towards a unified description of ground and excited state...
TRANSCRIPT
Towards a unified description of ground and excitedstate properties: RPA vs GW
Patrick Rinke
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin - Germany
Towards Reality in Nanoscale Materials V
Patrick Rinke (FHI) RPA/GW Levi 2012 1
Wish list for “optimum” electronic structure approach
d/f -electrons(e.g. Cerium)
(bio)molecules
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)
applicable across dimensionalities:
molecules/clusters
wires/tubes
surfaces/films
solids, extended systems
applicable across periodic table:
from light to heavy elements
including d/f electronphysics/chemistry
Patrick Rinke (FHI) RPA/GW Levi 2012 2
Wish list for “optimum” electronic structure approach
d/f -electrons(e.g. Cerium)
(bio)molecules
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)
ground/excited state:
consistent description
parameter freefree from pathologies e.g.:
◮ self-interaction error◮ absence of van der Waals◮ band-gap problem
computationally efficient
gradients ⇒ structure relaxation
. . .
Patrick Rinke (FHI) RPA/GW Levi 2012 2
Wish list for “optimum” electronic structure approach
d/f -electrons(e.g. Cerium)
(bio)molecules
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)
ground/excited state:
consistent description
parameter freefree from pathologies e.g.:
◮ self-interaction error◮ absence of van der Waals◮ band-gap problem
computationally efficient
gradients ⇒ structure relaxation
. . .
Patrick Rinke (FHI) RPA/GW Levi 2012 2
Treating van der Waals interactions
-+-
+
-+
-+
-+
-+
fluctuatingdipoles
difficult, because they require non-local treatment
fluctuating dipoles described by dielectric function ε(r, r′, ω)
correlation energy in random-phase approximation (RPA):
ERPAc =
1
2π
∫ ∞
0dωTr [ln (ε(iω)) + (1− ε(iω))]
Patrick Rinke (FHI) RPA/GW Levi 2012 3
Attractive features of the RPA
exact exchange + random phase approximation: (EX+cRPA)@reference
EEX+cRPAtot = Ts + Eext + EH + Eexact
x + ERPAc
ERPAc = + + · · ·
“Exact exchange” (with Kohn-Sham orbitals): OEPx◮ self-interaction error considerably reduced
vdW interactions included automatically and seamlessly
Screening taken into account◮ applicable to metals/small gap systems (in contrast to MP2)
Patrick Rinke (FHI) RPA/GW Levi 2012 4
Attractive features of the RPA
exact exchange + random phase approximation: (EX+cRPA)@reference
EEX+cRPAtot = Ts + Eext + EH + Eexact
x + ERPAc
ERPAc = + + · · ·
“Exact exchange” (with Kohn-Sham orbitals): OEPx◮ self-interaction error considerably reduced
vdW interactions included automatically and seamlessly
Screening taken into account◮ applicable to metals/small gap systems (in contrast to MP2)
For an overview of RPA, see
our recent review article:Ren, Joas, Rinke, Scheffler, arXiv:1203.5536v1
Patrick Rinke (FHI) RPA/GW Levi 2012 4
van der Waals bonding
3.0 4.0 5.0 6.0Bond length (Å)
-15
-10
-5
0
5
10
15
20B
indi
ng e
nerg
y (m
eV)
5.0 6.0 7.0 8.0-4
-2
0
C6/R
6
Ar2
PBE
RPA@PBE
Reference
✔ Correct asymptotic behavior ⇒ crucial for large molecules
✘ Underbinding around the equilibrium distance
Patrick Rinke (FHI) RPA/GW Levi 2012 5
The SOSEX correction to RPA
Digrammatic representation (motivated in coupled cluster context)
ERPA+SOSEXc = + + ...
+
2nd−order 3rd−order
+ ...+
= +
Arising from the anti-symmetric nature of the many-body wave function
RPA+SOSEX is one-electron self-correlation free
D. L. Freeman, Phys. Rev. B 15, 5512 (1977). A. Gruneis et al., J. Chem. Phys. 131, 154115
(2009). J. Paier et al., J. Chem. Phys. 132, 094103 (2010); Erratum: 133, 179902 (2010).
