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

11.2 Surface LCAO Calculations on TiO2 and SnO2

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In the case of a (110) surface the broadening of the DOS distribution towards the BG is wider and the separation of the Obr subband from the bulk VB is more prominent than those in the case of a (100) surface. This fact indicates [784] that the basicity of Obr is larger on a (110) surface than on a (100) surface.

Vertical displacements of the surface atoms relative to their positions in the bulk structure are reported in Tables 11.17 and 11.18.

Table 11.17. Vertical displacements (˚A) and atomic chargesa (e) on the SnO2 (110) surface (5 Sn2O4-layer slab results using LCAO DFT (B3LYP calculations; AH – associatively hydroxylated, DH – dissociatively hydroxylated), [816]

	Bare surface		AH surface		DH surface
Atomb	z-	charge	z-	charge	z-	charge
	shiftc		shift		shift
Sn6f , 6-fold surface tin	0.21	1.91	0.10	1.97	–0.05	1.96
	(0.22)
Sn5f , 5-fold surface tin	–0.12	1.76	–0.06	2.00	0.04	1.99
	(–0.11)
Obr, bridging oxygen	0.05	–0.91	0.05	–1.01	0.02	–0.79
	(0.09)
Os, 3-fold surface oxygen	0.14	–0.93	0.04	–1.02	–0.01	–1.03
	(0.18)
Ou6f , subbridging oxygen	0.04	–1.03	0.03	–1.03	–0.02	–1.04
	(0.07)
Ou5f , oxygen underneath Sn5f	–0.04	–0.93	–0.01	–1.04	0.02	–1.03
	(–0.04)
Oterm, oxygen in terminal	–	–	–	–0.66	–	–0.84
hydroxyl or water oxygen
Hterm, hydrogen at Oterm or	–	–	–	0.35	–	0.34
corresponding water hydrogen
Hbr, hydrogen at Obr or	–	–	–	0.39	–	0.38
corresponding water hydrogen

aCharges in the bulk SnO2: q(Sn) = 2.12, q(O) = –1.06; charges in the isolated water molecule: q(O) = –0.62, q(H) = 0.31.

bFor atom indexing see text.

cDFT result of [807] is given in parentheses.

As expected, the top-surface atoms have the largest displacements. However, there is a qualitative di erence between the two surfaces under consideration. Namely, in the case of a (110) surface the shifts of Sn5f (inward) and Sn6f together with Os (upward) are the most noticeable, while in the case of a (100) surface only the bridging oxygen has a marked positive displacement. In spite of the di erent calculation methods used, the agreement of these LCAO values with the PW data from [807] is excellent.

The Mulliken population analysis has been applied for the calculation of the atomic charges that are also included in Tables 11.17 and 11.18. The noticeable reduction of the absolute charges for the top surface atoms is seen. This e ect is relatively larger for a (100) surface than for a (110) surface. Thus, the charge of Sn5f decreases by

500 11 Surface Modeling in LCAO Calculations of Metal Oxides

Table 11.18. Vertical displacements (˚A) and atomic charges (e) on the SnO2 (100) surface (5 Sn2O4-layer slab results using LCAO DFT (B3LYP calculations)a, [816]

		Bare surface		AH surface		DH surface
	Atom	z-	charge	z-	charge	z-	charge
		shiftc		shift		-shift
	Sn5f , surface 5-fold tin	0.03	1.67	0.02	1.95	–0.01	1.97
		(0.05)
	Sn6f , surface 6-fold tin	0.00	2.07	0.00	2.10	–0.01	2.11
	Obr, bridging oxygen	0.13	–0.83	0.05	–1.03	0.00	–0.81
		(0.18)
	Os, 3-fold surface oxygen	0.07	–0.94	0.04	–1.02	–0.02	–1.05
		(0.09)
	Ou5f , oxygen underneath Sn5f	–0.02	–0.95	–0.01	–1.04	0.01	–1.03
	Ou6f , oxygen underneath Sn6f	0.00	–1.05	0.00	–1.06	–0.01	–1.06
	Oterm, oxygen in terminal	–	–	–	–0.65	–	–0.85
	hydroxyl or water oxygen
	Hterm, hydrogen at Oterm or	–	–	–	0.35	–	0.32
	corresponding water hydrogen
	Hbr, hydrogen at Obr or	–	–	–	0.40	–	0.40
	corresponding water hydrogen
aSee footnotes to Table 11.17.

