Abstract
The ab initio GW method is considered as the most accurate approach for calculating the band gaps of semiconductors and insulators. Yet its application to transition metal oxides (TMOs) has been hindered by the failure of traditional approximations developed for conventional semiconductors. In this work, we examine the effects of these approximations on the values of band gaps for ZnO, Cu2O, and TiO2. In particular, we explore the origin of the differences between the two widely used plasmon-pole models. Based on the comparison of our results with the experimental data and previously published calculations, we discuss which approximations are suitable for TMOs and why.
I. INTRODUCTION
Many-body perturbation theory within the GW approximation has been successfully used to describe the electronic spectra of sp-bonded semiconductors and insulators from first principles [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] . However, application of the GW methodology to materials with localized d-electrons, such as transition metal oxides (TMOs), has revealed some controversial results. One of the heavily debated topics is the GW band gap of ZnO for which values ranging from 2.1 to 3.9 eV have been reported [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] . This wide variation can be attributed to the use of different self-consistent schemes [12-14, 16, 28, 29] , plasmon-pole models (PPMs) [21, 23, 26] , and starting points [15, 17, 20] , as well as to a false convergence behavior as discussed in Ref. 17 and to the basis set convergence issues as discussed in Refs. 18 and 27. At the same time, it is difficult to pinpoint the contributions of each approximation (self-consistent scheme, PPM, and starting point) to the total difference, since the different results reported in the literature were obtained with different codes and with different sets of numerical parameters.
I. INTRODUCTION
The motivation behind the present study was to systematically isolate the contributions of these approximations. For that purpose we performed multiple GW calculations for three TMOs (wurtzite ZnO, cuprite Cu 2 O, and rutile TiO 2 ) using many possible combinations of these approximations. Analyzing the results of these calculations allowed us to collect valuable information about the validity and applicability of these approximations. We were able to show that the theoretically justified choice of approximations gives the best agreement with experiment for all the materials studied. We further discuss the origin of the differences between the two widely used PPMs, and we demonstrate how one of them can be modified to give better accuracy as compared to the results of higher level calculations.
I. INTRODUCTION
The paper is organized as follows. Sec. II gives the theoretical background, followed by the computational details in Sec. III. Sec. IV presents the results and a discussion thereof. The main findings of this work are summarized in Sec. V.
II. THEORETICAL BACKGROUND
Within the GW approximation, the electron selfenergy operator Σ is given by [1, 7, 9, 10, 30, 31] :
II. THEORETICAL BACKGROUND
where r is the spatial coordinate, ω is the energy, η is a positive infinitesimal, G is the Green's function, and W is the screened Coulomb potential. The expression for G is:
II. THEORETICAL BACKGROUND
where n is the band index, k is the Bloch wave vector, ψ QP nk (r) is the quasiparticle orbital, E QP nk is the quasiparticle energy, and η nk is a positive (negative) infinitesimal for occupied (unoccupied) states. The expression for W is:
II. THEORETICAL BACKGROUND
where ǫ is the microscopic dielectric function, v(r) = e 2 / |r| is the bare Coulomb potential, and e is an elementary charge. The expression for ǫ is:
II. THEORETICAL BACKGROUND
where δ is the Dirac delta function and P is the polarizability. The latter is evaluated within the random phase approximation (RPA):
II. THEORETICAL BACKGROUND
Calculations are performed in reciprocal space, for instance ǫ(r, r ′ ; ω) is Fourier transformed to ǫ GG ′ (q; ω), where G is the reciprocal lattice vector and q is the Bloch wave vector.
II. THEORETICAL BACKGROUND
In practice, the GW method is applied perturbatively on top of Kohn-Sham density functional theory (DFT) [32] calculations. It is often assumed that the KohnSham orbitals ψ KS nk (r) are good approximation for the quasiparticle orbitals ψ QP nk (r). Σ is then diagonal in the basis of ψ KS nk and the quasiparticle energies are expressed by [7] :
II. THEORETICAL BACKGROUND
where E KS nk are the Kohn-Sham energies, V xc is the exchange-correlation potential, and ρ scf is the selfconsistent charge density.
