D O U B L E E X C IT A T IO N S IN F IN IT E SY ST E M S
Transcript of D O U B L E E X C IT A T IO N S IN F IN IT E SY ST E M S
DOUBLE EXCITATIONS IN FINITE SYSTEMS
Pina RomanielloLSI, École Polytechnique, Palaiseau (France)
Benasque, 10-15 September 2008
Davide Sangalli, Luca Molinari, Giovanni Onida Univesita’ degli Studi di Milano (Italy)
Lucia Reining, Francesco Sottile, and Arjan Berger, École Polytechnique (France)
Thanks
Outline
Excitation energies in TDDFT: double excitations...problematic
Double excitations using BSE
Application to simple models
From BSE back to TDDFT
Conclusions
Double excitationsElectron excitations in one-electron picture
✴ open-shell ✴ closed-shell
energy reasonspin-symmetry reason
E|i↑a↓〉 ! E|v↑v↓〉
⤹⤹⤹
Double excitationsElectron excitations in one-electron picture
✴ open-shell ✴ closed-shell
energy reasonspin-symmetry reason
E|i↑a↓〉 ! E|v↑v↓〉
⤹⤹⤹
M. E. Casida et al., Lect. Notes Phys., 706, 243-257 (2006), R. J. Cave et al., Chem. Phys. Lett., 389, 39 (2004)
Casida’s equations
Excitation energies in TDDFTDyson-like response equation
χ(x1, x2,ω) = χs(x1, x2,ω)+
∫
dx3dx4χs(x1, x3,ω)
[
1
|x3 − x4|+ fxc(x3, x4,ω)
]
χ(x4, x2,ω)
projection in transition space
(
A B
−B∗−A∗
) (
X
Y
)
= ω
(
X
Y
)
Aiaσ,jbτ = δστδabδij(εaσ − εiτ ) + Kiaσ,jbτ
Biaσ,jbτ = Kiaσ,bjτ
Kiaσ,jbτ =
∫ ∫
ψ∗
iσ(r)ψaσ(r)
[
1
|r − r′|
+ fστxc (r, r′,ω)
]
ψ∗
bτ (r′)ψjτ (r′)drdr′
⤹
Casida’s equations
Excitation energies in TDDFTDyson-like response equation
χ(x1, x2,ω) = χs(x1, x2,ω)+
∫
dx3dx4χs(x1, x3,ω)
[
1
|x3 − x4|+ fxc(x3, x4,ω)
]
χ(x4, x2,ω)
projection in transition space
(
A B
−B∗−A∗
) (
X
Y
)
= ω
(
X
Y
)
Aiaσ,jbτ = δστδabδij(εaσ − εiτ ) + Kiaσ,jbτ
Biaσ,jbτ = Kiaσ,bjτ
Kiaσ,jbτ =
∫ ∫
ψ∗
iσ(r)ψaσ(r)
[
1
|r − r′|
+ fστxc (r, r′,ω)
]
ψ∗
bτ (r′)ψjτ (r′)drdr′
ALDA✴ only singly-excited configurations✴ n. eigenvalues=n. single-excitations
⤹
eigenvalue problem lower-dimensional onem × m
(
S C1
C2 D
) (
e1
e2
)
= ω
(
e1
e2
)
fxc(ω)
Reduced space ω-dependence
✴ Four-point BSE two-point TDDFT ( even if the BSE kernel can be static)
✴ Multiple-excitation space single-excitation space (D in double excitation space, S in single-excitation space (Casida’s within ALDA))
[S + C1(ω − D)−1C2]e1 = ωe1
e2 = (ω − D)−1C2e1
✴ Many-body hamiltonian single-particle hamiltonian
Bethe-Salpeter equation
