Relevance of Dispersion and the Electronic Spin in the DFT + U Approach for the Description of Pristine and Defective TiO₂ Anatase

Abstract

A density functional theory + U systematic theoretical study was performed on the geometry, electronic structure, and energies of properties relevant for the chemical reactivity of TiO₂ anatase. The effects of D3(BJ) dispersion correction and the Hubbard U value over the energies corresponding to the TiO₂/Ti₂O₃ reduction reaction, the oxygen vacancy formation, and transition-metal doping were analyzed to attain an accurate and well-balanced description of these properties. It is suggested to fit the Hubbard correction for the metal dopant atom by taking as reference the observed low spin–high spin (HS) energy difference for the metal atom. PBEsol-D3 calculations revealed a distinct electronic ground state for the yttrium-doped TiO₂ anatase surface depending upon the type of doping and interstitial or substitutional defects. Based on the calculations, it was found that a HS state explains the observed ferromagnetism in cobalt-substituted TiO₂ anatase. The results presented herein might be relevant for further catalytic studies on TiO₂ anatase using a large surface model that would be worthwhile for heterogeneous catalysis simulations.

1. Introduction

Titania (TiO₂) is a reducible oxide with semiconductor character that has the capability to exchange oxygen relatively easily under moderate conditions. In a reducible oxide, there exists a band gap between the valence and conduction bands that is smaller than that in an insulator and makes the system more reactive. Then, when the oxygen is removed through the formation of a vacant site, the excess electrons are redistributed on the metal d empty levels, changing the oxidation state of the neighboring metal ions from Ti⁴⁺ to Ti³⁺.¹ The formed defective sites might function as active sites for adsorption and dissociation of oxygen-containing molecules.² TiO₂ is also used as support for active metal nanoparticles. These materials have been widely studied as catalysts for the CO oxidation reaction for which the support not only stabilizes the nanoparticles but also plays an active role when the reaction progresses through the Mars–van Krevelen mechanism.³ By following this mechanism, the oxygen atoms from the metal oxide lattice react with CO to form CO₂; then, after the product desorption, the oxygen molecule activation takes place in an oxygen vacancy site. Recently, Zanella et al. reported a novel preparation method of gold-based bimetallic catalysts supported on TiO₂ conducted by the method of sequential deposition–precipitation with urea to study the catalytic oxidation of CO.^4,5 Particularly, CO oxidation is a widely used prototype reaction in heterogeneous catalysis to probe the catalytic activity of novel materials. It is a model reaction used for the methodological development of highly active catalysts, given that CO oxidation is a reaction with one rate-limiting step and a unique product.⁶ It allows exploring the nature of the employed catalyst and to get a physical insight into the chemical and structural transformations of the catalyst interacting with the reactants.

For these kinds of materials employed in heterogeneous catalysis, it has been assigned the important role of the support in the activation of the reacting molecules. One widely used strategy to stabilize supported gold (Au) nanoparticles is the addition of transition atoms to the TiO₂ support. In previous studies, it has been reported that yttrium doping favored the formation of oxygen vacancies which act as nucleating centers for Au atoms.^7,8 Zanella et al.⁸ prepared yttrium-doped TiO₂ supports that presented the formation of the anatase phase. The incorporation of yttrium enhanced the adsorption of Au nanoparticles and prevented their sintering. They found that the Au/Y–TiO₂ catalyst exhibited increased activity for CO oxidation compared to that of the undoped Au/TiO₂ system.⁸

Transition-metal doped TiO₂-based materials have been widely studied for the development of semiconductors with enhanced catalytic activity.⁸⁻¹¹ The incorporation of transition metals into the lattice of TiO₂ induces the formation of new energy levels near the conduction band. For instance, cobalt-modified TiO₂ anatase presented ferromagnetism with n-type semiconducting properties.^12,13 Cobalt might possibly occupy an interstitial position in the TiO₂ lattice because of its relatively larger size compared to that of titanium.¹⁴ Spectroscopic determinations revealed that cobalt-doped TiO₂ exhibits band gap narrowing.¹⁵ Moreover, recent investigations reported a superior catalytic performance of cobalt oxide–TiO₂ catalysts for soot combustion and CO oxidation.^16,17 In regard to theoretical results, it has been reported that there was a decrease of 1.7 eV in the projector augmented wave (PAW)-Perdew–Burke–Ernzerhof (PBE) computed energy gap (the energy difference between the highest occupied state and the lowest unoccupied state) of anatase after cobalt-substitutional doping and determined the presence of mid-gap impurity states that provided trapping potential wells for holes and electrons.¹⁸ A similar study showed a more significant decrease of 1.86 eV in the energy gap with cobalt incorporation by employing the HSE06 hybrid functional with a Hartree–Fock exchange contribution of 18%.¹⁹ However, in the mentioned studies, the electronic state of the dopant metal is not discussed or only the low spin (LS) state was explored.

