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Intro MBPT & GW Linear BSE TDDFT
Excited States Electronic Propertiesand Theoretical Spectroscopy
Francesco Sottile
Ecole de simulation numerique en matiere condensee
Jussieu, 9 June 2016
Intro MBPT & GW Linear BSE TDDFT
Outline
Introduction and reminder of ground-state results
Photoemission via MBPT: GW approximation
Absorption and Loss spectroscopies: linear response quantities
Neutral excitations within MBPT: Bethe-Salpeter Equation
Neutral excitations within DFT: Time-Dependent DensityFunctional Theory
Intro MBPT & GW Linear BSE TDDFT
Outline
Introduction and reminder of ground-state results
Photoemission via MBPT: GW approximation
Absorption and Loss spectroscopies: linear response quantities
Neutral excitations within MBPT: Bethe-Salpeter Equation
Neutral excitations within DFT: Time-Dependent DensityFunctional Theory
Intro MBPT & GW Linear BSE TDDFT
Density Functional Theory
E [n] ⇒
Total energy, phase stability,bulk modulus, lattice constant, etc.[
−∇2 + Vion + VH + Vxc
]φi (r) = εiφi (r)
n(r) =∑i
|φi (r)|2
Intro MBPT & GW Linear BSE TDDFT
Density Functional Theory
E [n] ⇒
Total energy, phase stability,bulk modulus, lattice constant, etc.[
−∇2 + Vion + VH + Vxc
]φi (r) = εiφi (r)
n(r) =∑i
|φi (r)|2
Intro MBPT & GW Linear BSE TDDFT
Density Functional Theory
E [n] ⇒
Total energy, phase stability,bulk modulus, lattice constant, etc.[
−∇2 + Vion + VH + Vxc
]φi (r) = εiφi (r)
n(r) =∑i
|φi (r)|2
Intro MBPT & GW Linear BSE TDDFT
Density Functional Theory
E [n] ⇒
Total energy, phase stability,bulk modulus, lattice constant, etc.[
−∇2 + Vion + VH + Vxc
]φi (r) = εiφi (r)
n(r) =∑i
|φi (r)|2
Intro MBPT & GW Linear BSE TDDFT
Density Functional Theory
Often reasonable results also for excitations
• ab initio
• qualitative estimate
• powerful analysis tool
• starting point for more accurate methods
Intro MBPT & GW Linear BSE TDDFT
Density Functional Theory
homo lumo
Intro MBPT & GW Linear BSE TDDFT
Density Functional Theory
Z A M Γ X R Z Γ A R Γ
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
En
erg
y(e
V)
Band Structure of SnO2
Intro MBPT & GW Linear BSE TDDFT
Density Functional Theory
Z A M Γ X R Z Γ A R Γ
-10
-8
-6
-4
-2
0
2
4
6
8
10
12E
ner
gy
(eV
)
Band Structure of SnO2
Intro MBPT & GW Linear BSE TDDFT
Density Functional Theory
calculated
gap(eV)
:LDA
HgTe InSb
,P,InAs
InN,Ge,GaS
b,CdO
SiInP,GaA
s,CdTe,AlSb
Se,Cu2O
AlAs,GaP
,SiC,AlP,CdS
ZnSe
,CuB
rZnO,GaN
,ZnS
diam
ond
SrO AlN
MgO
CaO
experimental gap (eV)
Intro MBPT & GW Linear BSE TDDFT
Optical properties
11 12 13 14 15 16ω(eV)
5
10
15
Im
εM
exp
DFT level
Solid Argon Absorption spectrum
Intro MBPT & GW Linear BSE TDDFT
Density Functional Theory
E [n] ⇒
Total energy, phase stability,bulk modulus, lattice constant, etc.[
−∇2 + Vion + VH + Vxc
]φi (r) = εiφi (r)
n(r) =∑i
|φi (r)|2
Intro MBPT & GW Linear BSE TDDFT
What is an electron ?