0.36 on a (110) surface, while on a (100) surface it decreases by 0.45 relative to the tin charge in the bulk crystal.

No such significant charge reduction was found in the case of rutile TiO2 [790] for the surface Ti atoms. The Obr charge on a (110) surface is more negative than that on a (100) surface (–0.91 vs. –0.83). This correlates with the conclusion about the larger basicity of the bridging oxygen on the (110) surface.

The results of LCAO calculations of the bare surface slabs validate the admitted approach (including single-slab model, LCAO basis, and BSSE correction) and indicate that the use of hybrid functionals to describe SnO2 surfaces provides more reasonable results.

PW-DFT calculations [811] predicted a significant di erence in the structure of adsorbed H2O on (110) TiO2 (rutile) and SnO2 surfaces, where the dissociative adsorption was favored on SnO2 vs. associative adsorption on TiO2. The di erence in H2O behavior on the surface was attributed [811] to the larger 2D unit-cell parameters of SnO2 compared to those of TiO2 (a = 3.186 ˚A, b = 6.699 ˚A; vs. a = 2.959 ˚A, b = 6.497 ˚A, respectively). The larger unit-cell dimensions caused H-bonding among the adsorbed H2O molecules on the surface to be less energetically important on SnO2 than on TiO2.

It was pointed out in [784] that molecular adsorption on an oxide surface can be understood as an acid–base process. From this point of view, the more covalent character of the Sn–O bonds compared to the corresponding Ti–O bonds might be the second factor that enforces the dissociation of water on the SnO2 surface. In fact, the electronegativity of tin (1.8) is markedly greater than the electronegativity of titanium (1.4). The significant reduction of the top surface Sn5f charges (see Tables

11.2 Surface LCAO Calculations on TiO2 and SnO2

501

11.17 and 11.18) correlates with the higher electronegativity of tin. As a result, the hydrated Sn5f ion perhaps is a stronger Brønsted acid than the hydrated Ti5f ion on the rutile surfaces. The water hydrolysis can obviously be promoted further by the larger basicity of the bridging oxygens (see discussion below).

To check the validity of the computational scheme used for describing the hydrogen bonds, a preliminary calculation of the water-water interactions has been made [816]. Full geometry optimization has been made for the isolated water dimer starting with the well-known most favorable relative orientation of the water molecules. The values of 30 kJ/mol using PBE and 26 kJ/mol using B3LYP for the H-bond energy (without zero-point correction and BSSE correction) have been obtained. This is not bad compared to the corresponding MP2 value, 25 kJ/mol. The optimal geometry is also very close to the MP2 result: the O–O distance is 2.86 ˚A, 2.88 ˚A, and 2.91 ˚A for PBE, B3LYP, and MP2 calculations, respectively. On the other hand, this example shows that the energy di erence of the order of 5 kJ/mol per H-bond is within the error bounds of the usual DFT calculations. It also gives evidence that the B3LYP functional can produce more correct values for H-bonded systems.

A single-slab LCAO approach was firstly used in [816] for modeling the water adsorption on SnO2 surfaces. A doubled (in the x direction) 2D unit cell was used to model the water adsorption on both surfaces. The Monkhorst–Pack 3 × 3 and 3 × 4 sets of special k-points were taken for the Brillouin-zone sampling in (110) and (100) slabs, respectively. Up to fifteen atomic planes were included in both types of SnO2 slabs for the adsorption-energy calculation. Two water molecules were placed on each side of the slab to simulate the full-monolayer coverage. Inversion symmetry was imposed on all systems containing water molecules to ensure the equivalence of both slab sides. The optimization of all atomic positions in the slabs has been made. The contribution of BSSE to the calculated adsorption energies has been estimated using the PBE functional. As in the case of the surface-energy calculation, the ghost atoms have been added to represent the water molecules above the optimized slabs and the total energy was recalculated. The energy of the isolated water molecule has also been recalculated with the 10–15 ghost atoms originated from the corresponding SnO2 surfaces. The resulting BSSE is approximately independent of the surface kind and reaches 18 kJ/mol of adsorbed water (see Table 11.19).

Although this is a noticeable contribution, it decreases all calculated adsorption energies by almost the same value, and does not influence the relative stability of associative and dissociative adsorption forms. The obtained adsorption energies for the (110) surface (Table 11.19) satisfactorily agree with the former periodic-slab PWDFT calculations [810, 811].

