ab initio Computational Methods for

Electronic Excitations: Density-

Functional and the Green’s Function

Zhigang Wu

Department of Physics

Colorado School of Mines

Golden, CO 80401

Lusk/Wu Group Meeting, CSM, 12/01/2009

zhiwu@mines.edu

Outline

Part 0: Introduction

Electronic excitations

Approaches

Part I: Density Functional Theory (DFT)

Part II: Orbital-Dependent DFT

Part III: Time-Dependent DFT

Part IV: GW/BSE

1

2

Photoemission: Quasiparticle Gap

Electron addition/removing

Eg = IP – EA

Electron Affinity

EA = E(N) – E(N+1)

Ionization Potential

IP = E(N-1) – E(N) Eg

VBM

CBM

Direct

Photoemission

Inverse

Photoemission

Quasiparticle

3

• Excitations of many-electron

system can often be described

in terms of weakly interacting

“quasiparticles”

• Quasiparticle (QP) = bare

particle + polarization clouds

- -

Bare particle Quasiparticle

EQP = E0+

: response of system to the

excitation(self-energy)

4

Optical Absorption: Neutral Excitation

Optical Gap

Electron addition

removing – exciton

binding energy

Eoptg = Eg – Ee-h

Ee-h ~e2

2 a0

CBM

VBM

5

Approaches (1)

Density Functional Theory (KS-states)

Serves as 0th-order approximation; very efficient

Unjustified theoretically; Underestimates gaps severely.

SCF Can be accurate for QP-gap, moderate accuracy for

neutral excitation; Difficult to get the full spectrum; Only

applicable for finite systems.

Ensemble DFT Can compute the excitation energy; But ensemble LDA

performs poorly - need better approximation.

LDA + DMFT: strongly-correlated materials (cuprate)

LDA + U: Mott insulators (NiO); U is ad hoc.

6

Approaches (2)

Orbital-Dependent DFT

Self-Interaction Correction (SIC)

Hybrid DFT: B3LYP, PBE0, HSE, etc

Exact Exchange (EXX)

Screened Exchange (sx)

Could predict good QP-gap, and more efficient than GW;

But the accuracy varies strongly for different materials.

Time-Dependent (TD) DFT Exact theory in principle for neutral excitations; In

practice, adiabatic LDA is OK for molecules, poor for

solids; No quasi-particles.

7

Approaches (3)

Green-function methods: GW/BSE State-of-the-art for both QP and neutral excitation;

scaling as N4-7.

Quantun chemistry post-HF methods:

CI, CC, MP2, etc. Accurate for QP and neutral excitation of molecules

But bad scaling of N5-7; Prohibitive for solids.

Quantum Monte Carlo

Scaling as N3: most accurate benchmarks for systems

of medium size; Essentially SCF for excitations:

difficult to get the full spectrum.

Balance Between Accuracy and Applicability

8

Carter, Science 321, 800 (2008)

PART I: Density Functional Theory

9

KS-States as Quasiparticles

10

2 2

2m+ vKS(

r )

i(

r ) = i i(

r ),

where vKS(

r ) = vext (

r ) + vH(

r ) + vxc(

r ),

with vH(

r ) =(

r ')

|

r

r ' |d

r ' and vxc(

r ) =

Exc[ (

r )](

r )

E / ni = i

Janak’s Theorem (PRB 18, 7165):

E (N ) E(N 1) = i (n )dn i (N 0.5)N 1N

E (N +1) E(N ) = i (n )dn i (N +0.5)NN+1

{

KS-States as Quasiparticles

11

EA = Emin (N +1) E(N ) N +1(N + 0.5)

IP = E(N ) Emin (N 1) N (N 0.5)

For finite KS systems (molecules):

KS eigenstates cannot be treated as

quasiparticles.

For infinite KS systems (solids):

E(N +1) E(N ) = i (N ) for i > N

E(N ) E(N 1) = i (N ) for i N

N

KS-Eigenvalues vs SCF

12

1/N

-12

-8

-4

0

4

E (

eV)

E(N+1)-E(N)E(N)-E(N-1)

CBM

VBM

1/2432 1/608 1/304 1/152

DFTQMC

Si slab

PRB 79, 201309 (2009)

KS-Eigenvalues vs SCF

13

Benzene

LUMOKS

= 1.16 eV EN +1 EN = 1.10 eV

HOMOKS

= 6.25 eV EN EN 1 = 9.12 eV

EgKS

= 5.1 eV EgSCF

= 10.22 eV

EgGW = 10.51 eV

Lack of Derivative Discontinuity in Exc

14

E / ni = i

CBM =E

n n=N +

VBM=E

n n=N

xc

Exc

n n=N +

Exc

n n=N

xcDFT= 0

The DFT band gap is due to the

derivative discontinuity in Ekin

Excni

=Excnj

=ExcnN

xc = EgQP Eg

KS

Underestimate the band gap by xc

iKS

jKS

< iQP

jQP

Underestimate the band width

KSN (N ) = IPKSN +1(N +1) = EA

Levy et al., PRA 30, 2745 (1984)

Other KS eigenvalues cannot be

rigorously assigned to the QP energies.

Presence of Self-Interaction

Too much Coulumb

repulsion (each

electron is repelled by

itself) pushes electrons

apart: delocalization of

the KS wavefunction

15

PRL 103, 176403 (2009)

Convex Behavior of DFT Energy

16

Science 321, 792 (2008)

PART II: Orbital-Dependent DFT

17

Why Adding Orbitals to Exc

Using orbitals in Ekin is precisely what made DFT

so successful.

Using orbitals allows more flexibility and accuracy

in construction of functionals.

Orbital dependence may introduce derivative

discontinuity in exchange-correlation.

Orbital dependence can compensate for self-

interaction.

