GW approximation in ABINIT

Transcription

1 GW approxmaton n ABINIT Servce de Recherches de Métallurge Physque CEA Saclay

2 Outlne I. Introducton: gong beyond DFT II. Introducton of the Green's functon III. Exact Hedn's equatons and the GW approxmaton IV. Calculatng the GW self-energy n practce V. Applcatons

3 Standard DFT has unfortunately some shortcomngs band gap Band gap problem! after van Schlfgaarde et al PRL (2008)

4 A pervasve problem Conductvty for charge transport n semconductors Optcal absorpton onset Defect formaton energy Charge transton level Photoemsson Exp.

5 Gap re-normalzaton by a (metallc) substrate Benzene deposted on copper, gold, graphte Neaton, Hybertsen, Loue PRL (2006)

6 How do go beyond wthn the DFT framework? Not easy to fnd mprovement wthn DFT framework There s no such thng as a perturbatve expanson Perdew's Jacob's ladder does not help for the band gap after J. Perdew JCP (2005). Need to change the overall framework!

7 Outlne I. Introducton: gong beyond DFT II. Introducton of the Green's functon III. Exact Hedn's equatons and the GW approxmaton IV. Calculatng the GW self-energy n practce V. Applcatons

8 Many-body perturbaton theory Hstorcally older than the DFT (from the 40-50's)! Bg names: Feynman, Schwnger, Hubbard, Hedn, Lundqvst Green's functons = propagator G r t, r ' t ' =

9 The Green's functon N,0 Exact ground state wavefuncton: Creaton, annhlaton operator: r t N,0 1 2 r ' t ' N,0 r t, r t s a (N+1) electron wavefuncton not necessarly n the ground state s another (N+1) electron wavefuncton Let's compare the two of them!

10 Green's functon defnton N,0 r t r ' t ' N, e = G r t, r ' t ' for t t ' Mesures how an extra electron propagates from (r't') to (rt).

11 Green's functon defnton N,0 r ' t ' r t N, h for = G r ' t ', r t t ' t Mesures how a mssng electron (= a hole) propagates from (rt) to (r't').

12 Fnal expresson for the Green's functon G r t, r ' t ' = N,0 T [ r t r ' t ' ] N, 0 tme-orderng operator e G r t, r ' t ' =G r t, r ' t ' h G r ' t ', r t Compact expresson that descrbes both the propagaton of an extra electron and an extra hole

13 Lehman representaton + G (r, r ', t t ' )= N, 0 T [ Ψ(r t)ψ (r ' t ') ] N, 0 Closure relaton M, M, M, Lehman representaton: G r, r ', = where f r f r ' ± E N 1, E N,0 = E N,0 E N 1, { Exact exctaton energes!

14 Related to photoemsson spectroscopy Ekn hn Energy conservaton: before after h E N,0 =E kn E N 1, Quaspartcle energy: =E N,0 E N 1, =E kn h

15 And nverse photoemsson spectroscopy hn Ekn Energy conservaton: before after E kn E N,0 =h E N 1, Quaspartcle energy: =E N 1, E N,0 =E kn h

16 Exact realzaton of the Lehman decomposton h m G (ω) m = + N 0 c^ m N 1 N 1 c^ m N 0 ω ϵ η N =2 N 1=1 m=1 s He He+ Quaspartcle peak Obtaned from FCI calculatons Satellte or shake-up structure

17 Satelltes n realty? Helum gas Thompson et al. J. Phys. B: At. Mol. Opt. Phys Slcon crystal Guzzo et al. PRL 2011

18 Other propertes of the Green's functon Get the electron densty: ρ(r )= G(r t, r, t ) Galtsk-Mgdal formula for the total energy: μ 1 Etotal = π d ω Tr [ ( ω h 0 ) ImG (ω)] Expectaton value of any 1 partcle operator (local or non-local) O =lm Tr [ OG ] t t '

19 Outlne I. Introducton: gong beyond DFT II. Introducton of the Green's functon III. Exact Hedn's equatons and the GW approxmaton IV. Calculatng the GW self-energy n practce V. Applcatons

20 Equaton of moton of Green's functons: Dyson equaton Let us start wth a non-nteractng Green's functon G 0 correspondng to a hamltonan h0 d r 2 δ (r 1 r 2 )[ ω h 0 (r 2 )] G 0 (r 2, r 3, ω)=δ (r 1 r 3) In short: [ ω h0 ] G0 =1 or 1 G0 = [ ω h0 ] Imagne h0 s Hartree and hks s Kohn-Sham [ ω hks ] G KS=1 Exercce [ ω h0 v xc ] G KS=1 [ G 1 0 v xc ] G KS=1 GKS =G0 +G0 v xc GKS GKS =G0 + G0 v xc G 0 + G0 v xc G0 v xc G0 +...