In simulations [816] the water molecules initially were placed above the fivefold tin. During the optimization both (on each side) water molecules spontaneously dissociated in the case of PBE calculations, whereas a stable associative structure has been obtained in the case of the B3LYP functional (Fig. 11.17a).

Hence, in contrast to previous DFT calculations the hybrid HF-DFT functional leads to stable molecular adsorption on this surface. Starting with the broken water molecules, a stable dissociated structure has been obtained where hydroxyls were attached to fivefold tin and protons were bonded to the bridging oxygens (Fig. 11.17b). The energy of the hydrolyzed structure is lower by about 35 kJ/mol than the energy of the molecular structure (see Table 11.19). It is di cult to compare directly these

502 11 Surface Modeling in LCAO Calculations of Metal Oxides

Table 11.19. Ab-initio adsorption energies per water molecule for H2O monolayer on SnO2 surfaces (N /M – Number of atomic planes/Sn2O4 layers), [816]

Surface	Cell	k-set	N /M	Method	Adsorption energy (kJ/mol)
					associative	dissociative
110	2×1	3×3	15/5	LCAO DFT	unstable	–179.7a
				(PBE)		(–162)
				LCAO DFT	–143.2	–176.6
				(B3LYP)
				PW DFT	unstable	–165.0
				(PW91) [810]
				PW DFT	unstable	–170.8
				(PW91) [811]
100	2×1	3×4	15/2.5	LCAO DFT	–130.2a	–147.0a
				(PBE)	(–112)	(–129)
				LCAO DFT	–116.86	–145.0
				(B3LYP)

aBSSE-corrected values are given in parentheses for LCAO DFT (PBE) calculations.

Fig. 11.17. Optimized monolayer structure for the water molecules adsorbed on a (110) SnO2 surface obtained using the B3LYP functional, [816]: a) stable structure for molecular adsorption; b) stable structure for dissociative adsorption. Large dark gray balls are O atoms; light gray balls are Sn atoms; small white balls are H atoms.

adsorption energies of water on SnO2 with those on TiO2 [790] due to the di erent methods used; however, the adsorption on a (110) surface of cassiterite seems to be more exothermic by approximately 30 kJ/mol than the adsorption on a (110) surface of TiO2. This conclusion is in accordance with the DFT result of [809], where it is found that the dissociative configuration gives an adsorption energy of 134 kJ/mol for SnO2(110) and 104 kJ/mol for TiO2(110). In [809] it has been roughly estimated that the experimental adsorption energy on SnO2(110) should be about 110 kJ/mol. The data in Table 11.19 show that most of the theoretical values are 50% larger than the proposed estimation.

Both resulting geometries are stabilized by the H-bonds between water hydrogen and bridging oxygen in the associative case (Fig. 11.17a) and bridging hydrogen and oxygen of the terminal hydroxyl in the dissociative case (Fig. 11.17b). It should

11.2 Surface LCAO Calculations on TiO2 and SnO2

503

be noted that the PBE functional produces shorter hydrogen bonds than the B3LYP functional. In contrast to the (110) TiO2 surface, there is no su ciently strong interaction between the neighboring water molecules on the (110) SnO2 surface. The distance Ow–Ow between the water oxygens is 3.1 ˚A and 3.4 ˚A on TiO2 and on SnO2(110) surfaces, respectively. This is in accordance with the results of Lindan [811], who pointed out the role of intermolecular interactions in the stabilizing of the molecular adsorption on the (110) TiO2 surface. On the other hand, the H-bond between water hydrogen and bridging oxygen proves to be very short, 1.53 ˚A, indicating the large basicity of Obr. Moreover, H-bonds, which could be hypothetically obtained with the PBE functional for molecular adsorption on a (110) surface, should be even shorter than those using the B3LYP functional, as occurs in all other cases with the real H-bonds. This can be the main reason why the molecular adsorption becomes unstable using the PBE variant of DFT. Also, this example shows that particular approximations influence the details of the calculated potential energy surface for the water–oxide interactions.

Taking into account the fact that the B3LYP approximation gives the better values of energy and bond distances for the H-bonded systems, it can be supposed that hybrid HF-DFT functionals can produce the more correct data for the water adsorption as well. This could be important not only in the case of SnO2 but also for other materials: applying the hybrid functionals may lead to the greater stability of molecular forms adsorbed on the oxide surfaces than predicted by plain DFT techniques.

