The GW Approximation. Lucia Reining, Fabien Bruneval
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1 , Faben Bruneval Laboratore des Soldes Irradés Ecole Polytechnque, Palaseau - France European Theoretcal Spectroscopy Faclty (ETSF) Belfast, June 2007
2 Outlne 1 Remnder 2 GW approxmaton 3 GW n practce 4 Easer? 5 More complcated? 6 More results 7 References
3 Outlne 1 Remnder 2 GW approxmaton 3 GW n practce 4 Easer? 5 More complcated? 6 More results 7 References
4 Towards Hedn s equatons rreducble vertex screened Coulomb nteracton delectrc functon rreducble polarzablty Σ = Gvε 1 Γ 1 δg Γ = δv = 1 + δσ GG Γ δg W = ɛ 1 v ɛ = 1 v χ χ = δρ δv = GG Γ
5 Hedn s equatons Σ = GW Γ Γ = 1 + δσ GG Γ δg W = ɛ 1 v ɛ = 1 v χ χ = GG Γ Σ W G Hedn s wheel χ ~ ~ Γ
6 Outlne 1 Remnder 2 GW approxmaton 3 GW n practce 4 Easer? 5 More complcated? 6 More results 7 References
7 Hedn s equatons Σ = GW Γ Γ = 1 + δσ GG Γ δg W = ɛ 1 v ɛ = 1 v χ χ = GG Γ Σ W G Σ (0) = 0 Γ (1) = 1 χ (1) = GG = χ RPA Σ (1) = GW W G Hedn s wheel χ ~ ~ Γ
8 Hedn s equatons Σ = GW Γ Γ = 1 + δσ GG Γ δg W = ɛ 1 v ɛ = 1 v χ χ = GG Γ Σ W G Σ (0) = 0 Γ (1) = 1 χ (1) = GG = χ RPA Σ (1) = GW W G Hedn s wheel χ ~ ~ Γ
9 GW orgns
10 Physcs of the GW approxmaton, I Splttng of the screened Coulomb nteracton: W (ω) = ɛ 1 (ω)v = (1 + vχ(ω))v = v + W p (ω) Splttng of the self-energy: Σ(ω) = GW (ω) = Gv + GW p (ω) = Σ x + Σ c (ω) Screenng beyond Hartree Fock
11 Physcs of the GW approxmaton, II Add a charge - relaxaton? Not f you smear t out n a Bloch functon! What do we add to the system? And where? δ(r r 0 ) Coulomb hole: 0.5(W (r 0, r 0 ) v(r 0, r 0 )) SCF n small systems! Add screened exchange: COHSEX approxmaton.
12 Physcs of the GW approxmaton, II Add a charge - relaxaton? Not f you smear t out n a Bloch functon! What do we add to the system? And where? δ(r r 0 ) Coulomb hole: 0.5(W (r 0, r 0 ) v(r 0, r 0 )) SCF n small systems! Add screened exchange: COHSEX approxmaton.
13 Physcs of the GW approxmaton, II Add a charge - relaxaton? Not f you smear t out n a Bloch functon! What do we add to the system? And where? δ(r r 0 ) Coulomb hole: 0.5(W (r 0, r 0 ) v(r 0, r 0 )) SCF n small systems! Add screened exchange: COHSEX approxmaton.
14 Physcs of the GW approxmaton, II Add a charge - relaxaton? Not f you smear t out n a Bloch functon! What do we add to the system? And where? δ(r r 0 ) Coulomb hole: 0.5(W (r 0, r 0 ) v(r 0, r 0 )) SCF n small systems! Add screened exchange: COHSEX approxmaton.
15 Physcs of the GW approxmaton, II Add a charge - relaxaton? Not f you smear t out n a Bloch functon! What do we add to the system? And where? δ(r r 0 ) Coulomb hole: 0.5(W (r 0, r 0 ) v(r 0, r 0 )) SCF n small systems! Add screened exchange: COHSEX approxmaton.
16 Physcs of the GW approxmaton, II Add a charge - relaxaton? Not f you smear t out n a Bloch functon! What do we add to the system? And where? δ(r r 0 ) Coulomb hole: 0.5(W (r 0, r 0 ) v(r 0, r 0 )) SCF n small systems! Add screened exchange: COHSEX approxmaton.
