Abstract
In this work, we systematically evaluate the accuracy in band gap prediction of rangeseparated hybrid functionals on a large set of semiconducting and insulating materials and carry out comparisons with the performance of their global counterparts. We observe that all the rangeseparated hybrid functionals that correctly describe the longrange dielectric screening significantly improve upon standard hybrid functionals such as PBE0 and HSE06. The choice of the shortrange Fock exchange fraction and the screening length can further reduce the predicted error. We then propose a universal expression for the selection of the inverse screening parameter as a function of the shortrange and longrange Fock exchange fractions, which results in a mean absolute error as small as 0.15 eV for band gap prediction.
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Introduction
Accurate prediction of materials band gaps is key to studying the electronic and optical properties of semiconductors and insulators. However, Kohn–Sham density functional theory (KSDFT)^{1,2} intrinsically underestimates the fundamental band gap E_{g}, defined as the difference between the ionization potential and the electron affinity^{3}. Even in exact KSDFT, the predicted band gap deviates from the fundamental band gap by Δ_{xc}, the discontinuity in the exchangecorrelation potential^{4}. In the localdensity approximation or in the generalized gradient approximation, this leads to the underestimation of the band gaps by about 50%^{5,6}. More recent developments in semilocal functionals, such as the modified Becke–Johnson potential^{7}, GLLBSC^{8}, and TASK^{9}, show improvement in band gap predictions. However, to date, an average error of 0.5 eV persists when applying the bestperforming semilocal functionals^{10,11}.
The generalized KS theory resolves the band gap prediction problem by admixing a fraction of Fock exchange with the semilocal DFT exchange, thus opening up the band gap^{12}. For example, the global hybridfunctional PBE0 includes a fraction α = 0.25 of Fock exchange, which was found to optimize the atomization energies of molecules^{13,14,15}. Heyd et al. developed the screened hybridfunctional HSE06^{16,17}, which has a mixing parameter of 0.25 in the short range and reproduces semilocal exchange in the long range. In spite of their widespread use, the hybrid functionals adopting fixed mixing parameters, such as PBE0 and HSE06, are not universally applicable. For solidstate systems, these functionals perform best on materials with intermediate band gaps^{18}. However, their accuracy greatly deteriorates for wide band gap materials, such as MgO and LiF, and narrow band gap materials, such as Si and Ge^{19,20}. This inadequacy led to the development of nonempirical hybrid functionals. In these functionals, the mixing parameters are determined nonempirically by enforcing certain exact constraints on the exchangecorrelation potential^{4}. Through adopting materialspecific fractions of Fock exchange, the nonempirical hybrid functionals are promising in achieving more uniform accuracy in band gap prediction, as well as in predicting other electronic, optical, and structural properties of semiconductors and insulators^{21,22,23,24,25}.
Based on the specific exact constraints imposed, nonempirical hybrid functionals are sought according to two lines of thought. The first group, often denoted dielectricdependent (DD) hybrid functionals, is built by connecting α with the macroscopic dielectric constant ϵ_{∞}^{19,26}. The simplest form of this group, DDPBE0, admixes a fraction α = 1/ϵ_{∞} of Fock exchange. These functionals correctly describe the longrange interaction, which asymptotically approaches \(1/({\epsilon }_{\infty } {{{\bf{r}}}}{{{{\bf{r}}}}}^{{\prime} } )\)^{27}. DDPBE0 allows for strong screening in the case of narrow band gap materials and weak screening in the case of wide band gap materials, and thus greatly enhances the uniformity of the achieved accuracy. This idea has subsequently been combined with the development of rangeseparated hybrid (RSH) functionals, in which different Fock fractions are admixed in the long range and in the short range, separated through the use of an inverse screening length μ^{28,29,30}. Hence, DDRSH functionals generally adopt a longrange Fock fraction α_{l} = 1/ϵ_{∞} and various differing strategies for determining the shortrange Fock fraction α_{s} and the inverse screening length μ^{20,31,32}.
