The effects of O deficiency on the electronic structure of rutile TiO 2

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1 Journal of Physics and Chemistry of Solids 68 (2007) The effects of O deficiency on the electronic structure of rutile TiO 2 Gunhild U. von Oertzen a, Andrea R. Gerson b, a Ian Wark Research Institute, University of South Australia, Mawson Lakes, SA 5095, Australia b Applied Center for Structural and Synchrotron Studies, University of South Australia, Mawson Lakes, SA 5095, Australia Received 12 April 2006; received in revised form 23 August 2006; accepted 30 September 2006 Abstract Ab initio density functional calculations (plane wave GGA, CASTEP) were performed to determine the effect of O deficiency on the electronic structure of rutile, TiO 2. O deficiency was introduced through either the removal of O or the insertion of interstitial Ti atoms. At physically realistic concentrations of O vacancies in the rutile lattice (i.e. 25% and less) O deficiency results in the population of the bottom of the conduction band, the location of the Ti 3d orbitals in the pure structure, increasingly with increasing vacancy concentration. We propose that this could be confused with the formation and population of gap states especially where O vacancies occur in isolated positions in the lattice. In contrast, Ti interstitials introduce a defect state into the energy gap, without an overall reduction in the size of the energy gap. O vacancies result in a spin polarized solution, whereas Ti interstitials do not. r 2006 Elsevier Ltd. All rights reserved. Keywords: A. Semiconductors; C. Ab initio calculations; D. Electronic structure 1. Introduction The versatility of rutile TiO 2 is well documented, its uses range from applications in solar cells [1] to water purification [2] and pigments [3]. Recently, renewed interest in rutile has arisen from its ability to split water, making it one of the materials of interest in the quest for a sustainable hydrogen based economy [4]. In the electrolytic splitting of water, the electron-hole pair formed on absorption by TiO 2 of a photon is used to facilitate the chemical transformation of water into hydrogen and oxygen gas. In applications that exploit the photocatalytic activity of rutile [5], the size of the band gap is of importance, as a decreased band gap more readily allows photo excitation of electrons from the valence band (VB) to the conduction band (CB), thus maximizing solar utilization. Rutile has a wide band gap of 3.0 ev [6] and therefore photo excitation occurs for pure rutile on irradiation by light of wavelength approximately 400 nm (UV region) or below. A decrease in the band gap of 0.75 ev, for instance, would enable photo excitation by the green light region (550 nm). Solar Corresponding author. Tel.: ; fax: address: Andrea.Gerson@unisa.edu.au (A.R. Gerson). irradiance at the Earth s surface is about 1.2 W m 2 nm 1 at wavelengths of around 550 nm as compared to 0.2 W m 2 nm 1 around 400 nm [7]. Rutile crystallizes in the tetragonal space group P4 2 / mnm (number 136). The conventional unit cell contains two formula units of TiO 2, with Ti cations at the corner (0, 0, 0) and center (1/2, 1/2, 1/2) of the unit cell, and O anions at the positions (17u, 17u, 0) and (1/27u, 1/27u, 1/2); u equals [8]. Bonding results from the interaction of the three lone electron pairs on the O anion with metal cations resulting in distorted trigonal planar O coordination. The Ti atoms are octahedrally coordinated (Fig. 1). The strongly ionic character of the bonds results in localization of charge around the O 2 anions, with the VB being dominated by the O 2p electrons, while the lowest CB is populated chiefly by the unoccupied Ti 3d electron states [9]. The electronic and optical properties of rutile may be altered considerably by doping and/or defect structures. The dopant may either be inserted substitutionally or interstitially, and defects may also be created by the removal of atoms from the lattice. Early experiments demonstrated that the reduction of rutile by heating, which effectively removes oxygen atoms from the crystal structure, results in a blue color [10]. If O atoms are removed /$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi: /j.jpcs

