Abstract
Although the GW approximation is recognized as one of the most accurate theories for predicting materials excited states properties, scaling up conventional GW calculations for large systems remains a major challenge. We present a powerful and simpletoimplement method that can drastically accelerate fully converged GW calculations for large systems, enabling fast and accurate quasiparticle calculations for complex materials systems. We demonstrate the performance of this new method by presenting the results for ZnO and MgO supercells. A speedup factor of nearly two orders of magnitude is achieved for a system containing 256 atoms (1024 valence electrons) with a negligibly small numerical error of ±0.03 eV. Finally, we discuss the application of our method to the GW calculations for 2D materials.
Introduction
Accurate predictions of excited states properties are critical for computational screening and design of materials for energy and electronics applications. While density functional theory (DFT)^{1} within the local density approximation (LDA)^{2} or the generalized gradient approximation (GGA)^{3} has been very successful in describing materials groundstate properties, the KohnSham (KS) eigenstates in principle cannot be compared directly with quasiparticle (QP) excitations. The GW approximation^{4,5,6}, systematically derived from the perturbative expansion of electron selfenergy, is widely recognized as one of the most accurate methods for predicting excited states properties of materials.
Unfortunately, despite the tremendous success, accurate and efficient predictions of excitedstates properties of complex solid systems remain a major challenge due to complication of the convergence issue^{7,8,9,10,11} and the unfavorable scaling of the computational cost (both in terms of the computational time and the storage and memory requirements) with respect to the system size. This is particularly true for systems containing substantially localized electronic states and/or with large unit cells (e.g., nanostructures, complex multinary compounds, surfaces and interfaces, and defects in solids). For example, it has been shown that for simple materials such as ZnO one has to include as many as 3,000 empty states in conventional GW calculations to acheive properly converged results^{9,11,12,13}. Although ZnO has probably received the most research attention, the convergence problem of GW calculations is not just limited to this single material. In fact, GW calculations for many systems, including MgO, CuCl, AgBr, and CdO (to name a few), suffer similar convergence problems. Recently, it was shown that about 6,000 bands are need to to converge the GW band gap for monolayer MoS_{2} to within 50 meV^{14}, and that for MoSe_{2} is as large as 10,000^{15}.
In this work, we present an efficient method that can drastically speed up GW calculations for large systems and is also easy to be implemented within available GW packages. Our method takes advantage of the fact that the the density of states (DOS) of all materials resembles that of the free electron gas at high energies, thus the summation over highenergy conduction in GW calculations can be approximated by a numerical integration on a sparse energy grid. Our method makes fully converged GW calculations for large systems feasible, eliminating the need for performing expensive and tedious convergence tests since one can now afford to include effectively all conduction bands in GW calculations at greatly reduced computational costs.
Results and Discussions
The GW approximation for the electron selfenergy Σ(r, r′; E) is
where δ = 0^{+}, G is the oneparticle Green’s function, and W is the screened Coulomb interaction, which is related to the dielectric function ε and the bare Coulomb interaction V_{c} as W = ε^{−1}V_{c}. In conventional firstprinciples GW methods^{5,6}, both the Green’s function and the dielectric functions are constructed using the KS eigenstates, and, in principle, all states in the Hilbert space of the KS Hamiltonian should be included in the summations in calculating G and W. In practical calculations, truncations are almost always applied. Since the convergence behavior of GW calculations could be very different from materials to materials, it is often difficult to predict a priori proper truncation parameters to ensure fully converged GW results. For large systems, it is extremely difficult (if possible at all) to perform fully converged GW calculations within this scheme.