Furthermore, it is crucial to describe accurately the electronic properties of TiO₂ support as a first step to model the catalytic processes in more complex systems for heterogeneous catalysis. The preferred method for the theoretical study of strongly correlated electrons in materials, as those occupying 3d bands in reduced TiO₂, is the density functional theory (DFT) under the Hubbard parameter approximation (DFT + U)²⁰ which enables the computation of large systems at a much lower computational cost than through hybrid functionals. The Hubbard term U, which introduces the on-site Coulombic repulsion between electrons U′, and the site exchange term J are applied to the effective potential which remedies importantly the self-interaction error present in generalized gradient approximation (GGA) methods that underestimates the band gap and fails to describe the localized states of defective moieties. This can also lead to a wrong prediction of metallic properties in materials that exhibit a gap such as Ti₂O₃. Dudarev’s approximation introduces an effective Coulomb interaction U_eff = U′ – J that incorporates the exchange correction J.²¹ The U_eff term simply called U will be the Hubbard parameter used throughout the present work. The U parameter has been determined semiempirically based on its accuracy in reproducing bulk properties such as the electronic band gap and structure parameters. However, in calculations aiming to understand the catalytic processes, it would seem plausible to fit the U parameter to a redox reaction energy. Therefore, in the present study, the U term was fitted with respect to the TiO₂ reduction reaction: 2TiO₂ + H₂ → Ti₂O₃ + H₂O. U recommended values for TiO₂ range from 2 to 10 through semiempirical fitting or linear response methods.²² Its value depends on the functionals and pseudopotentials, and then, it should be parameterized for each computational setup. The anatase phase of TiO₂ has been studied by Kitchin et al.²³ by parameterizing U to predict the relative energetic ordering between different phases of TiO₂ using different functionals and pseudopotentials. For anatase, the U values of 2.6 and 2.9 were determined using the linear response theory with PBE and PBEsol functionals, respectively, in conjunction with standard pseudopotentials.²³ PBE-D2 + U (U = 3) calculations have shown an accurate description of the electronic structure of reduced TiO₂ anatase using a TiO₂(001)-2 × 2 surface model.²⁴ It has been reported that including both the Hubbard U and van der Waals corrections is important for describing the interactions of reactive molecules with the anatase surface.²⁵

Therefore, it is important to consider an accurate and well-balanced description of some properties of the TiO₂ anatase model for further applications on surface simulations relevant for catalysis. Herein, we present a systematic study of the geometry, electronic structure, and the redox reaction energy for TiO₂ anatase and Ti₂O₃ through dispersion-corrected DFT + U calculations. The energetics of the formation of defects on a TiO₂(001) anatase surface were studied along with the analysis of the spin electronic state of the metal (Co and Y) dopant and the metal-doped structures.

2. Results and Discussion

2.1. TiO₂ Redox Reaction Energy, Geometry, and Electronic Structure

As can be seen from the results presented in Table 1, the reduction energy value computed at the PW91 level of theory exceeded the experimental reference value by 90 meV when U = 0. PBE-D3 and PBEsol computed values (Table 1) are close to the experimental value. Then, by applying the Hubbard correction (U ≠ 0), the reduction energy computed at the PBE-D3 level of theory exceeded the experimental value, thus being less suitable to be parameterized. For instance, the reduction energy calculated with the PBE-D3 (U = 1) method presented herein is overestimated in ∼59 meV with respect to the reference experimental value. The PBEsol-D3 method performed better, offering a wider range in the reduction reaction energy to fit the Hubbard correction and then it was selected to perform subsequent calculations.

Table 1. Comparative Analysis of the TiO₂/Ti₂O₃ Reduction Reaction Energies (E_r) Computed with Different Functionals without the Hubbard Correction (U = 0)^a.

functional	PWB91	PBE-D3	PBEsol	PBEsol-D3	ΔHrexp T=298K (eV)
Er (eV)	1.31	1.17	1.13	0.96	1.22

^aThe experimental value (ΔH_rexp T=298K) is presented for reference.

