Quasiparticle interfacial level alignment of highly hybridized frontier levels: H$_2$O on TiO$_2$(110)

Knowledge of the frontier levels' alignment prior to photo-irradiation is necessary to achieve a complete quantitative description of H$_2$O photocatalysis on TiO$_2$(110). Although H$_2$O on rutile TiO$_2$(110) has been thoroughly studied both experimentally and theoretically, a quantitative value for the energy of the highest H$_2$O occupied levels is still lacking. For experiment, this is due to the H$_2$O levels being obscured by hybridization with TiO$_2$(110) levels in the difference spectra obtained via ultraviolet photoemission spectroscopy (UPS). For theory, this is due to inherent difficulties in properly describing many-body effects at the H$_2$O-TiO$_2$(110) interface. Using the projected density of states (DOS) from state-of-the-art quasiparticle (QP) $G_0W_0$, we disentangle the adsorbate and surface contributions to the complex UPS spectra of H$_2$O on TiO$_2$(110). We perform this separation as a function of H$_2$O coverage and dissociation on stoichiometric and reduced surfaces. Due to hybridization with the TiO$_2$(110) surface, the H$_2$O 3a$_1$ and 1b$_1$ levels are broadened into several peaks between 5 and 1 eV below the TiO$_2$(110) valence band maximum (VBM). These peaks have both intermolecular and interfacial bonding and antibonding character. We find the highest occupied levels of H$_2$O adsorbed intact and dissociated on stoichiometric TiO$_2$(110) are 1.1 and 0.9 eV below the VBM. We also find a similar energy of 1.1 eV for the highest occupied levels of H$_2$O when adsorbed dissociatively on a bridging O vacancy of the reduced surface. In both cases, these energies are significantly higher (by 0.6 to 2.6 eV) than those estimated from UPS difference spectra, which are inconclusive in this energy region. Finally, we apply self-consistent QP$GW$ (scQP$GW$1) to obtain the ionization potential of the H$_2$O-TiO$_2$(110) interface.

ABSTRACT: Knowledge of the frontier levels' alignment prior to photo-irradiation is necessary to achieve a complete quantitative description of H 2 O photocatalysis on TiO 2 (110). Although H 2 O on rutile TiO 2 (110) has been thoroughly studied both experimentally and theoretically, a quantitative value for the energy of the highest H 2 O occupied levels is still lacking. For experiment, this is due to the H 2 O levels being obscured by hybridization with TiO 2 (110) levels in the difference spectra obtained via ultraviolet photoemission spectroscopy (UPS). For theory, this is due to inherent difficulties in properly describing many-body effects at the H 2 O-TiO 2 (110) interface. Using the projected density -10 0 ε − ε VBM (eV)

INTRODUCTION
The photooxidation activity of a surface is determined by the interfacial level alignment between the occupied adsorbate levels and those of the substrate. 1,2 Water photooxidation on TiO 2 has attracted enormous attention 3-10 for energy applications 11,12 based on H 2 production. 13 This reaction also plays an important role in photocatalytic environmental remediation and surface selfcleaning/sterilizing. 1,2,14 This is because the resulting hydroxyl radicals are the key intermediates in the oxidative degradation of organic species. 15,16 To understand water photooxidation, it is necessary to understand the interfacial level alignment between the occupied levels of H 2 O and the TiO 2 substrate. 17 Experimentally, the most common approach to access the adsorbate levels is to take the difference between the covered and clean surface spectra from photoemission spectroscopy. However, when the adsorbate and surface levels are strongly hybridized, it becomes difficult to disentangle the adsorbate and surface contributions to the UPS spectra using only the difference spectra. 18 For example, shifting of the surface levels due to hybridization or band bending may completely obscure the adsorbate levels. 18 Further, the adsorbate levels near the valence band maximum (VBM) are the most likely to be obscured. It is precisely these levels that are most important for photooxidation processes. Using a theoretical approach, one can directly disentangle the molecular levels by projecting the density of states (DOS) of the interface onto the atomic orbitals of the molecule. Altogether, this makes a robust theoretical approach necessary to accurately predict the alignment of the adsorbate and substrate levels, and separate the adsorbate and surface spectra.
A robust theoretical treatment requires quasiparticle (QP) G 0 W 0 to capture the anisotropic screening of the electron-electron interaction at the interface. [19][20][21] As previously demonstrated for CH 3 OH on TiO 2 (110), QP G 0 W 0 is necessary to obtain even a qualitative description of the level alignment. [22][23][24] For this interface, the occupied levels of the molecule are only weakly hybridized with the surface levels. This allowed an unambiguous comparison to the photoemission difference spectrum. 22 However, for H 2 O on rutile TiO 2 (110), this is not the case.