Intro MBPT & GW Linear BSE TDDFT
Spectroscopies :: charged excitations
Direct Photoemission
Intro MBPT & GW Linear BSE TDDFT
Spectroscopies :: charged excitations
Inverse Photoemission
Intro MBPT & GW Linear BSE TDDFT
Spectroscopies :: charged excitations
Direct and inverse photoemission spectroscopy
Intro MBPT & GW Linear BSE TDDFT
Spectroscopies :: neutral excitations
Absorption
Intro MBPT & GW Linear BSE TDDFT
Spectroscopies :: neutral excitations
i
unoccupied states
occupied states
j
Absorption
Intro MBPT & GW Linear BSE TDDFT
Spectroscopies :: neutral excitations
Electron Energy Loss Spectroscopy
Intro MBPT & GW Linear BSE TDDFT
Outline
Introduction and reminder of ground-state results
Photoemission via MBPT: GW approximation
Absorption and Loss spectroscopies: linear response quantities
Neutral excitations within MBPT: Bethe-Salpeter Equation
Neutral excitations within DFT: Time-Dependent DensityFunctional Theory
Intro MBPT & GW Linear BSE TDDFT
DFT vs MBPT
Density Functional Theory
O[n] ⇐ n(r) =occ∑i
|φi (r)|2
Many-Body Perturbation Theory
O[G ]
G (r1, t1, r2, t2) = −i⟨N∣∣∣T [ψ(r1, t1)ψ†(r2, t2)
]∣∣∣N⟩
Intro MBPT & GW Linear BSE TDDFT
DFT vs MBPT
Density Functional Theory
O[n] ⇐ n(r) =occ∑i
|φi (r)|2
Many-Body Perturbation Theory
O[G ]
G (r1, t1, r2, t2) = −i⟨N∣∣∣T [ψ(r1, t1)ψ†(r2, t2)
]∣∣∣N⟩
Intro MBPT & GW Linear BSE TDDFT
DFT vs MBPT
Density Functional Theory
O[n] ⇐ n(r) =occ∑i
|φi (r)|2
Many-Body Perturbation Theory
O[G ]
G (r1, t1, r2, t2) = −i⟨N∣∣∣T [ψ(r1, t1)ψ†(r2, t2)
]∣∣∣N⟩
Intro MBPT & GW Linear BSE TDDFT
Green’s Function G
G (r1, r2, ω) =∑n
An(r1)A∗n(r2)
ω − En + iη
En =
additional energy :: En > µ
removal energy :: En < µ
The poles of G are the excitation energies
Intro MBPT & GW Linear BSE TDDFT
Green’s Function G
G (r1, r2, ω) =∑n
An(r1)A∗n(r2)
ω − En + iη
En =
additional energy :: En > µ
removal energy :: En < µ
The poles of G are the excitation energies
Intro MBPT & GW Linear BSE TDDFT
Spectroscopies :: charged excitations
Direct Photoemission
Intro MBPT & GW Linear BSE TDDFT
Green’s Function G - spectral function
A(r1, r2, ω) ∝ ImG (r1, r2, ω)
interacting
non-interactingA( )
E
Intro MBPT & GW Linear BSE TDDFT
Green’s Function G - spectral function
A(r1, r2, ω) ∝ ImG (r1, r2, ω)
A( )
E
non-interacting
interacting
Intro MBPT & GW Linear BSE TDDFT
How to calculate G ?
G (1, 2) = −i⟨N∣∣∣T [ψ(1)ψ†(2)
]∣∣∣N⟩G (1, 2) = G 0(1, 2) +
∫d(34)G 0(1, 3)Σ(3, 4)G (4, 2)
Σ = self-energy
Intro MBPT & GW Linear BSE TDDFT
How to calculate G ?
G (1, 2) = −i⟨N∣∣∣T [ψ(1)ψ†(2)
]∣∣∣N⟩G (1, 2) = G 0(1, 2) +
∫d(34)G 0(1, 3)Σ(3, 4)G (4, 2)
Σ = self-energy
Intro MBPT & GW Linear BSE TDDFT
Hedin’s equations
Σ(1, 2) = i
∫d(34)G (1, 3)Γ(3, 2, 4)W (4, 1+)
G (1, 2) = G 0(1, 2) +
∫d(34)G 0(1, 3)Σ(3, 4)G (4, 2)
Γ(1, 2, 3) = δ(1, 2)δ(1, 3)+
∫d(4567)
δΣ(1, 2)
δG (4, 5)G (4, 6)G (7, 5)Γ(6, 7, 3)
P(1, 2) = −i∫
d(34)G (1, 3)G (4, 1+)Γ(3, 4, 2)
W (1, 2) = v(1, 2) +
∫d(34)v(1, 3)P(3, 4)W (4, 2)
Intro MBPT & GW Linear BSE TDDFT
Hedin’s equations
Σ(1, 2) = i
∫d(34)G (1, 3)Γ(3, 2, 4)W (4, 1+)
G (1, 2) = G 0(1, 2) +
∫d(34)G 0(1, 3)Σ(3, 4)G (4, 2)
Γ(1, 2, 3) = δ(1, 2)δ(1, 3)+
∫d(4567)
δΣ(1, 2)
δG (4, 5)G (4, 6)G (7, 5)Γ(6, 7, 3)
P(1, 2) = −i∫
d(34)G (1, 3)G (4, 1+)Γ(3, 4, 2)
W (1, 2) = v(1, 2) +
∫d(34)v(1, 3)P(3, 4)W (4, 2)
Intro MBPT & GW Linear BSE TDDFT
Hedin’s equations
Σ(1, 2) = i
∫d(34)G (1, 3)Γ(3, 2, 4)W (4, 1+)
G (1, 2) = G 0(1, 2) +
∫d(34)G 0(1, 3)Σ(3, 4)G (4, 2)
Γ(1, 2, 3) = δ(1, 2)δ(1, 3)+
∫d(4567)
δΣ(1, 2)
δG (4, 5)G (4, 6)G (7, 5)Γ(6, 7, 3)
