GW Many-Body Theory for Electronic Structure. Rex Godby

Size: px

Start display at page:

Download "GW Many-Body Theory for Electronic Structure. Rex Godby"

Avice Logan
6 years ago
Views:

1 GW Many-Body Theory for Electronic Structure Rex Godby

2 Outline Lecture 1 (Monday) Introduction to MBPT The GW approximation (non-sc and SC) Implementation of GW Spectral properties Lecture 2 (Tuesday) GW total energy Self-consistency in GW Vertex corrections beyond GW Using MBPT to get better DFT functionals Workshop talk GW-like approaches to quantum transport GW Many-Body Theory for Electronic Structure - Benasque, August

3 The Basic Problem Nucleus α (fixed) Electron i Electrons interact with each other as well as with the nuclei! GW Many-Body Theory for Electronic Structure - Benasque, August

Quantum Mechanics One-electron Schrödinger equation: 2 ħ 2m 2 ψ ( r) + V ( r) ψ ( r) Eψ ( r) Account for other electrons through an effective potential felt by each

4 Quantum Mechanics One-electron Schrödinger equation: 2 ħ 2m 2 ψ ( r) + V ( r) ψ ( r) Eψ ( r) Account for other electrons through an effective potential felt by each electron Density-functional theory: V ( r) Veff ( r) Many-body perturbation theory: V r) Σ = ( ( r, r', E) GW Many-Body Theory for Electronic Structure - Benasque, August

5 Approaches to the electron-electron interaction Density-functional theory (for ground-state properties only): [ V ext (r) + V Hartree (r) + V xc (r) ε i ]ψ i (r) = 0 Many-body perturbation theory based on Green s functions: [ V ext + V Hartree + Σ xc (ω ) ω]g(r, r,ω) = δ(r r ) (which, inter alia, leads to the quasiparticle equation [ V ext + V Hartree + Σ xc (ε) ε]ψ (r) = 0) Note the two ways of describing exchange and correlation: In DFT: V xc (r) (local, energy-independent potential) In many-body theory: Σ xc (r, r,ω ) (non-local, energydependent potential) GW Many-Body Theory for Electronic Structure - Benasque, August

6 Why Not Just Leave it to DFT?

7 Some Pathologies of the KSDFT functional δe [ n] Band-gap problem: V ( r) xc xc = δn( r) is discontinuous (by a constant) when an electron is added to an insulator (Sham and Schlüter, Perdew and Levy 1983) Widely separated open-shell atoms: V xc(r) has a peculiar spatial variation between distant atoms so as to equalise highest occupied states (Almbladh and von Barth 1985) Exchange-correlation electric field accompanies real electric field in an insulator (Godby and Sham 1994, Gonze, Ghosez and Godby 1995) GW Many-Body Theory for Electronic Structure - Benasque, August

8 Why Not Just Leave it to DFT? DFT functionals are known to have important pathologies, meaning that density-based approximate functionals can be inadequate MPBT quantities better behaved MBPT role in analysing existing functionals and devising new ones (e.g. orbital functionals) Access to excited-state properties, some of which are inaccessible or inaccurate in (TD)DFT Accurate ground-state quantities also available GW Many-Body Theory for Electronic Structure - Benasque, August

9 MBPT vs. (TD)DFT

10 MBPT vs. TDDFT Many-body perturbation theory (e.g. GW) Based on Green s functions Self-energy theories give one-particle G, e.g. electron addition/removal Natural domain: quasiparticle energies, band structure, spectral function Similar diagrammatic theories for two-particle G, e.g. optical absorption Other applications: ground state total energy, etc. GW Many-Body Theory for Electronic Structure - Benasque, August

11 MBPT vs. TDDFT (2) DFT Natural domain (of ordinary DFT): ground state total energy TDDFT permits study of time-evolution (e.g. excited states) of N-electron system Treatment of exchange and correlation in TDDFT requires some approximation, often based on the HEG / WIEG GW Many-Body Theory for Electronic Structure - Benasque, August

12 Pros and Cons Method DFT / TDDFT Advantages Inexpensive (esp. density-based functionals) Disadvantages XC functional and f xc (r,r,ω) pathological and can be difficult to approximate sufficiently accurately [Intermediate approaches such as GW-based orbital functionals (KS/GKS)] GW (+ MBPT) Description of XC explicit diagrammatic series, better behaved Can be expensive GW Many-Body Theory for Electronic Structure - Benasque, August

13 Introduction to Many-Body Perturbation Theory

14 Electronic Excitations: G 1 and G 2 N,0 N+1,s N 1,s N,s GW Many-Body Theory for Electronic Structure - Benasque, August

