Quantum Chemistry of Solids / 24-Surface Modeling in LCAO Calculations of Metal Oxides

there are two possible OH orientations: inplane, with torsion angle H − O · · · H − O equal to 180◦ (structure 1, Fig. 11.12) and a conformation with smaller torsion angle (structure 2, Fig. 11.13).

O ter

Ti ter

Fig. 11.12. Dissociative adsorption, half-monolayer coverage: structure 1, view in (001) direction

Energy minimizations of the corresponding slabs led to similar adsorption energies for both structures, which were lower than the corresponding associative values for 1/2 monolayer coverage by ≈ 3–5 kcal/mol. Thus, these results are in accordance with previous DFT simulations for half-monolayer coverage that predict H2O dissociation on the rutile (110) surface.

A di erent conclusion was obtained from calculations on models corresponding to full-monolayer coverages. The distinct feature of these structures, in contrast to other cases, is the nearly equivalent displacements of the Ti and O atoms on the surface Ti layer such that they lie in one plane. Also, it is interesting to note that the overall arrangement of H and O atoms in the (110) surface is similar in both associative and dissociative adsorption. The obtained binding energies are stronger for associative adsorption on both the 3- and 5-layer slabs. This size of di erence (3.6 kcal/H2O molecule) in favor of associative adsorption was not obtained in previous DFT calculations.

In Table 11.11, the adsorption energies, obtained in PW calculations, are compared with those obtained in LCAO HF and DFT calculations for the structures, optimized

490 11 Surface Modeling in LCAO Calculations of Metal Oxides

Fig. 11.13. Dissociative adsorption, half-monolayer coverage: structure 2, view in (1–10) direction

in PW DFT calculations. The computational details of both PW and LCAO calculations are given in [790].

In Table 11.12 n Ti-L designates the number of Ti planes in the slab model, DB ECP means Durand–Barthelat e ective core potential for Ti atoms, [484]. As can be seen, the HF method using both all-electron and ECP bases gives the dissociative adsorption energies that are about 2 to 3 kcal/mole more favorable than those for associative adsorption. The DFT method results in the opposite picture: associative adsorption energies exceed the dissociative values by approximately 5 kcal/mole both for the 3 and 5 Ti-layer slabs.

The basis-set superposition error was estimated for ECP 3 Ti-layer hydroxylated slabs and it appears to be 6.4 kcal/mole per water molecule. The TiO2 slabs with the dissociated and molecular form of H2O have been calculated using exactly the same basis sets in all cases, so the large value of BSSE influences the absolute values of ∆E only, but not the relative energy of water dissociation on the TiO2 surface. Taking into account the value of BSSE, the absolute values of the obtained LCAO water-adsorption energies seem to lie in the interval of 25–35 kcal/mole that is slightly greater than the corresponding PW results.

The z-shifts of positions of the surface atoms on TiO2 slabs obtained by LCAO methods are given in Table 11.12.

The optimized geometry of a bare TiO2 surface and of hydroxylated (hydrated) slabs exhibits qualitatively the same relaxations in LCAO calculations as were found in PW calculations. But there are some quantitative di erences between the results of the DFT-PW, DFT-LCAO, and HF-LCAO approximations. It appears that the

Table 11.11. Calculated (without zero-point correction) at monolayer coverage and experimental adsorption energies (per one water molecule) for H2O/TiO2(110) (kcal/mol), [790].

a [809] b [803] c [798]

Table 11.12. z-shifts (˚A) of Ti and O atoms on the bare surface of (110) rutile 3 Ti-layer models obtained by LCAO calculations, [790]

492 11 Surface Modeling in LCAO Calculations of Metal Oxides

Kohn–Sham Hamiltonian (both on the PW and LCAO basis) produces the larger expansion of the slabs towards the vacuum, making them thicker and thus a ects the vertical displacements of atoms to be more positive relative to HF and experimental values. The same e ect was found for hydroxylated and hydrated species: horizontal and vertical distortions of top-surface oxygens and titaniums from their bulk positions are substantially larger for DFT-LCAO than for HF-LCAO. Also, the DFT method leads to more flexible hydroxyl groups on hydroxylated surfaces and as a consequence, to shorter H-bonds between them.

The LCAO basis gives the possibility to calculate atomic charges in a more direct way than the PW basis. In was found in [790] that the values of Mulliken charges lie in a reasonable region and reflect the partially covalent nature of chemical bonds in titanium oxides, although the HF method gives absolute values about 25% larger than the DFT method. The deviations of atomic charges on the surface of unhydroxylated slabs from their bulk values, generally, were less than 0.1 e both for Ti and O atoms, except the charge on the bridging oxygen obtained in DFT calculations that was reduced by more than 0.2 e. It is interesting to note that oxygen charges in hydroxylated and hydrated slabs become less negative in the order: |q(Obr)| > |q(O3f )| > |q(OH2O)|, which is correlated with the acidity of the corresponding hydroxyls.