P(1, 2) = −i∫
d(34)G (1, 3)G (4, 1+)Γ(3, 4, 2)
W (1, 2) = v(1, 2) +
∫d(34)v(1, 3)P(3, 4)W (4, 2)
Intro MBPT & GW Linear BSE TDDFT
Hedin’s equations
Σ(1, 2) = i
∫d(34)G (1, 3)Γ(3, 2, 4)W (4, 1+)
G (1, 2) = G 0(1, 2) +
∫d(34)G 0(1, 3)Σ(3, 4)G (4, 2)
Γ(1, 2, 3) = δ(1, 2)δ(1, 3)+
∫d(4567)
δΣ(1, 2)
δG (4, 5)G (4, 6)G (7, 5)Γ(6, 7, 3)
P(1, 2) = −i∫
d(34)G (1, 3)G (4, 1+)Γ(3, 4, 2)
W (1, 2) = v(1, 2) +
∫d(34)v(1, 3)P(3, 4)W (4, 2)
Intro MBPT & GW Linear BSE TDDFT
Hedin’s equations
Σ(1, 2) = i
∫d(34)G (1, 3)Γ(3, 2, 4)W (4, 1+)
G (1, 2) = G 0(1, 2) +
∫d(34)G 0(1, 3)Σ(3, 4)G (4, 2)
Γ(1, 2, 3) = δ(1, 2)δ(1, 3)+
∫d(4567)
δΣ(1, 2)
δG (4, 5)G (4, 6)G (7, 5)Γ(6, 7, 3)
P(1, 2) = −i∫
d(34)G (1, 3)G (4, 1+)Γ(3, 4, 2)
W (1, 2) = v(1, 2) +
∫d(34)v(1, 3)P(3, 4)W (4, 2)
Intro MBPT & GW Linear BSE TDDFT
Hedin’s pentagon
Intro MBPT & GW Linear BSE TDDFT
Hedin’s pentagon: possible strategy
Intro MBPT & GW Linear BSE TDDFT
Hedin’s pentagon: possible strategy
Intro MBPT & GW Linear BSE TDDFT
Hedin’s pentagon: possible strategy
Intro MBPT & GW Linear BSE TDDFT
Hedin’s pentagon: possible strategy
Intro MBPT & GW Linear BSE TDDFT
Hedin’s pentagon: possible strategy
Intro MBPT & GW Linear BSE TDDFT
Hedin’s pentagon: possible strategy
Intro MBPT & GW Linear BSE TDDFT
In practice: quasi-particle approximation
Quasi-particle equation
[−∇2 + Vion + VH + Σ(r, r′,E )
]ψi (r) = Eiψi (r)
Intro MBPT & GW Linear BSE TDDFT
In practice: quasi-particle approximation
Quasi-particle equation
[−∇2 + Vion + VH + Σ(r, r′,E )
]ψi (r) = Eiψi (r)
interacting
non-interactingA( )
E
Intro MBPT & GW Linear BSE TDDFT
In practice: quasi-particle approximation
Quasi-particle equation
[−∇2 + Vion + VH + Σ(r, r′,E )
]ψi (r) = Eiψi (r)
• Σ(r, r′, ω) non-local,non-hermitian, energydependent
• Ei complexquasiparticle
interacting
non-interactingA( )
EReE i
ImE i
Intro MBPT & GW Linear BSE TDDFT
In practice: quasi-particle approximation
Quasi-particle equation
[−∇2 + Vion + VH + Σ(r, r′,E )
]ψi (r) = Eiψi (r)
DFT-KS equation
[−∇2 + Vion + VH + Vxc
]φi (r) = εiφi (r)
Intro MBPT & GW Linear BSE TDDFT
In practice: quasi-particle approximation
Quasi-particle equation
[−∇2 + Vion + VH + G 0W 0
]ψi (r) = Eiψi (r)
DFT-KS equation
[−∇2 + Vion + VH + V lda
xc
]φi (r) = εiφi (r)
Intro MBPT & GW Linear BSE TDDFT
In practice: quasi-particle approximation
interacting
non-interactingA( )
EReE i
ImE i
ilda
Intro MBPT & GW Linear BSE TDDFT
LDA band gap
calculated
gap(eV)
:LDA
HgTe InSb
,P,InAs
InN,Ge,GaS
b,CdO
SiInP,GaA
s,CdTe,AlSb
Se,Cu2O
AlAs,GaP
,SiC,AlP,CdS
ZnSe
,CuB
rZnO,GaN
,ZnS
diam
ond
SrO AlN
MgO
CaO
experimental gap (eV)
Intro MBPT & GW Linear BSE TDDFT
GW results :: band gap
calculated
gap(eV)
:LDA:GW(LDA)
HgT
e InSb
,P,In
AsInN,Ge,GaS
b,CdO
SiInP,GaA
s,CdT
e,AlSb
Se,Cu2
OAlAs
,GaP
,SiC,AlP,CdS
ZnSe
,CuB
rZn
O,GaN
,ZnS
diam
ond
SrO AlN
MgO
CaO
experimental gap (eV)
van Schilfgaarde et al., PRL 96, 226402 (2006)
Intro MBPT & GW Linear BSE TDDFT
GW results :: band gap
experimental gap (eV)
QPs
cGW
gap(eV)
MgOAlN
CaO
HgT
e InSb
,InAs
InN,GaS
bInP,GaA
s,CdT
eCu2
O ZnTe
,CdS
ZnSe
,CuB
rZn
O,GaN
ZnS
P,Te
SiGe,CdO
AlSb,SeAlAs,GaP,SiC,AlP
SrOdiamond
van Schilfgaarde et al., PRL 96, 226402 (2006)
Intro MBPT & GW Linear BSE TDDFT
Outline
Introduction and reminder of ground-state results
Photoemission via MBPT: GW approximation
Absorption and Loss spectroscopies: linear response quantities
Neutral excitations within MBPT: Bethe-Salpeter Equation
Neutral excitations within DFT: Time-Dependent DensityFunctional Theory