The consideration of a chemical reactivity parameter such as the redox energy for the functional selection and U fitting to describe the physical properties, such as the adsorption energy, seems to be less important. Indeed, PBE-D2 U = 3 calculations were carried out for the analysis of the defective TiO₂(001) surface and to study the adsorption properties.²⁴ However, it is unclear whether this may affect the oxygen vacancy formation energy because there is an additional limiting factor that must be taken into account related to the overestimation of the oxygen-binding energy when the GGA method is used.²⁶ This may be considered, however, for studies on chemical reactivity in catalysis, given the relevance of the reaction energy values on determining a preferred reaction pathway.²²

The reduction reaction energy was computed at different U values and comparatively analyzed with and without D3 dispersion correction. As can be seen from Table 2, TiO₂/Ti₂O₃ reduction reaction energies calculated by including D3 dispersion correction exhibit lower values compared to those without van der Waals correction (differences ranging from 0.13 to 0.18 eV). Indeed, when Hubbard U = 1 correction is considered in the PBEsol calculation, the reaction energy is overestimated with respect to the experimental reference value of 1.22 eV. Interestingly, dispersion-corrected computed values present smaller differences with respect to the experimental values with U = 2 and U = 3 as can be seen from the relative errors displayed in Table 2. By plotting the U parameter versus the PBEsol-D3 computed reaction energies and fitting the data to a second-degree polynomial equation (R² = 0.990), the U value of 2.4 is obtained for reproducing the experimental reduction reaction energy.

Table 2. Comparative Analysis and D3 Dispersion Correction Effect on the TiO₂/Ti₂O₃ Reduction Reaction Energies Computed with Different U Parameter Values at the PBEsol Level of Theory.

Hubbard parameter value	method/reaction energy (eV)
U	PBEsol	PBEsol-D3	% ϵ absolute percentage relative error PBEsol-D3
0	1.14	0.96	21.4
1	1.26	1.08	11.4
2	1.37	1.19	2.3
3	1.44	1.27	4.5
4	1.49	1.33	9.0
6	1.52	1.39	13.9

To analyze the effect of D3 correction on the electronic structure properties, the band gap (E_g) is computed from the band structures and is presented in Table 3.

Table 3. D3 Dispersion Correction Effect on the TiO₂ and Ti₂O₃ Band Gap (in eV) Calculated with Different U Values at the PBEsol Level of Theory.

	TiO2		Ti2O3
Hubbard parameter value	PBEsol	PBEsol-D3	PBEsol	PBEsol-D3
U	Eg (TiO2)	Eg (TiO2)	Eg (Ti2O3)	Eg (Ti2O3)
0	2.11	2.11	2.20 × 10–3	3.40 × 10–3
1	2.24	2.22	2.30 × 10–3	3.60 × 10–3
2	2.24	2.34	2.90 × 10–3	2.10 × 10–3
3	2.36	2.46	2.19 × 10–1	1.61 × 10–1
4	2.61	2.60	5.93 × 10–1	5.49 × 10–1
6	2.88	2.87	1.29	1.26
Eg experimental value (eV)	3.26a		1.00 × 10–1b

^aThis value was taken from ref (27).

^bTaken from ref (28).

As can be seen from the results, the D3 correction has a minor effect on the band gap values. As it is well known, the computed band gap for TiO₂ can hardly reach the experimental value at small U values using GGA functionals.^29,30 For the narrow band gap system Ti₂O₃, when using the dispersion-corrected method, the band gap computed value with U = 3 is close to the experimental reference value.²⁸ An increase in the band gap with the Hubbard parameter as expected is observed. The band plots are shown in Figure 1 for both TiO₂ and Ti₂O₃ structures.

Figure 1.

PBEsol-D3 (U = 3) band structures for (a) TiO₂ and (b) Ti₂O₃.

The TiO₂ anatase structure exhibits an indirect band gap of 2.46 eV (between M and Γ points) at the PBEsol-D3(U3) level of theory, in agreement with previously reported results.³¹ Similarly, Ti₂O₃ shows an indirect band gap (between ∼T and F points) of 0.16 eV (Figure 1), in agreement with the experimental value.²⁸

