Properties of Single Oxygen Vacancies on a Realistic (TiO2)84 Nanoparticle: A Challenge for Density Functionals

Based on all electron relativistic density functional theory (DFT) calculations, the properties of single oxygen vacancies in TiO2 nanoparticles have been obtained using a suitable representative model consisting of an octahedral (TiO2)84 nanoparticle of ~3 nm size terminated with (101) facets. This nanoparticle can be safely considered at the onset of the so-called scalable regime where properties scale linearly with size towards bulk like limit and hence results can be more directly compared to experiment. A set of reduced Ti84O167 nanoparticles are selected to investigate the geometrical, energetical and electronical properties by using PBE semi-local functional with three different amounts of Fock exchange 0% (PBE), 12.5% (PBEx) and 25% (PBE0). In particular, using the PBEx hybrid functional, previously validated for bulk anatase and rutile, it is predicted that the highly (three)-coordinated oxygen atom, located in the subsurface, and the least coordinated one at top sites are energetically the most suitable candidate for generating the oxygen vacancy. The subsurface case is in line with conclusions from experiments carried out on (101) single crystal anatase surfaces. The electronic structure of the reduced particles suggests that these would have better photocatalytic activity than their stoichiometric counterparts. Nevertheless, several properties of reduced TiO2 NPs are strongly affected by the choice of the exchange-correlation functional implying that, in absence of validation by comparison to experiment, predictions must be taken with caution


Introduction
Titanium dioxide (TiO2), a semiconducting metal oxide, is widely used as a photocatalyst because of its chemical stability, low cost, and nontoxicity. [1][2][3][4] Both rutile and, especially, anatase nanostructures catalyze the overall splitting of water into molecular hydrogen and oxygen when illuminated with ultraviolet (UV) light. [5][6][7][8] This photocatalytic process represents a promising clean and sustainable alternative to fossil fuels. 9,10 However, applications of TiO2 show a low quantum yield (QY) and limited harvesting of visible light which is a direct consequence of the too large band gap of this material. Moreover, fast recombination of photogenerated electron-hole pairs represents an additional problem. To tackle the above obstacles, several strategies have been applied to modify TiO2, like depositing noble metals (e.g. Pt), 11 doping with transition-metal ions (e.g. Cu) or nonmetal elements (e.g. N), 12 adding electron carriers (e.g. graphene) 13 or coupling to small band gap quantum dots (e.g. PdS). 14 In addition, TiO2 nanostructuring with specific morphologies and crystal facets has emerged as a promising way to improve the QY with visible light. 15,16 Alternatively, engineering defects on the TiO2 nanostructures appears as another encouraging way to circumvent the aforementioned low QY of TiO2. 17 It has been reported that creating a highly disordered surface layer and a large amount of oxygen vacancies (Ov) in TiO2 not only improves visible and near-infrared light absorption but also leads to a superior activity in photocatalytic degradation of organic pollutants and H2 evolution from H2O. 18 From simple electron counting and assuming formal oxidation states, one neutral oxygen vacancy, Ov, provides two excess electrons to the TiO2 system, and these electrons are available for the reduction of Ti 4+ to Ti 3+ . Indeed, the formation of Ti 3+ center has been confirmed by experiments such as photoelectron spectroscopy and electron paramagnetic resonance (EPR). 19,20 Recently, it has been shown that Ti 3+ self-doped TiO2 nanocrystals with a 1:80 Ti 3+ :Ti 4+ ratio exhibit a remarkable enhancement of the photocatalytic efficiency; this is precisely ascribed to the presence of Ti 3+ centers and concomitant Ov point defects. 21 The combination of Ov and Ti 3+ centers leads to specific electronic structure features usually referred to as trapped electrons. These states appear around 1.0 eV below the conduction band minimum (CBM), which induces band gap narrowing. Engineering Ti 3+ /Ov sites can favorably promote CO2 activation and conversion to CO under visible light. 22 Several computational studies have focused on these systems and reported the properties of Ov in bulk, surface and subsurface TiO2 sites. [23][24][25] Experiments and calculations show that formation energy of neutral oxygen vacancy is lower when defect site is located in the subsurface of anatase (101) and (001) surfaces, whereas the surface site is the most stable site in rutile (110) surface. Similar 3 information on TiO2 nanostructures is so far lacking except for a recent study on a (TiO2)35 representation of an octahedral nanoparticle. 26 This is indeed the smallest TiO2 NP compatible with   Wulff construction, yet metastable with respect to non-crystalline structures 27 as further commented below.
The electronic properties of TiO2 nanoparticles (NPs) have been discussed in the light of experiments reporting that the control of the shape and size of titania NPs modulates the photoactivity. 15 The understanding of TiO2 NP properties is particularly attractive because its size and shape are appropriate to rationalize the experimental evidences. 28 However, experimentally, it is difficult to discern between the different effects of size and shape and those introduced by the synthetic conditions. The difficulties encountered by experiments to separate complex factors are not present when employing computational models in which one can represent different morphologies for a given composition or vary the composition for a given morphology. 27,29-31 For instance, the analysis of the atomic structure of TiO2 small clusters and NPs of increasing size allows one to estimate the dimensions for emergence of crystallinity and scrutiny of electronic properties of well defined nanoparticles allows one to identify the size-limit of titania NP at which the properties show an asymptotic trend towards bulk-like behavior. 30 In particular, it has been shown that the realistic (TiO2)84 anatase NP is at the start of the scalable regime limit above which properties of titania NPs scale linearly towards bulk limit values. Assuming spherical shape 32 one would get an estimate size of 3 nm for the octahedral (TiO2)84 NP, a value which nicely fits with the predicted height from a linear relationship between the largest vertex to vertex distance in octahedral TiO2 NPs and the cubic root of the number of TiO2 units. 30 From both arguments one can conclude that (TiO2)84 anatase NP is an appropriate model system to investigate the changes induced by the generation of Ov in their electronical and structural properties and hence in its photocatalytic activity.
Herein, using state of the art density functional theory (DFT) based methods, we systematically investigate the properties of oxygen vacancies in an initially stoichiometric (TiO2)84 anatase NP. To facilitate the study and handle the numerous possible isomers resulting from the creation of a single Ov in this quite large NP, we follow the strategy developed in a preliminary study on a smaller (TiO2)35 anatase NP. 26 We have already mentioned that this is the smallest size for a bulk cut octahedral NP exhibiting (101) facets although in terms of total energy per TiO2 unit it is clearly metastable. 27 This fact can affect the conclusions regarding reduction and ask for subsequent studies on a more realistic model. This is precisely the main reason to undertake the present work on the properties of oxygen vacancies on the (TiO2)84 NP. Several possible sites for 4 oxygen vacancy formation are evaluated according to the position and the coordination number (CN), using results for oxygen vacancy in (TiO2)35 as a convenient guide. 26

