Introduction

While the properties of many materials are controlled by their geometries and electronic structures, defects can also play a key role in many of their applications [1,2,3,4,5]. Manipulating the nature and the concentration of defects by defect engineering is thus fundamental, albeit challenging, since this can indeed not only lead to tuned materials properties in a controlled way but also to completely new properties in some cases [6].

The importance of oxygen vacancies (VO) as point defects in many metal oxides [5] has resulted in numerous investigations focusing at elucidating their role both experimentally and theoretically [3]. This is especially true for titanium dioxide (TiO2), where VO rather than Ti interstitials make the dominant contribution to the band gap states of TiO2 surfaces [7]. With applications in various fields such as photocatalysis, solar cells, electronic devices, and environmental cleanup [8,9,10,11,12,13,14,15] for instance, TiO2 is a technologically important material for which VO play an important role in many of these applications. For instance, they were found not only to enhance the visible-light absorption of TiO2 samples [16] but also to influence the adsorption and subsequent reactivity of molecules at TiO2 surfaces [17,18,19], with possible important outcomes in photovoltaics and photocatalysis. Despite the larger thermodynamic stability of TiO2 rutile compared to anatase, TiO2 nanoparticles are common of the anatase type, with a higher efficiency in photocatalysis and photovoltaics than rutile [20,21,22], especially for its most reactive (001) and its most stable (101) surfaces. A better understanding of the role of VO in TiO2 anatase is therefore crucial to better rationalize the defect engineering of such material.

From a modeling viewpoint, investigations dedicated to oxygen-defective TiO2 have mainly been carried out at the density functional theory (DFT) level, using both spin-restricted and -unrestricted formalisms and a variety of DFT models [4, 23,24,25,26,27,28,29,30,31,32,33,34], although more advanced formalisms such as GW [35] have also been recently applied [36, 37]. Focusing on DFT investigations on anatase, a high sensitivity of the results to the DFT model has been evidenced both for the bulk [29] and for the defective (101) surface [34]. Cheng and Selloni showed that VO are more stable subsurface than on the surface of anatase (101) using the generalized gradient approximation (GGA) [23], while a dependence of the relative stabilities of the surface and subsurface VO could later be evidenced at the DFT+U level as a function of the U parameter value for the same surface [38]. The importance and stability of surface VO was later confirmed with the hybrid HSE06 functional [39, 40] by Deák et al. [25] and Ha and Alexandrova [34] for instance, due to a possible interaction of the surface vacancy with a surface polaron [25]. In addition, the experimental spectral signature of reduced TiO2 samples, with localized defect states at about 1.1 eV below the conduction band minimum (CBM) for TiO2 anatase bulk [41] for instance, is not correctly reproduced by standard DFT models due to the overestimation of the delocalization of unpaired electrons and thus an underestimation of d electron localization in such models. A correct description indeed requires the use of either an on-site repulsion potential U with a sufficiently large, and potentially unphysical, value as in DFT+U approaches [38, 42,43,44] or of hybrid functionals [25, 32, 33]. These investigations have however generally considered different U values and/or different DFT models, supercell approaches with different supercell sizes, as well as singlet or triplet spin states, making a straightforward comparison of all computed data often difficult.

In this context, in this article, we present an investigation of stoichiometric and defective TiO2 anatase bulk and (101) surface, considering different hybrid DFT functionals with both effective core pseudopotentials (ECP) and all-electron (AE) atom-centered Gaussian basis sets, along with both singlet and triplet spin states. For the defective (101) surface, nine different VO positions have been considered. In all cases, we focus on local structural polaronic distortions around the VO sites, and electronic structures description, especially regarding the positions of the deep defect levels in the TiO2 band gap and the localizations of the excess electrons upon vacancy creation. This paper is organized as follows: in the “Computational details” section, we first introduce computational details, before presenting and discussing in the “Results and discussion” section the results obtained on stoichiometric and defective TiO2, first for the bulk and then for the (101) surface. Finally, conclusions are drawn in the “Conclusions” section.

