Testing variations of the GW approximation on strongly correlated transition metal oxides: hematite (a-fe 2 O 3 ) as a benchmark

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1 PCCP Dynamic Article Links Cite this: Phys. Chem. Chem. Phys., 2011, 13, PAPER Testing variations of the GW approximation on strongly correlated transition metal oxides: hematite (a-fe 2 O 3 ) as a benchmark Peilin Liao a and Emily A. Carter* b Received 19th March 2011, Accepted 21st June 2011 DOI: /c1cp20829b Quantitative characterization of low-lying excited electronic states in materials is critical for the development of solar energy conversion materials. The many-body Green s function method known as the GW approximation (GWA) directly probes states corresponding to photoemission and inverse photoemission experiments, thereby determining the associated band structure. Several versions of the GW approximation with different levels of self-consistency exist in the field. While the GWA based on density functional theory (DFT) works well for conventional semiconductors, less is known about its reliability for strongly correlated semiconducting materials. Here we present a systematic study of the GWA using hematite (a-fe 2 O 3 ) as the benchmark material. We analyze its performance in terms of the calculated photoemission/inverse photoemission band gaps, densities of states, and dielectric functions. Overall, a non-selfconsistent G 0 W 0 using input from DFT+U theory produces physical observables in best agreement with experiments. I. Introduction Technological applications of materials frequently exploit not only their ground state properties but also those of electronic excited states. For example, in photo-electrochemical reactions, excited states play an important role in determining if the reaction takes place at all. 1 As a result, knowledge about excited states of materials from both experimental measurements and theoretical predictions is crucial to gather. On the theory side, however, correct descriptions of excited states are more challenging to obtain compared to ground state information. For molecules, correlated wave function methods beyond Hartree Fock (HF) have been quite successful in shedding light on energies and characters of excitations; 2 6 local excitations at surfaces have also been treated using embedded correlated wavefunction theories. 7 For crystalline solids, researchers tend to apply density functional theory (DFT) for band structure calculations Despite of the success of DFT, it is intrinsically a ground state theory, although time-dependent versions of DFT are used increasingly to describe excited states in molecules and finite nanostructures. 11 One theory that goes beyond DFT for excited states in extended solidsisthegwapproximation(gwa) The GWA, originally formulated by Hedin, 12 uses the manybody Green s function to solve the quasiparticle (QP) equation, a Department of Chemistry, Princeton University, Princeton, NJ , USA b Department of Mechanical and Aerospace Engineering, Program in Applied and Computational Mathematics, and Andlinger Center for Energy and the Environment, Princeton University, Princeton, NJ , USA. eac@princeton.edu with a perturbative expansion for the self-energy operator. It has been applied primarily to conventional semiconductors for calculating band gaps and band structures Application of the GWA requires input of initial guess QP energies and wave functions, which are usually taken from DFT. Therefore, the GWA is typically referred to as a perturbation theory improvement to DFT. The QP band gaps predicted by the GWA are the theoretical counterparts of gaps derived from photoemission spectroscopy (PES) and inverse photoemission spectroscopy (IPES), in which the total number of electrons decreases or increases after the measurement, respectively. Quite a large number of versions of the GWA have been proposed in the literature, with varying degrees of selfconsistency, among other variations. Different levels of selfconsistency lead to different QP band gap values. 14,16,17 Predictions from one level of the GWA are also sensitive to numerical inputs: the number of empty bands, the density of k-point sampling, the choice of pseudopotentials, and the input wave function. In this work, we assess varying degrees of self-consistency within the GWA and their dependence on numerical inputs, using hematite as a benchmark material. Hematite (a-fe 2 O 3, a- is omitted henceforth) is a potential candidate for photo-electrochemical splitting of water. 18,19 It has an optical gap of ev, which is capable of absorbing B40% of the solar spectrum. 20 PES and IPES measure a band gap of ev. 21 Fe 2 O 3 crystallizes in the corundum structure, with oxygen anions hexagonally close-packed (hcp) and Fe cations filling two-thirds of the octahedral sites between hcp oxygen layers. Below the Ne el temperature (T N = 963 K), Fe 2 O 3 is antiferromagnetic with weak ferromagnetism. 22 This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13,