The exchange energy is formulated naturally in

terms of orbitals.

18

Optimized Effective Potential (OEP)

19

Exc

[ (r )] obeys the Honhenberg-Kohn principle: E

tot

(r )= 0

Exc

[{j}] obeys the Sharp-Horton principle:

Etot

vKS

(r )= 0

These two principles are equivalent

vxc

(r ) =E

xc

(r )

PR 90, 317 (1953)

PR 136, B864 (1964)

vOEP

xc (r ) =

Exc

[{j}]

(r )=

Exc

[{j}]

i(r ')i

occ

i(r ')

vKS

(r '')

vKS

(r '')

(r )dr 'dr '' + c.c.

Chain rule:

v {

j} {

j}= {

j}[ ]

Exc

[{j}] is within the framework of KS theorem

Optimized Effective Potential (OEP)

20

vOEP

xc (r ) =

Exc

[{j}]

i(r ')i

occ

i(r ')

vKS

(r '')

vKS

(r '')

(r )dr 'dr '' + c.c.

i(r ')

vKS

(r )=

j(r ') *

j(r )

i ji j

i(r )

First-order perturbation theory

vKS

(r ')

(r )=

(r )

vKS

(r ')

1

KS(r ,r ')

(r ')

vKS

(r )=

i

occi

*(r )i(r ')

j(r )

j

*(r ')

i jj i

+ c.c.

Response function

PRA 50, 196 (1994); PRB 59 10031 (1999); PRB 68, 035103 (2003)

The OEP Equation

21

i*(r')

i

occ

vxc (r ') uxci (r ') GKS

i (r ',r ) i (r )dr '+ c.c. = 0

where the KS Green's function GKSi (r ',r ) is

GKSi (r ',r ) = j (r ') j

*(r )

i ji j

and

uxci (r ) =

1

i*(r )

Exc[{ j}]

i (r )

Multiplying KS on both sides from the right, then integrating

Orbital-Specific Potential Approach

Orbital-specific potential (OSP) approaches:

Self-interaction correction (SIC)

Hybrid functionals (Mixing HF with LDA/GGA)

Meta-GGA

KS equation with OSP

KS equation with OEP

22

2 2

2m+ v

H(r ) + v

xc([ ];r )

i(r ) + F[

i(r )] =

i i(r )

2 2

2m+ v

H(r ) + v

xc([{

j}];r )

i(r ) =

i i(r )

OEP versus OSP

OEP is within KS theorem, thus no derivative discontinuity; while

OSP is not within KS, but still obeys the HK theorem, and it has

derivative discontinuity.

OEP orbitals are diagonal, while OSP orbitals are non-diagonal.

For unoccupied states, OEP is a significant improvement over OSP.

For localization phenomena, OSP are often advantageous.

The total energy and total electron density for both approaches are

similar if the second or higher order perturbation terms for OEP are

small.

23

Mixing LDA/GGA with HF

24

Hybrid Functional: Exchyb

= bExHF

+ (1 b)Exapprx

+ Ecapprx

Band Gap (eV) LDA HF Expt.

C 4.1 12.1 5.5

Si 0.5 6.3 1.2

Ge -0.1 4.2 0.7

LDA/GGA

Orbital delocalization

Convex behavior of total energy

Underestimation of band gap

HF

Orbital localization

Concave behavior of total energy

Overestimation of band gap

Adiabatic connection theorem (PRB 15, 2884; PRA 26, 1200)

Hybrid Functionals

25

ExcPBE0

= 0.25ExHF

+ 0.75ExPBE

+ EcPBE

ExcB3LYP

= 0.20ExHF

+ 0.80ExLDA

+ 0.72 ExB88

+ 0.81EcLYP

+ 0.19EcLDA

ExcHSE

= 0.25ExHF, SR

+ 0.75ExPBE, SR

+ ExPBE, LR

+ EcPBE

More: B3P86, B1B95, B1LYP, MPW1PW91, B97, BHHLYP

B98, B971, B972, PBE1PBE, O3LYP, BHH, BMK, etc

Band Gaps from Hybrids PBE0 and HSE03

26

JCP 124, 154709 (2006)

B3LYP Functional

27

ExcB3LYP

= 0.20ExHF

+ 0.80ExLDA

+ 0.72 ExB88

+ 0.81EcLYP

+ 0.19EcLDA

Most popular functional in computational chemistry: > 12,000 citations

Si Band Structure

: GW

: QMC

: Expt.

CPL 342, 397 (2001)

B3LYP Functional

28

ExcB3LYP

= 0.20ExHF

+ 0.80ExLDA

+ 0.72 ExB88

+ 0.81EcLYP

+ 0.19EcLDA

LDA/GGA

1.0

CPL 342, 397 (2001)

( gap)

APL 79, 368 (2001)

Screened Exchange (sx)

29

Ex

sx=

i

*(r )j

* (r ')ek

TF|r r '|

j(r )

j(r ')

r r 'i, j

occ

drdr '

The bare Coulomb potential is

replaced with a Yukawa potential, with kTF the Thomas-

Fermi screening factor.

An approximation of GW theory

sx-LDA approximation

Exact Exchange (EXX)

30 PRB 59 10031 (1999)

Ex=

1

2

i

*(r )j

*(r ')j(r )

i(r ')

| r r ' |dr 'dr

i, j

occ

OEP cohesive

energy

Is EXX Equivalent to HF?

31

Ex=

1

2

i

*(r )j

*(r ')j(r )

i(r ')

| r r ' |dr 'dr

i, j

occ

EXX satisfies HK theorem, but HF does not: EXX

optimizes Etot w.r.t. density, while HF w.r.t orbitals.

EXX potential is local and multiplicative (within KS)

while HF potential is non-local and orbital-specific.