The occupied molecular levels of H 2 O on single crystal rutile TiO 2 (110) have been probed via ultraviolet photoemission spectroscopy (UPS) 18,25,26 and metastable impact electron spectroscopy (MIES). 26 These experiments were performed under ultrahigh vacuum (UHV) conditions from low to room temperature, 25 from 0.01 to 100 L H 2 O exposure, 18 and for various surface preparations resulting in either reduced TiO 2−x (110) with surface oxygen defects or "nearly-perfect" TiO 2 (110). 18 Altogether, these experiments have addressed the long-standing controversy as to where and how H 2 O adsorbs and dissociates on TiO 2 (110). [27][28][29][30][31][32][33][34][35][36] At 150 K the photoemission difference spectrum between H 2 O covered and clean TiO 2 (110) surfaces consists of three peaks, which are attributed to intact H 2 O adsorbed on Ti coordinately unsaturated sites (Ticus). 25 Upon heating to 300 K, the difference spectrum's three-peak structure evolves into a two-peak structure, which is attributed to dissociated H 2 O adsorbed on bridging O vacancies (O vac br ), i.e., O br H surface species. 25 This assignment of the UPS spectra to intact (I) H 2 O@Ticus or dissociated (D) H 2 O@O vac br is based on the peak energy separations being consistent with those reported for H 2 O 37 in gas phase or OH − in NaOH. 38 A comparison to the H 2 O and OH − peaks is robust for the molecular levels that lie below and have little hybridization with the surface DOS. However, the adsorbate levels that lie within the surface valence band may significantly hybridize with the surface, with a single molecular level contributing to many interfacial levels. These interfacial levels are thus not easily associated with H 2 O and OH − levels. This is exacerbated by the mixing of the molecular levels due to symmetry breaking at the interface. As a result, "between 5 and 8 eV" below the Fermi level, experimentally they "are unable to produce reliable difference structures" from the UPS spectra obtained for "nearly-perfect" TiO 2 (110) exposed to H 2 O at 160 K. 18 Using the QP G 0 W 0 H 2 O projected DOS (PDOS), we have disentangled the adsorbate and surface contributions to the UPS spectra within this difficult energy range. This has been done as a function of H 2 O coverage and dissociation on stoichiometric and reduced surfaces. In so doing, we provide quantitative values for the energies of the highest H 2 O occupied levels, prior to photoirradiation, for a number of experimentally relevant 3,5-7,15 H 2 O-TiO 2 (110) structures.
To directly compare to red-ox potentials, the important quantities for determining photoelectrocatalytic activity, one needs the alignment relative to the vacuum level, Evac. 39,40 With this, one obtains the ionization potential directly from −ε PDOS peak + Evac. To obtain a more accurate absolute level alignment, we employ our recently introduced self-consistent QP GW 41-43 technique scQPGW1. 22 The presentation of the results is organized as follows. First, we focus on the H 2 O levels that lie below and have little hybridization with the substrate DOS. This is done for intact H 2 O@Ticus in Section 3.1 and dissociated H 2 O@O vac br in Section 3.2. Further, in Section 3.3, we shown that these results are rather independent of the choice of xc-functional. In so doing we provide evidence for a robust semi-quantitative agreement with the UPS difference spectra for the adsorbate levels for which an unambiguous comparison with the experiment is possible. For a more complete understanding of the UPS experiments, in Section 3.4 we analyze the H 2 O PDOS for a variety of other H 2 O structures on the stoichiometric and reduced surfaces. These may form under different experimental conditions and surface preparations. In Section 3.5 we focus on the highest H 2 O occupied levels, which are significantly hybridized with the substrate DOS. The success of the QP G 0 W 0 PDOS strategy for the lower-energy part of the UPS difference spectra provides support for our results in this difficult spectral region, where a straightforward comparison with experiment is not possible. Finally, in Sec-tion 3.6, we employ scQPGW1 to obtain an improved absolute level alignment relative to Evac, and thus estimate the ionization potential of the H 2 O-TiO 2 (110) interface.

METHODOLOGY
Our QP G 0 W 0 calculations [44][45][46] have been performed using vasp within the projector augmented wave (PAW) scheme. 47 The G 0 W 0 calculations are based on Kohn-Sham wavefunctions and eigenenergies from density functional theory (DFT) obtained using a generalized gradient approximation (PBE) 48 for the exchange correlation (xc)-functional. 49 The dependence of the QP G 0 W 0 DOS and PDOS on the DFT xc-functional has been tested for 1 ML intact H 2 O@Ticus of stoichiometric TiO 2 (110) and 1 ⁄2ML dissociated H 2 O@O vac br of defective TiO 2− 1 ⁄4 (110) with 1 ⁄2ML of O vac br . For these structures, G 0 W 0 calculations based on the local density approximation (LDA), 50 van der Waals (vdW-DF) 51 , and the rangeseparated hybrid (HSE) 52 xc-functionals have been carried out for comparison with the PBE based G 0 W 0 calculations. In particular, we use the HSE06 53 variant of the HSE xc-functional.
In the QP G 0 W 0 approach, the contribution to the Kohn-Sham (KS) eigenvalues from the exchange and correlation (xc)-potential Vxc is replaced by the self energy Σ = iGW, where G is the Green's function and W is the screening 44 based on the KS wavefunctions. 45 The dielectric function is obtained from linear response time-dependent (TD) density functional theory (DFT) within the random phase approximation (RPA), including local field effects. 46 From G 0 W 0 one obtains first-order QP corrections to the KS eigenvalues, but retains the KS wavefunctions. Since our aim is to compare the computed interfacial level alignment with measured UPS spectra, it is most consistent to align the QP G 0 W 0 levels with the VBM.
We find Evac, i.e., the effective potential far from the surface, from G 0 W 0 is essentially the same as the Evac from DFT. In other words, the effective potential is unchanged by G 0 W 0 . To obtain a more accurate absolute QP level alignment relative to Evac, we employ a self-consistent QP GW approach. 41 In particular, by employing the scQPGW1 approach, we obtain both a QP PDOS comparable to that from QP G 0 W 0 and an improved alignment relative to Evac. 22,23 Here, 25%, 25%, and 50%, of the QP self energies are "mixed" with the DFT xc-potential over three self-consistent QP GW cycles, 41 respectively. If, instead, 100% of the DFT xcpotential were replaced by QP self energy in a single self-consistent QP GW cycle, one would exactly obtain the QP G 0 W 0 eigenvalues. However, this mixing is required to obtain a smooth convergence of both the QP wavefunctions and the absolute QP level alignment. To fully converge our self-consistent QP GW calculations (scQPGW), we perform a further eight cycles, with each introducing a further 25% of the QP self energy.
The geometries have been fully relaxed using LDA, 50 PBE, 48 or vdW-DF 51 xc-functionals, with all forces ≲ 0.02 eV/Å. HSE calculations are performed for the relaxed geometries obtained with PBE. We employ a plane-wave energy cutoff of 445 eV, an electronic temperature k B T ≈ 0.2 eV with all energies extrapolated to T → 0 K, and a PAW pseudopotential for Ti which includes the 3s 2 and 3p 6 semi-core levels. All calculations have been performed spin unpolarized.
