The Bethe-Salpeter Equation

Transcription

An Introduction
Francesco Sottile
ETSF and Ecole Polytechnique, Palaiseau - France
Lyon, 14 December 2007
Francesco Sottile
Generalities of Linear Response Approach
The Bethe-Salpeter equation
Results
The EXC Code
Outline
1
2
Polarizability in MBPT
Definition of BSE
BSE in practice
3
Results
4
The EXC Code
Francesco Sottile
Results
The EXC Code
Outline
1
2
Definition of BSE
BSE in practice
3
Results
4
The EXC Code
Francesco Sottile
Results
The EXC Code
Linear Response Approach
System subject to an external perturbation
Francesco Sottile
Results
The EXC Code
Vtot = ε−1 Vext
Vtot = Vext + Vind
Francesco Sottile
Results
The EXC Code
Vtot = ε−1 Vext
Vtot = Vext + Vind
E = ε−1 D
Francesco Sottile
Results
Vtot = ε−1 Vext
Vtot = Vext + Vind
E = ε−1 D
Dielectric function ε
ε
The EXC Code
Results
Vtot = ε−1 Vext
Vtot = Vext + Vind
E = ε−1 D
ε
Abs
The EXC Code
Results
Vtot = ε−1 Vext
Vtot = Vext + Vind
E = ε−1 D
EELS
ε
Abs
The EXC Code
Results
Vtot = ε−1 Vext
Vtot = Vext + Vind
E = ε−1 D
EELS
ε
Abs
X-ray
The EXC Code
Results
The EXC Code
Vtot = ε−1 Vext
Vtot = Vext + Vind
E = ε−1 D
EELS
R index
ε
Abs
X-ray
Francesco Sottile
Results
The EXC Code
Definition of polarizability
ε−1 = 1 + v χ
χ is the polarizability of the system
Francesco Sottile
Results
The EXC Code
Polarizability
interacting system δn = χδVext
non-interacting system δnn−i = χ0 δVtot
Francesco Sottile
Results
The EXC Code
Polarizability
Single-particle polarizability
χ0 =
X φi (r)φ∗j (r)φ∗i (r0 )φj (r0 )
ij
ω − (i − j ) + iη
hartree, hartree-fock, dft, etc.
G.D. Mahan Many Particle Physics (Plenum, New York, 1990)
Francesco Sottile
Results
The EXC Code
Polarizability
j
unoccupied states
0
χ =
ij
ω − (i − j ) + iη
occupied states
i
Francesco Sottile
Results
The EXC Code
First approximation: IP-RPA
0
χ =
ij
ω − (i − j ) + iη
0 X
Abs = Im χ =
|hj|D|ii|2 δ(ω − (j − i ))
ij
Francesco Sottile
Results
The EXC Code
0
χ =
ij
ω − (i − j ) + iη
0 X
Abs = Im χ =
|hj|D|ii|2 δ(ω − (j − i ))
ij
Francesco Sottile
Results
The EXC Code
Francesco Sottile
Results
The EXC Code
How to go beyond χ0 ?
Francesco Sottile
Results
The EXC Code
Outline
1
2
Definition of BSE
BSE in practice
3
Results
4
The EXC Code
Francesco Sottile
Results
The EXC Code
Outline
1
2
Definition of BSE
BSE in practice
3
Results
4
The EXC Code
Francesco Sottile
Results
The EXC Code
Many Body Perturbation Theory
Hedin’s equations
Z
d(34)G (1, 3)Γ̃(3, 2, 4)W (4, 1+ )
Z
0
G (1, 2) = G (1, 2) + d(34)G 0 (1, 3)Σ(3, 4)G (4, 2)
Σ(1, 2) = i
Z
δΣ(1, 2)
Γ̃(1, 2, 3) = δ(1, 2)δ(1, 3)+ d(4567)
G (4, 6)G (7, 5)Γ̃(6, 7, 3)
δG (4, 5)
Z
P̃(1, 2) = −i d(34)G (1, 3)G (4, 1+ )Γ̃(3, 4, 2)
Z
W (1, 2) = v (1, 2) + d(34)v (1, 3)P̃(3, 4)W (4, 2)
Francesco Sottile
Results
The EXC Code
Hedin’s pentagon
Σ=
~
Γ
GW
Σ
G=
G
0
+G
0
ΣG
W
+(δΣ
~
~
P
Γ=1
~
+ vPW
/δG)
G
W=v
GΓ~
G
~
~
P = GGΓ
~
Γ
Francesco Sottile
Results
The EXC Code
Polarizability P̃ is irreducible
P̃ =
δn
δVtot
Γ̃ =
;
δG −1
δΣ