EXX orbitals are different from HF orbitals: EXX

orbitals are uniquely determined by density, HF not.

EXX and HF predicts different band gaps:

Band Gap (eV) LDA EXX HF Expt.

Si 0.5 1.4 6.3 1.2

Why EXX and HF Predict Different Gaps?

32

EXX predicts better gap than LDA/GGA and HF due to error cancellation

E

g

KS=

N +1(N )

N(N )

E

g= E(N +1) + E(N 1) 2E(N )

E

g= E

g

KS+

xc

For Si, using EXX:

Eg

KS= 1.4 eV

xc= 4.5 eV

Eg= E

g

KS+

xc= 5.9 eV

E

g

HF= 6.3 eV

The derivative discontinuity xc

in KS scheme can be evaluated

from the GW-operator and the KS eigenvalues and orbitals:

xc=

N +1(

N +1) v

xc N +1xc

GW

N(

N) v

xc Nxc

GW

PRB 70, 245155 (2004)

EXX is an approximation to GW due to screening of response-function in OEP

Derivative Discontinuity

33

JCP 124, 154108 (2006)

Exact Exchange (EXX) with Compatible Correlation

The hyper-generalized-gradient approximation (HGGA)

Perdew et al., JCP 123, 062201 (2005).

34 Jacob’s Ladder

Exact Exchange (EXX) with Compatible Correlation

Perturbation theory

Higher rung than HGGA; From construction to evaluation

Görling-Levy theory

PRB 47, 13105 (1993); PRA 50, 196 (1994); PRA 50, 4493 (1995)

Range-separated functionals

Underlies the random-phase approximation (RPA)

PRB 59, 10461 (1999); PRB 61, 16430 (2000); PRB 67, 045101 (2003)

Generalized hybrid functionals (PRA 72, 012510; CPL 415, 100)

Correlation factor model [Int. J. Quan. Chem. 61, 287 (1997)]

Model the coupling-constant-averaged xc hole

Interaction-strength interpolation [PRA 59, 51 (1999)]

Model the coupling-constant integrand:

The forefront of present-day density-functional research

Computationally much more demanding than LDA/GGA 35

Exc[ ] = Exc[ ]d0

1

Summary: Why DFT Fails to Predict Correct QP Gap?

Is the problem due to XC approximations (LDA/GGA),

or it is intrinsic due to electron density used instead of

orbitals?

It is intrinsic to the KS scheme, which lacks the necessary

derivative discontinuity in XC due to local and multiplicative

potential.

Do KS eigenvalues/states have any meaning at all?

KS states are QPs of the infinite KS system.

For real infinite systems, rigorously, only the highest

occupied KS state corresponds to VBM; practically, they can

be a good 0-order approximation to QPs.

For finite systmes (molecules), KS states cannot be assigned

to QP at all. 36

Summary: Why DFT Fails to Predict Correct QP Gap?

Are the underestimation in gap for solids (infinite

systems) and molecules (finite systems) due to the

same reason?

No. For solids, it is mainly due to lacking derivative

discontinuity in XC.

For molecules, KS eigenvalues can not be associated to QP

energies (orbital relaxation is needed for the N±1 systems).

If we have the holy-grail "exact XC functional", can

DFT predict correct gap for solids and molecules?

In principle, “exact DFT” can predict correct gaps for both

systems if go beyond KS scheme (even beyond HK).

Practically, it is doomed! Though some approaches based on

DFT are fortuitous to predict correct gap for many materials. 37

PART III: Time-Dependent DFT

38

JACS 127, 1438 (2005)

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

Optical Absorption of the C6H6 Molecule

56

PRA 60, 1271 (1999)

Optical Absorption of the Na4 Cluster

57

Solid line: TDLDA

Dotted line: TD-EXX

Gray solid: Experiment

Dashed line: BSE

Optical Absorption of the Na4 Cluster

58

Solid line: TDLDA

Dotted line: TD-EXX

Gray solid: Experiment

Vertical dashed: BSE

Vertical solid: QMC

The lowest

excitation

energies

Optical Absorption of Solid Si

59

In infinite system

V (q) =4

q2

f

xc(q) 1 / q2 for q 0

PRL 88, 066404 (2002)

But LDA/GGA fall

off exponentially

LRC: long range

correction

fxc

(r ,r ') =4

1

| r r ' |

Optical Absorption of Solid Si

60

TD-LDA TD-HSE

Exc

HSE= 0.25E

x

HF, SR (μ) + 0.75Ex

PBE, SR (μ) + Ex

PBE, LR (μ) + Ec

PBE

μ is the range separation : 1

r=

erfc(μr)

r+

erf(μr)

r

PRB 78, 121201 (2008)

Optical Absorption of Semiconductors

61

PRB 78, 121201 (2008)

Si

GaAs

C

SiC

LiF

PART IV: GW/BSE

62

References

Hedin, Physical Review 139, A796 (1965).

Hybertsen & Louie, PRB 34, 5390 (1986).

Rohlfing & Louie, PRB 62, 4927 (2000).

Aulbur, Jönsson, & Wilkins, Solid State Physics 54,

1 (1999).

Onida, Reining, & Rubio, RMP 74, 601 (2002).

Books on many-particle physics

Quantum Theory of Many-Particle Systems, by Fetter &

Walecka, McGraw-Hill, 1971.

Many-Particle Physics, by Mahan, Springer, 2000.

