Towards a unified description of ground and excited state ...helper.ipam.ucla.edu/publications/plws4/plws4_10647.pdfBond length (Å)-15-10-5 0 5 10 15 20 Binding energy (meV) 5.0 6.0

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




(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

. . .




(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

. . .


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))]


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)


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


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


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).


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

+ +


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)



ErSEc = + ia + ...

a

bi+


ia

i

f ai

ja

f ia

f ij

f

f ia

f biaj

ab

f

f


ESEc =

occ∑

i

unocc∑

a

|fia|2


“Diagonal rSE ” : including only terms with i = j = · · · and a = b = · · ·

ErSE-diagc =

occ∑

i

unocc∑

a

|fia|2

fii − faaProblematic for weak interactions



ErSEc = + ia + ...

a

bi+


ia

i

f ai

ja

f ia

f ij

f

f ia

f biaj

ab

f

f


ESEc =

occ∑

i

unocc∑

a

|fia|2


“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


Renormalized 2nd-order Perturbation Theory (r2PT)

ERPA+SOSEX+rSEc = + + ...

+ + ...+

( = RPA)

( = SOSEX)

+ + + ...+

2nd−order 3rd−order

( = rSE)

( = 2PT)

“r2PT” = “RPA+SOSEX+rSE”



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



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



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)



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


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)



58%

19%16%

PBE

MP2

(EX+cR

PA)@PBE

mean absolute percentage error



58%

19%16%

8%

PBE

MP2

(EX+cR

PA)@PBE

(EX+cR

PA+SE)@PBE


X. Ren, P. Rinke, A. Tkatchenko, M. Scheffler,Phys. Rev. Lett. 106, 153003 (2011)



58%

19%16%

7%8%

PBE

MP2

(EX+cR

PA)@PBE

r2PT@PBE

(EX+cR

PA+SE)@PBE


X. Ren, P. Rinke, A. Tkatchenko, M. Scheffler,Phys. Rev. Lett. 106, 153003 (2011)


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)



-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



-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)



-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)


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


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


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



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"



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"


Cerium - electronic structure

n(α)− n(γ) at a=4.6A

f -electrons more localized in γ-phase



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



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


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


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


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



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



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



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



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?


Band structures: photo-electron spectroscopy

- -

-

-

+

Photoemission

hn

GW

- -

-

Inverse Photoemission

-

-

hn

-

GW

- -

-+

Absorption

-

hn

BSETDDFT


Experimental: angle-resolved photoemission spectroscopy

ARPES

=+1qo

Binding Energy (eV)8 6 4 2 0

Masaki Kobayashi, PhD dissertation


Experimental: angle-resolved photoemission spectroscopy

ARPES

=+1qo

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

e



Quasiparticle:


Ak(ǫ) = ImGk(ǫ) ≈Zk

ǫ− (ǫk + iΓk)

ǫk : excitation energyΓk : lifetimeZk : renormalisation

A ( )k

e quasiparticle-peak

ek

qp

Gk

e



Quasiparticle:


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


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!


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


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


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)


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)


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)



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


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


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


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


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


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


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


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)


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!


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


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+)


Dipole moment (in Debye):

Exp. scGW CCSD HF PBE

0.11 0.07 0.06 −0.13 0.20


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



TTF/TCNQ dimer revisited

HOMO

LUMO

scGW opt !


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]


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?


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)


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


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


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


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


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

Documents

Towards a unified description of ground and excited state ...helper.ipam.ucla.edu/publications/plws4/plws4_10647.pdfBond length (Å)-15-10-5 0 5 10 15 20 Binding energy (meV) 5.0 6.0