QuasiParticle Self-consistent GW ApproximationMark van Schilfgaarde, Takao Kotani

Arizona State University

Review of GW approximationA new kind self-consistent (scGW) GWAApply to broad classes of materialssp compounds,d (f) metals, NiO, MnO, GaMnAs, …Total energy through Luttinger Ward functional

Cast of CharactersF. Aryasetiawan (AIST) “grandfather” original GW code from which existing method was developed.S. Faleev (Sandia): made scGW work!T. Miyake (TITech) Total energies w/ LW functional

Failures in the Local Density Approximation(Standard ab initio Approximation in solids)

bandgaps too small fails for correlated materials

Systematic overbinding

Significant errors in

mechanical properties

GW Approximation: Analog to Hartree-Fock

Start from some non-interacting hamiltonian H0.

1.

2.

3.

4.

( )2

0 00

1, ' 2 effH V G

Hω∇

= − + ⇒ =−

r r Example: H0 = HLDA

0 0D i G G= − × Linear response polarization function

1( , , ) ( , , ) ( , )W r r dr r r V r rω ε ω−′ ′′ ′′ ′′ ′= ∫

vD−=1ε Compute ε within the time-dependent Hartree approximation (aka RPA)

Dynamically screened exchangeRecover HF theory by ε→1: 1( , ') ( , ') 'W v −→ = −r r r r r r

GW and G LDA W LDA Approximation

5.

6. 0i G WΣ =Self-energy

( ) ( ) ( ) ( ) [ ]2

01, ', + , ',

2H extH V V G G G

Hω ω

ω∇

= − + + Σ ⇒ = =−

r r r r r r

( , , ) ( , ', ) ( , , )2

iir r d G r r e W r rδωω ω ω ω ωπ

′′ ′ ′ ′′ ′ ′Σ = +∫

Key points:Construct G from G0 .QP levels are at poles G.Σ is ω-dependent and non-hermitianH is nonlocal

If the non-interacting hamiltonian H0 is H0 = H LDA

and Σ = ΣLDA = G LDA W LDA .

G LDA W LDA ApproximationOften G LDA W LDA is referred to as the “GW”approximation.

Note that Σ depends on the LDAAdditional common approximations:

PseudopotentialPseudopotential or shape approximations.or shape approximations.Exceptions: Exceptions: Ku,EquiluzKu,Equiluz; Arnaud; ; Arnaud; KotaniKotani

LDA treatment for the core (LDA treatment for the core (Σ≠ΣΣ≠Σallall))Exceptions: Exceptions: AryasetiawanAryasetiawan; ; KotaniKotani

PlasmonPlasmon--pole approximationpole approximation

Implementations of GW differ in the following:skip

•Treatment of core – usually use LDA core•Use of plasmon-pole approximation: assume ε(ω) from ε(0)•Approximations to potential (e.g. PP; ASA; semilocal Σ)•Choice of basis for one-electron (usu. LDA) wave functions•Choice of basis for screened Coulomb interaction W•No Self-consistency

Present work is most accurate implementation to date:

•No plasmon-pole approximation•Full-potential treatment (features in common w/ LAPW, PAW)•LDA basis: smoothed Hankels + local orbitals (beyond linear)•W expanded in IPW + product basis inside MT spheres•Core treated on footing similar to valence (HF at lowest level)•New kind of self-consistency

Advantage: consistent treatment of RPA: Norm conservation

Drawback: GW is a perturbation theory based on ansatz of existence of quasiparticles.D and W have physical interpretation as “response” and “screened exchange;” also e-, h+, have physical meaning.scGW moves away from this picture … no well-defined e-, h+, D or W

Self-consistent GW

( ) ( )

1

/ 2 H extH T V V

H G D iGG W v iGW

G n iG d V H

ε

ω π

−

= + + + Σ

⇒ ⇒ = − ⇒ = ⇒ Σ =

⇒ = ⇒ →∫

Observed failings in self-consistent GW

Conventional wisdom : Something wrong with scGW.But G LDA W LDA accurate to 0.1 eV.Ergo, better to stick with G LDA W LDA.

NoninteractingGW

G0W0

Bandwidth of homogeneous electron gas widens relative to noninteracting case–when it should narrow(Holm and von Barth, PRB 57, 2108 (1998))

Si bandgap ~1.9 eV according to PP calculation: Schone and Eguiluz, PRL 81, 1662 (1998) (LAPW by Ku and Eguiluz, PRL 89, 126401 (2002),contradict this finding)

LDA

expt

Results of G LDA W LDA ApproximationBandgaps in sp systems Minority bands of Fe

←Cationsemicore dstates

Energy bands of NiO →

Many other examples …

QuasiParticle Self-Consistent GWKey principles:

Self-consistency is essential (show later)Stay within QP picture:⇒ Everything (Σ, true G, physical observables)

should be functional of a noninteracting G0.⇒ Poles of G0 should correspond to QP levelsWhy should QPscGW be better than scGW?