21 A frst contact wth dagrams GKS (1,2) = G0 (1,2) + d 3 G0 (1,3)v xc (3)GKS (3,2) Dyson equaton connects the Green's functons arsng from dfferent approxmatons What about the exact Green's functon?

22 Dyson equaton for the exact Green's functon Imagne there exsts an operator that generates the exact G G(1,2) = G0 (1,2) + d (34)G0 (1,3)Σ(3,4 )G(4,2) Ths operator s the famous self-energy : non-local n space tme-dependent non-hermtan Everythng else now deals wth fndng expressons for the self-energy!

23 A herarchy of equatons of moton In fact there s an exact expresson for the self-energy as a functon of the twopartcle Green's functon 1 G [ 0 Σ ] G=1 1 G [ 0 G2 ] G=1 + + G2 (1,2; 3,4)= N, 0 T [Ψ (1)Ψ (2) Ψ (3)Ψ ( 4)] N, 0 And try to guess the equaton of moton for the two-partcle Green's functon? G2 needs G3 G3 needs G 4 G4 needs G 5...

24 An expresson for the self-energy Trck due to Schwnger (1951): Introduce a small external potental U (that wll be made equal to zero at the end) Calculate the varatons of G wth respect to U δg (1,2) G2 (1,3 ; 2,3)= δ U (3) Obtan a perturbaton theory wth basc ngredents G and v 1st order s Hartree-Fock 2nd order s MP2 However MP2 dverges for metals! Trck due to Hubbard+Hedn (late 1950's early 1960's): Introduce the electrostatc response V to U Calculate the varatons of G wth respect to V V (1)=U (1) d 2 v (1,2)δG (2,2) Obtan a new renormalzed perturbaton theory wth basc ngredents G and W 1st order s GW

25 Shftng from U to V U (1)=ε δ(r r 1 )δ (t t 1 ) U (1)=ε δ(r r 1 )δ (t t 1 ) V (1)=U (1)+ d r v (r 1 r )δρ(r ) Everythng s functonal of U V also ncludes the electrostatc response Everythng s functonal of V G[U ] G[V ]

26 Hedn's coupled equatons 6 coupled equatons: 1= r 1 t 1 1 2= r 2 t 2 2 G(1,2)=G 0 (1,2)+ d 34 G 0 (1,3)Σ(3,4) G( 4,2) 1,2 = d34 G 1,3 W 1,4 4,2,3 1,2,3 = 1,2 1,3 d 4567 Dyson equaton self-energy 1,2 G 4,6 G 5,7 6,7,3 G 4,5 vertex 0 1,2 = d34 G 1,3 G 4,1 3,4,2 polarzablty 1,2 = 1,2 d3 v 1,3 0 3,2 delectrc matrx 1 W 1,2 = d3 1,3 v 3,2 screened Coulomb nteracton

27 Smplest approxmaton Σ(1,2)=G (1,2)v (1 +, 2) t Fock exchange Dyson equaton: v 1,2 G=G0 G 0 G G=G0 G 0 G0... G 1,2 Not enough: Hartree-Fock s known to perform poorly for solds

28 Hartree-Fock approxmaton for band gaps

29 Hedn's coupled equatons 6 coupled equatons: G 1,2 =G 0 1,2 d34 G0 1,3 3,4 G 4,2 1,2 = d34 G 1,3 W 1,4 4,2,3 1,2,3 = 1,2 1,3 d 4567 Dyson equaton self-energy 1,2 G 4,6 G 5,7 6,7,3 G 4,5 0 1,2 = d34 G 1,3 G 4,1 3,4,2 1,2 = 1,2 d3 v 1,3 0 3,2 1 W 1,2 = d3 1,3 v 3,2 screened Coulomb nteracton

30 Hedn's coupled equatons 6 coupled equatons: G 1,2 =G 0 1,2 d34 G0 1,3 3,4 G 4,2 1,2 = d34 G 1,3 W 1,4 4,2,3 1,2,3 = 1,2 1,3 d 4567 Dyson equaton self-energy 1,2 G 4,6 G 5,7 6,7,3 G 4,5 0 1,2 = d34 G 1,3 G 4,1 3,4,2 1,2 = 1,2 d3 v 1,3 0 3,2 1 W 1,2 = d3 1,3 v 3,2 screened Coulomb nteracton