2 The Fe cations within the same Fe layer are ferromagnetic, but are antiferromagnetically coupled to the Fe cations in the nextnearest Fe layers. Standard DFT predicts narrow eigenvalue gaps of ev for Fe 2 O 3 23,24 due largely to the selfinteraction error (SIE) incurred by using approximate exchange correlation (XC) functionals. To correct for the SIE, DFT+U 25,26 and hybrid functional methods that incorporate exact exchange were developed. For Fe 2 O 3, DFT+U predicts eigenvalue gaps fortuitously close to measurements, 24,30 but since eigenvalue gaps are not what is measured by PES/IPES, it is important to use the GWA to actually calculate the observable that can be directly compared to PES/IPES. Hence, here we use wave functions from DFT, DFT+U, and hybrid functionals as input to the GWA to evaluate which ground state method paired with which version of the GWA provides the most accurate excited state properties, such as the band gap. II. Theory a. DFT/DFT+U/hybrid functionals The Kohn Sham (KS) DFT equations can be written as hˆ0(r)j DFT i (r) +V xc (r)j DFT i (r) =e DFT i j DFT i (r) (1) where hˆ0 is the single-particle Hamiltonian containing the kinetic energy operator, the external potential (typically simply the terms due to the nuclei or pseudopotentials representing the nuclei plus core electrons) and the Hartree electronelectron repulsion potential, e DFT i is the eigenvalue for the KS orbital j DFT i (r) and V xc (r) is the XC functional. DFT SIE arises from the approximate nature of available XC functionals. The SIE is large for the tightly localized Fe 3d electrons in Fe 2 O 3 due to the large number and magnitude of intra-atomic exchange terms 31 that are being approximated. Although they both aim to introduce exact exchange contributions, DFT+U theory and hybrid XC functionals address the SIE through different strategies. DFT+U corrects the problem by explicitly including on-site intra-atomic HF-like interactions of the 3d (or any tightly localized) electrons and then subtracting the doubly-counted terms E DFT+U [r,{n j }] = E DFT [r] +E on-site [{n j }] E dc [{n j }] (2) where E DFT (E DFT+U ) is the total DFT (DFT+U) energy of the system with density r, and E on-site (E dc ) is the on-site (doubly counted) energy for a set of on-site orbital occupation numbers {n j }. In the rotationally-invariant DFT+U proposed by Dudarev et al., 32 the E DFT+U has the following expression: E DFTþU ½r; fn j gš ¼ E DFT ½rŠþ X j ðu JÞ ðn j n 2 j 2 Þ; where the on-site interactions are approximated by the average intra-atomic Coulomb (U) and exchange (J) energies. Hybrid functionals instead correct the SIE by including exact HF exchange explicitly into V xc (r). For PBE0, 27,33 the XC functional takes the form E PBE0 xc = 1 4 EHF x EPBE x ð3þ + E PBE c. (4) For HSE, 29 the exchange part is further split into long-range and short-range components to avoid calculating the longrange part of the HF exchange, which decays slowly. The HSE XC functional can be written as E HSE xc = 1 4 EHF,SR x (o) EPBE,SR x (o) EPBE,LR x (o) +E PBE (5) c, where SR ( LR ) denotes short-range (long-range), and o is the parameter determining the range separation via the (complementary) error function. In HSE06, o is set equal to 0.2 A b. GW approximation The QP energies e i and wave functions j i are obtained by solving the QP equations hˆ0(r)j i (r) + R P (r,r 0 ;e i /h)j i (r 0 )d 3 r 0 = e i j i (r) (6) where P is the non-local energy-dependent self-energy operator, which replaces the XC potential in the KS eqn (1). Hedin s equations are a closed set of integro-differential equations that relate the Green s function G, the polarizability P, the bare (v) and screened (W) Coulomb interaction, the selfenergy P, and the vertex function G. 12 Hedin s equations need to be solved self-consistently. Taking the vertex function only to zeroth order in Hedin s equations in the initial iteration produces the GW approximation. The set of equations then becomes 12 P (1,2) = ig(1,2)w(1 +,2) (7) W(1,2) = R e 1 (1,3)v(3,2)d3 (8) e(1,2) = d(1,2) R v(1,3)p(3,2)d3 (9) P(1,2) = ig(1,2 + )G(2,1) (10) G(1,2) = G 0 (1,2) + RR G 0 (1,3) P (3,4)G(4,2)d3d4 (11) G 0 ðr; r 0 ; eþ ¼ X i j i ðrþj i ðr0 Þ e e i þ izsgnðe i e F Þ : ð12þ A compressed notation is used in which 1 (r 1,t 1 ) are spacetime coordinates for quasiparticle 1 and 1 + (r 1,t 1 + d), where d is an infinitesimal positive time. The screened Coulomb interaction W can be calculated from the bare Coulomb interaction v and the inverse dielectric function e 1 via eqn (8). The polarizability P calculated via eqn (10) describes the mean-field response to the total potential, which corresponds to the random phase approximation for the dielectric matrix (see eqn (9)). The commonly used G 0 W 0 approach calculates G and W (and then the self-energy via eqn (7)) based on the KS eigenvalues {e DFT i } and wave functions {j DFT i (r)} from some form of DFT via eqn (12), where Z is an infinitesimal positive number. Eqn (11) is used in self-consistent forms of GW to update the Green s function. Two approaches can be employed to introduce selfconsistency beyond G 0 W 0. The first one is the energy-only self-consistent approach, in which the eigenvalues are updated only in G (GW 0 ) or in both G and W (GW), while the QP wave functions are kept fixed at the KS ones. Shishkin and Kresse 17 studied a series of semiconductors (Si, SiC, GaN, ZnO, PbS, etc.) and insulators (C, BN, MgO, and LiF) with Phys. Chem. Chem. Phys., 2011, 13, This journal is c the Owner Societies 2011