Computational details

All calculations have been performed with CRYSTAL [45, 46] which solves self-consistently both the Hartree-Fock and the Kohn-Sham equations as well as hybrid schemes, such as PBE0 [47] and HSE06 [39, 40], by expanding crystalline orbitals as a linear combination of atom-centered Gaussian type orbitals (GTO).

Five exchange-correlation functionals belonging to the GGA and hybrid families have been considered: PBE [48], PBE0 [47], B3LYP [49], B3PW [50], and HSE06 [39, 40]. Additional dispersive interactions have also been taken into account, using the empirical and semi-classical D2 and D3 corrections, as proposed by Grimme et al. [51, 52].

For all calculations, the ECP of Durand and Barthelat [53,54,55] have been combined into (1s1p5d)→[1s1p2d] and (4s4p)→[2s2p] contraction schemes for Ti and O [56], respectively, where the following cores have been used: Ti:(Ar) and O:(He). Additional AE basis sets with (20s,12p,4d)→[5s,4p,2d] and (14s,6p,1d)→[4s,3p,1d] contraction schemes [57] have also been used for Ti and O, respectively.

The infinite lattice sums of the Coulomb and exchange series were truncated with threshold values of 10-8, 10-8, 10-8, 10-8, and 10-24, and numerical integration has been performed with 75 radial points and 974 angular points, ensuring an error on the integrated electron density to an accuracy of 10-5 e per cell [58].

All structures have been fully optimized, relaxing both atomic positions and cell parameters. Convergence was determined from the root-mean-square and absolute values of the largest component of the forces and displacements, considering default values [58].

To study neutral O vacancies, an unrestricted open-shell formalism has been considered throughout, both in singlet and triplet spin states, and spin contamination has been monitored by calculating the expectation value of S2. Although the singlet open-shell case is not expected to be accurately described by a monodeterminental approach such as the one considered here, to allow for a straightforward comparison with previous works which have explored open-shell singlets at the DFT level [25, 32, 34, 37], a similar computational strategy has been considered here. Based on previous work [32], a (3×3×1) anatase bulk supercell of the conventional unit cell with 108 atoms was used. For the anatase (101) surface, a slab made of 12 Ti-layers, leading to converged surface properties, expanded in a (2×2) supercell with 144 atoms has been chosen. When creating an oxygen vacancy, the basis sets centered at the defect site have been maintained in the calculation. Only the D2 approach has been considered in this case by setting the C6 parameter related to VO to zero. The oxygen vacancy formation energy (\({E}_f^V\)) was computed with respect to the total energy of an oxygen molecule:

$${E}_f^V=E(D)+\frac{1}{2}E\left({O}_2\right)-E(H)$$

where E(D) is the energy of the defective system, E(O2) is the energy of an O2 molecule in its triplet ground state and E(H) is the energy of the stoichiometric host material.

The reciprocal spaces have been sampled with (6×6×6) and (4×4×4) Monkhorst-Pack k point meshes [59], corresponding to 30 and 36 points in the irreducible Brillouin zones (IBZ) of stoichiometric and defective bulk anatase, respectively. For the anatase (101) surface calculations, a (4×4×1) k-point mesh has been used, corresponding to 7 k points in the IBZ of the (2×2) supercell considered.

Results and discussion

Bulk TiO2

Stoichiometric bulk

Anatase has a tetragonal Bravais lattice with I41/amd space group, whose conventional unit cell is shown in Fig. 1a. Table 1 presents computed lattice parameters and band gaps, using all combinations of DFT models and basis sets considered.

