Excited States and GW/BSE · 2011. 7. 22. · Excited States and GW/BSE Patrick Rinke...

Excited States and GW /BSE

Patrick Rinke

Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin - Germany

Hands-on Tutorial on Ab Initio Molecular Simulations: Toward aFirst-Principles Understanding of Materials Properties and Functions

Patrick Rinke (FHI) Excited States Berlin 2011 1 / 54

Excited States in (Material) Science – Are Ubiquitous

Theoretical SpectroscopyExperiment/Spectroscopy

appropriate?

accurate?

Material Properties + Applications

appropriate?

accurate?

Excited States in (Material) Science – Are Ubiquitous

Theoretical SpectroscopyExperiment/Spectroscopy

appropriate?

accurate?

Material Properties + Applications

appropriate?

accurate?

photoemission

optical absorption DFT+ +BSEG W0 0

DFT+G W0 0

Band structures: photo-electron spectroscopy

Photoemission

Inverse Photoemission

Absorption

BSETDDFT

Photo-Electron Excitation Energies

Photoemission electron removal

removal energy

N-electrons

Photoemission

|N〉 : N -particle ground state

E(N, s) : total energy in state s

ψ(r)|N〉

removal energy

(N-1)-electrons

Photoemission

|N〉 : N -particle ground state

E(N, s) : total energy in state s

〈N − 1, s|ψ(r)|N〉

removal energy

E(N)− E(N − 1, s)

(N-1)-electrons

Photoemission

|N〉 : N -particle ground state|N − 1, s〉 : (N − 1)-particle excited state sE(N, s) : total energy in state s

ψs(r) = 〈N − 1, s|ψ(r)|N〉

removal energy

ǫs = E(N)− E(N − 1, s)

(N-1)-electrons

Photoemission

ψs(r) = 〈N − 1, s|ψ(r)|N〉

removal energy

ǫs = E(N)− E(N − 1, s)

Inverse Photoemission electron addition

ψs(r) = 〈N |ψ(r)|N + 1, s〉

addition energy

ǫs = E(N + 1, s)− E(N)

(N-1)-electrons

Photoemission

Single Particle Green’s Functions

Lehmann representation of single particle Green’s function G:

G(r, r′; ǫ) = limη→0+

ψs(r)ψ∗s(r

ǫ− (ǫs + iη sgn(Ef − ǫs))

Excitation energies are poles of the Green’s function!

spectroscopically relevant quantity: spectral function A:

A(ǫ) = −1

dr limr′→r

ImG(r, r′; ǫ)

Single Particle Green’s Functions

Lehmann representation of single particle Green’s function G:

G(r, r′; ǫ) = limη→0+

ψs(r)ψ∗s(r

ǫ− (ǫs + iη sgn(Ef − ǫs))

Excitation energies are poles of the Green’s function!

spectroscopically relevant quantity: spectral function A:

A(ǫ) = −1

dr limr′→r

ImG(r, r′; ǫ)

Experimental: angle-resolved photoemission spectroscopy

Binding Energy (eV)8 6 4 2 0

Masaki Kobayashi, PhD dissertation

Experimental: angle-resolved photoemission spectroscopy

Binding Energy (eV)8 6 4 2 0

Masaki Kobayashi, PhD dissertation

ARPES vs GW : the quasiparticle concept

Quasiparticle:

single-particle like excitation

A ( )k

e spectralfunction

Quasiparticle:

Ak(ǫ) = ImGk(ǫ) ≈Zk

ǫ− (ǫk + iΓk)

ǫk : excitation energyΓk : lifetimeZk : renormalisation

A ( )k

e quasiparticle-peak

Quasiparticle:

Ak(ǫ) = ImGk(ǫ) ≈Zk

ǫ− (ǫk + iΓk)

ǫk : excitation energyΓk : lifetimeZk : renormalisation

A ( )k

e quasiparticle-peak

A ( )k e quasiparticle-

Green’s function and screening

ejected

system polarizes

ejected

system polarizes

polarization is

time-dependent

hole is screened dynamically

ejected

system polarizes

polarization is

time-dependent

hole is screened dynamically

quasiparticle

GW Approximation - Screened Electrons

G: propagator

GS=GW :

W: screened

Coulomb

Self-Energy

ΣGW (r, r′, ω) = −i

dωeiωηG(r, r′, ω + ω′)W (r, r′, ω′)

Green’s function is solution to Hedin’s equations

Hedin’s equations - exact L. Hedin, Phys. Rev. 139, A796 (1965)

notation: 1 = (r1, σ1, t1)

P (1, 2) = −i

G(2, 3)G(4, 2+)Γ(3, 4, 1)d(3, 4)

