Quasiparticle interfacial level alignment of highly hybridized frontier levels: H{}_{\text{2}}O on TiO{}_{\text{2}}(110)

Quasiparticle interfacial level alignment of highly hybridized frontier levels: HO on TiO(110)

Annapaola Migani annapaola.migani@cin2.es [    Duncan J. Mowbray [    Jin Zhao [    Hrvoje Petek [

[.] \captionsetup[table]labelsep=period,labelfont=bf,rm \captionsetup[figure]labelfont=bf,labelsep=period ICN2]22footnotemark: 2  ICN2 - Institut Català de Nanociència i Nanotecnologia, ICN2 Building, Campus UAB, E-08193 Bellaterra (Barcelona), Spain \alsoaffiliation[CSIC]
33footnotemark: 3  CSIC - Consejo Superior de Investigaciones Científicas, ICN2 Building, Campus UAB, E-08193 Bellaterra (Barcelona), Spain UPV/EHU]
55footnotemark: 5  Nano-Bio Spectroscopy Group and ETSF Scientific Development Center, Departamento de Física de Materiales, Universidad del País Vasco UPV/EHU and DIPC, E-20018 San Sebastián, Spain USTC]
44footnotemark: 4  Department of Physics and ICQD/HFNL, University of Science and Technology of China, Hefei, Anhui 230026, China \alsoaffiliation[SICQI]
66footnotemark: 6  Synergetic Innovation Center of Quantum Information & Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China UP]
Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA

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1 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 has attracted enormous attention 3, 4, 5, 6, 7, 8, 9, 10 for energy applications 11, 12 based on H production 13. This reaction also plays an important role in photocatalytic environmental remediation and surface self-cleaning/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 HO and the TiO 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 spectra18. 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) to capture the anisotropic screening of the electron–electron interaction at the interface 19, 20, 21. As previously demonstrated for CHOH on TiO(110), QP 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 HO on rutile TiO(110), this is not the case.

The occupied molecular levels of HO on single crystal rutile TiO(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 HO exposure 18, and for various surface preparations resulting in either reduced TiO(110) with surface oxygen defects or “nearly-perfect” TiO(110) 18. Altogether, these experiments have addressed the long-standing controversy as to where and how HO adsorbs and dissociates on TiO(110) 27, 28, 29, 30, 31, 32, 33, 34, 35, 36.

At 150 K the photoemission difference spectrum between HO covered and clean TiO(110) surfaces consists of three peaks, which are attributed to intact HO adsorbed on Ti coordinately unsaturated sites (Ti) 25. Upon heating to 300 K, the difference spectrum’s three-peak structure evolves into a two-peak structure, which is attributed to dissociated HO adsorbed on bridging O vacancies (O), i.e., OH surface species 25. This assignment of the UPS spectra to intact (I) HO@Ti or dissociated (D) HO@O is based on the peak energy separations being consistent with those reported for HO37 in gas phase or OH in NaOH38.

A comparison to the HO 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 HO 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(110) exposed to HO at 160 K 18.

Using the QP HO 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 HO coverage and dissociation on stoichiometric and reduced surfaces. In so doing, we provide quantitative values for the energies of the highest HO occupied levels, prior to photo-irradiation, for a number of experimentally relevant6, 7, 15, 3, 5 HO–TiO(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, .39, 40 With this, one obtains the ionization potential directly from . To obtain a more accurate absolute level alignment, we employ our recently introduced self-consistent QP 41, 42, 43 technique scQP 22.

The presentation of the results is organized as follows. First, we focus on the HO levels that lie below and have little hybridization with the substrate DOS. This is done for intact HO@Ti in Section 3.1 and dissociated HO@O 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 HO PDOS for a variety of other HO 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 HO occupied levels, which are significantly hybridized with the substrate DOS. The success of the QP 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 Section 3.6, we employ scQP1 to obtain an improved absolute level alignment relative to , and thus estimate the ionization potential of the HO–TiO(110) interface.

2 Methodology

Our QP calculations44, 45, 46 have been performed using vasp within the projector augmented wave (PAW) scheme 47. The 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 DOS and PDOS on the DFT xc-functional has been tested for 1 ML intact HO@Ti of stoichiometric TiO(110) and \sfrac12ML dissociated HO@O of defective TiO(110) with \sfrac12ML of O. For these structures, calculations based on the local density approximation (LDA) 50, van der Waals (vdW-DF) 51, and the range-separated hybrid (HSE) 52 xc-functionals have been carried out for comparison with the PBE based calculations. In particular, we use the HSE06 53 variant of the HSE xc-functional.

In the QP approach, the contribution to the Kohn-Sham (KS) eigenvalues from the exchange and correlation (xc)-potential is replaced by the self energy , where is the Green’s function and 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 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 levels with the VBM.