Fig. 1
figure 1

a Conventional unit cell of bulk anatase, and b (3×3×1) supercell considered for the calculations of the defective bulk system. The unit cells are shown as blue dotted lines. Ti, O, and defective VO sites are shown as grey, red, and yellow spheres, respectively

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Table 1 Computed data for stoichiometric bulk anatase with different functionals, considering both ECP and AE basis sets. Lattice parameters (a and c, in Å), oxygen fractional coordinate components (u) along c, and band gaps (Eg, in eV) are reported. The values given in parentheses correspond to band gaps computed with AE basis sets using ECP geometries
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Clearly, all combinations perform well for the description of the a lattice parameter, with an error lower than 0.05 Å in all cases and only a very slight effect of dispersion effects no matter which dispersion model is considered. On the other hand, larger errors are usually obtained on the c lattice parameter, which might be related to the two axial Ti–O bonds along this direction. In particular, for all functionals, the addition of D2 slightly increases c with respect to the corresponding value without dispersion, while D3 significantly decreases it with values computed approaching the experimental parameter. Overall, although the AE basis sets perform better for the structural description than the ECP ones, the performances of these latter remain acceptable. In addition, the D3 model always leads to a better agreement with the experimental data when compared to the bare DFT model, no matter which function is considered. The computed structures are in line with previously-reported GTO calculations [57, 60].

Regarding the computed band gaps, although the same qualitative picture is obtained in all cases, as already well-known, PBE tends to underestimate it while hybrids tend to overestimate it. Overall, the best agreement is obtained with the range-separated HSE06 functional, ECP basis sets leading to an artificially better agreement with the experimental value than the AE ones. Only a slight effect of the inclusion of dispersion effects can be evidenced for all DFT models considered. In addition, we note that computed band gaps with AE basis sets using ECP geometries (referred to as AE//ECP level in the following and given in parentheses in Table 1) are very close to the AE data in all cases considered. This is especially true at the HSE06-D2 level, where only a 0.01 eV difference is obtained despite the ~ 0.2 Å difference obtained on the c lattice parameter. Since we found the large core pseudopotentials and related ECP basis sets not to be suitable to describe the electronic structures of defective systems, in the following, all calculations have therefore been performed at the HSE06-D2 AE//ECP level as a suitable compromise between accuracy and cost, considering the prohibitive computational costs of AE basis sets for the large supercells with low symmetry considered for defective systems.

The band structure, total and partial densities of states (DOS and PDOS) of stoichiometric bulk anatase computed at the HSE06-D2 level with AE basis sets using ECP geometries (referred to as HSE06-D2 AE//ECP level in the following) are shown in Fig. 2. The top of the valence band (VB) corresponds mainly to O-2sp states, with some slight contributions from Ti-3d states, while the bottom of the conduction band (CB) is largely composed of Ti-3d states. As already mentioned above, the computed indirect band gap between ~X and Γ of 3.96 eV is overestimated compared to the experimental value of 3.18 eV [62].

Fig. 2
figure 2

Band structure, total and partial densities of states of stoichiometric bulk anatase obtained at the HSE06-D2 AE//ECP level. The Fermi level (shown as thin dotted line) has been set to 0 eV

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Overall therefore, for stoichiometric TiO2 anatase, among the different combinations of DFT models and basis sets considered, the best results are obtained with the HSE06 functional, including dispersion effects (with D2 or D3 contributions) combined to AE basis sets. As mentioned above however, due to the prohibitive computational cost of AE basis sets for the comprehensive investigation of defective TiO2 carried out below, single-point AE energies computed on ECP geometries (AE//ECP data) offer a suitable tradeoff between accuracy and computational costs. In addition, due to the form of the D3 contributions to the total energy, only the D2 dispersion model has been considered in all calculations including oxygen vacancies (see Computational Details for more details).

Defective bulk

The formation of one VO leaves two excess electrons in the lattice which, according to experimental photoelectron spectroscopy (PES) data [41], localize at neighboring Ti sites at about 1.1 eV below the CBM, leading to Ti3+ species. Although different localization sites are possible (see Ref. 23, 29, 32 and for instance), in the following, we focus exclusively on the fully localized solutions on the two first nearest neighbor Ti atoms to the defective site, considering both singlet (S) and triplet (T) spin states in a (3×3×1) supercell (see Fig. 1b). This is in line with a recent study suggesting the importance of Ti3+–Ti3+ σ bond in defective TiO2 [37]. The structures obtained after geometry optimization are shown in Fig. 3. Tables 2 and 3 report selected distances and atomic displacements nearby the VO site after geometry relaxation, respectively, while relevant data regarding formation energies and electronic structures are collected in Table 4.

