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GW Calculations for Molecules and Solids: Theory and Implementation in CP2K GW calculations for molecules and solids: Theory and implementation in CP2K Jan Wilhelm [email protected] 13 July 2017 Jan Wilhelm GW calculations for molecules and solids 13 July 2017 1 / 41 Overview 1 Theory and practical G0W0 scheme 2 Physics of the GW approximation 3 Benchmarks and applications of G0W0 4 Canonical G0W0: Implementation in CP2K and input 5 Periodic G0W0 calculations: Correction scheme and input 6 Cubic-scaling G0W0: Formalism, implementation and input 7 Summary Jan Wilhelm GW calculations for molecules and solids 13 July 2017 2 / 41 1 Theory and practical G0W0 scheme 2 Physics of the GW approximation 3 Benchmarks and applications of G0W0 4 Canonical G0W0: Implementation in CP2K and input 5 Periodic G0W0 calculations: Correction scheme and input 6 Cubic-scaling G0W0: Formalism, implementation and input 7 Summary Jan Wilhelm GW calculations for molecules and solids 13 July 2017 3 / 41 Quasiparticle energies in GW : Theory Definition: A quasiparticle energy "n is defined as energy which is needed to remove an electron from the system to the vacuum or Energy is gained if one places an electron from the vacuum to the system In DFT and Hartree-Fock, there is no theoretical foundation that "vac =0 the eigenvalues "n from an SCF, "LUMO+1 2 ! r + vel-core(r) + vH(r) + vxc(r) n(r) = "n n(r) − 2 "LUMO have anything to do with quasiparticle energies. Theorem: A self-energy Σ(r; r0;" ) (non-local, energy-dependent) containing "HOMO exchange and correlation effects exists, such that the solution of 2 ! Z "HOMO-1 −∇ 0 0 0 + vel-core(r) + vH(r) n(r) + dr Σ(r; r ;"n) n(r ) = "n n(r) 2 Single-electron (quasiparticle) levels of gives the correct quasiparticle energies "n of the interacting a closed shell molecule many-electron system. In the GW approximation, the self-energy reads 1 Z GW 0 i 0 0 0 0 0 Σ (r; r ;") = d" G(r; r ;" " ; "n ; n ) W (r; r ;" ; "n ; n ) 2π − f g f g f g f g −∞ Jan Wilhelm GW calculations for molecules and solids 13 July 2017 4 / 41 Quasiparticle energies in GW : G0W0 formalism in practice DFT G0W0: Start from DFT MOs n (r) and compute first-order correction to DFT eigenvalues: 1 Converge DFT SCF (e.g. PBE functional for solids "G0W0@PBE", PBE0 for molecules) 2 DFT DFT DFT r + vel-core(r) + vHartree(r) + vxc(r) (r) = " (r) : − 2 n n n 2 Compute density response (most expensive step): occ virt "DFT "DFT 0 X X DFT 0 DFT 0 DFT DFT i a 4 χ(r; r ; i!) = 2 a (r ) i (r ) i (r) a (r) − : ( (N )) !2 + "DFT "DFT2 O i a i − a 3 Compute dielectric function with v(r; r0)=1= r r0 jZ− j (r; r0; i!) = δ(r; r0) dr00v(r; r00)χ(r00; r0; i!) : ( (N3)) − O 4 Compute screened Coulomb interaction Z W (r; r0; i!) = dr00−1(r; r00; i!)v(r00; r0) : ( (N3)) 0 O 5 Compute the self-energy 1 all Z d!0 X DFT(r0) DFT(r) Σ( ; 0; !) = ( ; 0; ! !0) ( ; 0; !0) ; ( ; 0; !) = m m : ( 3) r r i G0 r r i i W0 r r i G0 r r i DFT ( N ) − 2π − i! + "F "m O −∞ m − 6 Compute G0W0 quasiparticle energies (replace wrong XC from DFT by better XC from GW ) "G0W0 = "DFT + DFT Re Σ("G0W0 ) v xc DFT ( (N3)) n n h n j n − j n i O Jan Wilhelm GW calculations for molecules and solids 13 July 2017 5 / 41 Historical sketch of GW 1965: Proposition of the GW method Lars Hedin: New Method for Calculating the One-Particle Green’s Function with Application to the Electron-Gas Problem, Phys. Rev. 139, A796 (1965), ∼ 3700 citations 1986: First G0W0@LDA calculation for diamond, Si, Ge, and LiCl M. S. Hybertsen and S. G. Louie: Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies, Phys. Rev. B 34, 5390 (1986), ∼ 2700 citations 2005 – now: GW for solids in publicly available plane-waves codes Abinit: X. Gonze et al., Z. Kristallogr. 220, 558 – 562 (2005) VASP: M. Shishkin and G. Kresse, Phys. Rev. B 74, 035101 (2006) Yambo: A. Marini, C. Hogan, M. Grüning, D. Varsano, Comput. Phys. Commun. 180, 1392 – 1403 (2009) BerkeleyGW: J. Deslippe et al., Comput. Phys. Commun. 