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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
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)