31 Hedn's coupled equatons 6 coupled equatons: G 1,2 =G 0 1,2 d34 G0 1,3 3,4 G 4,2 2 1,4 2 4,2,3 1,2 = d34 G 1,3 W 1,2,3 = 1,2 1,3 d 4567 Dyson equaton self-energy 1,2 G 4,6 G 5,7 6,7,3 G 4,5 2 4, ,2 = d34 G 1,3 G 3,4,2 1,2 = 1,2 d3 v 1,3 0 3,2 1 W 1,2 = d3 1,3 v 3,2 screened Coulomb nteracton

32 Here comes the GW approxmaton Σ (1,2)= G (1,2) W (1,2) GW approxmaton χ 0 (1,2)= G (1,2) G(2,1) RPA approxmaton ε(1,2)=δ (1,2) d 3 v (1,3) χ0 (3,2) W (1,2)= d 3 ε 1 (1,3) v (3,2)

33 Let us play wth dagrams 0 1,2 = G 1,2 G 2,1 1,2 = 1,2 d3 v 1,3 0 3,2 W 1,2 = d3 1 1,3 v 3,2 1,2 = G 1,2 W 1,2 W = v + v χ0w = v + v χ0 v + v χ0 v χ0 v +... Infnte summaton over bubble (or rng) dagrams

34 What s W? Interacton between electrons n vacuum: 2 1 e v (r, r ' )= 4 πε0 r r ' Interacton between electrons n a homogeneous polarzable medum: 1 e2 W (r, r ' )= 4 π ε0 ε r r r ' Delectrc constant of the medum Dynamcally screened nteracton between electrons n a general medum: 2 1 e ε (r, r ' ', ω) W (r, r ', ω)= d r ' ' 4 π ε0 r ' ' r '

35 W s frequency dependent W can measured drectly by Inelastc X-ray Scatterng Im W (q=0.80 a.u,ω) Slcon Plasmon frequency ω [ev] Zero below the band gap H-C Wessker et al. PRB (2010)

36 Summary

37 Summary: DFT vs Green s functon Electronc densty r Local and statc exchange-correlaton potental v xc r Approxmatons: LDA, GGA, hybrds Green's functon G r t, r ' t ' Non-local, dynamc Depends onto empty states exchange-correlaton operator = self-energy Σ xc ( r t, r ' t ' ) HF, GW approxmaton GW r t, r ' t ' =G r t, r ', t ' W r t, r ' t '

38 GW vewed as a super Hartree-Fock GW Approxmaton Hartree-Fock Approxmaton x r 1, r 2 = Σ xc (r 1, r 2, ω)= d ' G r 1, r 2, ' v r 1, r 2 2 d ω ' G( r 1, r 2, ω+ ω' )W ( r 2, r 1, ω ' ) 2π = bare exchange Exercce x r 1, r 2 Bare exchange c r 1, r 2, + correlaton GW s nothng else but a screened verson of Hartree-Fock. Non Hermtan dynamc

39 GW approxmaton gets good band gap No band gap problem anymore! after van Schlfgaarde et al PRL (2008)

40 Outlne I. Introducton: gong beyond DFT II. Introducton of the Green's functon III. Exact Hedn's equatons and the GW approxmaton IV. Calculatng the GW self-energy n practce V. Applcatons

41 Hstorcal recap of GW calculatons 1965: Hedn's calculatons for the homogeneous electron gas Phys Rev 2201 ctatons 1967: Lundqvst's calculatons for the homogeneous electron gas Physk der Kondenserte Matere 299 ctatons 1982: Strnat, Mattausch, Hanke for real semconductors but wthn tght-bndng PRB 154 ctatons 1985: Hybertsen, Loue for real semconductors wth ab nto LDA PRL 711 ctatons & PRB 1737 ctatons 1986: Godby, Sham, Schlüter for real semconductors to get accurate local potental PRL 544 ctatons & PRB 803 ctatons ~2001: Frst publcly avalable GW code through the ABINIT project 2003: Arnaud, Alouan for extenson to Projector Augmented Wave PRB 102 ctatons 2006: Shshkn, Kresse for extenson to Projector Augmented Wave (agan) PRB 256 ctatons

42 GW approxmaton n practce For perodc solds: Abnt, BerkeleyGW, VASP, Yambo based on plane-waves (wth pseudo or PAW) For fnte systems: MOLGW, Festa, FHI-AIMS based on localzed orbtals (Gaussans or Slater or other)