Martínez-Casado et al.³² reported that the dispersion correction exerted a minor influence on the electronic structure of TiO₂ anatase computed with the hybrid functional HSE because the calculated band gap slightly approaches to the experimental value when the D3 correction is employed. However, to our knowledge, no studies have reported the reduction energy of TiO₂ anatase computed with the PBEsol-D3(BJ) method. The effect of the dispersion correction on the energies of reduction of CeO₂ was thoroughly analyzed by Paier et al.³³ They found that by adding a dispersion term to the hybrid functional HSE, the reduction energies are improved compared to the experimental values. Indeed, Conesa highlighted the effect of the dispersion correction in describing the correct relative stability between the TiO₂ polymorphs.³⁴ Standard PBE calculations predicted that anatase and brookite are more stable than rutile, while the dispersion-corrected values recover the correct stability of rutile over the other phases. The results presented herein have shown that the dispersion correction improved the values obtained for the TiO₂ reduction energies by hydrogenation and are in the same line as those discussed above for reducible oxides. In addition, the D3(BJ) scheme enabled a better Hubbard correction parameterization when the PBEsol functional was used.

The crystal structure parameters were determined at different U values, and the dispersion correction effect D3(BJ) was analyzed for the studied titanium oxides and is presented in Figure 2. The fractional deviation expressed as a percentage error is reported for the computed cell parameters in plots a and c, whereas the volume percentage error is presented in separate plots (b and d). The results show that at higher U values, the geometry parameters tend to be overestimated and the dispersion-corrected geometries exhibit less variation with respect to the experimental reference parameters. With U = 2, dispersion-corrected geometries for TiO₂ are closer to the reference parameters, while for Ti₂O₃ with U = 3/D3, the geometry is better described.

Figure 2.

Percentage error (% Δ) between the computed and experimentally determined geometry parameters, a or c lattice parameters (Å), and the angle α (degrees) or the cell volume (ų), for TiO₂ (a,b) and Ti₂O₃ (c,d).

Recently, Jovanović et al.³⁵ obtained an accurate description on the structure and electronic properties of pristine and doped V₂O₅ by using the Hubbard-corrected PBE-D2 method. They found a good agreement between the computed band gap and lattice parameters with the experimental values. It was reported that the dispersion correction is needed to account for the interlayer spacing in the metal oxide.

The optimized structural parameters of the TiO₂ unit cell are a = 3.81 Å and c = 9.59 Å at the PBEsol U = 3 level of theory, whereas when the PBEsol-D3 method is used, the computed lattice parameters are a = 3.80 Å and c = 9.50 Å. A slight improvement in the calculated geometry compared to the experimentally determined structure is reached when a similar level of theory is used with a higher U value (U = 8.5).³⁶ For Ti₂O₃, at PBEsol U = 3, the computed lattice parameters are a = 5.44 Å and α = 56.74°, while when the D3 correction is considered, these values are a = 5.41 Å and α = 56.71°.

In general, the TiO₂/Ti₂O₃ reduction reaction energies computed using the PBEsol functional with U = 1 and using the PBEsol-D3 method with U = 3 are closer to the experimental values. However, after the analysis of the band gap values obtained for both titanium oxides, it is found that the dispersion-corrected PBEsol calculations with a Hubbard parameter of U = 3 give better results, specifically for Ti₂O₃ which approaches sufficiently well to the experimental value. At the PBEsol (U = 1) level of theory, it is determined that there is a deviation of 1.02 and −0.097 eV in the calculated band gaps of TiO₂ and Ti₂O₃, respectively, compared to that of the experimental values. Through PBEsol-D3 (U = 3) calculations, these deviations are 0.8 and >0.06. From the optimized geometry analysis presented in Figure 2, it can be seen that the geometry parameters of both titanium oxides are better described through the PBEsol-D3 (U = 3) method than with the PBEsol (U = 1) theory approach because they exhibited a lower error with respect to the reference values.

After analyzing the reduction energy, the electronic structure, and geometry described using different Hubbard parameter values, U = 3 is selected to get an adequate description on the properties of the titanium oxides studied herein. Consequently, PBEsol-D3 U = 3 calculations were carried out for the analysis of the energetics and electronic structure of the defective TiO₂(001) anatase surface.

2.2. Analysis of Oxygen Vacancy Formation in the TiO₂(001) Anatase Surface

The PBEsol-D3 (U = 3) TiO₂ anatase-optimized unit cell was used to build a surface model to analyze oxygen vacancy formation energies. The studied defective structures present an oxygen vacancy at sites 1, 2, and 3, as shown in Figure 3. The selected defective sites have shown lower vacancy formation energies in previous studies.^37,38 The extracted oxygen atom was computed in its triplet ground state, while the defective surface was computed in the singlet state, given that it has shown to be the most stable one as reported previously.³⁷

Figure 3.

