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Theoretical Spectroscopy
Patrick Rinke
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin - Germany
Materials Theory Lecture
Patrick Rinke (FHI) Theoretical Spectroscopy TU Berlin 2012 1
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Excited States in (Material) Science – Are Ubiquitous
Theoretical SpectroscopyExperiment/Spectroscopy
appropriate?
accurate?
Material Properties + Applications
appropriate?
accurate?
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Excited States in (Material) Science – Are Ubiquitous
Theoretical SpectroscopyExperiment/Spectroscopy
appropriate?
accurate?
Material Properties + Applications
appropriate?
accurate?
photoemission DFT+G W0 0
from
first
principles
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Band structures: photo-electron spectroscopy
- -
-
-
+
Photoemission
hn
GW
- -
-
Inverse Photoemission
-
-
hn
-
GW
- -
-+
Absorption
-
hn
BSETDDFT
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Experimental: angle-resolved photoemission spectroscopy
photoemission: photon in – electron out
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Experimental: angle-resolved photoemission spectroscopy
ARPES
=+1qo
Binding Energy (eV)8 6 4 2 0
Masaki Kobayashi, PhD dissertation
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Experimental: angle-resolved photoemission spectroscopy
ARPES
=+1qo
Binding Energy (eV)8 6 4 2 0
Masaki Kobayashi, PhD dissertation
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ARPES vs GW : the quasiparticle concept
Quasiparticle:
single-particle like excitation
A ( )k
e spectralfunction
e
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ARPES vs GW : the quasiparticle concept
Quasiparticle:
single-particle like excitation
Ak(ǫ) = ImGk(ǫ) ≈Zk
ǫ− (ǫk + iΓk)
ǫk : excitation energyΓk : lifetimeZk : renormalisation
A ( )k
e quasiparticle-peak
ek
qp
Gk
e
The aim is to calculate the Green’s function G!
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Green’s function and screening
-
-
-
-
-
---
--
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Green’s function and screening
-
-
-
-
-
-
--- -
-
-
--
+
ejected
system polarizes
tt-
-- -
- -
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Green’s function and screening
-
-
-
-
-
-
---
-
-
-
-
-+- -
-
-
--
+
ejected
system polarizes
tt tt
polarization is
time-dependent
hole is screened dynamically
-
--
--
- -
- -
W (r, r′, t) =
∫
dr′′ε−1(r, r′′, t)
|r′′ − r′|
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Green’s function and screening
-
-
-
-
-
-
---
-
-
-
-
-+- -
-
-
--
+
ejected
system polarizes
tt tt
polarization is
time-dependent
hole is screened dynamically
-
--
--
- -
- -
quasiparticle
W (r, r′, t) =
∫
dr′′ε−1(r, r′′, t)
|r′′ − r′|
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GW Approximation - Screened Electrons
-
-
-
-
-
-+
-
--
G: propagator
W
GS=GW :
W: screened
Coulomb
Self-Energy
ΣGW (r, r′, t) = iG(r, r′, t)W (r, r′, t)
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G0W0 Approximation in Practise
1 start from Kohn-Sham calculation for ǫKSs and φKS
s (r)
2 KS Green’s function:
G0(r, r′; ǫ) = lim
η→0+
∑
s
φKSs (r)φKS∗
s (r′)
ǫ− (ǫKSs + iη sgn(Ef − ǫKS
s ))
3 GW : χ0 = iG0G0 → W0 = v/(1− χ0v) → Σ0 = iG0W0
4 G0W0: ǫqps = ǫKSs + 〈s| Σ(ǫqps ) |s〉 − 〈s|vxc|s〉
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GW Approximation - Screened Electrons
-
-
-
-
-
-+
-
--
G: propagator
W
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)
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On the importance of screening – Silicon
ǫqpnk = ǫLDAnk + 〈φnk|Σx +Σc(ǫ
qpnk)− vxc|φnk〉
exp: 1.17 eV
Hartree-Fock (exchange-only) gap much too large
Screening is very important in solids!