2 G.U. von Oertzen, A.R. Gerson / Journal of Physics and Chemistry of Solids 68 (2007) Fig. 1. The crystal structure of rutile. Dark gray spheres represent Ti atoms and light gray spheres represent O atoms. from the lattice, the two electrons previously associated with the anion have to be retained in the lattice in order to preserve neutrality. There is some debate as to whether these electrons are associated with the vacancy site to form helium-like donor sites, or whether two adjoining Ti 4+ atoms are transformed into Ti 3+ sites [11]. Some experimenters have also suggested that O deficiency may be a result of the presence of interstitial Ti [12,13]. Huntington and Sullivan [14] have argued that the anisotropic blue coloration of reduced rutile points towards the interstitial mechanism. The electronic structure of pure bulk rutile has been the subject of numerous ab initio quantum chemical investigations [9,15 23]. However, considerably fewer studies have addressed the changes in electronic structure due to O deficiency in bulk rutile. A phenomenological tight-binding method due to Vos [24,25] has been used by Halley et al. [26] to investigate multiple O vacancies in rutile. A tail of donor states tailing into the band gap below the conduction band was observed with O vacancy concentration of up to 10% of the total O in the cluster. It is not clear, however, whether their model resulted from the removal of charged or neutral atoms, and the position of the Fermi level is not indicated. A periodic Hartree-Fock calculation by Mackrodt et al. [27] finds electron-excess gap states associated with O vacancies both in the bulk and on the surface. According to their model, these gap states are spin-polarized states corresponding to excess electron density trapped at bulk vacancy sites. Semi empirical cluster calculations of rutile by Hagfeldt et al. [28] also show a gap state due to O vacancies in one-layer clusters. This observation must be interpreted as a surface effect however, as a one-layer cluster cannot adequately describe the bulk. O vacancies on the rutile {1 1 0} surface have been proposed to play an important role in dissociative water adsorption associated with the photocatalytic activity of rutile. Schaub et al. [29] find dissociation of water on the {1 1 0} surface to be energetically possible only at bridging O vacancies, indicating that the O vacancies enhance the reactivity of the surface. They propose a mechanism whereby oxygen vacancies in the surface layer dissociate H 2 O through the transfer of one proton to a nearby oxygen atom, thus forming two hydroxyl groups at an oxygen vacancy site. The application of different quantum chemical methodologies to different, and sometimes poorly defined, O defect structures of varying concentrations has made an overall assessment as to the effect of O deficiency in rutile difficult. A systematic investigation of the different forms that O deficiency may take in rutile over a wide concentration range has not previously been undertaken using a single quantum chemical methodology. In this paper, we shall discuss the effect of bulk O deficiency in the form of lattice vacancies, over wide concentration ranges, as well as Ti interstitials, with the aim of providing an encompassing assessment as to the effects of these defects on the electronic structure. 2. Computation method CASTEP [30] is a density functional (DFT) plane-wave pseudopotential method. We use the generalized gradient approximation (GGA) throughout, which provides a more accurate overall description of the electronic subsystem than the local density approximation (LDA); in particular we have chosen to use the PBE exchange-correlation functional due to Perdew et al. [31], which is the default functional used in GGA CASTEP calculations. The electronic structure is calculated by parameterizing all the atoms of the crystal using ultrasoft pseudopotentials for the core electrons. The geometry optimization of the non-defective rutile structure was performed with a basis set cut-off of 380 ev. All calculations involving defects were performed using fine accuracy with a basis-set plane wave cut-off energy of 340 ev or higher. This is sufficient to obtain a good geometry optimization of the structure, and gives accurate results for the electronic structure. It is necessary to adopt the supercell approach to successfully model low impurity concentrations. The largest size of the bulk supercell structures for the impurities was generally taken as 2 2 2, i.e. two conventional unit cells wide in each of the three spatial directions. This gives a reasonably low, 3.1%, concentration of a single impurity relative to the O content in the pure bulk, while still a manageable calculation. Even if physical vacancy concentrations are much lower than this, trends in the electronic structure on decreasing the vacancy concentration in the supercell approach can be used to