(a) Pristine TiO₂(001) anatase surface model and (b) oxygen vacancy sites. Color code: Ti, gray; O, red; and oxygen vacancy, yellow.

The formation energies calculated herein for the studied vacancy sites (see Table 4) are in good agreement with the previously computed values.³⁸ The PBE (U = 4) reported values by Illas et al.³⁸ exhibit a similar stability trend on these vacancy sites, with site one (VO₁) being the most stable.

Table 4. Oxygen Vacancy Formation Energies at Different Sites on the TiO₂(001) Anatase Surface.

oxygen vacancy site	Ef(VO) (eV)
TiO2 (VO1)	3.54
TiO2 (VO2)	5.98
TiO2 (VO3)	5.12

It is well known that the oxygen vacancy induces excess charge in the titanium atoms adjacent to the defect showing occupied states near the TiO₂ conduction band.³⁹ The density of states (DOS) corresponding to the defective structures presented the mid-gap states induced by this charge excess as shown in Figure 4. These mid-gap states in the TiO₂ oxygen-deficient surfaces with an oxygen vacancy at sites 1 and 3 lay ∼1.2 eV below the conduction band of the pristine TiO₂(001) surface (Figure 4a,c).

Figure 4.

DOS plot of the TiO₂(001) anatase surface with an oxygen vacancy at sites (a) 1, (b) 2, and (c) 3, as depicted in Figure 3. The DOS plot of the pristine surface can be found in the Supporting Information.

The fully localized solution for the oxygen vacancy defect presents the two localized states in the mid-gap region. The experimentally observed defect levels for TiO₂ anatase were determined at ∼1 eV below the conduction band.⁴⁰ The defect level in the TiO₂ (VO₁) structure is located at 1.2 eV below the conduction band (Figure 4a). In a previous study, the conduction band offsets of the defect level for a neutral oxygen vacancy on a (001) TiO₂ anatase surface were determined at HSE and DFT + U levels of theory.⁴¹ The HSE reported values were 0.98 and 0.81 eV, while the PBE + U (U_p = 5.25 eV for O and U_d = 4.2 eV for Ti) computed values are 1.48 and 1.39 eV. The corresponding PBE + U formation energy reported for that neutral oxygen vacancy in the anatase surface is 3.49 eV, which is close to that determined in the present work for the TiO₂ (VO₁) structure of 3.54 eV (Table 4).

2.3. Hubbard U Correction Effect on the Spin Electronic States Relative Energy for the Metal Dopant in the TiO₂(001) Anatase Surface

The substitutional and interstitial doping by cobalt and yttrium atoms were explored for the TiO₂(001) anatase surface model. As presented in Figure 5, the more stable doping sites on the surface as reported by Illas et al.³⁸ were considered to compute the defect formation energies. The U = 3 value was used for Ti while computing further results, whereas different U values were explored for the metal dopant atoms cobalt and yttrium.

Figure 5.

(a) Substitutional and (b) interstitial doping by the metallic atom (M = Co or Y) in the TiO₂(001) anatase surface. Color code: Ti, gray; O, red; and dopant atom, pink.

For exploring the possible electronic states of the doping atoms, the energies of the isolated metal atom with different spin multiplicity have been computed with U = 0, 1, 2, and 3. Interestingly, for cobalt, the computed energy difference between the doublet and quartet states with U = 0 of 0.91 eV is consistent with the experimental data (see Table 5). From the results presented in Table 5, the doublet–quartet energy difference (E_D–Q) increases with the U value and the ground state for the cobalt atom corresponds to a high spin (HS) quartet state. The lowest lying electronic states corresponding to the LS state with one unpaired electron (D) and the HS state with three unpaired electrons (Q) were studied in the metal-doped structures.

Table 5. Energy Difference between the Doublet and Quartet Electronic States for the Cobalt and Yttrium Atoms (in eV) at Different U Values^a.

U	Co ED–Q (eV)	Y ED–Q (eV)
0	0.91	–0.96
1	1.39	–1.19
2	1.89	–1.41
3	2.40	–1.48
ED–Q exp42	0.92	–1.36

^aThe experimental values (E_D–Q exp) are included from ref (42).