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ARPES – GW: wurtzite ZnO
H L A G K M G
Ener
gy r
elat
ive
to(e
V)
EF
0
-2
-4
-6
ZnO G W0 0
@OEPx(cLDA)
Yan, Rinke, Winkelnkemper, Qteish, Bimberg, Scheffler, Van de Walle,
Semicond. Sci. Technol. 26, 014037 (2011)
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G0W0 Band Gaps
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Experimental Band Gap [eV]
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
The
oret
ical
Ban
d G
ap [
eV]
LDAOEPx(cLDA)OEPx(cLDA) + G
0W
0
CdS
zb-GaNZnO
ZnS
CdSe
ZnSe
InNGe
wz-GaN
Rinke et al. New J. Phys. 7, 126 (2005), phys. stat. sol. (b) 245, 929 (2008)
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Nitrides (Al,In,Ga)N – Areas of (potential) Application
LEDs and Laser Diodes
High Frequency, High Power(e.g. HEMT)
Photovoltaics
Water Splitting
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Group-III nitrides
(potential) applications
solid state lighting
photovoltaics
RGB projectors
until ∼2006
debate over gap of InN
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Band gap of InN
1016
1017
1018
1019
1020
1021
carrier concentration (cm-3
)
0.40.60.81.01.21.41.61.82.02.22.4
Ega
p+E
F (eV
)
Wu et alHaddad et alIoffe InstituteBriot et alTansley and FoleyRF sputteringDC sputteringButcher et alSugita et al
Figure adapted from Butcher and TansleySuperlattices Microstruct. 38, 1 (2005)
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How can first principles calculations contribute?
Proposed reasons for band gap variation e.g. Butcher and Tansley Superlattices Microstruct. 38 (2005)
high carrier concentration ⇒ Moss-Burnstein effect
impurities, point defects, trapping centers
non-stoichiometry
formation of oxides and oxynitrides
metal inclusions, formation of metal clusters
First principles approach:
Density-functional theory (DFT) (ground state)
◮ atomistic control◮ stoichiometric, defect and impurity free structures
many-body perturbation theory in the GW approximation (excited state)
◮ method of choice for calculation of band gaps in solids
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InN – GW band structure and Moss-Burstein effect
M/8|k
[110]|
0.0
0.5
1.0
1.5E
nerg
y [e
V]
Γ A/4|k
[001]|
w-InN
1020
1019
1018
1017
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InN – GW and Moss-Burstein effect
1016
1017
1018
1019
1020
1021
carrier concentration (cm-3
)
0.60.81.01.21.41.61.82.02.2
Eg(n
) (e
V)
parabolic (meff
=0.07 m0)
G0W
0@OEPx(cLDA)
P. Rinke et al. APL 89, 161919 (2006)
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Group-III nitrides
(potential) applications
solid state lighting
photovoltaics
RGB projectors
now
debate over gap of InNsettled
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Future Technologies – Organic Electronics
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Hybrid inorganic/organic systems (HIOS)
ZnO/PTCDA organic electronics
organic field effect transistors (OFET)
organic light emitting diodes (OLED)
organic photovoltaic cells (OPVC)
. . .
Hybrid inorganic/organic interfacesare omnipresent!
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Hybrid inorganic/organic systems (HIOS)
ZnO/PTCDA
ZnO/p-sexiphenyl(courtesy of S. Blumstengel)
organic electronics
organic field effect transistors (OFET)
organic light emitting diodes (OLED)
organic photovoltaic cells (OPVC)
. . .
Hybrid inorganic/organic interfacesare omnipresent!
hybrid electronics
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Hybrid inorganic/organic systems (HIOS)
ZnO/p-sexiphenyl(courtesy of S. Blumstengel)
Combine the best of two worlds...
inorganic materials:
stable crystal structures
good growth control
high charge carrier mobilities
organic materials:
strong light-matter coupling
large chemical space gives diverserange of properties
... or more!
great potential for:
synergies between different materials
new physics at interface
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Potential for new physics at HIOS
solid organic
HIOS
Potential for new interface morphologies.
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Open questions at HIOS
-
polaronic-
band-like?
What is the nature of charge carriers?
New quasiparticles?
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Open questions at HIOS
valence band
conduction band
molecular states
-
HOMO
LUMO
-free carrier
What does the notion of “state” imply?
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Open questions at HIOS
valence band
conduction band
molecular states
-
HOMO
LUMO
- polaron
??
-
-
polaron?
free carrier?
+ polaron ??
What does the notion of “state” imply?
How do “states” line up energetically?
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Open questions at HIOS
valence band
conduction band
molecular states
-
HOMO
LUMO
- polaron
??
-
-
polaron?
free carrier?
hybrid state?
+ polaron ??
What does the notion of “state” imply?
How do “states” line up energetically?
Do hybrid “states” form?
