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Comparative Hybrid Hartree-Fock-DFT Calculations of WO_{2}-Terminated Cubic WO_{3} as Well as SrTiO_{3}, BaTiO_{3}, PbTiO_{3} and CaTiO_{3} (001) Surfaces

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## Abstract

**:**

_{2}-terminated cubic WO

_{3}(001) surface and analyzed the systematic trends in the WO

_{3}, SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}(001) surface ab initio calculations. According to our first principles calculations, all WO

_{2}or TiO

_{2}-terminated WO

_{3}, SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaZrO

_{3}(001) surface upper-layer atoms relax inwards towards the crystal bulk, while all second-layer atoms relax upwards. The only two exceptions are outward relaxations of first layer WO

_{2}and TiO

_{2}-terminated WO

_{3}and PbTiO

_{3}(001) surface O atoms. The WO

_{2}or TiO

_{2}-terminated WO

_{3}, SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}(001) surface-band gaps at the Γ–Γ point are smaller than their respective bulk-band gaps. The Ti–O chemical bond populations in the SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}bulk are smaller than those near the TiO

_{2}-terminated (001) surfaces. Conversely, the W–O chemical bond population in the WO

_{3}bulk is larger than near the WO

_{2}-terminated WO

_{3}(001) surface.

## 1. Introduction

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}(001) surfaces have been broadly explored theoretically and experimentally [1,2,3,4,5,6,7,8,9,10]. At the same time, to the best of our knowledge, there are no reports of ab initio calculations dealing with the atomic relaxation and electronic structure of the pristine WO

_{2}-terminated WO

_{3}(001) surface in the cubic perovskite-like structure. Nevertheless, a large amount of experimental studies exist dealing with WO

_{3}(001) surfaces [11,12,13,14]. Recent theoretical studies have been devoted, for example, to hydrogen adsorption on the WO

_{3}(001) surface [15], understanding the water splitting process on the WO

_{3}(001) surface [16], and H

_{2}O adsorption on the WO

_{3}(001) surface [17].

_{3}, PbTiO

_{3}and CaTiO

_{3}perovskites have attracted huge fundamental interest in these materials mostly for their phase transitions. Historically the ABO

_{3}perovskites were highly promising low-cost energy materials. They have been used for numerous optoelectronic and photonic device applications [18]. SrTiO

_{3}perovskite thin films are important for a large amount of technologically important applications [19,20]. For example, they are used for catalysis, optical wave guides, high-capacity memory cells as well as substrates for high-temperature cuprate superconductor growth [19,20]. Barium titanate (BaTiO

_{3}) is an excellent photorefractive material [18]. Ferroelectric PbTiO

_{3}thin films have been applied to large numbers of electronic devices, such as non-volatile memory FET [19] and Si monolithic ultrasonic sensors [18]. CaTiO

_{3}is used worldwide in technologically important electronic ceramic materials [18]. Tungsten trioxide (WO

_{3}) and its thin films exhibit a large number of novel properties useful for high-technology applications [21]. In particular, WO

_{3}undergoes phase transitions, which are explored for their potential in industrial applications, display systems and microelectronics [21]. It is worth noting that the predictive power of ab initio calculations makes possible the design of new materials for high-technology applications on paper. Nowadays, consumer electronics mostly use lithium-ion batteries containing LiCoO

_{2}cathode, which was discovered in 1980 by J. Goodenough, one of the 2019 Nobel Prize winners for Chemistry [22]. The experimentally detected LiCoO

_{2}average intercalation voltage is 4.0–4.1 V [23]. Based on ab initio calculations by Eglitis and Borstel [24,25,26], it was demonstrated that a novel Li

_{2}CoMn

_{3}O

_{8}battery cathode material can lead to a high-energy lithium-ion battery working at the 5 V regime.