Fig. 3
figure 3

a Fully relaxed structures along with computed spin densities, and b DOS and PDOS of bulk anatase (3x3x1) supercell with one VO both in singlet (S) and triplet (T) spin states. All data obtained at the HSE06-D2 AE//ECP level. The Ti1 and Ti2 atoms correspond to the two Ti3+ sites where the excess electrons localize. The Fermi level has been set to 0 eV. For spin densities plots: isosurface values of |0.005| e.bohr-3. Ti, O and defective VO site shown as grey, red, and yellow spheres, respectively. The O5 atom (see Fig. 1b) is highlighted in purple

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Table 2 Selected distances (in Å) of stoichiometric and defective bulk anatase (3x3x1) supercell, considering both singlet (S) and triplet (T) spin states. All data obtained at the HSE06-D2 ECP level. See Fig. 1b for atom labeling.
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Table 3 Atomic displacements (in Å, from bulk-truncated positions) nearby the VO site for the defective bulk anatase (3x3x1) supercell along different directions during geometry relaxation, both in singlet (S) and triplet (T) spin states. All data obtained at the HSE06-D2 ECP level. See Fig. 1b for atom labeling
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Table 4 Formation energies (\({E}_f^V\)) of one VO, band gaps (Eg) of α/β electrons, and offsets of the two deep defect levels (Ea and Eb) for the defective bulk anatase (3x3x1) supercell with respect to the CBM, along with the corresponding mean values (Ē), both in singlet (S) and triplet (T) spin states. All data (in eV) obtained at the HSE06-D2 AE//ECP level
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Significant differences have been obtained between the S and T solutions regarding the local structural polaronic distortions around the VO site, with a larger asymmetry in the T case than in the S one. Indeed, while all the Ti–Ti distances reported in Table 2 for the T spin state are larger than those obtained for the stoichiometric systems, the S solution presents either smaller or larger values. In particular, all distances involving Ti1 (see Fig. 1b) are smaller than in the stoichiometric case, while the Ti3-Ti2 one is larger. This can be related to the large displacements of the O5 atom both in the [100] and [001] directions, which relaxes to almost occupy the original position of the VO site, with a final computed d(Ti1–O5) value of 2.069 Å, that is about half of the one (3.977 Å) obtained in the T case. The computed distances reported in Table 2 are generally in qualitative agreement with a previous HSE/PAW investigation [32], although in this latter work: (i) no fully localized solution in the T spin state could be obtained, (ii) identical Ti1–Ti2 and Ti3–Ti2 distances were obtained, in clear contrast to our calculations where symmetry breaking around the VO site occurs. According to the formation energies, \({E}_f^V\) reported in Table 4, the S spin state is found about 0.2 eV more stable than the T one. We note that the \({E}_f^V\) values obtained here are higher than those previously-reported at the HSE/PAW [32] or B3LYP/GTO [29] levels for instance, mainly due to the AE//ECP computational strategy adopted in our work. Nevertheless, based on the spin densities and densities of states reported in Fig. 3, we note that the two excess electrons left in the lattice by the neutral VO localize almost exclusively at the two nearest-neighbor Ti sites Ti1 and Ti2, with absolute values of the magnetic moments of these atoms of at least 0.85 μB in all cases. In addition, based on the data reported in Table 4, when averaging the two Ti1 and Ti2 levels positions Ea and Eb in the DOS, offsets of 0.90 and 0.98 eV with respect to the CBM are obtained in the S and T spin states, respectively, in very good agreement with the 1.1 eV value reported experimentally by PES [41].

Finally, we note that the relaxed lattice parameters were found to be always within 1% of the stoichiometric values, indicating that the structural distortions around the VO site are mainly due to strong local atomic relaxations rather than strong lattice relaxations. In addition, the spin contamination is high for the S state, while it is low for the T state, with computed expectation values of S2 of 1.01 and 0.03 for the S and T solutions, respectively. While the former is related to the mono-determinantal nature of the approach considered, the low value obtained in the T case further confirms the validity of the solution obtained.