L(1, 2, 1′, 2′) = L0(1, 2, 1′, 2′) +
∫d3d4d5d6L0(1, 4, 1′, 3)Ξ(3, 5, 4, 6)L(6, 2, 5, 2′)
Ξ(3, 5, 4, 6) = δ(3, 4)δ(5, 6)v(3, 6) + iδΣ(3, 4)
δG(6, 5)
L0(1, 2, 1′, 2′) = −iG(1, 2′)G(2, 1′)
χ(1, 2) = L(1, 2, 1, 2)
Bethe-Salpeter equation
L(1, 2, 1′, 2′) = L0(1, 2, 1′, 2′) +
∫d3d4d5d6L0(1, 4, 1′, 3)Ξ(3, 5, 4, 6)L(6, 2, 5, 2′)
Ξ(3, 5, 4, 6) = δ(3, 4)δ(5, 6)v(3, 6) + iδΣ(3, 4)
δG(6, 5)
L0(1, 2, 1′, 2′) = −iG(1, 2′)G(2, 1′)
Σ = GW
δΣ
δG= W + G
δW
δG! Wapproximations
W (t − t′) " W δ(t − t
′)
χ(1, 2) = L(1, 2, 1, 2)
⤷
Bethe-Salpeter equation
G(1, 2) = −i〈N |T [ψ(1)ψ†(2)]|N〉
G(1, 2, 1′, 2′) = (−i)2〈N |T [ψ(1)ψ(2)ψ†(2′)ψ†(1′)]|N〉
Two-particle propagator
✴ 1-particle GF
✴2-particle GF
L(1, 2, 1′, 2′) = −G(1, 2, 1′, 2′) + G(1, 1′)G(2, 2′)
G(τ = [(t1 + t1′)/2 − (t2 + t2′)/2], τ1 = t1 − t1′ , τ2 = t2 − t2′)
Bethe-Salpeter equation
G(1, 2) = −i〈N |T [ψ(1)ψ†(2)]|N〉
G(1, 2, 1′, 2′) = (−i)2〈N |T [ψ(1)ψ(2)ψ†(2′)ψ†(1′)]|N〉pphhph
Two-particle propagator
✴ 1-particle GF
✴2-particle GF
L(1, 2, 1′, 2′) = −G(1, 2, 1′, 2′) + G(1, 1′)G(2, 2′)
G(τ = [(t1 + t1′)/2 − (t2 + t2′)/2], τ1 = t1 − t1′ , τ2 = t2 − t2′)
G. Csanak, H. S. Taylor, and R. Yaris, Adv. At. Mol. Phys., 7, 289, (1971)
1′ 2
1 2′
1 2′ 1 3 6 2′
1′ 2 1′ 4 5 2
Ξ
= +
1
1 2′
1 2′
L-
1
1′ 2 1′ 2
1
3=4
1′
5=6
3=6
4=5 2′
2
( )
Bethe-Salpeter equation✴ BSE
✴ BSE with Ξ = v − W
= +
1
L L
= +
1
L
1′ 2
1 2′
1 2′ 1 3 6 2′
1′ 2 1′ 4 5 2
= +
1
L LΞ
= +
1
= +
1
1 2′
1 2′
L L-
1
1′ 2 1′ 2
1
3=4
1′
5=6
3=6
4=5 2′
2
( )
Bethe-Salpeter equation✴ BSE
✴ BSE with Ξ = v − W
1′ 2
1 2′
1 2′ 1 3 6 2′
1′ 2 1′ 4 5 2
= +
1
L LΞ
= +
1
= +
1
1 2′
1 2′
L L-
1
1′ 2 1′ 2
1
3=4
1′
5=6
3=6
4=5 2′
2
x1 x2′
x1 x2′
-
1
x1′ x2 x1′ x2
x1
x3=x4
x1′
x3=x6
x4=x5
x2′
=
+
1
=
+
1
=
+
1
=
+
1
= +
1
x2
=
+
1
=
+
1
= +
1
Lt 1=t1′ t 2=t2′ t 1=t1′ t 2=t2′ t 1=t1′
t 3=t4=t 5=t6x5=x6
t 2=t2′L
( )
( )
Bethe-Salpeter equation✴ BSE
✴ BSE with Ξ = v − W
✴ BSE with W δ(t − t′)
L(t1 − t2) → L(ω)
Bethe-Salpeter equationBSE in frequency space
L(ω,ω′,ω′′) = L0(ω,ω′,ω′′) − iG(ω′ + ω/2)G(ω′− ω/2)×
[
v
∫
dω
2πL(ω, ω,ω′′) −
∫
dω
2πW (ω′
− ω)L(ω, ω, ω′′)
]
Bethe-Salpeter equationBSE in frequency space
L(ω,ω′,ω′′) = L0(ω,ω′,ω′′) − iG(ω′ + ω/2)G(ω′− ω/2)×
[
v
∫
dω
2πL(ω, ω,ω′′) −
∫
dω
2πW (ω′
− ω)L(ω, ω, ω′′)
]
L0(ω,ω′,ω′′) = −2π iδ(ω′− ω′′)G(ω′ + ω/2)G(ω′′
− ω/2)
G(x1, x2,ω) =∑
n
φn(x1)φ∗
n(x2)
ω − εn + iηsign(εn − µ)