Pole of G (QP weight) is reduced by Z=1-∂Σ/∂ω(shifted to high plasmon energies)Bare polarization D=G×G reduced by Zocc ×Zunocc

D is too small ⇒ ε too small ⇒ W=V/ε too large

( )0

0 0 0 0

1 1( ) ( ) / ( )

GH H

ωω ω ω ω ω ω ω

= ≈− − Σ − − Σ − ∂Σ ∂ −

QuasiParticle Self-Consistent GW II

Guiding principleFind a prescription for a noninteracting G0where poles of G0 = poles of G (QP levels)

No unique choice for G0 !Solution to exact many-body S-eqn does not depend

on choice, but the closer G0 is to G, the more rapidly the perturbation series converges.

How can we construct the optimum G0?

Given G0 we can construct

What is the best way to make ?

or ...0

GWG G⎯⎯⎯⎯→

?0G G⎯⎯→

QuasiParticle Self-Consistent GW III

Use Landau’s QP picture to deduce form of H0, G0

True G(r, r’, ω) should written as

Landau’s QP picture says there should be a one-to-one correspondence between QP excitations and the true G.

The optimum H0, G0 defined by {Ei0, ψi

0 } should reproduce the true {Ei , ψi} as well as possible

*( ) ( ')( , ', ) residual part( )

i i i

i i i

ZGE i

ψ ψωω

= +− + Γ∑ r rr r

QuasiParticle Self-Consistent GW IV

Need a prescription to generate an energy-independent and hermitian potential Vxc(r,r’) that is“as close as possible” to Σ(r, r’, Ei)

Solution : minimize norm Where

Can show that (with some modest approximations)

Replaces LDA Vxc in making {Ei ,ψi} : Ei(Vxc)= Ei(Σ)

2( ) xc

ij i ijij

E VΣ −∑( ) ( ) xc xc

ij i i i j ij i jE E V Vψ ψ ψ ψΣ = Σ =

Re ( ) Re ( )2

i jxcij i j

E EV ψ ψ

Σ + Σ=

Recall Results of G LDA W LDA Approximation …Bandgaps in sp systems Minority bands of Fe

←Cationsemicore dstates

Energy bands of NiO →

Many other examples …

Results of scGW theory I: sp bonded systems

LDA: broken blueQPscGW: greenGLDAWLDA: Dotted redO: Experiment

m* (scGW) = 0.073m* (LDA) = 0.022m* (expt) = 0.067

Ga d level well described Na bandwidth reduced by 15%

Gap too large by ~0.3 eVBand dispersions ~0.1 eV

Results of scGW theory II: sp bonded systems

• = Direct gap= Indirect gap

Errors are small, highly systematic (not corrected for phonon contribution to gap!)

Γ−Γ transitionsoverestimated by 0.2±0.1 eVOther transitionsoverestimated by 0.1±0.1 eVGaps for Si and Ge within ~0.1 eV

of scGW using LAPW, Ku and Equiluz, PRL 89, 126401 (2002) Sole exception: MgO

Origin of errors in QPscGW levels (Semiconductors)

1. Exact many-body theory has vertex correction Γ : Σ = GW Γ = G ε−1V Γ

2. And we compute ε only within the RPA (bubble diagrams)…time-dependent Hartree approximation: response is purely

electrostatic -- no correlations included in the screening⇒ε should be underestimated

This is what happens:comparing ε∞ to experiment …

Proposal: ε∞ too small ⇒ W too large ⇒Eg too large.

QPscGW theory applied to simple d bonded systems

Valence d bandwidths Wd, relative position of s and d band bottoms, exchange splittings Ex, and magnetic moments in

3d compounds. Wd (eV) εsd (eV) LDA scGW Expt LDA scGW Expt Ti 6.0 5.7 3.5 4.3 Cr 6.6 6.2 3.5 4.3 Fe 5.2 4.6 4.6 3.6 4.4 4.6 Co 4.1 3.8 3.7 4.6 5.3 4.9±1 Ni 4.4 4.0 4.0 4.4 5.0 5.5

moment (µB) Ex (eV) LDA scGW Expt LDA scGW Expt Fe 2.18 2.20 2.2 1.95 1.67 1.75 Co 1.59 1.68 1.6 1.70 1.21 1.08 Ni 0.63 0.74 0.57 0.6 0.5 0.3 Cr 0.64 1.38(?) — MnO 4.48 4.76 4.6 NiO 1.28 1.72 1.9 MnAs 3.0 3.5 3.4

Fe (minority)

QPscGW theory in TM oxidesBroadening of valence band½−1eV error in fundamental gap

NiO: AFM insulatorFaleev, MvS, Kotani,PRL 2004

TiO2 : insulator with d-like CB

Magnetic systems

maj GdP minGd:7f+3spdEr: 11f+3spd(ErAs, min spin). Orbital-dependent exchange splits f; only sp at Fermi s.