3 326 ARTICLE IN PRESS G.U. von Oertzen, A.R. Gerson / Journal of Physics and Chemistry of Solids 68 (2007) infer the expected behavior in physical systems. However, defect-defect distances in the largest supercells are only about 6 A, not large enough to fully eliminate interactions between defects. In particular, the defective lattices of smaller supercells would represent the clustering of defects, as the defect defect distances can become as small as the length of the unit cell. All the calculations involving O vacancies have been performed in spin polarized mode, even though the total number of electrons involved is even, suggesting spin pairing. Both the fractional coordinates of the atomic positions as well as the size of the primitive unit cell are optimized with respect to the energy, starting from the experimentally determined structure. This unit cell size was then applied thereafter. Geometry optimization of the supercell structures consisted of only the optimization of the position of atoms, not the size of the supercell, as defects are supposed to be localized features not affecting the overall crystal symmetry and geometry. In fact, if the cell size and geometry is allowed to relax as well, the local symmetry is destroyed, particularly in the case of the smaller supercells. In the case of Ti interstitials, only a supercell was studied. Band structure diagrams and density of states are then obtained from the optimized structure. Because of the size of the supercells, band structure diagrams are very crowded and it is more instructive to study the density of states. 3. Results 3.1. Pure bulk rutile The experimental and calculated structural data for rutile are presented in Table 1. The optimized cell size and structural parameters agree very well with the experimental values, with deviations of 1% or less. The result for the bulk modulus is smaller than the experimental value by 6.5%, in contrast to the LDA result obtained by Glassford and Chelikowsky [16], which is larger than the experimental value by 11%. This is as expected, as the overbinding of atoms characteristic of the LDA functionals is generally over-corrected for when using GGA functionals [30]. There is no difference between a spin polarized and a spin unpolarized calculation for bulk rutile: the spin-polarized calculation converges to a zero spin solution (even if a highspin configuration is chosen as a starting value), and the pure bulk can thus be safely regarded as unpolarized. The band structure of pure bulk rutile is shown in Fig. 2. The band gap is direct at symmetry point G, and with a gap of 1.85 ev, in close agreement with the 2.0 ev obtained by the LDA calculation of Glassford and Chelikowsky [16]. The smaller than experimentally derived band gap is as expected for density functional calculations which generally underestimate the band gaps of insulators and semiconductors. Table 1 Crystal structure data for pure bulk rutile TiO 2 Experiment Theory (GGA-PBE, %) Lattice parameter a (A ) [39] (+0.9) Lattice parameter c (A ) [39] (+0.02) Bulk modulus (GPa) 216 [40] 202 ( 6.5) T O bond length (A ) [39] (+0.3) [39] (+1.0) O Ti O bond angle [39] (+0.5) [39] Optical band gap 3.0 ev [6] 1.85 ev Structure parameter u [8] (+0) The density of states of pure bulk rutile is shown in Fig. 3. The DOS plot is produced with a smearing width of 0.2 ev in order to more closely resemble experimental data such as the X-ray absorption spectrum characterizing the CB [32] or the photoemission spectrum characterizing the VB [33]. The smearing width results in an apparent band gap of approximately 1.6 ev in the DOS plot. Thus the exact value of the band gap (1.85 ev) must be obtained directly from the band structure diagram, with the DOS plot showing only the overall distribution of states across the energy range. For the defect DOS plots, we use a smearing of only 0.05 ev in order to clearly identify possible defect states in the band gap. In the VB, the O states are almost exclusively 2p, while the Ti states are predominantly 3d. The VB has a band width of 5.55 ev, in excellent agreement with the experimental value of 5.4 ev [34], and is clearly split into O 2p bonding (below 3.3 ev) and nonbonding (from 3.3 to 0 ev) contributions. In the CB, Ti states are almost exclusively d-like and O states are predominantly p-like. As can be seen from the band structure diagram (Fig. 2), the CB is split into two Ti 3d regions, which can be identified with the lower-lying Ti-t 2g (six bands, three from each Ti atom in the primitive cell) and the higher energy Tie g (four bands, two from each Ti atom). Although the octahedral coordination of the rutile structure is