Intro MBPT & GW Linear BSE TDDFT
Absorption
Beer Law
I (x) = I0e−αx
α⇐⇒ ε
Intro MBPT & GW Linear BSE TDDFT
Absorption
Ellipsometry Experiments
ε = sin2Φ + sin2Φtan2Φ
(1− Er
Ei
1 + ErEi
)
Intro MBPT & GW Linear BSE TDDFT
Absorption
Creation of an electron-hole pair
i
unoccupied states
occupied states
j
Intro MBPT & GW Linear BSE TDDFT
Absorption
Lautenschlager et al., PRB 36, 4821 (1987)
Intro MBPT & GW Linear BSE TDDFT
Absorption
Izumi et al., Anal.Chem. 77, 6969 (2005)
Intro MBPT & GW Linear BSE TDDFT
Spectroscopy: Electron Scattering
Intro MBPT & GW Linear BSE TDDFT
Spectroscopy: Electron Scattering
Energy Loss Function
d2σ
dΩdE∝ Im
ε−1
Intro MBPT & GW Linear BSE TDDFT
Spectroscopy: Electron Scattering
Intro MBPT & GW Linear BSE TDDFT
Spectroscopy: Electron Scattering
Intro MBPT & GW Linear BSE TDDFT
Spectroscopy: Electron Scattering
Intro MBPT & GW Linear BSE TDDFT
Spectroscopy: Electron Scattering
Intro MBPT & GW Linear BSE TDDFT
Spectroscopy: Electron Scattering
Intro MBPT & GW Linear BSE TDDFT
Spectroscopy: Electron Scattering
Intro MBPT & GW Linear BSE TDDFT
Spectroscopy: X-ray Scattering
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
System submitted to an external perturbation
Vtot = ε−1Vext
Vtot = Vext + Vind
E = ε−1D
Dielectric function ε
Abs
EELS
εX-ray
R index
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
System submitted to an external perturbation
Vtot = ε−1Vext
Vtot = Vext + Vind
E = ε−1D
Dielectric function ε
Abs
EELS
εX-ray
R index
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
System submitted to an external perturbation
Vtot = ε−1Vext
Vtot = Vext + Vind
E = ε−1D
Dielectric function ε
Abs
EELS
εX-ray
R index
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
System submitted to an external perturbation
Vtot = ε−1Vext
Vtot = Vext + Vind
E = ε−1D
Dielectric function ε
Abs
EELS
εX-ray
R index
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
System submitted to an external perturbation
Vtot = ε−1Vext
Vtot = Vext + Vind
E = ε−1D
Dielectric function ε
Abs
EELS
εX-ray
R index
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
System submitted to an external perturbation
Vtot = ε−1Vext
Vtot = Vext + Vind
E = ε−1D
Dielectric function ε
Abs
EELS
εX-ray
R index
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
System submitted to an external perturbation
Vtot = ε−1Vext
Vtot = Vext + Vind
E = ε−1D
Dielectric function ε
Abs
EELS
εX-ray
R index
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
System submitted to an external perturbation
Vtot = ε−1Vext
Vtot = Vext + Vind
E = ε−1D
Dielectric function ε
Abs
EELS
εX-ray
R index
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Definition of polarizability
not polarizable ⇒ Vtot = Vext ⇒ ε−1 = 1polarizable ⇒ Vtot 6= Vext ⇒ ε−1 6= 1
ε−1 = 1 + vχ
χ is the polarizability of the system
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Definition of polarizability
not polarizable ⇒ Vtot = Vext ⇒ ε−1 = 1polarizable ⇒ Vtot 6= Vext ⇒ ε−1 6= 1
ε−1 = 1 + vχ
χ is the polarizability of the system
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Definition of polarizability
not polarizable ⇒ Vtot = Vext ⇒ ε−1 = 1polarizable ⇒ Vtot 6= Vext ⇒ ε−1 6= 1
ε−1 = 1 + vχ
χ is the polarizability of the system
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Definition of polarizability
not polarizable ⇒ Vtot = Vext ⇒ ε−1 = 1polarizable ⇒ Vtot 6= Vext ⇒ ε−1 6= 1
ε−1 = 1 + vχ
χ is the polarizability of the system