II. THEORETICAL BACKGROUND
The Kohn-Sham ansatz is often used in conjunction with ab initio pseudopotentials [33] assuming separation of electrons into core and valence states. This implies that the Σ and V xc terms of Eq. (6) only include contributions from the valence states, while contributions from the core states are treated at the DFT level in the E KS nk term of Eq. (6), and the core-valence interaction is neglected [2, 7] . The latter is of particular concern when core and valence orbitals overlap, such as would occur in Zn if 1s 2 2s 2 2p 6 3s 2 3p 6 states were treated as core states and 3d 10 4s 2 states as valence states. The core-valence interaction can be included at the DFT level using the non-linear core correction (NLCC) [34] which introduces the partial core charge density ρ core in the evaluation of the exchange-correlation potential, V xc [ρ core + ρ scf ]. The GW method on the other hand requires the entire shell of semicore states (such as 3s 2 3p 6 3d 10 states in Zn) to be explicitly treated as valence states in order to eliminate errors due to neglecting the core-valence interaction [35] [36] [37] [38] [39] . All calculations in this work are performed treating the entire third shells of Zn, Cu, and Ti as valence states.
II. THEORETICAL BACKGROUND
The core-valence partitioning brings up another issue, namely that the charge density used for the evaluation of the V xc term in Eq. (6) must be consistent with the orbitals used in the construction of the Σ operator in the said equation [31, [40] [41] [42] . In particular, it was shown that if the NLCC is used in the DFT calculation, ρ core must be set to zero when evaluating the V xc term of Eq. (6) [41] . To study the effect of imbalance between the Σ and V xc terms in Eq. (6), we use ρ core derived from the deep core states (such as 2s 2 2p 6 states in Zn). Even though there is negligible overlap between the deep core and semicore orbitals (such as the second and third shells of Zn), the integrated partial core charge q core = drρ core (r) is not small (q core = 7.67e in Zn). In what follows we examine how keeping ρ core in the V xc term of Eq. (6) affects the results of GW calculations as compared to the case of zeroing out ρ core in the V xc term.
II. THEORETICAL BACKGROUND
Several different approaches have been developed for constructing Σ and calculating its matrix elements entering Eq. (6):
II. THEORETICAL BACKGROUND
• Non-self-consistent G 0 W 0 scheme [7] when G and P are obtained by plugging ψ • Eigenvalue self-consistent GW scheme [13] when G and P are constructed from ψ KS nk and E QP nk , the latter being determined iteratively starting from E KS nk .
II. THEORETICAL BACKGROUND
• Eigenvalue self-consistent GW 0 scheme [13] when G is calculated using ψ KS nk and E QP nk while P is calculated using ψ KS nk and E KS nk .
II. THEORETICAL BACKGROUND
• Eigenvector self-consistent GW schemes [28, 29, 43] when ψ QP nk are constructed iteratively using offdiagonal matrix elements of Σ in the basis of ψ KS nk .
II. THEORETICAL BACKGROUND
It was shown that the self-consistent GW scheme without the vertex correction in Σ (beyond the GW approximation) overestimates the experimental band gaps [44] . Better agreement with experiment is obtained using the GW 0 scheme because the effects of self-consistency in W and of vertex correction in Σ largely cancel out [45] [46] [47] . It should be noted that the self-consistency in G without the vertex correction in Σ violates the Ward-Takahashi identity representing the local electron number conservation law [48] . For the purpose of this work, we employ non-self-consistent G 0 W 0 and eigenvalue self-consistent GW 0 schemes.
II. THEORETICAL BACKGROUND
The energy integral in Eq. (1) can be evaluated by direct numerical integration [30] , employing the Hilbert transform [49] , the contour deformation technique [50] , or using a plasmon-pole model (PPM) to approximate the ω dependence of ǫ −1 . The first three methods are thereafter referred as non-PPM. Two popular choices for PPM are the Hybertsen-Louie (HL) PPM [7, 51] and the Godby-Needs (GN) PPM [52] . The HL PPM takes as input the static inverse dielectric function ǫ −1 at ω = 0 and the charge density ρ ppm which is used to compute the effective bare plasma frequencies. The GN PPM takes as input ǫ −1 at two frequencies, ω = 0 and ω = iΩ, where Ω is a parameter. The HL PPM recently came under criticism for poorly reproducing the ω dependence of the RPA ǫ −1 as compared to the GN PPM [21, 23, 26] .