3 G 0 W 0,GW 0 and GW, starting from pure DFT inputs. They found that while G 0 W 0 underestimated the gaps, GW 0 yielded values in good agreement with experiments, and GW overestimated the gap values. Subsequent to the development of DFT+U, DFT+U wave functions have been tested as inputs for G 0 W 0, with success for NiO, MnO, V 2 O 3, 35 and lanthanide oxides, 36 while results for ZnS were found to be insensitive to initial wave functions. 37 Recently, Jiang et al. 38 reported G 0 W 0 and GW 0 results using local density approximation (LDA)+U inputs for d-electron-containing materials (MnO, FeO, CoO, etc.). They found a strong U J dependence in the LDA+U and GW results, and the U J dependence also differs for different materials. Another approach is the quasiparticle self-consistent GW (QSGW), 16,39,40 in which the QP eigenvalues and wave functions are both updated. While the QSGW provides satisfactory descriptions of the character of valence and conduction bands, the QSGW systematically overestimates the band gaps. 40,41 Since there is no unanimous agreement in the literature on which levels of GW theory or which types of starting wavefunctions to use for first row transition metal oxides, in what follows, we compare all four GWA schemes, with input from four different ground state electronic structure methods (DFT, DFT+U, HSE, and PBE0). III. Computational details The VASP program 42,43 is used for all DFT/DFT+U/hybrid functional and GW calculations reported. Both the LDA of Perdew and Zunger 44 and the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE) 45 XC functionals are tested. All-electron frozen core projector augmented wave (PAW) potentials are employed for the ion-electron terms ( ion refers to a nucleus screened by core electrons). 46,47 We use the standard version of the PAW potential for O, which keeps the 1s electrons in the PAW core and treats the 2s2p electrons as valence. For Fe, two types of potentials are tested: one with 1s2s2p3s in the PAW core resulting in 14 electrons treated as valence (labeled by (3p) ), and the other with 1s2s2p3s3p in the PAW core leaving 8 electrons treated as valence. QSGW calculations with Fe 3p electrons in the PAW core encountered numerical errors, and therefore QSGW results are only available for calculations using the Fe (3p) PAW potential. A rhombohedral unit cell containing ten atoms (two Fe 2 O 3 formula units) are used in the calculations. Initial geometries are optimized at the DFT, DFT+U, or hybrid functional level. For DFT+U calculations, we use the rotationallyinvariant DFT+U formalism proposed by Dudarev et al. 32 and implemented by Bengone et al. 48 We use the ab initio value of U J = 4.3 ev that was derived for Fe 2 O 3 using an electrostatically-embedded cluster unrestricted HF scheme. 30 A planewave basis kinetic energy cutoff of 650 ev and k-point grid of 444 are used, since they converge the total energy to within 5 mev per formula unit. Gaussian smearing is used to integrate over the Brillouin zone and improve convergence. A smearing width of 0.1 ev is used during electronic optimization of the ground state, but a smaller width of 0.05 ev is used to obtain the initial wave function used as input for the GWA. All calculations were performed spin-polarized with maximum symmetry imposed. For the Fe PAW calculations with a larger frozen core, a total of 80 bands are used, with 46 of them empty. For the Fe (3p) PAW calculations, a total of 96 bands are used, with 50 of them empty. To evaluate response functions, 100 frequency points are included and the k-point grid is reduced by half to speed up the calculations while maintaining accuracy (QP band gap at the G 0 W 0 level changes by o0.02 ev upon k-mesh reduction). The QP band gaps are converged to o0.1 ev for the settings described above. IV. Results and discussion a. DFT, DFT+U, and hybrid functional predictions Table 1 summarizes geometric and electronic structure predictions for ground state properties of hematite. In terms of the crystal structure, PBE predictions are close to the experimental lattice constant and volume. 