GW approximation: One- and two-particle excitations

in solids and molecules

Friedhelm BechstedtFriedrich-Schiller-Universität JenaGermany

Goaltheoretical / computational approach to calculate optical and other excitation spectra including (i) quasiparticle effects due to excitation

(ii) electron-hole atraction & exchange

� �� � � � � �

� �

� � � �

� � � �� � � � � �

c

v

GW

DFT-LDA

c

v

Eg

� � � � � � � �

� � � �

� � � � � � � �

applicable tosystems of arbitrary dimension

0D3D

O

N 6

C 5

N 7

N 4

N 10

C 9

C 3

C 2

C 1

N 8

Guanine

Jena: e.g. organic functionalization

Si bulk

2DIn/Si(111)4x1

Spectra

Absorption Reflectance Anisotropy (RA)

� �

� �

��

� � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � �� � � � � � �

)(RR

ω∆)(Im

)(cn)( ω∈

ωω

=ωαd)(e~ ωα−

System of fundamental equationsHedin/Lundqvist (69): Many-body perturbation theory (MBPT)

dielectric function (length gauge)

• polarization function

• one-electron Green function ⇒ Dyson equation

• XC self-energy

• screened Coulomb potential

• vertex function ⇒ Bethe-Salpeter equation

vP1−∈=

Γ= GGP

RPA (independent particle) vertex corrections (excitons)

( ) 1GGHt/ H1 =Σ−−∂∂

Γ=Σ GW

vPWvW +=

ΓδΣδ

+=Γ GGG

1

Outline: Unified treatment of solids and molecules

• GW: reformulation

• single-particle excitations

• two-particle excitations

• summary

Single-particle problem

Quasiparticle QP approachone-electron Green function ⇒ Dyson equation XC self-energy

screened Coulomb potential

GW approximation

Standard treatment First-order perturbation theory(start wave functions = KS wave functions)diagonal self-energy only energy shifts

Update of wave Wrong energetical ordering, band crossingsfunctions → off-diagonal elements

( ) 1GGHt/ H1 =Σ−−∂∂ Γ=Σ GW

( ) ν−εΣν=∆ νν XCQP V

Γ = 1, W = WRPA

Computational method

• start: DFT-LDA, Kohn-Sham equationnc pseudopotentials, supercell approach (Bloch picture) ⇒ also for confined systems

two exceptions: all-electron PAW wave functions ⇒ futurenc pp + plane waves

• real-space multigrid methodE.L. Briggs, D.J. Sullivan, J. Bernholc, PRB 14362 (1996)

- Basis functions: no plane waves instead: 3D mesh in real space GaP: 0.238 Å (= 24 Ry)

• advantage (i): massive parallelization(ii): matrix elements

- ⟨ ck | grad | vk ⟩ → 6-point method- real-space cutoff for surface optical properties

• screened potential: model or Ehrenreich-Cohen formula for matrix

Special treatments

• Blomberg-Bergersen method for G in GWRPA

⇒ correct positions of first satellites (no plasmaron!)

• Avoid sum over intermediate states

∑ν ν

νν

ε−ϕϕ

= QP

*

z)'()()z;',(G

h

xxxx

( )[ ] ( )

( ) ⎥⎦

⎤⎢⎣

⎡ω

−ωε−ε

++ω++∈πω

+

⎥⎦

⎤⎢⎣

⎡++−−δ−++∈+

δ++⎩⎨⎧−

++π

=εΣ

νµν

ννν

−∞

µνν

νννµ−

µνν

νννµ

∑∫

∑

∑∑

hmhm

h 11)'(B)(B;',d

)'(B)(B'B210;',

)'(B)(B'

e4V1)(

QP'

*'

''

1

0

*'

occ

'''GG

1

'*

'

occ

''

',,

2

GqGqGqGq

GqGqGGGqGq

GqGqGqGq GG

GGq

a

EX

COHSEX-EX

dyn

main problem

Wrong ordering of bands:Example: 2H-InN

• too shallow In4d electrons• overestimation of pd repulsion• “negative gap“ (Γ6v, Γ1v, Γ1c)

• In4d frozen in the core

→ automatically gap opening

Ene

rgy

(eV

)

Γ K H A Γ M L A-8

-4

0

4

8

Ene

rgy

(eV

)

23

23

33

1,2

13

1,3

3

5

165,6

1

1,5

3

65,6 3

31

1,3

1,3

41,3

3

22,4

1

1,33

1

Kohn-Sham

KS + Quasiparticle

Wrong ordering of bands:Example: C(111)2×1 π-bonded chain model without

buckling and dimerizationKohn-Sham Quasiparticle

⇐ diagonal GW fails (more or less equal shifts)

⇐ non-perturbation GW fails(matrix elements small)

M. Marsili et al., PRL (submitted)

⇒ self-consistent GW with respect to occupation(updating the quasiparticle energies including their occupation till self-consistency)⇒ gap of 0.8 eV (≈1.0 eV: onset of EELS)

Diagonal quasiparticle approach:Example: Silan SiH4 (Td)

Computation:dSi-H = 1.477 ÅLsupercell = 11.9 Å48 grid pointsdiel. Matrix: 4v, 256c

|G| = 90eV^

Comparison: Rohlfing/Louie, PRB 62, 4927 (2000)

exp.: Itoh, J. Chem. Phys. 85, 4867 (1986)Quasiparticle energies (eV):

LDA HF GW *) GW **) Exp.