II. THEORETICAL BACKGROUND
As we show in this paper, the poor performance of the HL PPM stems from the improper choice of ρ ppm . One sensible choice for ρ ppm is the charge density of the valence electrons (oxygen 2p 6 states), ρ ppm = ρ val , owing to the fact that the dielectric screening is dominated by the valence electrons [53] . This choice for ρ ppm was implicitly assumed in the original derivation of the HL PPM [7] . Another common choice for ρ ppm is the selfconsistent charge density, ρ ppm = ρ scf , which includes the core electrons treated as valence in the construction of the pseudopotentials (oxygen 2s 2 states and the transition metal third shell). Our calculations demonstrate that the HL PPM approaches the GN PPM and the RPA results when ρ ppm is set to ρ val . At the same time, the poor performance of the HL PPM discussed in the literature [21, 23, 26 ] is attributed to setting ρ ppm equal to ρ scf . 
III. COMPUTATIONAL DETAILS
To examine the effects of different approximations discussed in Sec. II on the quasiparticle band gaps and band edges of TMOs, we perform a series of calculations for wurtzite ZnO, cuprite Cu 2 O, and rutile TiO 2 using Quantum ESPRESSO [59] and BerkeleyGW [30] codes for the DFT and GW parts, respectively. Calculations are carried out for the spin-unpolarized case with the lo-TABLE III. Structural parameters of wurtzite ZnO, cuprite Cu2O, and rutile TiO2 measured by X-ray diffraction [55] [56] [57] and calculated using DFT with LDA and GGA exchangecorrelation functionals.
Wurtzite ZnO Cuprite Cu2O
Rutile TiO2 cal density approximation (LDA) in the PW form [60] and the generalized gradient approximation (GGA) in the PBE form [61] for the exchange-correlation functional. Norm-conserving pseudopotentials are generated in a separable non-local form [62] using the RRKJ scheme [63] and including scalar relativistic corrections and nonlinear core corrections (NLCC) [34] . The pseudopotential parameters are summarized in Table I studies with respect to the size of the Monkhorst-Pack grid [54] , kinetic energy cutoffs, and the number of unoccupied Kohn-Sham bands used in the calculation of ǫ and Σ are reported elsewhere [17, 18, 21, 64, 65] . The parameters used in our calculations are summarized in Table II . The Monkhorst-Pack grids for ρ scf , ρ val , and Σ are Γ-centered and the ones for ǫ are shifted by half a grid spacing in all directions. A small wave vector along the (111) direction in crystal coordinates is used to calculate ǫ at the Γ point. The convergence of Σ with respect to the size of the Monkhorst-Pack grid is accelerated by averaging v and W inside the Voronoi cells of the (k + G)-points near the Γ-point [30] . The convergence of Σ with respect to the number of unoccupied Kohn-Sham bands is accelerated by using the static remainder correction [64] . To ensure convergence of the stress tensor, structural relaxations are performed using 3 times higher kinetic energy cutoffs than those listed in Table II . The experimental and theoretical structural parameters (thereafter referred to as ES and TS, respectively) are listed in Table III. Special consideration is required when constructing ρ val used in the HL PPM. Given the two formula units per primitive cell and the electronic valence configurations listed in Table I 6 states. For that purpose we employ the DFT+U method with the following parameters: U = 8.0 eV and J = 0.9 eV for ZnO [17] ; U = 7.5 eV and J = 0.98 eV for Cu 2 O [66] . Note that the DFT+U method is only used for constructing ρ val , while GW calculations are carried out starting from DFT orbitals. To quantify the effect of U , we perform two sets of GW calculations, one using DFT ρ val and another using DFT+U ρ val . It is found that the inclusion of U in ρ val only changes the GW band gaps by 10 meV and the GW band edges by 40 meV. The much larger effect of using ρ scf in the HL PPM will be discussed in Sec. IV.
Wurtzite ZnO Cuprite Cu2O
Let us now describe the implementation of the eigenvalue self-consistent GW 0 scheme. Iterations on E [67] and calculated using DFT and GW . DFT and GW band gaps are obtained using different exchange-correlation functionals (LDA and GGA), experimental and theoretical structural parameters (ES and TS), non-self-consistent G0W0 and eigenvalue self-consistent GW0 schemes, HL PPM with ρppm = ρ val and ρ scf , and matrix elements of Vxc without and with NLCC (ρcore = 0 and = 0). The DFT and GW band gaps are direct at the Γ point. All values are in eV. 