The cobalt-doped structures exhibit a HS ground electronic state with a larger LS–HS energy difference for the substitutional doping. The LS–HS energy difference values for the cobalt-substituted (Co_s) and interstitially doped (Co_i) systems with U = 0 are 0.52 and 0.01 eV, respectively. There is a negligible difference by no more than 0.06 eV between these U = 0 LS–HS gap values and the U = 3 computed values (Co_sE_D–Q = 0.46 eV and Co_iE_D–Q = 0.02 eV). This result might explain the ferromagnetism detected in cobalt-doped TiO₂ anatase, and it agrees with the experimental determinations by Singh et al.¹³

For the yttrium-isolated atom, the computed energy difference between the doublet and quartet states with U = 2 of −1.41 eV is closer to the experimental value (see Table 5) and exhibits a doublet ground state. The yttrium-substituted (Y_s) structure exhibited a LS ground state as that determined for the yttrium atom. The LS–HS computed energy difference value with U = 2 for the yttrium-substituted doped system is −2.04 eV, and it only differs in 0.01 eV with respect to the U = 0 computed value (Y_sE_D–Q = −2.03 eV).

Unlike the yttrium atom, a HS electronic ground state is determined for the yttrium interstitially doped system, the HS and LS are quasi-degenerated, and they differ only in 3 meV with U = 0 or 2.

The metal doping formation energies are displayed in Figure 6, and as can be seen, they tend to increase with the Hubbard value. From the results, it can be inferred that the interstitial doping impurity (regardless of the metal atom) is energetically favored over the substitutional doping evidenced by the negative formation energy values shown in Figure 6a.

Figure 6.

Metal-doping energies (eV) for (a) interstitial (i subindex) and (b) substitutional defects (s subindex) in the TiO₂(001) anatase surface. The dotted line shows the cobalt doping data, and the solid line represents the computed data for yttrium-doped structures [the dark line corresponds to the HS (Q) ground state while the gray line corresponds to the LS state (D)].

The structure comparisons between the cobalt- and yttrium-doped structures are shown in the Supporting Information (Figure S3). Given the sizes of the substituted and doping atoms (the atomic radii of titanium, cobalt, and yttrium are 2.30, 2.16, and 2.47 Å, respectively),⁴³ it is expected that the yttrium atom induces more local distortion in the TiO₂ anatase crystal network than the cobalt atom, and this is depicted in Figure S3 as a greater change for yttrium in the average bond distances between the U = 0 and U = 3 computed defective structures. The interstitial doping produced the greatest local distortion in the TiO₂ crystal surface, as can be evidenced in Figure S3a,b.

The local atom position displacements in the defective structure increased with the Hubbard U value, thus inducing a more widespread charge spin density for cobalt (Figure 7b,d).

Figure 7.

Spin charge density differences of the metal-doped TiO₂(001) anatase surface plotted at an iso-value of 2.2 × 10^–3 e/Bohr³. Blue and yellow colors represent an increase (blue color) or a decrease (yellow color) in the electronic charge distribution. Color code: Ti, gray; O, red; Co, blue; and Y, pink.

As can be seen from Figure 7, the spin charge density difference is located over the doping site. It is worth mentioning that with U = 0, for which the observed LS–HS states energy difference of the cobalt atom is well reproduced by the calculation (Table 5), the spin charge density is more localized over the defective site regardless of the doping-type (see Figure 7a,c). In the interstitial doping, this is accompanied by the formation of mid-gap states through the contribution of Co 3d orbitals and oxygen p orbitals as can be seen from the partial DOS (PDOS) presented in Figure 8. For U_Coi = 3, the spin charge density spreads mainly more over the nearest titanium atoms (Figure 7d) and additional states appear close to the valence band (VB) like a band tail as depicted in the PDOS plot (Figure 8b).

Figure 8.

PDOS for the interstitially doped structures for the metal dopant (M, where M = Co and Y) with U_M = 0 or U_M = 3. (a) U_Coi = 0, (b) U_Coi = 3, (c) U_Yi = 0, and (d) U_Yi = 3. The mid-gap states are shown in the inset of each PDOS plot. The orbital contributions (d, p) for each atom (Y, Ti, O, and Co) are displayed in the plots. The dark line corresponds to the total DOS.

On the other hand, as mentioned above for yttrium, the U = 2 value allowed us to compute an energy difference between the LS–HS electronic states of the atom closer to the experimental reported value (Table 5). Interestingly, for yttrium-substitutional doping with U = 3, the spin charge density is more localized over the doping site than the defective structure, for which the Hubbard U parameter is not considered (U_Ys = 0). For the yttrium interstitial structure computed with U = 3, the spin charge density is slightly more delocalized than with U = 0 and seems to be related with the major local geometry distortion induced by the dopant atom. The spin charge density difference mapping for yttrium-doped structures with U = 2 resembled that displayed for U = 3. The PDOS for yttrium exhibited mid-gap states with contributions from Ti d orbitals and O p orbitals. The dopant did not contribute significantly to these intermediate states. With the increase in U, the mid-gap states close to the VB, mainly from oxygen, are more localized (see Figure 8d).