The second group of nonempirical hybrid functionals is constructed by imposing the piecewise linearity condition, which asserts that the groundstate energy E(N) as a function of electron number N must be linear upon electron occupation between integer electron numbers^{33}. Through Janak’s theorem^{34}, this constraint translates to the singleparticle energy level of the highest occupied state being constant irrespective of its occupation, a constraint known as the generalized Koopmans’ condition. To construct a piecewise linear hybrid functional, the mixing parameters can be found by enforcing Koopmans’ condition on a localized electronic state. These functionals were first applied to organic molecules^{35,36} and more recently to extended systems^{22,24,37,38,39,40,41,42,43,44,45}. They were demonstrated to be especially useful for materials with heterogeneous dielectric screening, for example, for twodimensional materials^{21,46} and interfaces^{47}.
Despite these recent developments of nonempirical hybrid functionals, the methods generally adopt different ways of choosing α and μ values and there has been a lack of systematic comparison among these choices. The average errors in band gap predictions are reported on different sets of materials, with different material structures, pseudopotentials, or convergence parameters. Furthermore, some of these functionals require a rather costintensive construction process, hindering their widespread use^{20,23,37,38,41,43}. In this work, we present a comprehensive comparison of the performance of six nonempirical hybrid functionals by evaluating their accuracy in predicting the band gaps for a variety of semiconducting and insulating materials. We show that nonempirical hybrid functionals significantly outperform standard hybrid functionals such as PBE0 and HSE06. We then provide a detailed analysis on how the fraction of Fock exchange and the inverse screening parameter affect the predicted band gaps. We show that available methods for determining the inverse screening parameter do not lead to an improvement in the overall accuracy compared to adopting a fixed value. In light of this observation, we further propose an analytical expression for setting the inverse screening parameter as a function of the fractions of Fock exchange in the short and long range. The optimal functional constructed in this way further reduces the average error in the band gap prediction to 0.15 eV.
Results and discussion
Table 1 gives a summary of the hybrid functionals considered in this work. We start with global hybrid functionals in which the fraction of Fock exchange is defined by a single parameter α. In the commonly used PBE0 functional, α is set to 0.25^{14}. In DDPBE0, α is set to 1/ϵ_{∞}.
RSH functionals adopting the Coulomb attenuating method (CAM)^{48} separate the nonlocal exchange potential into shortrange and longrange parts through an error function with inverse screening length μ:
In this way, the exchange potential is defined as follows:
where \({v}_{x}^{{{{\rm{PBE}}}}}\) and \({v}_{x}^{{{{\rm{Fock}}}}}\) are the semilocal and the nonlocal exchange potentials, respectively, with their shortrange and longrange component fractions specified by α_{s} and α_{l}. The parameter α_{l} is generally set to 1/ϵ_{∞} to comply with the exact condition of the asymptotic potential in the long range, as done in DD functionals^{4,27}. Depending on how α_{s} and μ are chosen, various versions of RSH functionals can be constructed. Here, we group them into two main classes based on the choice of α_{s}. In the first class, α_{s} is set to 0.25, like in PBE0. The widely used hybridfunctional HSE06 belongs to this class, with α_{l} set to 0 and μ to 0.106 bohr^{−1 }^{16,17}. Another common choice of μ is the ThomasFermi (TF) screening parameter^{31,49}, which is defined as follows:
where n is the valence electron density. Here, all the electrons in the outer shell are counted as valence electrons^{32,50}. For example, we take two valence electrons for Ca and thirteen valence electrons for Ga.
In the second class, α_{s} is set to 1. Two recently proposed functionals belong to this class: the DDCAM^{20} and the doubly screened hybrid (DSH) functional^{32}. The two functionals use the same settings for α_{s} and α_{l}, but adopt different settings for μ. In the former, μ is nonempirically determined through fitting the dielectric function calculated from linear response^{20}. In the latter, μ is defined by the analytical expression:
in which γ is empirically set to 1.563. To determine how the materialspecific values of μ influence the overall accuracy of band gap predictions, we also consider setting μ to a fixed value of 0.71 bohr^{−1} for both classes with α_{s} = 0.25 and α_{s} = 1 (\({\mu }_{{{{\rm{fix}}}}}^{{\alpha }_{s} = 0.25}\) and \({\mu }_{{{{\rm{fix}}}}}^{{\alpha }_{s} = 1}\)). This value for the inverse screening parameter has been determined in ref. ^{20} from an average over a large variety of materials.