Whereas many groups have studied the water adsorption on TiO2 surfaces, there have been few experimental water-adsorption studies for SnO2. In [822] thermal desorption spectroscopy and ultraviolet photoemission spectroscopy have been used to investigate perfect and defective surfaces of SnO2. In this study it was suggested that the amount of dissociation was about 10–15% on the stoichiometric (110) surface and the dissociation increased to 35% on the defective surface. Although additional experimental and theoretical work may be needed to confirm these conclusions, these data at least show that the molecular form of water can be stable on the cassiterite (110) surface, which is in accordance with B3LYP results under consideration.

The DOS for both types of water adsorption on a (110) surface are displayed in Fig. 11.18.

In the molecular adsorption case, depicted in Fig. 11.18a, the Obr subband tightly adjoins the bulk VB states, and a new Ow subband (corresponding to the water electronic states) was formed at the bottom of the VB. For comparison, the clear surface DOS projected to bridging oxygen states is also plotted in Fig. 11.18. In the dissociative case, Fig. 11.18b, the Obr subband almost completely disappeared and three new peaks originated from electronic states of the terminal hydroxyl group. One of these peaks is very sharp and is prominently separated from the top of the VB. As in the case of the (110) surface, two di erent initial states were chosen for the (100) surface. In the first, the water molecules were placed near the fivefold tin. The optimization did not change considerably the positions of water molecules using both PBE and B3LYP functionals (Fig. 11.19a).

Thus, the molecular form of adsorption is obviously stable on the (100) surface. The relative positions of fivefold tin and bridging oxygen favor the formation of H- bonds between the water hydrogen and bridging surface oxygen (Fig. 11.19a), although these bonds are not so short as in the (110) case. A stable structure for the

504 11 Surface Modeling in LCAO Calculations of Metal Oxides

a
	Obr clean
DOS(arb.un.)	Total
	Water
	Obr
-10.0	-8.0	-6.0	-4.0	-2.0	0.0	2.0

E (eV)

b
	Obr clean
DOS(arb.un.)	Total
	(OH)term
	(OH)br
-10.0	-8.0	-6.0	-4.0	-2.0	0.0	2.0

E (eV)

Fig. 11.18. Total and O-projected valence-band DOS in the hydroxylated (110) SnO2 surface systems (see text for explanation of energy scale), [816]. a) Molecular adsorption: total DOS (fine line), projection onto the water states (bold dark line), projection onto the Obr states (bold gray line); b) dissociative adsorption: total DOS (fine line), projection onto the (OH)term states (bold dark line), projection onto the (OH)b states (bold gray line). The projection onto the Obr states in the clean (110) surface is shown by a dotted line.

11.2 Surface LCAO Calculations on TiO2 and SnO2

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Fig. 11.19. Optimized monolayer structure for the water molecules adsorbed on a (100) SnO2 surface obtained using the B3LYP functional, [816]: a) stable structure for molecular adsorption; b) stable structure for dissociative adsorption. Large dark gray balls are O atoms, light gray balls are Sn atoms; small white balls are H atoms.

dissociative adsorption (Fig. 11.19b) was obtained from another initial state with the broken water molecules. There are no straight-line H-bonds in this structure. Nevertheless, as in the case of the (110) surface, the dissociative adsorption is more favorable (by about 30 kJ/mol) for the (100) SnO2 surface. The absolute value of the adsorption energy on a (100) surface is lower than that on a (110) surface. This is in accordance with the PW-DFT investigation of Bandura et al. [791].

As may be expected, the atomic displacements for the hydroxylated surfaces are much smaller (especially for the dissociative adsorption) than those in the case of the bare surfaces (Tables 11.17 and 11.18), due to saturation of the vacant coordination places for the surface tin up to the total coordination number 6. Data in Tables 11.17 and 11.18 show that oxygen charges in the hydroxyl groups are noticeably less negative than charges on the other surface oxygens. The hydrogen charges were arranged in the order q(Hbr) > q(Hterm) > q(HH2O), which correlates with the relative acidity of the corresponding O–H bonds [823].

Figure 11.20 reports the DOS distributions for hydroxylated (100) surfaces. The DOS for the associative adsorption (Fig. 11.20a) di ers from that for the clean (100) surface mainly by the presence of the Ow subband at the bottom of the VB.