It follows from the previously presented results that it might seem reasonable to guide the U value selection for each dopant metal atom (cobalt or yttrium) trying to reproduce the energy difference between the ground state and the first higher spin electronic state of the isolated atom, thereby making the U choice relatively straightforward.

The U = 0 doping energies of the cobalt-doped TiO₂ anatase surface for substitutional and interstitial doping are 5.51 and −4.98 eV, respectively (see Figure 6). For yttrium calculations with U = 2, for the substitutional and interstitial defects the corresponding doping energy values are −0.86 and −6.51 eV (Figure 6).

3. Conclusions

From the obtained results, it can be concluded that it is important to include the D3 dispersion correction in the TiO₂ anatase and Ti₂O₃ DFT + U calculations. Herein, it is shown that a better geometry description is obtained for the titanium oxides when the D3 dispersion correction is applied. From the reactivity point of view, a parameter such as the redox reaction energy should be considered along with the electronic structure for the selection of the exchange–correlation density functionals aiming to get a reasonable theoretical model for further catalytic studies and to perform the U fitting. The Hubbard U = 3 value is suggested for a proper and well-balanced description of the geometry, electronic structure, and TiO₂/Ti₂O₃ reduction reaction energy in conjunction with the PBEsol-D3 method. By using the suggested method, the oxygen vacancy sites in a TiO₂(001) surface were studied. A surface oxygen vacancy site coordinated to two titanium ions was found as the most stable on the basis of the computed oxygen vacancy formation energy and an adequate description on the electronic structure of the defective structure. The results for the cobalt or yttrium doping in the TiO₂ surface might suggest that the HS–LS energy difference for the metal atom might be useful to guide the choice of the Hubbard parameter for the dopant atom. The U = 0 and U = 2 values were found for cobalt and yttrium atoms, respectively, attempting to fit these Hubbard parameters with respect to the experimentally determined LS–HS energies. The U parameter does not affect the ground electronic state determination in the metal-doped structures; however, higher values tend to increase the metal-doping energy value. The cobalt-doped TiO₂ anatase surface exhibited a HS (with three unpaired electrons) ground state. Besides, for the yttrium-doped structure with a substitutional impurity, a HS state was determined (with three unpaired electrons), while a LS state (with one unpaired electron) was detected for the interstitially doped TiO₂ surface. Based on the calculated metal-doping energies, it was found that the interstitial doping is energetically favored. The Hubbard U = 3 parameter in PBEsol-D3 calculations enabled an accurate description of the electronic structure for the oxygen vacancy sites and metal doping (Co and Y) in a TiO₂(001) anatase surface by using a large model (196 atoms) that would be worthwhile for heterogeneous catalysis simulations.

4. Theoretical Methods

Spin-polarized DFT-based calculations were carried out within the PAW method,^44,45 as implemented in the Vienna ab initio simulation package (VASP) code (version 5.4.4).⁴⁶⁻⁴⁹ The use of different GGA functionals, PBE,^50,51 PBE for solids (PBEsol),⁵² and Perdew–Wang PW91^53,54 was explored in the U fitting calculations. A plane-wave basis set was used with a kinetic cutoff energy of 600 eV. The Dudarev implementation of the Hubbard U (DFT + U) corrections²¹ and the D3(BJ) Grimme dispersion correction⁵⁵ were used in the PBE and PBEsol calculations. An electronic smearing method was incorporated to carry out the DFT calculations, and a 0.01 eV/Å force tolerance was considered on each atom. The full optimization of the TiO₂ unit cell was performed with an 8 × 8 × 8 Monkhorst–Pack k-point sampling set,⁵⁶ while a 12 × 12 × 12 k-point set was used for Ti₂O₃. QuantumATK (NanoLab, v R-2020.09)⁵⁷ and Vesta⁵⁸ software packages were used for the visualization of the structural models and the mapping of electronic properties. The VASP calculated data were postprocessed through the VASPKIT code.⁵⁹

The GGA functionals PBE, PBEsol, and PW91 were tested to compute the TiO₂ anatase reduction energy. The geometry of the TiO₂ and Ti₂O₃ unit cells was initially optimized by performing a procedure similar to that reported by Gulans et al. and Kitchin et al.^23,60 through sequential Birch–Murnaghan EOS calculations followed by polynomial fit calculations, which can be found in detail in the Supporting Information section. After that, a full optimization was performed over each unit cell, verifying in all cases the convergence of the preoptimized lattice parameters (see Supporting Information). The TiO₂ unit cell with a tetragonal crystal lattice (space group I4₁/amd) containing 12 atoms was taken from the work by Howard et al.⁶¹ The crystal parameters are shown in Figure 9. For Ti₂O₃ the primitive cell with rhombohedral structure and R3c symmetry was taken from Straumanis et al.⁶² This primitive cell contains 10 atoms, and the lattice parameters are shown in Figure 9.