GW calculations involve two computationally expensive summations over conduction bands in calculating the dielectric function ε and the Coulombhole part of selfenergy operator Σ_{COH}(E)^{5}. Within the random phase approximation (RPA), the dielectric function is related to the electron polarizability χ^{0} via ε = 1 − V_{c}χ^{0}. The (irreducible) electron polarizability is constructed using KS eigenstates:
where M_{vc}(k, q, G) = 〈v, k + qe^{i(q+G)}c, k〉, and v and c index valence and conduction bands. The band summations in the calculations of dielectric function should in principle include all conduction (empty) states in the Hilbert space of the KS Hamiltonian. The calculation of the Coulombhole (COH)^{5} selfenergy also involves a summation over conduction bands^{5,16}:
where ε^{r} (ε^{a}) is the retarded (advanced) dielectric function. The screened exchange (SEX)^{5} part of the selfenergy, on the other hand, involves a summation only with occupied states. The convergence of GW results with respect to the number of conduction bands included in the above summations can sometimes be extremely slow, which has become both of a burden and a source of confusion since underconverged results reported by different groups can vary significantly. Our new method effectively allows including all conduction bands in the above two summations as explained below.
We first use MgO as an example to illustrate the difficulty of scaling up conventional GW calculations for large systems. We then propose our new approach and demonstrate the performance of the newly developed method by showing results for MgO and ZnO. Finally, we discuss the application of our method to GW calculations for 2 dimensional (2D) materials. Due to the slow convergence of the GW results with respect to the kpoint sampling density, interlayer spacing, and the number of conduction bands included in the calculations, reported quasiparticle band gap of monolayer MoS_{2} ranges from 2.41 to 2.84 eV^{14,17}, underlining the importance of affordable and efficient GW approaches for reliable predictions of the excited states properties of 2D systems.
The convergence issue in GW calculations
To better illustrate the convergence issue in GW calculations, we show in Fig. 1 the convergence behavior of the calculated quasiparticle band gap of MgO (a twoatom primitive cell) as a function of the number of conduction bands included in the COH self energy summation and the kinetic energy cutoff for the dielectric matrices. Depending on the choice of these cutoff parameters, the calculated QP band gap varies from 7.25 to 7.90 eV. Once fully converged, we obtain a QP band gap of about 7.90 eV, in good agreement with the experimental value of 7.78 eV^{18} and a previous calculation^{11}. As it is shown in Fig. 1, a high kinetic cutoff (G_{cut}^{2}/2 ~ 40 Ry) for the dielectric matrices ε_{G,G′}(q, ω) and a large number of conduction bands (N_{c} ~ 800) in the COH summation are required. If a small kinetic energy cutoff for the dielectric matrices is used, the band gap converges quickly but prematurely with respect to the number of band included in the summation.
With today’s massively parallel computers, fully converged GW calculations for small systems such as 2atom MgO can be done easily. The problem, however, quickly becomes intractable when one attempts to scale up the calculations for large systems containing hundreds of atoms because the number of conduction bands required in the COH and dielectric function calculations scales linearly with the system size (i.e., number of atoms). To put this problem in perspectives, suppose now we would like to carry out GW calculations for a 128atom MgO supercell containing a defect (e.g., an Fcenter), the required number of bands would increase by 64 (i.e., 128/2) times. Not only it is extremely difficult to generate so many wave functions, storing these wave functions would also pose a serious challenge, not to mention performing the band summations in the GW calculations.
Considering the importance of predicting materials excited states properties, it is not surprising that there has been much work on improving the efficiency of the GW formalism for large systems beyond simple crystals. One direction is to introduce better (optimized) bases for representing the dielectric matrix and/or the selfenergy operator beyond plane waves and KS eigen states^{19,20,21}. The second direction is to reduce the number of conduction bands N_{c} by replacing the summation over highenergy states with computationally tractable terms^{7,8,11,22,23}, or completely removing the necessity of summing over unoccupied states^{24,25,26}. Other efforts include applications of the Lanczos algorithm for calculating the dielectric functions^{27,28}, the development of the stochastic GW approach^{29}, and the separation of the dynamical and nonlocal correlations^{30} in dynamical mean field theory (DMFT) calculations which may also be applied to GW calculations. These efforts underline the urgency of developing efficient GW methods to meet the challenge of fast and accurate predictions of the excited states properties of large systems such as nanostructures, defects in solids, and surfaces and interfaces. Some of these approaches, however, may require substantial modifications (or completely rewriting) existing GW codes, and the study of the convergence behavior of these methods is still in the early stage.