For the clean stoichiometric TiO 2 (110) surface 23 we have used a four layer slab and an orthorhombic 1×1 unit cell of 6.497×2.958× 40 Å 3 , i.e., where D ≈ 27 Å is the vacuum thickness and a and c are the ex-perimental lattice parameters for bulk rutile TiO 2 (a = 4.5941 Å, c = 2.958 Å). 54 We have employed a Γ-centered 4 × 8 × 1 k-point mesh, and 320 bands = 9 1 ⁄3 unoccupied bands per atom, i.e. including all levels up to 26 eV above the valence band maximum (VBM). For the clean reduced TiO 2− 1 4 (110) surface we have used a monoclinic 1 × 2 unit cell of 6.497 × 5.916 × 40 Å 3 , i.e., to maximize the separation between the O vac br . For the H 2 O covered surfaces, we have employed a four layer slab with adsorbates on both sides and an orthorhombic 1 × 2 unit cell of 6.497 × 5.916 × 47 Å 3 , i.e., where D ≈ 34 Å. We employed a Γ centered 4 × 4 × 1 k-point mesh, with approximately 9 1 ⁄6 unoccupied bands per atom, i.e. including all levels up to 30 eV above the VBM, an energy cutoff of 80 eV for the number of G-vectors, and a sampling of 80 frequency points for the dielectric function. The G 0 W 0 parameters are consistent with those previously used for describing rutile TiO 2 bulk, TiO 2 (110) clean surface and interfaces. 22,23 These parameters have been shown to provide accurate descriptions of bulk optical absorption spectra, and both clean surface and interfacial level alignment. 22,23 To model H 2 O in the gas phase, we employed a unit cell with C 2v symmetry and 16 Å of vacuum in each direction. At the G 0 W 0 level, we used a smaller energy cutoff of 40 eV for the number of Gvectors, which has previously shown to provide an accurate description of the optical absorption spectra for isolated molecules. 55,56 To obtain DFT total energies and the relaxed structure of the clean reduced TiO 2− 1 8 (110) we have used a monoclinic 1 × 4 unit cell of 6.497 × 11.832 × 28 Å 3 , i.e., where D ≈ 15 Å, and employed a Γ-centered 4 × 2 × 1 k-point mesh.
In this study, we have performed PBE and subsequent singlepoint RPBE 57 based DFT calculations for the H 2 O adsorption energies E ads on the stoichiometric and reduced surfaces. The RPBE xc-functional was especially developed for the prediction of adsorption properties on metal surfaces. 57 The H 2 O adsorption energy on the Ticus site of a stoichiometric TiO 2 (110) surface is given by  (e) H 2 O molecular orbitals, G 0 W 0 calculated eigenenergies marked in cyan, and experimental gas phase spectrum aligned with the 1b 1 level of (c). 37 Energies are relative to the VBM (ε VBM ). Intensity references are provided for ε > ε VBM when available. Figure 1 we disentangle adsorbate and substrate contributions to the spectrum of intact H 2 O@Ticus, and compare the H 2 O PDOS to the theoretical and experimental difference DOS. Specifically, we model a monolayer (ML) of H 2 O molecules with parallel (⇉) interfacial hydrogen bonds aligned along the [001] direction ( Figure 1(b)). 58,59 Note that 1ML of intact H 2 O is the most stable coverage and structure on the stoichiometric rutile TiO 2 (110) surface. 32 The theoretical difference DOS is the difference between the total DOS of the H 2 O covered (H 2 O@Ticus) and clean stoichiometric (TiO 2 (110)) surfaces, as shown schematically in Figure 1(a). Turquoise areas in the H 2 O@Ticus and difference DOS indicate regions of greater density for the H 2 O covered versus clean stoichiometric surface. The gray area indicates the DOS energy range for the clean stoichiometric TiO 2 (110) surface. Figure 1(c) and (d) show two sets of UPS difference spectra obtained either by raising the temperature (from 150 K to 190 K) for a consistent exposure to H 2 O (0.2 L) for an annealed TiO 2 (110) surface 25 (Figure 1(c)), or by increasing the H 2 O dose (from 0.01 L to 1 L) at low temperature (160 K) for a nearly perfect surface 18 (Figure 1d). The experimental spectra have been referenced to the VBM, which is positioned 3.2 eV below the experimental Fermi level. 23 Comparing the difference DOS to the H 2 O PDOS, we find the peaks lying outside the TiO 2 (110) DOS energy range are clearly attributable to H 2 O levels. As shown in Figure 1(b), these levels are related to the 1b 2 and 3a 1 H 2 O orbitals shown in Figure 1e. This is not the case within the TiO 2 (110) DOS region, where the adsorbate levels are broadened by hybridization with the surface. This hybridization with the surface has been severely underestimated by previous cluster-based MP2 calculations. 60 Within the TiO 2 (110) DOS region, the peaks in the H 2 O PDOS have corresponding peaks in the difference DOS, although the relative peak intensities differ substantially between the two methods. More importantly, the difference DOS has dips centered at  61 respectively, as marked in Figure 1(a). These peaks split due to mixing with the 3a 1 and 1b 1 H 2 O orbitals. This splitting is the origin of the observed dips in the difference DOS, which are also seen experimentally in Figure 1(c) and d.