Patrick Rinke (FHI) RPA/GW Levi 2012 6
The concept of single excitation (SE) correctionsRayleigh-Schrodinger perturbation theory:
H = H0 + H ′
Ground-state energy: E0 = E(0)0 + E
(1)0 + E
(2)0 + · · ·
zeroth-order: E(0)0 = 〈Φ0|H
0|Φ0〉
1st order: E(1)0 = 〈Φ0|H ′|Φ0〉
2nd order:
E(2)0 =
∑
n 6=0
|〈Φ0|H ′|Φn〉|2
E(0)0 − E
(0)n
=∑
i,a
|〈Φ0|H ′|Φi,a〉|2
E(0)0
− E(0)i,a
︸ ︷︷ ︸
Single excitations
+∑
ij,ab
|〈Φ0|H ′|Φij,ab〉|2
E(0)0
− E(0)ij,ab
︸ ︷︷ ︸
Double excitations
= ESEc MP2
SE accounts for orbitalrelaxations
+ +
Patrick Rinke (FHI) RPA/GW Levi 2012 7
Renormalized single excitations (rSE)
ErSEc = + ia + ...
a
bi+
3rd−order2nd−order
ia
i
f ai
ja
f ia
f ij
f
f ia
f biaj
ab
f
f
fpq = 〈p|f |q〉 : Hartree-Fock operator evaluated within Kohn-Sham orbitals
ESEc =
occ∑
i
unocc∑
a
|fia|2
ǫi − ǫa(ǫi, ǫa : Kohn-Sham orbital energies)
Patrick Rinke (FHI) RPA/GW Levi 2012 8
Renormalized single excitations (rSE)
ErSEc = + ia + ...
a
bi+
3rd−order2nd−order
ia
i
f ai
ja
f ia
f ij
f
f ia
f biaj
ab
f
f
fpq = 〈p|f |q〉 : Hartree-Fock operator evaluated within Kohn-Sham orbitals
ESEc =
occ∑
i
unocc∑
a
|fia|2
ǫi − ǫa(ǫi, ǫa : Kohn-Sham orbital energies)
“Diagonal rSE ” : including only terms with i = j = · · · and a = b = · · ·
ErSE-diagc =
occ∑
i
unocc∑
a
|fia|2
fii − faaProblematic for weak interactions
Patrick Rinke (FHI) RPA/GW Levi 2012 8
Renormalized single excitations (rSE)
ErSEc = + ia + ...
a
bi+
3rd−order2nd−order
ia
i
f ai
ja
f ia
f ij
f
f ia
f biaj
ab
f
f
fpq = 〈p|f |q〉 : Hartree-Fock operator evaluated within Kohn-Sham orbitals
ESEc =
occ∑
i
unocc∑
a
|fia|2
ǫi − ǫa(ǫi, ǫa : Kohn-Sham orbital energies)
“Diagonal rSE ” : including only terms with i = j = · · · and a = b = · · ·
ErSE-diagc =
occ∑
i
unocc∑
a
|fia|2
fii − faaProblematic for weak interactions
“full rSE ” : ErSE-fullc =
occ∑
i
unocc∑
a
∣∣∣fia
∣∣∣
2
fi − fa
Diagonalize fij and fab blocks separately → fi, fa, and transform fia → fia
Patrick Rinke (FHI) RPA/GW Levi 2012 8
Renormalized 2nd-order Perturbation Theory (r2PT)
ERPA+SOSEX+rSEc = + + ...