43 Workflow of a typcal GW calculaton G0 W 0 vs GW One-shot GW DFT KS ϕ,ϵ KS occuped AND empty states KS ϕ,ϵ KS calculate W If self-consstent GW GW ϕ, ϵ Egenvalues calculate G * W GW ϵ (k)

44 Workflow of a typcal GW calculaton G0 W 0 vs GW One-shot GW Ø DFT KS ϕ,ϵ scf -2 nband 100 KS occuped AND empty states KS ϕ,ϵ optdrver 3 calculate W If self-consstent GW optdrver 4 GW ϕ, ϵ KS Egenvalues calculate G * W GW k

45 How to get G? From Kohn-Sham DFT Remember [ ω hks ] G KS=1 whch means KS KS * KS ϕ (r)ϕ (r ') GKS (r, r ',ω)= ω ϵ ± η Ths expresson wll be used to get W and Σ

46 How to get W? From the RPA equaton χ 0 (1,2)= G KS (1,2)G KS (2,1) whch translates nto Exercce j χ 0 (r 1, r 2, ω)= ϕ ( r 1) ϕ (r 2 )ϕ j (r 2) ϕ (r 1) occ j vrt [ 1 1 ω (ϵ j ϵ) η ω (ϵ ϵ j)+ η ] Ths s the Alder-Wser formula or the SOS formula It nvolves empty states! Then W = v + v χ0w 1 W = (1 v χ0 ) v geometrc seres

47 Dealng wth two-pont functons n recprocal space Remember 1-pont functons are 1 (k +G ). r ϕ k (r ) = c k (G)e Ω kg 1 vector of coeffcents per k-pont n the Brlloun zone Then 2-pont functons are 1 (q +G ).r (q+g ' ).r W ( r 1, r 2 ) = e W ( q) e GG ' Ω q G G' 1 2 a matrx of coeffcents per q-pont n the BZ due to translatonal symmetry: W ( r 1, r 2 ) = W (r 1 + R, r 2+ R)

48 W n plane-waves and frequency space χ 0 (r 1, r 2, ω)= ϕ (r 1)ϕ (r 2 )ϕ j (r 2) ϕ j ( r 1) (1) occ j vrt [ (2) 1 1 ω (ϵ j ϵ ) η ω (ϵ ϵ j)+ η ] ε(1,2)=δ (1,2) d 3 v (1,3) χ0 (3,2) (3) W (1,2)= d 3 ε 1 (1,3)v (3,2) (1) (q+ G). r 1 χ 0 G G ' (q,ω)= j k q e k k e (q +G ' ). r j k q 2 k occ j vrt (2) (3) [ 1 1 ω (ϵ j ϵ ) η ω (ϵ ϵ j )+ η εg G ' (q, ω)=δg, G ' v G G ' ' (q)χ 0 G ' ' G ' (q, ω) G' ' W G G ' (q, ω)=ε 1 G G ' ( q, G ' )v G ' (q) ] nband ecuteps q the same regular grd as k but Γ-centered 4π v G G '' (q)= δ 2 G,G ' ' q+g matrx nverson

49 Self energy evaluaton n GW Correlaton part of the GW self energy requres a convoluton n frequency: + Σ c (ω)= d ω ' G(ω+ω ' )W p (ω ') 2π G(ω)= nband ϕ (r ) ϕ (r ' ) ω ϵ± η? How to deal wth the frequency dependence n W? How do we perform the convoluton? How do we treat the frequency dependence n W?

50 Analytc structure of W(ω) Tme ordered response functon: Many poles whch go by pars: Plasmon-pole model: One par of poles: ±( ~ ω η) ±(~ ω η) Complex plane: Eg Eg Slcon: For a gven q+g:

51 Plasmon-Pole Models n GW Correlaton part of the GW self energy requres a convoluton n frequency: + Σ c (ω)= d ω ' G(ω+ω ' ) W p (ω ') 2π Generalzed Plasmon-Pole Model: Ω ε (ω ' ) 1 = ~ 2 ω ω ' ~ ω+ η ω ' +~ ω η [ 1 ppmodel Ampltude of the pole Poston of the pole 2 parameters need two constrants: - Hybertsen-Loue (HL): - Godby-Needs (GN): 1 ε (0) ε 1 (0) ] small real number + and f sum rule and 1 1 π ω2 ω Imε (ω)= p 2 0 ε ( ω)jouvence, ppmfrq Quantum Materals 2018