✴ Single-particle Green’s function
✴Independent quasiparticle response function
Bethe-Salpeter equationBSE in frequency space
L(ω,ω′,ω′′) = L0(ω,ω′,ω′′) − iG(ω′ + ω/2)G(ω′− ω/2)×
[
v
∫
dω
2πL(ω, ω,ω′′) −
∫
dω
2πW (ω′
− ω)L(ω, ω, ω′′)
]
L0(ω,ω′,ω′′) = −2π iδ(ω′− ω′′)G(ω′ + ω/2)G(ω′′
− ω/2)
G(x1, x2,ω) =∑
n
φn(x1)φ∗
n(x2)
ω − εn + iηsign(εn − µ)
✴ Single-particle Green’s function
Self-energy dynamical effects neglected
✴Independent quasiparticle response function
⤷
Bethe-Salpeter equationBSE in frequency space
L(ω,ω′,ω′′) = L0(ω,ω′,ω′′) − iG(ω′ + ω/2)G(ω′− ω/2)×
[
v
∫
dω
2πL(ω, ω,ω′′) −
∫
dω
2πW (ω′
− ω)L(ω, ω, ω′′)
]
Bethe-Salpeter equationBSE in frequency space
L(ω,ω′,ω′′) = L0(ω,ω′,ω′′) − iG(ω′ + ω/2)G(ω′− ω/2)×
W (ω) ! W (ω = 0)
L(ω) = L0(ω) + L0(ω)KL(ω)⤹[
v
∫
dω
2πL(ω, ω,ω′′) −
∫
dω
2πW (ω′
− ω)L(ω, ω, ω′′)
]
Bethe-Salpeter equationBSE in frequency space
L(ω,ω′,ω′′) = L0(ω,ω′,ω′′) − iG(ω′ + ω/2)G(ω′− ω/2)×
W (ω) ! W (ω = 0)
L(ω) = L0(ω) + L0(ω)KL(ω)⤹Two-particle Hamiltonian
L(n1′
n1)(n2n2′
) =
∫dx1dx2dx1′dx2′L(x1, x2, x1′ , x2′ ;ω)φ∗
n1(x1)φn
1′(x1′)φn
2′(x2′)φ∗
n2(x2)
[
v
∫
dω
2πL(ω, ω,ω′′) −
∫
dω
2πW (ω′
− ω)L(ω, ω, ω′′)
]
Bethe-Salpeter equationBSE in frequency space
L(ω,ω′,ω′′) = L0(ω,ω′,ω′′) − iG(ω′ + ω/2)G(ω′− ω/2)×
W (ω) ! W (ω = 0)
L(ω) = L0(ω) + L0(ω)KL(ω)
S. Albretcht, Ph.D. thesis, Ecole Polytechnique, France (1999);F. Sottile, Ph.D. thesis, Ecole Polytechnique, France (2003)
⤹Two-particle Hamiltonian
L(n1′
n1)(n2n2′
) =
∫dx1dx2dx1′dx2′L(x1, x2, x1′ , x2′ ;ω)φ∗
n1(x1)φn
1′(x1′)φn
2′(x2′)φ∗
n2(x2)
L(n1′
n1)(n2n2′
) =[
H2p− Iω
]
−1
(n1′
n1)(n2n2′
)(fn2
− fn2′
)
H2p
(n1′
n1)(n2n2′
) = (εn1− εn
1′)δn1,n
2′δn
1′,n2
+ (fn1′− fn1
)K(n1′
n1)(n2′n2)
⤹
L0(n1′
n1)(n2n2′
) =δn
1′n2
δn1n2′
(fn1′− fn1
)
ω − (εn1− εn
1′) + iηsign(εn1
− εn1′
)
[
v
∫
dω
2πL(ω, ω,ω′′) −
∫
dω
2πW (ω′
− ω)L(ω, ω, ω′′)
]
Bethe-Salpeter equationExcitonic Hamiltonian
H2p,exc =
(
H2p,reso(vc)(v′c′) K
coupling(vc)(c′
v′)
−[Kcoupling(vc)(c′
v′)]
∗−[H2p,reso
(vc)(v ′c′)]
∗
)
H2p,exc(n
1′n1)(n2n
2′)A
(n2n2′
)λ = ωλA
(n1′
n1)λ
Bethe-Salpeter equationExcitonic Hamiltonian
H2p,exc =
(
H2p,reso(vc)(v′c′) K
coupling(vc)(c′
v′)
−[Kcoupling(vc)(c′
v′)]
∗−[H2p,reso
(vc)(v ′c′)]
∗
)
H2p,exc(n
1′n1)(n2n
2′)A
(n2n2′
)λ = ωλA
(n1′
n1)λ
L(x1, x2, x1′ , x2′ ,ω) =∑
λλ′
[
∑
n1n1′
A(n
1′n1)
λφn1
(x1)φ∗
n1′
(x1′)