+ + ...+
( = RPA)
( = SOSEX)
+ + + ...+
2nd−order 3rd−order
( = rSE)
( = 2PT)
“r2PT” = “RPA+SOSEX+rSE”
Patrick Rinke (FHI) RPA/GW Levi 2012 9
van der Waals bonding
3.0 4.0 5.0 6.0Bond length (Å)
-15
-10
-5
0
5
10
15
20B
indi
ng e
nerg
y (m
eV)
5.0 6.0 7.0 8.0-4
-2
0
C6/R
6
Ar2
PBE
RPA@PBE
Reference
✔ Correct asymptotic behavior ⇒ crucial for large molecules
✘ Underbinding around the equilibrium distance
Patrick Rinke (FHI) RPA/GW Levi 2012 10
van der Waals bonding
3.0 4.0 5.0 6.0Bond length (Å)
-15
-10
-5
0
5
10
15
20B
indi
ng e
nerg
y (m
eV)
5.0 6.0 7.0 8.0-4
-2
0
C6/R
6
Ar2
PBE
RPA@PBERPA+SOSEX
Reference
SOSEX has almost no effect
Patrick Rinke (FHI) RPA/GW Levi 2012 10
van der Waals bonding
3.0 4.0 5.0 6.0Bond length (Å)
-15
-10
-5
0
5
10
15
20B
indi
ng e
nerg
y (m
eV)
5.0 6.0 7.0 8.0-4
-2
0
C6/R
6
Ar2
PBE
RPA@PBERPA+SOSEX
RPA+SEReference
much improved binding and asymptotics
Ren, Tkatchenko, Rinke, and Scheffler, Phys. Rev. Lett. 106, 153003 (2011)
Patrick Rinke (FHI) RPA/GW Levi 2012 10
van der Waals bonding
3.0 4.0 5.0 6.0Bond length (Å)
-15
-10
-5
0
5
10
15
20B
indi
ng e
nerg
y (m
eV)
5.0 6.0 7.0 8.0-4
-2
0
C6/R
6
Ar2
PBE
RPA@PBERPA+SOSEX
RPA+SEReference
r2PT
best overall performance
Patrick Rinke (FHI) RPA/GW Levi 2012 10
Van der Waals interactions - the S22 benchmark set
S22 set of binding energies
22 representative dimers:
◮ 7 hydrogen bonded◮ 8 dispersion bonded◮ 7 mixed character
CCSD(T) (“gold standard”) referencevalues
Jurecka et al., PCCP 8,1985 (2006)
Patrick Rinke (FHI) RPA/GW Levi 2012 11
Van der Waals interactions - the S22 benchmark set
58%
19%16%
PBE
MP2
(EX+cR
PA)@PBE
mean absolute percentage error
Patrick Rinke (FHI) RPA/GW Levi 2012 11
Van der Waals interactions - the S22 benchmark set
58%
19%16%
8%
PBE
MP2
(EX+cR
PA)@PBE
(EX+cR
PA+SE)@PBE
mean absolute percentage error
X. Ren, P. Rinke, A. Tkatchenko, M. Scheffler,Phys. Rev. Lett. 106, 153003 (2011)
Patrick Rinke (FHI) RPA/GW Levi 2012 11
Van der Waals interactions - the S22 benchmark set
58%
19%16%
7%8%
PBE
MP2
(EX+cR
PA)@PBE
r2PT@PBE
(EX+cR
PA+SE)@PBE
mean absolute percentage error
X. Ren, P. Rinke, A. Tkatchenko, M. Scheffler,Phys. Rev. Lett. 106, 153003 (2011)
Patrick Rinke (FHI) RPA/GW Levi 2012 11
Atomization energies
-0.8
-0.4
0.0
0.4
0.8M
ean
erro
r (e
V) Overbinding
Underbinding
G2 set Solids
PB
E
RP
A@
PB
E
PB
E0
PB
EPB
E0
RP
A@
PB
E
RPA underestimates bond strengths
Paier et al., J. Chem. Phys. 132, 094103 (2010); 133, 179902 (2010)Harl, Schimka, and Kresse, Phys. Rev. B 81, 115126 (2010)
Patrick Rinke (FHI) RPA/GW Levi 2012 12
Atomization energies
-0.8
-0.4
0.0
0.4
0.8M
ean
erro
r (e
V) Overbinding
Underbinding
G2 set Solids
PB
E
RP
A@
PB
ER
PA
+SO
SE
X
PB
E0
PB
EPB
E0
RP
A+S
OS
EX
RP
A@
PB
E
SOSEX alleviates the underbinding problem
Paier et al., J. Chem. Phys. 132, 094103 (2010); 133, 179902 (2010)Harl, Schimka, and Kresse, Phys. Rev. B 81, 115126 (2010)Paier et al., arXiv:1111.0173