=1+
δVtot
δVtot
Irreducible P̃ and Reducible χ
δn
δn
P̃ =
;
χ=
δVtot
δVext
χ = P̃ + P̃v χ
Vtot = Vext + VH
Different quantities
P̃, Γ̃, G
χ0 , χ
=
=
time-ordered
retarded
Francesco Sottile
Results
The EXC Code
P̃ =
δn
δVtot
Γ̃ =
;
δG −1
δΣ
=1+
δVtot
δVtot
δn
δn
P̃ =
;
χ=
δVtot
δVext
χ = P̃ + P̃v χ
Vtot = Vext + VH
P̃, Γ̃, G
χ0 , χ
=
=
time-ordered
retarded
Francesco Sottile
Results
The EXC Code
P̃ =
δn
δVtot
Γ̃ =
;
δG −1
δΣ
=1+
δVtot
δVtot
δn
δn
P̃ =
;
χ=
δVtot
δVext
χ = P̃ + P̃v χ
Vtot = Vext + VH
P̃, Γ̃, G
χ0 , χ
=
=
time-ordered
retarded
Francesco Sottile
Results
The EXC Code
P̃ =
δn
δVtot
Γ̃ =
;
δG −1
δΣ
=1+
δVtot
δVtot
δn
δn
P̃ =
;
χ=
δVtot
δVext
χ = P̃ + P̃v χ
Vtot = Vext + VH
P̃, Γ̃, G
χ0 , χ
=
=
time-ordered
retarded
Francesco Sottile
Results
The EXC Code
Spectra in MBPT
Spectra in IP picture
j
unoccupied states
IP-RPA
Abs = Im χ0
occupied states
i
Francesco Sottile
Results
The EXC Code
Spectra in MBPT
Spectra in GW approximation
j
unoccupied (GW corrected) states
GW-RPA
Abs = Im χ0GW
χ0GW = P = −iGG
occupied states
i
Francesco Sottile
Results
The EXC Code
Spectra in MBPT
GW pentagon
~
Γ
Σ
W
=G
Σ
G=
G
0
+G
0
ΣG
G
W=v
G
+(δΣ
/δG)
~ =G
P
~
~
P = GGΓ
~
Γ
~
~
P
Γ=1
~
+ vPW
GGΓ~
W
Francesco Sottile
Results
The EXC Code
Spectra in MBPT
Spectra in GW-RPA
0
χ =
ij
ω − (i − j )
⇓
0
χGW =
X
ij
φi (r)φ∗j (r)φ∗i (r0 )φj (r0 )
h
i
GW
GW
ω − (i + ∆i ) − j + ∆j
Francesco Sottile
Results
The EXC Code
Spectra in MBPT
Spectra in GW-RPA
0
χ =
ij
ω − (i − j )
⇓
0
χGW =
X
ij
φi (r)φ∗j (r)φ∗i (r0 )φj (r0 )
h
i
GW
GW
ω − (i + ∆i ) − j + ∆j
Francesco Sottile
Results
The EXC Code
Spectra in MBPT
Spectra in GW-RPA
Francesco Sottile
Results
The EXC Code
Spectra in MBPT
GG Polarizability
P̃(1, 2) = −i
G (1, 2)G (2, 1+ )
Francesco Sottile
Results
The EXC Code
Spectra in MBPT
GGΓ Polarizability
Z
P̃(1, 2) = −i
d(34)G (1, 3)G (4, 1+ )Γ̃(3, 4, 2)
Francesco Sottile
Results
The EXC Code
Outline
1
2
Definition of BSE
BSE in practice
3
Results
4
The EXC Code
Francesco Sottile
Results
The EXC Code
Bethe-Salpeter Equation
Γ̃(1, 2, 3) = δ(1, 2)δ(1, 3)+
Z
δΣ(1, 2)
+ d(4567)
G (4, 6)G (7, 5)Γ̃(6, 7, 3)
δG (4, 5)
Francesco Sottile
Results
The EXC Code
Towards the Bethe-Salpeter
Γ̃ = 1 +
δΣ
GG Γ̃
δG
GG Γ̃ = GG + GG
L̃ = L0 + L0
δΣ
GG Γ̃
δG
δΣ
L̃
δG
L̃(1234) = L0 (1234) + L0 (1256)
δΣ(56)
L̃(7834)
δG (78)
Francesco Sottile
Results
The EXC Code
Γ̃ = 1 +
δΣ
GG Γ̃
δG
GG Γ̃ = GG + GG
L̃ = L0 + L0
δΣ
GG Γ̃
δG
δΣ
L̃
δG
L̃(1234) = L0 (1234) + L0 (1256)
δΣ(56)
L̃(7834)
δG (78)
Francesco Sottile
Results
The EXC Code
Γ̃ = 1 +
δΣ
GG Γ̃
δG
GG Γ̃ = GG + GG
L̃ = L0 + L0
δΣ
GG Γ̃
δG
δΣ
L̃
δG
L̃(1234) = L0 (1234) + L0 (1256)
δΣ(56)
L̃(7834)
δG (78)
Francesco Sottile
Results
The EXC Code
Γ̃ = 1 +
δΣ
GG Γ̃
δG
GG Γ̃ = GG + GG
L̃ = L0 + L0
δΣ
GG Γ̃
δG
δΣ
L̃
δG
L̃(1234) = L0 (1234) + L0 (1256)
δΣ(56)
L̃(7834)
δG (78)
Francesco Sottile
Results
The EXC Code
Towards the Bethe-Salpeter Equation
From electron and hole propagation ...
L0 (1234) = G (13)G (42)
..
...