HOMOLUMOQP gap

-8.42 (-8.42)-0.50 (0.57)7.93 (7.85)

12.77 (12.98)1.13 (1.49)13.90 (14.47)

-11.680.7212.40

-12.56 (-12.69)0.50 (1.10)13.06 (13.79)

-12.6

8.8

*) unclever treatment of intermediate states:1028 states shifted by 0.7eV**) use of completeness in static part

increase supercell: further shift by 0.1eV

HFHOMOε

Problem ⇔ Off-diagonal elements of δΣ- in contrast to LDA, LUMO above Evac in QP (diag)

- KS wave function too localized- indeed mixing of LUMO with LUMO + 4 of same symmetry + drastic lowering of its energy by 0.63eV

however: depends on description of “continuum“ above Evac(supercell size, number of states)

preliminarysolution: only 8 bands

Energy0 = E vac

LUMO

HOMO

KS QP (diag) QP (non-diag)

⇒ (with Coulomb effects) reliable reproduction of pair excitation energies

H2O molecule: QP approachdH-O = 0.966 Å , < = 104.49°)

Lowest ionization energies (eV): (512 bands, 6000 G vectors, Lcell = 10 Å, 64 grid points)

MO LDA HF GW Exp

1b1

3a1

1b2(HOMO)

EgQP

13.119.277.21

6.2

19.1415.4713.16

14.1

18.79 (18.51)14.42 (14.33)11.94 (12.04)

12.5

18.72, 18.5514.83, 14.7312.78, 12.61

comparison: Shigeta, Int. J. Quant. Chem. 85, 411 (2001)

exp: Handbook of He I Photoelectron Spectra of Fundamental MoleculesBrundle/Turner, Proc. R. Soc. London Ser. A 307, 27 (1968)

Pair excitations(Optical spectra)

Two-particle problem (quasielectron-quasihole pair excitations, excitons)

_P

1 1 12 2 23 4

1' 2' 1' 2' 1' 2'3' 4'=

=

+

+3 43' 4'

3 4

3' 4'

3

3'

4

4'

_P Ξ

Ξ

• Macroscopic optical polarizability → polarization function

• BSE

0q→=α Pv

PLLP 00 Ξ+=

vG/ +δΣδ=Ξ

GGL0 ⋅=v - short range (electron-

hole exchange, local fields)• Standard approximations

(i) GW: δΣ/δG = W + GδW/δG = ladder approximationcommon believe, for extended systems (Strinati, Nuov. Cim. 11, 1 (1988))

(ii) neglect of dynamical screening (K. Shindo, J. Phys. Soc. J. 29, 287 (1970))

(iii) neglect of non-particle conserving terms and coupling between resonant and non-resonant terms ⇒ no effect on optical absorption of

Si (S. Albrecht, Ph.D. thesis)

^

Bloch-Kohn-Sham representation of two-particle problem

approximate BSE

( ) ( ){ } ( )

''vv'cc

''k,''v,''c''''vv''cc ;''v'c,''''v''cPi''''v''c,cvH

kk

kk kkkk

δδδ−=

ωδδδγ+ω−∑ h

Pair Hamiltonian

(i) independent-particle approximation (RPA)

(ii) independent-quasiparticle approximation

(iii) Coulomb-correlated electron-hole pairs

( ) [ ] ''vv'ccvc )()(''v'c,cvH kkkkkk δδδε−ε=

)()()( nnQPn kkk ∆+ε=ε kkkk nV))((n)( xc

QPn n

−εΣ=∆

[ ])''v'c,cv(v2)''v'c,cv(W

)()()''v'c,cv(H ''vv'ccQPv

QPc

kkkkkkkk kk

+−δδδε−ε=

Representation for polarizability• eigenvalue problem (homogeneous BSE)

(Albrecht et al, PRL 80, 4510 (1998); Rohlfing, Louie, PRB 62, 4927 (2000))

∑λ λ

λλλλ γ+ω−

φµ=ωα→φ=φ

)i(E)(EH

2

h

small surface slab ≈ 28 atoms→ rank of H < NcNvNk ≈ 400,000 !!