Patrick Rinke (FHI) RPA/GW Levi 2012 12
Atomization energies
-0.8
-0.4
0.0
0.4
0.8M
ean
erro
r (e
V) Overbinding
Underbinding
G2 set Solids
PB
E
RP
A@
PB
ER
PA
+SO
SE
XR
PA
+SE
PB
E0
PB
EPB
E0
RP
A+S
OS
EX
RP
A@
PB
EH
F+R
PA
“HF+RPA”: EX@HF + RPA@PBE similar to“RPA+SE”: (EX+RPA+SE)@PBE
Paier, Ren, Rinke, Scuseria, Gruneis, Kresse, and Scheffler, NJP 14, 043002 (2012)Ren, Tkatchenko, Rinke, and Scheffler, Phys. Rev. Lett. 106, 153003 (2011)
Patrick Rinke (FHI) RPA/GW Levi 2012 12
Atomization energies
-0.8
-0.4
0.0
0.4
0.8M
ean
erro
r (e
V) Overbinding
Underbinding
G2 set Solids
PB
E
RP
A@
PB
ER
PA
+SO
SE
XR
PA
+SE
PB
E0
r2P
T
PB
EPB
E0
RP
A+S
OS
EX
RP
A@
PB
EH
F+R
PA
HF+
RP
A+S
OS
EX
“HF+RPA”: EX@HF + RPA@PBE similar to“RPA+SE”: (EX+RPA+SE)@PBE
Paier, Ren, Rinke, Scuseria, Gruneis, Kresse, and Scheffler, NJP 14, 043002 (2012)Ren, Tkatchenko, Rinke, and Scheffler, Phys. Rev. Lett. 106, 153003 (2011)
Patrick Rinke (FHI) RPA/GW Levi 2012 12
The f -electron system Cerium
iso-structural (fcc-fcc)α-γ phase transition
accompanied by large(15%-17%) volumecollapse
LDA/GGA capture only low-volume (α) phase
LDA+U/SIC-LDA capture only high-volume (γ) phase
LDA+DMFT so far restricted to high temperature
Patrick Rinke (FHI) RPA/GW Levi 2012 13
Cerium from first principles
All electrons are treated on the same quantum mechanical level
hybrid functional PBE0◮ 25% exact exchange◮ reduces self-interaction error
(EX+cRPA)@PBE0◮ physically meaningful screening◮ compatible with GW approach◮ cluster extrapolation approach
Patrick Rinke (FHI) RPA/GW Levi 2012 14
Cerium - cohesive energy
4.0 4.4 4.8 5.2 5.6 6.0 6.4a0 (Å)
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0E
coh (
eV)
α γLDA
PBE
Patrick Rinke (FHI) RPA/GW Levi 2012 15
Cerium - cohesive energy
4.0 4.4 4.8 5.2 5.6 6.0 6.4a0 (Å)
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0E
coh (
eV)
α γLDA
PBE
PBE0 "α-phase"
PBE0 "γ-phase"
Patrick Rinke (FHI) RPA/GW Levi 2012 15
Cerium - cohesive energy
4.0 4.4 4.8 5.2 5.6 6.0 6.4a0 (Å)
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0E
coh (
eV)
α γLDA
PBE
PBE0 "α-phase"
PBE0 "γ-phase"
Patrick Rinke (FHI) RPA/GW Levi 2012 15
Cerium - electronic structure
n(α)− n(γ) at a=4.6A
f -electrons more localized in γ-phase
Patrick Rinke (FHI) RPA/GW Levi 2012 16
Cerium - cohesive energy
4.0 4.4 4.8 5.2 5.6 6.0 6.4a0 (Å)
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
Eco
h (eV
)
α γLDA
PBE
PBE0 "α-phase"
PBE0 "γ-phase"
wrong phase ordering
Patrick Rinke (FHI) RPA/GW Levi 2012 17
Cerium - cohesive energy
4.2 4.4 4.6 4.8 5 5.2 5.4
a0 (Å)
-4.80
-4.75
-4.70
-4.65
-4.60
-4.55
Eco
h (eV
)
EX+cRPA@PBE0 α-phaseEX+cRPA@PBE0 γ-phase
-0.88 GPa
correct phase ordering
results for a Ce19 cluster
Patrick Rinke (FHI) RPA/GW Levi 2012 17
TTF/TCNQ dimer at infinite separation