Francesco Sottile
Results
The EXC Code
Towards the Bethe-Salpeter Equation
From electron and hole propagation to the electron-hole interaction
L̃(1234) = L0 (1234) + L0 (1256)
δΣ(56)
L̃(7834)
δG (78)
Francesco Sottile
Results
The EXC Code
Irreducible form of the Bethe-Salpeter equation
L̃(1234) = L0 (1234) + L0 (1256)
δΣ(56)
L̃(7834)
δG (78)
Reducible quantity
L = L̃ + L̃vL
Francesco Sottile
Results
The EXC Code
δΣ(56)
L(1234) = L (1234) + L (1256) v (57)δ(56)δ(78)+
L(7834)
δG (78)
0
0
Francesco Sottile
Results
The EXC Code
Comparison with Linear Response quantities
χ(12) =
δn(1)
δVext (2)
Francesco Sottile
Results
The EXC Code
χ(12) =
δn(1)
δVext (2)
L(1234) =
Francesco Sottile
Results
The EXC Code
χ(12) =
L(1234) =
δn(1)
δVext (2)
δG (12)
δ Ṽext (34)
Francesco Sottile
Results
The EXC Code
χ(12) =
L(1133) =
δn(1)
δVext (2)
δG (11)
δ Ṽext (33)
Francesco Sottile
Results
The EXC Code
χ(12) =
δn(1)
δVext (2)
L(1133) =
δn(1)
δVext (3)
Francesco Sottile
Results
The EXC Code
We have the (4-point)
Bethe-Salpeter equation.
And now ?
Francesco Sottile
Results
The EXC Code
First point: Choosing Σ
δΣ(56)
L(1234) = L (1234) + L (1256) v (57)δ(56)δ(78)+
L(7834)
δG (78)
0
0
BSE
Francesco Sottile
Results
The EXC Code
δΣ(56)
L(1234) = L (1234) + L (1256) v (57)δ(56)δ(78)+
L(7834)
δG (78)
0
0
Coulomb term
Σx (1, 2) = iG (12)v (21)
⇒ Time-Dependent Hartree-Fock
BSE
Francesco Sottile
Results
The EXC Code
δΣ(56)
L(1234) = L (1234) + L (1256) v (57)δ(56)δ(78)+
L(7834)
δG (78)
0
0
Screened Coulomb term
ΣGW (1, 2) = iG (12)W (21)
⇒ Standard Bethe-Salpeter equation
(Time-Dependent Screened Hartree-Fock)
BSE
Francesco Sottile
Results
The EXC Code
Choice of Σ = GW
Everything should be coherently chosen
⇒
⇒
ground state calculation −→ φi , i
⇒
⇒
Ei = i + ∆GW
i
0
GKS
;
0
ε−1
RPA (r, r , ω) ;
;
W (r, r0 , ω) = ε−1
RPA v
ψi ' φ i
G GW (r, r0 , ω)
Francesco Sottile
Results
The EXC Code
Choice of Σ = GW
⇒
⇒
⇒
⇒
Ei = i + ∆GW
i
0
GKS
;
0
ε−1
RPA (r, r , ω) ;
;
W (r, r0 , ω) = ε−1
RPA v
ψi ' φ i
G GW (r, r0 , ω)
Francesco Sottile
Results
The EXC Code
Choice of Σ = GW
⇒
⇒
⇒
⇒
Ei = i + ∆GW
i
0
GKS
;
0
ε−1
RPA (r, r , ω) ;
;
W (r, r0 , ω) = ε−1
RPA v
ψi ' φ i
G GW (r, r0 , ω)
Francesco Sottile
Results
The EXC Code
Choice of Σ = GW
⇒
⇒
⇒
Σ(r, r0 , ω) =
⇒
⇒
Ei = i + ∆GW
i
0
GKS
;
0
ε−1
RPA (r, r , ω) ;
R
W (r, r0 , ω) = ε−1
RPA v
0
dω 0 GKS
(r, r0 , ω 0 )W (r, r0 , ω + ω 0 )
;
ψi ' φ i
G GW (r, r0 , ω)
Francesco Sottile
Results
The EXC Code
Choice of Σ = GW
⇒
⇒
⇒
Σ(r, r0 , ω) =
⇒
⇒
Ei = i + ∆GW
i
0
GKS
;
0
ε−1
RPA (r, r , ω) ;
R
W (r, r0 , ω) = ε−1
RPA v
0
dω 0 GKS
(r, r0 , ω 0 )W (r, r0 , ω + ω 0 )
;
ψi ' φ i
G GW (r, r0 , ω)
Francesco Sottile
Results
The EXC Code
Choice of Σ = GW
⇒
⇒
⇒
Σ(r, r0 , ω) =
⇒
⇒
Ei = i + ∆GW
i
0
GKS
;
0
ε−1
RPA (r, r , ω) ;
R
W (r, r0 , ω) = ε−1
RPA v
0
dω 0 GKS
(r, r0 , ω 0 )W (r, r0 , ω + ω 0 )
;
ψi ' φ i
G GW (r, r0 , ω)
Francesco Sottile
Results
The EXC Code
Choice of Σ = GW
⇒
⇒
⇒
Σ(r, r0 , ω) =
⇒
⇒
Ei = i + ∆GW
i
0
GKS
;
0
ε−1
RPA (r, r , ω) ;
R
W (r, r0 , ω) = ε−1
RPA v
0
dω 0 GKS
(r, r0 , ω 0 )W (r, r0 , ω + ω 0 )
;
ψi ' φ i
G GW (r, r0 , ω)
Francesco Sottile
Results
The EXC Code
Choice of Σ = GW
⇒
⇒
⇒
Σ(r, r0 , ω) =
⇒
⇒
Ei = i + ∆GW
i
0
GKS
;
0
ε−1
RPA (r, r , ω) ;
R
W (r, r0 , ω) = ε−1
RPA v
dω 0 G GW (r, r0 , ω 0 )W (r, r0 , ω + ω 0 )
;
ψi ' φ i
G GW (r, r0 , ω)
Francesco Sottile
Results
The EXC Code
δΣ
L = L0 + L0 v +
L
δG
δW
⇒ Approx.