Replacing the bandbyband summation with an energy integration in GW calculations
Our approach to speeding up the calculation is inspired by the observation that the electron DOS of all materials can be divided into two regions: a low energy region in which the behavior of electronic states is strongly affected by the crystal potential (thus materials dependent) and a high energy region in which kinetic energy dominates and the DOS scales as as that for free electron systems, where V_{xc}(0) is the averaged exchangecorrelation potential of the material. Figure 2 compares the DOS of MgO and ZnO calculated within the LDA and that of the freeelectron gas scaled with the volume of the respective unit cells. The DOS below certain energy (E_{0} shown in left panel of Fig. 2) is what distinguishes one material from another. Above E_{0}, the DOS can be wellapproximated by that of freeelectron gas.
This observation leads us to propose that the band summation in the GW calculations be divided into two regions: a low energy region (shaded area in Fig. 2) in which the band summation is carried out explicitly and a highenergy region in which the band summation is replaced (approximated) by an energy integration. For example, the static polarizability χ^{0} is rearranged in the following form:
where is the approximated DOS for the system with a cell volume Ω, E_{0} corresponds to the energy of band N_{0}, and M_{vE}(k, q, G) are the matrix elements between the valance band v and the conduction band at (or near) the energy E. In practical calculations, we find that the number of bands N_{0} can be limited to less than 10 per atom for most systems. The auxiliary integral part can be carried out using simple numerical integration techniques by sampling the electronic states on an energy grid as illustrate in left panel of Fig. 2:
where N_{E} is the number of sampling points used in the numerical integration as shown schematically in the left panel of Fig. 2 with large blue dots. The same approach is applicable to the calculation of the COH selfenergy. The integration can be carried on a uniform or nonuniform energy grid, and a typical energy step ΔE can range from 1 to 5 eV. Higher order numerical integration techniques may also be applied.
Demonstration of the accuracy and efficiency of the new method
We now examine the accuracy of this new method. We first compare the GW band structure of MgO (2atom primitive cell) calculated using the conventional approach and our new method as shown in left panel of Fig. 3. The difference is unnoticable at this scale. We have also tested the result for ZnO. The right panel of Fig. 3 shows the calculated GW band gap of ZnO as a function of number bands included in the COH selfenergy summation (and in the dielectric matrix calculations). The difference between the results calculated using the two methods is again negligibly small (~0.02 eV). For the calculation using our new method, the number of integration points plus the number of bands within the energy window E_{0} is about 350, to be compared with 3,000 bands in conventional GW calculations.
We then test the stability and performance of our method using a reasonably small (but not too small) system, an MgO supercell containing 16 atoms, as an example. For this system, we choose N_{0} to be 128. In other words, 8 conduction bands per atom are included in the explicit bandbyband summations in the calculation of the dielectric matrices and the COH selfenergy. The auxiliary integral part is calculated on a uniform energy grid with a step ΔE ranging from 1.5 to 4.0 eV, and the integration is carried out up to an energy which is equivalent to including 8,000 (or 1,000 per 2atom cell) conduction bands. The kinetic energy cutoff for the dielectric matrix is set at 40 Ry.
For such a small system, we can compare directly the results calculated using the new method and the conventional bandbyband summation approach, as shown in Table 1. As the energy integration step varies from 1.5 to 4 eV (while keeping N_{0} fixed at 128), the number of conduction bands plus the number of the energy integration points decreases from 895 to 320, to be compared with 8,000 required in conventional calculations to achieve the same level of convergence, representing a speedup factor of 9 to 25. It is rather encouraging that the results calculated using this new method are insensitive to the energy integration step ΔE, showing the stability of the this method. The GW band gap calculated with the conventional method is 7.86 eV, and that obtained from the new method only deviates by a negligible small amount of ±0.02 eV.