Intact H 2 O on the Stoichiometric Surface. In
The peak at −9.4 eV in the H 2 O PDOS, which has 1b 2 molecular character, agrees semi-quantitatively with the most strongly bound experimental peaks at −9.8 eV (Figure 1 The assignment of the peaks located within the TiO 2 (110) DOS is much more complicated. The assumption that the highest peak in the experimental spectra originates solely from the H 2 O 1b 1 level 25,26 is an oversimplification. In fact, both the 3a 1 and 1b 1 molecular levels contribute within this region ( Figure 1(b)). While the levels with intermolecular 3a 1 bonding character give rise to a distinct peak below the TiO 2 (110) DOS region, those with intermolecular 3a 1 antibonding character are pushed to higher energies and mixed with the 1b 1 molecular levels ( Figure 1(b)). The latter is due to symmetry breaking at the interface. Consequently, the H 2 O PDOS is broadened into several peaks between −5 and −1 eV. These levels have interfacial (3a 1 /1b 1 -O 2pσ/2pπ) bonding and antibonding character (not visible at the isosurface value used).

Dissociated H 2 O on Reduced Surfaces.
To see how dissociation of H 2 O@O vac br affects the spectrum, we now consider 1 ⁄2ML of H 2 O dissociated on a reduced TiO 2− 1 4 (110) surface (Figure 2). Here, we have used TiO 2− 1 4 (110) to denote a surface consisting of 1 ⁄2ML of O vac br defects. This structure corresponds to the staggered O br H surface species, shown in Figure 2 The theoretical difference DOS is the difference between the total DOS of the H 2 O covered (H 2 O@O vac br ) and the clean reduced (TiO 2− 1 4 (110)) surfaces, shown schematically in Figure 2(a). Turquoise areas in the H 2 O@O vac br and difference DOS indicate regions of greater density for the H 2 O covered versus clean reduced surface. The gray area indicates the DOS energy range for the clean reduced TiO 2− 1 4 (110) surface. The O vac br defects give rise to occupied levels with Ti 3d character that are just below the conduction band minimum and outside the energy range shown. 62  for T = 300 K after between 0.01 and 100 L exposure, 18 and (e) for T = 120 K after 0.14, 0.3, 0.4, 0.5, and 0.7 L exposure. 26 Peak positions 18,25,26 are marked in brown. Energies are relative to the VBM (ε VBM ). Intensity references are provided for ε > ε VBM when available. make up the O br H species. In this way the PDOS is provided in terms of H 2 O formula units.
The peak in the difference DOS and PDOS at −7.0 eV has O br H σ character, as shown in Figure 2(b). Note that the peak intensity in the PDOS is about half that in the difference DOS, as the PDOS includes half the O br atoms. This peak's position agrees semiquantitatively with the experimental peaks at −7.1 (Figure 2 Much of the theoretical difference DOS's structure is attributable to the defect healing of O vac br , as seen from the difference DOS between TiO 2 (110) and TiO 2− 1 4 (110) in Figure 3. This suggests that the observed features in the experimental difference spectra over-    generalized gradient approximation (PBE), 48 long-ranged van der Waals interactions (vdW-DF) 51 , or a range-separated hybrid (HSE06) 53 are employed for the xc-functional. This is consistent with the previously reported similarities between PBE and HSE based G 0 W 0 PDOS for CH 3 OH on TiO 2 (110). 23 Figure 5(d,e) shows that scQPGW1 provides a similar H 2 O PDOS level alignement to G 0 W 0 . This is consistent with what was previously reported for the CH 3 OH-TiO 2 (110) interface. 22,23 We clearly see that the differences between the DFT and G 0 W 0 PDOS, i.e., the QP energy shifts, are far from simply being rigid. For instance, we find for PBE that the QP energy shifts for the levels that contribute to the highest-energy PDOS peak ε PDOS peak are almost negligible (cf. Figures 4(d,e) and 5(b,e)) . As a result, the QP G 0 W 0 ε PDOS peak is only ∼ 0.1 eV lower compared to DFT. On the other hand, we find significant QP shifts to stronger binding for the levels that contribute to the most strongly bound PDOS peak with 1b 2 σ molecular character. For example, with PBE the QP G 0 W 0 lowest energy peak is shifted by ∼ −1.7 eV compared to DFT (cf. Figures 4(d,e) and 5(b,e)).
As previously shown for the CH 3 OH-TiO 2 (110) interface, these differences in the shifts of the peaks are directly related to differences in the spatial distribution of the wave functions for the levels contributing to the peaks. [22][23][24] This is because the QP G 0 W 0 corrections to the DFT eigenenergies for interfaces are directly correlated with the spacial distribution of the wave functions. [22][23][24] The negligible shift of the DFT highest-energy PDOS peak (Figures 4  (b,d,f) and 5(b,c)) is due to its strong hybridization with the surface, i.e., weight on TiO 2 (110), for the levels contributing to this peak. [22][23][24] On the other hand, the levels that contribute to the most strongly bound PDOS peak have little weight on TiO 2 (110), and have σ character. Both their localized H 2 O character as well as their σ nature explain why these levels have large QP energy shifts to stronger binding. [22][23][24] Oxygen defective and hydroxylated (h−)TiO 2 surfaces have occupied 3d levels which are associated with reduced Ti 3+ atoms. 