We find , i.e., the effective potential far from the surface, from is essentially the same as the from DFT. In other words, the effective potential is unchanged by . To obtain a more accurate absolute QP level alignment relative to , we employ a self-consistent QP approach41. In particular, by employing the scQP1 approach, we obtain both a QP PDOS comparable to that from QP and an improved alignment relative to . 22, 23. Here, 25%, 25%, and 50%, of the QP self energies are “mixed” with the DFT xc-potential over three self-consistent QP cycles 41, respectively. If, instead, 100% of the DFT xc-potential were replaced by QP self energy in a single self-consistent QP cycle, one would exactly obtain the QP 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 calculations (scQP), we perform a further eight cycles, with each introducing a further 25% of the QP self energy.

The geometries have been fully relaxed using LDA50, PBE48, or vdW-DF51 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 eV with all energies extrapolated to K, and a PAW pseudopotential for Ti which includes the 3 and 3 semi-core levels. All calculations have been performed spin unpolarized.

For the clean stoichiometric TiO(110) surface 23 we have used a four layer slab and an orthorhombic unit cell of Å, i.e.,

(1)

where  Å is the vacuum thickness and and are the experimental lattice parameters for bulk rutile TiO ( Å, Å) 54. We have employed a -centered k-point mesh, and 320 bands = 9\sfrac13 unoccupied bands per atom, i.e. including all levels up to 26 eV above the valence band maximum (VBM).

For the clean reduced TiO(110) surface we have used a monoclinic unit cell of Å, i.e.,

(2)

to maximize the separation between the O. For the HO covered surfaces, we have employed a four layer slab with adsorbates on both sides and an orthorhombic unit cell of  Å, i.e.,

(3)

where  Å. We employed a centered k-point mesh, with approximately 9\sfrac16 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 parameters are consistent with those previously used for describing rutile TiO bulk, TiO(110) clean surface and interfaces22, 23. These parameters have been shown to provide accurate descriptions of bulk optical absorption spectra, and both clean surface and interfacial level alignment22, 23.

To model HO in the gas phase, we employed a unit cell with C symmetry and 16 Å of vacuum in each direction. At the level, we used a smaller energy cutoff of 40 eV for the number of G-vectors, 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(110) we have used a monoclinic unit cell of  Å, i.e.,

(4)

where  Å, and employed a -centered k-point mesh.

In this study, we have performed PBE and subsequent single-point RPBE57 based DFT calculations for the HO adsorption energies on the stoichiometric and reduced surfaces. The RPBE xc-functional was especially developed for the prediction of adsorption properties on metal surfaces 57. The HO adsorption energy on the Ti site of a stoichiometric TiO(110) surface is given by

(5)

where is the number of adsorbed HO functional units in the supercell, and , , and are the total energies of the covered and clean stoichiometric surfaces and gas phase water molecule, respectively. Similarly, the HO adsorption energy on the O site of a reduced TiO(110) surface is given by

(6)

where and are the total energies of the covered and clean reduced surfaces, respectively.

3 Results and Discussion

3.1 Intact HO on the Stoichiometric Surface

In Figure 1

Figure 1: Intact HO adsorbed with parallel () interfacial hydrogen bonds on coordinately unsaturated Ti sites (HO@Ti). (a) DOS for 1 ML of intact HO covered (turquoise regions) or clean (gray region) stoichiometric TiO(110), their total DOS difference (dashed line), and the HO PDOS. (b) Selected molecular orbitals at and their energies (dotted lines). UPS difference spectra for HO covered TiO(110) (c) after 0.2 L exposure for 150, 160, 175, and 190 K 25 and (d) for 160 K after 0.05, 0.1, 0.3, 0.7, and 1 L exposure 18. Peak positions25, 18 are marked in brown. (e) HO molecular orbitals, calculated eigenenergies marked in cyan, and experimental gas phase spectrum aligned with the 1b level of (c)37. Energies are relative to the VBM (). Intensity references are provided for when available.

 

we disentangle adsorbate and substrate contributions to the spectrum of intact HO@Ti, and compare the HO PDOS to the theoretical and experimental difference DOS. Specifically, we model a monolayer (ML) of HO molecules with parallel () interfacial hydrogen bonds aligned along the [001] direction (Figure 1(b))58, 59. Note that 1ML of intact HO is the most stable coverage and structure on the stoichiometric rutile TiO(110) surface 32.

The theoretical difference DOS is the difference between the total DOS of the HO covered (HO@Ti) and clean stoichiometric (TiO(110)) surfaces, as shown schematically in Figure 1(a). Turquoise areas in the HO@Ti and difference DOS indicate regions of greater density for the HO covered versus clean stoichiometric surface. The gray area indicates the DOS energy range for the clean stoichiometric TiO(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 HO (0.2 L) for an annealed TiO(110) surface25 (Figure 1(c)), or by increasing the HO dose (from 0.01 L to 1 L) at low temperature (160 K) for a nearly perfect surface18 (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 HO PDOS, we find the peaks lying outside the TiO(110) DOS energy range are clearly attributable to HO levels. As shown in Figure 1(b), these levels are related to the 1b and 3a HO orbitals shown in Figure 1e. This is not the case within the TiO(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 calculations60. Within the TiO(110) DOS region, the peaks in the HO 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 , , and  eV, where there are adsorbate levels in the PDOS, and a peak at  eV, where there are no adsorbate levels in the PDOS. The dips at and  eV correspond to the O 2p and O 2p peaks in the TiO(110) DOS 61, respectively, as marked in Figure 1(a). These peaks split due to mixing with the 3a and 1b HO 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.