QPscGW theory in closed f-shell systemsGd: FM metalCeO2 : insulator with f-like CB

Need for full self-consistency Breakdown of RPA:f levels split by ~16 eV.Experiment: ~12 eV

(−)

(+)

Maj Min

Ce

O

Systematics of errors in QPscGW QP levelsSimple sp systems very well describedFundamental gap Eg systematically overestimated by ~0.2 eV.Small errors in m* for large Eg; scale with gap errors for small EgSmall errors in all known band dispersionsBandwidths widen when they should (e.g. diamond, oxides), narrow when

they should (e.g. Na) and stay unchanged when they shouldSimple transition metals also very well desribed.εsd, d bandwidth, exchange splitting systematically better than LDAMagnetic moment slightly overestimated

TM oxides show slightly larger errors.Error in Eg is ~ ½−1eVValence band well described in GW; poor in LDA

f-shell materials somewhat less well describedCeO2 demonstrates importance of full self-consistencyGd, GdN, GdAs, GdP seriously overestimate f-f splitting

Different errors for spin, charge degrees of freedom

Total Energies: Luttinger-Ward Functional

T. Miyake (TITech)F. Aryasetiawan (AIST)T. Kotani (Osaka):

Compare Luttinger-Ward, density-functionals:

( ) ( ) ( )

( ) ( ) ( )

*

*

'[ ][ ]; 0; , ',

[ ][ ]; 0;

LWn nLW

n nDF

DFn n

n

E GE E G G iG i

E nE E n nn

σ σ

σ σσ

ψ ψδ ωδ ω ε

δ ψ ψδ

= = =−

= = =

∑

∑

r rr r

r r r

Both exact, both satisfy variational principle.Both must be approximated to be tractable.

Luttinger-Ward Functional

( )( )

10 0 0

0 0 ee

[ ] 1 ln ln [ ]

, Energy and Green's function for V 0

"Exchange-correlation energy"1 ln 1 14

LW

RPAx

E G E Tr G G Tr G G G

E G

Tr Pv vP vP Pv

−⎡ ⎤ ⎡ ⎤= + − − − + Φ⎣ ⎦ ⎣ ⎦= =

Φ =

⎡ ⎤≅ Φ = Φ + − − + +⎣ ⎦

Key difference with DFT: G has much more variational freedom: orbitals, orbital-dependent (screened) exchange. Functional much less pathological

So far, use G=GLDA. (Aryasetiawan et al,PRL 88, 166401). Then

, [ ] [ ] [ ]LW RPA LDA LDA LDA LDA RPA LDAxc x cE G E E G G= − + Φ + Φ

Volume dependence of total energy for Na, LW functional(Takheshi Miyake), using

, [ ] [ ] [ ]LW RPA LDA LDA LDA LDA RPA LDAxc x cE G E E G G= − + Φ + Φ

Accuracy of functional not yet known (calculation of Φ difficult to do)

Propose a new QPscGW theoryPerturbations from a QP picture, as distinct from the conventional

scGW theoryA formal justification providedApparently improves on true scGW in practice

QPscGW theory provides an accurate description of QP levels for most of periodic table. Checks for:

• Simple sp systems• Elemental d systems• Complex d systems, including a variety of TM oxides• Closed f-shell systems

Dramatic improvements over LDA for many systemsErrors are:Consistently small, except for exchange splitting of Gd f statesConsistent with higher order diagrams left out of the theory

Starting point for simplifications (larger scale) and many-body perturbation theory (approach to exact theory)Total energy from Luttinger-Ward functional (in progress)

Conclusions

QuasiParticle Self-Consistent GW III0 0

0

0

Bare QP: ,

Turn on Dressed QP: ,

i i

i i

H E

H HH E

ψ

ψ−↓ ↓

Perturbation theory best when {Ei , ψI} = {Ei

0, ψI0 } as closely as possible.

LMTO Basis for All-Electron GW methodskip

Eigenstates expanded as generalized Linear Muffin-Tin Orbitals(both efficient and accurate).

0 1.0 2.0 3.00

0.2

0.4

0.6

Standard LMTO basis:envelopes → r -l as r→0Solves S-eqn for flat V=VMTZ.