Anatase (101) surface

Stoichiometric surface

The anatase (101) surface is a stoichiometric non-polar surface with a characteristic sawtooth structure (see Fig. 4) with two and threefold coordinated O atoms (O2c and O3c) as well as five and sixfold coordinated Ti atoms (Ti5c and Ti6c). The computed surface energy of the (2×2) supercell considered with 12 Ti-layers is of 0.83 J/m2 at the HSE06-D2 AE//ECP level, and 0.78 J/m2 when fully relaxed at the HSE06-D2 AE level. These values are in line with previously-reported ones obtained with various hybrid functionals with the same AE basis sets and the surface primitive cell [63]. In addition, the computed band gaps are 4.06 and 4.00 eV at the AE//ECP and AE levels, respectively, outlining the suitability of the computational strategy considered.

Fig. 4
figure 4

a Anatase (101) slab model expanded in a (2×2) supercell with 12 Ti-layers showing VO sites positions numbering (P1 to P9), and corresponding atom labelings of the b P1 and c P4 cases. The unit cell is shown as blue dotted lines. Ti, O, and defective VO site shown as grey, red, and yellow spheres, respectively

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The computed DOS and PDOS shown in Fig. 5 are very similar to those previously discussed in the bulk case: the top of the VB corresponds mainly to O-2sp states, with some slight contributions from Ti-3d states, while the bottom of the CB is largely composed of Ti-3d states.

Fig. 5
figure 5

Total and partial densities of states of the (2×2) supercell with 12 Ti-layers of anatase (101) computed at the HSE06-D2 AE//ECP level. The Fermi level (shown as thin dotted line) has been set to 0 eV

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Defective surface

As already done in Ref. [34], nine different positions of the defective VO sites have been considered (P1 to P9, see Fig. 4), going deeper and deeper in the slab, both in the S and T spin states. This allows to investigate surface (P1 to P3), subsurface (P4 to P6), and “bulk-like” (P7 to P9) vacancies. All optimized structures are shown in Figures S1 and S2 of the Supplementary Material, while relevant data regarding formation energies and electronic structures are collected in Table 5. As already done in the bulk case and supported by a recent study suggesting the importance of Ti3+-Ti3+ σ bond in defective TiO2 anatase (101) [37], we focus exclusively on fully localized solutions, that is solutions where the two excess electrons due to the VO formation are localized on two first nearest neighbors Ti atoms of the VO site.

Table 5 Formation energies (\({E}_f^V\)) of one VO, band gaps (Eg) of α/β electrons, and positions of the two deep defect levels (Ea and Eb) for the defective (2×2) supercell of anatase (101) with respect to the CBM, along with the corresponding mean values (Ē), both in singlet (S) and triplet (T) spin states, as a function of the VO position (P). All data (in eV) was obtained at the HSE06-D2 AE//ECP level. The data obtained on the bulk system, already reported in Table 4, are also given
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Based on the data of Table 5, the S solution is generally found more stable than the T one like in the bulk, although differences remain generally small, with a maximal difference of formation energies lower than 0.3 eV among the different VO positions considered. In addition, all computed values are lower than in the bulk case except the P2 case, in line with the importance of VO in TiO2 surface processes [18, 19]. More precisely, the \({E}_f^V\) values range between 5.04 (at P1) and 6.73 eV (at P2) in the S case, while they range between 5.17 (at P1) and 6.78 eV (at P2). The relative stability order of the different VO positions considered is the same in the S and T cases: P1 > P4 > P7 > P3 > P5 > P9 > P8 > P6 > P2. The greatest stability of the surface P1 position and the least stability of the surface P2 position are in line with recently-published data obtained in S spin state at the HSE06 level with ultra-soft pseudopotentials (USPP) [34], although the relative stability of the other positions differ. We note that this is also in agreement with recent a recent STM investigation which suggests that subsurface VO clusters might convert to surface VO for anatase (101) [64]. Figure 6 presents the computed highest occupied crystalline orbitals (HOCO) in the most stable S spin state for the two most P1 and P4 positions. The difference in stability between these two sites can be related to a fully bonding interaction between the d orbitals centered on Ti1 and Ti2 and the p orbitals of O7 in P1, while a mixed bonding/antibonding character can be evidenced between the corresponding atoms in the P4 case.