63

64

Photoemission: Quasiparticle Gap

Electron addition/removing

Eg = IP – EA

Electron Affinity

EA = E(N) – E(N+1)

Ionization Potential

IP = E(N-1) – E(N) Eg

VBM

CBM

Direct

Photoemission

Inverse

Photoemission

65

Optical Absorption: Neutral Excitation

Optical Gap

Electron addition

removing – exciton

binding energy

Eoptg = Eg – Ee-h

Ee-h ~e2

2 a0

CBM

VBM

Corresponding Green’s Functions

66

• Quasiparticle gap One-particle Green’s function

Motion of added/removed electron

• Optical gap Two-particle Green’s function

Motion of added/removed electron-hole pair

G 1,2, 1 , 2 ( ) = i( )2N T[ (1) (2) +( 2 ) +( 1 )]N

G 1, 1 ( ) = i N T[ (1) +( 1 )]N

1, 1 ( ) rt, r t ( )where

Why Green’s Functions?

Physically-motivated approach, based on responses to

well-defined excitations of the system, that includes exact

exchange and nonlocal, dynamical correlation effects.

First-principles treatment allows correlation effects to be

assessed in a variety of environments, including solids,

surfaces, interfaces, clusters, and molecules.

Computationally, more intensive than DFT, but scales

better (~NG4, where NG is number of plane waves) than

quantum chemistry methods.

67

Single-Particle Green’s Functions

G(xt,x' t ') = i < N |T[ (x,t) +(x', t ')] |N >

Consider a system of N electrons in its ground state and define a one-electron

Green’s function

which can be rewritten in terms of eigenstates of H for the N+1 or N-1

particles, i.e.

G(x,x ', t t') = i i(x) i*(x ')exp[ i i(t t')]

i

( (t t ') ( i μ) (t' t) (μ μ))

where μ is the Fermi level (i.e. defining minimum energies for injection of

electrons or holes)

68

i)

Quasiparticle

69

Quasiparticle energies and states

Occupied states: i = E0(N) Ei(N 1),

i(r) =< N 1,i | (r) |N,0 >

Empty states:i = Ei(N +1) E0(N),

i(r) =< N,0 | (r) |N +1,i >

where

E0(N)

Ei(N 1)

(r)

Ground state energy of N-particle system

Excited state i of (N-1)-particle system

Field operator

Quasiparticle

70

- -

Bare particle Quasiparticle

EQP = E0+

: response of system to the

excitation(self-energy)

Excitations of many-electron

system can often be described

in terms of weakly interacting

“quasiparticles”.

Quasiparticle (QP) = bare

particle + polarization clouds.

Quasiparticle Equation

±=

)(

)'()(),',(

*

i

xxxxG

i

ii

Equation for G:

(1

22+V 0)G(x,x', ) + (x,x", )G(x",x ', )dx" = (x x ')

Schrödinger-like equation for quasiparticles

(1

22+V 0) i(x) + (x,x", i) i(x")dx" = i i(x)

Fourier transform of G

71

(Derived from the Dyson’s equation)

Quasiparticle Equation

72

Schrödinger-like Kohn-Sham equation

)()()2

1( 2

rrVVViii

extxcH=+++

Where the self-energy is independent of i and has the form

(x, x ; i) = V xc ( x ) (x x )

Schrödinger-like equation for quasiparticles

(1

22

+ V 0) i(x) + (x, x , i) i( x )d x = i i(x)

DFT

GW Approximation for Self-Energy

The self-energy can be regarded as a dynamical, nonlocal

exchange-correlation potential

(x,x ', ) =V XC (x,x', )

DFT (Static, local self-energy):

Hartree approximation:

(x,x ', ) =V XC (n(x)) (x,x ')(x,x ', ) = 0

In GWA, the self-energy is approximated by a product of the Green’s

function (G) and the dynamically screened Coulomb interaction (W):

GW (x,x ', ) =i

2W (x,x '; ')G(x,x '; + ')exp(i ')d

W

G

73

Quasiparticle Energies

Green s functionfrom KS wavefunctions

Dielectric functionwithin the RPA and the GPP model

where the self-energy operator is approximated as

= iG0W = iG01V

G = nk (r) nk* (r')

E nk

and G is constructed from Kohn-Sham eigenvalues & eigenstates(independent particle G)

74

E

i

QPE

i

KS+ Z

i i

KS (x,x ';i

KS ) VXC

KS

i

KS

with Zi= 1

d (x,x ', )

d=

i

KS

1

Quasiparticle energies are computed to first order as

Green’s function from

KS wavefunctions

Dielectric function

within the RPA and the GPP model

From Hedin’s Equation to GW Approximation

(12;3) = (12) (13) +(12)

G(45)G(46)G(75) (67;3)d(4567)

(12) = i W (1+3)G(14) (42;3)d(34)

W (12) = v(12) + W (13)P(34)v(42)d(34)

P(12) = i G(23)G(42) (34;1)d(34)

(1

22+V 0)G(x,x', ) + (x,x", )G(x",x ', )dx" = (x x ')

Integral equations relating G, , , and W:

75

Equation of the Green’s function G:

Hardin, PR139, A796 (1965)

From Hedin’s Equation to GW Approximation

76

(12;3) (12) (13)

(12) iG(12)W (1+2)

W (12) v(12) + W (13)P(34)v(42)d(34)

P(12) iG(12)G(21)

Hedin’s GW approximation

Neglect vertex

RPA

First-order expansion in screened Coulomb interaction W

Core electron polarization, and core electron screening,

are neglected

.