0 2 4
Distance [Å]
-0.3
0.0
0.3
0.6
0.9
1.2
Ebi
nd [e
V]
MP2
HF
(EX+cRPA)@LDA
Patrick Rinke (FHI) RPA/GW Levi 2012 18
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
Patrick Rinke (FHI) RPA/GW Levi 2012 19
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
Patrick Rinke (FHI) RPA/GW Levi 2012 19
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
Patrick Rinke (FHI) RPA/GW Levi 2012 20
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
Patrick Rinke (FHI) RPA/GW Levi 2012 20
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
Patrick Rinke (FHI) RPA/GW Levi 2012 20
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?
Patrick Rinke (FHI) RPA/GW Levi 2012 20
Band structures: photo-electron spectroscopy
- -
-
-
+
Photoemission
hn
GW
- -
-
Inverse Photoemission
-
-
hn
-
GW
- -
-+
Absorption
-
hn
BSETDDFT
Patrick Rinke (FHI) RPA/GW Levi 2012 21
Experimental: angle-resolved photoemission spectroscopy
ARPES
=+1qo
Binding Energy (eV)8 6 4 2 0
Masaki Kobayashi, PhD dissertation
Patrick Rinke (FHI) RPA/GW Levi 2012 22
Experimental: angle-resolved photoemission spectroscopy
ARPES
=+1qo
Binding Energy (eV)8 6 4 2 0
Masaki Kobayashi, PhD dissertation
Patrick Rinke (FHI) RPA/GW Levi 2012 22
ARPES vs GW : the quasiparticle concept
Quasiparticle:
single-particle like excitation
A ( )k
e spectralfunction
e
Patrick Rinke (FHI) RPA/GW Levi 2012 23
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
Patrick Rinke (FHI) RPA/GW Levi 2012 23
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
Density-Functional Theory: G0
GW : χ0 = iG0G0 → W0 = v/(1− χ0v) → Σ0 = iG0W0
G−1 = G−10 − Σ0
Patrick Rinke (FHI) RPA/GW Levi 2012 23
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!
Patrick Rinke (FHI) RPA/GW Levi 2012 24
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
Patrick Rinke (FHI) RPA/GW Levi 2012 24
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 ǫDFTnk :
ǫqpnk = ǫDFTnk +ΣG0W0
nk (ǫqpnk)− vxcnk
includes image effect
Patrick Rinke (FHI) RPA/GW Levi 2012 25
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)
Patrick Rinke (FHI) RPA/GW Levi 2012 26
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 [e
V]
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)
Patrick Rinke (FHI) RPA/GW Levi 2012 27
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)
Patrick Rinke (FHI) RPA/GW Levi 2012 28
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)G
0W
0@PBE0(α)
optimal α
choose α to match G0W0
Patrick Rinke (FHI) RPA/GW Levi 2012 29
TTF/TCNQ dimer – RPA binding curve
0 2 4
Distance [Å]
-0.3
0.0
0.3
0.6
0.9
1.2
Ebi
nd [e
V]
(EX+cRPA)@PBE0(α*)
PBE0(α*)
(EX+cRPA)@LDA
Patrick Rinke (FHI) RPA/GW Levi 2012 30
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
Patrick Rinke (FHI) RPA/GW Levi 2012 31
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
Can we do the same in GW ?