Computational Details
First of all, it must be recalled that standard density functionals within the Generalized Gradient Approach (GGA) fail to provide even a qualitatively correct description of electronic properties in stoichiometric and reduced TiO2 systems due to an exceedingly large effect of the socalled self-interaction error inherent to the Kohn-Sham practical implementation of DFT. 33,34 To correctly describe the partial occupation and appropriate localization of states arising from Ti 3+ sites hybrid functionals are required where a given percent of Fock exchange is mixed into a given exchange GGA functional. Assuming that the Kohn-Sham energy levels provide an estimate of the binding energies of quasiparticles, previous studies have shown that modifying the PBE0 (25% Fock) 35,36 functional so as to contain 12.5% of Fock exchange (PBEx) reproduces successfully experimental band gap of anatase and rutile polymorphs whereas the commonly used PBE0 hybrid functional 35 significantly overestimates them. 23 In addition, PBEx functional describes properly the properties of the Ov in both rutile and anatase TiO2 bulk phases. 23 In the present study DFT based calculations have been carried with the PBEx hybrid functionals. However, in the view of the strong influence of the exchange-correlation functional on the outcome of the calculation, results from more traditional approaches such as PBE 36 and PBE0 35 are included for comparison. Another aspect regarding the accuracy of the density functional methods chosen concerns the contribution of dispersion terms. For the atomic structure of bulk anatase and rutile, Deringer and Scányi 37 found rather small changes in the predicted lattice parameters when taking into account these effects on top of results obtained from PBE. The changes are similar those encountered when going from PBE to PBEx. Consequently, the contribution of dispersion has not been further considered.
All calculations explicitly include all electrons and the electron density is described through a numerical atom-centered (NAO) orbital basis set, as implemented in the Fritz Haber Institute ab initio molecular simulations (FHI-aims) package. 38 The light grid and tier-1 basis set are selected, the numerical accuracy of this basis set for TiO2 systems is similar to that of a valence triple-ζ plus polarization Gaussian type orbitals (GTO) basis. 30 Recent studies have further assessed the quality of the light tier-1 NAO basis set on the stability and electronic properties of TiO2 NPs. 31 The convergence threshold for atomic forces in relaxation of pristine and reduced (TiO2)84 NPs is set to 5 to properly ensure sufficient accuracy for calculated properties. In the present work, relativistic effects are included through the zero order regular approximation (ZORA). 39,40 A series of reduced Ti84O167 NPs has been considered which differ in the position where the oxygen vacancy has been created. The considered NPs have all formally two Ti 3+ centers and hence involve two excess electrons leading to different electronic states¾ closed shell singlet, open shell singlet and triplet¾ depending on the spin coupling. Hereby, closed-shell singlet and open-shell triplet states are considered as in previous work. 26 The oxygen vacancy formation energy, " # , is calculated as: where.