Screened Coulomb Interaction W

Short hand W =1VCoul

(r,r', ) = (r r') V (r,r")P(r' ',r', )dr"

where V is the bare Coulomb potential and the (irreducible) polarizability is

P(r,r', ) = nk* (r) n 'k' (r) n 'k'

* (r') nk (r')nk,n'k '

(fnk (1 fn'k ' )

nk n 'k' + + i+

fn'k ' (1 fnk )

n'k ' nk + i)

where epsilon is the full frequency dependent dielectric function

77

Screened Coulomb Interaction W

78

In the Berkeley code, the dielectric function is calculated in reciprocal

space, within the Random Phase Approximation (RPA):

Short hand W =1V 1

=1(q, )where

WG G (q, ) = G G 1 (q, )V G (q)

Frequency-dependent dielectric matrix Coulomb interaction

V G (q) =4

q + G 2

The dielectric function is calculated in reciprocal space, within the

Random Phase Approximation (RPA):

Screened Coulomb Interaction W

79

Random Phase Approximation (RPA)

G G 0 (q, ) = 2 n,k e i(q +G) r n ,k + q

n n k

n ,k + q ei(q + G ) r n,kf ( n ,k +q ) f ( n,k )

n ,k +q n,k i

Ingredients: Kohn-Sham eigenvalues and eigenstates

G G RPA (q, ) = G G VG (q) G G

0 (q, )

Independent particle polarizability (Adler-Wiser form)

Static Dielectric Function

In practice, we first compute the static polarizability

G G 0 (q, = 0) = 2 n,k e i(q +G) r n ,k + q

n n k

n ,k + q ei(q + G ) r n,kf ( n ,k +q ) f ( n,k )

n ,k +q n,k

cutting off the sum to include only a finite (but sufficient) number of empty

bands. We then construct

invert this matrix in G and G’, and extend it to finite frequency using the

generalized plasmon-pole model.

G G RPA (q,0) = G G VG (q) G G

0 (q,0)

80

Dielectric Function with Finite Frequencies

Static dielectric function (q, =0) is a ground-state property

obtainable from DFT or DFPT within RPA or beyond.

To evaluate the GW self-energy, we need W, the dynamically-screened

Coulomb interaction. This is a convolution of -1(q, ) and the bare

Coulomb potential V(q).

Thus, we need to capture the frequency dependence of the screening.

Evaluating the frequency-dependence is computationally-expensive,

and in practice, we find that (q, =0) can be extended to finite

frequency using a plasmon-pole model (PPM).

81

Plasmon-Pole Model Physical motivation: The loss function is a peaked function of

The peaks are dictated by the plasmon dispersion, (q)

Generalized plasmon-pole: Hybertsen-Louie (1986)

Parameters A, , and are fixed by KK relations and the f-sum rule.

Im G, G 1 (q, ) = AG, G (q) ( ˜ G, G (q)){ }

Re G, G 1 (q, ) = G, G + G, G

2

2 ˜ G, G 2 (q)

S(q, ) Im 1(q, ) = Aq ( ˜ q ){ }

82 Hybertsen & Louie, PRB 34, 5390 (1986)

KK relations & f-sum rule

83

For a complex function ( ) = 1( ) + i 2( ) of the complex

variable , analytic in the upper half plane of and which

vanishes faster than 1/| | as | | , the Kramers–Kronig relations are given by

and

where P denotes the Cauchy principal value.

1( ) =

1P

2( ')

'd ' and

2( ) =

1P

1( ')

'd '

Thomas-Reiche-Kuhn (f-)sum rule

Oscillator strength f:

(E0

En)

n

n | x | 02

=

2

2m

f12=

2me

3 2(E

2E

1) 1m

1| R | 2m

2

2

= x , y ,xm2

Plasmon-Pole Model

84

Plasmon-pole frequency

Bare plasma frequency

˜ G G 2 (q) = G G

2 (q)

Re G G 1 (q,0)

G G 2 (q) = p

2 (q + G) (q + G )

q + G2

(G G )(0)

AGG '

(q) =2

GG '(q)

GG '(q)

The Amplitude A can be determined as

GW Self-Energy

nkSX (Enk ) = n,k nk

SX (r, r ;Enk ) n,k

= n,k e i(q +G) r n1,k q n1,k q e i(q + G ) r n,kn1 ,q,G, G

1+ G G 2 (q)

(Enk n1k q )2 ˜ G G 2 (q)

v(q + G )

nkCH (Enk ) = n,k nk

CH (r, r ;Enk ) n,k

= n,k e i(q +G) r n1,k q n1,k q e i(q + G ) r n,kn1 ,q,G, G

G G 2 (q)

2 ˜ G G (q)(Enk n1k q ˜ G G (q))

v(q + G )

Coulomb-Hole (CH)

Screened-exchange (X+SX)

= x + sx + ch

Screened exchange: poles of G

Coulomb-hole: poles of W

Bare exchange: Fock term

GW Self-Energy

85

Bare exchange: Fock term Screened exchange:

Poles of G

Coulomb-hole:

Poles of W

Comments on GW Approximation

The self-energy is, in general, a complex non-Hermitian operator.

The real part gives quasiparticle correction to the DFT energies, and the

imaginary part is related to the quasiparticle lifetime. Here, we focus only on

Re[ ]. The GPP model prevents us from inferring much about Im[ ].

In principle, the quasiparticle equations are coupled nonlinear equations and

should be solved self-consistently.

In practice, off-diagonal corrections are relatively small, and we just

compute the first-order correction to the Kohn-Sham energies.

Self-consistent GW calculations can be done self-consistently, although it is

very expensive, and its merits are an issue of current debate in the literature.

GW with the vertex correction: computationally demanding and often

overestimates the band gap.

