Many-Body I - TDDFT.org

Many-Body Theory 

and DFT / TDDFT 

Rex Godby

Outline 

• Introduction to many-body perturbation 

theory 

• The GW approximation (non-SC and SC) 

• Implementation of GW 

• Spectral properties 

• GW total energy 

• Vertex corrections beyond GW 

Many-body theory and TDDFT - Benasque, 30/31 Aug 2006 2 

+

The Basic Problem 

Nucleus α (fixed) 

Electron i 

( H ˆ − E)Ψ ( r 1 , r 2 ,...r N ) = 0, where 

∑ 

H ˆ = − 1 2 ∇ i 2 

i 

+ 1 2 

i≠ j 

− 

+ 1 2 

∑ 

∑ 

1 

r i − r j 

Z α 

electronic K. E 

e - e interaction 

r i − R α 

electronic P.E. 

Z α Z β 

∑ 

α ≠β R α − R β 

nucleus - nucleus P. E. 

iα 

• The second term makes the problem interesting! 

(Otherwise see undergraduate courses: Bloch’s 

theorem, energy bands, etc.) 

Many-body theory and TDDFT - Benasque, 30/31 Aug 2006 3

Approaches to the 

electron-electron interaction 

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

[ − 1 2 ∇2 + V ext (r) + V Hartree (r) + V xc (r) − ε i ]ψ i (r) = 0 

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

[ − 1 2 ∇2 + V ext + V Hartree + Σ xc (ω ) − ω]G(r, r ′,ω) = δ(r − r ′) 

(which, inter alia, leads to the quasiparticle equation 

[ − 1 2 ∇2 + 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) 


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) 


Introduction to Many-Body 

Perturbation Theory

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 


+

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 


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 


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) 


Lehmann Representation for G 

G( 

rt; 

r' 

t') 

= 

f 

s 

⎪⎧ 

( r) 

= ⎨ 

⎪⎩ 

∑ 

s 

Nψˆ ( r) 

fs 

( r) 

fs 

ω −ε 

N + 1, s 

N −1, 

sψˆ ( r) 

N 

s 

*( r') 

∓ iδ 

, ε 

, ε 

s 

s 

⎧< 

⎫ 

, ε 

s ⎨ ⎬µ 

⎩> 

⎭ 

= 

= 

E 

E 

N + 1, s 

N 

− 

− 

E 

E 

N 

N −1, 

s 

> µ 

< µ 

• 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 


Spectral function and 

quasiparticles from Im G 

Im G non-interacting 

interacting 

E E µ 

1 2 

ω 

[ − 1 2 ∇2 + V ext + V Hartree + Σ xc (ε) − ε]ψ (r) = 0) 


Equation of Motion of G 

• Using equation of motion of field 

operators, G can be shown to satisfy 

⎛ 

⎜i 

⎝ 

∂ 

∂t 

− 

⎞ 

h( 

x) 

⎟G( 

xt; 

⎠ 

x' 

t') 

+ 

i 

∫ 

v( 

r, 

r') 

N T[ 

ψˆ 

ψψψ ˆ ˆ ˆ 

† 

† 

] 

N 

d 

4 

x 

− 

∫ ΣG 

= δ ( x 

− 

x') 

δ ( t 

−t') 

- defines Σ 


⎛ 

⎜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 

⎞ 

⎟ 

⎠ 

KS 

− x') 

δ ( t 


Hedin’s Equations 

• Exact closed equations for G, Σ etc. 


The GW Approximation

The GW approximation 

• Iterate Hedin’s equations once starting 

with Σ=0 


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 


Vertex Corrections beyond GW 

Arno Schindlmayr, Thomas Pollehn and RWG 


GW 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 

• Similar diagrammatic theories for two-particle G, 

e.g. optical absorption 

• Natural domain: quasiparticle energies, band 

structure, spectral function 

• Other applications: ground state total energy, 

etc. 


GW vs. TDDFT (2) 

DFT 

• Natural domain (of ordinary DFT): ground 

state total energy 

• TDDFT permits study of excited states of 

N-electron system 

• TDDFT also allows formulation of groundstate 

total energy through ACFDT 


GW vs. DFT / TDDFT 

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 


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. 

Space-time method: 

H.N. Rojas, RWG and 

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

1827 (1995) 

Total energy 

Updated 

self-energy 


Self-Consistency 

Fully self-consistent 

Σ=iGW: Conserving 

Partially self-consistent 

Σ=iGW 0 : Conserving 

Non-self-consistent 

Σ=iG 0 W 0 : Non-conserving 

Arno Schindlmayr, Pablo García-González and RWG 

Kris Delaney 


Spectral Properties: 

Effect of self-consistency 

and pseudopotentials

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 

http://www.physics.ohiostate.edu/~wilkins/vita/publications.htm 

l#reviews] 


Tests for Be atom (QP energies) 

K.T. Delaney, P. García-González, Angel Rubio, Patrick Rinke and 

RWG, Phys. Rev. Lett. 2004 


SC+Pseudopotentials: 

Tests for Be atom 




Tests for Be atom (3) 




Total Energy from Many- 

Body Perturbation Theory

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 


Total Energy of the H.E.G. 

ε XC 

(Ha) 

-0.6 

-0.5 

-0.4 

-0.3 

-0.2 

-0.1 

-0.30 

-0.25 

-0.20 

-0.15 

-0.10 

2 3 4 5 6 

Exact 

G 0 

W 0 

GW 0 

GW 

0.0 

2 4 6 8 10 

r s 

(a.u.) 

Pablo García-González and RWG, PRB 2001 


Van der Waals forces 

GW 

Pablo García-González and RWG, PRL 2002 


GW Total Energy for Real Matter 

• Bulk silicon (Tim Gould) 

• E tot in mHa per electron: 

QMC –995 

LDA –991 = QMC + 4 

G 0 W 0 –1002 = QMC – 7 

GW –997 = QMC – 2 


+

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) 

• 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 


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 V xc 

LDA 


ALDA Vertex (a.k.a. “GWΓ”) 

• ALDA vertex in W (G 0 W 0 

LDA 

) yields slight improvement 

over G 0 W 0 for atoms and jellium. However, this vertex is 

unphysical when added into Σ (G 0 W 0 Γ LDA ) 

Morris, Stankovski, Rinke, Delaney, RWG 


ALDA Vertex 2 

Morris, Stankovski, Rinke, Delaney, RWG 


Acknowledgements and 

Summary

Collaborators 

• Pablo 

García-González 

• Jeil Jung 

• Kris Delaney 

• Tim Gould 

• 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

Many-Body I - TDDFT.org

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?