need a GW charge density
self-consistent GW
Patrick Rinke (FHI) RPA/GW Levi 2012 31
G0W0 for water
-13.0
-12.5
-12.0
-11.5
-11.0
HO
MO
[eV
] G0W
0@PBE
G0W
0@HF
Experiment
10% EX20% EX
30%
50%
water molecule
Patrick Rinke (FHI) RPA/GW Levi 2012 32
GW – The Issue of Self-Consistency
Hedin’s GW equations:
G(1, 2) = G0(1, 2)
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)
self-
consistency
self-consisten
cy
Patrick Rinke (FHI) RPA/GW Levi 2012 33
scGW for water
-13.0
-12.5
-12.0
-11.5
-11.0
HO
MO
[eV
] G0W
0@PBE
G0W
0@HF
Experiment
10% EX20% EX
30%
50%
scGW
water molecule
Patrick Rinke (FHI) RPA/GW Levi 2012 34
scGW and ionization potentials
4 6 8 10 12 14 16Experimental IEs [eV]
4
6
8
10
12
14
16
The
oret
ical
IEs
[eV
]
4 6 8 10 12 14 16
4
6
8
10
12
14
16 LDA eigenvalueG
0W
0@LDA
Self-Consistent GWSelf-Consistent GW
0
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)
Patrick Rinke (FHI) RPA/GW Levi 2012 35
Dependence on the starting point: N2
Galitskii-Migdal formula: Etot = −i
2
∫
Tr [(ω + h0)G(ω)]dω
2π
1 2 3 4 5 6 7Iteration
-109.8
-109.6
-109.4
-109.2T
otal
Ene
rgy
[Ha]
scGW@PBEscGW@HF
N2
-20 -18 -16 -14Energy [eV]
Spe
ctru
m G0W0@HFG0W0@PBEscGW@HFscGW@PBE
N2
At self-consistency no dependence on input Green’s function!
Patrick Rinke (FHI) RPA/GW Levi 2012 36
The scGW density
ρ(CCSD)− ρ(HF ) (54)
ρ(scGW )− ρ(HF ) (55)ρ(PBE)− ρ(HF ) (56)
C
O
density from Green’s function: ρ(r) = −i∑
σ Gσσ(r, r, τ = 0+)
Caruso, Ren, Rinke, Rubio, Scheffler: arXiv:1202.3547v1
Patrick Rinke (FHI) RPA/GW Levi 2012 37
The scGW density
ρ(CCSD)− ρ(HF ) (54)
ρ(scGW )− ρ(HF ) (55)ρ(PBE)− ρ(HF ) (56)
C
O
density from Green’s function: ρ(r) = −i∑
σ Gσσ(r, r, τ = 0+)
Caruso, Ren, Rinke, Rubio, Scheffler: arXiv:1202.3547v1
Dipole moment (in Debye):
Exp. scGW CCSD HF PBE
0.11 0.07 0.06 −0.13 0.20
Patrick Rinke (FHI) RPA/GW Levi 2012 37
CO – ground state properties
CO d νvib µ Eb
Exp. (NIST database) 1.128 2169 0.11 11.11sc-GW 1.118 2322 0.07 10.19G0W0@HF 1.119 2647 – 11.88G0W0@PBE 1.143 2322 – 12.16(EX+cRPA)@HF 1.116 2321 – 10.19(EX+cRPA)@PBE 1.135 2115 – 10.45PBE 1.137 2128 0.20 11.67HF 1.102 2448 -0.13 7.63
Galitskii-Migdal formula:
EGM = −i
∫dω
2πTr {[ω + h0]G(ω)} + Eion
Caruso, Ren, Rinke, Rubio, Scheffler: arXiv:1202.3547v1
Patrick Rinke (FHI) RPA/GW Levi 2012 38
TTF/TCNQ dimer revisited
HOMO
LUMO
scGW opt !