slightly distorted, it has been shown by Sorantin and Schwarz [15] that the classification into t 2g and e g is justified, as the DOS of the Ti-t 2g have almost no admixture of the Ti-e g bands and vice versa. Both the VB and CB contain contributions from O 2p and Ti 3d, indicating hybridization between these states, which also means that transitions across the band gap will involve both O 2p and Ti 3d states. Therefore, both of these bands are involved in bonding. The planar arrangement of the three Ti atoms coordinated to each O atom suggests 2sp 2 hybridization of the central O atom. The two different bond lengths (two bonds of A and one of A in our model) suggest the formation of three s bonds in the plane, involving sp 2 orbitals, as well as one p bond perpendicular to the plane, involving the remaining p orbital, i.e. the shorter bond length involves the p bond as well as a s bond. Metal atoms have six ligands positioned

4 G.U. von Oertzen, A.R. Gerson / Journal of Physics and Chemistry of Solids 68 (2007) Fig. 2. Band structure of pure bulk rutile along selected symmetry directions in the Brillouin zone. A zero energy corresponds to the highest occupied state. The gray horizontal line marks the size of the band gap, 1.85 ev. Fig. 3. Total and partial density of states of bulk rutile, showing the total density of states (bottom), the Ti projected density of states (top), both with s-, p- and d-like contributions, and O projected density of states (middle) with s- and p-like contributions. calculated using neutral atoms, each with a single O vacancy. The number of occupied states in what used to be the CB in the pure bulk, increases with increasing O vacancy concentration, as expected. The details of the band structure diagrams are summarized in Table 2. The population of the bottom of the CB must be interpreted as the excess electron gap states reported by Mackrodt et al. [27] as these states constitute the states occupied by additional electrons neutralizing the lattice after the removal of neutral O atoms. The fact that no defect level is found within in the energy gap itself is consistent with the findings of Cho et al. [35], who find that the band associated with the defect is located within the CB and not in the energy gap. The energy gaps of all the impurity supercells are larger than the pure bulk band gap, at between 1.98 and 2.33 ev. For 25% O vacancy, the band gap is smaller than for the structures with either 12.5% or 6.25% O vacancy, and the smallest band gap is exhibited by the pure vacancy free structure. There is a decrease in VB width on increase of the O vacancy concentration with the VB width for the supercell with one O vacancy being 4.6 ev and the largest width, 5.55 ev, exhibited by the pure vacancy free structure, as well as the lowest vacancy concentration with 3.14% impurity concentration. It must be noted, however, that the population of the bottom of the CB by the excess electrons states left over after the removal of O ions, constitutes the presences of states in the band gap, so that the band gap for these impure structures is really just a gap in the VB, no longer a band gap in the conventional meaning of the word. Spin polarization of the supercells increases on an increase of supercell size, i.e. a decrease in O vacancy concentration (Table 2). The excess electrons, which are left behind when O is removed from the crystal, are to some around them, suggesting e 2 g 4sp 3 hybridization for the unoccupied valence states, which would form s bonds with ligand 2sp 2 orbitals, while p bonds are formed between the t 2g orbitals and the remaining unhybridized O 2p orbital. According to Sorantin and Schwarz [15], the bonding is predominantly between Ti-t 2g and O-2p y on the one hand, and Ti-e g and O-2p x and O-2p z on the other hand (charge contour maps can be found therein) O vacancies The O deficient crystal will be charge neutral. This means that 2 e remaining per O atom removed will have to be accommodated in the band structure, which results in the bottom of the CB being populated. This is demonstrated in Fig. 4, which shows the DOS of varying sized supercells Fig. 4. Density of states for charge neutral TiO 2 supercells containing a single O 0 vacancy, with the pure bulk shown at the bottom, followed successively above by an eight unit (2 2 2), a four unit (2 2 1) and a two unit (2 1 1) supercell, and a unit cell with one O vacancy (top). Black vertical lines mark the amount of occupation of the CB, and the DOS were shifted respectively by, from top to bottom, 2.70, 2.65, 2.38, 2.17 ev, in order to align the top of the VBs.