Results and Discussion
It is necessary to insist on the fact that the contribution of non-local, exact, Fock exchange to the exchange-correlation potential has a marked influence in predicted properties of oxides in general and of TiO2 related systems, in particular. Precisely, this is an especially delicate issue for the properties of Ov containing systems exhibiting unpaired electrons strongly localized at Ti 3+ centers but with the degree of localization depending on the choice of the exchange-correlation potential. To rigorously tackle this problem and to avoid any particular bias, this section is divided into two parts. In the first one, properties of Ov on reduced Ti84O167 NPs are discussed in light of results obtained with the PBEx functional which, as pointed out above, properly describes stoichiometric and reduced bulk TiO2. In the second part the influence of the Fock exchange contribution in the exchange-correlation potential is analyzed in some details so as to reach unbiased conclusions regarding the interpretation on vacancy defect states in the Ti84O167 NPs. To this end, results have been obtained from the PBE and PBE0 functionals. Results from the later are explicitly discussed here whereas the former are reported in the Supporting Information. 6 Properties of Ov@Ti84O168 as described from the PBEx functional The optimized structure of the realistic (TiO2)84 NP obtained from a Wulff construction 41,42 and featuring the most stable (101) facets 43 is shown in Figure 1. To systematically examine the effect of a single oxygen vacancy one would need to consider all possibilities which, even exploiting symmetry, faces a combinatorial explosion. Therefore, we rely on results obtained from an earlier systematic study on the smaller (TiO2)35 NP 26 and choose the most favorable sites for Ov as candidates for (TiO2)84 NP. This is justified because both NPs have the same octahedral shape, symmetry and exposed (101)  domain. Note also that an additional digit (1 or 2) is used to distinguish two I-3 analog sites.
We start analyzing two different optimized structures of the pristine (TiO2)84 NP, which just depend on the orientation of apical oxygen atoms: (i) almost linear, straight tip, or (ii) bent to opposite sides, bent tip (see Figure 3 in Ref. [30]). The bent tip structure depicted in Fig. 1a is energetically more stable by around 0.011 eV per TiO2 unit (0.96 eV in absolute terms) for all employed DFT functionals, consistent with the previous analysis over (TiO2)35 NP where a similar favorable energetic stability for bent tip structure was found (1.06 eV in absolute terms). 26 We assume that the difference between straight and bent tip structures tends to be negligible by increasing the size of TiO2 NP although this is not an important issue. In fact, despite their slight structural difference, the electronic band gap (Egap), evaluated as the difference between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) is almost the same. This value, however, strongly depends on the exchange-correlation potential used to describe the electronic structure and PBEx functional nicely reproduces 23  To avoid any possible artifacts, the properties of Ov in (TiO2)84 NP are studied by removal of one neutral oxygen atom from the bent tip structure (Figure 1). Six oxygen sites are selected and 7 systematically removed one at a time. This set of Ti84O167 reduced nanoparticles represents the structures that have the lowest " # in each one of the domains based on our previous study of (TiO2)35 NP 26 having the same morphology and exhibing the same facets as commented above. Table 1 reports the values of Ev, Erel, " # and the energy level of the oxygen vacancy defect state below LUMO (Sv-1 and Sv-2) calculated at the PBEx level. Note that the oxygen vacancy sites with just a Sv-1 energy correspond to a closed-shell singlet state which is found to be the electronic ground state after geometry optimization. This is quite an unexpected result since spin polarized  Table 1). Note, however, that the T-1 domain is energetically competitive showing an energy difference of 0.02 eV only. Another highlighted energetic parameter is the vertical energy Ev (Table 1) According to experimental evidences, surface oxygen vacancies are not present on freshly cleaved anatase (101) samples. 45,46 A reduced anatase (101) crystal shows isolated as well as ordered intrinsic subsurface defects in scanning tunneling microscopy (STM), consistent with DFT calculations for surface slab models which predict that Ov at subsurface and bulk sites are significantly more stable than on the surface. 