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}perovskite cubic unit cells contain five atoms. The A type atom (A = Sr, Ba, Pb or Ca) has the coordinates (0, 0, 0), and it is located in the cube corner position. The Ti atom has the coordinates (½, ½, ½), and it is located in the cube body center position. The 3 O atoms have the coordinates (½, ½, 0), (½, 0, ½), (0, ½, ½), and they are located in the cube face centered positions. All SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}cubic perovskites have the same space group Pm3m with the space group number equal to 221. WO

_{3}in its cubic perovskite-like structure has exactly the same space group as ATiO

_{3}perovskites Pm3m, and also the same space group number 221. The only striking difference between the SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}cubic perovskites as well as WO

_{3}in its cubic perovskite-like structure is that WO

_{3}has an empty A cation position. Thereby, the cubic perovskite-like unit cell of WO

_{3}contain only four atoms.

_{2}-terminated polar WO

_{3}(001) surfaces in the cubic perovskite-like structure. We compared our WO

_{2}-terminated WO

_{3}(001) surface-atomic and electronic-structure ab initio calculations with our results for the related structure TiO

_{2}-terminated SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}cubic perovskite (001) surfaces. We carefully compared our calculation results for all five of our calculated materials and detected systematic common trends. The results for WO

_{2}-terminated WO

_{3}and TiO

_{2}-terminated SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}(001) surfaces were summarized and analysed in a way easily readable for a broad audience of scientists.

## 2. Computational Methods and Surface Models

_{3}, BaTiO

_{3}and PbTiO

_{3}in [30]. The Hay–Wadt small-core, effective-core pseudopotentials (ECP) were adopted for Ca and Ti atoms [31,32,33]. The small-core ECPs replaced only the inner-core orbitals, while orbitals for subvalence electrons as well as for valence electrons were calculated self-consistently. Oxygen atoms were treated with the all-electron BS. Finally, for the W atom we used BS developed by Cora et al. [34]. Our calculations were performed by means of the B3LYP [35] or B3PW [36,37,38] hybrid exchange–correlation functionals. For all WO

_{3}, SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}materials we performed the reciprocal space integration with an 8 × 8 × 8 and 8 × 8 × 1 extension of Pack–Monkhorst mesh for the bulk and (001) surfaces of these materials. The CRYSTAL computer program package [27] makes possible the calculation of isolated 2D slabs perpendicular to the Oz direction. In order to compare the performance of different exchange–correlation functionals and choose the best method for our calculations, we calculated the SrTiO

_{3}, SrZrO

_{3}, BaZrO

_{3}, MgF

_{2}and CaF

_{2}bulk Γ–Γ band gaps [30,39,40,41,42] (Table 1 and Figure 1). The experimentally detected SrTiO

_{3}, SrZrO

_{3}, BaZrO

_{3}, MgF

_{2}and CaF

_{2}bulk band gaps at the Γ-point are mentioned in Table 1 for comparison purposes as well as depicted in Figure 1 [43,44,45,46,47].

_{3}bulk Γ–Γ band gap. Even HF calculated MgF

_{2}bulk Γ–Γ band gap overestimates the experimental value 1.51 times (Table 1 and Figure 1).

_{3}perovskites as well as MgF

_{2}and CaF

_{2}. For example, the PWGGA (6.94 eV) and PBE (6.91 eV) calculated MgF

_{2}bulk band gap at Γ-point is 1.87 and 1.88 times, respectively, smaller than the experimental MgF

_{2}bulk Γ–Γ band gap value of 13.0 eV [37].

_{3}, SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}bulk and (001) surface calculations by means of the B3PW [36,37,38] or B3LYP [35] hybrid exchange–correlation functionals. The hybrid functional incorporates a portion of exact exchange energy density from HF theory (20%) while the rest of the exchange–correlation part is a mixture of different approaches (both exchange and correlation). It is obvious, that the B3PW and B3LYP hybrid exchange–correlation functionals, since they are a superposition of HF and DFT methods as implemented in the CRYSTAL computer code [27], allowed us to achieve as good an agreement as possible between the first principles calculated and the experimentally detected Γ–Γ band gaps for WO

_{3}, SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}bulk and their (001) surfaces.