86

Dynamic Dependence of GW Self-Energy

First-order correction depends on the quasiparticle eigenvalue

EnkQP

= EnkDFT

+ nk (r, r ; = EnkQP ) Vxc

DFT (r) nk

In principle, this is a nonlinear equation that should be solved self-consistently. However, in practice, linearization is sufficient:

nk (EnkQP ) = nk (Enk

DFT ) + (EnkQP Enk

DFT )d nk ( )

d EnkDFT

For this discussion, we neglect off-diagonal corrections

87

E

i

QPE

i

KS+

i

KS (x,x '; Ei

QP ) VXC

KS

i

KS

i(E

i

QP )i(E

i

KS ) + Ei

QPE

i

KS( )d

i( )

dE

i

KS

Dynamic Dependence of GW Self-Energy

88

Defining the renormalization constant Znk, we have

EnkQP

= EnkDFT

+ Znk nk (r, r ; = EnkDFT ) Vxc

DFT (r) nk

The self-energy can be expressed in terms of the DFT & energies as

Znk = 1d nk ( )

d =EnkDFT

1

nk

Comments• E in Znk is evaluated by finite differences, and is typically quite small.• The Green s function can be iterated, and recomputed, to further checkthat this procedure results in small changes in QP energies

Zi

Zi= 1

d (x,x ', )

d=E

i

KS

1

E

i

QPE

i

KS+ Z

i i

KS (x,x ';i

KS ) VXC

KS

i

KS

E in Znk is evaluated by finite differences, and is typically quite small.The Green s function can be iterated, and recomputed, to further check

Zi,

states & energies as

Zi d

d

Comments

Neutral (Electron-Hole) Excitations

89

2( ) =16 e2

2 MS

2( S )

Im part of the transverse dielectric function

hv

0

S

S

Optical gap Two-particle Green’s function

0 S

Optical excitations

G 1,2, 1 , 2 ( ) = i( )2N T[ (1) (2) +( 2 ) +( 1 )]N

Bethe-Salpeter Formalism

where is taken as the product of occupied & empty

quasiparticle states in the Tamm-Dancoff approximation.

This puts the BSE in the following form:

where s is the excitation energy of the state S

(Eck Evk )AvckS

+ vck v c k

K eh v c k A v c k S

=S Avck

S

S = AvckS

vck

vck

Seek solutions to the BSE of the form

vck

90

Bethe-Salpeter Formalism

G2 1,2; 1 , 2 ( ) = i( )2N T[ (1) (2) +( 2 ) +( 1 )]N

Bethe-Salpeter equation (BSE)

L(1,2; 1 , 2 ) =G1(1, 2 )G1(2, 1 ) + G1(1,3)G1(4, 1 )Keh (3,5;4,6)L(6,2;5, 2 )d(3456)

L(1,2; 1 , 2 ) = G2(1,2; 1 , 2 ) +G1(1, 1 )G1(2, 2 )

Keh (3,5;4,6) =V (3) (3,4) + (3,4)[ ]

G1(6,5)

where

G1 1, 1 ( ) = i N T[ (1) +( 1 )]N

Electron-Hole interaction kernel

Electron-Hole correlation function

91

Effective Two-Particle Interaction

Keh (3,5;4,6) Kx (3,5;4,6) + Kd (3,5;4,6), = iGW

“Exchange” term

Bare Coulomb interaction

(spin-singlet, spin-triplet

splitting, etc)

“Direct” term

Screened Coulomb interaction

Leads to e-h attraction

Comments• As before, W is the fully dynamically-screened Coulomb interaction.However, in practice, since the exciton binding energies are much smaller

than the typical plasmon energies, only static screening 1(q,0) is used.

Electron-hole interaction kernel

92

As before, W is the fully dynamically-screened Coulomb interaction.

Simplification of the Bethe-Salpeter Equation

S (x, x ) =< N,0 | +( x ) (x) |N,S >

S (x, x ) = AvcS

c (x)cv

v*( x ) + Bvc

Sv (x) c

*( x )

L(1,2; 1 , 2 ) S (x, x )

2-particle Green’s function

Define the expectation value

G 1,2, 1 , 2 ( ) = i( )2N T[ (1) (2) +( 2 ) +( 1 )]N

Expand these functions in terms of single-particle states

Rewrite the ‘L’s’ in the BSE

93 Rohlfing & Louie, PRB 62, 4927 (2000)

Simplification of the Bethe-Salpeter Equation

where is taken as the product of occupied & empty

quasiparticle states in the Tamm-Dancoff approximation.

This puts the BSE in the following form:

where s is the excitation energy of the state S

(Eck Evk )AvckS

+ vck v c k

K eh v c k A v c k S

=S Avck

S

S = AvckS

vck

vck

Seek solutions to the BSE of the form

vck

94

Optical Absorption ASpectra Imaginary part of the transverse dielectric function:

2( ) =16 e2

2

r 0

r v S

S

2

( S )

2( ) =16 e2

2

r v

r v c

vc

2

( (Ec Ev ))

r = A /

r A

r v = i /h H,

r r [ ]

Polarization vector

Single-particle velocity

operator

Same thing, without electron-hole interactions:

95

Polarization

vector

Single-particle

velocity operator

BSE optical transition matrix elements: 0r v S = Avc

S vr v c

cv

Coherent sum of matrix elements of contributing electron-hole transitions

GW/BSE Implementation: Summary

96

GW code is often implemented in the frequency-reciprocal

space with a plane wave basis and norm-conserving

pseudopotentials.

Applicability: Bulk systems, surfaces, clusters; ~100

atoms.

Input: KS wavefunctions (of occupied and many

unoccupied bands) for a converged ground state system.

Approximations: GW, RPA and plasmon-pole model for

the dielectric function.