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability
interacting system δn = χδVext
non-interacting system δnn−i = χ0δVtot
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability
interacting system δn = χδVext
non-interacting system δnn−i = χ0δVtot
Single-particle polarizability
χ0 =∑ij
φi (r)φ∗j (r)φ∗i (r′)φj(r
′)
ω − (εi − εj)
hartree, hartree-fock, dft, etc.
G.D. Mahan Many Particle Physics (Plenum, New York, 1990)
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability
interacting system δn = χδVext
non-interacting system δnn−i = χ0δVtot
χ0 =∑ij
φi (r)φ∗j (r)φ∗i (r′)φj(r
′)
ω − (εi − εj)
i
unoccupied states
occupied states
j
Intro MBPT & GW Linear BSE TDDFT
Spectra within DFT
Loss spectrum of Graphite
Intro MBPT & GW Linear BSE TDDFT
Spectra within DFT
Intro MBPT & GW Linear BSE TDDFT
Spectra within DFT
11 12 13 14 15 16ω(eV)
5
10
15
Im
εM
exp
DFT level
Solid Argon Absorption spectrum
Intro MBPT & GW Linear BSE TDDFT
Outline
Introduction and reminder of ground-state results
Photoemission via MBPT: GW approximation
Absorption and Loss spectroscopies: linear response quantities
Neutral excitations within MBPT: Bethe-Salpeter Equation
Neutral excitations within DFT: Time-Dependent DensityFunctional Theory
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability
interacting system δn = χδVext
non-interacting system δnn−i = χ0δVtot
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability
interacting system δn = χδVext
non-interacting system δnn−i = χ0δVtot
Single-particle polarizability
χ0 =∑ij
φi (r)φ∗j (r)φ∗i (r′)φj(r
′)
ω − (εi − εj)
hartree, hartree-fock, dft, etc.
G.D. Mahan Many Particle Physics (Plenum, New York, 1990)
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability
interacting system δn = χδVext
non-interacting system δnn−i = χ0δVtot
χ0 =∑ij
φi (r)φ∗j (r)φ∗i (r′)φj(r
′)
ω − (εi − εj)
i
unoccupied states
occupied states
j
Intro MBPT & GW Linear BSE TDDFT
Spectra within DFT
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
How to go beyond χ0 ?
Intro MBPT & GW Linear BSE TDDFT
Hedin’s pentagon
Intro MBPT & GW Linear BSE TDDFT
Hedin’s pentagon
Intro MBPT & GW Linear BSE TDDFT
Hedin’s pentagon
Intro MBPT & GW Linear BSE TDDFT
Hedin’s pentagon
Intro MBPT & GW Linear BSE TDDFT
Spectra in MBPT
Spectra in IP picture
IP-RPA
Abs = Im χ0
i
unoccupied states
occupied states
j
Intro MBPT & GW Linear BSE TDDFT
Spectra in MBPT
Spectra in GW approximation
GW-RPA
Abs = Im χ0GW
χ0GW = P = −iGG
ioccupied states
j
unoccupied (GW corrected) states
Intro MBPT & GW Linear BSE TDDFT
Spectra in MBPT
Spectra in GW-RPA
χ0 =∑ij
φi(r)φ∗j (r)φ∗i (r′)φj(r′)
ω − (εi − εj)
⇓
χ0GW =
∑ij
φi(r)φ∗j (r)φ∗i (r′)φj(r′)
ω −[
(εi + ∆GW
i )−(εj + ∆GW
j
)]
Intro MBPT & GW Linear BSE TDDFT
Spectra in MBPT
Spectra in GW-RPA
χ0 =∑ij
φi(r)φ∗j (r)φ∗i (r′)φj(r′)
ω − (εi − εj)
⇓
χ0GW =
∑ij
φi(r)φ∗j (r)φ∗i (r′)φj(r′)
ω −[
(εi + ∆GW
i )−(εj + ∆GW
j
)]
Intro MBPT & GW Linear BSE TDDFT
Spectra in MBPT
Spectra in GW-RPA
Intro MBPT & GW Linear BSE TDDFT
Spectra in MBPT
GG Polarizability
P(1, 2) = −i G (1, 2)G (2, 1+)
Intro MBPT & GW Linear BSE TDDFT
Spectra in MBPT
GGΓ Polarizability
P(1, 2) = −i∫
d(34)G (1, 3)G (4, 1+)Γ(3, 4, 2)
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter Equation
Γ(1, 2, 3) = δ(1, 2)δ(1, 3)+
+
∫d(4567)
δΣ(1, 2)
δG (4, 5)G (4, 6)G (7, 5)Γ(6, 7, 3)
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter Equation
Towards the Bethe-Salpeter Equation
From electron and hole propagation .....