Our method reduces the computational time of fully converged GW calculations for a 16atom MgO supercell from about 5 hours to 15 minutes using 64 computing cores (8 Intel Xeon E52650 processors). We would like to mention that within this new approach, one only needs to calculate (and store) the wave functions of the N_{0} low energy states plus selected highenergy conduction states at or near predefined energy grid points. The foldedspectrum method developed by Wang and Zunger^{31} and an alternative approach proposed by Tackett and Ventra^{32} are ideal for this purpose. Using these methods, one can efficiently calculate KS eigenfunctions at or near specified energy grid points without the need to calculate all KS states. Therefore, the speedup factor shown in Table 1 also indicates the reduction in the required memory and disk space associated with storing the wave functions in GW calculations.
The most interesting and important aspect of our method is that the speedup factor actually increases with increasing system size, enabling fully converged GW calculations for large systems containing hundreds of atoms. This is because with a given integration step size ΔE_{i}, the weight of a single integration point, g(E_{i})ΔE_{i} in Eq.5, which measures effectively the number of states contributing to the summation represented by a single state at E_{i}, scales linearly with the system size. To demonstrate that our method can indeed be scaled up for large systems, we have carried out GW calculations for MgO systems containing 2 to 256 atoms.
Table 2 shows the performance of our method, both in terms of accuracy and the speedup factor. A speed up factor of over 80 times is achieved for the largest system containing 256 atoms (1,024 valence electrons), and the numerical integration error, measured by the calculated band gap, is only ±0.03 eV for all systems. Thus the greater speedup for larger systems does not compromise the accuracy of this method, and there is still room for improvement (for example, with improved numerical integration techniques). We would like to mention that such fully converged GW calculations for large oxide systems containing over 200 atoms would be extremely difficult using the conventional approach due to the vast amount of computational resources required (both in terms of the computational time and the memory and storage requirements).
Applications to 2D materials
GW calculations for 2D materials pose additional challenges due to the analytical behavior of the 2D electronic screening, which is different from 3 dimensional (3D) systems, and the need to remove interactions between period images^{33,34,35,36}. The rapid change in the 2D dielectric function as requires rather dense kpoint sampling even with clever integration techniques^{33,36}; the spurious interlayer interaction can be reduced with increasing interlayer distance and with the use of truncated Coulomb potential^{33,34}. Unfortunately, even with these creative methods, fully converged GW calculations for 2D materials remain a serious challenge since the number of conduction bands required in the calculation scales linearly with cell volume. For example, Qiu et al. concluded that at least 6,000 bands are needed for properly converged GW results of monolayer MoS_{2}.
Our method can be directly applied to 2D materials in the same way as 3D bulk materials. Using this new method, we thoroughly studied the convergence behavior of monolayer MoSe_{2}. The left panel of Fig. 4 shows the convergence behavior of the calculated COH selfenergy for the top of the valence band of MoSe_{2} as a function of the number of conduction bands and the kinetic energy cutoff of the dielectric function. Depending on the the choices of these parameters, the calculated Σ_{COH} varies from less than −11.5 eV to about −9.5 eV. The right panel of Fig. 4 shows a similar convergence behavior of calculated quasiparticle band gap E_{g} of MoSe_{2}. The variation in band gap, although not as large as that in the COH enegy due to cancellation of errors between valence and conduction bands, is still significant. These results again demonstrate the importance of the band convergence issue in GW calculations, and often seemingly conflicting GW results can be traced to the convergence issue.