5 328 ARTICLE IN PRESS G.U. von Oertzen, A.R. Gerson / Journal of Physics and Chemistry of Solids 68 (2007) Table 2 Band structure and spin details for charge neutral rutile supercells with one O vacancy Integrated spin density per supercell Number of vacancies %O vacancy Energy gap VB width Bottom of CB extent unpaired, the degree of which is inversely proportional to the ratio of O vacancies to bulk Ti atoms, i.e. at low vacancy concentrations the excess electrons have more empty Ti 3d orbitals over which they may spread. Thus Ti 4+ ions will be converted to Ti 3+ ions. In contrast Lindan et al. [36] have found that the lowest energy state for all of the 1 1 1, 2 1 1, and supercells is spin polarized, whereas Cho et al. [35] find spin moments close to zero, maybe because the latter use the local density approximation. As has been noted by Cho et al. [35], an increase in supercell size leads to a shift of the localized defect levels. As indicated by the shift from a spin-paired to a spinpolarized solution on an increase of the supercell size from 2 1 1to2 2 1, the smaller supercells may not give an accurate description of the localized levels. In particular, the authors have noted that fixing the geometry in the pure bulk position caused the creation of a gap state within the energy gap. Therefore, even larger supercells may be necessary to accurately characterize the defect states. The shift between a spin-paired and a spin-polarized scenario is highlighted when comparing the integrated spin densities per supercell in Table 2: while the integrated spin density in the supercell is , it is 2.43 in the case of the supercell and remains at the level of about 2 per supercell in the larger supercell. The difference is illustrated in Fig. 5, where the spinpolarized DOS for the 2 1 1and2 2 1 supercells is compared. In the case of the supercell, spin-up and spin-down electron states are similar, showing spin pairing, whereas in the case of the supercell, the CB contains additional spin-down states for which there are no counterpart spin-up states. The spin-unpaired electrons in the larger supercells (2 2 1 and 2 2 2) appear localized around a Ti atom neighboring the O vacancy, making the state ferromagnetic. Errico et al. [37] have performed ab inito calculations of a supercell with an O vacancy, finding a magnetic moment of 1 m B per supercell in this simple example, and confirm a large distortion of the local geometry around the impurity. They point out that the experimental evidence for ferromagnetism in rutile and anatase is still controversial and depends strongly on the growth conditions of the samples, and is linked to metallic impuritites which may be present in the samples. Fig. 5. Spin-polarized density of states of a (top) and (2 1 1) supercell (bottom) containing a single O vacancy, respectively, showing the spin-up and spin-down electron states separately Interstitial Ti The presence of either interstitial Ti or O vacancies constitutes an O deficiency. The DOS for these two systems is compared in Fig. 6, which shows a comparison between two charge neutral supercells. While the O vacancy does not introduce a defect state into the energy gap, the introduction of an interstitial Ti atom does, as can be seen from the DOS plot. However, this does not affect either the energy gap or the VB width significantly, as can be seen from the summary of band gaps in Table 3. Rather, the effect of the Ti interstitial is to shift the VB in such a way as to maintain practically the same energy gap as is found in pure rutile. The band gap in the case of Ti interstitials is increased over the pure bulk band gap by only 2% too small to be significant. As only neutral supercells have been investigated in this case, no statement can be made about trends with changes of defect concentration or charge on the size of the band gap. A notable difference between the two scenarios comes in the spin polarization: the structure containing the interstitial Ti converges, effectively, to a low spin solution whereas in contrast the O vacancy structure converges to a high-spin solution, at least in the supercell investigated here. This is due probably due to the fact that ati 4+ interstitial donates four extra electrons to the lattice, rather than two as is the case in an O vacancy, thus giving additional opportunity for spin pairing.