The peak at  eV in the HO PDOS, which has 1b molecular character, agrees semi-quantitatively with the most strongly bound experimental peaks at  eV (Figure 1(c)) or  eV (Figure 1d). The peak at  eV in the HO PDOS, which has intermolecular 3a bonding character, agrees semi-quantitatively with the experimental peaks at  eV (Figure 1(c)) or  eV (Figure 1d). Note that the theoretical average deviation is within that amongst the experiments. This may reflect differences in sample preparation, which result in a variety of different HO configurations, i.e., HO coverages, O concentrations, and mixtures of intact and dissociated HO. As we will show in Section 3.4, by considering a variety of HO structures a more complete description of the experiment is obtained. Altogether, this agreement for the and  eV PDOS peaks lends confidence to our results for regions where the experimental results are unclear.

The assignment of the peaks located within the TiO(110) DOS is much more complicated. The assumption that the highest peak in the experimental spectra originates solely from the HO 1b level 25, 26 is an oversimplification. In fact, both the 3a and 1b molecular levels contribute within this region (Figure 1(b)). While the levels with intermolecular 3a bonding character give rise to a distinct peak below the TiO(110) DOS region, those with intermolecular 3a antibonding character are pushed to higher energies and mixed with the 1b molecular levels (Figure 1(b)). The latter is due to symmetry breaking at the interface. Consequently, the HO PDOS is broadened into several peaks between and  eV. These levels have interfacial (3a/1b– O 2p/2p) bonding and antibonding character (not visible at the isosurface value used).

3.2 Dissociated HO on Reduced Surfaces

To see how dissociation of HO@O affects the spectrum, we now consider \sfrac12ML of HO dissociated on a reduced TiO(110) surface (Figure 2).

Figure 2: HO dissociated on bridging O vacancies (HO@O). (a) DOS for \sfrac12 ML of dissociated HO covered (turquoise regions) or clean (gray region) defective TiO(110) with \sfrac12ML of O, their total DOS difference (dashed line), and the HO PDOS. (b) Selected molecular orbitals and their energies (dotted lines). UPS difference spectra for HO on reduced TiO(110) (c) after 0.2 L exposure for 260 and 300 K 25,(d) for 300 K after between 0.01 and 100 L exposure 18, and (e) for 120 K after 0.14, 0.3, 0.4, 0.5, and 0.7 L exposure 26. Peak positions25, 18, 26 are marked in brown. Energies are relative to the VBM (). Intensity references are provided for when available.

 

Here, we have used TiO(110) to denote a surface consisting of \sfrac12ML of O defects. This structure corresponds to the staggered OH surface species, shown in Figure 2(b).

The theoretical difference DOS is the difference between the total DOS of the HO covered (HO@O) and the clean reduced (TiO(110)) surfaces, shown schematically in Figure 2(a). Turquoise areas in the HO@O and difference DOS indicate regions of greater density for the HO covered versus clean reduced surface. The gray area indicates the DOS energy range for the clean reduced TiO(110) surface. The O 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, 63, 64 Note that the HO PDOS includes half the O atoms and all the H atoms that make up the OH species. In this way the PDOS is provided in terms of HO formula units.

The peak in the difference DOS and PDOS at  eV has OH 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 atoms. This peak’s position agrees semi-quantitatively with the experimental peaks at (Figure 2(c)), (Figure 2d), or eV (Figure 2e). The PDOS has a broader feature between and eV, due to hybridization with the surface. This feature is associated with contributions coming from the bonding and antibonding combinations of two distinct p orbitals of the OH species (Figure 2(b)): one perpendicular to the OH bonds (the so-called OH level of NaOH 38); the other in the plane of the OH bonds. The lowest of these peaks at eV corresponds to the bonding combination of the OH levels. This peak’s position agrees semi-quantitatively with the consistently observed experimental peaks at , , and  eV in Figures 2(c), 2d, and 2e, respectively. However, the antibonding OH levels are shifted to much higher energies ( eV), as shown in Figure 2(b).

Much of the theoretical difference DOS’s structure is attributable to the defect healing of O, as seen from the difference DOS between TiO(110) and TiO(110) in Figure 3.

Figure 3: O difference DOS between (red) stoichiometric TiO(110) and (black) reduced TiO(110) with \sfrac12ML of O defects. Red areas indicate defect healing of O, i.e., regions of greater density for the stoichiometric versus reduced surfaces, shown in Figures 7 and 8, respectively.

This suggests that the observed features in the experimental difference spectra overlapping with the reduced surface’s DOS are simply O levels reintroduced by dissociated HO@O. In particular, the peak which is usually attributed to OH levels is actually composed of O surface levels unrelated to the presence of H atoms.