Tables 11.11 and 11.12 demonstrate the relatively large dependence of the calculated water-adsorption energies on the nature of the quantum-mechanical approximation (HF or DFT) and type of basis set (PW or LCAO) within the same periodic-slab model. It should be noted that the results for the equivalent 2D and 3D slab models coincide if the su ciently large vacuum gap for 3D model is used, see Sect. 11.1.3. Nevertheless, we can conclude that the Kohn–Sham Hamiltonian using both PW and LCAO basis sets gives the order of the adsorption energy that is in better agreement with the experimental observations. However, the relaxation of the surface atoms seems to be more appropriate to experimental data [804] in the case of the HF calculations.

The long-term goal of research [790] was to model the interface of TiO2 and bulk water, so solvation forces will need to be included to predict adsorption energies and surface structures. Solvation forces are di cult to include in periodic DFT calculations because a large number of H2O molecules must be included in layers between rutile slabs to simulate bulk water. On the other hand, the HF embedded-cluster approach in the program CECILIA [801] can include H-bonding to H2O molecules not directly bonded to the surface and long-range solvation via a dielectric continuum half-space representing bulk water. The embedded-cluster approach is one obvious way to develop a force field for the Ti–O–H system representing the real TiO2–H2O interface [791], so consistency between this approach and the results of periodic calculations would be a useful step toward reliable molecular-dynamics simulations of the larger-scale rutile–water system.

Embedded-cluster calculations have been made for comparison with periodic studies using the model developed in [801]. This model utilizes a three-level interaction approach and takes into account the long-range forces properly. The central stoichiometric part of the cluster is treated by the ab-initio LCAO method using pseudopotential cores of extra Ti atoms to saturate the dangling bonds of the outermost oxygen atoms. A large stoichiometric grid of point charges represents the surrounding ions, and a special array of point charges distributed on a closed surface around the cluster

is constructed to reproduce the remainder of the long-range Coulomb interactions (i.e. the Madelung potential of the bulk crystal at the surface). The neutrality condition is satisfied separately for the stoichiometric part of the cluster and lattice array of ions plus extra pseudopotential cores.

Embedded-cluster calculations were carried out on stoichiometric clusters of different size: Ti7O14, Ti13O26 and Ti17O34 (the largest cluster is shown in Figures 11.14 and 11.15).

Fig. 11.14. Embedded-cluster results for the HF geometry of Ti17O34 − H2O: associative adsorption, [790]. Small-sized balls represent the Ti pseudopotential cores

In the Ti7O14 and Ti13O26 −H2O clusters only one Ti, one bridging O and the H2O atoms were allowed to relax during energy minimization. In contrast, the additional 4 oxygen atoms (nearest to the central Ti) were free to move in the Ti17O34 − H2O cluster. The larger cluster and greater surface relaxation were considered necessary to adequately describe the possible H-bonding arrangements of associatively and dissociatively adsorbed H2O on the TiO2 (110) surface. About 1000 full-point charges (i.e. +4 and -2) from the 4 Ti-layer slab and up to 310 partial charges fitted to the Madelung potential of the rest of the crystal were introduced to represent the electrostatics of the bulk TiO2. The positions of cluster atoms and lattice ions were chosen in accordance with the experimentally determined surface relaxation [804]. For embedded-cluster HF LCAO calculations the Hay–Wadt LANL1 or LANL2 [483] pseudopotentials for Ti atoms and SBK [485] pseudopotential on O atoms and corresponding basis sets have been used. In the case of LANL1, the 3s and 3p electrons of the Ti atom are included in the atomic core, whereas in the case of LANL2 the corresponding orbitals are treated as semicore states. For the pseudopotential cores of Ti atoms saturating the dangling bonds, only the semicore basis functions were included in the case of the LANL2 pseudopotential, whereas in the case of the LANL1

494 11 Surface Modeling in LCAO Calculations of Metal Oxides

Fig. 11.15. Embedded-cluster results for the HF geometry of Ti17O34 − H2O: dissociative adsorption, [790]. Small-sized balls represent the Ti pseudopotential cores

pseudopotential the basis functions were not included. This trick provides the same net charge on the extra pseudopotential Ti cores as on the lattice ions.

Table 11.13. Embedded-cluster results (without BSSE), [790] and experimental adsorption energies for H2O/TiO2 (110) (kJ/mol)

aEstimated BSSE is about 35 kJ/mol. bEstimated BSSE is about 29 kJ/mol.