We also include in our comparison two functionals satisfying the piecewise linearity condition, KPBE0 and KCAM^{36,37,51}. The KPBE0 functional is a global one, for which the mixing parameter α is determined by inserting an atomic probe into the material system^{38,43}. One then systematically varies α until the localized electronic state of the probe is constant irrespective of its occupation. Thus, the value α = α_{K} found in this way satisfies the piecewise linearity condition. The KCAM functional is rangeseparated with α_{l} = 1/ϵ_{∞} and μ = 0.106 bohr^{−1} as in HSE06. The shortrange mixing parameter α_{s} is determined by enforcing the piecewise linearity condition on a localized potential probe, in the same way as for KPBE0.
In the following sections, we give a detailed analysis of how the functional forms with their mixing parameters influence the achieved accuracy in predicting band gaps. Specifically, we focus on the dependence on α for global hybrid functionals, and on the combined dependence on α_{s} and μ for RSH functionals. Following this analysis, we propose a universal formulation for choosing the inverse screening parameter μ as a function of α_{s} and α_{l}. Last, we give a comprehensive comparison of the various functionals in terms of their accuracy and discuss strategies for optimizing RSH functionals.
Global hybrid functionals
First, we consider the global hybrid functionals PBE0(α) and the dependence of the predicted band gaps on α. Figure 1 shows the band gaps as obtained with PBE (α = 0), PBE0 (α = 0.25), DDPBE0 (α = 1/ϵ_{∞}) and KPBE0 (α = α_{K}) as a function of the respective α values for all the materials considered in this work. As clearly seen in Fig. 1, the calculated band gaps closely follow a linear relationship with α. This linearity allows us to fit the band gap as a function of α and to find the fraction α_{expt} that reproduces the experimental band gap, thus providing a visual guidance for comparing the errors of each functional.
We first observe that the PBE band gaps are systematically smaller than the experimental ones, demonstrating the notorious band gap underestimation problem of semilocal functionals. As α increases, the band gaps become larger. The α values reproducing the experimental band gaps also tend to increase with increasing band gap. For PBE0, which includes a fixed Fock fraction of 0.25, the band gaps are overestimated in the small band gap regime and underestimated in the large band gap regime. A severe underestimation is observed for wide band gap materials such as Ar and LiF. This problem is greatly mitigated by adopting materialspecific α values. Indeed, for both DDPBE0 and KPBE0, the respective α values fall much closer to α_{expt}, yielding uniform accuracy over the whole band gap range. Between these two, KPBE0 has a slight advantage over DDPBE0 in terms of accuracy, producing a mean absolute error (MAE) of 0.34 eV compared to 0.41 eV for DDPBE0 when compared for the same set of materials (see Supplementary Table 4).
This analysis of the role of α also sheds some light on the choice of α_{s} for RSH functionals. Going back to Eqs. (1) and (2), in the limit of μ → ∞, the RSH functional falls back to PBE0(α_{l}). In the limit of μ → 0, it falls back to PBE0(α_{s}). In other words, tuning the value of μ essentially modulates the predicted band gap between PBE0(α_{l}) and PBE0(α_{s}). If we consider the class of rangeseparated functionals with α_{l} set to 1/ϵ_{∞} and α_{s} to 0.25, the tunable range of the predicted band gap is limited by the values from PBE0 and DDPBE0. At variance, by setting α_{s} to 1, the tunable range is between the band gap values predicted by DDPBE0 and PBE0(α = 1). Considering that PBE0(α = 1) largely overestimates the band gaps with respect to experimental values, selecting α_{s} = 1 yields a much larger tunable range of band gaps compared to that of α_{s} = 0.25. This observation helps us to better understand the influence of μ on the calculated band gaps for RSH functionals in the next section.
Rangeseparated hybrid functionals
In this section, we examine how the choice of μ and α_{s} influence the accuracy of RSH functionals. Similar to the previous analysis for α, we show in Fig. 2 how the calculated band gaps depend on μ for the two classes of functionals with α_{s} = 0.25 and α_{s} = 1. Also in this case, we assume that E_{g} depends linearly on μ and find the μ_{expt} values that reproduce the experimental band gaps. The relationship can well be approximated as being linear (cf. Fig. 2).
