Science China Mathematics

, Volume 59, Issue 8, pp 1593–1612 | Cite as

Low rank approximation in G 0 W 0 calculations

  • MeiYue Shao
  • Lin Lin
  • Chao YangEmail author
  • Fang Liu
  • Felipe H. Da Jornada
  • Jack Deslippe
  • Steven G. Louie


The single particle energies obtained in a Kohn-Sham density functional theory (DFT) calculation are generally known to be poor approximations to electron excitation energies that are measured in transport, tunneling and spectroscopic experiments such as photo-emission spectroscopy. The correction to these energies can be obtained from the poles of a single particle Green’s function derived from a many-body perturbation theory. From a computational perspective, the accuracy and efficiency of such an approach depends on how a self energy term that properly accounts for dynamic screening of electrons is approximated. The G 0 W 0 approximation is a widely used technique in which the self energy is expressed as the convolution of a noninteracting Green’s function (G 0) and a screened Coulomb interaction (W 0) in the frequency domain. The computational cost associated with such a convolution is high due to the high complexity of evaluating W 0 at multiple frequencies. In this paper, we discuss how the cost of G 0 W 0 calculation can be reduced by constructing a low rank approximation to the frequency dependent part of W 0. In particular, we examine the effect of such a low rank approximation on the accuracy of the G 0 W 0 approximation. We also discuss how the numerical convolution of G 0 and W 0 can be evaluated efficiently and accurately by using a contour deformation technique with an appropriate choice of the contour.


density functional theory G0W0 approximation Sternheimer equation contour deformation low rank approximation 