49 LDA+4.3 (shorthand notation for DFT+U using the LDA XC functional with U J = 4.3 ev) overbinds the material, underestimating the lattice constant (volume) by B2% (5%), while PBE+4.3 underbinds the material, overestimating the lattice constant (volume) by B1% (4%). Hybrid functionals produce the best structures of all, with both lattice constant and volume in excellent agreement with experiments. Magnetic moments on Fe 50 are underestimated regardless of the theory applied, by B1.3 m B from PBE and by B0.8 m B using either DFT+4.3 or hybrid functionals. Because of the SIE and lack of a proper derivative discontinuity, 51 PBE predicts a very small eigenvalue gap of 0.6 ev. DFT+4.3, with on-site corrections for Fe 3d electrons, Table 1 Fe 2 O 3 bulk ground state properties predicted by DFT, DFT+4.3, and hybrid XC functionals. Values outside parentheses are obtained using Fe (3p) PAW potentials (see section III for details). Values in parentheses are obtained using Fe PAW potentials with 3p electrons included in the frozen core Rhombohedral lattice constant a/a Equilibrium volume V/A 3 Magnetic moment m [m B ] Eigenvalue (Band) gap E gap [ev] Bader charge Fe/O PBE (5.446) (100.2) 3.6 (3.5) 0.6 (0.6) 1.64/ 1.09 (1.58/ 1.05) LDA (5.338) 96.9 (95.5) 4.1 (4.1) 1.9 (1.9) 1.73/ 1.16 (1.66/ 1.11) PBE (5.470) (102.9) 4.2 (4.2) 2.2 (2.3) 1.81/ 1.21 (1.74/ 1.16) HSE (5.424) (100.4) 4.1 (4.1) 3.5 (3.5) 1.91/ 1.28 (1.85/ 1.23) PBE (5.423) (100.4) 4.1 (4.1) 4.2 (4.2) 1.91/ 1.28 (1.85/ 1.23) Expt a a 4.9 b c d a Ref. 49. b Ref. 50. c Optical gap, see ref. 20. d PES/IPES gap, see ref. 21. This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13,

4 gives eigenvalue gaps near experimental values. Hybrid functionals produce grossly overestimated eigenvalue gaps. The eigenvalue gap from HSE06, which only includes short-range HF exchange similar in spirit to DFT+U, is smaller than that produced by PBE0, which includes both short- and long-range HF exchange. We can understand these trends by examining how the electron density distributions change as HF exchange is added to the functional. One metric is provided by Bader charges, obtained from integrating electron density distributions around nuclei inside zero flux surfaces 52 (Table 1). While the absolute differences in Bader charges from different levels of theory are o0.3, it is clear that going from PBE to DFT+U and finally to hybrid functionals, the Fe and O ions become more ionic, suggesting more localized electron densities. The least ionic character predicted by PBE is expected, due to artificial over-delocalization of the electron density caused by the extra repulsion due to SIE in pure DFT. Both the overestimations of gaps and the most ionic ions predicted by hybrid functionals are manifestations of the HF character of the electron distribution, coming from the HF exchange included in the XC functionals. Since HF theory includes no electron correlation, it overemphasizes ionic character, leading to overestimated band gaps. Overall, the effect of treating the Fe 3p electrons explicitly as valence electrons is small, with a maximum difference o2% in lattice constant and equilibrium volume, 0.1 m B in magnetic moment, 0.1 ev in eigenvalue gap, and 0.1 in Bader charge. Therefore, at these levels of theory for prediction of ground state properties, the Fe 3p electrons can be safely treated as frozen, though below we continue to report properties calculated with the Fe 3p treated as valence to check if the same holds for excited states properties. Projected densities of states (PDOS) predicted by different levels of theory using the Fe (3p) PAW potential are displayed at the top of Fig The PDOS with Fe 3p electrons in the PAW frozen core are very similar and hence are not shown. In all cases, a gap opens between the occupied and unoccupied states, and the lowest unoccupied states are mostly Fe 3d minority states that split into t 2g and e g peaks. The main differences among the PDOS are the degree of gap opening between the occupied and unoccupied states and the character of the occupied states. The magnitudes of the gaps were already discussed above (Table 1). PBE predicts that the Fe 3d and O 2p states are strongly hybridized and spread out more or less evenly over the energy range shown. DFT+4.3 and hybrid functional predictions for the occupied states are similar to each other and different from pure DFT (PBE): while the highest occupied states have mixed Fe 3d and O 2p character, the majority of the Fe d states are greatly stabilized and located at lower energy. b. GWA predictions of band gaps QP band gap values using various KS wave function inputs for different levels of the GWA are summarized in Table 2 and plotted in Fig. 4. Regardless of which level of the GWA is applied, we find that the QP band gaps are always larger than the corresponding eigenvalue gaps from pure DFT, DFT+U, or hybrid functional DFT. The corrections are largest for Fig. 1 Projected densities of states (PDOS) (top) and QP energy shifts with respect to DFT eigenvalues (bottom) for PBE (3p). In the PDOS plot, the majority (minority) spin is set to have positive (negative) occupations for ease of viewing. The DFT energies are shifted such that the Fermi energy is set to be zero, which is highlighted in the PDOS plot by a solid vertical line. LDA+4.3 and PBE+4.3, intermediate for PBE and HSE06, and smallest for PBE0. For the energy-only self-consistent GW, the QP band gaps follow the