V0

V

Ψ0

Ψ

RMT

r

Solves Schrodinger eqn for this potential

( )2 2

0

4( ) /2

( , ) 4 ( , ) 4 ( ) ( , )4( , , ) ( )s

L s L s L s

q r iL s L

H r G r Y g r r

H r e Y i eε

ε π ππε

ε− − ⋅

∆ + = − = − −∇

−= −

−q R

r r

q qq

( ) 4 ( , ) / ( , )MTZ L s L sV V G r H rπ= −r r r

1( , ) rsH r e

rεε − −=Smoothed

Hankels:envelopes→ r l as r→0.

Basis sets for GW

1. Orbital basis χ for wave functions. Then

Eigenstates ψikn are expanded in basis functions χi

ψikn and εn

k are found from solutions of the Schrodinger equation

( )( , ) ( , ) ( ) ( )2

i t tij i j

ij

dG t t e Gωω ω χ χπ

′− −′ ′ ′= ∑ ∑∫ k k

kr r k r r

*

( , )n n

i jij

n n

Gi

ψ ψω

ω ε δ=

−∑k k

kkm

Two independent basis sets are required.

2

ext H ( , )2

n nij ij j n ij

v Vm

ω ψ ε ψ⎛ ⎞∇

− + + + Σ =⎜ ⎟⎝ ⎠

∑ k k k kk

Muffin-Tin-Orbitals theory:Eigenfunctions ψkn expanded in MTO’s χs

( ) ( )n ns s

s

uψ χ= ∑k kr r Local orbital

Labels site, l,other attributes

•local functions φRLi inside augmentation spheres, i=1..2 or 1..3

•Plane waves in the interstitial

( )0 if any MT sphere

( ) exp otherwise

Pi

∈⎧= ⎨ ⎡ ⎤+ ⋅⎣ ⎦⎩

kG

rr

k G r

Then

( ) ( ) ( )n n n nRLi RLi

RLiPψ α φ β= +∑ ∑k k k k k

G GG

r r r

Linear method has two φRLi per RL. Local orbitals have 3.With local orbitals, LDA energy bands accurate over a wide range.

skip

Example : GaAsBlue : this method

(Methfessel and van S.) Red: old FP-LMTO method

(Methfessel and van S.)Green: QMTO-ASA

(Andersen) — bands from 2nd gen. ASA V(r).

2. For GW, need ψψ products to compute M.E. of coulomb

M = intermediate basis for expansion of products ψψ.= product basis B={φ×φ} inside MT spheres (Aryasetiawan)= Plane waves P×P→P in the interstitial (conventional methods)

Therefore:A complete basis M for products ψψ is:

Now

v M M v M Mψψ ψψ ψψ ψψ=

{ }( ), ( )IM P B≡ k kG r r

1 1 2 2 1 2

1 2

( ) ( ) ( )

( ) ( ) ( )

+ ( )

n n n nai ai

ai

n nRI

ai

P

B B

P P PP

ψ α φ β

ψ ψ φφ α α

β β

+

+

= +

= × × ×

× × ×

∑ ∑

∑

∑

k k k k kG G

G

k k k k

k kG

G

r r r

r r r

r

For a given potential and basis, make these quantities:

Eigenfunctions ψkn and eigenvalues εkn

Coulomb matrix

Eigenfunction products

Now we can carry out GW cycle. Make : ΣX , D, W, ΣC:

Exchange part ΣX of self-energy

Where the M must be orthogonalized

( ) , { , }IJ I Jv M v M I RLi= =k kk G

, { , }j i IM I RLiψ ψ − =kq q k G

( )BZ occ

X j i I IJ J i ji

j j M v Mψ ψ ψ ψ− −Σ = ∑∑ k kq q k q k q

k

q q k% %

-1

, ,( , ') ( ) =

BZ

I IJ J I J J II J J

v M v M M M M M= ∑ ∑k k k k k k

kr r k% % %

skip

Polarization function D

Important technical point:Fast integration contour for D: (Faleev)•Tetrahedron method ⇒ImD on real axis. •Hilbert transform to get ReD.

( ) 1( , ) 1IJW vD vω −= −qScreened Coulomb interaction:

skipCorrelation part ΣC of self-energy

( )

All

C ''

' 1, '2 '

BZ

n n I J n nn

IJi

n n M M

i d Wi

ψ ψ ψ ψ

ω ωπ ω ω ε δ

− −

∞

−∞−

Σ =

×− − ±

∑∑

∫

k kq q k q k q

k

q k

q q

k

% %

Standard integration contour for Σ:

(Use –iδ for occupied,+iδ for unoccupied states)

GW starting from LDA (non self-consistent)LDA LDA

x c( ) and , , , ( )nn nnn n D Wψ ε ω→ Σ Σk kr

Need diagonal part Σnn of Σ at QP energies Ekn.

Actually make Σ at LDA εkn. Correct by using Z factor.