3.3 XC-Functional and Methodology Dependence of HO Spectra for Stoichiometric and Reduced Surfaces

To assess the robustness of the calculated QP HO PDOS, we consider its dependence on the xc-functional and methodology. Specifically, we compare the HO PDOS from DFT, scQP1, and for 1ML intact HO@Ti with parallel () and antiparallel () interfacial hydrogen bonds and \sfrac12ML dissociated HO@O in Figures 4, 5, and 6, respectively.

Figure 4: (a) 1ML intact HO adsorbed with parallel () interfacial hydrogen bonds (black dashed lines) on coordinately unsaturated Ti sites (HO@Ti). Total (maroon) and HO projected (blue) DOS computed with (b,d,f) DFT and (c,e,g) using the (b,c) local density approximation (LDA)50 (d,e) generalized gradient approximation (PBE)48 and (f,g) long-ranged van der Waals interactions (vdW-DF)51 for the xc-functional. The calculated HO PDOS are compared with the UPS spectrum at 150 K after 0.2 L exposure25 (black). Energies are relative to the valence band maximum, .

 

Figure 5: (a) 1ML intact HO adsorbed with antiparallel () interfacial hydrogen bonds on coordinately unsaturated Ti sites (HO@Ti). Total (maroon) and HO projected (blue) DOS computed with (b,c) DFT, (d) scQP1, and (e,f) using the (b,d,e) generalized gradient approximation (PBE)48 and (c,f) range-separated hybrid (HSE06)53 for the xc-functional. The calculated HO PDOS are compared with the UPS spectrum at 150 K after 0.2 L exposure25 (black). Energies are relative to the valence band maximum, .
Figure 6: (a) \sfrac12ML HO dissociated on bridging O vacancies (HO@O) of defective TiO(110) with \sfrac12ML of O. Total (maroon) and HO projected (blue) DOS computed with (b,c) DFT and (d,e) using the (b,d) generalized gradient approximation (PBE)48 and (c,e) range-separated hybrid (HSE06)53 for the xc-functional. Filling denotes occupation. The calculated HO PDOS are compared with the UPS spectrum at 300 K after 0.2 L exposure25 (black). Energies are relative to the valence band maximum, .

 

Method (Å)
SXPS65, 66 2.210
LDA 2.180
PBE 2.367
vdW-DF 2.434

 

Table 1: Height of HO Above TiO(110) for 1ML Intact HO@Ti Measured with SXPS and Calculated with LDA, PBE, or vdW-DF XC-Functionals.

We find the observed structure of the HO PDOS is independent of whether the local density approximation (LDA)50, 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 PDOS for CHOH on TiO(110)23. This is despite the greater differences observed amongst the DFT HO PDOS, which all differ qualitatively from the experiments. Furthermore, the HO PDOS is robust to the resulting changes in the HO height above the surface, i.e., the distance between HO and Ti , shown in Table 1. Furthermore, Figure 5(d,e) shows that scQP1 provides a similar HO PDOS level alignement to . This is consistent with what was previously reported for the CHOH–TiO(110) interface 22, 23.

We clearly see that the differences between the DFT and 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 are almost negligible (cf. Figures 4(d,e) and 5(b,e)) . As a result, the QP is only  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 molecular character. For example, with PBE the QP lowest energy peak is shifted by  eV compared to DFT (cf. Figures 4(d,e) and 5(b,e)).

As previously shown for the CHOH–TiO(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 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(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(110), and have character. Both their localized HO character as well as their nature explain why these levels have large QP energy shifts to stronger binding.22, 23, 24

Oxygen defective and hydroxylated ()TiO surfaces have occupied 3d levels which are associated with reduced Ti atoms 64. One such example is the \sfrac12ML dissociated HO@O on reduced TiO(110) with \sfrac12ML of O 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  K the 3d density exhibits an asymmetric localized character.67

A localized description of the Ti 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 spin-polarized DFT calculations with a hybrid xc-functional on such systems, one obtains localized Ti 3d levels between 0.7 and 1.6 eV below the CBM, along with a structural deformation of the TiO(110) surface64, 63. 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 level based on PBE, this shoulder evolves into a distinct peak about 0.6 eV below the Fermi level, . This effect is even more pronouced when the calculation is based on HSE06 (cf. Figure 6(d,e)), which yields peaks at 0.6 and 0.9 eV below . As compared to PBE, 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 in the UPS spectra of HO@O 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 overestimates by about 1 eV the VBM position relative to as compared with UPS experiments.18

This result is completely independent of the wavefunction’s spacial distribution, i.e., localization, as the calculations are based on the KS wavefunctions. This is different from previous findings, which showed DFT with either PBE or hybrid xc-functionals is only giving distinct peaks for the occupied 3d levels provided the relaxed spin-polarized distorted structure is used in the calculations.64, 63

While for based on PBE and HSE06 one sees noticeable differences in the description of the 3d occupied levels, the QP HO PDOS and its alignment relative to the VBM are unchanged. Although localization of the Ti occupied levels and associated structural deformations are absent from our approach, such features should not significantly alter the QP HO PDOS. This is because the Ti levels are too far above the VBM ( eV64) to hybridize with the HO. Moreover, as we will show in Section 3.4, the QP HO PDOS is rather robust to local deformations of the surface structure, e.g., due to changes in coverage.