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Molecules on surfaces – a challenge for first principles
The objective is:
to develop an atomistic understanding of (nano)materials
!"#"$%&'()*$$
$$$$$$$$$$$$$+,$$
$$$$$$$$$$$$$$$$-,.$
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Molecules on surfaces – a challenge for first principles
!"#$%&'(#'&)*+,#"&-)*(./+)#0-)*(1')2',#3(
4'5'6(%671*2'*,( 8%*(/'#(9%%6+(7*,'#%&-)*+(
:;%#1'(,#%*+$'#(
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Molecules on surfaces – a challenge for first principles
State-of-the-art for interfaces: density-functional theory (DFT)
local-density approximation (LDA)
generalized gradient approximation, e.g. PBE functional
!"#$%&'(#'&)*+,#"&-)*(./+)#0-)*(1')2',#3(
4'5'6(%671*2'*,( 8%*(/'#(9%%6+(7*,'#%&-)*+(
:;%#1'(,#%*+$'#(
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Molecules on surfaces – a challenge for first principles
Beyond state-of-the-art:
advanced density functionals e.g. hybrid functionals or RPA
Green’s function methods e.g. GW
!"#$%&'(#'&)*+,#"&-)*(./+)#0-)*(1')2',#3(
4'5'6(%671*2'*,( 8%*(/'#(9%%6+(7*,'#%&-)*+(
:;%#1'(,#%*+$'#(
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Molecules on surfaces – a challenge for first principles
!"#$%&'(#'&)*+,#"&-)*(./+)#0-)*(1')2',#3(
4'5'6(%671*2'*,( 8%*(/'#(9%%6+(7*,'#%&-)*+(
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Patrick Rinke (FHI) Theoretical Spectroscopy TU Berlin 2012 27
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Level alignment at HIOS
conductionband
valenceband
molecularstates
EF
EIB: lectrone i bnjection arrier
HIB: oleh i bnjection arrier
injection limited current:
j ∝ AT 2 exp
(
−charge injection barrier
kBT
)
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Molecular levels at a surface
gas phasesurface
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Molecular levels at a surface
gas phasesurface
+
electronaffinity (EA)
ionizationpotential (IP)
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Molecular levels at a surface
gas phasesurface
+
EA
IP
-
-
-
-
image effect
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Molecular levels at a surface
gas phasesurface
+
EA
IP
-
-
-
-
image potentials
classical image potential:
metal: −1
4zsemiconductor/insulator: −
ε− 1
4(ε+ 1)
1
zε: dielectric constant
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Molecular levels at a surface
gas phaseadsorbed
+
EA
IP
-
-
-
-
renormalization
+
classical image potential:
metal: −1
4zsemiconductor/insulator: −
ε− 1
4(ε+ 1)
1
zε: dielectric constant
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The screened Coulomb interaction W
Work with screened coulomb interaction:
W (r, r′, t) =
∫
dr′′ε−1(r, r′′, t)
|r′′ − r′|
ε(r, r′′, t): dielectric function
The challenge is to calculate W (r, r′′, t) from first principles!
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The screened Coulomb interaction W
Work with screened coulomb interaction:
W (r, r′, t) =
∫
dr′′ε−1(r, r′′, t)
|r′′ − r′|
ε(r, r′′, t): dielectric function
The challenge is to calculate W (r, r′′, t) from first principles!
we use random-phase approximation (RPA) for W :
= + + + ...
v: bare Coulomb interaction
c0: Kohn-Sham polarizability
W
formally scales with (system size)4
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From screening to quasiparticle energies
-
-
-
-
-
-+
-
--
G: propagator
W
GS=GW :
W: screened
Coulomb
Quasiparticle energies: G0W0 scheme
electron addition and removal energies
correction to DFT eigenvalues ǫDFT
nk :
ǫqpnk = ǫDFT
nk +ΣG0W0
nk (ǫqpnk)− vxcnk
includes image effect
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GW for surfaces – CO on NaCl
Na ClOC
HOMO-LUMO gap of CO:
gap/eV LDA G0W0 Exp.@LDA
free CO : 6.9 15.1 15.8∗
CO@NaCl: 7.4 13.1∗Constants of Diatomic Molecules (1979),
Phys. Rev. Lett. 22, 1034 (1969)
surface polarization significant here
C. Freysoldt, P. Rinke, and M. Scheffler, Phys. Rev. Lett. 103, 056803 (2009)
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GW for surfaces – CO on Ultrathin NaCl on Ge
CO
NaCl
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?