64 One such example is the 1 ⁄2ML dissociated H 2 O@O vac br on reduced TiO 2− 1 ⁄4 (110) with 1 ⁄2ML of O vac br shown in Figure 6(a). The spacial distribution of the 3d density for O defective surfaces has been characterized by low temperature scanning tunneling microscopy (STM). 63,67 STM measurements find at 77 K the 3d density is homogeneously distributed along the [001] direction, 63 while at ∼ 5 K the 3d density exhibits an asymmetric localized character. 67 A localized description of the Ti 3+ occupied 3d levels is not obtained from DFT with standard xc-functionals. For example, the occupied 3d levels obtained with PBE are highly delocalized, as clearly shown in Figure 6(b). This is due to self-interaction errors which are inherent in such xc-functionals. If one performs spinpolarized DFT calculations with a hybrid xc-functional on such systems, one obtains localized Ti 3+ 3d 1 levels between 0.7 and 1.6 eV below the CBM, along with a structural deformation of the TiO 2 (110) surface. 63,64 However, spin-paired calculations with HSE06 on the PBE relaxed geometry only yield an occupied shoulder at the CBM ( Figure 6(c)). At the QP G 0 W 0 level based on PBE, this shoulder evolves into a distinct peak about 0.6 eV below the Fermi level, ε F . This effect is even more pronouced when the G 0 W 0 calculation is based on HSE06 (cf. Figure 6(d,e)), which yields peaks at 0.6 and 0.9 eV below ε F . As compared to G 0 W 0 PBE, G 0 W 0 HSE06 shifts the unoccupied 3d levels further up in energy revealing the double peak structure. These energies are in very good agreement with the peak at 0.8 eV below ε F in the UPS spectra of H 2 O@O vac br of Figure 2(d). This peak is not shown in Figure 2(d) as it is slightly above 2 eV with respect to VBM. 18 However, note that G 0 W 0 overestimates by about 1 eV the VBM position relative to ε F as compared with UPS experiments. 18 This result is completely independent of the wavefunction's spacial distribution, i.e., localization, as the G 0 W 0 calculations are based on the KS wavefunctions. This is different from previous findings, which showed DFT with either PBE or hybrid xcfunctionals is only giving distinct peaks for the occupied 3d levels provided the relaxed spin-polarized distorted structure is used in the calculations. 63,64 While for G 0 W 0 based on PBE and HSE06 one sees noticeable differences in the description of the 3d occupied levels, the QP H 2 O PDOS and its alignment relative to the VBM are unchanged. Although localization of the Ti 3+ occupied levels and associated structural deformations are absent from our approach, such features should not significantly alter the QP H 2 O PDOS. This is because the Ti 3+ levels are too far above the VBM (∼ 2 eV 64 ) to hybridize with the H 2 O. Moreover, as we will show in Section 3.4, the QP H 2 O PDOS is rather robust to local deformations of the surface structure, e.g., due to changes in coverage. br defects, respectively. The relative importance of these geometries is illustrated in Figure 10(a) and 10(b) by the average absorption energy E ads per H 2 O molecule on the stoichiometric or reduced surfaces 68 with either PBE 48 or RPBE 57 xcfunctionals. In so doing, the contribution of different structures to the measured spectra can be disentangled. Note that an intact 1 ⁄2ML of H 2 O@O vac br (Figure 8(b)) is probably only a transient locally stable state of the reduced H 2 O-TiO 2− 1 ⁄4 (110) interface, 29 which may easily evolve into the ∼ 0.7 eV more stable dissociated 1 ⁄2ML H 2 O@O vac br (Figure 8(c)). For this reason, we only consider dissociated H 2 O@O vac br structures in Figure 10d. By comparing to lower coverage H 2 O structures ( 1 ⁄2ML [30][31][32]69 to 1ML [30][31][32]69 in Figure 7 and 1 ⁄4ML 70 in Figure 9 to 1 ⁄2ML 71 in Figure 8), we can disentangle the effect of interaction between the   H 2 O molecules on the spectra. Further, these structures allow us to probe the isolated molecule limit.

Coverage and Dissociation Dependence of H 2 O Spec
As shown in Figure 10, at lower coverages the overall width of the spectra is reduced with fewer distinct peaks. When the coverage is increased to include intermolecular interactions between adjacent species, the molecular levels hybridize into bonding and antibonding intermolecular levels. This produces additional peaks above and below those present at low coverage. As a result, the peak with intermolecular bonding 3a 1 character at −6.3 eV for 1ML of H 2 O@Ticus is absent for a 1 ⁄2ML coverage. This reinforces the assignment of the experimental spectra shown in Figure 1 to an intact 1ML H 2 O@Ticus geometry with interacting molecules.
To see how the spectra for dissociation of H 2 O@Ticus compare to H 2 O@O vac br , we have considered the half-dissociated ( 1 ⁄2D) and fully dissociated (D) H 2 O structures shown in Figure 7. As shown in Figure 10(c), the peak at −7.0 eV with O br H σ character for H 2 O@O vac br splits into two peaks for dissociated H 2 O@Ticus. The lower energy peak has both OcusH and O br H σ character, while the higher energy peak is mostly OcusH in character. Furthermore, we find a similar couple of peaks for 3 ⁄4ML mixtures of dissociated H 2 O@Ticus and H 2 O@O vac br shown in Figure 10d. This means one may recognize dissociated H 2 O@Ticus by both the presence of two peaks at about −7.0 and −6.3 eV, and the absence of the low-energy peak with 1b 2 character for intact H 2 O@Ticus.