Figure 9.

TiO₂ and Ti₂O₃ unit cells. Color code: Ti, gray and O, red.

The TiO₂ reduction energy was computed by considering the following reaction: 2TiO₂ + H₂ → Ti₂O₃ + H₂O and by using the next equation

1

where E(Ti₂O_3(bulk)) is the energy of the TiO₂ unit cell and it is divided by the number of formula units present in its unit cell (4), E(H₂O) is the energy of the water molecule, E(Ti₂O_3(bulk)) is the energy of the Ti₂O₃ unit cell and it is divided by the number of formula units present in its unit cell (2), and E(H₂) is the energy of the hydrogen molecule. The water and hydrogen molecules were optimized in a supercell with vacuum and a lattice parameter of 20 Å with each different functional.

The experimental reduction energies were calculated from the reported enthalpies of formation at 298.15 K.^63,64 The theoretical reduction energies were not subjected to thermal corrections, given that it is expected that this correction lies within the error inherent to the theoretical method, as it is shown in previous studies.^64,65

In recent years, efforts have been made for the theoretical description of supported multimetallic nanoparticles. Earlier studies reported PAW-PBE calculations without further improvement in the treatment of electron correlation or dispersion corrections on the adsorption and electronic properties of supported bimetallic clusters on metal oxides.⁶⁶⁻⁶⁸ Notwithstanding that progress, a proper method accounting for an accurate description of the electronic properties of the support and the metal–support interactions are needed to rationalize the reactivity of these systems as catalysts for oxidation reactions. An improved description of the correlation effects for reducible metal oxides demands a higher level of approximation within the DFT framework. Moreover, periodic DFT calculations on these heterogeneous catalysts require the adsorption of metal cluster models with sizes ranging from 3 to 6 Å of diameter. Then, a large surface supercell, such as that used herein for TiO₂(001) surface calculations, is needed to prevent interactions between periodic images of the supported clusters.

In view of the foregoing and based on the optimized TiO₂ unit cell optimized at the PBEsol-D3(BJ) + U = 3 level of theory, a TiO₂ anatase (001) surface model containing 192 atoms has been constructed by a 4 × 4 × 1 supercell with lattice parameters a = 15.21 Å, b = 15.21 Å, and c = 29.00 Å including a vacuum >15 Å.

For the geometry optimization, the lattice parameters were fixed at the optimized values for bulk anatase and then the four bottom layers were fixed, while the positions of the atoms in the outermost layers were fully optimized. A 2 × 2 × 1 k-points set was used to perform the surface calculations. The pristine and defective surfaces were optimized, and the oxygen vacancy formation energy was computed using the following equation

2

where E_f(VO) corresponds to the vacancy formation energy, E_O-vacancy is the supercell energy with a single vacancy site, E_perfect refers to the energy of the pristine supercell, μO is the oxygen potential computed as the half of the total energy of an oxygen molecule, and n is the number of vacancies which has a value of one in the present contribution.

The substitutional or interstitial metal-doping energies are calculated using the following equations

3

4

E_fs is the formation energy of the TiO₂-doped surface with yttrium or cobalt at a substitutional site, E_TM is the energy of the doped system, E_T is the energy of the TiO₂ pristine surface, μ_Ms corresponds to the chemical potential of the substitutional element, and μ_Ti is the chemical potential of titanium. E_fi is the formation energy of the TiO₂-doped surface with the transition metal at an interstitial lattice site, in which μ_Mi is the chemical potential of the interstitial doping atom..

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.1c02761.

• Geometry optimization procedure; converged geometries of TiO₂; converged geometries of Ti₂O₃; calculation parameters; TiO₂(001) anatase surface; geometry comparisons; CO₂ adsorption energy over the TiO₂(001) anatase surface; and DOS plot of the pristine TiO₂(001) anatase surface (PDF)

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

The authors declare no competing financial interest.