_{2}-terminated WO

_{3}as well as TiO

_{2}-terminated SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}(001) slabs containing 9 alternating layers. First our calculated WO

_{3}(001) slab was terminated by WO

_{2}planes from both sides (WO

_{2}–O–WO

_{2–}O–WO

_{2}–O–WO

_{2}–O–WO

_{2}) from a 19-atom supercell (Figure 2). Another of our calculated SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}(001) slabs was terminated by TiO

_{2}planes from both sides (TiO

_{2}–AO–TiO

_{2}–AO–TiO

_{2}–AO–TiO

_{2}–AOvTiO

_{2}) and consisted of a 23-atom supercell (Figure 3). Both our calculated slabs were non-stoichiometric and had unit-cell equations W

_{5}O

_{14}as well as A

_{4}Ti

_{5}O

_{14}, respectively. To analyse the chemical bonds, effective atomic charges and covalency effects for WO

_{3}and ATiO

_{3}perovskite bulk and (001) surfaces, we used the well-known Mulliken population analysis [48,49,50,51,52].

## 3. Ab initio Calculation Results for WO_{3}, SrTiO_{3}, BaTiO_{3}, PbTiO_{3} and CaTiO_{3} Bulk

_{3}bulk lattice constant (3.775 Å). Our calculated cubic WO

_{3}constant (3.775 Å) was only slightly larger than the experimental value of a

_{0}= 3.71–3.75 Å [51] (Table 2); nevertheless, it was in almost perfect agreement with the earlier calculation result for the WO

_{3}cubic structure bulk lattice constant calculated by the full-potential linear muffin-tin (FP-LMTO) code equal to 3.78 Å [52]. Our B3PW-calculated SrTiO

_{3}bulk lattice constant (3.904 Å) was only slightly overestimated with respect to the experimental SrTiO

_{3}bulk lattice constant (3.89 Å) extrapolated to 0 K [53] (Table 2). Our ab initio calculation of the BaTiO

_{3}bulk lattice constant (4.008 Å) was in an outstanding agreement with the experimental value of 4.00 Å [53,54,55]. Our B3PW-calculated PbTiO

_{3}bulk lattice constant (3.936 Å) [54,55,56] was only 0.86% under the experimental value of 3.97 Å [57]. Finally, our calculated CaTiO

_{3}bulk lattice constant (3.851 Å) was 1.17% smaller than the experimentally detected (3.8967 Å) [58,59,60] (Table 2).

_{3}bulk were (+3.095e) for the W atom, and (−1.032e) for each of the three O atoms (Table 3). Our B3LYP-calculated effective W atomic charge (+3.095e) was almost two times smaller than the generally accepted classical ionic charge for the W(+6e) atom. In addition, our calculated effective atomic charge for the O (−1.032e) atom was almost two times smaller than the generally accepted O atom classical ionic charge (−2e). In addition, for the SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}perovskites, our calculated A atomic charges (+1.871e, +1.797e, +1.354e and +1.782e, respectively) were considerably smaller than those of the classical Sr, Ba, Pb, Ca atom ionic charges (+2e) (Table 3) [61,62,63,64,65,66]. Our B3PW-calculated O atom Mulliken charges in SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}perovskites (−1.407e, −1.388e, −1.232e and −1.371e, respectively) are also at least 29.65% smaller than the classical ionic O atomic charge (−2e) [67,68,69]. Finally, our ab initio-calculated SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}Ti atomic charges (+2.351e, +2.367e, +2.341e and 2.330e) are more than one-and-a-half times smaller than the formal Ti atom ionic charge (+4e). Our calculated chemical bond population between W and O atoms in WO

_{3}bulk (0.142e) is approximately one-and-a-half times larger than the Ti–O atom chemical bond population in SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}perovskites (+0.088e, +0.098e, +0.098e and +0.084e, respectively).