GW Implementation

DFT { nk,Enk}

RPA 1(q,0)

=iGW

• PARATEC code

• SCF for ground-state (PW, NC-PPs)

• NSCF for 100’s of empty states

• xi0 code• Computes 0(q) for G, G’

• Inverts for -1(q) for each G, G’

• sigma code• Forms matrix elements to compute ,

including GPP• Outputs Vxc for specified nk

97

RPA 1(q,0)

=iGW

Keh, 2( )

• xcton/diag codes

• Forms electron-hole kernel using staticlongitudinal dielectric function -1(q)

• Outputs two-particle excitation energiesand transverse 2( )

• xi0 code• Computes 0(q) for G, G’

• Inverts for -1(q) for each G, G’

• sigma code• Forms matrix elements to compute ,

including GPP• Outputs Vxc for specified nk

(eqp.corrections file)

BSE Implementation

98

GW/BSE Scaling Within a Planewave Implementation

Generating 0 ~ NG

Inverting G,G’ ~ (NG)3

Total scaling ~ (NG)4

99

GW: the Main cost is generating the dielectric function

Diagonalization of BSE Hamiltonian

Hamiltonian ~ Nk Ncond Nval, in principle, Ncond and

Nval are small)

Need a fine k-mesh; cost of diagonalization ~ (Nk)3

BSE: the QP corrections

Band Gap and Band Energy Differences

100

LDA GWA(a) Exp

diamond 3.9 5.6 5.48

Silicon 0.52 1.29 1.17

Germanium ~0 0.75 0.74

LiC 6.0 9.1 9.4

GW improves on the DFT gap problem Gaps in

garden variety semiconductors and insulators are

under-estimated by 30~50%

(eV)

ELDA (eV)

Hybertsen & Louie, PRB 34, 5390 (1986)

LDA GWA

Ge

8v 7c -0.26 0.85

8v X5c 0.55 1.09

8v L6c -0.05 0.73

GaAs

8v 7c 0.13 1.42

8v X5c 1.21 1.95

8v L6c 0.70 1.75

GWA

0.85

1.09

0.73

1.42

1.95

1.75

Exp

0.89

1.10

0.74

1.52

2.01

1.84

Band energy difference between

high symmetry points (in eV)

W. G. Aulbur, L. Jonsson, and

J. W. Wilkins, 2000

Band Structure of Copper

101

Solid line: GW

Dashed-line: DFT-LDA

Circle: Experiment

Onida, Reining, Rubio, RMP 74, 6015 (2002)

Benzene/Graphite Interface

102

Fermi Energy

E

z

Affinity Level

Ionization Level

metal adsorbedmolecule

vacuum

Ph

Pe

gas phasemolecule

Fermi Energy

E

z

Affinity Level

Ionization Level

metal adsorbedmolecule

vacuum

Ph

Pe

gas phasemolecule

FIG. 1 (color online). Schematic energy level diagram indicat-ing polarization shifts in the frontier energy levels (ionizationand affinity) of a molecule upon adsorption on a metal surface.

Benzene/Graphite Interface

103

(c)

ππππ*

ππππ

(a) (b)

ΓΓΓΓ K´

E –

EF

(eV

)

-4

-2

0

2

4

K´ΓΓΓΓ

ππππ

π∗∗π∗ππ∗

LDA GW GW

(c)

ππππ*

ππππ

(a) (b)

ΓΓΓΓ K´

E –

EF

(eV

)

-4

-2

0

2

4

K´ΓΓΓΓ

ππππ

π∗∗π∗ππ∗

LDA GW GW

PRL 97, 216405 (2006)

Gas phase

Optical Absorption Spectra

104

PRL 88, 066404 (2002)

Si

Optical Absorption Spectra

105 PRL 95, 156401 (2005)

SiO2

Si Nanowires

106

[111] [110]

FIG. 1. Ball-and-stick models of silicon nanowires (SiNWs)along the [110] (right) and [111] (left) directions viewed fromthe top (upper panels) and the side (lower panels). Filled circlesstand for Si atoms, and open circles for H atoms.

0 1 2 3 4 5 6 7 8Diameter (nm)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Eg (

eV

)

[110] [111] [110], GW [112], expt. (Ref. 6) [110], expt. (Ref. 6)

FIG. 2 (color online). LDA band gaps calculated for [110](empty squares) and [111] (empty circles) wires, and theGW-corrected gaps (filled squares) for the two thinnest [110]wires and bulk Si, compared with the measured gaps for [112]wires (+) and a [110] wire (*). The dotted, dashed, and long-dashed lines are fitted to the data points (see text). The LDAband gap of bulk Si is indicated by the solid line, and the bulkGW gap is marked around d � 8:0 nm for convenience.

Zhao et al., PRL92, 236805 (2004)

Si Nanowires

107 Yan et al., PRB76, 115319 (2007)

Solid symbols: GW

Open symbols: LDA

Si Nanowires

108 Yang et al., PRB75, 186401 (2008)

FIG. 3. Absorption spectrum of the silicon nanowire calculatedwith a smearing width of 0.1 eV. The arrows marked by A and A1

indicate the fundamental gap and the lowest-energy exciton loca-tions, respectively. The extremely small absorption for polarizationperpendicular to the axis illustrates the depolarization effect.

FIG. 4. �Color online� The density of the electron for severalexcitonic states with the hole fixed at the center of the wire betweentwo nearest-neighbor Si atoms. From top to bottom, four represen-tative states are shown with energy values of 2.1 eV, 3.3 eV,4.2 eV, and 4.2 eV, respectively. Corresponding cross-section plotson the xz plane are shown on the right, while the three-dimensionalplots for the same data are shown on the left. Lengths are in atomicunits.

Graphene Nanoribbon

109

z

y

w wArmchair Zigzag

Yang et al., PRL99, 186801 (2007)

Graphene Nanoribbon

110

FIG. 2 (color online). (a) Optical absorption spectra,(b) density of excited states, (c) absorbance of graphene withand without excitonic effects included; and (d) comparison withexperiments. In (d), rough colored curves within the smallrectangular box are measurements from Ref. [11], and opencircles are from those from Ref. [10].

FIG. 4 (color online). (a) Absorbance of bilayer graphene and(b) imaginary part of the dielectric function of graphite with andwithout excitonic effects included. Experiment data of graphite[29] are included in (b).