We mention that GW calculations for 2D materials usually converge much slower with respect to kpoint sampling than 3D systems^{33,35}. if one employs truncated Coulomb potentials, which is usually required to avoid spurious influences from periodic images. As shown in Fig. 5, for monolayer MoS_{2} and MoSe_{2}, one has to increase the kgrid sampling to at least 18 × 18 × 1 in order to converge the quasiparticle band gap. In contrast, 6 × 6 × 6 kgrids are usually sufficient for typical bulk materials with small unit cells containing a few atoms. Our calculations make use of the truncation technique proposed by IsmailBeigi^{33}. This issue significantly increases the computational costs for GW calculations for 2D systems and makes the convergence test even more trickier and difficult due to possible cancellations of errors arising from various cutoff parameters. Our new method enables including effectively all conduction bands in the GW calculations with greatly reduced computational costs (by one to two orders of magnitude) compared with conventional methods. Using this new method, we have calculated GW band gap of freestanding monolayer MoSe_{2} to be 2.28 eV, consistent with previously reported value 2.26 eV^{15}, and the band gap of monolayer MoS_{2} to be 2.64 eV, which is also in good agreement with a previously calculation^{37}. These previous results were obtained with highly converged (but computationally expensive) conventional GW calculations.
Conclusion
In conclusion, we present an efficient integration method that can drastically speed up fully converged GW quasiparticle calculations for large systems. Our method takes advantage of the fact that the DOS of all materials resembles that of the free electron gas at high energies, thus the summation over highenergy conduction bands in both the dielectric function and selfenergy calculations within the GW approximation can be well approximated by a numerical integration on a sparse energy grid. Using this new method, we can now carry out fully converged GW calculations for large systems without the need to perform tedious convergence tests. We have demonstrated a nearly two orders of magnitude speedup of GW calculations for a 256atom MgO model system while achieving a numerical accuracy of ±0.03 eV for the predicted band gap. Although we have presented results calculated within the socalled G^{0}W^{0} approximation and using the HLGPP model for the dielectric function, we believe that our approach can be applied to various levels of selfconsistent GW calculations and/or GW calculations without the use of plasmonpole models. Our new method also works equally well for 2D systems. Our approach is conceptually simple and is very easy to be implemented in welldeveloped GW packages.
Methods
All GW results presented in this work are calculated within the G^{0}W^{0} approach using a modified version of the BerkeleyGW^{16} package. The HyberstenLouie generalized plasmonpole (HLGPP) model^{5} is used to extend the static dielectric function to finite frequencies. The new method presented in this work, however, can be applied to other GW implementations including various selfconsistent GW approaches. The DFT calculations within the LDA are carried out using the PARATEC^{38} code. Normconserving TroullierMartin pseudopotentials^{39} are used. The kinetic energy cutoff for the planewave expansion of the KS wave functions is set at 80 Ry for MgO, 300 Ry for ZnO, and 125 Ry for monolayer MoSe_{2} and MoS_{2}. Experimental crystal structures are used for all calculations. For monolayer MoSe_{2} and MoS_{2}, the interlayer distance is kept at 25 Å. All semicore electrons (4s and 4p for Mo and 3s and 3p for Zn) are included in the calculations. The GW quasiparticle band structures for MgO 2atom cell are obtained using a Wannier interpolation technique^{40} based on results calculated on a 8 × 8 × 8 kgrid.
Additional Information
How to cite this article: Gao, W. et al. Speeding up GW Calculations to Meet the Challenge of Large Scale Quasiparticle Predictions. Sci. Rep. 6, 36849; doi: 10.1038/srep36849 (2016).
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
Hohenberg, P. & Kohn, W. Inhomogeneous electron gas. Phys. Rev. 136, B864–B871 (1964).
Kohn, W. & Sham, L. J. Selfconsistent equations including exchange and correlation effects. Phys. Rev. 140, A1133–A1138 (1965).
Perdew, J. P. & Yue, W. Accurate and simple density functional for the electronic exchange energy: Generalized gradient approximation. Phys. Rev. B 33, 8800–8802 (1986).
Hedin, L. New method for calculating the oneparticle green’s function with application to the electrongas problem. Phys. Rev. 139, A796–A823 (1965).
Hybertsen, M. S. & Louie, S. G. Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies. Phys. Rev. B 34, 5390–5413 (1986).