3.4 Coverage and Dissociation Dependence of HO Spectra for Stoichiometric and Reduced Surfaces

As different experimental conditions and surface preparations have been employed, there are expected to be different HO structures on the surface. To evaluate how strongly the DOS depends on the adsorption geometry, we now consider a variety of coverages of intact and dissociated HO on rutile stoichiometric TiO(110) (Figure 7) and reduced TiO(110) (Figure 8)

Figure 7: Schematics of HO adsorbed intact (I) or dissociated (D) on coordinately unsaturated Ti sites (Ti) of stoichiometric TiO(110). Higher coverages are obtained by the addition of second-layer HO. Coverage is the number of HO formula units per (110) unit area of the clean stoichiometric surface. Dissociation is the fraction of HO molecules which are dissociated, i.e., one minus the ratio of intact HO molecules to HO formula units. Colored frames encompass regions of common fractional dissociation. Charge transfer of about accompanying deprotonation22 of intact HO adsorbed at Ti is represented by arrows, while intermolecular (gray) and interfacial (black) hydrogen bonds are denoted by dotted lines.

 

Figure 8: Schematics of reduced TiO(110) with \sfrac12ML of bridging O vacancies (O) (a) clean, covered with \sfrac12ML (b) intact and (c) dissociated HO@O, and with an additional (d) \sfrac12ML or (e) 1ML of intact HO adsorbed on coordinately unsaturated Ti sites (Ti). Coverage is the number of HO formula units per (110) unit area of the clean reduced surface. Dissociation is the fraction of HO molecules which are dissociated, i.e., one minus the ratio of intact HO molecules to HO formula units. Charge transfer of about accompanying deprotonation22 is represented by arrows, while intermolecular (gray) and interfacial (black) hydrogen bonds are denoted by dotted lines.

 

Figure 9: Schematics of HO adsorbed dissociated (D) on \sfrac14ML of bridging O vacancies (O) on reduced TiO(110) ( \sfrac18). Higher coverages are obtained by the addition of HO@Ti. Coverage is the number of HO formula units per (110) unit area of the clean stoichiometric or reduced surface. Dissociation is the fraction of HO molecules which are dissociated, i.e., one minus the ratio of intact HO molecules to HO formula units. Colored frames encompass regions of common fractional dissociation. Charge transfer of about accompanying deprotonation22 of intact HO adsorbed at Ti or O is represented by arrows, while intermolecular (gray) and interfacial (black) hydrogen bonds are denoted by dotted lines.

 

and TiO(110) (Figure 9) with \sfrac12ML and \sfrac14ML of O defects, respectively. The relative importance of these geometries is illustrated in Figure 10(a) and 10(b) by the average absorption energy per HO molecule on the stoichiometric or reduced surfaces68 with either PBE48 or RPBE57 xc-functionals. In so doing, the contribution of different structures to the measured spectra can be disentangled. Note that an intact \sfrac12ML of HO@O (Figure 8(b)) is probably only a transient locally stable state of the reduced HO–TiO(110) interface29, which may easily evolve into the  eV more stable dissociated \sfrac12ML HO@O (Figure 8(c)). For this reason, we only consider dissociated HO@O structures in Figure 10d.

By comparing to lower coverage HO structures (\sfrac12ML30, 31, 69, 32 to 1ML30, 31, 69, 32 in Figure 7 and \sfrac14ML70 in Figure 9 to \sfrac12ML71 in Figure 8), we can disentangle the effect of interaction between the HO molecules on the spectra. Further, these structures allow us to probe the isolated molecule limit.

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 character at  eV for 1ML of HO@Ti is absent for a \sfrac12ML coverage. This reinforces the assignment of the experimental spectra shown in Figure 1 to an intact 1ML HO@Ti geometry with interacting molecules.

To see how the spectra for dissociation of HO@Ti compare to HO@O, we have considered the half-dissociated (\sfrac12D) and fully dissociated (D) HO structures shown in Figure 7. As shown in Figure 10(c), the peak at  eV with OH character for HO@O splits into two peaks for dissociated HO@Ti. The lower energy peak has both OH and OH character, while the higher energy peak is mostly OH in character. Furthermore, we find a similar couple of peaks for \sfrac34ML mixtures of dissociated HO@Ti and HO@O shown in Figure 10d. This means one may recognize dissociated HO@Ti by both the presence of two peaks at about and  eV, and the absence of the low-energy peak with 1b character for intact HO@Ti.

The absence of a peak at about  eV in the experimental spectra shown in Figure 2(c) reinforces its attribution to dissociated HO@O rather than dissociated HO@Ti. This is further supported by the calculated HO absorption energies (Figure 10(a) and 10(b)). These are generally weaker for dissociated HO@Ti, and stronger for HO@O, as in previous calculations 29.

To check whether changes in the absorption geometry of HO affect the spectra for the same coverage, we compare 1ML of HO {I, \sfrac12D, 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 HO 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 and 3a molecular character. In particular, for intact HO, the lowest energy peak with molecular 1b character splits with a separate peak at  eV, which is closer to the peaks at 25 (Figure 1(c)) and  eV18 (Figure 1d) observed experimentally.