_{3}bulk Γ–Γ band gap (4.95 eV) overestimated by 1.21 eV the experimental direct WO

_{3}bulk band gap value at Γ-point of 3.74 eV [70] (Table 4). Moreover, our B3PW-calculated bulk Γ–Γ band gaps for SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}perovskites (3.96 eV, 3.55 eV, 4.32 eV and 4.18 eV, respectively) were always slightly overestimated with respect to the experimentally measured direct band gap values at Γ-point for SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}perovskites (3.75 eV [43], 3.2 eV [71], 3.4 eV [72] and 3.5 eV [73], respectively) (Table 4).

## 4. Ab Initio Calculation Results for the WO_{2}-Terminated WO_{3} as Well as TiO_{2}-Terminated SrTiO_{3}, BaTiO_{3}, PbTiO_{3} and CaTiO_{3} (001) Surfaces

_{2}-terminated WO

_{3}and the TiO

_{2}-terminated SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}(001) surface upper-three or two layers are presented in Table 5. According to our B3LYP or B3PW calculations, all atoms of the first (upper) surface layer relaxed inwards, while all second-layer atoms relaxed outwards (Table 5). The only two exceptions to this systematic trend were the outward relaxation of the first layer O atom of the WO

_{2}-terminated WO

_{3}(001) surface (+0.42% of a

_{0}) and the outward relaxation of the TiO

_{2}-terminated PbTiO

_{3}(001) surface first-layer O atom by (0.31% of a

_{0}) (Table 5). The first layer metal atom relaxation magnitudes range from −1.71% of a

_{0}for the TiO

_{2}-terminated CaTiO

_{3}(001) surface to −3.08% of a

_{0}for the TiO

_{2}-terminated BaTiO

_{3}(001) surface (Table 5). The first- and second-layer metal atom displacement magnitudes for WO

_{2}-terminated WO

_{3}and TiO

_{2}-terminated SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}(001) surfaces were always considerably larger than the respective first- and second-layer O atom displacement magnitudes (Table 5).

_{2}-terminated WO

_{3}(001) surface (+2.49) is in qualitative agreement with our B3PW-calculated surface rumpling amplitudes s for TiO

_{2}-terminated BaTiO

_{3}, PbTiO

_{3}, CaTiO

_{3}and SrTiO

_{3}(001) surfaces (+2.73, +3.12, +1.61 and +2.12, respectively) (Table 6). Our B3PW-calculated surface rumpling amplitude s for TiO

_{2}-terminated SrTiO

_{3}(001) surface (+2.12) is in fair agreement with available RHEED (+2.6 [74]) and LEED (+2.1 ± 2 [75]) experimental data (Table 6). Unfortunately, our B3PW-calculated interlayer distance Δd

_{12}for the TiO

_{2}-terminated SrTiO

_{3}(001) surface (−5.80) had the opposite sign to the experimentally measured RHEED (+1.8 [74]) and LEED (+1 ± 1 [67]) interlayer distances (Table 6). Finally, our B3PW-calculated interlayer distance Δd

_{23}for the TiO

_{2}-terminated SrTiO

_{3}(001) surface (+3.55) is in qualitative agreement with the RHEED experiment result (+1.3), but had the opposite sign to that of the LEED experimental result (−1 ± 1). Nevertheless, it is worth noting that the RHEED (+1.3) and the LEED experiments (−1 ± 1) had opposite signs for the interlayer distance Δd

_{23}(Table 6).

_{2}-terminated WO

_{3}and TiO

_{2}-terminated SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}(001) surfaces with an analysis of charge redistribution in the top-three surface planes (Table 7). The ab initio-calculated atomic displacements, bond populations between the nearest metal and oxygen atoms and the effective atomic charges are collected in Table 7. For example, the effective static atomic charges on WO

_{2}-terminated WO

_{3}as well as TiO

_{2}-terminated SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}(001) surface upper-layer W and Ti atoms are always reduced in comparison to the bulk WO

_{3}, SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}crystal charges (−0.312e, −0.06e, −0.06e, −0.062e and −0.052e, respectively). We recently observed a similar effect: the reduction of surface upper-layer metal atomic charges near the ReO

_{2}-terminated ReO

_{3}and the ZrO

_{2}-terminated SrZrO

_{3}, BaZrO

_{3}, PbZrO

_{3}and CaZrO

_{3}(001) surfaces [76]. According to our ab initio calculations, the largest upper-layer metal atom displacement was observed for the TiO

_{2}-terminated BaTiO

_{3}(001) surface Ba atom (−0.123 Å). Nevertheless, the TiO

_{2}-terminated PbTiO

_{3}(001) surface second-layer Pb atom outward displacement (+0.209 Å) was even larger.