52 GW obtaned as a frst-order perturbaton G = G0 + G0 ΣG 1 G 1 = G KS (Σ v xc ) GKS = G 0 + G 0 v xc G KS Approxmaton : 1 G = ϕ GW ϕ ) ϕ ϕ (ω ϵgw G 1 KS = KS KS GW ϵ GW = ϵ + ϕ Σ( ϵ ) v xc ϕ ϕ (ω ϵks) ϕ

53 Lnearzaton of the energy dependance ϵ GW KS ϵ = ϕ KS [ Σ(ϵ GW ) v xc ] ϕ KS Not yet known Taylor expanson: Σ Σ(ϵ )=Σ(ϵ )+(ϵ ϵ ) ϵ +... GW KS GW KS Fnal result: ϵ GW KS =ϵ + Z ϕ where KS [ Σ (ϵ KS ) v xc ] ϕ Σ Z =1/ 1 ϵ ( ) KS nomegasrd

54 Typcal GW output n ABINIT nkptgw kptgw bdgw k = Band E0 <VxcLDA> SgX SgC(E0) E^0_gap E^GW_gap GW E-E E KS ϵ =ϵ +Z φ Z dsgc/de Sg(E) KS Σxc (ϵ KS ) v xc φ KS

55 Full quaspartcle soluton Im Σ (ω) 1 A (ω) = π Im G(ω) = KS 2 2 (ω ϵ +Re Σ (ω) v xc ) +Im Σ (ω)

56 Outlne I. Introducton: gong beyond DFT II. Introducton of the Green's functon III. Exact Hedn's equatons and the GW approxmaton IV. Calculatng the GW self-energy n practce V. Applcatons

57 GW approxmaton gets good band gap No more a band gap problem! van Schlfgaarde et al PRL (2008)

58 Spectral functon / π

59 Exctaton lfetme

60 Exact realzaton of the Lehman decomposton h m G (ω) m = + N 0 c^ m N 1 N 1 c^ m N 0 ω ϵ η N =2 N 1=1 m=1 s He He+ Quaspartcle peak Obtaned from FCI calculatons Satellte or shake-up structure

61 Clusters de sodum Na 4 e Na 4 4 E 0 Na 4 E 0 Na = { HOMO, Na4 LUMO, Na4 + Na4 /Na4 Bruneval PRL (2009)

62 What s the best startng pont for G0W0? Ionzaton of small molecules Hybrds perform better, preferably wth a large content of EXX ~ 50 % & MAL Marques, JCTC (2013)

63 Defect calculaton wthn GW approxmaton Up to 215 atoms Cubc slcon carbde

64 Photolumnescence of VS Bruneval and Roma PRB (2011)

65 3d metal band structure

66 Band Offset at the nterface between two semconductors Very mportant for electroncs! Example: S/SO2 nterface for transstors GW correcton wth respect to LDA R. Shaltaf PRL (2008).

67 Summary The GW approxmaton solves the band gap problem! The calculatons are extremely heavy, so that we resort to many addtonal techncal approxmatons: method named G0W0 The complexty comes from Dependence upon empty states Non-local operators Dynamc operators that requres freq. convolutons

68 Revews - Lnks Revews: L. Hedn, Phys. Rev. 139 A796 (1965). L. Hedn and S. Lunqdvst, n Sold State Physcs, Vol. 23 (Academc, New York, 1969), p. 1. 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). and M. Gatt, Quaspartcle Self-Consstent GW Method for the Spectral Propertes of Complex Materals, Top. Curr. Chem (2014) 347: Codes:

69 Exercce: H2 n mnmal bass: GW@HF Fnd the locaton of the poles of the self-energy Szabo-Ostlung book chapter 3 teaches how to perform HF n ths example: Bass: STO-3G r(h-h) = 1.4 bohr 2 bass functons 2 egenstates: LUMO ant-bondng HOMO bondng In egenvector bass: Hamltonan: CT H C = Coulomb nteracton: 1 ( ) = r 1 r 2 1 ( ) = r 1 r 2 1 ( ) = r 1 r 2 ( ) Atomc unts

70 Exercce: H2 n mnmal bass: GW@HF Fnd the locaton of the poles of W Dagonalze the RPA equaton k l ω (ϵ j ϵ ) f f j 1 χ (ω) = 1 ( j k l ) r j Δ ϵ = ϵ 2 ϵ1 = v = (12 1/r 1 2) = ω Δ ϵ ω+δ ϵ 2 v v v v Ω = ± Δ ϵ v Δ ϵ = ±1.569