Yang et al., PRL103, 186802 (2009)

GW Band Gaps

111

W. G. Aulbur, L. Jonsson,

and J. W. Wilkins, 2000

Self-Consistency in GW

112

(1

22+V 0)G(x,x', ) + (x,x", )G(x",x ', )dx" = (x x ')

GW (x,x ', ) =i

2W (x,x '; ')G(x,x '; + ')exp(i ')d

G0W0: NO self-consistency, using G0 and W0 constructed

from KS wave functions and eigenvalues.

Why NOT going beyond G0W0:

Large computational cost: a formidable task.

A full self-consistent solution often worsens band structures.

Better results from G0W0 than self-consistent GW is due to an

error cancellation between the dressing of Green functions and

the vertex corrections.

Levels of self-consistency: G0W0 GW0 GW

GW Vertex Correction

113

(12;3) = (12) (13) +(12)

G(45)G(46)G(75) (67;3)d(4567)

GW : including the vetex gamma, which is expressed

in terms of functional derivative

Vxc

/

Onida, Reining, Rubio, RMP 74, 6015 (2002)

��1

1�fxcP0,

W�W1��1�fxcP0�v

1��v�fxc�P0, G�G0

��iG0 W1 ��iG0v

1��v�fxc�P0

DFT-Based

GW :

GW Vertex Correction

114

Systematic GW : iterative solution of Hedin’s equations.

Schindlmayr and Godby, PRL 80, 1702 (1998)

�(n�1)�123����12���13��� d�4567���(n)�12�

�G(n�1)�45�

�G(n)�46�G(n)�75��(n�1)�673�

�(2)�123����12���13��� d�4567���(1)�12�

�G(0)�45�

�G(1)�46�G(1)�75��(2)�673�.

The lowest-order version:

The second equation is the starting point for the Bethe-Salpeter equation.

Practically, very few systematic GW calculations for real materials.

Self-Consistent GW Calculations

115

PBE G0W0 GW0 GW Expt. a SO

PbSe −0.17 0.10 0.15 0.19 0.15a 6.098b 0.40

PbTe −0.05 0.20 0.24 0.26 0.19c 6.428b 0.73

PbS −0.06 0.28 0.35 0.39 0.29d 5.909b 0.36

Si 0.62 1.12 1.20 1.28 1.17e 5.430f

GaAs 0.49 1.30 1.42 1.52 1.52e 5.648f 0.10

SiC 1.35 2.27 2.43 2.64 2.40g 4.350g

CdS 1.14 2.06 2.26 2.55 2.42h 5.832h 0.02

AlP 1.57 2.44 2.59 2.77 2.45h 5.451h

GaN 1.62 2.80 3.00 3.32 3.20i 4.520i 0.00

ZnO 0.67 2.12 2.54 3.20 3.44e 4.580h 0.01

ZnS 2.07 3.29 3.54 3.86 3.91e 5.420h 0.02

C 4.12 5.50 5.68 5.99 5.48g 3.567g

BN 4.45 6.10 6.35 6.73 6.1–6.4j 3.615h

MgO 4.76 7.25 7.72 8.47 7.83k 4.213l

LiF 9.20 13.27 13.96 15.10 14.20m 4.010n

Ar 8.69 13.28 13.87 14.65 14.20o 5.260p

Ne 11.61 19.59 20.45 21.44 21.70o 4.430p

MARE 45% 9.9% 5.7% 6.1%

MRE 45% −9.8% −3.6% 4.7%

PRB 8075, 235102 (2007)

Si/SiO2 Interface

116

TABLE I. Quasiparticle corrections (in eV) at the VBM (�Ev),at the CBM (�Ec), and for the band gap (�Eg) for Si and c-SiO2.The corrections are calculated within GW using the PPMsproposed by Hybertsen and Louie (HL) [27], von der Lindenand Horsch (vdLH) [28], Godby and Needs (GN) [29], Engel andFarid (EF) [30], and without PPM.

HL vdLH GN EF no PPM

Si �Ev �0:6 �0:6 �0:4 �0:6 �0:4�Ec �0:1 �0:1 �0:2 �0:1 �0:2�Eg �0:7 �0:7 �0:6 �0:7 �0:6

c-SiO2 �Ev �2:6 �2:5 �2:0 �2:3 �1:9�Ec �1:3 �1:1 �1:5 �1:2 �1:5�Eg �3:9 �3:6 �3:5 �3:5 �3:4

TABLE II. Quasiparticle corrections (in eV) at the VBM(�Ev), at the CBM (�Ev), and for the band gap (�Eg) for Si,c-SiO2, and s-SiO2. The corrections are calculated within GW,GW�, and QSGW.

Si c-SiO2 s-SiO2

GW GW� QSGW GW GW� QSGW GW GW� QSGW

�Ev �0:4 �0:1 �0:6 �1:9 �1:3 �2:8 �1:9 �1:3 �2:8�Ec �0:2 �0:7 �0:2 �1:5 �1:8 �1:3 �1:4 �1:8 �1:1�Eg �0:6 �0:6 �0:8 �3:4 �3:1 �4:1 �3:3 �3:1 �3:9

TABLE III. Quasiparticle band offsets (eV) for cubic andstrained SiO2 using GW, GW�, and QSGW.

Cubic Strained

Model DFT GW GW� QSGW GW GW� QSGW Expt.

VBO I 2.6 4.1 4.0 4.8 4.1 4.0 4.8 4.3II 2.5 4.0 3.9 4.7 4.0 3.9 4.7

CBO I 1.6 2.9 2.7 2.7 2.8 2.7 2.5 3.1II 1.8 3.1 2.9 2.9 3.0 2.9 2.7

QSGW: quasiparticle self-consistent GW

GW : GW with the vertex correction

PRL 100, 186401 (2008)