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}bulk Ti-O chemical bond covalency (+0.088e, +0.098e, +0.098e and +0.084e, respectively) were always smaller than near the TiO

_{2}-terminated SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}(001) surfaces (0.118e, 0.126e, 0.114e, 0.114e, respectively) (Table 8 and Figure 4). Just opposite situation was obtained for the WO

_{3}crystal: the B3LYP-calculated W-O chemical bond population in the WO

_{3}bulk (0.142e) was larger than near the WO

_{2}-terminated WO

_{3}(001) surface (0.108e) (Table 8 and Figure 4). Nevertheless, it is worth noting that the W–O chemical bond population between the W atom on the top layer of WO

_{2}-terminated WO

_{3}(001) surface and the O atom on the second layer (0.278e) is the largest one (Table 8), which was in agreement with our previous B3LYP calculations dealing with ReO

_{2}-terminated ReO

_{3}(001) surfaces [76].

_{3}, SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}bulk band gaps at the Γ–Γ point were always reduced near the WO

_{2-}or TiO

_{2}-terminated WO

_{3}, SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}(001) surfaces. The B3PW-calculated SrTiO

_{3}bulk band gap at Γ–Γ point near the TiO

_{2}-terminated SrTiO

_{3}(001) surface at Γ–Γ point was reduced only by 0.01 eV. At the same time, our B3LYP-calculated WO

_{3}bulk band gap (4.95 eV) at the Γ–Γ point near the WO

_{2}-terminated WO

_{3}(001) surface was reduced by 3.79 eV to 1.16 eV (Table 9 and Figure 5).

## 5. Conclusions

_{2-}or TiO

_{2}-terminated WO

_{3}, SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}(001) surfaces, as a rule, all first-layer surface atoms relax inwards, whereas all second-layer surface atoms relax upwards. The only two exceptions from this systematic trend are the upward relaxation of WO

_{2-}or TiO

_{2}-terminated WO

_{3}and PbTiO

_{3}(001) surface first-layer O atoms. As a result of our ab initio-calculated atomic relaxation, TiO

_{2}-terminated SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}(001) surfaces exhibited a reduction of the interlayer distance Δd

_{12}(−5.80, −5.59, −8.13, −4.46% of a

_{0}, respectively) as well as an expansion of Δd

_{23}(+3.55, +2.51, +5.32, +2.75% of a

_{0}, respectively). It is worth noting that after geometry optimization, it is very useful to perform ab initio molecular dynamics computations to ensure the stability of the structures over time [77].

_{12}) were always larger than between the second and third layer (Δd

_{23}) for all our calculated perovskites SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}.

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}bulk was always smaller than near their TiO

_{2}-terminated (001) surface (see Figure 4). In contrast, the W–O chemical bond population in the WO

_{3}bulk (0.142e) was larger than near the WO

_{2}-terminated WO3 (001) surface (0.108e). Nevertheless, the largest W–O chemical bond population, according to our ab initio calculations, is between the W atom located on the WO

_{2}-terminated WO

_{3}(001) surface and the second-layer O atom (0.278e). It was worth noting, that also for the related material ReO

_{3}, according to our calculations [76], the situation was similar. Namely, the Re–O chemical bond population in the ReO

_{3}bulk (0.212e) was larger than near the ReO

_{2}-terminated ReO

_{3}(001) surface (0.170e). Nevertheless, the Re–O chemical bond population between the Re atom located on the ReO

_{2}-terminated ReO

_{3}(001) surface upper-layer and O atom located on the ReO

_{2}-terminated ReO

_{3}(001) surface second layer was the largest (0.262e).

