2145-9

Spring College on Computational Nanoscience

P. UMARI

17 - 28 May 2010

CNR-IOM DEMOCRITOS Theory at Elettra Group, Basovizza

Trieste Italy

Applications of GW. GW quasi-particle spectra from occupied states only: latest developments.

Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

GW quasi-particle spectra from occupied states only:latest developments

Paolo Umari

CNR-IOM DEMOCRITOS Theory@Elettra Group,Basovizza-Bazovica, Trieste, Italy

May 24, 2010

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Outline

Introduction

Optimal polarizability basis

GW without empty states

Examples

Polarizability basis: optimal vs plane waves

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Photoemission spectroscopy

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Many-Body Perturbation Theory

G.

Onida, L. Reining, A. Rubio,

Rev. Mod. Phys.74, 601 (2002)

Quasi-particle energies

N → N ± 1

E∗N±1 − EN

Many-Body Perturbation Theory (MBPT)

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

GWA

M.S. Hybertsen and S.G. Louie, Phys. Rev. Lett 55, 1418 (1985)

En ' εn + 〈ΣGW (En)〉n − 〈Vxc〉n

ΣGW (r, r′;ω) =i

2π

∫dω′G(r, r′;ω − ω′)W (r, r′;ω′)

W = v + v · Π · v where Π = P · (1− v · P )−1

P (r, r′;ω) =1

2π

∫dω′G(r, r′;ω − ω′)G(r, r′;ω′)

G(r, r′;ω) =∑

i

ψi(r)ψ∗i (r

′)

ω − εi ± iδ

For accurate(?) calculations: analytic continuation methodM.M. Rieger, L. Steinbeck, I.D. White, H.N. Rojas and R.W. Godby, Comp. Phys. Comm. 117 211 (1999)

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

GWA

Two big challenges:

Computational cost:

We must represent operators O(r, r′)prohibitive for large systems

Sums over empty states

In principle sums over all empty-statesprohibitive for large systemsanalogous to DFPT

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Polarization basis

If an optimal representation of P can be found:

P (r, r′;ω) '∑αβ

Φα(r)P αβ(ω)Φβ(r′)

Π(r, r′;ω) '∑αβ

Φα(r)Παβ(ω)Φβ(r′)

W (r, r′;ω) '∫dr′′dr′′′

∑αβ

v(r, r′′)Φα(r′′)Παβ(ω)Φβ(r′′′)v(r′′′, r′)

then a huge speed-up can be achieved

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Polarization basis

The same optimal basis for Π(r, r′;ω) and P (r, r′;ω)with:

P (r, r′;ω) =∑v,c

ψv(r)ψc(r)ψv(r′)ψc(r′)εc − εv + ω

we want to build a basis for the products in real space of valence andconduction states

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Old idea: building an optimal representation

1. Wannier transformation:

ψv → wv

ψc → wc

2. Reject the small overlaps:

wv(r)wc(r) → Φvc(r)

3. Orthonormalization

Φvc → Φµ

we use a treshold s2 for rejecting almost linear dependent terms

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Implementation

Analytic continuation approachM.M. Rieger, L. Steinbeck, I.D. White, H.N. Rojas and R.W. Godby, Comp. Phys. Comm. 117 (1999) 211

Γ-sampling only, real wavefunctions

Implemented in the Quantum-ESPRESSO code; a community

project for high-quality quantum simulation software, coordinated by P.

Giannozzi. See http:/www.quantum-espresso.org

See: PU, G.Stenuit, S.Baroni, Phys. Rev. B 79, 201104(R) (2009)

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Old idea: Benzene molecule

Isolated benzene molecule

C6H6

Convergence within 10 meV is achieved with E2c ≥ 30 eV (300 states) and a

polarizability basis set of only 340 elements (s2 ≤ 0.1).