The absence of a peak at about −6.3 eV in the experimental spectra shown in Figure 2(c) reinforces its attribution to dissociated H 2 O@O vac br rather than dissociated H 2 O@Ticus. This is further supported by the calculated H 2 O absorption energies (Figure 10(a) and 10(b)). These are generally weaker for dissociated H 2 O@Ticus, and stronger for H 2 O@O vac br , as in previous calculations. 29 To check whether changes in the absorption geometry of H 2 O affect the spectra for the same coverage, we compare 1ML of H 2 O {I, 1 ⁄2D, D} adsorbed with either parallel (⇉) or antiparallel (⇄) interfacial hydrogen bonds 58 (black dashed lines in Figure 7). Overall, the two sets of spectra are consistent, and demonstrate the general robustness of the DOS to minor changes in the water absorption geometry. However, as the H 2 O molecules are no longer equivalent when the interfacial hydrogen bonds are antiparallel, there is a greater splitting between bonding and antibonding contributions for the peaks with 1b 2 and 3a 1 molecular character. In particular, for intact H 2 O, the lowest energy peak with molecular 1b 2 character splits with a separate peak at −9.6 eV, which is closer to the peaks at −9.8 25 (Figure 1(c)) and −10.0 eV 18 (Figure 1d) (Figure 7), and consider the effect of additional H 2 O@Ticus to 1 ⁄4ML ( Figure 9) and 1 ⁄2ML (Figure 8) H 2 O@O vac br . 69 In this way we can can see how robust the observed features in the individual spectra for isolated species are to screening by H 2 O layers, 6,7 and probe the liquid water limit. 73 When a second layer of H 2 O is added to the low coverage intact 1 ⁄2ML H 2 O@Ticus structure, the levels with H 2 O 1b 2 character are unchanged, while the levels with 3a 1 and 1b 1 second layer character are more localized and weakly hybridized with the surface. These levels are seen as the two most intense peaks at −4.3 and −2.2 eV (Figure 10(c)). The former coincides with the peak at −4.2 eV observed experimentally at low temperatures (Figure 1(c)), suggesting multilayer H 2 O structures may be present under these experimental conditions. The intermolecular H bonding between the layers delocalizes the molecular levels of the first layer. This is seen from the peak at −6.1 eV with antibonding 3a 1 character on the first layer. We saw the same behavior when increasing the first layer's coverage from 1 ⁄2ML to 1 ML. This is further confirmation that the peak observed experimentally at −6.4 eV has intermolecular character. When a second 1 ⁄2 layer of H 2 O is added to the 1ML H 2 O@Ticus { 1 ⁄3D, 2 ⁄3D} structures, 72 a denser network of intermolecular and interfacial hydrogen bonds is formed, as shown in Figure 7. This causes a stronger hybridization between the OH and H 2 O σ levels. For the 1 ⁄3D structure, this results in the four distinct σ peaks shown in Figure 10(c). On the one hand, the peaks at −9.1 and −6.2 eV have predominantly intact H 2 O and OcusH character, as was the case for 1ML of 1 ⁄2D H 2 O@Ticus. On the other hand, the peaks at −7.9 and −7.4 eV are most related to the second layer. In effect, the H 2 O σ level of the second-layer H 2 O, which is fully saturated with four hydrogen bonds, is upshifted by more than an eV. This is not the case for the 2 ⁄3D structure (Figure 7), where the peak at −9.1 eV instead has mostly intact second-layer H 2 O 1b 2 character. As was the case for intact 1 1 ⁄2ML H 2 O@Ticus, the addition of a second 1 ⁄2 layer of H 2 O induces a stronger hybridization of the O br H levels, and introduces an additional intense peak at −4.4 eV (Figure 10(c)). This again suggests the experimentally observed peak at −4.2 eV (Figure 1(c)) may be due to multilayer H 2 O.
Overall, we find the addition of second-layer H 2 O affects the resulting spectrum qualitatively. We find both additional features and a redistribution of those due to the first H 2 O layer. When we instead add H 2 O@Ticus to the 1 ⁄4ML and 1 ⁄2ML H 2 O@O vac br structures (Figures 9, and 8) we find the resulting spectrum is the sum of the separate spectra to within 0.2 eV (Figure 10). For example, the 1 1 ⁄2ML 1 ⁄3D spectrum (Figure 10d) for 1ML of intact H 2 O added to 1 ⁄2ML H 2 O@O vac br (Figure 8) is basically the sum of the 1ML in-tact H 2 O@Ticus (Figure 1(a)) and 1 ⁄2ML H 2 O@O vac br (Figure 2(a)) PDOS spectra downshifted by 0.2 eV. This explains the ease with which the experimental single-layer H 2 O spectra may be analyzed for levels outside the surface DOS region.

Alignment of the Highest H 2 O Occupied Levels.
So far, we have concentrated our analysis on the lower energy peaks observed in the experimental spectra. This was done to demonstrate the robustness of the calculated QP DOS. Having established this, we now focus on the adsorbate levels near the VBM, which play an important role in photooxidation processes. In this respect, the highest H 2 O occupied levels' alignment for 1ML intact and dissociated H 2 O@Ticus, and 1 ⁄2ML dissociated H 2 O@O vac br is of utmost importance. The former structure corresponds to the reactant species on stoichiometric surfaces, 32 which undergoes photoirradiation. The latter structures act as hole traps and are thus the main oxidizing agents on TiO 2 (110). 74,75 We have shown that the experimental peak at −4.2 eV 25 is not, in fact, the highest energy peak of H 2 O@Ticus. We instead find the highest-energy PDOS peak, ε PDOS peak , for 1ML intact H 2 O@Ticus at −1.1 eV relative to the VBM (Figure 10(c)). This is 0.6 eV closer to the VBM than the ∼ −1.7 eV estimate 15 deduced from the onsets of the UPS difference spectra in Ref. 18. Moreover, as 1ML H 2 O@Ticus dissociates, ε PDOS peak moves up to −1.0 eV ( 1 ⁄2D) and −0.9 eV (D) (Figure 10(c)). This is again significantly higher than the ∼ −1.8 eV estimate 7 based on UPS difference spectra for the TiO 2 (100) surface from Ref. 76. As was the case for CH 3 OH on TiO 2 (110), 22 this raising of ε PDOS peak can be related to the charge transfer of −0.4e that accompanies deprotonation (arrows in Figure 7). We find for the 1ML intact structure on TiO 2 (110) ε PDOS peak is 0.2 eV closer to the VBM for H 2 O than for CH 3 OH, 22-24 while for the 1ML 1 ⁄2D structures ε PDOS peak is the same. However, the highest PDOS peak is both less intense and broader for H 2 O compared to CH 3 OH, due to the stronger hybridization with the surface. This is why, as discussed in Section 3.1, the QP G 0 W 0 ε PDOS peak is only ∼ 0.1 eV lower compared to DFT [22][23][24] (Figure 4). After adding second-layer H 2 O, ε PDOS peak is unchanged with weight mostly remaining on the first layer.