_{3}, SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}bulk Γ–Γ band gap values (4.95, 3.96, 3.55, 4.32, 4.18 eV, respectively) were always reduced with respect to the bulk near the WO

_{2}- or TiO

_{2}-terminated WO

_{3}, SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}(001) surfaces (1.16, 3.95, 2.96, 3.18, 3.30 eV, respectively) (see Figure 5).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Ab initio calculated and experimentally measured bulk Γ–Γ band gaps for SrTiO

_{3}, SrZrO

_{3}, BaZrO

_{3}, MgF

_{2}and CaF

_{2}obtained by means of different exchange–correlation functionals: (1) PWGGA; (2) B3LYP; (3) B3PW; (4) Experiment; (5) HF.

**Figure 4.**Our ab initio-calculated bulk (1) as well as BO

_{2}-terminated (001) surface (2). B–O chemical bond populations (in e) for WO

_{3}, SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3.}

**Figure 5.**Our ab initio-calculated WO

_{2}-terminated WO

_{3}and TiO

_{2}-terminated SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}(001) surface Γ–Γ band gaps (line 1). Experimentally measured bulk Γ–Γ band gaps (line 2). Our ab initio-calculated WO

_{3}, SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}bulk Γ–Γ band gaps (line 3).

**Table 1.**By means of different exchange–correlation functionals calculated SrTiO

_{3}, SrZrO

_{3}, BaZrO

_{3}, MgF

_{2}and CaF

_{2}bulk Γ–Γ band gaps (eV). Experimental bulk band gaps at the Γ-point are listed for comparison.

Method | SrTiO_{3} [30] | SrZrO_{3} [39] | BaZrO_{3} [40] | MgF_{2} [41] | CaF_{2} [42] |
---|---|---|---|---|---|

Experiment | 3.75 [43] | 5.6 [44] | 5.3 [45] | 13.0 [46] | 12.1 [47] |

B3PW | 3.96 | 5.30 | 4.93 | 9.48 | 10.96 |

B3LYP | 3.89 | 5.31 | 4.79 | 9.42 | 10.85 |

HF | 12.33 | 13.54 | 12.96 | 19.65 | 20.77 |

PWGGA | 2.31 | 3.53 | 3.24 | 6.94 | 8.51 |

PBE | 2.35 | 3.52 | - | 6.91 | 8.45 |

Material | Method | Theory | Experiment |
---|---|---|---|

WO_{3} | B3LYP | 3.775 | 3.71–3.75 [51] |

FP-LMTO | 3.78 [52] | ||

SrTiO_{3} | B3PW | 3.904 | 3.89 [53] |

BaTiO_{3} | B3PW | 4.008 | 4.00 [53,54] |

PbTiO_{3} | B3PW | 3.936 | 3.97 [54,57] |

CaTiO_{3} | B3PW | 3.851 | 3.8967 [58,59,60] |

**Table 3.**Our calculated atomic charges Q(e) as well as bond populations P(e) in WO

_{3}, SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}bulk materials.

Bulk Materials | WO_{3} | SrTiO_{3} | BaTiO_{3} | PbTiO_{3} | CaTiO_{3} | |
---|---|---|---|---|---|---|

Ion | Property | B3LYP | B3PW | B3PW | B3PW | B3PW |

A | Q | - | +1.871 | +1.797 | +1.354 | +1.782 |

P | - | −0.010 | −0.034 | +0.016 | +0.006 | |

O | Q | −1.032 | −1.407 | −1.388 | −1.232 | −1.371 |

P | +0.142 | +0.088 | +0.098 | +0.098 | +0.084 | |

B | Q | +3.095 | +2.351 | +2.367 | +2.341 | +2.330 |

**Table 4.**Ab initio-calculated WO

_{3}, SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}bulk band gaps for the cubic phase at the Γ–Γ point. Our B3LYP and B3PW calculations were compared with the relevant experimental data.

