. Computational resources and assistancewere provided by the “Laboratorio Interdipartimentale diChimica Computazionale” (LICC) at the Department of Chem-istry of the University of Padova and by CINECA (Bologna,Italy). Surface models were created and displayed with GDIS.28

The Wulff shape was obtained with the Wulffman code.29 Wethank G. Granozzi for useful discussions.

Supporting Information Available: Extended tables ofatomic relaxations, tables reporting selected bond distances andangles of the relaxed/unrelaxed surfaces, and CIF files containingthe coordinates of the entire optimized slab model. This materialis available free of charge via the Internet at http://pubs.acs.org.

References and Notes

(1) Marchand, R.; Brohan, L.; Tournoux, M. Mater. Res. Bull. 1980,15, 1129–1133.

(2) Banfield, J. F.; Veblen, D. R.; Smith, D. J. Am. Mineral. 1991, 76,343–353.

(3) Banfield, J. F.; Veblen, D. R. Am. Mineral. 1992, 77, 545–557.(4) Armstrong, G.; Armstrong, A. R.; Canales, J.; Bruce, P. G. Chem.

Commun. 2005, 2454–2456.(5) Suzuki, Y.; Yoshikawa, S. J. Mater. Res. 2004, 19, 982–985.(6) Morgado, E.; Jardim, P. M.; Marinkovic, B. A.; Rizzo, F. C.; De

Abreu, M. A. S.; Zotin, J. L.; Araujo, A. S. Nanotechnology 2007, 18,495710.

(7) Tournoux, M.; Marchand, R.; Brohan, L. Prog. Solid State Chem.1986, 17, 33–52.

(8) Armstrong, A. R.; Armstrong, G.; Canales, J.; Bruce, P. G. J. PowerSources 2005, 146, 501–506.

(9) Brousse, T.; Marchand, R.; Taberna, P. L.; Simon, P. J. PowerSources 2006, 158, 571–577.

(10) Yahia, M. B.; Lemoigno, F.; Beuvier, T.; Filhol, J.-S.; Richard-Plouet, M.; Brohan, L.; Doublet, M.-L. J. Chem. Phys. 2009, 130, 204501–11.

(11) Panduwinata, D.; Gale, J. D. J. Mater. Chem. 2009, 19, 3931–3940.

(12) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. ReV. Lett. 1996, 77,3865.

(13) Baroni, S.; dal Corso, A.; de Gironcoli, S.; Giannozzi, P.; Cavazzoni,C.; Ballabio, G.; Scandolo, S.; Chiarotti, G.; Focher, P.; Pasquarello, A.;Laasonen, K, Trave, A.; Car, R.; Marzari, N.; Kokalj, A. http://www.quan-tum-espresso.org/.

(14) Vanderbilt, D. Phys. ReV. B 1990, 41, 7892–7895.(15) Lazzeri, M.; Vittadini, A.; Selloni, A. Phys. ReV. B: Condens. Matter

2001, 63, 155409.(16) Gong, X.-Q.; Selloni, A. Phys. ReV. B: Condens. Matter 2007, 76,

235307–11.(17) Bernasconi, M.; Chiarotti, G. L.; Focher, P.; Scandolo, S.; Tosatti,

E.; Parrinello, M. J. Phys. Chem. Solids 1995, 56, 501–505.(18) Muscat, J.; Swamy, V.; Harrison, N. M. Phys. ReV. B: Condens.

Matter 2002, 65, 224112.(19) Labat, F.; Baranek, P.; Domain, C.; Minot, C.; Adamo, C. J. Chem.

Phys. 2007, 126, 12.(20) In this respect, the LDA functional gives somewhat better results,

as the enthalpy of the B phase is computed to be higher by 1.7, 5.0, and5.0 kJ mol-1 with respect to rutile, brookite, and anatase, respectively. Onthe other hand, the LDA functional brings an underestimation of the latticeconstants (a ) 12.076 Å, b ) 3.699 Å, c ) 6.648 Å, � ) 107.1°).

(21) Labat, F.; Baranek, P.; Adamo, C. J. Chem. Theory Comput. 2008,4, 341–352.

(22) Vittadini, A.; Casarin, M. Theor. Chem. Acc. 2008, 120, 551–556.(23) Casarin, M.; Vittadini, A.; Selloni, A. ACS Nano 2009, 3, 317–

324.(24) Wallenberg, L. R.; Mehri, S.; Arne, A. Microsc. Microanal.

Microstruct. 1990, 1, 357–364.(25) Feist, T. P.; Davies, P. K. J. Solid State Chem. 1992, 101, 275–

295.(26) Wulff, G. Z. Kristallogr. Mineral. 1901, 34, 31.(27) Sedona, F.; Sambi, M.; Artiglia, L.; Rizzi, G. A.; Vittadini, A.;

Fortunelli, A.; Granozzi, G. J. Phys. Chem. C 2008, 112, 3187–3190.(28) Fleming, S.; Rohl, A. Z. Kristallogr. 2005, 220, 580–584.(29) Roosen, A. R.; McCormack, R. P.; Carter, W. C. Comput. Mater.

Sci. 1998, 11, 16–26.

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