15 Key ideas of many-body perturbation theory Electronic and optical experiments often measure some aspect of the one-particle Green s function The spectral function, Im G, tells you about the singleparticle-like approximate eigenstates of the system: the quasiparticles Im G non-interacting interacting E 1 E 2 µ G obtained from self-energy Σ(r,r,E) Σ = GW + O(W 2 ); W = screened Coulomb interaction ω GW Many-Body Theory for Electronic Structure - Benasque, August

16 Second Quantisation Field operators add/remove an electron at r with spin σ (x=r,σ) at time t ψ ˆ ( x, t) ˆ ( x, t) Defined rigorously in terms of operation on Slater determinants Have equations of motion in the Heisenberg representation that follows from the TDSE ; ψ GW Many-Body Theory for Electronic Structure - Benasque, August

17 1-particle Green's Function G One-particle Green's function G ( x, x'; t t') i N T[ ψ ˆ ( x, t) ψˆ ( x', t' )] N + where x ( r, ξ) is space+spin, N = N-electron ground state, T=time-ordering operator GW Many-Body Theory for Electronic Structure - Benasque, August

18 GW Many-Body Theory for Electronic Structure - Benasque, August

19 G as a Propagator G( x, x'; t t') i N T[ ψ ˆ( x, t) ψˆ ( x', t')] + N ψˆ ( r' t') ψˆ ( rt) GW Many-Body Theory for Electronic Structure - Benasque, August

20 Equation of Motion of G Using equation of motion of field operators, G can be shown to satisfy i i = t v( r, r') δ ( x h( x) G( xt; x' t') + N T[ ψˆ x') δ ( t t') Then write Σ V H +Σ xc ψψψ ˆ ˆ ˆ ] N d 4 x ΣG - defines Σ GW Many-Body Theory for Electronic Structure - Benasque, August

21 i i Green's Functions and Self-Energies G obeys a very similar equation to the Green s function of the Kohn-Sham electrons in DFT, but with the non-local, time-dependent Σ replacing V xc : t t { h + V + Σ } H G( xt; x' t') = δ ( x t') { h + V + V } G ( xt; x' t') = δ ( x x') δ ( t t' ) H xc xc KS x') δ ( t GW Many-Body Theory for Electronic Structure - Benasque, August

22 GW Many-Body Theory for Electronic Structure - Benasque, August Lehmann Representation for G ( ) ( ) < = > = + = <> = + µ ε ψ µ ε ψ µ ε δ ε ω ω s N N s N s N s s s s s s s E E N s N E E s N N f i f f G 1, 1,, ) ˆ ( 1,, 1, ) ˆ ( ) (, ') *( ) ( ) ',, ( r r r r r r r s labels the excited states of the N+1 or N-1- electron systems G has its poles at each energy with which an electron can be added/removed

23 Our Cast of Players G Green s function Σ xc Self-energy (related to G as discussed) W Dynamically screened Coulomb interaction P Irreducible polarisation propagator Γ Vertex function GW Many-Body Theory for Electronic Structure - Benasque, August

24 Hedin s Equations With Σ/G relation, exact closed equations for G, Σ etc. GW Many-Body Theory for Electronic Structure - Benasque, August

25 The GW Approximation

26 The GW Approximation Iterate Hedin s equations once starting with Σ=0 GW Many-Body Theory for Electronic Structure - Benasque, August

27 The GW Approximation Σ ( x, x'; ω) i = W ( x, x'; ω') G ( x, x'; ω + ω') e 2π where x ( r, ξ ) is space+spin iδω ' d ω' In the time domain Σ( x, x'; t t') = iw ( x, x'; t t') G( x, x'; t t') W is the dynamically screened Coulomb interaction between electrons GW Many-Body Theory for Electronic Structure - Benasque, August

28 Implementation of GW

29 Local-density-approximation calculation of one-electron wavefunctions and eigenvalues, using ab initio pseudopotentials Compute Green's function G Compute screened Coulomb interaction W S(q,ω) Compute self-energy Σ(r,r',ω) QP energies E Solve Dyson equation to obtain updated Green's function G(r,r',ω) Spectral function Charge density + momentum distn. Space-time method: H.N. Rojas, RWG and R.J. Needs, Phys. Rev. Lett (1995) Total energy Updated self-energy GW Many-Body Theory for Electronic Structure - Benasque, August

30 Spectral Properties

31 G 0 W 0 Band Structures of Insulators From "Quasiparticle calculations in solids", W.G. Aulbur, L. Jönsson and J.W. Wilkins, Solid State Physics 54 1 (2000) [also available in preprint form at l#reviews] GW Many-Body Theory for Electronic Structure - Benasque, August