Wurtzite

IV. RESULTS AND DISCUSSION
GW calculations for wurtzite ZnO, cuprite Cu 2 O, and rutile TiO 2 are performed using LDA and GGA starting points, experimental and theoretical structural parameters (ES and TS), non-self-consistent G 0 W 0 and eigenvalue self-consistent GW 0 schemes, HL PPM with ρ ppm set to DFT+U ρ val and DFT ρ scf , and matrix elements of V xc without and with NLCC (ρ core = 0 and = 0). In the latter case, values of integrated partial core charge q core = drρ core (r) are listed in Table I . Fig. 1 shows the real parts of ǫ −1 00 (0; ω) for the three TMOs in case of the LDA starting point and experimental structural parameters (ES) calculated within the RPA and constructed using the HL PPM with ρ ppm = ρ val and ρ scf . Fig. 2 shows the quasiparticle band structures calculated using the LDA starting point, experimental structural parameters (ES), the eigenvalue self-consistent GW 0 scheme, the HL PPM with ρ ppm = ρ val , and matrix elements of V xc without NLCC (ρ core = 0). Fig. 3 shows the quasiparticle band gaps plotted as a function of the starting point (obtained from the LDA or GGA calculations) and of the structural parameters (either experimental or theoretical, labeled as ES and TS, respectively). Different symbols indicate the values calculated using different flavors of the GW method. In this context, flavor refers to the choice of self-consistent scheme, ρ ppm , and ρ core . The experimental band gaps taken from Refs. 67-69 are shown for comparison. Tables IV-VI give the experimental and calculated band gaps plotted in Fig. 3 as well as the Kohn-Sham values not shown in Fig. 3 . Kohn-TABLE V. Band gaps of cuprite Cu2O measured using optical absorption spectroscopy (OAS) [68] and calculated using DFT and GW . DFT and GW band gaps are obtained using different exchange-correlation functionals (LDA and GGA), experimental and theoretical structural parameters (ES and TS), non-self-consistent G0W0 and eigenvalue self-consistent GW0 schemes, HL PPM with ρppm = ρ val and ρ scf , and matrix elements of Vxc without and with NLCC (ρcore = 0 and = 0). The DFT and GW band gaps are direct at the Γ point. All values are in eV. TABLE VI. Band gaps of rutile TiO2 measured using photoemission spectroscopy (PES) [69] and calculated using DFT and GW . DFT and GW band gaps are obtained using different exchange-correlation functionals (LDA and GGA), experimental and theoretical structural parameters (ES and TS), non-self-consistent G0W0 and eigenvalue self-consistent GW0 schemes, HL PPM with ρppm = ρ val and ρ scf , and matrix elements of Vxc without and with NLCC (ρcore = 0 and = 0). The DFT and GW band gaps are direct at the Γ point (in regular font) and indirect between the Γ point at the VBM and the R point at the CBM (in cursive font Fig. 4(d) of Ref. 26 , we find that in the case of ZnO, the HL PPM becomes similar to the GN PPM when ρ ppm is set to ρ val . One can see from Fig. 1 that for all three oxides, the HL PPM with ρ ppm = ρ val gives a better fit to the RPA results than the HL PPM with ρ ppm = ρ scf . We note that the HL PPM suffers from the ambiguity of constructing the proper ρ ppm . This problem is absent in the GN PPM, suggesting that the HL PPM is more difficult to use for studying TMOs than the GN PPM.
IV. RESULTS AND DISCUSSION
Several conclusions can be drawn from statistical analysis of the data presented in Fig. 3 and Tables IV-VI. • Comparing the values in (G 0 W 0 , ρ val , 0) row at (LDA, ES) and (GGA, ES) columns for ZnO, we find that the band gap varies by 3.21 − 2.82 = 0.39 eV depending on the starting point (obtained from the LDA or GGA calculations). Averaging this quantity over different ρ core = 0 rows for ZnO gives the mean variation in the band gap with different starting points as equal to 0.44 eV. Repeating this procedure for Cu 2 O and TiO 2 yields the values of 0.06 eV and 0.04 eV, respectively. The large variation in the case of ZnO indicates that neither LDA nor GGA provides a good starting point for GW calculations. At the same time, small variations for Cu 2 O and TiO 2 imply that LDA and GGA give similar (but not necessarily good) starting points for GW calculations. Other starting points were tried in GW calculations for ZnO including DFT+U [17] , the screened hybrid functional [15] , and the exact exchange optimized effective potential [20] . Note that the latter starting point can present some challenges in the subsequent GW calculations [71] . Overall, the problem of the starting point in GW calculations for ZnO may require further research to give the full picture.