183, 1269 – 1289 (2012) GPAW: F. Hüser, T. Olsen, and K. S. Thygesen, Phys. Rev. B 87, 235132 (2013) WEST: M. Govoni and G. Galli, J. Chem. Theory Comput. 11, 2680 –2696 (2015) 2011 – now: GW with localized basis in publicly available codes FHI-aims: X. Ren et al., New J. Phys. 14, 053020 (2012) Turbomole: M. van Setten, F. Weigend, and F. Evers, J. Chem. Theory Comput. 9, 232 – 246 (2012) molgw: F. Bruneval et al., Comput. Phys. Commun. 208, 149 – 161 (2016) CP2K: J. Wilhelm, M. Del Ben, and J. Hutter, J. Chem. Theory Comput. 12, 3623 – 3635 (2016) Recent trend: Numerically converged results and agreement between codes J. Klimeš, M. Kaltak, and G. Kresse: Predictive GW calculations using plane waves and pseudopotentials, Phys. Rev. B 90, 075125 (2014) M. van Setten et al.: GW 100: Benchmarking G0W0 for Molecular Systems, JCTC 11, 5665 – 5687 (2015) Jan Wilhelm GW calculations for molecules and solids 13 July 2017 6 / 41 1 Theory and practical G0W0 scheme 2 Physics of the GW approximation 3 Benchmarks and applications of G0W0 4 Canonical G0W0: Implementation in CP2K and input 5 Periodic G0W0 calculations: Correction scheme and input 6 Cubic-scaling G0W0: Formalism, implementation and input 7 Summary Jan Wilhelm GW calculations for molecules and solids 13 July 2017 7 / 41 Hedin’s equations Hedin’s equation: Complicated self-consistent equations which give the exact self-energy. Notation: (1) = (r1; t1), G0: non-interacting Green’s function, e.g. from DFT Z Self-energy: Σ(1; 2) = i d(34)G(1; 3)Γ(3; 2; 4)W (4; 1+) Z Green’s function: G(1; 2) = G0(1; 2) + d(34)G0(1; 3)Σ(3; 4)G(4; 2) Z Screened interaction: W (1; 2) = V (1; 2) + d(34)V (1; 3)P(3; 4)W (4; 2) Bare interaction: V (1; 2) = δ(t t )= r r 1 − 2 j 1 − 2j Z Polarization: P(1; 2) = i d(34)G(1; 3)G(4; 1+)Γ(3; 4; 2) − Z @Σ(1; 2) Vertex function: Γ(1; 2; 3) = δ(1; 2)δ(1; 3) + d(4567) G(4; 6)G(7; 5)Γ(6; 7; 3) @G(4; 5) It can be shown that Σ(r ; t ; r ; t )=Σ(r ; r ; t t ). After a Fourier transform of Σ from 1 1 2 2 1 2 2 − 1 time t t t to frequency (= energy), the self-energy Σ(r; r0;!) can be used to compute ≡ 2 − 1 the quasiparticle levels "n using 2 ! Z 0 0 0 −∇ + vel-core(r) + vH(r) n(r) + dr Σ(r; r ;"n) n(r ) = "n n(r) 2 Jan Wilhelm GW calculations for molecules and solids 13 July 2017 8 / 41 Hedin’s equations: Hartree-Fock Hartree-Fock is GV : Z Self-energy: Σ(1; 2) = i d(34)G(1; 3)Γ(3; 2; 4)W (4; 1+)= G(1; 2)V (2; 1) Z Green’s function: G(1; 2) = G0(1; 2) + d(34)G0(1; 3)Σ(3; 4)G(4; 2) Z Screened interaction: W (1; 2) = V (1; 2) + d(34)V (1; 3)P(X3;X4)X!0W (4; 2)= V (1; 2) Bare interaction: V (1; 2) = δ(t t )= r r 1 − 2 j 1 − 2j ((h hZ hh (( hh (((+ Polarization: P(1; 2) = i d(34)(G((1; (h3)Gh(4h; 1h)Γ(3; 4; 2)= 0 (− ((( hhhh h ((h Z hhh ((( @Σ(h1h; 2)(h((( Vertex function: Γ(1; 2; 3) = δ(1; 2)δ(1; 3) + d(4567)((( G(h4;h6)Gh(7h; 5)Γ(6; 7; 3) (((( @G(4; 5) hhhh Jan Wilhelm GW calculations for molecules and solids 13 July 2017 9 / 41 Hedin’s equations: GW Z Self-energy: Σ(1; 2) = i d(34)G(1; 3)Γ(3; 2; 4)W (4; 1+)= iG(1; 2)W (2; 1+) Z Green’s function: G(1; 2) = G0(1; 2) + d(34)G0(1; 3)Σ(3; 4)G(4; 2) Z Screened interaction: W (1; 2) = V (1; 2) + d(34)V (1; 3)P(3; 4)W (4; 2) Bare interaction: V (1; 2) = δ(t t )= r r 1 − 2 j 1 − 2j Z X Polarization: P(1; 2) = i d(34)G(1; 3)G(4; 1+)Γ(3X; 4X; 2)= G(1; 2)G(2; 1+) − X hh (((h Z hh@Σ( ; ) ((( h1h2 (hh(( Vertex function: Γ(1; 2; 3) = δ(1; 2)δ(1; 3) + d(4567)((( G(4;h6)Gh(7h; 5)Γ(6; 7; 3) (((( @G(4; 5) hhhh Jan Wilhelm GW calculations for molecules and solids 13 July 2017 10 / 41 Screening In GW , the screened Coulomb interaction is appearing: Z W (r; r0;!) = dr00−1(r; r00;!)v(r00; r0) where v(r; r0) = 1= r r0 . j − j Compare to screened Coulomb potential with local and static [(r; r0;!) = δ(r r0)] r − dielectric function using the dielectric constant r (in SI units): 1 1 W (r; r0) = 4π r r0 0 r j − j screening charge which has been added to the system internal mobile charge carriers (e.g.
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