We first look at how the changes in μ determine the band gaps. As has been established, varying μ tunes the predicted band gap between the values produced by PBE0(α_{l}) and PBE0(α_{s}). This leads to a major difference between the cases of α_{s} = 0.25 and α_{s} = 1. For α_{s} = 0.25 (Fig. 2a), the band gaps decrease with increasing μ for materials having ϵ_{∞} > 4 (mostly in the small band gap regime), and the reverse occurs for materials having ϵ_{∞} < 4 (mostly in the large band gap regime). This is because α_{s} is larger than α_{l} (1/ϵ_{∞}) in the former group and α_{s} is smaller than α_{l} in the latter group. It also leads to the peculiar observation that for materials with ϵ_{∞} close to 4, changing μ has little effect on the predicted band gap, as manifested by the cases of CaO (ϵ_{∞} = 3.3), BN (ϵ_{∞} = 4.5), and ZnO (ϵ_{∞} = 3.74). For these materials, it is not possible to reproduce the experimental band gaps with reasonable values of μ. However, for α_{s} = 1 (Fig. 2b), it is ensured that α_{s} is larger than α_{l}. As a result, the calculated band gaps always decrease with increasing μ. Generally, the selection of α_{s} = 1 creates a larger difference between α_{s} and α_{l} and thus a stronger dependence of the band gaps on μ.
With the general E_{g}vsμ relationship established, we now take a closer look at the specific choices of μ values. When α_{s} is set to 0.25 (cf. Fig. 2a), we observe that the μ_{TF} values generally fall in the range of 0.6–0.8 bohr^{−1}, close to the average μ value of 0.71 bohr^{−1}. Consequently, the overall band gap accuracy of adopting μ_{TF} is almost the same as that of adopting the fixed value of 0.71 bohr^{−1}. The MAEs of both functionals are 0.41 eV, and the mean absolute relative errors (MAREs) are 14.3% for the former and 14.6% for the latter, demonstrating little advantage of using materialspecific μ values. The μ values reproducing experimental band gaps (henceforth referred to as μ_{expt}) are in fact much more scattered. Considering functionals with α_{s} = 1 (cf. Fig. 2b), we find that DDCAM and DSH perform better in terms of overall accuracy, producing MAEs of 0.23 and 0.24 eV, respectively (cf. Table 2). In Fig. 2b, we see that μ_{DDCAM} and μ_{DSH} are also relatively close to the average value of 0.71 bohr^{−1}. The MAE obtained with a fixed μ of 0.71 bohr^{−1} is 0.23 eV, again showing no advantage of using materialspecific μ values.
We now turn to the KCAM functional in which α_{s} values are determined in a materialspecific way by enforcing the generalized Koopmans’ condition. In this case, μ is fixed to 0.106 bohr^{−1}, like in HSE06. When compared for the same set of materials, the KCAM functional produces an MAE of 0.37 eV, which does not improve upon the MAE of 0.34 eV pertaining to the KPBE0 functional (cf. Supplementary Table 4). This agrees with previous investigations adopting the same strategy for determining α_{s}^{37,39}. A recent study shows that it is possible to achieve a better accuracy by fixing α_{s} and determine μ through the enforcement of the generalized Koopmans’ condition^{24}. However, we did not obtain such a higher accuracy when following an analogous strategy but with localized potential probes (see Supplementary information for more discussion).
Optimizing the inverse screening parameter
With the insight into the E_{g}vsμ relationship achieved above, we now inquire whether it is possible to devise a strategy for selecting μ that could further improve the accuracy of RSH functionals. In Fig. 2, we have seen that the TF, DDCAM, and DSH functionals adopt μ values that fall close to the average value of 0.71 bohr^{−1}, whereas the μ_{expt} values appear to be more scattered. We have also established that the dependence of E_{g} on μ is largely determined by the difference between α_{s} and α_{l}. In particular, when α_{s} = α_{l}, the change of μ has no effect on the calculated band gap. Prompted by this insight, we derive a relationship between μ_{expt}, α_{s}, and α_{l}. Assuming that E_{g} depends linearly on μ as in Fig. 2, we have
Considering that \({E}_{g}({\mu }_{{{{\rm{expt}}}}})={E}_{g}^{{{{\rm{expt}}}}}\), it follows that
where E_{g}(0) is the band gap value obtained with μ = 0, which coincides with the value obtained with PBE0(α = α_{s}). Using the properties of the exchange potential, it can analytically be shown that dE_{g}(μ)/dμ for a given material is proportional to α_{s} − α_{l} (cf. Section 6 in the Supplementary information). As seen in Fig. 3a, the proportionality constant is approximately constant for the materials considered in this work. To produce this figure, we set α_{s} = 0.25 and determine the derivative dE_{g}(μ)/dμ by finite differences. Next, we focus on the numerator \({E}_{g}^{{{{\rm{expt}}}}}{E}_{g}(0)\) in Eq. (6). From the success of DDH functionals, we can assume that \({E}_{g}^{{{{\rm{expt}}}}}\approx {E}_{g}[{{{\rm{PBE0}}}}(1/\epsilon )]\). Since E_{g}(0) = E_{g}[PBE0(α_{s})], we then infer that \({E}_{g}^{{{{\rm{expt}}}}}{E}_{g}(0)\) relates to 1/ϵ_{∞} and thus to α_{l}. In Fig. 3b, we show that this relationship can be closely approximated by a linear dependence of \({E}_{g}^{{{{\rm{expt}}}}}{E}_{g}(0)\) on α_{s} − α_{l}.
