gw many-body theory for electronic structure · 2019. 10. 2. · • mpbt quantities better behaved...

GW Many-Body Theoryfor Electronic Structure

Rex Godby

GW Many-Body Theory for Electronic Structure - Benasque, August 2008 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

+


The Basic Problem

• Electrons interact with each other as well as with the nuclei!

Nucleus α (fixed)

Electron i


• One-electron Schrödinger equation:

• Account for other electrons through an effective potential felt by each electron

• Density-functional theory:

• Many-body perturbation theory:

Quantum Mechanics

)()()()(2

22

rrrr ψψψ EVm

=+∇−ℏ

)()( eff rr VV →

),',()( EV rrr Σ→


Approaches to the electron-electron interaction

Note the two ways of describing exchange and correlation:

In DFT: Vxc (r) (local, energy-independent potential)

In many-body theory: Σxc (r, ′ r ,ω ) (non-local, energy-

dependent potential)

• Density-functional theory (for ground-state properties only):

− 12

∇2 + Vext (r) + VHartree (r) + Vxc (r) − εi[ ]ψ i (r) = 0

• Many-body perturbation theory based on Green’s functions:

− 12

∇2 + Vext + VHartree + Σxc (ω ) − ω[ ]G(r, ′ r ,ω ) = δ (r − ′ r )

(which, inter alia, leads to the quasiparticle equation

− 12

∇2 + Vext + VHartree + Σxc (ε ) − ε[ ]ψ (r) = 0)

Why Not Just Leave it to DFT?


Some Pathologies of the KSDFT functional

• Band-gap problem: )(

][)(

rr

n

nEV xc

xc δ

δ=

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: )(rxcV 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)


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

MBPT vs. (TD)DFT


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.


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


Pros and Cons

Can be expensiveDescription of XC explicit diagrammatic series, better behaved

GW (+ MBPT)

XC functional and

fxc(r,r′,ω) pathological and can be difficult to approximate sufficiently accurately

Inexpensive (esp. density-based functionals)

DFT / TDDFT

DisadvantagesAdvantagesMethod

[Intermediate approaches such as GW-based orbital functionals (KS/GKS)]

Introduction to Many-Body Perturbation Theory


Electronic Excitations: G1 and G2

| N,0 ⟩ | N+1,s ⟩ | N–1,s ⟩ | N,s ⟩


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 single-particle-like approximate eigenstates of the system: the quasiparticles

E E µ

Im G

ω

non-interacting

interacting

1 2

• G obtained from self-energy Σ(r,r´,E) • Σ = GW + O(W2); W = screened Coulomb interaction

+


Second Quantisation

• Field operators add/remove an electron at r with spin σ (x=r,σ) at time t

• Defined rigorously in terms of operation on Slater determinants

• Have equations of motion in the Heisenberg representation that follows from the TDSE

),(ˆ;),(ˆ †tt xx ψψ


1-particle Green's Function G

• One-particle Green's function

NttTNittG )]','(ˆ),(ˆ[)';',( xxxx+−≡− ψψ

where ),( ξrx ≡ is space+spin, =N N-electron ground state,

T=time-ordering operator


G as a Propagator

NttTNittG )]','(ˆ),(ˆ[)';',( xxxx+−≡− ψψ

)''(ˆ †trψ

)(ˆ trψ


Equation of Motion of G

• Using equation of motion of field operators, Gcan be shown to satisfy

• Then write Σ≡VH +Σxc

)'()'(

]ˆˆˆˆ[)',(

)'';()(

4††

ttxx

xdNTNrrvi

txxtGxht

i

−−=

+

−

∂

∂

∫δδ

ψψψψ ∫Σ− G

- defines Σ


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 Vxc:

{ }

{ } )'()'()'';(

)'()'()'';(

KSxcH

xcH

ttxxtxxtGVVht

i

ttxxtxxtGVht

i

−−=

++−

∂

∂

−−=

Σ++−

∂

∂

δδ

δδ


Lehmann Representation for G

( )( )

<−=−

>−=+=

<>−

=

−

+

∑

µεψ

µεψ

µεδεω

ω

sNNs

NsNs

s

s

s s

ss

EENsN

EEsNNf

i

ffG

,1

,1

,)(ˆ,1

,,1)(ˆ)(

,)'(*)(

),',(

r

rr

rrrr

∓

• 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


Our “Cast of Players”

• G – Green’s function

• Σxc – Self-energy (related to G as discussed)

• W – Dynamically screened Coulombinteraction

• P – Irreducible polarisation propagator

• Γ – Vertex function


Hedin’s Equations

• With Σ/G relation, exact closed equations for G, Σ etc.