=0
δG
Francesco Sottile
Results
The EXC Code
δ [GW ]
L = GG + GG v −
L
δG
δW
⇒ Approx.
=0
δG
Francesco Sottile
Results
The EXC Code
L = GG + GG [v − W ] L
⇒ Approx.
δW
=0
δG
Francesco Sottile
Results
The EXC Code
L = L0 + L0 [v − W ] L
Francesco Sottile
Results
The EXC Code
L(1234) = L0 (1234)+
+ L0 (1256) [v (57)δ(56)δ(78) − W (56)δ(57)δ(68)] L(7834)
Francesco Sottile
Results
The EXC Code
L(1234) = L0 (1234)+
+ L0 (1256) [v (57)δ(56)δ(78) − W (56)δ(57)δ(68)] L(7834)
Intrinsic 4-point equation
Correct!
It describes the (coupled) progation of
two particles, the electron and the hole !
Francesco Sottile
Results
The EXC Code
L(1234) = L0 (1234)+
+ L0 (1256) [v (57)δ(56)δ(78) − W (56)δ(57)δ(68)] L(7834)
Exercise
Show 4-point
that, if equation
W = 0, the equation
Intrinsic
for L(1133) is a two-point
equation!
Correct!
It describes the (coupled) progation of
two particles, the electron and the hole !
Francesco Sottile
Results
The EXC Code
Bethe-Salpeter equation (4-points - space and time)
L(1234) = L0 (1234)+
+ L0 (1256) [v (57)δ(56)δ(78) − W (56)δ(57)δ(68)] L(7834)
W (12) = W (r1, r2)δ(t1t2)
L(1234) =⇒ L(r1, r2, r3, r4; t − t 0) =⇒ L(1234, ω)
Francesco Sottile
Results
The EXC Code
L(1234) = L0 (1234)+
+ L0 (1256) [v (57)δ(56)δ(78) − W (56)δ(57)δ(68)] L(7834)
W (12) = W (r1, r2)δ(t1t2)
L(1234) =⇒ L(r1, r2, r3, r4; t − t 0) =⇒ L(1234, ω)
Francesco Sottile
Results
The EXC Code
L(1234) = L0 (1234)+
+ L0 (1256) [v (57)δ(56)δ(78) − W (56)δ(57)δ(68)] L(7834)
W (12) = W (r1, r2)δ(t1t2)
L(1234) =⇒ L(r1, r2, r3, r4; t − t 0) =⇒ L(1234, ω)
Francesco Sottile
Results
The EXC Code
Macroscopic Quantity from the contracted L
1
L(1234, ω) =⇒ L(r, r, r0 , r0 , ω)
2
R
0
εM (ω) = 1 − limq→0 v (q) drdr0 e iq·(r−r ) L(r, r, r0 , r0 , ω)
εM (ω) = 1 − limq→0 v (q)L(G = 0, G’ = 0, ω)
Francesco Sottile
Results
The EXC Code
1
L(1234, ω) =⇒ L(r, r, r0 , r0 , ω)
2
R
0
εM (ω) = 1 − limq→0 v (q)L(G = 0, G’ = 0, ω)
Francesco Sottile
Results
The EXC Code
1
L(1234, ω) =⇒ L(r, r, r0 , r0 , ω)
2
R
0
εM (ω) = 1 − limq→0 v (q)L(G = 0, G’ = 0, ω)
Francesco Sottile
Results
The EXC Code
BSE (4 space points - 1 frequency)
L(1234, ω) = L0 (1234, ω)+
+ L0 (1256, ω) [v (57)δ(56)δ(78) − W (56)δ(57)δ(68)] L(7834, ω)
How to solve it ?
Really invert 4-point function for every frequency?
Francesco Sottile
Results
The EXC Code
BSE (4 space points - 1 frequency)
L(1234, ω) = L0 (1234, ω)+
+ L0 (1256, ω) [v (57)δ(56)δ(78) − W (56)δ(57)δ(68)] L(7834, ω)
How to solve it ?
Really invert 4-point function for every frequency?
Francesco Sottile
Results
The EXC Code
Outline
1
2
Definition of BSE
BSE in practice
3
Results
4
The EXC Code
Francesco Sottile
Results
The EXC Code
L(1234, ω) = L0 (1234, ω)+
+ L0 (1256, ω) [v (57)δ(56)δ(78) − W (56)δ(57)δ(68)] L(7834, ω)
Francesco Sottile
Results
The EXC Code
L(1234, ω) = L0 (1234, ω) + L0 (1256, ω)K (5678)L(7834, ω)
Results
The EXC Code
L(1234, ω) = L0 (1234, ω) + L0 (1256, ω)K (5678)L(7834, ω)
We work in transition space...