Godby, R. W., Schlüter, M. & Sham, L. J. Selfenergy operators and exchangecorrelation potentials in semiconductors. Phys. Rev. B 37, 10159–10175 (1988).
Bruneval, F. & Gonze, X. Accurate gw selfenergies in a planewave basis using only a few empty states: Towards large systems. Phys. Rev. B 78, 085125 (2008).
Berger, J. A., Reining, L. & Sottile, F. Ab initio calculations of electronic excitations: Collapsing spectral sums. Phys. Rev. B 82, 041103 (2010).
Shih, B.C., Xue, Y., Zhang, P., Cohen, M. L. & Louie, S. G. Quasiparticle band gap of zno: High accuracy from the conventional G^{0}W^{0} approach. Phys. Rev. Lett. 105, 146401 (2010).
Samsonidze, G., Jain, M., Deslippe, J., Cohen, M. L. & Louie, S. G. Simple approximate physical orbitals for gw quasiparticle calculations. Phys. Rev. Lett. 107, 186404 (2011).
Deslippe, J., Samsonidze, G., Jain, M., Cohen, M. L. & Louie, S. G. Coulombhole summations and energies for gw calculations with limited number of empty orbitals: A modified static remainder approach. Phys. Rev. B 87, 165124 (2013).
Friedrich, C., Müller, M. C. & Blügel, S. Band convergence and linearization error correction of allelectron GW calculations: The extreme case of zinc oxide. Phys. Rev. B 83, 081101 (2011).
Stankovski, M. et al. G^{0}W^{0} band gap of zno: Effects of plasmonpole models. Phys. Rev. B 84, 241201 (2011).
Qiu, D. Y., da Jornada, F. H. & Louie, S. G. Optical spectrum of mos2: Manybody effects and diversity of exciton states. Phys. Rev. Lett. 111, 216805 (2013).
Ugeda, M. M. et al. Giant bandgap renormalization and excitonic effects in a monolayer transition metal dichalcogenide semiconductor. Nature Material 13, 1091–1095 (2014). Letter.
Deslippe, J. et al. Berkeleygw: A massively parallel computer package for the calculation of the quasiparticle and optical properties of materials and nanostructures. Computer Physics Communications 183, 1269–1289 (2012).
MolinaSánchez, A., Sangalli, D., Hummer, K., Marini, A. & Wirtz, L. Effect of spinorbit interaction on the optical spectra of singlelayer, doublelayer, and bulk mos2 . Phys. Rev. B 88, 045412 (2013).
Whited, R. C. & Walker, W. C. Exciton spectra of cao and mgo. Phys. Rev. Lett. 22, 1428–1430 (1969).
Umari, P., Stenuit, G. & Baroni, S. Optimal representation of the polarization propagator for largescale gw calculations. Phys. Rev. B 79, 201104 (2009).
Caruso, F., Rinke, P., Ren, X., Rubio, A. & Scheffler, M. Selfconsistent gw: Allelectron implementation with localized basis functions. Phys. Rev. B 88, 075105 (2013).
Pham, T. A., Nguyen, H.V., Rocca, D. & Galli, G. Gw calculations using the spectral decomposition of the dielectric matrix: Verification, validation, and comparison of methods. Phys. Rev. B 87, 155148 (2013).
Tiago, M. L. & Chelikowsky, J. R. Optical excitations in organic molecules, clusters, and defects studied by firstprinciples green’s function methods. Phys. Rev. B 73, 205334 (2006).
Kang, W. & Hybertsen, M. S. Enhanced static approximation to the electron selfenergy operator for efficient calculation of quasiparticle energies. Phys. Rev. B 82, 195108 (2010).
Giustino, F., Cohen, M. L. & Louie, S. G. Gw method with the selfconsistent sternheimer equation. Phys. Rev. B 81, 115105 (2010).
Lambert, H. & Giustino, F. Ab initio sternheimergw method for quasiparticle calculations using plane waves. Phys. Rev. B 88, 075117 (2013).