The di erence between the DOS for the slab dissociatively adsorbing water (Fig. 11.20b) and the DOS for the clean (100) surface is less significant than that in the (110) case. The terminal OH states also contribute to the top of the VB, but corresponding peaks are not as sharp as for the hydroxylated (110) surface. Except for the narrow zone at the top of the VB, the total DOS in the last case resembles the total DOS for the bulk crystal, which is also plotted in Fig. 11.20b for comparison.

Comparing the results for the clean and hydroxylated surfaces, one can conclude that a (110) surface exhibits stronger hydrophilic properties than a (100) surface due both to the more favorable geometrical structure and the more basic nature of the bridging oxygen.

The discussion above demonstrates that use of the hybrid HF-DFT approach allowed new results for the clean and hydroxylated oxide surfaces compared with plain DFT techniques to be obtained. The comparison of plain DFT functionals with the HF-DFT B3LYP hybrid functional shows interesting di erences, which could be im-

506 11 Surface Modeling in LCAO Calculations of Metal Oxides

a
		Obr clean
		Total
DOS(arb.un.)		Water
		Obr
-10.0	-8.0	-6.0	-4.0	-2.0	0.0	2.0
b			E (eV)

		bulk
		Obr clean
DOS(arb.un.)		Total
		(OH)term
		(OH)br
-10.0	-8.0	-6.0	-4.0	-2.0	0.0	2.0

E (eV)

Fig. 11.20. Total and O-projected valence-band DOS in the hydroxylated (100) SnO2 surface systems. a) Molecular adsorption; b) dissociative adsorption. (The sense of line types and energy scale are the same as in Fig. 11.19.) The projection onto the Obr states for the clean (100) surface is shown by the dotted lines, and the total bulk DOS is shown in b) by open circles.

portant not only in the case of SnO2 but also for other materials: in contrast to DFT plane-wave calculations (spontaneous dissociation), an associated adsorption of the water molecules becomes possible not only in the case of the (100) surface but also at the most stable (110) surface judging by the hybrid functional. This fact is probably due to the shortening of the H-bonds by the plain DFT methods.

LCAO HF-DFT investigation, as well as previous plane-wave calculations, shows that water dissociation on SnO2 surfaces is more favorable than on the similar TiO2 surfaces. Not only may the geometrical factors favor the hydrolysis of water on cassi-

11.3 Slab Models of SrTiO3, SrZrO3 and LaMnO3 Surfaces 507

terite surfaces but also the more covalent nature of Sn–O bonds and the larger basicity of bridging oxygens in SnO2 comparably to TiO2.

One can conclude that the arrangement of H-bonds between the hydroxyl hydrogens and surface oxygens, as well as between the water molecules themselves, plays an important role in stabilizing the water adsorption on the metal-oxide surfaces. Thus, one of the reasons why the absolute value of adsorption energy on a (100) surface is lower than that on a (110) surface is the relative positions of oxygen and tin atoms being less profitable for the H-bond formation.

The general agreement between the results of calculations using the di erent bases (LCAO and PW) or the di erent slab models (2D and 3D) justifies the validity of the various ab-initio methods to study the molecular adsorption on the crystalline surfaces. However, the BSSE correction may be needed to obtain more precise absolute adsorption energies within the LCAO calculations.

In the next section we consider the surface modeling in cubic perovskites.

11.3 Slab Models of SrTiO3, SrZrO3 and LaMnO3 Surfaces

11.3.1 Hybrid HF-DFT Comparative Study of SrZrO3 and SrTiO3 (001) Surface Properties

A variety of practical applications of perovskite systems ABO3 (the piezoelectrical and electro-optical devices, fuel cells, microelectrodes) have stimulated experimental and theoretical investigations of their surfaces.