(n n )
L(1234, ω) ⇒ L(n31 n42 ) (ω) =
Z
= d(1234)L(1234, ω)φn1 (1)φ∗n2 (2)φn3 (3)φ∗n4 (4) = L The Bethe-Salpeter Equation
Francesco Sottile
Results
The EXC Code
L(1234, ω) = L0 (1234, ω) + L0 (1256, ω)K (5678)L(7834, ω)
(n n )
0 (n n )
0 (n n )
(n n ) (n n )
L(n31 n42 ) (ω) = L(n1 n32 )4 (ω) + L(n1 n52 )6 (ω)K(n57n68) L(n73 n84 ) (ω)
(n n )
L(1234, ω) ⇒ L(n31 n42 ) (ω) =
Z
= d(1234)L(1234, ω)φn1 (1)φ∗n2 (2)φn3 (3)φ∗n4 (4) = L The Bethe-Salpeter Equation
Francesco Sottile
Results
The EXC Code
L(1234, ω) = L0 (1234, ω) + L0 (1256, ω)K (5678)L(7834, ω)
(n n )
0 (n n )
0 (n n )
(n n ) (n n )
(n n )
L(1234, ω) ⇒ L(n31 n42 ) (ω) =
Z
= d(1234)L(1234, ω)φn1 (1)φ∗n2 (2)φn3 (3)φ∗n4 (4) = L Clever choice of the basis φn
Francesco Sottile
Results
The EXC Code
Bethe-Salpeter Equation in transition space
(n n )
0 (n n )
0 (n n )
(n n ) (n n )
... some “trivial” mathematical arzigogoli ...
h
i
(n n )
(n n ) −1
L(n31 n42 ) (ω) = (En2 − En1 − ω)δn1 n3 δn2 n4 + K(n13n24)
(n n )
L(n13 n24 ) (ω) = [(En2 − En1 − ω)δn1 n3 δn2 n4 + v − W ]−1
Francesco Sottile
Results
The EXC Code
(n n )
0 (n n )
0 (n n )
(n n ) (n n )
h
i
(n n )
(n n ) −1
(n n )
Francesco Sottile
Results
The EXC Code
(n n )
0 (n n )
0 (n n )
(n n ) (n n )
h
i
(n n )
(n n ) −1
(n n )
Francesco Sottile
Results
The EXC Code
(n n )
0 (n n )
0 (n n )
(n n ) (n n )
h
i
(n n )
(n n ) −1
(n n )
Francesco Sottile
Results
The EXC Code
The Excitonic Hamiltonian
(n n )
(n n )
L(n31 n42 ) (ω) =
1
H exc − ω
H exc = (En2 − En1 )δn1 n3 δn2 n4 + v − W Resonant vs Coupling
n1 , n2 , n3 , n4 = v , c (k)
H reso = (Ec − Ev )δvv 0 δcc 0 + v − W The Bethe-Salpeter Equation
Francesco Sottile
Results
The EXC Code
(n n )
(n n )
L(n31 n42 ) (ω) =
1
H exc − ω
n1 , n2 , n3 , n4 = v , c (k)
Francesco Sottile
Results
The EXC Code
(n n )
(n n )
L(n31 n42 ) (ω) =
1
H exc − ω
n1 , n2 , n3 , n4 = v , c (k)
Francesco Sottile
Results
The EXC Code
Francesco Sottile
Results
The EXC Code
Francesco Sottile
Results
The EXC Code
(n n )
L(n31 n42 ) (ω) =⇒ L(r, r0 , ω) =
X
(n n )
L(n31 n42 ) (ω)φn1 (r)φ∗n2 (r)φn3 (r0 )φ∗n4 (r0 )
(n1 n2 )
(n3 n4 )
X (n n ) Z
0 0
(n n )
L(n31 n42 ) (ω) =⇒ L(G, G’, ω) = L(n31 n42 ) (ω) φn1 (r)e i(q+G)r φ∗n2 (r)φn3 (r0 )e i(q+G )r φ∗n4 (r0 )
(n1 n2 )
(n3 n4 )
Francesco Sottile
Results
The EXC Code
(n n )
L(n31 n42 ) (ω) =⇒ L(r, r0 , ω) =
X
(n n )
(n1 n2 )
(n3 n4 )
X (n n ) Z
0 0
(n n )
(n1 n2 )
(n3 n4 )
Francesco Sottile
Results
The EXC Code
(n n )
L(n31 n42 ) (ω) =⇒ L(r, r0 , ω) =
X
(n n )
(n1 n2 )
(n3 n4 )
X (n n ) Z
0 0
(n n )
(n1 n2 )
(n3 n4 )
Francesco Sottile
Results
The EXC Code
(n n )
(n n )
L(n31 n42 ) (ω) = [H exc − ω]−1
H exc = [(En2 − En1 )δn1 n3 δn2 n4 + v − W ]
Diagonalization
Iterative inversion
Francesco Sottile
Results
The EXC Code
(n n )
(n n )