W (1, 2) = v(1, 2) +

v(1, 3)P (3, 4)W (4, 2)d(3, 4)

Σ(1, 2) = i

G(1, 4)W (1+, 3)Γ(4, 2, 3)d(3, 4)

Γ(1, 2, 3) = δ(1, 2)δ(1, 3) +

∫δΣ(1, 2)

δG(4, 5)G(4, 6)G(7, 5)Γ(6, 7, 3)d(4, 5, 6, 7)

Dyson’s equations

G−1(1, 2) = G−10 (1, 2)− Σ(1, 2)

links non-interacting (G0) with interacting (G) system

Hedin’s equations - exact L. Hedin, Phys. Rev. 139, A796 (1965)

P (1, 2) = −i

G(2, 3)G(4, 2+)Γ(3, 4, 1)d(3, 4)

W (1, 2) = v(1, 2) +

v(1, 3)P (3, 4)W (4, 2)d(3, 4)

Σ(1, 2) = i

G(1, 4)W (1+, 3)Γ(4, 2, 3)d(3, 4)

Γ(1, 2, 3) = δ(1, 2)δ(1, 3) +

∫δΣ(1, 2)

δG(4, 5)G(4, 6)G(7, 5)Γ(6, 7, 3)d(4, 5, 6, 7)

Dyson’s equations

G−1(1, 2) = G−10 (1, 2)− Σ(1, 2)

links non-interacting (G0) with interacting (G) system

exact : yes

tractable: no

But, do not despair!

Hedin’s GW equations L. Hedin, Phys. Rev. 139, A796 (1965)

Γ(1, 2, 3) = δ(1, 2)δ(1, 3)

P (1, 2) = −i

G(2, 3)G(4, 2+)Γ(3, 4, 1)d(3, 4)

= −iG(1, 2+)G(2, 1)

W (1, 2) = v(1, 2) +

v(1, 3)P (3, 4)W (4, 2)d(3, 4)

Σ(1, 2) = i

G(1, 4)W (1+, 3)Γ(4, 2, 3)d(3, 4)

= iG(1, 2)W (1+ , 2)

G0W0 Approximation in Practise

1 start from Kohn-Sham calculation for ǫKSs and φKS

2 KS Green’s function:

G0(r, r′; ǫ) = lim

η→0+

φKSs (r)φKS∗

s (r′)

ǫ− (ǫKSs + iη sgn(Ef − ǫKS

3 Polarisability:

χ0(r, r′; ǫ) = −

dǫ′G0(r, r′; ǫ′ − ǫ)G0(r

′, r; ǫ′)

4 Dielectric function:

ε(r, r′, ǫ) = δ(r − r′)−

dr′′v(r− r′′)χ0(r

′′, r′; ǫ)

G0W0 Approximation in Practise

5 Screened interaction:

W0(r, r′, ǫ) =

dr′′ε−1(r, r′′; ǫ)v(r′′ − r′)

6 G0W0 self-energy:

Σ(r, r′; ǫ) =i

dǫ′eiǫ′δG0(r, r

′; ǫ+ ǫ′)W0(r, r′; ǫ′)

7 Quasiparticle equation:

h0(r)ψs(r) +

dr′Σ(r, r′; ǫqps )ψs(r′) = ǫqps ψs(r)

solve for quasiparticle energies ǫqps and wavefunctions ψs(r)

perturbation theory: ψs(r) = φKSs (r)

⇒ ǫqps = ǫKSs + 〈s|Σ(ǫqps )|s〉 − 〈s|vxc|s〉

GW Approximation - Screened Electrons

G: propagator

GS=GW :

W: screened

Coulomb

Self-Energy: Σ = Σx + Σc

Σx = iGv: exact (Hartree-Fock) exchange

Σc = iG(W − v): correlation (screening due to other electrons)

On the importance of screening – Silicon

ǫqpnk = ǫLDA

nk + 〈φnk|Σx +Σc(ǫqpnk)− vxc|φnk〉

exp: 1.17 eV

Hartree-Fock (exchange-only) gap much too large

Screening is very important in solids!