Neither experimental nor theoretical investigations of the (001) surface structure of SrZrO3 are known. However, a large number of experimental studies of SrTiO3 (001) surfaces have been reported, see [824, 825] and references therein. In some of these investigations the reconstructed surfaces have been observed. A reconstruction of the SrTiO3 surface mostly relates to surface defects, e.g., oxygen vacancies that can be created by annealing in O2 atmosphere at high temperature [826]. At the same time, the relaxation of a perfect titanate surface (no oxygen vacancies) has been experimentally observed and investigated [827]. This confirms the fact that perovskite surfaces with regular stoichiometry can be stable in some conditions. The displacements of atoms for the surface relaxation have been found in [827] by medium-energy ion scattering (MEIS) method. Charlton et al. [828] have used room-temperature surface X-ray di raction (SXRD) to investigate the 300 K structure of SrTiO3(001) with 78% terminated TiO2 and 22% terminated SrO. For the TiO2 surface, there is good agreement with MEIS data in the position of the top-layer Ti, with both techniques pointing to an essentially bulk-terminated position. The data in [828] indicated that a lateral ferroelectric distortion is absent at 300 K on both terminations, consistent with some theoretical calculations [829]. However, a reflection high-energy electron di raction (RHEED) study of the SrTiO3 (001) surface structure in a temperature region from 300 down to 5 K [830] gave evidence for a surface phase transition of a soft-mode type, promoted by the surface symmetry.

The properties of the SrTiO3 (001) surface have been examined in many quantummechanical studies [824, 829, 831–833, 836, 837]. Calculations of (001) surfaces of BaTiO3, PbTiO3, [775, 838–841] and defective (001) and (110) SrTiO3 surfaces

508 11 Surface Modeling in LCAO Calculations of Metal Oxides

[842, 843] have also been made. A large number of cited calculations of surface properties of the strontium titanate are based on density-functional theory (DFT) and plane-wave basis (PW) set. In DFT LDA simulations [829] of the surface relaxation it has been found that relaxations account for 0.18 eV of the surface energy per surface unit cell, compiling around 15% of the total surface energy (1.36 J/m2). Examining slabs with di erent surface terminations, the authors conclude that the bandgap (BG) for the SrO surface almost does not change with respect to the bulk value, and no ingap state occurs. For the TiO2 surface, there is a substantial reduction of the BG. However, there are also no deep-gap surface states, in accord with experimental reports. It should be noted that the LDA approach utilized in [829] (as well as the generalized-gradient approximation, GGA) tends to underestimate the BG and leads to a substantial discrepancy between calculated (1.85 eV) and experimental (3.30 eV) BG for the bulk SrTiO3.

In [833, 836] the results of GGA calculations for SrTiO3 surface systems are reported. In general, the data obtained in these studies do not di er considerably from LDA results except for the values reported in [836] for atomic displacements that seem to be on the order of a half of the values reported by all other DFT studies and may be attributed to some systematic error. Again, no midgap surface state for either TiO2- or SrO-terminated surfaces was found in the band structure, [836]. However, a clear small peak appears below the energy gap in the electronic density of states (DOS) of the TiO2-terminated surface, which has a tendency to move into the midgap. In [833] the electron redistribution in the surface layers has been analyzed using charge-density decomposition based on the Bader criteria [844]. The authors concluded that relatively strong hybridization between the Ti and O atoms leads to a noticeable charge transfer in the SrTiO3 surface systems. This enhanced charge transfer correlates with the strength of the surface relaxation.

In [824,831,832] HF and DFT LCAO calculations of (001) SrTiO3 surface employing a number of di erent exchange-correlation functionals have been performed. Prior to investigation of the surface properties, these authors tested the di erent quantummechanical methods on some bulk characteristics such as the lattice constant, bulk modulus, and BG. They obtained the best agreement between theoretical and experimental data for the hybrid HF-DFT B3PW method. The surface structure, surface energies, and electronic properties of the SrTiO3 (001) surface has been calculated by di erent HF and DFT LCAO methods. It was concluded that it is very di - cult to choose a method reproducing all properties equally well, but hybrid HF-DFT techniques B3PW and B3LYP turned out to be the most reasonable.

In [845] the comparison has been made between the calculated relaxed structures of the SrTiO3 surface obtained in various DFT PW studies and the data of several experimental investigations. Good agreement was found between the most theoretical studies, whereas the accordance of theoretical data with the available experimental results proved to be low. This fact can be explained by the poor agreement between the di erent experimental studies themselves [829, 831]. It was suggested in [828] that a possible reason for this discrepancy is the influence of soft vibrational modes, which are thought [799] to give rise to a 0.2˚A – disagreement between 0 K theory and 300 K experiment for TiO2 (110). Whatever the origin of the discrepancy, it presumably a ects the TiO2-terminated surface and the second-layer atoms on both surface types of SrTiO3 crystal.