32 Self-Consistency Fully self-consistent Σ=iGW: Conserving Partially self-consistent Σ=iGW0: Conserving Non-self-consistent Σ=iG0W0: Non-conserving Arno Schindlmayr, Pablo García-González and RWG Kris Delaney GW Many-Body Theory for Electronic Structure - Benasque, August

33 Tests for Be atom (QP energies) K.T. Delaney, P. García-González, Angel Rubio, Patrick Rinke and RWG, Phys. Rev. Lett GW Many-Body Theory for Electronic Structure - Benasque, August

34 Total Energy from Many- Body Perturbation Theory

35 Ab initio GW total energy Galitskii-Migdal expression for ground-state total energy E = µ 1 3 dω d r ( ω + h r) ) A( r, r', ω r' r + 2 ) ( V, where Start with LDA calculation Use GW space-time method to calculate self-energy Σ ( r, r', it) using a double grid in real space, and a grid in imaginary time LDA 0 xc ) ( ) Calculate new Green s function using G = 1 G ( Σ V G0 Recalculate Σ using new G and new W (based on new G) Repeat to self-consistency Calculate total energy using self-consistent G nucl 1 GW Many-Body Theory for Electronic Structure - Benasque, August

36 Total Energy of the H.E.G ε XC (Ha) ε XC (Ha) Exact G 0 W 0 GW 0 GW r s (a.u.) Pablo García-González and RWG, PRB 2001 Exact G 0 W 0 GW 0 GW GW Many-Body Theory for Electronic Structure - Benasque, August

37 Van der Waals forces GW Pablo García-González and RWG, PRL 2002 GW Many-Body Theory for Electronic Structure - Benasque, August

38 Vertex Corrections Beyond GW

39 The Vertex Σ=GWΓ is exact by definition Vertex function Γ needs to be approximated; in GW start with Σ=0 Can also start with Σ = simplified self-energy; simplest of all is V xc LDA GW Many-Body Theory for Electronic Structure - Benasque, August

40 Morris, Stankovski et al. DFT-based Vertex (A)LDA vertex in W (G 0 W 0 LDA ) yields slight improvement over G 0 W 0 If ALDA vertex is also added into Σ, things get much worse! Morris, Stankovski, Rinke, Delaney, RWG, PRB 2007 GW Many-Body Theory for Electronic Structure - Benasque, August

41 DFT-based Vertex (2) Averaged Density Approximation (ADA) produces convenient non-local vertex, which gives good total energies and quasiparticle properties for jellium [Stankovski, Robinson, Morris, Rinke, Delaney, RWG] GW Many-Body Theory for Electronic Structure - Benasque, August

42 DFT-based Vertex (3) and restores the good results of G 0 W 0 for the I.P. of atoms! GW Many-Body Theory for Electronic Structure - Benasque, August

43 DFT-based Vertex (4) Future question: Interplay between self-consistency and vertex GW Many-Body Theory for Electronic Structure - Benasque, August

44 Using self-energy approaches to shed light on DFT functionals

45 Using self-energy approaches to shed light on DFT functionals G.s. total energy already discussed (e.g. van der Waals) Orbital functions based on GW self-energies Beyond exact exchange DFT From Σ can calculate g.s. density ρ(r), then determine V xc such that the same density is reproduced: Sham-Schlüter equation (see e.g. Godby Schlüter Sham PRL 1986): relevant to Band-gap problem Exchange-correlation electric field in polarised systems (and TDDFT current functionals à la R. van Leeuwen et al.) GW Many-Body Theory for Electronic Structure - Benasque, August

46 Acknowledgements and Further Reading

47 Collaborators Pablo García-González Kris Delaney Patrick Rinke Martin Stankovski Peter Bokes Angel Rubio GW Many-Body Theory for Electronic Structure - Benasque, August

48 Some starting points for further reading "Quasiparticle calculations in solids", W.G. Aulbur, L. Jönsson and J.W. Wilkins, Solid State Physics 54 1 (2000), available in preprint form on the web Many-particle theory, E.K.U. Gross, Runge and Heinonen, Adam Hilger (approx. 1992) Electron Correlations in Molecules and Solids, P. Fulde, Springer (1993) "Density functional theories and self-energy approaches", R. W. Godby and P. García-González, chapter in "A Primer in Density Functional Theory", ed. Carlos Fiolhais, Fernando Nogueira and M. A. L. Marques, Lecture Notes in Physics vol. 620, Springer (Heidelberg), 2003 Further info at GW Many-Body Theory for Electronic Structure - Benasque, August

GW-like approaches to quantum transport. Rex Godby

GW-like approaches to quantum transport Rex Godby Outline Introduction to the quantum transport problem Ab initio quantum conductance in the presence of e e interaction (TDDFT / MBPT) 2 + Bothersome aspects