ARPES – GW: wurtzite ZnO

H L A G K M G

ZnO G W0 0

@OEPx(cLDA)

Yan, Rinke, Winkelnkemper, Qteish, Bimberg, Scheffler, Van de Walle,

Semicond. Sci. Technol. 26, 014037 (2011)

wurtzite ZnO – convergence of G0W0

0 1000 2000 3000number of bands

5 10 15 20 25 30 35band cutoff energy (Ha)

wz-ZnO

plane wave cutoff: 35 Ha

0@OEPx(cLDA)

see also related work by Shih, Xue, Zhang, Cohen and Louie, PRL 105, 146401 (2010)

LDA and G0W0 – the band gap question

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0Experimental Band Gap (eV)

selected group:IV, III-V, IIa-VI, IIb-VIcompounds

courtesy of Ricardo Gomez-Abal

DFT eigenvalues and excitation energies

Density-functional theory:

in exact DFT: ionization potential given byKohn-Sham eigenvalue of highest occupied state

IKS = −ǫN (N)

otherwise Janak’s theorem: (PRA 18, 7165 (1978)))

∂ns= ǫs

rearranging and making mid point approximation:

E(N + 1, s)−E(N) =

dn ǫs(n) ≈ ǫs(0.5)

Ionisation Potential, Electron Affinity and (Band) Gaps

Could use total energy method to compute

ǫs = E(N ± 1, s)− E(N)

Ionisation potential: minimal energy to remove an electron

I = E(N − 1)− E(N)

Electron affinity: minimal energy to add an electron

A = E(N)− E(N + 1)

(Band) gap: Egap = I −A

Ionisation Potential, Electron Affinity and Band Gap

He Be Mg Ca Sr Cu Ag Au Li Na K Rb Cs Zn-60.0

∆I in

E(N-1)-E(N)-ε

Ionisation potential in the LDA

Reference: NIST – Atomic reference data

∆SCF versus eigenvalues for finite systems

∞≈≈

data courtesy of Max Pinheiro

Band Gap of Semiconductors

Band gap: Egap = I −A = E(N + 1)− 2E(N) + E(N − 1)

For solids E(N + 1) and E(N − 1) cannot be reliable computed

In Kohn-Sham the highest occupied state is exact

Egap = ǫKSN+1(N + 1)− ǫKS

= ǫKSN+1(N + 1)− ǫKS

N+1(N)︸︷︷︸

+ ǫKSN+1(N)− ǫKS

N (N)︸︷︷︸

EKSgap

For solids: N ≫ 1 ⇒ ∆n(r) → 0 for N → N + 1

⇒ discontinuity in vxc upon changing the particle number

∆xc =

(δExc[n]

δn(r)

∣∣∣∣N+1

−δExc[n]

δn(r)

∣∣∣∣N

Ionisation Potential, Electron Affinity and Band Gap

so what about ∆SCF for excitations?

∆SCF: ǫs = EDFT(N)− EDFT(N ± 1, s)

reasonably good for I and A of small finite systems

most functionals suffer from delocalization or self-interaction error(Science 321, 792 (2008))

⇒ the more delocalized the state, the larger the error

only truely justified for differences of ground states ionisation potential, electron affinity excited states that are ground states of particular symmetry

difficult to find excited state wavefunction (density)

excited state density is not unique

separate calculation for every excitation needed not practical for large systems or solids

Can we also describe localized d- or f -electrons?

-8 -6 -4 -2 0 2 4 6 8 10Energy (eV)

XPS+BISXPS+XAS

DFT-LDA erroneously predicts metallic state

Exp: J. W. Allen, J. Magn. Magn. Mater. (1985), D. R. Mullins et al., Surf. Sci. (1998)

Ce2O3 in LDA+U

-8 -6 -4 -2 0 2 4 6 8 10Energy (eV)

XPS+BISXPS+XAS

3 LDA+U

LDA+U splits f -peaks and opens gap

f -Electron Systems – Lanthanide Oxides

f -Electrons

localized

itinerant

localized and itinerant electrons

⇒ rich physics and chemistry

LDA/GGA often not adequate

often termed strongly correlated

Lanthanide Oxides

Ln2O3 (Ln=lanthanide series)

technologically importantmaterials e.g. catalysis

f -Electron Systems – Lanthanide Oxides

f -Electrons

localized

itinerant

localized and itinerant electrons

⇒ rich physics and chemistry

LDA/GGA often not adequate

often termed strongly correlated

Lanthanide Oxides

Ln2O3 (Ln=lanthanide series)

technologically importantmaterials e.g. catalysis

How well will GW perform?

exact exchange essential for localized electrons

screening essential for itinerant electrons

answers from new LAPW-based GW code

Ce2O3 in G 0W 0@LDA+U

-8 -6 -4 -2 0 2 4 6 8 10Energy (eV)

XPS+BISXPS+XAS

0@LDA+U

G 0W 0@LDA+U reproduces main peaks and gaps

H. Jiang, R. Gomez-Abal, P. Rinke and M. Scheffler, Phys. Rev. Lett. 102, 126403 (2009)

GW for surfaces – CO on NaCl

gas phasechemisorbed/

physisorbed

change upon absorption: chemical bonding polarization of surface polarization of molecule