Govoni, M. & Galli, G. Large scale gw calculations. Journal of Chemical Theory and Computation 11, 2680–2696 (2015).
Soininen, J. A., Rehr, J. J. & Shirley, E. L. Electron selfenergy calculation using a general multipole approximation. Journal of Physics: Condensed Matter 15, 2573 (2003).
Laflamme Janssen, J., Rousseau, B. & Côté, M. Efficient dielectric matrix calculations using the lanczos algorithm for fast manybody G0W0 implementations. Phys. Rev. B 91, 125120 (2015).
Neuhauser, D. et al. Breaking the theoretical scaling limit for predicting quasiparticle energies: The stochastic gw approach. Phys. Rev. Lett. 113, 076402 (2014).
Schäfer, T., Toschi, A. & Tomczak, J. M. Separability of dynamical and nonlocal correlations in three dimensions. Phys. Rev. B 91, 121107 (2015).
Wang, L. & Zunger, A. Solving schrödinger’s equation around a desired energy: Application to silicon quantum dots. The Journal of Chemical Physics 100, 2394–2397 (1994).
Tackett, A. R. & Ventra, M. D. Targeting specific eigenvectors and eigenvalues of a given hamiltonian using arbitrary selection criteria. Phys. Rev. B 66, 245104 (2002).
IsmailBeigi, S. Truncation of periodic image interactions for confined systems. Phys. Rev. B 73, 233103 (2006).
Rozzi, C. A., Varsano, D., Marini, A., Gross, E. K. U. & Rubio, A. Exact coulomb cutoff technique for supercell calculations. Phys. Rev. B 73, 205119 (2006).
Hüser, F., Olsen, T. & Thygesen, K. S. How dielectric screening in twodimensional crystals affects the convergence of excitedstate calculations: Monolayer mos2 . Phys. Rev. B 88, 245309 (2013).
Rasmussen, F. A. & Thygesen, K. S. Efficient gw calculations on twodimensional materials using accurate long wavelength limit of the screened potential. arXiv:1511.00129 (2015).
Qiu, D. Y., da Jornada, F. H. & Louie, S. G. Erratum: Optical spectrum of mos2: Manybody effects and diversity of exciton states [phys. rev. lett.111, 216805 (2013)]. Phys. Rev. Lett. 115, 119901 (2015).
Paratec. http://www.nersc.gov/users/software/applications/materialsscience/paratec/ (2015).
Troullier, N. & Martins, J. L. Efficient pseudopotentials for planewave calculations. Phys. Rev. B 43, 1993–2006 (1991).
Mostofi, A. A. et al. An updated version of wannier90: A tool for obtaining maximallylocalised wannier functions. Computer Physics Communications 185, 2309–2310 (2014).
Acknowledgements
We thank Steven G. Louie for his helpful discussions. This work is supported in part by the US NSF (Grant No. DMR1506669 and DMR1626967) and NSFC (No. 11328401 and 11628407). We acknowledge the computational support provided by the Center for Computational Research at the University at Buffalo, SUNY, the Beijing Computational Science Research Center, the National Supercomputer Center in Guangzhou (NSCCGZ) and the Special Program for Applied Research on Super Computation of the NSFCGuangdong Joint Fund (the second phase).
Author information
Authors and Affiliations
Contributions
W.G. was responsible for most of the code development and calculations. P.Z. supervised the study. W.G. and P.Z. drafted the manuscript. All authors participated in the discussion and contributed to the manuscript writing.
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Gao, W., Xia, W., Gao, X. et al. Speeding up GW Calculations to Meet the Challenge of Large Scale Quasiparticle Predictions. Sci Rep 6, 36849 (2016). https://doi.org/10.1038/srep36849
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/srep36849
This article is cited by

Prediction of protected band edge states and dielectric tunable quasiparticle and excitonic properties of monolayer MoSi2N4
npj Computational Materials (2022)

Combined subsampling and analytical integration for efficient largescale GW calculations for 2D systems
npj Computational Materials (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.