17 Outlne 1 Remnder 2 GW approxmaton 3 GW n practce 4 Easer? 5 More complcated? 6 More results 7 References
18 Full GW calculaton Calculate the GW self-energy: Σ(1, 2) = G(1, 2)W (1 +, 2) whch s Fourer transformed to frequences Σ(r 1, r 2, ω) = dω G(r 1, r 2, ω + ω )W (r 1, r 2, ω )
19 Schematc GW calculaton Start s recpe by Hybertsen and Loue, PRL (1985) called G 0 W 0 or best G best W
20 Schematc GW calculaton Start s recpe by Hybertsen and Loue, PRL (1985) called G 0 W 0 or best G best W
21 GW for realstc materals Assumpton φ GW φ KS Quaspartcle equatons h 0 (r 1 )φ GW (r 1 ) + dr 2 Σ(r 1, r 2, ɛ GW )φ GW (r 2 ) = ɛ GW φ GW (r 1 ) Kohn-Sham equatons Dfferences h 0 (r 1 )φ KS (r 1 ) + v xc (r 1 )φ KS (r 1 ) = ɛ KS φ KS (r 1 ) φ KS Σ(ɛ GW ) v xc φ KS = ɛ GW ɛ KS
22 GW for realstc materals Assumpton φ GW φ KS Quaspartcle equatons h 0 (r 1 )φ KS (r 1 ) + dr 2 Σ(r 1, r 2, ɛ GW )φ KS (r 2 ) = ɛ GW φ KS (r 1 ) Kohn-Sham equatons Dfferences h 0 (r 1 )φ KS (r 1 ) + v xc (r 1 )φ KS (r 1 ) = ɛ KS φ KS (r 1 ) φ KS Σ(ɛ GW ) v xc φ KS = ɛ GW ɛ KS
23 GW for realstc materals Assumpton φ GW φ KS Quaspartcle equatons h 0 (r 1 )φ KS (r 1 ) + dr 2 Σ(r 1, r 2, ɛ GW )φ KS (r 2 ) = ɛ GW φ KS (r 1 ) Kohn-Sham equatons Dfferences h 0 (r 1 )φ KS (r 1 ) + v xc (r 1 )φ KS (r 1 ) = ɛ KS φ KS (r 1 ) φ KS Σ(ɛ GW ) v xc φ KS = ɛ GW ɛ KS
24 G 0 W 0 calculaton To calculate the GW self-energy: Σ(1, 2) = G(1, 2)W (1 +, 2) whch s Fourer transformed nto frequences Σ(r 1, r 2, ω) = dω G(r 1, r 2, ω + ω )W (r 1, r 2, ω ) We need the followng ngredents: The KS Green s functon: G(r 1, r 2, ω) = RPA 1 The RPA delectrc matrx: εgg (q, ω) φ KS (r 1 )φ KS (r 2 ) ω ɛ KS ±η
25 G 0 W 0 calculaton To calculate the GW self-energy: Σ(1, 2) = G(1, 2)W (1 +, 2) whch s Fourer transformed nto frequences Σ(r 1, r 2, ω) = dω G(r 1, r 2, ω + ω )W (r 1, r 2, ω ) We need the followng ngredents: The KS Green s functon: G(r 1, r 2, ω) = RPA 1 The RPA delectrc matrx: εgg (q, ω) φ KS (r 1 )φ KS (r 2 ) ω ɛ KS ±η
26 Calculaton of RPA screenng We need to know ε 1 G,G (q, ω) for all ω s, n order to get Σ. and the frequency convoluton may be problematc because both G and W have poles along the axs W(ω ) G(ω + ω ) µ + ω numercally compute the convoluton accurate, but expensve use a model to mmc ω-behavor of ε 1 rough, cheap
27 Plasmon-Pole model Plasmon-Pole Model ε 1 G,G (q, ω) = δ G,G + Ω2 GG (q) ω 2 ω G,G 2 (q) The two parameters Ω GG (q) and ω G,G (q) are ft on ab nto calculaton of ε 1 G,G (q, ω) at two frequences. We choose ω 1 = 0 ω 2 ω plasma pure magnary frequency
28 Quaspartcle energy ɛ GW = ɛ KS Taylor expanson of Σ(ɛ) around ɛ KS + Σ(ɛ GW ) v xc Fnal formula used by Abnt [ ] ɛ GW = ɛ KS + Z Σ(ɛ KS ) v xc where Z = 1/(1 Σ/ ɛ) Output from Abnt for the band gap of slcon k = Band E0 <VxcLDA> SgX SgC(E0) Z dsgc/de Sg(E) E-E E^0_gap E^GW_gap 3.158
29 Quaspartcle energy ɛ GW = ɛ KS Taylor expanson of Σ(ɛ) around ɛ KS + Σ(ɛ GW ) v xc Fnal formula used by Abnt [ ] ɛ GW = ɛ KS + Z Σ(ɛ KS ) v xc where Z = 1/(1 Σ/ ɛ) Output from Abnt for the band gap of slcon k = Band E0 <VxcLDA> SgX SgC(E0) Z dsgc/de Sg(E) E-E E^0_gap E^GW_gap 3.158