PRL 103, 056803 (2009), PRL 102, 046802 (2009), PRL 97 216405 (2006)

gas phasechemisorbed/

physisorbed

image effect

PRL 103, 056803 (2009), PRL 102, 046802 (2009), PRL 97 216405 (2006)

Na ClOC

HOMO-LUMO gap of CO:

gap/eV LDA G0W0@LDA

free CO : 6.9 15.1CO@NaCl: 7.4 13.1

surface polarization significant here

PRL 103, 056803 (2009), PRL 102, 046802 (2009), PRL 97 216405 (2006)

GW for surfaces – CO on Ultrathin NaCl on Ge

Supported ultrathin films are novel nano-systems withunexpected features (PRL 99, 086101 (2007))

Additional electrons/holes on CO see two interfaces

Will the CO gap depend on NaCl thickness?

GW for surfaces – CO on Ultrathin NaCl on Ge

12.512.612.712.8

2 3 4NaCl layers

7.47.57.67.77.8

DFT-LDA

π∗ spl

3.03.13.23.33.4

2π∗

2 3 4NaCl layers

-2.1-2.0-1.9-1.8-1.7

)Polarization at NaCl/Ge interface gives layer-dependent CO gap

⇒ molecular levels can be tuned by polarization engineering: interesting for molecular electronics and quantum transport

C. Freysoldt, P. Rinke, and M. Scheffler, PRL 103, 056803 (2009)

Can we Eliminate Periodic Boundary Conditions in GW for

Finite Systems?

FHI-aims – FHI ab initio molecular simulations package

Local atom-centered basis

ϕi[lm](r) =ui(r)

rYlm(Ω)

Standard DFT (LDA, GGA)

all-electron

periodic, cluster systems on equalfooting

favorable scaling (system size andCPUs)

Beyond DFT (only for finite systems)

Hartree-Fock, hybrid functionals,MP2, RPA

quasiparticle self-energies: GW ,MBPT2

V. Blum, R. Gehrke, F. Hanke, P. Havu, V. Havu, X. Ren, K. Reuter, and M. Scheffler,

Comp. Phys. Comm. 180, 2175 (2009).

CO revisited – Comparison of FHI-aims and GWST

G0W0@LDA HOMO-LUMO gap of gas phase CO:

GWST† : 15.10 eVFHI-aims : 15.05 eVexperiment∗ : 15.80 eV

†GWST: GW space-time code (real-space/plane-waves)(Steinbeck, Godby et al. CPC 117 (1999), CPC 125 (2000))

∗Constants of Diatomic Molecules (1979), PRL 22, 1034 (1969)

CO revisited – G0W0 with Different Starting Points

CO/eV ǫHOMO ǫLUMO

LDA -9.12 -2.25PBE -9.04 -2.00PBE0 -10.75 -0.75

G0W0@LDA -13.31 1.74G0W0@PBE -13.17 1.84G0W0@PBE0 -13.76 2.02

experiment∗ -14.00 1.80

PBE0 improves over LDA and PBE here

G0W0@PBE0 gives the best performance(X. Ren, P. Rinke, and M. Scheffler, Phys. Rev. B 80,

045402 (2009))

GW – The Issue of Self-Consistency

Hedin’s GW equations:

G(1, 2) = G0(1, 2) notation: 1 = (r1, σ1, t1)

Γ(1, 2, 3) = δ(1, 2)δ(1, 3)

P (1, 2) = −iG(1, 2)G(2, 1+)

W (1, 2) = v(1, 2) +

v(1, 3)P (3, 4)W (4, 2)d(3, 4)

Σ(1, 2) = iG(1, 2)W (2, 1)

Dyson’s equation:

G−1 (1, 2) = G−10 (1, 2) − Σ (1, 2)

consistency

self-consisten

CO – self-consistent GW

-20 -15 -10 -5 0 5Energy [eV]

-14 -13

CO revisited – self-consistent GW

CO/eV ǫHOMO ǫLUMO

G0W0@LDA -13.31 1.74G0W0@PBE -13.17 1.84G0W0@PBE0 -13.76 2.02scGW -13.63 2.16

experiment∗ -14.00 1.80

scGW independent of starting point

G0W0@PBE0 outperforms scGW in this case

Dependence on the starting point: N2

Galitskii-Migdal formula: Etot = −i

Tr [(ω + h0)G(ω)]dω

1 2 3 4 5 6 7Iteration

-109.8

-109.6

-109.4

-109.2T

scGW@PBEscGW@HF

-20 -18 -16 -14Energy [eV]

G0W0@HFG0W0@PBEscGW@HFscGW@PBE

At self-consistency no dependence on input Green’s function!

scGW and ionization potentials

4 6 8 10 12 14 16Experimental IEs [eV]

4 6 8 10 12 14 16

16 LDA eigenvalueG

Self-Consistent GWSelf-Consistent GW

deviation from exp.