P0(1234) = G (13)G (42) ...
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter Equation
Towards the Bethe-Salpeter Equation
From electron and hole propagation to the electron-holeinteraction
P(1234) = P0(1234) + P0(1256)
[v +
δΣ(56)
δG (78)
]P(7834)
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter Equation
P(1234)=P0(1234)+P0(1256)
[v(57)δ(56)δ(78)+
δΣ(56)
δG (78)
]P(7834)
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter Equation
P = P0 + P0
[v +
δΣ
δG
]P
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter Equation
P = GG + GG
[v − δ [GW ]
δG
]P
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter Equation
P = GG + GG [v −W ]P
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter EquationBethe-Salpeter Equation
P = P0 + P0 [v −W ]P
Intrinsic 4-point equation
Correct!It describes the (coupled) progation
of two particles, the electron andthe hole !
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter EquationBethe-Salpeter Equation
P(1234) = P0(1234)+
+ P0(1256) [v(57)δ(56)δ(78)−W (56)δ(57)δ(68)]P(7834)
Intrinsic 4-point equation
Correct!It describes the (coupled) progation
of two particles, the electron andthe hole !
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter EquationBethe-Salpeter Equation
P(1234) = P0(1234)+
+ P0(1256) [v(57)δ(56)δ(78)−W (56)δ(57)δ(68)]P(7834)
Intrinsic 4-point equation
Correct!It describes the (coupled) progation
of two particles, the electron andthe hole !
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter EquationBethe-Salpeter Equation
P(1234) = P0(1234)+
+ P0(1256) [v(57)δ(56)δ(78)−W (56)δ(57)δ(68)]P(7834)
Intrinsic 4-point equation
Correct!It describes the (coupled) progation
of two particles, the electron andthe hole !
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter equation results: Semiconductors
Albrecht et al., PRL 80, 4510 (1998)
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter equation results: Insulators
Sottile et al., PRB 76, 161103 (2007).
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter equation results: Molecule (Na4)
Onida et al., PRL 75, 818 (1995)
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter equation results: Silicon Nanowires
Bruno et al., PRL 98, 036807 (2007)
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter equation results: Hexagonal Ice
Hahn et al., PRL 94, 37404 (2005)
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter equation results: EELS of Silicon
Olevano and Reining, PRL 86, 5962 (2001)
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter equation results: Surface
Rohlfing et al., PRL 85, 005440 (2000)
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter equation results: Surface
Rohlfing et al., PRL 85, 005440 (2000)
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter equation: State-of-the-art
• DFT - ground state
• GW - quasiparticle energies
• BSE - optical and dielectric properties
√several spectroscopies
√variety of systems
× Cumbersome Calculations
Intro MBPT & GW Linear BSE TDDFT
Bethe-Salpeter equation: State-of-the-art
• DFT - ground state
• GW - quasiparticle energies
• BSE - optical and dielectric properties
√several spectroscopies
√variety of systems
× Cumbersome Calculations
Intro MBPT & GW Linear BSE TDDFT
References and Literature
GW and BSE
• Hedin, Lundqvist, Solid State Physics 23, 1 (1969)
• Onida, Reining, Rubio, RMP 74, 601 (2002)
• Strinati, Riv Nuovo Cimento 11, 1 (1988)
TDDFT
• Runge, Gross, Kohn, PRL 52, 997 (1984), PRL 55, 2850(1985)
• Marques et al eds, Time Dependent Density FunctionalTheory, Springer (2006).