We find for 1 ⁄2ML dissociated H 2 O@O vac br ε PDOS peak ≈ −1.1 eV rel-ative to the VBM (Figure 10d), the same as for intact H 2 O@Ticus. This is much higher than the previous estimate of ∼ −3.7 eV 7 for O br H based on the UPS difference spectra in Ref. 25. Our corrected ε PDOS peak value agrees with the recently demonstrated photocatalytic importance of O br H sites as the main oxidizing species on TiO 2 (110). 74 Based on ε PDOS peak for 1ML intact H 2 O@Ticus, vertical excitations from the highest H 2 O occupied levels to the TiO 2 (110) conduction band require photon energies that exceed the electronic band gap for bulk rutile TiO 2 (3.3 ± 0.5 eV 77 ) by ≳ 1 eV. However, the hole generated by such supra-band gap excitations should be mostly located on TiO 2 (110) O 2pπ rather than H 2 O O 2p levels. This is because the H 2 O highest levels are hybridized with TiO 2 (110) and are predominantly TiO 2 (110) in character.
The fact that the highest H 2 O levels are ∼ 1 eV below the VBM does not necessarily mean that they cannot be photooxidized by holes photogenerated within the TiO 2 (110) valence band. A recent DFT study with HSE06 found trapped holes at surface O sites, i.e., three-fold coordinated O 3fold , are shared with nearyby HO-Ticus groups. 5 Moreover, it has been suggested that H 2 O can only be photooxidized, i.e., trap a hole, upon deprotonation. 78,79 In other words, hole transfer to the HO-Ticus site should be mediated by the deprotonation of intact H 2 O@Ticus to the nearest O br site. Altogether, this suggests that H 2 O@Ticus photooxidation should be initiated by band-to-band and supra-band photo-excitations, which result in the generation of holes within the TiO 2 (110) valence band. These TiO 2 (110) free holes may then be trapped at O 3fold sites, and partially transferred to nearby HO-Ticus upon H 2 O deprotonation.
3.6. Vacuum Level Alignment. So far, we have considered the level alignment of the interfacial levels relative to the VBM of the substrate. This allows a direct comparison of the occupied PDOS with the measured UPS spectra. However, to assess the photoelectrocatalytic activity of the interface, one needs the absolute level alignment relative to the vacuum level Evac.
In Figure 11 we show the level alignment for gas phase H 2 O and 1ML intact H 2 O@Ticus relative to Evac from DFT, scQPGW1, and G 0 W 0 based on PBE and HSE xc-functionals. These are compared to the measured CBM for the liquid H 2 O-TiO 2 (110) interface, 40,86 and the measured and coupled-cluster (CCSD(T)) gas phase H 2 O ionization potential. 80 Our calculated IP values for H 2 O in gas phase are consistent with those reported previously in the literature. 80,[87][88][89] Although the relative energies of the 1b 1 , 3a 1 , and 1b 2 H 2 O levels are consistent over all five levels of theory, the levels are rigidly downshifted. We observe a clear ordering in increasing IP of PBE DFT (7.2 eV) < HSE DFT ≪ PBE scQPGW1 < PBE G 0 W 0 ≲ HSE G 0 W 0 ≲ PBE scQPGW (12.8 eV) < Hartree Fock (HF 13.9 eV 88 ).
To understand the origin of this ordering, we have probed the dependence of the IP on the fraction of Hartree-Fock exact exchange included in the range-separated HSE xc-functional via the parameter α in Figure 12. On the one hand, for DFT, we find a strong linear dependence of IP on α, i.e., IP ≈ IP PBE + (IP α=1 − IP PBE )α ≈ 7.2 + 5.9α, with α ∼ 0.9 providing a quantitative agrement with experiment and CCD(T) calculations. Overall, this linear dependence is not surprising, as α may be interpreted as the amount of electron-electron screening, i.e., the inverse dielectric constant ε −1 ∞ . 90,91 In other words, the fraction of exact exchange α included, determines the amount of screening, ε −1 ∞ , incorporated within the xc-functional. The quantitative agreement of the IP for α ∼ 0.9 is because small molecules, e.g., H 2 O, are weakly screened in the gas phase (ε∞ ∼ 1).
On the other hand, for G 0 W 0 , the calculated IP has a much weaker dependence on α, i.e., the starting xc-functional, with IP ≈ IP α=1 − ∆IP(10 α−1 − 1) ≈ 13.4 − 1.2 × 10 −α . Further, the G 0 W 0 and  DFT IP coincide when α → 1. For G 0 W 0 based on PBE (α = 0), the IP already agrees semi-quantitatively with experiment, with full quantitative agreement obtained for G 0 W 0 based on HSE06 (α = 0.25). This is because the RPA ε∞ ∼ 1, independently of α. Essentially, the calculated G 0 W 0 IPs would also be obtained from DFT using an HSE xc-functional with 0.84 < α < 1.0, i.e., 1 < ε∞ < 1.2. Overall, this implies G 0 W 0 is a predictive method for the IP of small molecules. However, the scQPGW technique has the added advantage of being completely independent of the starting xc-functional, 23,88 while providing a nearly quantitative IP.
For the H 2 O-TiO 2 (110) interface, e.g., 1ML intact H 2 O@Ticus, the highest energy H 2 O PDOS peak, ε PDOS peak , is pinned ∼ 1 eV below the VBM across PBE DFT, HSE DFT, PBE scGW1, PBE G 0 W 0 , and HSE G 0 W 0 . For this reason, the IP of the H 2 O interfacial levels is controlled by the alignment of the VBM with respect to the vacuum. This means we only need to consider the absolute VBM level alignment of the interface, i.e., the interface's IP = −ε VBM + Evac, as a descriptor of photoelectrocatalytic activity.