L(n31 n42 ) (ω) = [H exc − ω]−1
Diagonalization
Iterative inversion
Francesco Sottile
Results
The EXC Code
(n n )
(n n )
L(n31 n42 ) (ω) = [H exc − ω]−1
Diagonalization
Iterative inversion
Francesco Sottile
Results
The EXC Code
Diagonalization case (only resonant approx)
0 0
Lvvcc = [(Ec − Ev ) δvv 0 δcc 0 − ω+ v − W ]−1
X |Aλ >< Aλ |
1
=
H − ωI
Eλ − ω
λ
Spectrum within BSE
0
L(r, r , ω) =
X
λ
P
vc
(vc)
∗(vc)
Aλ φv (r)φ∗c (r)φ∗v (r0 )φc (r0 )Aλ
ω − Eλexc + iη
Francesco Sottile
Results
The EXC Code
0 0
X |Aλ >< Aλ |
1
=
H − ωI
Eλ − ω
λ
Spectrum within BSE
0
L(r, r , ω) =
X
λ
P
vc
(vc)
∗(vc)
ω − Eλexc + iη
Francesco Sottile
Results
The EXC Code
0 0
X |Aλ >< Aλ |
1
=
H − ωI
Eλ − ω
λ
Spectrum within BSE
0
L(r, r , ω) =
X
λ
P
vc
(vc)
∗(vc)
ω − Eλexc + iη
Francesco Sottile
Results
The EXC Code
Spectrum in BSE (only resonant)
X P A(vc) φv (r)φ∗ (r)φ∗ (r0 )φc (r0 )A∗(vc)
c
v
vc λ
λ
L(r, r , ω) =
exc
ω − Eλ + iη
0
λ
0
0
χ (r, r , ω) =
X φv (r)φ∗ (r)φ∗ (r0 )φc (r0 )
c
vc
v
ω − (c − v ) + iη
Francesco Sottile
Results
The EXC Code
X P A(vc) φv (r)φ∗ (r)φ∗ (r0 )φc (r0 )A∗(vc)
c
v
vc λ
λ
L(r, r , ω) =
exc
ω − Eλ + iη
0
λ
0
0
χ (r, r , ω) =
X φv (r)φ∗ (r)φ∗ (r0 )φc (r0 )
c
vc
v
ω − (c − v ) + iη
Francesco Sottile
Results
The EXC Code
2
X X (vc)
BSE
Abs (ω) = Im hL(ω)i =
Aλ hc|D|v i δ (ω − Eλexc )
vc
λ
X
AbsIP-RPA (ω) = Im χ0 (ω) =
|hc|D|v i|2 δ (ω − (c −v ))
vc
Francesco Sottile
Results
The EXC Code
BSE - creating and diagonalizing a (big) matrix
exc
⇒ H(vc)(v
0c 0)
0 0
⇒
(v 0 c 0 )
L(vc) (ω)
X A(vc) A∗(v c )
λ
λ
=
exc
ω − Eλ + iη
λ
Francesco Sottile
Results
The EXC Code
0 0
0 0
exc
vc
⇒ H(vc)(v
− Wvcv c
0 c 0 ) = (Ec − Ev ) δvv 0 δcc 0 + vvc
0 0
⇒
(v 0 c 0 )
L(vc) (ω)
X A(vc) A∗(v c )
λ
λ
=
exc
ω − Eλ + iη
λ
Francesco Sottile
Results
The EXC Code
0 0
(v c )
(vc)
exc
⇒ H(vc)(v
= Eλexc Aλ
0 c 0 ) Aλ
0 0
⇒
(v 0 c 0 )
L(vc) (ω)
X A(vc) A∗(v c )
λ
λ
=
exc
ω − Eλ + iη
λ
Francesco Sottile
Results
The EXC Code
0 0
(v c )
(vc)
exc
⇒ H(vc)(v
= Eλexc Aλ
0 c 0 ) Aλ
0 0
⇒
(v 0 c 0 )
L(vc) (ω)
X A(vc) A∗(v c )
λ
λ
=
exc
ω − Eλ + iη
λ
Francesco Sottile
Results
The EXC Code
Standard Approximations for BSE
Ground-state
pseudopotential
Vxc local density approximation
Quasi-particle Many-Body Theory
GW approximation for Σ
W rpa, plasmon-pole model
ψGW = φKS
Bethe-Salpeter equation
δW
=0
δG
W rpa, static
only resonant term
Francesco Sottile
Results
The EXC Code
The Bethe-Salpeter Soup
Francesco Sottile
Results
The EXC Code
Outline
1
2
Definition of BSE
BSE in practice
3
Results
4
The EXC Code
Francesco Sottile
Results
The EXC Code
Bethe-Salpeter equation results: Semiconductors
Albrecht et al., PRL 80, 4510 (1998)
Francesco Sottile
Results
The EXC Code
Bethe-Salpeter equation results: Insulators
Sottile, Marsili, et al., PRB (2007).