To see how increasing the HO coverage impacts the spectra, we compare monolayer (\sfrac12ML or 1ML) to multilayer (1\sfrac12ML) HO {I, \sfrac13D, \sfrac23D}72 (Figure 7), and consider the effect of additional HO@Ti to \sfrac14ML (Figure 9) and \sfrac12ML (Figure 8) HO@O69. In this way we can can see how robust the observed features in the individual spectra for isolated species are to screening by HO layers6, 7, and probe the liquid water limit73.

Figure 10: Structure and coverage dependence of (a,b) adsorption energy and (c,d) PDOS for HO adsorbed intact (I) or dissociated (D) on (a,c) coordinately unsaturated Ti sites (Ti) of stoichiometric TiO(110) (Figure 7) and (b,d) bridging O vacancies (O) of reduced TiO(110), with = \sfrac18 (thin lines, Figure 9) or \sfrac14 (thick lines, Figure 8). (a,b) calculated with PBE () and RPBE () xc-functionals for (white) low (\sfrac14 and \sfrac12ML), (turquoise) medium (\sfrac34 and 1ML), and (blue) high (1\sfrac14 and 1\sfrac12ML) coverage. UPS difference spectra at (c) 150 K and (d) 300 K after 0.2 L exposure are from Ref. 25. (c,d) Energies are relative to the VBM (). Gray regions denote the clean surface DOS. Red dashed lines denote the highest PDOS peaks () for 1ML HO@Ti and \sfrac12ML HO@O.

 

When a second layer of HO is added to the low coverage intact \sfrac12ML HO@Ti structure, the levels with HO 1b character are unchanged, while the levels with 3a and 1b second layer character are more localized and weakly hybridized with the surface. These levels are seen as the two most intense peaks at and  eV (Figure 10(c)). The former coincides with the peak at  eV observed experimentally at low temperatures (Figure 1(c)), suggesting multilayer HO 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  eV with antibonding 3a character on the first layer. We saw the same behavior when increasing the first layer’s coverage from \sfrac12ML to 1 ML. This is further confirmation that the peak observed experimentally at  eV has intermolecular character.

When a second \sfrac12 layer of HO is added to the 1ML HO@Ti {\sfrac13D, \sfrac23D} 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 HO levels. For the \sfrac13D structure, this results in the four distinct peaks shown in Figure 10(c). On the one hand, the peaks at and  eV have predominantly intact HO and OH character, as was the case for 1ML of \sfrac12D HO@Ti. On the other hand, the peaks at and eV are most related to the second layer. In effect, the HO level of the second-layer HO, which is fully saturated with four hydrogen bonds, is upshifted by more than an eV.

This is not the case for the \sfrac23D structure (Figure 7), where the peak at  eV instead has mostly intact second-layer HO 1b character. As was the case for intact 1\sfrac12ML HO@Ti, the addition of a second \sfrac12 layer of HO induces a stronger hybridization of the OH levels, and introduces an additional intense peak at  eV (Figure 10(c)). This again suggests the experimentally observed peak at  eV (Figure 1(c)) may be due to multilayer HO.

Overall, we find the addition of second-layer HO affects the resulting spectrum qualitatively. We find both additional features and a redistribution of those due to the first HO layer. When we instead add HO@Ti to the \sfrac14ML and \sfrac12ML HO@O 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\sfrac12ML \sfrac13D spectrum (Figure 10d) for 1ML of intact HO added to \sfrac12ML HO@O (Figure 8) is basically the sum of the 1ML intact HO@Ti (Figure 1(a)) and \sfrac12ML HO@O (Figure 2(a)) PDOS spectra downshifted by 0.2 eV. This explains the ease with which the experimental single-layer HO spectra may be analyzed for levels outside the surface DOS region.

3.5 Alignment of the Highest HO 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 HO occupied levels’ alignment for 1ML intact and dissociated HO@Ti, and \sfrac12ML dissociated HO@O is of utmost importance. The former structure corresponds to the reactant species on stoichiometric surfaces32, which undergoes photo-irradiation. The latter structures act as hole traps and are thus the main oxidizing agents on TiO(110).74, 75

We have shown that the experimental peak at  eV25 is not, in fact, the highest energy peak of HO@Ti. We instead find the highest-energy PDOS peak, , for 1ML intact HO@Ti at  eV relative to the VBM (Figure 10(c)). This is 0.6 eV closer to the VBM than the  eV estimate15 deduced from the onsets of the UPS difference spectra in Ref. 18. Moreover, as 1ML HO@Ti dissociates, moves up to  eV (\sfrac12D) and  eV (D) (Figure 10(c)). This is again significantly higher than the  eV estimate7 based on UPS difference spectra for the TiO(100) surface from Ref. 76. As was the case for CHOH on TiO(110)22, this raising of can be related to the charge transfer of that accompanies deprotonation (arrows in Figure 7). We find for the 1ML intact structure on TiO(110) is 0.2 eV closer to the VBM for HO than for CHOH22, 23, 24, while for the 1ML \sfrac12D structures is the same. However, the highest PDOS peak is both less intense and broader for HO compared to CHOH, due to the stronger hybridization with the surface. This is why, as discussed in Section 3.1, the QP is only  eV lower compared to DFT22, 23, 24 (Figure 4). After adding second-layer HO, is unchanged with weight mostly remaining on the first layer.