• Botti et al, Rep. Prog. Phys. 70, 357 (2007)
Matteo Gatti, PhD Thesis, http://etsf.polytechnique.fr/sites/default/files/users/matteo/matteo_thesis.pdf
Intro MBPT & GW Linear BSE TDDFT
Outline
Introduction and reminder of ground-state results
Photoemission via MBPT: GW approximation
Absorption and Loss spectroscopies: linear response quantities
Neutral excitations within MBPT: Bethe-Salpeter Equation
Neutral excitations within DFT: Time-Dependent DensityFunctional Theory
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability
interacting system δn = χδVext
non-interacting system δnn−i = χ0δVtot
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability
interacting system δn = χδVext
non-interacting system δnn−i = χ0δVtot
Single-particle polarizability
χ0 =∑ij
φi (r)φ∗j (r)φ∗i (r′)φj(r
′)
ω − (εi − εj)
hartree, hartree-fock, dft, etc.
G.D. Mahan Many Particle Physics (Plenum, New York, 1990)
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability
interacting system δn = χδVext
non-interacting system δnn−i = χ0δVtot
χ0 =∑ij
φi (r)φ∗j (r)φ∗i (r′)φj(r
′)
ω − (εi − εj)
i
unoccupied states
occupied states
j
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability
interacting system δn = χδVext
non-interacting system δnn−i = χ0δVtot
m
Density Functional Formalism
δn = δnn−i
δVtot = δVext + δVH + δVxc
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability
χδVext = χ0 (δVext + δVH + δVxc)
χ = χ0
(1 +
δVH
δVext+δVxc
δVext
)δVH
δVext=δVH
δn
δn
δVext= vχ
δVxc
δVext=δVxc
δn
δn
δVext= fxcχ
with fxc = exchange-correlation kernel
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability
χδVext = χ0 (δVext + δVH + δVxc)
χ = χ0
(1 +
δVH
δVext+δVxc
δVext
)δVH
δVext=δVH
δn
δn
δVext= vχ
δVxc
δVext=δVxc
δn
δn
δVext= fxcχ
with fxc = exchange-correlation kernel
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability
χδVext = χ0 (δVext + δVH + δVxc)
χ = χ0
(1 +
δVH
δVext+δVxc
δVext
)δVH
δVext=δVH
δn
δn
δVext= vχ
δVxc
δVext=δVxc
δn
δn
δVext= fxcχ
χ = χ0 + χ0 (v + fxc)χwith fxc = exchange-correlation kernel
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability
χδVext = χ0 (δVext + δVH + δVxc)
χ = χ0
(1 +
δVH
δVext+δVxc
δVext
)δVH
δVext=δVH
δn
δn
δVext= vχ
δVxc
δVext=δVxc
δn
δn
δVext= fxcχ
χ =[1− χ0 (v + fxc)
]−1χ0
with fxc = exchange-correlation kernel
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability
χδVext = χ0 (δVext + δVH + δVxc)
χ = χ0
(1 +
δVH
δVext+δVxc
δVext
)δVH
δVext=δVH
δn
δn
δVext= vχ
δVxc
δVext=δVxc
δn
δn
δVext= fxcχ
χ =[1− χ0 (v + fxc)
]−1χ0
with fxc = exchange-correlation kernel
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability χ in TDDFT
1. DFT ground-state calc. → φi , εi [Vxc ]
2. φi , εi → χ0 =∑
ij
φi (r)φ∗j (r)φ∗i (r′)φj (r
′)
ω−(εi−εj )
3.δVH
δn= v
δVxc
δn= fxc
variation of the potentials
4. χ = χ0 + χ0 (v + fxc)χ
A comment
• fxc =
δVxc
δn“any” other function
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability χ in TDDFT
1. DFT ground-state calc. → φi , εi [Vxc ]
2. φi , εi → χ0 =∑
ij
φi (r)φ∗j (r)φ∗i (r′)φj (r
′)
ω−(εi−εj )
3.δVH
δn= v
δVxc
δn= fxc
variation of the potentials
4. χ = χ0 + χ0 (v + fxc)χ
A comment
• fxc =
δVxc
δn“any” other function
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability χ in TDDFT
1. DFT ground-state calc. → φi , εi [Vxc ]
2. φi , εi → χ0 =∑
ij
φi (r)φ∗j (r)φ∗i (r′)φj (r
′)
ω−(εi−εj )
3.δVH
δn= v
δVxc
δn= fxc
variation of the potentials
4. χ = χ0 + χ0 (v + fxc)χ
A comment
• fxc =
δVxc
δn“any” other function
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability χ in TDDFT
1. DFT ground-state calc. → φi , εi [Vxc ]