Francesco Sottile
Results
The EXC Code
Bethe-Salpeter equation results: Molecule (Na4 )
Onida et al., PRL 75, 818 (1995)
Francesco Sottile
Results
The EXC Code
Bethe-Salpeter equation results: Silicon Nanowires
Bruno et al., PRL 98, 036807 (2007)
Francesco Sottile
Results
The EXC Code
Bethe-Salpeter equation results: Hexagonal Ice
Hahn et al., PRL 94, 37404 (2005)
Francesco Sottile
Results
The EXC Code
Bethe-Salpeter equation results: Cu2 O
Bruneval et al., PRL 97, 267601 (2006)
Francesco Sottile
Results
The EXC Code
Bethe-Salpeter equation results: EELS of Silicon
Olevano and Reining, PRL 86, 5962 (2001)
Francesco Sottile
Results
The EXC Code
Bethe-Salpeter equation results: Surface
Rohlfing et al., PRL 85, 005440 (2000)
Francesco Sottile
Results
The EXC Code
Bethe-Salpeter equation results: Surface
Rohlfing et al., PRL 85, 005440 (2000)
Francesco Sottile
Results
The EXC Code
Bethe-Salpeter equation results: liquid Water
Garbuio et al., PRL 97, 137402 (2006)
Francesco Sottile
Results
The EXC Code
I The EXC code - Sottile, Reining, Olevano, Onida, Albrecht
http://www.bethe-salpeter.org
Francesco Sottile
Results
The EXC Code
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
Francesco Sottile
Results
The EXC Code
DFT - ground state
GW - quasiparticle energies
BSE - optical and dielectric properties
√
several spectroscopies
√
variety of systems
× Cumbersome Calculations
Francesco Sottile
Results
The EXC Code
Some references
Hanke and Sham, PRB 21, 4656 (1980)
Onida, Reining, Rubio, RMP 74, 601 (2002)
Strinati, Riv Nuovo Cimento 11, 1 (1988)
Francesco Sottile
Results
The EXC Code
Outline
1
2
Definition of BSE
BSE in practice
3
Results
4
The EXC Code
Francesco Sottile
Results
The EXC Code
What do we calculate ?
Dielectric function (only resonant case)

(vc) 2 
X X c|e −iq·r |v Aλ 
εM (ω) = 1 − lim v0 (q)


q→0
Eλexc − ω − iη
λ
(vc)
diagonalize excitonic Hamiltonian
Hamiltonian:
(v 0 c 0 )
2p,exc
H(vc)(v
0 c 0 ) Aλ
=
(vc)
Eλexc Aλ
(v 0 c 0 )
2p
H(vc)(v
0 c 0 )= (Ec − Ev )δvv 0 δcc 0 + [v − W ](vc)
(v’c’)
2
...
6
H = (vc) 6
6
6
4 ...
...
..
.
3
..
7
7
7
7
5
.
Ei
=
quasiparticle energies (GW)
W
= ε−1 v screened Coulomb interaction
...
Francesco Sottile
Results
The EXC Code

(vc) 2 
εM (ω) = 1 − lim v0 (q)


q→0
λ
(vc)
Hamiltonian:
(v 0 c 0 )
2p,exc
H(vc)(v
0 c 0 ) Aλ
=
(vc)
Eλexc Aλ
(v 0 c 0 )
2p
H(vc)(v
(v’c’)
2
...
6
H = (vc) 6
6
6
4 ...
...
..
.
3
..
7
7
7
7
5
.
Ei
=
W
...
Francesco Sottile
Results
The EXC Code

(vc) 2 
εM (ω) = 1 − lim v0 (q)


q→0
λ
(vc)
Hamiltonian:
(v 0 c 0 )
2p,exc
H(vc)(v
0 c 0 ) Aλ
=
(vc)
Eλexc Aλ
(v 0 c 0 )
2p
H(vc)(v
(v’c’)
2
...
6
H = (vc) 6
6
6
4 ...
...
..
.
3
..
7
7
7
7
5
.
Ei
=
W
...
Francesco Sottile
Results
The EXC Code
What do we need ?
structure, screening, quasiparticle files + input file
|v i
gs (LDA) wfs
Ev
Quasi-Particle energies
ε−1
for the screened interaction
kss file
gw file
scr file
Francesco Sottile
Results
The EXC Code
The excitonic Hamiltonian
2p
H(vc)(v
=
E
−
E
c
v δvv 0 δcc 0 + [v − W ](vc)(v 0 c 0 )
0c 0)
2p
H(vc)(v
0 c 0 ) = (c − v ) δvv 0 δcc 0
RPA-NLF
2p
H(vc)(v
0 c 0 ) = (c − v ) δvv 0 δcc 0 + v(vc)(v 0 c 0 )
RPA
2p
H(vc)(v
0 c 0 ) = (Ec − Ev ) δvv 0 δcc 0 + v(vc)(v 0 c 0 )
GW
2p
H(vc)(v
0c 0)
BSE
= (Ec − Ev ) δvv 0 δcc 0 + v(vc)(v 0 c 0 ) + W(vc)(v 0 c 0 )
Francesco Sottile
Results
The EXC Code
2p
GW
H(vc)(v
(c + ∆GW
)
−
(
+
∆
)
δvv 0 δcc 0 + [v − W ](vc)(v 0 c 0 )
v
0c 0) =
c
v
2p
H(vc)(v
0 c 0 ) = (c − v ) δvv 0 δcc 0
RPA-NLF
2p
H(vc)(v
0 c 0 ) = (c − v ) δvv 0 δcc 0 + v(vc)(v 0 c 0 )
RPA
2p
H(vc)(v
GW
2p
H(vc)(v
0c 0)
BSE
Francesco Sottile
Results
The EXC Code
2p
GW
H(vc)(v
(c + ∆GW
)
−
(
+
∆
)
δvv 0 δcc 0 + [v − W ](vc)(v 0 c 0 )
v
0c 0) =
c
v
2p
H(vc)(v
0 c 0 ) = (c − v ) δvv 0 δcc 0
RPA-NLF
2p
H(vc)(v