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GW for surfaces – CO on Ultrathin NaCl on Ge
12.512.612.712.8
G0W
0@LDA
2 3 4NaCl layers
7.47.57.67.77.8
DFT-LDA
5σ-2
π∗ spl
ittin
g (e
V)
3.03.13.23.33.4
2π∗
2 3 4NaCl layers
-2.1-2.0-1.9-1.8-1.7
5σ
G0W
0 cor
rect
ions
(eV
)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)
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Molecules on surfaces – a challenge for first principles
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4'5'6(%671*2'*,( 8%*(/'#(9%%6+(7*,'#%&-)*+(
:;%#1'(,#%*+$'#(
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Electron gas at TTF/TCNQ interface
TTF and TCNQ crystals have
large band gap
but 2 dimensional electron gasobserved at interface
Nat. Mat. 7, 574 (2008)
What is the origin?
TTF
TCNQ
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Electron gas at TTF/TCNQ interface
-3 -2 -1 0 1 2 3E-E
F [eV]
0
20
40
DO
S
DFT-PBE density of states
DFT-PBE:
predicts metallic interface
0.12 e/molecule charge transferto TCNQ
in seeming agreement withexperiment
TTF
TCNQ
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TTF/TCNQ dimer at infinite separation
TTF TCNQ
EAIP
exp PBE expPBE
4.0
6.7
9.5
7.0
5.6
2.81.0
!
IP(TTF) > EA(TCNQ)
erroneous charge transfer
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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
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∆SCF versus eigenvalues for finite systems
∞≈≈
data courtesy of Max Pinheiro
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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
but:
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
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TTF/TCNQ dimer at infinite separation
culprit: self-interaction error in PBE
add Hartree-Fock (HF) exchange to removeself-interaction
⇒ PBE0-like hybrid functional:
Exc(α) = α(EHFx − EPBE
x ) + EPBEc
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TTF/TCNQ dimer at infinite separation
0 0.2 0.4 0.6 0.8 1 α
-2
0
2
4
∆[e
V]
Experiment
PBE0(α)
artificial charge transfer
PBE0-like hybrid functional:
Exc(α) = α(EHFx − EPBE
x ) + EPBEc
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TTF/TCNQ dimer at infinite separation
0 0.2 0.4 0.6 0.8 1 α
-2
0
2
4
∆[e
V]
Experiment
PBE0(α)
artificial charge transfer
How to choose α?
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Performance of G0W0
TTF TCNQ
S
CH
H
NC
EA
IP
expG0W0 exp G0W0
6.7 6.7
9.5 9.7
3.7
-0.7
2.8
C6H4S4
all values in eV
C12H4N4
4.6
Exp.: JACS 112, 3302 (1990), J. Chem. Phys. 60, 1177 (1974),Adv. Mat. 21,1450 (2009),Mol. Cryst. Liquid Cryst. Inc. Nonlin. Opt. 171, 255 (1989)
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TTF/TCNQ dimer – implications for interface
no charge transfer, but small charge rearrangement
2D electron gas not due to charge transfer
HOMO
LUMO
PBE optimal !
electron density difference
optimal-α calculations for interface in progress
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Molecules on surfaces – a challenge for first principles
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.,)27&(+,#"&,"#'(
=6'&,#)*7&(+,#"&,"#'(
87>#%-)*%6(+,#"&,"#'(
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!
!
!
!
Patrick Rinke (FHI) Theoretical Spectroscopy TU Berlin 2012 44
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Molecules on surfaces – a challenge for first principles
!"#$%&'(#'&)*+,#"&-)*(./+)#0-)*(1')2',#3(
4'5'6(%671*2'*,( 8%*(/'#(9%%6+(7*,'#%&-)*+:(
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.,)27&(+,#"&,"#'(
=6'&,#)*7&(+,#"&,"#'(
87>#%-)*%6(+,#"&,"#'(
!
!
!
!
!
Patrick Rinke (FHI) Theoretical Spectroscopy TU Berlin 2012 44
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Acknowledgements
GW for surfaces
Christoph Freysoldt
Nitrides
Abdallah QteishMomme WinkelnkemperJorg Neugebauer
GW in FHI-aims
Xinguo Ren
Support and Ideas
Matthias Scheffler
TTF-TCNQ
Viktor AtallaMina Yoon
gwst code
Rex Goby
FHI-aims code
Volker Blumthe whole FHI-aims developer team
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