71 Exercce: H2 n mnmal bass: GW@HF + Σ c (ω)= d ω ' G(ω+ω ' )W p (ω ') 2π G(ω)= ϕ (r )ϕ (r ' ) ω ϵ ± η W p (ω)= s Ls ( r) R s (r ' ) ω Ω s± η + β α Σ c (ω)= d ω' 2 π {1, 2} s {1 2, 2 1} ω+ ω ' ϵ ± η ω ' Ω± η Integraton n the complex plane: Ω ϵ 1 ω ϵ1 Pole table: Ω Ω ϵ2 ϵ 2 ω Ω ϵ 2+ Ω ϵ 1 Ω

72 Exercce: H2 n mnmal bass: GW@HF ϵ 2+ Ω = ϵ 1 Ω = Real part of the self-energy from MOLGW ϵ GW HOMO = ev ϵ GW ev LUMO =

73 Exercce: H2 n mnmal bass: GW@HF Same conclusons hold for a many-state case: Bulk slcon Plasmon frequency ~ 17 ev Occuped states ~ -5-0 ev Empty states ~ +2 - ev

74 Exercse 1 Green's functon n frequency doman G (r 1 t 1, r 2 t 2 )=θ (t 1 t 2 ) ϕ ( r 1 )ϕ (r 2 )e ϵ (t t ) 1 2 vrt ϵ (t 2 t 1) θ (t 2 t 1 ) ϕ (r 2 ) ϕ (r 1) e occ G(r 1, r 2, ω)= d (t 1 t 2 )e ω(t t ) G (r 1 t 1, r 2 t 2 ) 1 2 G( r 1, r 2, ω)= ϕ (r 1 ) ϕ (r 2) ω ϵ± η

75 Exercse 2: Fock exchange from Green's functons + Σ x (1,2)=G (1,2)v (1, 2) Σ x (r 1, r 2, ω)= occ ϕ (r 1 )ϕ (r 2) r 1 r 2

76 Exercse 3: let's play wth Dyson equatons 1) The multple faces of the Dyson equaton [ ω hks ] G KS=1 [ ω h0 v xc ] G KS=1 [ G 1 0 v xc ] G KS=1 GKS =G0 +G0 v xc GKS GKS =G0 + G0 v xc G 0 + G0 v xc G0 v xc G GKS = G0 v xc 2) Combnng the Dyson equatons 1 G 1 1 = G0 Σ 1 GKS = G0 v xc 1 G 1 = G KS (Σ v xc ) 1= [ G KS (Σ v xc ) ] G 1 1= [ ω h 0 Σ ] G

77 Exercse 4 Derve the standard Adler-Wser formula (1963): χ 0 (1, 2)= G(1, 2)G(2, 1) χ 0 (r 1, r 2, ω) = d ω ' G (r 1, r 2, ω+ω ')G (r 2, r 1, ω' ) 2π χ 0 (r 1, r 2, ω)= ϕ (r 1) ϕ (r 2 )ϕ j (r 2) ϕ j ( r 1) occ j vrt [ 1 1 ω (ϵ j ϵ) η ω (ϵ ϵ j)+ η ]

78 Exercse 5 Derve that the product n tme becomes a convoluton n frequency: Σ (r 1, r 2, t 1 t 2)=G (r 1, r 2, t 1 t 2) W (r 2, r 1, t 2 t 1) ω(t 1 t 2) G(r 1, r 2, ω)= d (t 1 t 2 ) e G(r 1, r 2, t 1 t 2)= Σ(r 1, r 2, ω)= G (r 1 t 1, r 2 t 2 ) 1 ω (t t ) d ωe G(r 1, r 2, ω) 2π 1 2 d ω ' G( r 1, r 2, ω+ω ' )W (r 2, r 1, ω ' ) 2π

79 Exercce 6: Feynman dagram drawng a) Draw all the 1st order dagrams for the self-energy b) Draw all the 2nd order dagrams for the self-energy c) What s the dfference between the proper and the mproper self-energy d) How self-consstency can smplfy the expanson? Self-energy dagram drawng rules: 1. Dagrams are combnatons of arrows (Green s functon) and horzontal lnes (Coulomb nteracton). Upward arrows are electrons, downward arrows are hole. 2. Dagrams should be connected. 3. Self-energy have an entry pont and an ext pont (possbly the same). 4. Each ntersecton should conserve the partcle numbers. 5. A vald dagram cannot be cut (by removng an arrow) nto another lower order self-energy.