65F15 65Z05 


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  1. 1.
    Aryasetiawan F, Gunnarsson O. The GW method. Rep Prog Phys, 1998, 61: 237–312CrossRefGoogle Scholar
  2. 2.
    Baroni S, de Gironcoli S, Corso A D. Phonons and related crystal properties from density-functional perturbation theory. Rev Mod Phys, 2001, 73: 515–562CrossRefGoogle Scholar
  3. 3.
    Baroni S, Giannozzi P, Testa A. Green’s-function approach to linear response in solids. Phys Rev Lett, 1987, 58: 1861–1864CrossRefGoogle Scholar
  4. 4.
    Bruneval F. Exchange and correlation in the electronic structure of solids: GW approximation and beyond. PhD thesis. Palaiseau: Ecole Polytechnique, 2005Google Scholar
  5. 5.
    Bruneval F, Gonze X. Accurate GW self-energies in a plane-wave basis using only a few empty states: Towards large systems. Phys Rev B, 2008, 78: 085125CrossRefGoogle Scholar
  6. 6.
    Cancès E, Gontier D, Stoltz G. A mathematical analysis of the GW 0 method for computing electronic excited energies of molecules. ArXiv:1506.01737, 2015zbMATHGoogle Scholar
  7. 7.
    Casida M E. Time-dependent density functional response theory for molecules. In: Recent Advances in Density Functional Methods. Singapore: World Scientific, 1995, 155–192CrossRefGoogle Scholar
  8. 8.
    Demmel J W. Applied Numerical Linear Algebra. Philadelphia: SIAM, 1997CrossRefzbMATHGoogle Scholar
  9. 9.
    Deslippe J, Samsonidze G, Strubbe D A, et al. Berkeley GW: A massively parallel computer package for the calculation of the quasiparticle and optical properties of materials and nanostructures. Comput Phys Commun, 2012, 183: 1269–1289CrossRefGoogle Scholar
  10. 10.
    Farid B. Ground and low-lying excited states of interacting electron systems: a survey and some critical analyses. In: Electron Correlation in the Solid State. Singapore: World Scientific, 1999, 103–261CrossRefGoogle Scholar
  11. 11.
    Foerster D, Koval P, Sánchez-Portal D. An O(N3) implementation of Hedin’s GW approximation for molecules. J Chem Phys, 2011, 135: 074105CrossRefGoogle Scholar
  12. 12.
    Freund R W, Nachtigal N M. QMR: A quasi-minimal residual method for non-Hermitian linear systems. Numer Math, 1991, 60: 315–339MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Giustino F, Cohen M L, Louie S G. GW method with the self-consistent Sternheimer equation. Phys Rev B, 2010, 81: 115105CrossRefGoogle Scholar
  14. 14.
    Godby R W, Schlüter M, Sham L J. Self-energy operators and exchange-correlation potentials in semiconductors. Phys Rev B, 1988, 37: 10159CrossRefGoogle Scholar
  15. 15.
    Govoni M, Galli G. Large scale GW calculations. J Chem Theory Comput, 2015, 11: 2680–2696CrossRefGoogle Scholar
  16. 16.
    Hedin L. New method for calculating the one-particle Green’s function with application to the electron-gas problem. Phys Rev, 1965, 139: A796–A823CrossRefGoogle Scholar
  17. 17.
    Horn R A, Johnson C R. Topics in Matrix Analysis. New York: Cambridge University Press, 1991CrossRefzbMATHGoogle Scholar
  18. 18.
    Hybertsen M S, Louie S G. Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies. Phys Rev B, 1986, 34: 5390–5413CrossRefGoogle Scholar
  19. 19.
    Knyazev A V. Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method. SIAM J Sci Comput, 2001, 23: 517–541MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lebègue S, Arnaud B, Alouani M, et al. Implementation of an all-electron GW approximation based on the projector augmented wave method without plasmon pole approximation: Application to Si, SiC, AlAs, InAs, NaH, and KH. Phys Rev B, 2003, 67: 155208CrossRefGoogle Scholar
  21. 21.
    Liu F, Lin L, Vigil-Fowler D, et al. Numerical integration for ab initio many-electron self energy calculations within the GW approximation. J Comput Phys, 2015, 286:1–13MathSciNetCrossRefGoogle Scholar
  22. 22.
    Nguyen H-V, Pham T A, Rocca D, et al. Improving accuracy and efficiency of calculations of photoemission spectra within the many-body perturbation theory. Phys Rev B, 2012, 85: 081101CrossRefGoogle Scholar
  23. 23.
    Paige C C, Saunders M A. Solution of sparse indefinite systems of linear equations. SIAM J Numer Anal, 1975, 12: 617–629MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Parks M L, de Sturler E, Mackey G, et al. Recycling Krylov subspaces for sequences of linear systems. SIAM J Sci Comput, 2006, 28: 1651–1674MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ping Y, Rocca D, Galli G. Electronic excitations in light absorbers for photoelectrochemical energy conversion: First principles calculations based on many body perturbation theory. Chem Soc Rev, 2013, 42: 2437–2469CrossRefGoogle Scholar
  26. 26.
    Ren X, Rinke P, Blum V, et al. Resolution-of-identity approach to Hartree-Fock, hybrid density functionals, RPA, MP2, and GW with numeric atom-centered orbital basis. New J Phys, 2012, 14: 053020CrossRefGoogle Scholar
  27. 27.
    Saad Y. Numerical Methods for Large Eigenvalue Problems: Revised Edition. Classics in Applied Mathematics. Philadelphia: SIAM, 2011Google Scholar
  28. 28.
    Saad Y, Schultz M H. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput, 1986, 7: 856–869MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Shishkin M, Kresse G. Implementation and performance of the frequency-dependent GW method within the PAW framework. Phys Rev B, 2006, 74: 035101CrossRefGoogle Scholar
  30. 30.
    Sogabe T, Hoshi T, Zhang S-L, et al. Solution of generalized shifted linear systems with complex symmetric matrices. J Comput Phys, 2012, 231: 5669–5684MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Soodhalter K M, Szyld D B, Xue F. Krylov subspace recycling for sequences of shifted linear systems. Appl Numer Math, 2014, 81: 105–118MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Stewart G W. On the early history of the singular value decomposition. SIAM Rev, 1993, 35: 551–566MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Tiago M L, Chelikowsky J R. Optical excitations in organic molecules, clusters, and defects studied by first-principles Green’s function methods. Phys Rev B, 2006, 73: 205334CrossRefGoogle Scholar
  34. 34.
    Umari P, Stenuit G, Baroni S. GW quasiparticle spectra from occupied states only. Phys Rev B, 2010, 81: 115104CrossRefGoogle Scholar
  35. 35.
    van der Vorst H A. Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J Sci Stat Comput, 1992, 13: 631–664MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    van Setten M J, Weigend F, Evers F. The GW-method for quantum chemistry applications: Theory and implementation. J Chem Theory Comput, 2013, 9: 232–246CrossRefGoogle Scholar
  37. 37.
    Yang C, Meza J C, Lee B, et al. KSSOLV—a MATLAB toolbox for solving the Kohn-Sham equations. ACM Trans Math Software, 2009, 36: Article 10MathSciNetCrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • MeiYue Shao
    • 1
  • Lin Lin
    • 1
    • 2
  • Chao Yang
    • 1
    Email author
  • Fang Liu
    • 3
  • Felipe H. Da Jornada
    • 4
    • 5
  • Jack Deslippe
    • 6
  • Steven G. Louie
    • 4
    • 5
  1. 1.Computational Research DivisionLawrence Berkeley National LaboratoryBerkeleyUSA
  2. 2.Department of MathematicsUniversity of California at BerkeleyBerkeleyUSA
  3. 3.School of Statistics and MathematicsCentral University of Finance and EconomicsBeijingChina
  4. 4.Department of PhysicsUniversity of California at BerkeleyBerkeleyUSA
  5. 5.Materials Science DivisionLawrence Berkeley National LaboratoryBerkeleyUSA
  6. 6.NERSCLawrence Berkeley National LaboratoryBerkeleyUSA

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