Fig. 6
figure 6

α HOCO of the defective (2x2) supercell of anatase (101) with one VO obtained in the S spin state for the a P1/(1/2;1/4) k point and b P4/(0;1/2) k point cases. Isosurface value of |0.02| e1/2.bohr-3/2. All data obtained at the HSE06-D2 AE//ECP level. Ti, O, and defective VO site shown as grey, red, and yellow spheres, respectively

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As already found in the bulk case, we note that computed formation energies obtained in this work with the AE//ECP strategy are overestimated compared to published data. Depending on the reference values considered, however, this overestimation is more or less important. For instance, at the HSE06/USPP level, Ha and Alexandrova [34] reported significantly different VO formation energies of 2.91, 3.41, and 3.30 eV at the P1, P4, and P5 positions, respectively, while Deák et al. [25] reported values of 4.81, 5.03, and 5.46 eV, in much better agreement with the values reported in Table 5. Nevertheless, we note that in these two previous works and the work presented here, the surface vacancy position P1 is found to be the most stable one.

In all T cases considered, the two reduced Ti3+ sites contribute to the DOS at different offsets from the CBM (see Ea and Eb values, Table 5), as already found in the bulk case. On the other hand, in the S case, only the P4 and P6 cases lead to Ti3+ contributions at significantly different offsets. This can be related to a much larger structural distortion in these latter two cases than in the other ones. For instance, from Table 6 where some relevant geometrical parameters of the two most stable cases P1 and P4 are reported, we can note the larger distortion around the VO site in the P4 case compared to the P1 one. This is confirmed by the computed atomic displacements upon relaxation from the relaxed stoichiometric supercell reported in Table 7. Indeed, while in P1 the largest displacements are obtained for the two reduced Ti3+ atoms (Ti1 and Ti2), a much more pronounced atomic reorganization is obtained in the P4 case. In particular, while the Ti1 and Ti2 atoms still relax substantially, the two-fold coordinated surface O9 atom displays very large displacements along the [−101] and [101] directions and migrates to fill the initial VO lattice position (see Fig. 8). This effect has already been mentioned by Ha and Alexandrova [34], although it was only obtained with DFT+U approaches and specific values of the U parameter.

Table 6 Selected geometrical parameters of the defective anatase (101) (2x2) supercell with one VO, both in singlet (S) and triplet (T) spin states. Distances (d) in Å and angles (θ) in degrees. All data obtained at the HSE06-D2 AE//ECP level. Atom labeling given in Fig. 4. The corresponding values of the stoichiometric (2x2) supercell are also reported
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Table 7 Atomic displacements (in Å, from relaxed stoichiometric supercell positions) around defective sites P1 and P4 for the (2×2) supercell of the anatase (101) surface with one VO, considering both single (S) and triplet (T) spin states. All data obtained at the HSE06-D2 ECP level. See Fig. 4 for atom labeling
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Fig. 7
figure 7

a Computed spin densities, and b DOS and PDOS of the (2×2) supercell of anatase (101) with one VO at position P1 in singlet (S) and triplet (T) spin states. All data obtained at the HSE06-D2 AE//ECP level. The Ti1 and Ti2 atoms correspond to the two Ti3+ sites where the excess electrons localize. The Fermi level has been set to 0 eV. For spin densities plots: isosurface values of |0.005| e.bohr-3. Ti, O, and defective VO site shown as grey, red, and yellow spheres, respectively