LDA 40.0%G0W0@LDA 5.6%scGW 2.9%GW0@LDA 1.5%

set taken from Rostgaard, Jacobsen, and Thygesen, PRB 81, 085103, (2010)

The scGW density

density from Green’s function: ρ(r) = −i∑

σ Gσσ(r, r, τ = 0+)

Photoemission

Absorption

BSETDDFT

Optical absorption – independent particles

Absorption spectrum = Im εM (ω) (macroscopic dielectric constant)

Optical absorption – independent particles

Absorption spectrum = Im εM (ω) (macroscopic dielectric constant)

Fermi’s Golden rule:

Im εM (ω) =16π

occ∑

unocc∑

|〈ψv|v|ψc〉|2 δ(ǫc − ǫv − ω)

v : velocity operator

Optical absorption – response function

Dielectric constant:

Im εM (ω) = limq→0

ε−1

G=0,G′=0(q, ω)

Dielectric function:

ε−1(r, r′, ω) = δ(r− r′)−

dr′′v(r− r′′)χ(r′′, r′, ω)

Response function (in TDDFT):

χ = χ0 + χ0

v + fxc

Response function (in RPA):

χ = χ0 + χ0

v +fxc

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0ω (eV)

Exp.TDLDARPA

absorption spectrum of silicon

from Sottile, Olevano, and Reining, PRL 91, 056402 (2003)

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0ω (eV)

Exp."GW"TDLDARPA

Optical absorption – concept of excitons

-exciton

electron-hole interaction lowers the energy

electron-hole pair (exciton) forms

excitonic state

electron-hole interaction lowers the energy

electron-hole pair (exciton) forms

G(1,2) G(1,2,1’,2’)

one particle

Green’s function

two particle

Green’s function

related quantity: two particle response function S(1, 2, 1′, 2′)

Optical absorption – Bethe-Salpeter equation

Approximations for G(1, 2, 1′, 2′) (or S(1, 2, 1′, 2′))

GW approximation for Σ

approximateδΣ

δGby iW

take W to be static

⇒ effective eigenvalue problem in electron-hole basis(like Casida’s equation)

HeffAλ = EλAλ

Optical absorption – Bethe-Salpeter equation

Bethe-Salpeter equation (BSE):

HeffAλ = EλAλ

Heffvv′cc′ = (ǫc − ǫv)δvv′δcc′ + 〈vc|v|v′c′〉 − 〈vc|W |v′c′〉

v ≡ Coulomb potential without the long-range part ǫv,c are G0W0 quasiparticle energies

macroscopic dielectric constant

ImεM (ω) =8π2

∣∣∣∣∣

occ∑

unocc∑

〈v|p|c〉

ǫc − ǫv

∣∣∣∣∣

δ(ω−Eλ)

p ≡ dipole operator

Optical absorption – Recipe for BSE

1 perform a DFT calculation

2 perform a G0W0 calculation

3 calculate 〈vc|v|v′c′〉 and 〈vc|W |v′c′〉

4 solve HeffAλ = EλAλ

5 calculate Im εM (ω)

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0ω (eV)

Exp.BSETDLDA

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0ω (eV)

Exp.BSE"GW"

BSE creates bound exciton

BSE shifts spectral weightPatrick Rinke (FHI) Excited States Berlin 2011 52 / 54

Photoemission

Absorption

BSETDDFT

Acknowledgements

GW surface calculations

Christoph Freysoldt

GW for f -electrons

Hong JiangRicardo Gomez-Abal

G0W0 in FHI-aims

Xinguo Ren

scGW in FHI-aims

Fabio CarusoAngel Rubio

Support and Ideas

Matthias Scheffler

Rex Goby

FHI-aims

Volker Blumthe whole FHI-aims developer team

Excited States and GW/BSE · 2011. 7. 22. · Excited States and GW/BSE Patrick Rinke...

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GW/BSE methods in chemistry: Computational aspects

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44 CE - Gewiss 44 ce_rev2.1... · GW 44 257 190 140 140 200 154 148 144 98- 94 GW 44 218 GW 44 418 GW 44 438 GW 44 258 240 190 160 254 200 167,5 194 140 - 97,5 GW 44 219 GW 44 419

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