By combining the results from this analysis in Eq. (6), we propose a universal formula for selecting μ as a function of α_{s} and α_{l}:
where f(x) is a linear function. Based on this formula, we fit the μ_{expt} values obtained previously and arrive at the following expressions for μ_{u}:
The \({\mu }_{{{{\rm{u}}}}}^{{\alpha }_{s} = 0.25}\) and \({\mu }_{{{{\rm{u}}}}}^{{\alpha }_{s} = 1}\) values are shown on the respective panels of Fig. 2 and listed in Table 3. We observe that this expression correctly captures the divergence in the E_{g}vsμ relationship at α_{s} = α_{l} and follows the scatter of μ_{expt}. As such, these μ_{u} values further improve the accuracy in predicting band gaps compared to the previous functionals, with MAEs of 0.15 eV for \({\mu }_{{{{\rm{u}}}}}^{{\alpha }_{s} = 0.25}\) and of 0.18 eV for \({\mu }_{{{{\rm{u}}}}}^{{\alpha }_{s} = 1}\). Similarly, the respective MAREs reduce to 3.8% and 5.6%. The predicted band gaps are shown in Fig. 4 and are provided in Table 3. As shown in Fig. 4, the two functionals adopting μ_{u} values yield a uniform accuracy over the full range of band gaps. In the case α_{s} = 0.25, we remark that Eq. (8) leads to \({\mu }_{{{{\rm{u}}}}}^{{\alpha }_{s} = 0.25}\) values lying close to the divergence for materials with ϵ_{∞} ≈ 4. Nevertheless, the band gaps in these cases depend only very weakly on μ, and μ can thus be set to 0.
Furthermore, we test the proposed formulas for μ on several materials that are not part of the set studied. To verify that the proposed functionals with \({\mu }_{{{{\rm{u}}}}}^{{\alpha }_{s} = 0.25}\) and \({\mu }_{{{{\rm{u}}}}}^{{\alpha }_{s} = 1}\) do not spuriously open up a band gap for metallic systems, we consider graphite, sodium, and aluminum and observe no band gap opening for any of these metals. In addition, we remark that although the proposed μ_{u} formulas perform consistently on a large set of materials, there can be outliers for which DD functionals are less successful. For example, it has been shown in literature that DD hybrid functionals may lead to inaccurate band gaps in the case of correlated antiferromagnetic transitionmetal oxides^{52}. In the case of NiO, we indeed find that the functionals with \({\mu }_{{{{\rm{u}}}}}^{{\alpha }_{s} = 0.25}\) and \({\mu }_{{{{\rm{u}}}}}^{{\alpha }_{s} = 1}\) proposed here noticeably overestimate the band gap (see Supplementary information for more discussion).
Conclusion
To sum up, we have performed a comprehensive evaluation of the performance of available nonempirical hybrid functionals in predicting band gaps for a varied set of semiconductors and insulators. In Fig. 5, we provide a comparison of the MAEs for the functionals considered in this work. First, we have shown that the Fock fractions required for producing the experimental band gaps are materialspecific. In most cases, they lie close to 1/ϵ_{∞}. As a result, standard hybrid functionals such as PBE0 and HSE06 generally perform better for materials with medium band gaps, whereas they severely underestimate the band gaps of wide band gap materials. Adopting materialspecific α values, as in DDPBE0 and KPBE0, greatly improves the uniformity of the accuracy over the band gap range. Between these two, KPBE0 performs slightly better in terms of overall accuracy (MAE 0.34 eV compared to 0.44 eV). Going from KPBE0 to KCAM shows little improvement in the MAE. These three functionals also consistently show a better performance for sp materials compared to materials with 3d electrons.
