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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