45,46 Note that Ov at T-1 site is essentially as stable as at I-3-2 site but, it is a feature that (i) is not present in extended crystalline (101) anatase surfaces and (ii) unique to the morphology of the nanoparticle, thus it would not be found in extended surfaces. 8 The prediction from the present all electron relativistic DFT based calculations for realistic TiO2 NP are in agreement with the experimental evidence and provides further support to the claim that (TiO2)84 corresponds to a particle size in the scalable regime and hence representative of the large NPs used in the experiments. This provides additional support to the choice of the PBEx hybrid functional as appropriate to investigate the properties of stoichiometric and reduced realistic TiO2 NP as suggested from previous studies. 23,26,31 We discuss now in some detail the structural changes around the first neighbor titanium atoms to each of the investigated Ov sites (Figure 3) domain increases in 0.7 and 2.18 eV for Ev and Erel, respectively. This is indeed the main energetic term that leads to " # below 3.0 eV (see Figure 2 and Table 2 remove. This is also in agreement with the results obtained with the PBEx functionals which predicts that Ov is almost equally favorable at T-1 and I-3-2 sites. The F-2 case is harder to understand and it is not consistent with the experimental findings for extended surfaces perhaps due 10 to its proximity to the nanoparticle edge (see Figure 1). 45,46 In any case, it is important to point out that the differences in " # for the different most stable sites are of the order of 0.3 eV only and this is perhaps in the error bar of this type of DFT based calculations for this property. Nevertheless, the final " # calculated values are much smaller with PBE0 functional implying that TiO2 appears to be more reducible than predicted with PBEx. Unfortunately, without a guide from experiment it is hard to assess which of the two functionals better describes TiO2 reduced nanoparticles and, in the view of the better performance of PBEx functional for stoichiometric and reduced rutile and anatase one is tempted to suggest that this will be also the case for the reduced NPs even if some feature is counterintuitive.
Although the experimental indications are not completely corresponding with predictions from the PBE0 functional, this approach reproduces the degree of localization on the neighboring Ti atoms, which are reduced to Ti 3+ in the I-3-2 site, which is not always observed by using PBEx functional (see Figure 4). Note that whereas PBEx functional gives the most favorable site at I-3-2 site consistent with the experiments for extended anatase (101) surfaces, it does not appropriately reproduce the oxygen vacancy defect state below LUMO ( Figure 4) promoting a doubly (singlet state) occupied molecular orbital. This issue is solved when the percentage of Fock contribution increases (PBE0 functional) giving singly (triplet state) occupied molecular orbitals as shown in Figure 5.
Therefore, one must conclude that the appropriate selection of the exchange-correlation functional to investigate the properties of reduced TiO2 NPs requires a thorough examination of a range of properties and to analyze how the amount of Fock exchange affects them. In the case of Ov in TiO2 NPs there is a certain degree of incertitude on determining exactly the most stable one although one can for sure make more reliable predictions by focusing on a range of formation energies including various possibilities. This is supported by analysis of Figure 2 clearly showing that PBEx and PBE0 functionals exhibit similar trends regarding the energy formation of different Ov and that this is clearly different from the trend predicted by PBE functional. Also, describing properly the stability and the electronic structure of Ov related states is far from being a solved problem since the amount of Fock exchange required to accurately describe the gap is insufficient to produce the expected localization. The latter remains, nevertheless, to be confirmed by experiments on suitable nanoparticles.

Conclusions
The

Supporting Information
The Supporting Information is available free of charge on the ACS Publications website at DOI: The description of energetic scheme is shown in Figure S1. Oxygen vacancy formation energies, Kohn-Sham orbital energy level diagrams and structural properties are described in Table S1 and Figures S2 and S3 for PBE level. Whereas, structural properties at PBE0 level are depicted at Figure S4.

Notes
The authors declare no competing financial interest.

ACKNOWLEDGEMENTS
The authors are indebted to Prof. Ulrike Diebold for pointing out the similarity of the present results to those measured in her group for anatase TiO2 (101)