The embedded-cluster results for water-adsorption energies are given in Table 11.13. This table demonstrates the dependence of the embedded-cluster results on the cluster size, basis set and Hamiltonian used. When the size of the cluster is increased, the energy di erence between the associative and dissociative adsorption mechanisms decreases.

The results for embedded-cluster calculations are di erent from that for the periodic models. Adsorption energies of a water molecule in the cluster case were much more favorable for the associative mechanism (Table 11.13), which is in better agreement with experimental observations. The predicted adsorption energies for the asso-

ciative mechanisms are higher than the measured values. There is no significant di erence between the optimized geometry for water adsorption in Ti13O26 and Ti17O34, the structures of the largest one are shown in Figures 11.14 and 11.15. No evidence of H-bond interactions was found in either the dissociative or associative adsorption structures in contrast to periodic PW and LCAO calculations. Bridging oxygen atoms in the cluster model appear to exhibit less relaxation than in periodic slabs.

The reason for such divergence between the periodic and cluster approaches is not entirely clear. The most probable reason is the di erences in symmetry and boundary conditions: the absence of periodic boundary conditions may lead to artificial rigidity of the Ti–O bonds. Also, assigning of formal charges to lattice ions may overestimate the Madelung potential of the bulk crystal in the cluster model. On the other hand, the 3- and 5-layer periodic models apparently overestimate the surface relaxation and the thicker-slab models are needed for improved quantities.

The results obtained in [790] correspond well with the experimental observation that the probability of H2O dissociation is increased with decreasing coverage. One explanation of this change in mechanism with coverage is that H2O molecules can readily align to provide the maximum H-bonding interaction in associatively adsorbed structures. At low coverages, H-bonding energies are not as significant as at higher coverages, so the additional energy of forming stronger Ti–OH bonds (as compared to Ti–OH2 bonds) outweighs the H-bonding term and dissociation may dominate.

Applying the di erent ab-initio quantum-mechanical approximations (DFT-PW, DFT-LCAO or HF-LCAO) within the periodic models produces qualitatively close results, but disagreement may exist in reproducing the di erence between close values if this di erence is about several kcal/mole. In particular, the DFT-LCAO method gives the order of the associative and dissociative water adsorption energy that is in better agreement with the experimental observations. Additional investigations of H-bonding between water molecules and oxygen atoms on the TiO2 surface using both DFT and HF methods should be made to resolve the disagreement between the periodic and embedded-cluster calculations.

The additional information about the water adsorption mechanism can be obtained in calculations of the oxides with the same bulk structure but di erent metal atom. The results of water adsorption on SnO2 surfaces is discussed in the next section.

11.2.3 Single-slab LCAO Calculations of Bare and Hydroxylated SnO2

Surfaces

Cassiterite (SnO2 in rutile structure)-based materials are extensively studied due to their gas-sensing properties. The theoretical modeling of the surface processes plays an important role in this research.

The first-principles methods have made an increasingly significant contribution to understanding the nature of clean SnO2 surfaces [806–808] and the interaction of these surfaces with adsorbed water [809–811], methanol [812], CO [813, 814], and O2 [815] molecules. Many of these calculations use the DFT PW periodic-slab model. The LCAO single-slab approach has been successfully applied for investigation of cassiterite [806, 812–815] A review of both PW and LCAO studies of adsorption on the perfect and reduced surfaces of metal oxides can be found in [784].

496 11 Surface Modeling in LCAO Calculations of Metal Oxides

We discuss here the results of LCAO calculations [816] based on both plain DFT PBE and hybrid B3LYP Hamiltonians to investigate the surface relaxation and adsorption of water molecules on the stoichiometric (110) and (100) SnO2 surfaces. The aims of this study can be summarized as follows: (1) comparison of the plain DFT functionals with the B3LYP hybrid functional for the surface modeling; (2) analysis of the energetic and structural adsorption properties of the stoichiometric (110) and (100) SnO2 surfaces (the latter was not studied before); and (3) finding the di erence in the adsorption of water on (110) and (100) surfaces.

Table 11.14 gives the comparison of calculated and experimental structure parameters (a, c, u), valence band (VB) width and bandgap (BG) for bulk SnO2 in rutile structure. As can be seen in Table 11.14, the LCAO-calculated values are in reason-