We find for \sfrac12ML dissociated HO@O  eV relative to the VBM (Figure 10d), the same as for intact HO@Ti. This is much higher than the previous estimate of  eV7 for OH based on the UPS difference spectra in Ref. 25. Our corrected value agrees with the recently demonstrated photocatalytic importance of OH sites as the main oxidizing species on TiO(110). 74

Based on for 1ML intact HO@Ti, vertical excitations from the highest HO occupied levels to the TiO(110) conduction band require photon energies that exceed the electronic band gap for bulk rutile TiO ( eV77) by  eV. However, the hole generated by such supra-band gap excitations should be mostly located on TiO(110) O 2p rather than HO O 2p levels. This is because the HO highest levels are hybridized with TiO(110) and are predominantly TiO(110) in character.

The fact that the highest HO levels are  eV below the VBM does not necessarily mean that they cannot be photooxidized by holes photogenerated within the TiO(110) valence band. A recent DFT study with HSE06 found trapped holes at surface O sites, i.e., three-fold coordinated O, are shared with nearyby HO–Ti groups.5 Moreover, it has been suggested that HO can only be photooxidized, i.e., trap a hole, upon deprotonation78, 79. In other words, hole transfer to the HO–Ti site should be mediated by the deprotonation of intact HO@Ti to the nearest O site. Altogether, this suggests that HO@Ti photooxidation should be initiated by band-to-band and supra-band photo-excitations, which result in the generation of holes within the TiO(110) valence band. These TiO(110) free holes may then be trapped at O sites, and partially transferred to nearby HO–Ti upon HO 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 .

Figure 11: Absolute level alignment for 1ML intact HO adsorbed with antiparallel () interfacial hydrogen bonds on coordinately unsaturated Ti sites (HO@Ti). Total (maroon) and HO projected (blue) DOS computed with DFT, scQP1, and using the generalized gradient approximation (PBE) and hybrid (HSE) xc-functionals. Energies are relative to the vacuum level . The measured from Ref. 40 (thick gray line), measured and coupled-cluster (CCSD(T)) HO gas phase ionization potentials IP from Ref. 80 (thin gray line), and for each level of theory the calculated gas phase 1b, 3a, and 1b HO levels (marked in cyan) are provided.
Figure 12: Ionization potential IP versus exact exchange fraction included in the HSE xc-functional and equivalent dielectric constant from DFT (filled symbols), (open symbols), scQP (green filled symbols), and scQP (brown filled symbols) for HO in gas phase (circles), a clean 23 (diamonds) and a 1ML of intact HO@Ti (squares) on the stoichiometric TiO(110) surface, and a \sfrac12ML of dissociated HO@O (triangles) on the defective TiO(110) surface with \sfrac12ML of O. The measured IP for HO in gas phase80, the stoichiometric TiO(110) surface 23, 81, 82, 83, 84, the % hydroxylated TiO(110) surface 83, and the liquid HO–TiO(110) interface40 are shown in gray. The self-consistent QP IP for HO in gas phase is indicated by the horizontal dashed line. The experimental dielectric constant of bulk TiO, 85, averaged over the (110) surface is marked in red. A linear fit to the DFT IP (blue), and an exponential fit to the IP (cyan) for HO in gas phase are provided for comparison.

 

In Figure 11 we show the level alignment for gas phase HO and 1ML intact HO@Ti relative to from DFT, scQP1, and based on PBE and HSE xc-functionals. These are compared to the measured CBM for the liquid HO–TiO(110) interface40, 86, and the measured and coupled-cluster (CCSD(T)) gas phase HO ionization potential 80.

Our calculated IP values for HO in gas phase are consistent with those reported previously in the literature80, 87, 88, 89. Although the relative energies of the 1b, 3a, and 1b HO 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 scQP PBE  HSE  PBE scQP (12.8 eV) Hartree Fock (HF 13.9 eV88).

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 in Figure 12. On the one hand, for DFT, we find a strong linear dependence of IP on , i.e., , with providing a quantitative agrement with experiment and CCD(T) calculations. Overall, this linear dependence is not surprising, as may be interpreted as the amount of electron-electron screening, i.e., the inverse dielectric constant 90, 91. In other words, the fraction of exact exchange included, determines the amount of screening, , incorporated within the xc-functional. The quantitative agreement of the IP for is because small molecules, e.g., HO, are weakly screened in the gas phase ().