As for the DDRSH functionals, the class with α_{s} = 1 shows an overall advantage over the class with α_{s} = 0.25. The functional adopting α_{s} = 0.25 and μ = μ_{TF} produces an MAE of 0.41 eV, whereas DDCAM and DSH have MAEs of 0.23 and 0.24 eV, respectively. It is also noteworthy that the accuracy of the latter group does not deteriorate as much for 3d materials compared to the global hybrid functionals, or to functionals with α_{s} = 0.25. In addition, we show that previous methods based on μ_{TF}, μ_{DSH}, and μ_{DDCAM}, all produce μ values fairly close to the average value of 0.71 bohr^{−1}. In fact, functionals using a fixed μ of 0.71 bohr^{−1} are as accurate as methods adopting materialspecific μ, consistent with previous findings by Chen et al.^{20}.
Last, we demonstrate that a suitable choice of μ improves the accuracy of rangeseparated functionals even further. The μ values reproducing the experimental band gaps are far more scattered than any of the available schemes for determining μ. Based on this observation, we propose a new formula μ_{u}, which correctly captures the divergence of μ at α_{s} = α_{l}. This formula produces surprisingly good MAEs of 0.15 eV for α_{s} = 0.25 and 0.18 eV for α_{s} = 1, demonstrating the potential of further lowering the band gap errors achieved with RSH functionals. The RSH functionals constructed either with fixed μ values (0.71 bohr^{−1}) or with μ values given by a simple analytical equation (μ_{u}) provide a scheme that is much simplified with respect to the DDCAM method^{20} or to the Koopmans construction process^{23,37,38,41,43}. With these findings, we have established that hybrid functionals with materialspecific parameters can approach the accuracy of stateoftheart GW calculations with no greater computational cost than that of standard hybridfunctional calculations, making these functionals ideal candidates for widespread use in predicting electronic properties of solidstate materials.
Methods
Computational details
All DFT calculations are performed with the Quantum ESPRESSO suite^{53}. Planewave basis sets for expanding the wave functions are used in conjunction with normconserving pseudopotentials including semicore d electrons^{54,55}. The lattice parameters are taken from experimental values, as given in refs. ^{20} and ^{50}. Planewave energy cutoffs and kpoint grids are individually set for each material to ensure band gap convergence within 1 meV. Details of the material structures, convergence parameters, and specific α and μ values used for each functional can be found in the Supplementary information. For the DDCAM functional, we take the α_{l} and μ values from ref. ^{20}. For the KPBE0 and KCAM functional, α_{K} values are taken from ref. ^{43}, but the accuracy of these schemes is here determined using the same experimental references as for the other functionals. The band gap calculations are repeated for all the functionals considered in this work to eliminate any effect resulting from the use of different pseudopotentials or materials structures.
The accuracy of the functionals considered in this work is determined with respect to experimental values corrected for zeropoint renormalization. The sources of these values are given in Supplementary information. For BN and diamond, it is difficult to correct the measured optical band gaps for the excitonic effect, because these materials have indirect band gaps^{56,57}. Thus, we use stateoftheart GW calculations as a reference in these two cases^{58}. Experimental errors still affect the accuracy determined for the various functionals, but the comparison between theory and experiment remains meaningful provided that the set of materials considered is large.
Data availability
The data associated with this work is available on Materials Cloud^{59}.
Code availability
The opensource Quantum ESPRESSO suite^{53} is freely available. The other relevant codes in this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
This work is supported by Grant No. 200020152799 of the Swiss National Science Foundation (SNSF). The calculations were performed at the Swiss National Supercomputing Center (CSCS) under project s1122 and at SCITASEPFL.
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J.Y. and A.P. conceived the project. J.Y. performed the calculations involved in this work. S.F. contributed to the derivations. All authors contributed to the writing of the manuscript.
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Yang, J., Falletta, S. & Pasquarello, A. Rangeseparated hybrid functionals for accurate prediction of band gaps of extended systems. npj Comput Mater 9, 108 (2023). https://doi.org/10.1038/s4152402301064x
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DOI: https://doi.org/10.1038/s4152402301064x
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