Table 11.14. Calculated equilibrium crystallographic parameters, valence-band width, and bandgap for the bulk rutile SnO2 crystal, [816]

aCalculated Fermi-energy level: –4.73 eV (B3LYP) and –4.01 eV (PBE). Experimental estimation: –4.35 eV [819]

able agreement with the experimental data and the results of DFT PW computations. The B3LYP form of GGA slightly underestimates the cell dimensions, whereas the PBE functional gives values that are closer to the experimental data. The minor discrepancies between the results of B3LYP calculations [816] and [815] may be due to the di erent form of pseudopotentials (Hay–Wadt in the first case and Durand– Barthelat in the second case) taken. The theoretical valence-band (VB) width (of 8.56 eV using B3LYP and 8.24 eV using PBE) agrees well with the experimental data [821] of 7.5–9.0 eV. As expected, the B3LYP method gives the bandgap (BG) of 4.0 eV that is close to the experimental value of 3.6 eV [821]. The di erences between the various PW-DFT calculations reported in Table 11.14 are obviously due to the computational details (choice of pseudopotentials, cuto energy, and others). In all cases, the plain DFT methods produce a bandgap that is about 2 to 3 times narrower than the experimental value. The valence-band density of electronic states (DOS) in bulk crystal calculated using the B3LYP functional [816] shows the well-known fact that oxygen p-states give the main contribution to the VB DOS, whereas the tin s- and p-states form the bottom of the conduction band.

From the results of PW-DFT periodic slab calculations [807], the cassiterite (110) and (100) type-2 surfaces are the most stable of the low-index faces and, hence, they should be the dominant crystallite arrangement of SnO2.

To check this by the single-slab approach LCAO calculations of the stoichiometric slabs (consisting of a whole number of SnO2 formula units) have been used for surface modeling [816]. As in the rutile TiO2 case the distribution of the bulk unit cell six atoms over atomic planes depends on the surface chosen: (110) O–Sn2O2–O; (100) O–Sn–O–O–Sn–O (see Fig. 11.3). The bridging oxygen atoms Obr terminate both surfaces; two bonds connect them with the sixfold Sn (Sn6f ) atoms on the rutile structure surfaces. There are fivefold tin atoms (Sn5f ) with one unsaturated bond and threefold oxygens (Os) in the atomic plane next to Obr, see Sect. 11.2.2 and Fig. 11.6. The positions of all the atoms in slabs were allowed to relax at a fixed dimension of the 2D surface unit cell that was taken from the bulk optimization result. The surface energy was calculated using (11.8) for threeand five-layer slabs for a (110) surface (9 and 15 atomic planes), and twoand 2.5-layer (12 and 15 atomic planes) slabs for a (100) surface to check the convergence of the surface energy with n in the DFT PBE LCAO calculations. The influence of the basis-set superposition error (BSSE) on the calculated values has also been estimated. For this purpose, three extra atomic layers of ghost atoms have been added on each side of the relaxed slabs and the total energy was recalculated.

Table 11.15. Calculated energy per unit area of the (110) and (100) surfaces of SnO2 crystal, [816]

aBSSE-corrected values are given in parentheses.

b5–6 Sn2O4–layers have been used in [807] for estimation of the surface energy.

The obtained surface energies are given in Table 11.15. Table 11.15 demonstrates that for both DFT PBE LCAO and hybrid B3LYP LCAO methods a (110) surface is more stable than a (100) surface, in accordance with PW-DFT calculations [807]. Comparing the surface energies in Table 11.15 before relaxation, we see that they are approximately the same for both surfaces. So, apparently, it is relaxation that provides the larger relative stability of the (110) surface. The B3LYP surface energies are larger than the corresponding PBE values by about 0.1 J/m2. On the other hand, the BSSE correction reduces the surface energies by about 0.1 J/m2, making them closer to the PW result [807]. However, it should be noted that the di erence between (110) and (100) surface energies is smaller using the B3LYP functional.

498 11 Surface Modeling in LCAO Calculations of Metal Oxides

In Table 11.16 we compare the bandwidths for the di erent surface models; the calculated Fermi energy is also included.

Table 11.16. Widths of electronic bands and Fermi energy (eV) for bulk SnO2 and for the di erent surface models (B3LYP results; AH – associatively hydroxylated, DH – dissociatively hydroxylated), [816]

Table 11.16 shows that a clean (110) surface has a wider VB and a narrower BG than a clean (100) surface. This fact can be attributed to the di erence in distribution of the electronic states corresponding to bridging oxygens on these two surfaces. It is known that just the bridging oxygens primarily contribute to the density of electronic states at the top of the VB. The LCAO calculations confirm this conclusion. We compare the obtained total DOS of the valence band for bulk, (110), and (100) surfaces in Fig. 11.16.

Fig. 11.16. Total valence-band DOS of bulk SnO2 crystal (fine line), its (110) (bold dark line) and (100) (bold gray line) surfaces calculated using the B3LYP functional (see text for explanation of energy scale)

Here, and in all figures below, the zero energy is taken at the Fermi level of the bulk crystal and all curves were shifted in such a way that the centers of O 1s bands coincide. All DOS values have been calculated per Sn2O4 formula unit in the solid phase to permit comparison on an equal scale. It is clearly seen in Fig. 11.16 that the new Obr subband appears in the surface systems at the top of the bulk VB.