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Example: Si3N4

amorphous Si3N4

model obtained byCar-Parrinello MD

152 atoms at exp.density

344 valence states

USpseudopotentials

Neutron, IR, Raman

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Example: Si3N4

amorphous Si3N4

EG(DOS)=2.9 eV

EG(GW)=4.4 eV

EG(exp)=∼5 eV

See: L.Giacomazzi and P.U. PRB 80, 144201 (2009)

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

New idea: without empty states

An optimal polarizability basis can be found solving:∫dr′P (r, r′; t = 0)Φµ = qµΦµ(r) with: qµ > q∗

where:

P (r, r′; t = 0) = Qv(r, r′)Qc(r, r′) = Qv(r, r′)(δ(r− r′)−Qv(r, r′))

we can approximate:

P (r, r′; t = 0) ≈ Qv(r, r′)Qe(r, r′)

withQe ≈

∑G,G′

Qc|G〉R−1G,G′〈G′|QcΘ(G2 − E)Θ(G′2 − E)

with:

RG,G′ =

∫dr〈G|Qc|r〉〈r|Qc|G′〉

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

New idea: benzene

IP of the benzene molecule

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Caffeine

IP of the caffeine molecule

IP(Ec) = IP(∞)− α

Ec

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

The solution I

We work on the imaginary frequency axisM.M. Rieger, L. Steinbeck, I.D. White, H.N. Rojas and R.W. Godby, Comp. Phys. Comm. 117 211 (1999)

Polarizability basis

we work with real wavefunctions (Γ-point)

Let Φµ be a basis for the irreducible polarizability P

Φµ is also a basis for the reducible polarizability Π

plane waveslocalized basis setsoptimal basis sets

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

The solution II

Sternheimer approach for P

The polarizability matrix P µν(iω) :

P µν(iω) = −4<

∑v,c

∫drdr′Φµ(r)ψv(r)ψc(r)ψv(r′)ψc(r′)Φν(r′)

εc − εv + iω.

the projector over the conduction manifold Qc:

Qc(r, r′) =∑

c

ψc(r)ψc(r′) = δ(r− r′)−∑

v

ψv(r)ψv(r′),

with the notation:〈r|ψiΦν〉 = ψi(r)Φν(r).

We can now eliminate the sum over c :

P µν(iω) = −4<

∑v

〈Φµψv|Qc(H − εv + iω)−1Qc|ψvΦν〉,

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

The solution III

The computational load can be hugely reduced:

with an optimal basis:

〈r|Qc|ψvΦµ〉 ≈∑α

t0α(r)Tα,vµ,

We can easily solve:

〈t0α|(H − εv + iω)−1|t0β〉

for every εv and every ωFor each t0α: Lanczos-chain: t1α, t

2α, t

3α, ..

〈tiα|H − εv + iω|tjα〉 = δi,j(di − εv + iω) + δi,j+1f

i + δi,j−1fj .

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

The solution III

Equivalent Lanczos approach for the self-energy:

〈r|ψn(vΦµ)〉 ≈∑α

s0α(r)Sα,nµ,

with:

〈r|(vΦµ)〉 =

∫dr′v(r, r′)Φµ(r′)

Implemented in the quantum-Espresso packagewww.quantum-espresso.org

See: PU, G. Stenuit, and S. Baroni, Phys. Rev. B 81, 115104 (2010)

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Caffeine

Convergence with respect to Lanczos steps

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Caffeine

IP of the caffeine molecule

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Benzene

IP of the benzene molecule

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Benzene

IPs of the benzene molecule

E = 10 Ry q∗ = 0.035 a.u.N = 2900

E = 10 Ry q∗ = 14.5 a.u.N = 500

Extrapolations

Plane waves E = 5 RyN = 1500

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Benzene

Extension to extended systems

Head(G = 0,G′ = 0) and wings (G = 0,G′ 6= 0) of the symmetricdielectric matrix are calculated using Lanczos chains (k-pointssampling implemented)

Wings are projected over the polarizability basis vectors

Element G = 0 added to the polarizability basis

v(G) = 1Ω

∫dq 1

|G+q|2

Grid on imaginary frequency can be denser around ω = 0

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Benzene

Extension to extended systems: test

Bulk Si: 64 atoms cubic cell

Optimal polarizability basis:E∗=2Ry, q∗=2.7 a.u. (#2000)

k-points sampling for: head andwings, DFT charge density

state LDA GW Expt.