Patrick Rinke (FHI) RPA/GW Levi 2012 39
RPA and GW – a consistent pair
GWmany-body perturbation theory
Dyson’s equation:
G−1 = G−10 − Σ[G]
total energy:
E = E[G]
e.g. Galitskii-Migdal
RPAdensity-functional theory
optimized effective potential:
G0
[Σ− vOEP
xc
]G0 = 0
Adiabatic connection fluctuationdissipation theorem (ACFDT):
Exc = Exc[χ0] = Exc[−iG0G0]
Patrick Rinke (FHI) RPA/GW Levi 2012 40
RPA and GW – a consistent pair
GWmany-body perturbation theory
Dyson’s equation:
G−1 = G−10 − Σ[G]
total energy:
E = E[G]
e.g. Galitskii-Migdal
RPAdensity-functional theory
optimized effective potential:
G0
[Σ− vOEP
xc
]G0 = 0
Adiabatic connection fluctuationdissipation theorem (ACFDT):
Exc = Exc[χ0] = Exc[−iG0G0]
scGW versus scRPA:
RPA also iterated to self-consistency (scRPA)
Do MBPT and DFT differ for the same diagrams?
Patrick Rinke (FHI) RPA/GW Levi 2012 40
The most strongly correlated system
1 2 3 4 5 6
Bond Length [Å]
-1.20
-1.15
-1.10
-1.05
-1.00
-0.95
-0.90
-0.85T
otal
Ene
rgy
[Ha]
H2 scGW
full CI
scRPA
full CI: L. Wolniewicz, J. Chem. Phys. 99, 1851 (1993)
Patrick Rinke (FHI) RPA/GW Levi 2012 41
Kinetic energy
correlation energy in G0W0:
EG0W0
c = −
∫ ∞
0
dω
2πTr
[∞∑
n=2
(χ0(iω)v)n
]
correlation energy in RPA:
ERPAc = −
∫ ∞
0
dω
2πTr
[∞∑
n=2
1
n(χ0(iω)v)
n
]
adiabatic connection recovers kinetic energy
Patrick Rinke (FHI) RPA/GW Levi 2012 42
Kinetic and correlation energy
RPA kinetic energy contribution:
TRPAc = ERPA
c − EG0W0
c
RPA kinetic energy:
TRPA = T0 + TRPAc
RPA Coulomb correlation:
URPAc = ERPA
c − TRPAc
Patrick Rinke (FHI) RPA/GW Levi 2012 43
H2 – energy contributions
1 2 3 4Bond Length [Å]
-0.05
0.00
0.05
0.10
0.15
Ene
rgy
diffe
renc
e [H
a]
∆(Etot-Ec)
∆Etot
∆Ec
1 2 3 4Bond Length [Å]
-0.2
-0.1
0.0
0.1
Ene
rgy
diffe
renc
e [H
a]
∆Eext
∆Ex
∆T
∆EH
Coulomb correlation energy makes the difference
Patrick Rinke (FHI) RPA/GW Levi 2012 44
Acknowledgements
RPA+SOSEX/SE/R2PT
Xinguo RenAlex TkatchenkoJoachim PaierAndreas GruneisGeorg KresseGus Scuseria
TTF-TCNQ
Viktor AtallaMina Yoon
scRPA
Daniel RohrMaria Hellgren
scGW/Cerium
Fabio Caruso/Marco CasadeiXinguo RenAngel Rubio
Support and Ideas
Matthias Scheffler
FHI-aims code
Volker Blumthe whole FHI-aims developer team
Patrick Rinke (FHI) RPA/GW Levi 2012 45
Cerium - spectroscopy
DO
S
UPS+BISPBEPBE0 α-phase
-4 -2 0 2 4 6E-E
f (eV)
DO
S
UPS+BISPBE0 γ-phase
description of γ-phase ok – α-phase problematic
UPS: Wieliczka et al. PRB 29, 3028 (1984) – BIS: Wuilloud et al. PRB
28, 7354 (1983)Patrick Rinke (FHI) RPA/GW Levi 2012 46