IV. RESULTS AND DISCUSSION
• Comparison of the values in ρ core = 0 rows at (LDA, ES) and (LDA, TS) columns, as well as at (GGA, ES) and (GGA, TS) columns, shows that the variation of the band gap with the structural parameters is 0.12 eV for ZnO, 0.10 eV for Cu 2 O, and 0.09 eV for TiO 2 . This suggests that the band gaps are fairly insensitive to the structural parameters.
IV. RESULTS AND DISCUSSION
• Comparing the values in (G 0 W 0 , ρ val , 0) and (GW 0 , ρ val , 0) rows, as well as in (G 0 W 0 , ρ scf , 0) and (GW 0 , ρ scf , 0) rows, we find that the eigenvalue self-consistency in G increases the band gap by 0.37 eV for ZnO, 0.16 eV for Cu 2 O, and 0.24 eV for TiO 2 . This is consistent with previous studies [13] .
IV. RESULTS AND DISCUSSION
• Comparison of the values in (G 0 W 0 , ρ val , 0) and (G 0 W 0 , ρ scf , 0) rows, as well as in (GW 0 , ρ val , 0) and (GW 0 , ρ scf , 0) rows, shows that the inclusion of core electrons in ρ ppm increases the band gap of ZnO and Cu 2 O by 0.51 and 0.15 eV, respectively, and decreases the band gap of TiO 2 by 0.22 eV. This is because for ZnO and Cu 2 O, the VBM is lowered by a larger amount than the CBM, while the opposite scenario takes place for TiO 2 , as follows from Supplemental Material [70] .
IV. RESULTS AND DISCUSSION
• Comparing the values in ρ core = 0 and ρ core = 0 rows, we find that the inclusion of NLCC in the matrix elements of V xc decreases the band gap of ZnO and Cu 2 O by 0.70 and 0.69 eV, respectively, and increases the band gap of TiO 2 by 0.27 eV.
IV. RESULTS AND DISCUSSION
This is due to the fact that for ZnO and Cu 2 O, the VBM is raised by a larger amount than the CBM, while the opposite holds for TiO 2 , as one can see from Supplemental Material [70] .
IV. RESULTS AND DISCUSSION
Overall, the largest variation of the band gap comes from the inclusion of NLCC in the matrix elements of V xc . This inclusion introduces significant errors in the calculated band gaps. Fair agreement is found when comparing our results to those of previous GW calculations for each oxide and specific flavor. In line with the criticism of the HL PPM [21, 23, 26] , previous HL PPM calculations are compared to our ρ ppm = ρ scf results, while previous GN PPM and non-PPM calculations to our ρ ppm = ρ val results.
IV. RESULTS AND DISCUSSION
• For ZnO, we focus on (LDA, ES) column in Fig. 3(a) or • For Cu 2 O, let us look at (LDA, ES) column in Fig. 3(b) or Table V . The (G 0 W 0 , ρ val , 0) band gap of 1.56 eV compares with non-PPM G 0 W 0 value of 1.34 eV [72] . The (GW 0 , ρ val , 0) band gap of 1.77 eV is close to non-PPM eigenvalue self-consistent GW band gap of 1.80 eV [72] .
IV. RESULTS AND DISCUSSION
• For TiO 2 , we start with (LDA, ES) column in Fig. 3(c) or Table VI . The (G 0 W 0 , ρ val , 0) band gap of 3.44 eV is close to non-PPM G 0 W 0 value of 3.34 eV [38] . We now move on to (GGA, TS) column. The (G 0 W 0 , ρ val , 0) band gap of 3.34 eV is comparable to the GN PPM G 0 W 0 value of 3.59 eV [73] . The (G 0 W 0 , ρ scf , 0) band gap of 3.14 eV is close to the HL PPM G 0 W 0 value of 3.13 eV [65] .