Γ1v -11.94 -11.63 -12.5X1v -7.80 -8.77X4v -2.88 -2.90 -2.9,-3.3Γ25v 0. 0. 0.X1c 0.67 1.36 1.25

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Tetraphenylporphyrin

C44H30N4

IPexp=6.4 eVIPLDA=5.0 eVIPGWA=6.7 eV

Experimental PS spectrum: N.E. Gruhn, et al. Inor Chem (1999)

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Tetraphenylporphyrin

Experimental PS spectrum: C. Cudia Castellarin and A. Goldoni

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Tetraphenylporphyrin

Tetraphenylporphyrin: analysis

-3

-2

-1

0

-25 -20 -15 -10 -5 0

∆ E

= E

GW

A-E

PB

E (

eV)

EPBE (eV)

-10

-8

-6

-4

-2

0

2

-30 -25 -20 -15 -10 -5 0E

corr =

EH

F-E

GW

A (

eV)

EGWA (eV)

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Tetraphenylporphyrin

Tetraphenylporphyrin: analysis

0

0.1

0.2

0.3

0.4

-30 -25 -20 -15 -10 -5 0

Eco

rr/E

X

EGWA (eV)

0

10

20

30

40

Den

sity

of S

tate

s (e

V-1

)

TPP

Tot DOSC (s)

C (p: σ + π)C (p: π)

0

2

4

6N (s)N (p)

0

4

8

12

-30 -25 -20 -15 -10 -5 0

Energy (eV)

H (s)

G. Stenuit, C. Castellarin-Cudia,O.Plekan, V. Feyer, K.C. Prince, A. Goldoni,

and PU, PCCP (accepted) (2010).

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Indene

Indene moleculeGWW is now a standard tool for PS analysis:

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

H2O molecule

Polarizability basis for H2O: optimal

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-20 -10 0 10 20

Im Σ

(iω)

(ryd

)

ω (ryd)

300400500600700

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

-20 -10 0 10 20

Re

Σ(iω

) (r

yd)

ω (ryd)

300400500600700

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

H2O molecule

Polarizability basis for H2O: plane-waves

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-60 -40 -20 0 20 40 60

Im Σ

(iω)

(ryd

)

ω (ryd)

5 ryd7 ryd9 ryd

11 rydOpt. Bas. #200Opt. Bas. #300

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

-60 -40 -20 0 20 40 60

Re

Σ(iω

) (r

yd)

ω (ryd)

5 ryd7 ryd9 ryd

11 rydOpt. Bas. #200Opt. Bas. #300

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

H2O molecule

VIP convergence: plane-waves

-12.4

-12.2

-12

-11.8

-11.6

-11.4

-11.2

-11

5 6 7 8 9 10 11

V.I.

P (

eV)

pmat-cutoff (Ryd)

PW basisConverged Opt. Pol. Bas.

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Conclusions

Concept and importance of optimal polarizability basis

Lanczos chain approach

Large systems affordable without loss of accuracy

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Acknowledgments

G. Stenuit (CNR-INFM DEMOCRITOS)

S. Baroni (CNR-INFM DEMOCRITOS & SISSA)

L. Giacomazzi (CNR-INFM DEMOCRITOS & ICTP)

X.-F. Qian (MIT)

A. Goldoni (Elettra)

C. Cudia Castellarin (Elettra)

V Feyer Elettra

K.C. Prince (Elettra)

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Introduction Polarizability basis GW without empty states Lanczos chains Results Polarizability basis Conclusion

Don’t go to the beach!

Dr. L. Martin Samos’s lecture: today 14:00

TDDFT&GWW hands on: tomorrow 15:30 (Adriatico)

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