Crystal | Method | Optical Bulk Band Gap at Γ–Γ Point | |
---|---|---|---|

Ab Initio Calculations | Experimental Results | ||

WO_{3} | B3LYP | 4.95 | 3.74 [70] |

SrTiO_{3} | B3PW | 3.96 | 3.75 [43] |

BaTiO_{3} | B3PW | 3.55 | 3.2 [71] |

PbTiO_{3} | B3PW | 4.32 | 3.4 [72] |

CaTiO_{3} | B3PW | 4.18 | 3.5 [73] |

**Table 5.**WO

_{2}-terminated WO

_{3}as well as TiO

_{2}-terminated SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}(001) surface atom relaxation for upper-three surface layers (in percent of the bulk lattice constant).

Surfaces, (001) | WO_{3} | SrTiO_{3} | BaTiO_{3} | PbTiO_{3} | CaTiO_{3} | |
---|---|---|---|---|---|---|

Layer | Ion | WO_{2}-Term. | TiO_{2}-Term. | TiO_{2}-Term. | TiO_{2}-Term. | TiO_{2}-Term. |

Method | B3LYP | B3PW | B3PW | B3PW | B3PW | |

1 | B | −2.07 | −2.25 | −3.08 | −2.81 | −1.71 |

O | +0.42 | −0.13 | −0.35 | +0.31 | −0.10 | |

2 | A | Absent | +3.55 | +2.51 | +5.32 | +2.75 |

O | +0.11 | +0.57 | +0.38 | +1.28 | +1.05 | |

3 | B | −0.01 | - | - | - | - |

O | 0.00 | - | - | - | - |

**Table 6.**Our B3LYP- or B3PW-calculated surface rumplings s and relative displacements Δd

_{ij}between the three near-surface planes for the WO

_{2}-terminated WO

_{3}and the TiO

_{2}-terminated BaTiO

_{3}, PbTiO

_{3}, CaTiO

_{3}and SrTiO

_{3}(001) surfaces as a percent of the bulk material lattice constant. The available experimental data are listed for comparison purposes.

Material | Method | WO_{2}- or TiO_{2}-Terminated (001) Surface | ||
---|---|---|---|---|

s | Δd_{12} | Δd_{23} | ||

WO_{3} | B3LYP | +2.49 | - | - |

BaTiO_{3} | B3PW | +2.73 | −5.59 | +2.51 |

PbTiO_{3} | B3PW | +3.12 | −8.13 | +5.32 |

CaTiO_{3} | B3PW | +1.61 | −4.46 | +2.75 |

SrTiO_{3} | B3PW | +2.12 | −5.80 | +3.55 |

RHEED exp. [74] | +2.6 | +1.8 | +1.3 | |

LEED exp. [75] | +2.1 ± 2 | +1 ± 1 | −1 ± 1 |

**Table 7.**Our B3LYP- or B3PW-calculated atomic shift magnitudes D (in Å) as well as the effective atomic charges Q (in e) and nearest atomic chemical bond populations P (in e) for the WO

_{2}- and TiO

_{2}-terminated WO

_{3}, SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}(001) surfaces.

WO_{2} and TiO_{2}-Term. (001) Surfaces | WO_{3} | SrTiO_{3} | BaTiO_{3} | PbTiO_{3} | CaTiO_{3} | ||
---|---|---|---|---|---|---|---|

Layer | Property | Ion | WO_{2}-Ter. | TiO_{2}-Ter. | TiO_{2}-Ter. | TiO_{2}-Ter. | TiO_{2}-Ter. |