In Figure 11 we see that the IP of the interface follows a different ordering across the methodologies from that of gas phase H 2 O. In particular, we find PBE G 0 W 0 (6.0 eV) ∼ PBE DFT < HSE06 G 0 W 0 ≈ PBE scQPGW1 < HSE06 DFT (7.3 eV). Figure 12 shows that, as was the case for H 2 O in gas phase, the IP of the H 2 O@Ticus interface across the various methods is ordered according to the method's description of the screening, ε −1 ∞ . As discussed above, for hybrid xc-functionals such as HSE, the effective screening is determined by the fraction of exact exchange α included. Essentially, α plays the role of the effective screening within the method, ε −1 ∞ . Although HSE06 incorporates less screening (ε∞ ≈ 4) than experiment for rutile TiO 2 (ε TiO2 ∞ ≈ 7.6), 85 the HSE06 IP for the interface is in agreement with the experimental estimate of IP ≈ 7.1 eV. 40,86 If one performs G 0 W 0 based on HSE06, a stronger screening is applied, i.e., ε∞ ≈ 5.7, yielding a lower IP for the interface. In fact, as indicated by the red arrow in Figure 12, a similar IP to HSE06 G 0 W 0 should be obtained from HSE DFT by setting the fraction of exact exchange to the inverse dielectric constant of bulk TiO 2 , i.e., α = 1 ε TiO2 ∞ . Adjusting α to the measured inverse dielectric constant has been previously found to give improved band gaps. 90 From PBE scQPGW1, one obtains an IP consistent with that of HSE06 G 0 W 0 . This is because we find the screening in scQPGW decreases from PBE RPA with each self-consistent cycle. Essentially, the final screening incorporated in scQPGW1 is similar to that of HSE06 RPA.
As shown in Figure 11, PBE G 0 W 0 gives an IP slightly lower than PBE DFT for the interface, while the PBE G 0 W 0 CBM is shifted up by about 2 eV. This is surprising, since PBE DFT already yields a CBM level alignment for the interface in excellent agreement with experiment. This is partially due to PBE RPA's overestimation of the screening of TiO 2 (ε∞ ∼ 8.3). Although HSE06 G 0 W 0 has a weaker screening than PBE G 0 W 0 , the resulting absolute alignment of the CBM is quite similar. If instead, the self energy corrections are applied self-consistently via PBE scQPGW1, the absolute alignment of the CBM is significantly lower, but still greater than that of PBE DFT or HSE06 DFT. This is again related to decreases in the dielectric constant with each self-consistent cycle. For this reason, scQPGW1 tends to provide reasonable band gaps for TiO 2 (110) interfaces. Overall, we observe an ordering in increasing band gap of PBE DFT < HSE06 DFT ≲ PBE scQPGW1 < PBE G 0 W 0 ≈ HSE06 G 0 W 0 , with HSE06 DFT providing the best absolute alignment of the CBM and VBM for the H 2 O@Ticus interface.
HSE06 DFT provides the most accurate description of the IP of the clean and H 2 O@Ticus covered stoichiometric TiO 2 (110) surfaces. Although the HSE06 DFT IP for H 2 O@O vac br is significantly lower than the one measured for h−TiO 2 (110), in both cases, the IP is shifted to lower energies relative to the clean stoichiometric surface. Differences in the magnitude of the shifts are probably due to the differences in defect coverage between the experiment (6-9%) 83 and the calculation (50%).
The similarty between HSE06 DFT and scQPGW based on either PBE or HSE06 for the clean TiO 2 (110) surface, 23 points to a similar screening from these two techniques. This also demonstrates the starting point independence of the scQPGW technique.
To summarize, although scQPGW provides accurate IPs, the band gap is greatly overestimated, as reported previously. 22,23,41,92 While scQPGW1 provides a more accurate band gap, it achieves only a qualitative description of the IP. HSE06 achieves a quantitative description of both the IP and band gap, but provides a poor description of the molecular level alignment relative to the VBM. 22,23,92 However, since the highest occupied H 2 O levels are significantly hybridized with the substrate, this is not a major drawback in this case. In general, for TiO 2 (110), a more effective strategy is to combine the calculated IP from HSE06 with the occupied interfacial levels' alignment from G 0 W 0 or scQPGW1.

CONCLUSIONS
The level alignment prior to photo-irradiation is an important piece of the puzzle needed to get a complete atomistic picture of photocatalytic processes. Here we have shown that the complex UPS spectra for the H 2 O-TiO 2 interface may be disentangled using QP G 0 W 0 PDOS. We have firmly established the robustness of the QP G 0 W 0 H 2 O PDOS by: (1) demonstrating its xc-functional (PBE, LDA, vdW-DF, and HSE06) independence, (2) comparing to selfconsistent QP GW techniques (scQPGW1), and (3) considering its dependence on surface coverage and dissociation. Altogether, these calculations provide an accurate interpretation of the complex UPS and MIES experiments 18,25,26 for the H 2 O-TiO 2 (110) interface, and provide accurate estimates of the highest H 2 O occupied levels' alignment relative to the VBM.
Our results provide two important pieces of the puzzle: (1) the molecular structure of the photocatalytic interface and (2) the molecular alignment of the doubly occupied levels near the VBM responsible for hole trapping prior to irradiation. To complete the picture, the molecular structure and level alignment in the presence of the photo-generated hole is also needed. Previous DFT studies using the hybrid HSE xc-functional have found a hole can be trapped at surface O 2pπ levels of O br and HO-Ticus sites. 5 However, the screening of such localized levels may not be well described by HSE, which tends to underbind localized interfacial levels. 23 This underbinding is corrected upon inclusion of manybody effects via QP G 0 W 0 . 23 Having demonstrated the capability of G 0 W 0 for the description of level alignment prior to irradiation, this work points the way forward via future QP G 0 W 0 studies of level alignment for trapped hole levels.

Supporting Information
Total energies and optimized geometries. This material is available free of charge via the Internet at http://pubs.acs.org.