IV. RESULTS AND DISCUSSION
We now compare the calculated band gaps with the experimental data. The absolute difference of the experimental band gap and the value in (G 0 W 0 , ρ val , 0) row at (LDA, ES) column for ZnO is equal to 0.23 eV. Averaging this quantity over the four columns in Fig. 3(a) or Table IV gives the value of 0.43 eV. Further averaging over the three TMOs gives the mean deviation from experiment of 0.40 eV. This procedure is repeated for each flavor represented by different row in Fig. 3 and Tables IV-VI. Among the eight flavors, the smallest deviation of 0.18 eV is found for (GW 0 , ρ val , 0) flavor, followed by the 0.30 eV deviation for (GW 0 , ρ scf , 0) flavor, the 0.31 eV deviation for (G 0 W 0 , ρ scf , 0) flavor, and the 0.40 eV deviation for (G 0 W 0 , ρ val , 0) flavor. Corresponding deviations for ρ core = 0 rows fall within the 0.49 to 0.78 eV range. We conclude that the best overall agreement with experiment is obtained for (GW 0 , ρ val , 0) flavor. On the other hand, we note from Fig. 3 that (GW 0 , ρ val , 0) values irregularly underestimate and overestimate the experimental band gaps, while (G 0 W 0 , ρ val , 0) values always underestimate the experiment, suggesting that the latter flavor may be preferable to the former. Yet this conclusion may be deceiving given that the HL PPM with ρ val overestimates the non-PPM band gap of ZnO by 0.34-0.38 eV (see the comparison with the previous calculations above) and that the experimental band gaps are renormalized by electron-phonon interaction not included in our calculations. Overall, it may be premature to conclude which flavor is preferable for TMOs until the effect of the vertex correction on GW band gaps of these materials is thoroughly studied.
V. SUMMARY
In summary, we quantify the effects of different approximations used in the GW method on the band gaps and band edges for three TMOs: wurtzite ZnO, cuprite Cu 2 O, and rutile TiO 2 . It is found that the GW band gap of ZnO is sensitive to the starting point obtained from the LDA or GGA calculations, suggesting that the Kohn-Sham orbitals differ from the quasiparticle orbitals. It is shown that the HL PPM becomes similar to the GN PPM and gives better agreement with the RPA when ρ ppm is set to ρ val , that is, only the valence electrons are used to determine the effective bare plasma frequencies for the HL PPM. It is demonstrated that the theoretically justified choice of approximations, namely eigenvalue self-consistent GW 0 scheme, ρ val in the HL PPM, and the proper treatment of the V xc term, give the best overall agreement between the calculated and measured band gaps.
(Grant No. NRF-2013R1A1A1076141). This research used resources of the Oak Ridge Leadership Computing
Facility located in the Oak Ridge National Laboratory, which is supported by the Office of Science of the Department of Energy under Contract DE-AC05-00OR22725.
(Grant No. NRF-2013R1A1A1076141). This research used resources of the Oak Ridge Leadership Computing
Supplemental material: Insights and challenges of applying the GW method to transition metal oxides ) and matrix elements of Vxc and Σ at the valence band maximum (VBM) and conduction band minimum (CBM) of wurtzite ZnO calculated within DFT and GW using different exchange-correlation functionals (LDA and GGA), experimental and theoretical structural parameters (ES and TS), non-selfconsistent G0W0 and eigenvalue self-consistent GW0 schemes, HL PPM with ρppm = ρ val and ρ scf , and matrix elements of Vxc without and with NLCC (ρcore = 0 and = 0). The VBM and CBM are at the Γ point. All values are in eV. The zero reference for the energy scale is the average (G = 0 component) electrostatic (ionic plus Hartree) potential. ) and matrix elements of Vxc and Σ at the valence band maximum (VBM) and conduction band minimum (CBM) of cuprite Cu2O calculated within DFT and GW using different exchange-correlation functionals (LDA and GGA), experimental and theoretical structural parameters (ES and TS), non-selfconsistent G0W0 and eigenvalue self-consistent GW0 schemes, HL PPM with ρppm = ρ val and ρ scf , and matrix elements of Vxc without and with NLCC (ρcore = 0 and = 0). The VBM and CBM are at the Γ point. All values are in eV. The zero reference for the energy scale is the average (G = 0 component) electrostatic (ionic plus Hartree) potential. ) and matrix elements of Vxc and Σ at the valence band maximum (VBM) and conduction band minimum (CBM) of rutile TiO2 calculated within DFT and GW using different exchange-correlation functionals (LDA and GGA), experimental and theoretical structural parameters (ES and TS), non-selfconsistent G0W0 and eigenvalue self-consistent GW0 schemes, HL PPM with ρppm = ρ val and ρ scf , and matrix elements of Vxc without and with NLCC (ρcore = 0 and = 0). The VBM and CBM are at the Γ point (in regular font) and at the R point (in cursive font). All values are in eV. The zero reference for the energy scale is the average (G = 0 component) electrostatic (ionic plus Hartree) potential. 

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