order of G 0 W 0 o GW 0 o GW, similar to what was found for conventional main group semiconductors by Shishkin and Kresse 17 and for MnO, CoO, and NiO (though not FeO) comparing only G 0 W 0 and GW 0 by Jiang et al. 38 Among them, G 0 W (shorthand notation for G 0 W 0 calculation using LDA+4.3 wave functions and energies as input) and G 0 W predictions are in best agreement with PES/IPES measurements. QP band gaps from QSGW are overestimated except in the case of QSGW@PBE. Regarding the effects of different treatments of the Fe 3p electrons, the QP band gaps from the Fe (3p) PAW potential description are consistently larger than those from the Fe PAW potential with 3p electrons in the frozen core, with a maximum difference of 0.4 ev, suggesting that outer core relaxation effects are not negligible. Next, we analyze the effects of using different input wave functions and eigenvalues. Despite GWA widening the band gap, because the PBE eigenvalue gap is so much smaller than measured values, the QP band gaps are still smaller than measurements except in the case of QSGW. However, the QP band gap from QSGW differs by a very large amount (2.3 ev) from the PBE gap, which suggests that PBE is not a good starting point for QSGW, given that the GWA is a perturbation theory and hence the corrections should be relatively small. This is not surprising, since we know that PBE provides a poor description of strongly correlated Phys. Chem. Chem. 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5 Fig. 2 PDOS (top) and QP energy shifts with respect to DFT eigenvalues (bottom) for (a) LDA+4.3 (3p) and (b) PBE+4.3 (3p). See caption in Fig. 1 for details. Fig. 3 PDOS (top) and QP energy shifts with respect to DFT eigenvalues (bottom) for (a) HSE06 (3p) and (b) PBE0 (3p). See caption in Fig. 1 for details. materials such as hematite. Turning to DFT+4.3 theory, the only results that fall within the range of the PES/IPES gap come from G0W0@DFT+4.3. All other GW results overestimate the gap. The fact that the first order perturbation This journal is c the Owner Societies 2011 result from G0W0 produces band gaps that agree with experiment suggests that the ground state wavefunctions and eigenvalues from DFT+4.3 are quite accurate. Among the energy-only GWA, QP band gaps from PBE+4.3 are consistently Phys. Chem. Chem. Phys., 2011, 13,

6 Table 2 Fe 2 O 3 DFT eigenvalue gaps and QP band gaps in ev. The first column indicates type of initial wave function and eigenvalue inputs. The values outside the parentheses are obtained using Fe (3p) PAW potentials. The values in parentheses are obtained using Fe PAW potentials with 3p electrons included in the frozen core Input DFT G 0 W 0 GW 0 GW QSGW PBE (3p) 0.6 (0.6) 1.3 (1.1) 1.7 (1.4) 1.8 (1.6) 2.9 LDA+4.3 (3p) 1.9 (1.9) 2.8 (2.6) 3.3 (3.0) 4.0 (3.6) 3.5 PBE+4.3 (3p) 2.2 (2.3) 3.1 (2.9) 3.6 (3.3) 4.3 (3.9) 3.5 HSE06 (3p) 3.5 (3.5) 4.0 (3.8) 4.4 (4.1) 4.7 (4.4) 4.2 PBE0 (3p) 4.2 (4.2) 4.5 (4.3) 4.7 (4.4) 4.8 (4.5) 4.3 Experiment a b a Optical gap, see ref. 20. b PES/IPES gap, see ref. 21. Fig. 4 DFT eigenvalue gaps and QP band gaps for Fe 2 O 3. Labels for input wave functions and eigenvalues are given next to the plot. The grey shaded area indicates the measured optical gap range, while the orange shaded area indicates the range for the PES/IPES gap; the latter is the relevant comparison. larger than those from LDA+4.3 by B0.3 ev, which appears to be due to the difference of eigenvalue gaps between LDA+4.3 and PBE+4.3. The converged QP band gaps from QSGW@LDA+4.3 and QSGW@PBE+4.3 are the same, which indicates that the LDA+4.3 and PBE+4.3 wave functions are similar in character, such that fully self-consistent perturbations to them converge to the same solution. In the case of hybrid functionals, since these functionals already overestimate the gap, applying the GWA opens up the gaps further, producing results that deviate even more from measurements. Therefore, in the following, we use GW results from PBE+4.3 input wavefunctions (results from LDA+4.3 input wavefunctions are similar, and hence omitted) to compare with experimental spectra. Fig. 5 compiles experimental PES/IPES and X-ray emission spectroscopy/x-ray absorption spectroscopy (XES/XAS) data, along with the PDOS from PBE+4.3 and various GW calculations using the PBE+4.3 input wavefunction. The PDOS reveal that GW corrections have little effect on the occupied hybridized Fe 3d and O 2p states from 5 evto 0 ev, with only slightly increased densities of states (DOS) around the tails at 5 ev. Compared to the PBE+4.3 PDOS, the occupied Fe 3d states from 7 evto 5 ev are shifted to the right (destabilized) with G 0 W 0 /GW 0 /GW, but slightly to the left for QSGW. The dominant changes to the PDOS by applying the GW approximation are to the positions of the unoccupied Fe 3d states, which determine the magnitude Fig. 5 Experimental spectra and PDOS from PBE+4.3 and various GW calculations using the PBE+4.3 input wavefunction. The topmost PES/IPES data is from ref. 21, the XES/XAS data is from ref. 54, and the site-specific PES data is from ref. 53. The PDOS from different calculations are shifted vertically for ease of viewing, with n(e) = 0atE = 8 ev for all PDOS, so the vertical axis absolute values are not meaningful. The Fe 3d states are plotted as solid lines, while the O 2p states are plotted as dotted lines. The Fermi energy is set to be zero, highlighted by a black solid vertical line. The black vertical dashed line is used to reference positions of Fe 3d states. The orange vertical dashed line is positioned at 2.6 ev, referenced to the measured PES/IPES gap Phys. Chem. Chem. Phys., 2011, 13, This journal is c the Owner Societies 2011