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The spin densities computed for the P1 and P4 cases and reported in Figs. 7 and 8 confirm these trends, with spin densities localizing almost exclusively on the two nearest neighbors Ti sites to the VO, as indicated by the absolute values of the magnetic moments of these two atoms of at least 0.83 μB in all cases. Like in the bulk case, the spin contamination of the T spin state was found to be small (below 0.01), but close to 1.00 for the S spin state. The computed average offsets from the CBM of the Ti3+ contributions to the DOS for the two most stable positions found (P1 and P4) are in excellent agreement with the experimental PES value for anatase (101) surface samples of 1.3 eV [65], as well as the 1.3 eV value reported by Deák et al. at the HSE06/PAW level [65] in the S spin state. In addition, we note a larger delocalization of the defect levels in the DOS in the subsurface P4 case than in the P1 case, in line with a recent publication [37], as well as the HOCO plots presented in Fig. 6, where this CO is found more delocalized when the defective site is P4 than P1.

Fig. 8
figure 8

a Computed spin densities, and b DOS and PDOS of the (2×2) supercell of anatase (101) with one VO at position P4 in singlet (S) and triplet (T) spin states. All data obtained at the HSE06-D2 AE//ECP level. The Ti2 and Ti3 atoms correspond to the two Ti3+ sites where the excess electrons localize. The Fermi level has been set to 0 eV. For spin densities plots: isosurface values of |0.005| e.bohr-3. Ti, O, and defective VO site shown as grey, red, and yellow spheres, respectively. The O9 atom (see Fig. 4c) is highlighted in purple

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Conclusions

In this work, five exchange-correlation functionals (PBE and four hybrids PBE0, B3LYP, B3PW, and HSE06), combined with two dispersion models (D2 and D3) have first been considered to investigate the structural and electronic properties of stoichiometric bulk TiO2 anatase, taking into account both ECP and AE basis sets within a GTO-based periodic formalism. Although the best results have been obtained with the HSE06 functional including additional dispersive interactions (with D2 or D3 contributions) and AE basis sets, an AE//ECP computational strategy was found to be a suitable tradeoff between accuracy and computational cost. Indeed, the investigation of the defective bulk system with one VO both in singlet (S) and triplet (T) spin states has proven ECP basis sets to be inadequate for the description of the positions of the two defect levels in the stoichiometric material band gap. When considering the HSE06-D2 AE//ECP computational strategy however, a very good agreement with experimental PES data indicating an offset of about 1.1 eV from the CBM for these levels could be obtained, with computed values of 0.90 and 0.98 eV in the S and T spin states, respectively. Significantly different local structural polaronic distortions around the VO site between the S and T spin state solutions could be evidenced however, with a larger asymmetry in the T case than in the S one. In both cases, the vacancy creation led to the localization of the two excess electrons on the two first nearest neighbor Ti3+ sites and lattice relaxation was found to be small, compared to the substantial local structural modifications around the VO site.

The same computational strategy was then considered to investigate a defective anatase (101) surface, taking into account nine different vacancy positions and both S and T spin states. In all cases, computed vacancy formation energies were found to be lower than in the bulk case, and significantly different values were obtained between the different positions. The surface twofold coordinated oxygen site was found to be the most stable one, while the surface threefold coordinated oxygen site was found to be the least stable one, in agreement with recently published theoretical works with hybrid functionals. For the most stable sites found, the computed offsets of the defect levels positions from the CBM were found to be in excellent agreement with the 1.3 eV value obtained experimentally by PES, with computed values ranging between 1.1 and 1.3 eV. While the two first nearest neighbor Ti3+ sites where the excess electrons localize relax substantially upon vacancy creation, significant O atoms migration up to the initial vacancy site were also found in some cases, in excellent agreement with other hybrid functional-based computational works.

Overall therefore, the very good agreement obtained with available experimental data regarding the offsets from the CBM of the defect levels positions both for the bulk and the (101) surface of TiO2 anatase is encouraging for the application of the proposed HSE06-D2 AE//ECP computational strategy to TiO2 surface-related processes such as TiO2-based photocatalysis where oxygen vacancies are known to play a key role.