30 If you want to do GW calculatons... GW space-tme-code SELF: fsca.unroma2.t/ self
31 Outlne 1 Remnder 2 GW approxmaton 3 GW n practce 4 Easer? 5 More complcated? 6 More results 7 References
32 Easer?
33 Observe, I M. Rohlfng, P. Krueger, and J. Pollmann, Electronc structure of S(100)2X1, Phys. Rev. B 52, 1905 (1995).
34 Observe, II O. Pulc, G. Onda, R. Del Sole, and L. Renng, Ab-nto calculaton of self-energy effects on electronc and optcal propertes of GaAs(110), Phys. Rev. Lett. 81, 5347 (1998).
35 Observe, III V. Garbuo, M. Cascella, L. Renng, R. Del Sole, and O. Pulc, Ab nto calculaton of optcal spectra of lquds: Many-body effects n the electronc exctatons of water, Phys. Rev. Lett. 97, (2006) The sample Lqud water s a dsordered system Huge unt cell 20 molecular dynamcs snapshots and average of Confguratons of results. 17 molecules n a box wth 15 a.u. sde obtaned wth classcal molecular dynamcs smulatons*
36 Observe, III Electronc gap GGA LDA Experme ntal gap 8.7 ± 0.5 ev Bernas et al., Chem. Phys. 222, 151 (1997) One-shot perturbatve GW
37 Observe, III GW correctons ndependent of confguraton!!! DFT gap GW HOMO GW LUMO GW gap Ε Ε Ε
38 Approxmate GW correcton = 9.1/ɛ V. Forentn and A. Balderesch, Phys. Rev. B 51, (1995).
39 Outlne 1 Remnder 2 GW approxmaton 3 GW n practce 4 Easer? 5 More complcated? 6 More results 7 References
40 Bass set dependence Slcon band gap LMTO Kotan 0.84 LAPW Ku 0.85 PAW Arnaud 0.92 PAW Kresse 1.05 FLAPW Schndlmayr 1.07 PP+PW me 1.14 Expt PP+PW Godby PP+PW Godby PP+PW Hybertsen 1.29
41 Self-consstency??? Σ Γ W G χ ~ ~ Σ Γ W G χ ~ ~ Σ Γ W G χ ~ ~
42 Results for jellum Spectral functon k = k F k = 0 dashed: G 0 W 0 sold: self-consstent GW from B. Holm and U. von Barth, PRB (1998).
43 Self-consstency on the QP wavefunctons and energes?
44 Band gaps of semconductors expermental gap (ev) from M. van Schlfgaarde et al., PRL (2006).
45 QP self-consstency from M. van Schlfgaarde et al. PRL (2006).
46 QP self-consstency Slcon 0.10 electron densty ( e / a.u. 3 ) LDA HF QPscGW 0.00 ( ) drecton Bruneval et al. PRB 74, (2006).
47 Other ssues...for example, the sem-core! see e.g. Copper: A. Marn, G. Onda, R. Del Sole, Phys. Rev. Lett. 88, (2002) F. Bruneval, N. Vast, L. Renng, M. Izquerdo, F. Srott, and N. Barrett, Exchange and correlaton effects n electronc exctatons of Cu2O, Phys. Rev. Lett. 97, (2006)
48 Outlne 1 Remnder 2 GW approxmaton 3 GW n practce 4 Easer? 5 More complcated? 6 More results 7 References
49 More results
50 Result for a complex metal Nckel from F. Aryasetawan, PRB (1992).
51 Motvaton to go beyond... Sodum, photoemsson
52 Outlne 1 Remnder 2 GW approxmaton 3 GW n practce 4 Easer? 5 More complcated? 6 More results 7 References
53 References L. Hedn, Phys. Rev. 139 A796 (1965). L. Hedn and Lunqdvst,Sold State Physcs 23, (1969) F. Aryasetawan and O. Gunnarsson, Rep. Prog. Phys (1998). W.G. Aulbur, L. Jonsson, and J.W. Wlkns, Sol. State Phys (2000). G. Strnat, Rv. Nuovo Cmento 11 1 (1988). G. Onda, L. Renng, and A. Rubo, Rev. Mod. Phys. 74, 601 (2002) theory.polytechnque.fr
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