6 G.U. von Oertzen, A.R. Gerson / Journal of Physics and Chemistry of Solids 68 (2007) Table 4 Reaction energies for pure and defective bulk rutile, in kj mol 1 per unit cell Reaction energy E r (kj mol 1 per unit cell) Pure rutile supercell with one O vacancy supercell with one interstitial Ti Fig. 6. Density of states of pure rutile (gray), compared to the DOS of a rutile supercell containing an interstitial Ti 0 atom (black solid line) and of a supercell with an O 0 vacancy (dotted black line). The DOS of the latter two were shifted to align the top of the VB for the three graphs. Table 3 Band gap and spin information for a supercell that is either stoichiometric or contains one O vacancy or a Ti interstitial Band gap VB width Integrated spin density per supercell Bottom of CB O vacancy Interstitial Ti Pure bulk In order to determine the relative stability of the two scenarios, we calculate the reaction energy E r for the conversion of the gas phase reactants into defective TiO 2 and gases [38]. For pure bulk rutile, the unit cell contains 2 formula units of TiO 2,so 2TiCl 4 þ 2O 2! 4Cl 2 þ 2TiO 2 (1) yielding the reaction energy per unit cell E r ¼ Eð2TiO 2 Þþ4EðCl 2 Þ 2EðTiCl 4 Þ 2EðO 2 Þ, (2) where E(X) is the total energy of compound X. For ease of comparison, the energies of TiCl 4, Cl 2 and O 2 were computed by inserting one molecule of each into a cubic cell with a cell parameter of 8 A, then calculating the total energy of the structure with its geometry optimized. For O deficiency, the reaction is nticl 4 þ 2n 1 O 2! 2nCl 2 þ Ti n O 2n 1 (3) 2 and the reaction energy is E r ¼ EðTi n O 2n 1 Þþ2nEðCl 2 Þ neðticl 4 Þ 2n 1 EðO 2 Þ, 2 (4) where for the supercell, n ¼ 16. For the Ti interstitials ðn þ 1ÞTiCl 4 þ no 2! Ti nþ1 O 2n þð2n þ 1ÞCl 2 (5) leading to the reaction energy E r ¼ EðTi nþ1 O 2n Þþð2n þ 1ÞEðCl 2 Þ ðn þ 1ÞEðTiCl 4 Þ neðo 2 Þ. ð6þ The reaction energies for pure bulk rutile, rutile with an O vacancy and rutile with an interstitial Ti atom are summarized in Table 4. The largest negative value for the reaction energy signifies the most stable structure, so stability is decreased in the table going down, from pure bulk rutile (most stable) via rutile with an O vacancy to rutile with an interstitial Ti atom (least stable). Of course, the DFT calculations are performed at zero temperature, not at the physically realistic temperatures applicable to the gas phase reactants or the actual reaction temperatures. Because the zero temperature reaction energies calculated here differ from each other by only about 15%, this could be sufficient to reverse the result for physically realistic temperatures. However, Cho et al. [35], although using a different method, arrive at the same conclusion regarding the relative stability of O vacancies and Ti interstitials. 4. Conclusions At realistic concentrations of O vacancies in the rutile lattice (i.e. 25% and less), we find no evidence for the formation of states in the band gap, as has been reported by some authors, e.g. Halley et al [26]. However, the population of states in what was formerly the CB, is equivalent to the formation and population of gap states, as the effects on the optical properties will be the same, especially as O vacancies will occur in isolated positions in the lattice. In these cases the band gap (valence band to occupied conduction band) is larger or at least equal to that of the non-defective lattice. The width of the VB gradually decreases on increasing O vacancy concentration. The bottom of the CB, populated by the Ti 3d orbitals, is populated increasingly with increasing vacancy concentration. The occupation of the bottom of the CB by electrons would lead to a significant reduction in the photon energy required for electron excitation and hence associated photocatalytic reactions. The degree of spin polarization