7 of the gap. They systematically shift upward with increasing degrees of self-consistency in the self-energy expression. Compared with experiments, the occupied Fe 3d states in all PDOS are within the experimental range ( 9 to 4.5 ev) in the site-specific PES. The QP gap from G 0 W is closest to the PES/IPES gap. The splitting of the unoccupied Fe 3d t 2g and e g peaks from XAS is 1.39 ev, 53 while the splitting calculated as the energy difference between the center of the t 2g and e g peaks are 0.81 ev, 1.14 ev, 1.17 ev, 1.11 ev and 1.30 ev going from the bottom PBE+4.3 to the top QSGW@PBE+4.3 PDOS in Fig. 5. Overall, the best agreement with experimental spectra of hematite is achieved with G 0 W especially for the unoccupied states. In what follows, we focus on analyzing the performance of different variations of the GW approximations. Fig. 6 illustrates how the QP band gaps converge with respect to the number of iterations. QP band gaps from GW 0 and GW converge the fastest (o0.01 ev change in QP band gap between iterations), within 7 iterations, while the convergence of QP band gaps from QSGW is slower and depends on the input wave function. QSGW@PBE requires about 19 iterations, QSGW@DFT+4.3 takes about 13 iterations, and QSGW@hybrid functionals converges with less than 9 iterations, indicative of how close the ground state wavefunctions and energies are to the fully self-consistent GW solution. QP energy shifts with respect to DFT eigenvalues using the Fe (3p) PAW potential are plotted at the bottom of Fig The shifts with the Fe 3p electrons in the PAW frozen core are very similar and hence omitted. The dominant correction is a jump at the gap. Except for QSGW, the QP energy shifts are positive for both occupied states and unoccupied states, with the unoccupied states shifted more, leading to QP band gaps larger than corresponding DFT eigenvalue gaps. For the PBE input wave functions and energies (Fig. 1), the occupied states are shifted upwards except for those predicted by QSGW. The occupied states in the 1 to 0 ev region exhibit larger positive shifts than the states below 1.5 ev, such that a gap between these occupied states opens up. For the DFT+4.3 input wave functions and energies (Fig. 2), there is a discontinuity in the shifts within the occupied states between the Fe 3d states in the 7 to 5 ev region and the hybridized O 2p and Fe 3d states in the 5 to 0 ev region. The net effect is that G 0 W 0 /GW 0 /GW increases the energy of the states in the 7 to 5 ev region, while QSGW further stabilized those states (see also PDOS in Fig. 5). In the case of hybrid functional derived wave functions and energies (Fig. 3), the QP shifts among the occupied states from G 0 W 0 /GW 0 /GW are quite uniform, with the largest shifts for Fe 3d states in the 8 to 6 ev region. The net effect is that the Fe 3d states are again shifted towards the hybridized O 2p and Fe 3d states. For QSGW, the QP shifts among the occupied states have many scattered points off the trend line. To understand this effect, we plot the total DOS from the HSE06 wavefunction and QSGW@HSE06 in Fig. 7. The QSGW corrections separate the hybridized states in the 6 to 0 ev region and the majority of Fe 3d states in the 7.5 to 6 ev region, such that there is a distinctive decrease in the number of states near 6 ev. We suggest that the scattering points in Fig. 3 are results of rearranging energy levels for separating the originally mixed states. Overall, besides opening Fig. 6 Fe 2 O 3 QP band gaps from different levels of the GW approximation versus the number of iterations. See caption of Fig. 4 for detailed descriptions of labels and shaded areas. the gap, applying G 0 W 0 /GW 0 /GW corrections to DFT+U or hybrid functionals DFT wavefunctions increases the overlap of Fe 3d and O 2p states, while QSGW corrections decrease the overlap by stabilizing the majority of Fe 3d states. Fig. 7 Total densities of states from HSE06 and QSGW@HSE06. The Fermi energy is set to be zero, highlighted by a black solid vertical line. This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13,