0 c 0 ) = (c − v ) δvv 0 δcc 0 + v(vc)(v 0 c 0 )
RPA
2p
H(vc)(v
GW
2p
H(vc)(v
0c 0)
BSE
Francesco Sottile
Results
The EXC Code
2p
GW
GW
H(vc)(v
0 c 0 ) = ((c + ∆c ) − (v + ∆v )) δvv 0 δcc 0 + [v − W ](vc)(v 0 c 0 )
2p
H(vc)(v
0 c 0 ) = (c − v ) δvv 0 δcc 0
RPA-NLF
2p
H(vc)(v
0 c 0 ) = (c − v ) δvv 0 δcc 0 + v(vc)(v 0 c 0 )
RPA
2p
H(vc)(v
GW
2p
H(vc)(v
0c 0)
BSE
Francesco Sottile
Results
The EXC Code
2p
GW
GW
H(vc)(v
0 c 0 ) = ((c + ∆c ) − (v + ∆v )) δvv 0 δcc 0 + [v − W ](vc)(v 0 c 0 )
2p
H(vc)(v
0 c 0 ) = (c − v ) δvv 0 δcc 0
RPA-NLF
2p
H(vc)(v
0 c 0 ) = (c − v ) δvv 0 δcc 0 + v(vc)(v 0 c 0 )
RPA
2p
H(vc)(v
GW
2p
H(vc)(v
0c 0)
BSE
Francesco Sottile
Results
The EXC Code
2p
H(vc)(v
=
E
−
E
c
v δvv 0 δcc 0 + [v + W ](vc)(v 0 c 0 )
0c 0)
2p
H(vc)(v
0 c 0 ) = (c − v ) δvv 0 δcc 0
RPA-NLF
2p
H(vc)(v
0 c 0 ) = (c − v ) δvv 0 δcc 0 + v(vc)(v 0 c 0 )
RPA
2p
H(vc)(v
GW
2p
H(vc)(v
0c 0)
BSE
Francesco Sottile
\
|
|
|
/
Results
The EXC Code
__
____
___
__
___) \ \ / / / __)
(__
\ \/ / | /
__)
>
<
| |
(___
/ /\ \ | \__
)_/ /__\ \__\
)_
http://www.bethe-salpeter.org
program EXC version 2.3.4
built: 06 Dec 2007
calculate dielectric properties
Bethe-Salpeter equation code in frequency domain
reciprocal space on a transitions basis
Copyright (C) 1992-2007, Lucia Reining, Valerio Olevano,
Francesco Sottile, Stefan Albrecht, Giovanni Onida.
This program is free software; you can redistribute it
and/or modify it under the terms of the GNU General
Public License.
This program is distributed in the hope that it will be
ANY WARRANTY;
useful,
but Equation
WITHOUT
The
Bethe-Salpeter
Francesco Sottile
Results
The EXC Code
The input file
rpa, gw, exc (default) . . . . . . . . . . . . . . . . . . . . . . . type of calculation
nlf, lf (default) . . . . . . . . . . . . . . . . . . . . . . . . . . with or w/o local fields
wdiag (default), wfull . . . . . . . . . . . . . . . . . . . . . . . . . . WGG0 δGG0 or WGG0
resonant (default), coupling . . . . . . . . coupling reso-antireso or not
soenergy <value> (default=0.0) . . . . . . scissor operator energy [eV]
matsh <value> (default=all) . . . . . . . . number of G-shell for the εGG0
wfnsh <value> (default=all) . . . . . plane waves shells for the ψnk (G)
nbands <value> (default=all) last band included in the calculation
lomo <value> (default=1) . . . . . first band included in the calculation
omegai <value> (default=0.0) . . . . . . . . . frequency initial point [eV]
omegae <value> (default=10.0) . . . . . . . . . . frequency end point [eV]
domega <value> (default=0.01) . . . . . . . . . . . . . . . frequency step [eV]
broad <value> (default=domega) . . . . . . . lorentzian broadening [eV]
haydock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iterative diagonalization scheme
niter <value> (default=100) . . . . . . . . . iterations for haydock scheme
Francesco Sottile
Results
The EXC Code
The input file
Francesco Sottile
Results
The EXC Code
The input file
Francesco Sottile
Results
The EXC Code
The input file
Francesco Sottile
Results
The EXC Code
The input file
Francesco Sottile
Results
The EXC Code
The input file
Francesco Sottile
Results
The EXC Code
The input file
Francesco Sottile
Results
The EXC Code
The input file
Francesco Sottile
Results
The EXC Code
The input file
Francesco Sottile
Results
The EXC Code
The input file
Francesco Sottile
Results
The EXC Code
The input file
Francesco Sottile
Results
The EXC Code
The input file
Francesco Sottile
Results
The EXC Code
The input file
Francesco Sottile
Results
The EXC Code
The input file
Francesco Sottile
Results
The EXC Code
The input file
Francesco Sottile

The Bethe-Salpeter Equation

Transcription

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BBH simulations and their applications: an update