7 330 ARTICLE IN PRESS G.U. von Oertzen, A.R. Gerson / Journal of Physics and Chemistry of Solids 68 (2007) increases with reduction in vacancy concentration as the excess electrons can spread over more unoccupied Ti 3d orbitals. In contrast to an O vacancy, an interstitial Ti atom in a supercell introduces a defect state into the band gap, without a change in energy gap or VB width. The formation of O vacancies in the lattice is slightly favored over the formation of Ti interstitial atoms energetically, but the difference is small enough to suggest both may occur concurrently. However, the supercell with an O vacancy converges to a spin polarized solution, whereas the one with an interstitial Ti atoms does not. Thus it may be possible to differentiate between these two forms of defect via electron paramagnetic resonance spectroscopy. Acknowledgements We acknowledge the use of CPU time under the Australian Partnership for Advanced Computing (APAC) s Merit Allocation Scheme, and at the South Australian Partnership for Advanced Computing (SAPAC). References [1] B. O Regan, M. Grätzel, Nature 353 (1991) 737. [2] M. Lazzeri, A. Vittadini, A. Selloni, Phys. Rev. B 63 (2001) /1. [3] J.H. Braun, J. Coatings Technol. 69 (1997) 59. [4] A. Fujishima, K. Honda, Nature 238 (1972) 37. [5] U. Gesenhues, J. Photochem. Photobiol. A: Chem. 139 (2001) 243. [6] A. Amtout, R. Leonelli, Phys. Rev. B 51 (1995) [7] G.J. Rottman, G. Mount, G. Lawrence, T. Woods, J. Harder, S. Tournois, Metrologia 35 (1997) 707. [8] R. Restori, D. Schwarzenbach, J.R. Schneider, Acta Crystallogr. B 43 (1987) 251. [9] N. Daude, C. Gout, C. Jouanin, Phys. Rev. B 15 (1977) [10] D.C. Cronemeyer, Phys. Rev. 113 (1959) [11] F.A. Grant, Rev. Mod. Phys. 31 (1959) 646. [12] C. Meis, J.L. Fleche, Solid State Ion (1997) 333. [13] C. Sayle, C.R.A. Catlow, M.A. Perrin, P. Nortier, J. Phys. Chem. Solids 56 (1995) 799. [14] H.B. Huntington, G.A. Sullivan, Phys. Rev. Lett. 14 (1965) 177. [15] P.I. Sorantin, K. Schwarz, Inorg. Chem. 31 (1992) 567. [16] K.M. Glassford, R. Chelikowsky, Phys. Rev. B 46 (1992) [17] A.R. Gerson, R. Jones, D. Simpson, G. Pacchioni, T. Bredow, Ionics 7 (2001) 290. [18] S.-D. Mo, W.Y. Ching, Phys. Rev. B 51 (1995) [19] B. Poumellec, P.J. Durham, G.Y. Guo, J. Phys.: Condensed Matter 3 (1991) [20] L.B. Lin, S.D. Mo, D.L. Lin, J. Phys. Chem. Solids 54 (1993) 907. [21] T. Bredow, K. Jug, Chem. Phys. Lett. 223 (1994) 89. [22] R.A. Evarestov, A.V. Leko, V.A. Veryazov, Phys. Stat. Sol. B 203 (1997) R3. [23] A. Fahmi, C. Minot, B. Silvi, M. Causa, Phys. Rev. B 47 (1993) [24] K. Vos, J. Phys. Chem. Solids 10 (1977) [25] K. Vos, H.J. Krusemeyer, J. Phys. Chem. Solids 10 (1977) [26] J.W. Halley, M.T. Michalewicz, N. Tit, Phys. Rev. B 41 (1990) [27] W.C. Mackrodt, E.A. Simson, N.M. Harrison, Surf. Sci. 384 (1997) 192. [28] A. Hagfeldt, H. Siegbahn, S. Lindquist, S. Lunell, Int. J. Quant. Chem. 44 (1992) 477. [29] R. Schaub, P. Thostrup, N. Lopez, E. Lægsgaard, I. Stensgaard, J.K. Nørskov, F. Besenbacher, Phys. Rev. Lett. 87 (2001) [30] Accelrys, Materials Studio CASTEP, Accelrys Inc., San Diego, [31] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) [32] G. van der Laan, Phys. Rev. B 41 (1990) [33] Z. Zhang, S.-P. Jeng, V.E. Henrich, Phys. Rev. B 43 (1991) [34] S.P. Kowalczyk, F.R. Mcfeely, L. Ley, V.T. Gritsyna, A. Schirley, Solid State Commun. 23 (1977) 161. [35] E. Cho, S. Han, H.-S. Ahn, K.-R. Lee, S.K. Kim, C.S. Hwang, Phys. Rev. B 73 (2006) [36] P.J.D. Lindan, N.M. Harrison, M.J. Gillan, J.A. White, Phys. Rev. B 55 (1997) [37] L.A. Errico, M. Renteria, M. Weissman, Phys. Rev. B 72 (2005) [38] M. Steveson, T. Bredow, A.R. Gerson, Phys. Chem. Chem. Phys. 4 (2002) 358. [39] T. Cromer, K.J. Herrington, Am. Chem. Soc. 77 (1955) [40] M.H. Manghnani, S. Fisher, W.S.J. Brower, J. Phys. Chem. Solids 33 (1972) 2149.