1 | D | B | −0.078 | −0.088 | −0.123 | −0.111 | −0.066 |

Q | +2.783 | +2.291 | +2.307 | +2.279 | +2.278 | ||

P | +0.108 | +0.118 | +0.126 | +0.114 | +0.114 | ||

D | O | +0.016 | −0.005 | −0.014 | +0.012 | −0.004 | |

Q | −1.146 | −1.296 | −1.280 | −1.184 | −1.267 | ||

P | −0.014 | −0.014 | −0.038 | +0.044 | +0.016 | ||

2 | D | A | - | +0.139 | +0.101 | +0.209 | +0.106 |

Q | - | +1.850 | +1.767 | +1.275 | +1.754 | ||

P | - | −0.008 | −0.030 | +0.008 | +0.006 | ||

D | O | +0.004 | +0.022 | +0.015 | +0.050 | +0.041 | |

Q | −0.925 | −1.365 | −1.343 | −1.167 | −1.324 | ||

P | +0.064 | +0.080 | +0.090 | +0.080 | +0.086 | ||

3 | D | B | −0.0004 | - | - | - | - |

Q | +3.001 | +2.348 | +2.365 | +2.335 | +2.326 | ||

P | +0.144 | +0.096 | +0.104 | +0.108 | +0.090 | ||

D | O | 0.000 | - | - | - | - | |

Q | −1.037 | −1.384 | −1.371 | −1.207 | −1.354 | ||

P | −0.032 | −0.010 | −0.034 | +0.018 | +0.008 |

**Table 8.**Our B3LYP or B3PW calculated W–O or Ti–O chemical bond populations for WO

_{3}, SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}bulk and for WO

_{2}- or TiO

_{2}-terminated (001) surfaces (in e).

Material | Method | W-O or Ti-O Chemical Bond Populations | |
---|---|---|---|

Bulk | WO_{2}, TiO_{2}-Term. (001) | ||

WO_{3} | B3LYP | +0.142 | +0.108 (W(I)—O(I)) |

+0.142 | +0.278 (W(I)—O(II)) | ||

SrTiO_{3} | B3PW | +0.088 | +0.118 |

BaTiO_{3} | B3PW | +0.098 | +0.126 |

PbTiO_{3} | B3PW | +0.098 | +0.114 |

CaTiO_{3} | B3PW | +0.084 | +0.114 |

**Table 9.**B3LYP- or B3PW-calculated Γ–Γ band gaps for WO

_{3}, SrTiO

_{3}, BaTiO

_{3}, PbTiO

_{3}and CaTiO

_{3}bulk as well as their WO

_{2}or TiO

_{2}-terminated (001) surfaces.

Material | Method | Calculated Band Gap at Γ–Γ Point | |
---|---|---|---|

Bulk | WO_{2}, TiO_{2}-Term. (001) | ||

WO_{3} | B3LYP | 4.95 | 1.16 |

SrTiO_{3} | B3PW | 3.96 | 3.95 |

BaTiO_{3} | B3PW | 3.55 | 2.96 |

PbTiO_{3} | B3PW | 4.32 | 3.18 |

CaTiO_{3} | B3PW | 4.18 | 3.30 |

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Eglitis, R.I.; Purans, J.; Jia, R. Comparative Hybrid Hartree-Fock-DFT Calculations of WO_{2}-Terminated Cubic WO_{3} as Well as SrTiO_{3}, BaTiO_{3}, PbTiO_{3} and CaTiO_{3} (001) Surfaces. *Crystals* **2021**, *11*, 455.
https://doi.org/10.3390/cryst11040455

**AMA Style**

Eglitis RI, Purans J, Jia R. Comparative Hybrid Hartree-Fock-DFT Calculations of WO_{2}-Terminated Cubic WO_{3} as Well as SrTiO_{3}, BaTiO_{3}, PbTiO_{3} and CaTiO_{3} (001) Surfaces. *Crystals*. 2021; 11(4):455.
https://doi.org/10.3390/cryst11040455

**Chicago/Turabian Style**

Eglitis, R. I., Juris Purans, and Ran Jia. 2021. "Comparative Hybrid Hartree-Fock-DFT Calculations of WO_{2}-Terminated Cubic WO_{3} as Well as SrTiO_{3}, BaTiO_{3}, PbTiO_{3} and CaTiO_{3} (001) Surfaces" *Crystals* 11, no. 4: 455.
https://doi.org/10.3390/cryst11040455