On the other hand, for , the calculated IP has a much weaker dependence on , i.e., the starting xc-functional, with . Further, the and DFT IP coincide when . For based on PBE (), the IP already agrees semi-quantitatively with experiment, with full quantitative agreement obtained for based on HSE06 (). This is because the RPA , independently of . Essentially, the calculated IPs would also be obtained from DFT using an HSE xc-functional with , i.e., . Overall, this implies is a predictive method for the IP of small molecules. However, the scQP technique has the added advantage of being completely independent of the starting xc-functional 88, 23, while providing a nearly quantitative IP.

For the HO–TiO(110) interface, e.g., 1ML intact HO@Ti, the highest energy HO PDOS peak, , is pinned  eV below the VBM across PBE DFT, HSE DFT, PBE sc1, PBE , and HSE . For this reason, the IP of the HO 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 = , 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 HO. In particular, we find PBE  (6.0 eV) PBE DFT HSE06  PBE scQP1 HSE06 DFT (7.3 eV). Figure 12 shows that, as was the case for HO in gas phase, the IP of the HO@Ti interface across the various methods is ordered according to the method’s description of the screening, .

As discussed above, for hybrid xc-functionals such as HSE, the effective screening is determined by the fraction of exact exchange included. Essentially, plays the role of the effective screening within the method, . Although HSE06 incorporates less screening () than experiment for rutile TiO ()85, the HSE06 for the interface is in agreement with the experimental estimate of  eV 40, 86.

If one performs based on HSE06, a stronger screening is applied, i.e., , yielding a lower IP for the interface. In fact, as indicated by the red arrow in Figure 12, a similar IP to HSE06 should be obtained from HSE DFT by setting the fraction of exact exchange to the inverse dielectric constant of bulk TiO, i.e., . Adjusting to the measured inverse dielectric constant has been previously found to give improved band gaps90. From PBE scQP1, one obtains an IP consistent with that of HSE06 . This is because we find the screening in scQP decreases from PBE RPA with each self-consistent cycle. Essentially, the final screening incorporated in scQP1 is similar to that of HSE06 RPA.

As shown in Figure 11, PBE gives an IP slightly lower than PBE DFT for the interface, while the PBE 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 (). Although HSE06 has a weaker screening than PBE , the resulting absolute alignment of the CBM is quite similar. If instead, the self energy corrections are applied self-consistently via PBE scQP1, 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, scQP1 tends to provide reasonable band gaps for TiO(110) interfaces. Overall, we observe an ordering in increasing band gap of PBE DFT HSE06 DFT PBE scQP PBE HSE06 , with HSE06 DFT providing the best absolute alignment of the CBM and VBM for the HO@Ti interface.

In Figure  12, we show that a similar correlation between IP and the method’s description of screening holds for clean and hydroxylated TiO(110). Specifically, we consider clean stoichiometric TiO(110) 23, and dissociated HO@O on defective TiO(110) with \sfrac12ML of O. Overall, for all systems considered. We again find that the IP of PBE PBE DFT, HSE06 HSE() PBE , and PBE scQP HSE06 scQP HSE06 DFT.

HSE06 DFT provides the most accurate description of the IP of the clean and HO@Ti covered stoichiometric TiO(110) surfaces. Although the HSE06 DFT IP for HO@O is significantly lower than the one measured for TiO(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 scQP based on either PBE or HSE06 for the clean TiO(110) surface23, points to a similar screening from these two techniques. This also demonstrates the starting point independence of the scQP technique.

To summarize, although scQP provides accurate IPs, the band gap is greatly overestimated, as reported previously 22, 23, 41, 92. While scQP 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 HO levels are significantly hybridized with the substrate, this is not a major drawback in this case. In general, for TiO(110), a more effective strategy is to combine the calculated IP from HSE06 with the occupied interfacial levels’ alignment from or scQP.

4 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 HO–TiO interface may be disentangled using QP PDOS. We have firmly established the robustness of the QP HO PDOS by: (1) demonstrating its xc-functional (PBE, LDA, vdW-DF, and HSE06) independence, (2) comparing to self-consistent QP techniques (scQP), and (3) considering its dependence on surface coverage and dissociation. Altogether, these calculations provide an accurate interpretation of the complex UPS and MIES experiments18, 26, 25 for the HO–TiO(110) interface, and provide accurate estimates of the highest HO 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 2p levels of O and HO–Ti 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 many-body effects via QP 23. Having demonstrated the capability of for the description of level alignment prior to irradiation, this work points the way forward via future QP studies of level alignment for trapped hole levels.

Associated Content

Supporting Information

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

Author Information

Corresponding Author

E-mail: annapaola.migani@cin2.es (A.M.)

Notes

The authors declare no competing financial interest.

Acknowledgments

We ackowledge fruitful discussions with Angel Rubio, and we thank Stefan Krischok for providing experimental data. We acknowledge funding from Spanish Grants (FIS2012-37549-C05-02, RYC-2011-09582, JCI-2010-08156); Generalitat de Catalunya (2014SGR301, XRQTC); Grupos Consolidados UPV/EHU del Gobierno Vasco (IT-578-13); NSFC (21003113 and 21121003); MOST (2011CB921404); and NSF Grant CHE-1213189; and computational time from BSC Red Espanola de Supercomputacion and EMSL at PNNL by the DOE.

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