The GW Approximation



• Iterate Hedin’s equations once starting

with Σ=0



∫ +=Σ ')';',()';',(2

);',( ' ωωωωπ

ω δωdeGW

i ixxxxxx

where ),( ξrx ≡ is space+spin

In the time domain

)';',()';',()';',( ttGttiWtt −−=−Σ xxxxxx

W is the dynamically screened Coulomb interaction between electrons

Implementation of GW


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.

Total energy Updated self-energy

Space-time method:

H.N. Rojas, RWG and

R.J. Needs, Phys. Rev. Lett. 74

1827 (1995)

Spectral Properties


G0W0 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

http://www.physics.ohio-

state.edu/~wilkins/vita/publications.htm

l#reviews]


Self-Consistency

Arno Schindlmayr, Pablo García-González and RWG

Fully self-consistent

Σ=iGW: Conserving

Partially self-consistent

Σ=iGW0: Conserving

Non-self-consistent

Σ=iG0W0: Non-conserving

Kris Delaney


Tests for Be atom (QP energies)

K.T. Delaney, P. García-González, Angel Rubio, Patrick Rinke and RWG, Phys. Rev. Lett. 2004

Total Energy from Many-Body Perturbation Theory


Ab initio GW total energy

• Start with LDA calculation

• Use GW space-time method to calculate self-energy ),',( itrrΣ using a

double grid in real space, and a grid in imaginary time

• Calculate new Green’s function using ( ) 0

1LDAxc0 )(1 GVGG

−−Σ−=

• Recalculate Σ using new G and new W (based on new G)

• Repeat to self-consistency

• Calculate total energy using self-consistent G

• Galitskii-Migdal expression for ground-state total energy

( )∫ ∫∞−

→ ++=

µ

ωωω nuclVAhddE rrrrrr '3

21 ),',()( , where


Total Energy of the H.E.G.

2 4 6 8 100.0

-0.1

-0.2

-0.3

-0.4

-0.5

-0.6

Exact

G0W

0

GW0

GW

ε XC (

Ha)

rs (a.u.)

2 3 4 5 6-0.10

-0.15

-0.20

-0.25

-0.30

Pablo García-González and RWG, PRB 2001-0.2

-0.3

-0.4

-0.5

-0.6

Exact

G0W

0

GW0

GW

ε XC (

Ha) 2 3 4 5 6

-0.10

-0.15

-0.20

-0.25

-0.30


Van der Waals forces

Pablo García-González and RWG, PRL 2002

GW

Vertex Corrections Beyond GW


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 Vxc

LDA


DFT-based Vertex

• (A)LDA vertex in W (G0W0LDA) yields slight improvement

over G0W0

• If ALDA vertex is also added into Σ, things get much worse!

Morris, Stankovski, Rinke, Delaney, RWG, PRB 2007

Morris, Stankovski et al.


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]


DFT-based Vertex (3)… and restores the good results of G0W0 for

the I.P. of atoms!


DFT-based Vertex (4)

• Future question:

– Interplay between self-consistency and vertex

Using self-energy approaches to shed light on

DFT functionals


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 Vxc 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.)

Acknowledgements and Further Reading


Collaborators

• Pablo

García-González

• Kris Delaney

• Patrick Rinke

• Martin

Stankovski

• Peter Bokes

• Angel Rubio


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 http://www-users.york.ac.uk/~rwg3

gw many-body theory for electronic structure · 2019. 10. 2. · • mpbt quantities better behaved...

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