8 The QP energy shifts from different levels of the GWA exhibit a mostly consistent pattern, with a few exceptions. Overall, the shifts for occupied states follow the trend QSGW o G 0 W 0 E GW o GW 0, while the shifts for unoccupied states follow the trend G 0 W 0 o QSGW o GW 0 o GW. Among G 0 W 0,GW 0 and GW, the shifts are parallel to each other. The positive shifts of the unoccupied states dominate, such that the QP band gaps follow the same trend as the shift for the unoccupied states: G 0 W 0 o GW 0 o GW. QSGW is distinct from the rest in that QSGW predicts the smallest or even negative shifts for the occupied states. To understand the effect of updating the wavefunction in QSGW, we compare the electron density from DFT/DFT+ U/hybrid functionals calculations to the corresponding electron density from converged QSGW calculations. In all cases, the electron densities from QSGW are nearly identical to those from the input wavefunctions. To illustrate the similarity, Fig. 8 plots the plane-integrated electron density of PBE+4.3 and QSGW. The maximum absolute difference between the black and orange dots in Fig. 8 is o0.007e, while there are 92 valence electrons in the rhombohedral simulation cell (including Fe 3p as valence electrons). The small change in electron density implies that the wavefunction changes little in QSGW. Therefore, updating the wavefunctions does not play an important role in the reported QSGW calculations for hematite. The static macroscopic dielectric constants e predicted within the random phase approximation are summarized in Table 3 to compare the screening predicted by different GW approximations. By comparing e from G 0 W 0 and GW, we see that updating the eigenvalues in W has the effect of decreasing e. By contrast, the values of e predicted by QSGW are significantly larger than those from G 0 W 0, especially in the case of PBE, where e is 4130, about 20 times larger than experimental values. This result provides further evidence that PBE gives a poor description of hematite. The GWA using PBE input grossly overestimates e in all cases. The origin of this overestimation is the SIE in the PBE XC functional, which causes electron distributions to be too delocalized, i.e., more metallic in nature and recall that e becomes infinite for metals. Thus, even though the QP band gap from QSGW@PBE falls Fig. 8 Plane-integrated electron density from the PBE+4.3 wavefunction and the QSGW@PBE+4.3 wavefunction. Each point in the plot represents the integrated electron density of a specific (111) plane. The values sum up to the total number of valence electrons in the rhombohedral simulation cell. The grey (red) circles on the x-axis indicate the locations of Fe (O) layers within the (111) planes. Table 3 Static macroscopic dielectric constants e predicted by different levels of the GW approximation. The levels of self-consistency are given by the column headings. Input wave functions and energies are listed in the first column. Values outside the parentheses are obtained using Fe (3p) PAW potentials. Values in parentheses are obtained using Fe PAW potentials with 3p electrons included in the frozen core Input G 0 W 0 GW 7 iterations QSGW 20 iterations PBE (3p) 18.5 (21.8) 12.7 (14.0) LDA+4.3 (3p) 7.8 (7.9) 5.6 (5.9) 11.2 PBE+4.3 (3p) 6.8 (6.9) 4.9 (5.2) 14.7 HSE06 (3p) 4.9 (4.9) 4.3 (4.4) 8.5 PBE0 (3p) 4.1 (4.1) 3.9 (4.0) 7.9 Expt a a Ref in the experimental range, it does not come from a physically correct description of Fe 2 O 3. On the other hand, e from G 0 W falls within the measured range, supporting the fact that QP band gaps from G 0 W are closest to experimental measurements and are based on physically reasonable dielectric response behavior. By contrast, e from G 0 W functionals and GW@hybrid functionals are all underestimated, because the hybrid functionals tend to produce electron densities that are too localized, which gives rise to small dielectric constants. The complex dielectric functions from GW using PBE+4.3 (representative of DFT+4.3) and HSE06 (representative of hybrid functionals DFT) input wavefunctions are shown in Fig. 9 along with measured dielectric functions. The real part of the dielectric functions from the HSE06 wavefunction are smaller than the corresponding ones from PBE+4.3, which is a result of the more localized electron densities as discussed above. The imaginary part of the dielectric functions is directly proportional to the optical absorption spectrum. Since the onsets of the peaks are not well-defined, positions at half peak height are used for comparison (see vertical dotted/dashed lines in the bottom panels of Fig. 9). Positions at half peak height of -Ime from the HSE06 wavefunction are at higher energy than the corresponding ones from PBE+4.3, which is consistent with the larger QP gap predicted from HSE06 wavefunction. Within PBE+4.3 or HSE06 starting wavefunctions, the positions at half peak height follow the ordering of G 0 W 0 o QSGW o GW, which also agrees with the trend of QP gaps. The position at half peak height of the G 0 W curve is nearly overlapping the experimental curves. One interesting feature in the Ree is that the curves for QSGW are nearly parallel to those for GW, with a constant positive shift. Both G and W are updated in QSGW and GW, but the wavefunction is updated only in QSGW. Since the wavefunctions in QSGW do not change that much, as discussed above, one might expect the QSGW and GW results to be similar. These data suggest that updating G and W in QSGW and GW has different effects on the screening properties, with the larger Ree from QSGW giving rise to consistently smaller QP gaps from QSGW than those from GW (see Table 2). However, this trend does not hold for comparing QSGW to G 0 W 0 or GW 0, where W is not updated and the Ree curves have different shapes. Overall, the Phys. Chem. Chem. Phys., 2011, 13, This journal is c the Owner Societies 2011