• initial value problem (inhomogeneous BSE)(Hahn et al., PRL 88, 016402 (02); Schmidt et al., PRB 67, 084307 (2003))

)()(vc

)0(;)t(H)t(dtdi :with

)t(edtei)(

vc

0

tti

kkkvkε−ε

=µ=ΨΨ=Ψ

Ψµ=ωα ∫∞

γ−ω

h

h

central-difference („leap-frog“) method

• boundary-value problem: (Benedict et al., PRL 80, 4514 (1998))

Solve initial-value problem...

… with central difference method:

• matrix diagonalization → matrix-vector multiplications

tnt);t(Ht2

)t()t(i n1nn2n ∆=Ψ=

∆Ψ−Ψ

++h

( ) ( )4at

6at NONO →•

• parallelization

Example: CPU time for bulk Si (Pentium PC)

HP9000 with 8 CPUs / 32 GB sufficient for surface calculation

Standard approachExample: ionic compound InP

(200 random k-points, 8-atom sc supercell)

Exp.: P. Lautenschläger et al., PRB36, 4813 (87)Calc.: P.H. Hahn, PhD Thesis, Jena (04)

Influence of wave function representationReal-space grid versus PAW

Example: Si bulk

good agreementpreliminary: advantage: first-row elements

disadvantage: too slow

Kernel beyond ladder approximation

Example: Si crystal(sc supercell, 25 random points)

[ ])31(W)24(W)41(W)23(W)43(Gi)34(G)12(W

+−=δδ

h

standard approximations (dynamics, particle conservation)

- reduction of e-h attraction- zero for Wannier-Mott exciton

here:- reduction of E2 → E1 redistribution- shift by 0.1 eV toward higher energies

Influence of dynamical screeningProblem: no closed BSE

{ }∫ ωωω−ωω−ωω+ω≈ωω )',''(P~)''(W''d1)(G)'(G),'(P~

SilanShindo approximation

(Shindo, J. Phys. Soc. Japan 29, 287 (1970))

[ ] )(P~),'(P~)''(G)''(G''d

)(G)'(G ω∫

≈ωωω−ω+ωω

ω−ω+ω

only influence on higher transitions

partial compensation of dynamical effects and resonant-nonresonant coupling

Monohydride Si(001)2×1-H:Model system for a passivated surface

no π/π* states in the gapSaturation of one dangling bond per atom

only electron-hole pairs in the energy range of bulk optical transitions

RA of Si(100)2×1-Hcalculation: 12-layer slab, 200 k points,

spectrum < 6eV (50c, 50v) ≈ 5×105 pair states

• GW: not only rigid shift

• local fields: surprisingly small

• screened electron-hole attraction:

- enhancement of optical anisotropynear E1

- redistribution of oscillator strengthE2 → E1

⇒ surface modification of bulk excitons(analog Si(110)1×1:H)

Hexagonal ice (Ih)

phase diagram of water structural model

Frenkel versus Wannier-Mott exciton

Absorption |Ψ(re, rh)|2 with rh = R0

Exp.: Kobayashi, J. Phys. Chem. 87, 4317 (1983)Calc.: P.H. Hahn et al., PRL 94, 037404 (2005)

• Compensation of QP and excitonic shifts• but Coulomb enhancement ⇒ bound state

(Frenkel exciton)

Virtual J. Biological Physics Research, February 1, 2005

Many-body versus condensation effects (H2O)

lowest pair excitation (eV)

phase DFT-GGA QP QP + BSE Exp.molecule

solid6.28.0

12.512.5

7.29.2

7.48.7

peak (not onset)

• strange result: E (molecule) < E (solid)

• reduction of many-body effects by 30 (QP) or40 (Ex) % in solid

• reason: increased screening delocalization of excitonrex = 2.27 → 4.02 Å

Exp: Chan, Chem. Phys. 178, 387 (93)Kerr, PRA 5, 2523 (72)

Conclusions• in principle: MBPT in GW applicable independent of dimensionality

even using supercellsneed: screening with dynamics and local fields

• shown for 3D (bulk semiconductors), 2D (C and Si surfaces), and 0D (SiH4 and H2O molecules)

• critical: • wrong energetical ordering of states• supercell size• description of states above ionization edge?• completeness of KS basis set (self-energy,

screening function) • in general: convergence

We are on the right way!?

more or less correct oscillator& spectral strengths

but accuracy on energy scale 0.1 ... 0.2 eV

Acknowledgements

Collaboration: W.G. Schmidt, P.H. Hahn, M. Marsili, O. Pulci, R. Del Sole, J. Furthmüller, L.E. Ramos, H.-Ch. Weissker

Grants: EU: RTN NANOPHASE, NoE NANOQUANTA

Federal Government: Junior scientist groupComputational Materials Science

DFG: Project Si & Ge nanocrystals