2. φi , εi → χ0 =∑
ij
φi (r)φ∗j (r)φ∗i (r′)φj (r
′)
ω−(εi−εj )
3.δVH
δn= v
δVxc
δn= fxc
variation of the potentials
4. χ = χ0 + χ0 (v + fxc)χ
A comment
• fxc =
δVxc
δn“any” other function
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability χ in TDDFT
1. DFT ground-state calc. → φi , εi [Vxc ]
2. φi , εi → χ0 =∑
ij
φi (r)φ∗j (r)φ∗i (r′)φj (r
′)
ω−(εi−εj )
3.δVH
δn= v
δVxc
δn= fxc
variation of the potentials
4. χ = χ0 + χ0 (v + fxc)χ
A comment
• fxc =
δVxc
δn“any” other function
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Polarizability χ in TDDFT
1. DFT ground-state calc. → φi , εi [Vxc ]
2. φi , εi → χ0 =∑
ij
φi (r)φ∗j (r)φ∗i (r′)φj (r
′)
ω−(εi−εj )
3.δVH
δn= v
δVxc
δn= fxc
variation of the potentials
4. χ = χ0 + χ0 (v + fxc)χ
A comment
• fxc =
δVxc
δn“any” other function
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Most important approximation for fxc
fxc = 0 RPA
f ALDAxc (r, r′) = δVxc (r)
δn(r′) δ(r − r′) ALDA
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Role of v
χ(r, r′, ω) = χ0(r, r′, ω) + χ0(r, r′′, ω)v(r′′, r′′′)χ(r′′′, r′, ω)
⟨χ(r, r′, ω)
⟩⇒⟨χ0(r, r′, ω)
⟩spectrum
(χ(|r − r′|, ω)
)spectrum
(χ(r, r′, ω)
)v contains all the information of the crystal localfield effects (local dishomogeneity of the system)
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Role of v
χ(r, r′, ω) = χ0(r, r′, ω)
⟨χ(r, r′, ω)
⟩⇒⟨χ0(r, r′, ω)
⟩spectrum
(χ(|r − r′|, ω)
)spectrum
(χ(r, r′, ω)
)v contains all the information of the crystal localfield effects (local dishomogeneity of the system)
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Role of v
χ(r, r′, ω) = χ0(r, r′, ω)
⟨χ(r, r′, ω)
⟩⇒⟨χ0(r, r′, ω)
⟩spectrum
(χ(|r − r′|, ω)
)spectrum
(χ(r, r′, ω)
)v contains all the information of the crystal localfield effects (local dishomogeneity of the system)
Intro MBPT & GW Linear BSE TDDFT
Linear Response Approach
Role of v
χ(r, r′, ω) = χ0(r, r′, ω) + χ0(r, r′′, ω)v(r′′, r′′′)χ(r′′′, r′, ω)
⟨χ(r, r′, ω)
⟩⇒⟨χ0(r, r′, ω)
⟩spectrum
(χ(|r − r′|, ω)
)spectrum
(χ(r, r′, ω)
)v contains all the information of the crystal localfield effects (local dishomogeneity of the system)
Intro MBPT & GW Linear BSE TDDFT
Spectra within DFT and TDDFT-RPA
Loss spectrum of Graphite
A.Marinopoulos et al. Phys.Rev.Lett 89, 76402 (2002)
Intro MBPT & GW Linear BSE TDDFT
ALDA: Achievements and Shortcomings
Inelastic X-ray scattering of Silicon
H-C.Weissker et al., Physical Review Letters 97, 237602 (2006)
Intro MBPT & GW Linear BSE TDDFT
ALDA: Achievements and Shortcomings
Photo-absorption cross section of Benzene
K.Yabana and G.F.Bertsch Int.J.Mod.Phys.75, 55 (1999)
Intro MBPT & GW Linear BSE TDDFT
ALDA: Achievements and Shortcomings
Absorption Spectrum of Silicon
Intro MBPT & GW Linear BSE TDDFT
ALDA: Achievements and Shortcomings
Absorption Spectrum of Argon
Intro MBPT & GW Linear BSE TDDFT
ALDA: Achievements and Shortcomings
Good results
• Photo-absorption ofsmall molecules
• ELS of solids
Bad results
• Absorption of solids
Why?
f ALDAxc is short-range
fxc(q→ 0) ∼ 1
q2
Intro MBPT & GW Linear BSE TDDFT
ALDA: Achievements and Shortcomings
Good results
• Photo-absorption ofsmall molecules
• ELS of solids
Bad results
• Absorption of solids
Why?
f ALDAxc is short-range
fxc(q→ 0) ∼ 1
q2
Intro MBPT & GW Linear BSE TDDFT
ALDA: Achievements and Shortcomings
Good results
• Photo-absorption ofsmall molecules
• ELS of solids
Bad results
• Absorption of solids
Why?
f ALDAxc is short-range
fxc(q→ 0) ∼ 1
q2
Intro MBPT & GW Linear BSE TDDFT
ALDA: Achievements and Shortcomings
Absorption of Silicon fxc = αq2
L.Reining et al. Phys.Rev.Lett. 88, 66404 (2002)