9 Fig. 9 Complex dielectric functions from ellipsometric spectroscopy 57 and theory. The hollow and solid circles on the black experimental curves come from two polycrystalline specimens of hematite sintered at different temperatures. In the bottom panels for Ime, vertical dotted (dashed) lines are referenced to the positions at half peak height for experimental (theoretical) curves, following the same color as the corresponding curves. dielectric functions from G 0 W are in best agreement with experiments. V. Conclusions Various levels of the GWA using different input wave functions and energies have been tested to see how well they are able to capture the photoemission/inverse photoemission band gap and dielectric response of a strongly correlated electron material, namely hematite (Fe 2 O 3 ). Since hematite is under consideration as a solar energy conversion material, realistic modeling of excited states in hematite requires testing of various theories such as we have performed here. In general, the QP band gaps from the GWA are larger than the corresponding DFT eigenvalue gaps, as expected since the latter tend to underestimate the true band gap due to SIE, among other shortcomings of approximate XC potentials. The GWA predictions depend significantly on the input wave functions. The PBE wave functions suffer from SIE, such that the GWA, even the fully self-consistent one, does not give physically reasonable predictions of band gaps and dielectric constants with PBE inputs. The SIE in PBE produces an overly spread out electron distribution, leading to too metallic an environment, which results in a huge overestimate of the dielectric constant (recall that the dielectric constant of a perfect metal is infinite). The DFT+U wave functions are more physically correct since they approximately ameliorate the SIE; this is manifested by the GWA with DFT+U inputs yielding the most accurate band gaps and dielectric constants. Since hybrid functionals predict eigenvalue gaps that are already too large, applying GWA with hybrid functional inputs only worsens the predictions. The hybrid functionals lead to overly localized electron distributions, which results in poor screening and a too small dielectric constant. Analysis of the QP energy shifts shows that different levels of the GWA correct input wave functions and energies differently. Among the energy-only GWA, introducing self-consistency increases the positive energy shifts of the unoccupied states, and thus opens up the QP band gaps. The energy-only GWA also shifts the Fe 3d bands towards the Fermi level, and thus favors overlaps of O 2p and Fe 3d states. For QSGW, the positive energy shifts of the unoccupied states increase the QP band gaps, while the energy shifts for occupied states are less positive than those from the energy-only GWA and often even become negative, leading to the very large, overestimated band gaps. In addition, QSGW disfavors overlaps of O 2p and Fe 3d states by stabilizing the majority of Fe 3d states. The effect of the semicore Fe 3p electrons is negligible for ground state properties described by DFT, DFT+U, or hybrid functional, but is non-negligible for excited states described by the GWA; this is to be expected since ionization/electron addition events will polarize the outer core and therefore its relaxation should be included. In sum, comparison of the measured dielectric function and photoemission/inverse photoemission spectra leads us to conclude that the G 0 W level of theory is the method of choice for examining excitations in hematite, and possibly other first row, mid-to-late transition metal oxides. In some sense, this should not be a surprise. The GWA is meant to be applied perturbatively; the fact that G 0 W furnishes the highest accuracy is entirely consistent with the fact that high quality input wavefunctions and energies which the ab initio DFT+U theory 30 provides for hematite paired with a first order perturbative correction provides a high fidelity output. The ab initio DFT+U theory provides an accurate electron distribution which produces a dielectric response that is physically correct. Thus, taking into account both accuracy and cost of computing resources, the G 0 W method is recommended for calculating the quasiparticle band structures and dielectric constants of This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13,

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