ωλ − ω + iηsign(εn1′− εn1
)×
S−1λλ′
∑
n2n2′
A∗(n2n
2′)
λ′ φn2(x2)φ
∗
n2′
(x2′)(fn2′− fn2
)
]
⤹
Bethe-Salpeter equationExcitonic Hamiltonian
H2p,exc =
(
H2p,reso(vc)(v′c′) K
coupling(vc)(c′
v′)
−[Kcoupling(vc)(c′
v′)]
∗−[H2p,reso
(vc)(v ′c′)]
∗
)
H2p,exc(n
1′n1)(n2n
2′)A
(n2n2′
)λ = ωλA
(n1′
n1)λ
L(x1, x2, x1′ , x2′ ,ω) =∑
λλ′
[
∑
n1n1′
A(n
1′n1)
λφn1
(x1)φ∗
n1′
(x1′)
ωλ − ω + iηsign(εn1′− εn1
)×
S−1λλ′
∑
n2n2′
A∗(n2n
2′)
λ′ φn2(x2)φ
∗
n2′
(x2′)(fn2′− fn2
)
]
Static kernel✴ only singly-excited configurations✴ n. eigenvalues=n. single-excitations⤹
S. Albretcht, Ph.D. thesis, Ecole Polytechnique, France (1999);F. Sottile, Ph.D. thesis, Ecole Polytechnique, France (2003)
Bethe-Salpeter equationExcitonic Hamiltonian: W (ω)
G. Strinati, Rev. Nuovo Cimento 11, 1, (1988)
H2p,exc(n
1′n1)(n2n
2′)(ωλ)A(n2n
2′)
λ (ωλ) = ωλA(n
1′n1)
λ (ωλ)
H2p,exc(n
1′n1)(n2n
2′)(ωλ) = (εn1
−εn1′
)δn1′
n2δn1n
2′+(fn
1′−fn1
)[
v(n1′
n1)(n2n2′
) − W(n1′
n1)(n2n2′
)(ωλ)]
Bethe-Salpeter equationExcitonic Hamiltonian: W (ω)
G. Strinati, Rev. Nuovo Cimento 11, 1, (1988)
H2p,exc(n
1′n1)(n2n
2′)(ωλ)A(n2n
2′)
λ (ωλ) = ωλA(n
1′n1)
λ (ωλ)
H2p,exc(n
1′n1)(n2n
2′)(ωλ) = (εn1
−εn1′
)δn1′
n2δn1n
2′+(fn
1′−fn1
)[
v(n1′
n1)(n2n2′
) − W(n1′
n1)(n2n2′
)(ωλ)]
✴ ω-dependent screening
W(n1′
n1)(n2n2′
)(ωλ) = i
∫
dω
2πW(n
1′n1)(n2n
2′)(ω)
[
1
ωλ − ω − (εn2′− εn
1′) + iη
+1
ωλ + ω − (εn1− εn2
) + iη
]
Bethe-Salpeter equationExcitonic Hamiltonian: W (ω)
W = v + vχv
⤹
G. Strinati, Rev. Nuovo Cimento 11, 1, (1988)
H2p,exc(n
1′n1)(n2n
2′)(ωλ)A(n2n
2′)
λ (ωλ) = ωλA(n
1′n1)
λ (ωλ)
W(n1′
n1)(n2n2′
)(ω) = v(n2′
n1)(n2n1′
)+∑
λ
∑
vc,v′c′
v(n2′
n1)(vc)A
(vc),staticλ A
∗(v′c′),staticλ
ω − ωstaticλ + iη
v(v′c′)(n2n1′
)
−
∑
λ
∑
cv,c′v′
v(n2′
n1)(cv)A
(cv),staticλ A
∗(c′v′),staticλ
ω + ωstaticλ − iη
v(c′v′)(n2n1′
)
H2p,exc(n
1′n1)(n2n
2′)(ωλ) = (εn1
−εn1′
)δn1′
n2δn1n
2′+(fn
1′−fn1
)[
v(n1′
n1)(n2n2′
) − W(n1′
n1)(n2n2′
)(ωλ)]
✴ ω-dependent screening
W(n1′
n1)(n2n2′
)(ωλ) = i
∫
dω
2πW(n
1′n1)(n2n
2′)(ω)
[
1
ωλ − ω − (εn2′− εn
1′) + iη
+1
ωλ + ω − (εn1− εn2
) + iη
]
Bethe-Salpeter equationExcitonic Hamiltonian: W (ω)
H2p,exc(n
1′n1)(n2n
2′)(ωλ) = (εn1
−εn1′
)δn1′
n2δn1n
2′+(fn
1′−fn1
)[
v(n1′
n1)(n2n2′
) − W(n1′
n1)(n2n2′
)(ωλ)]
✴ ω-dependent screening
H2p,exc(n
1′n1)(n2n
2′)(ωλ)A(n2n
2′)
λ (ωλ) = ωλA(n
1′n1)
λ (ωλ)
Bethe-Salpeter equation
⤹
Excitonic Hamiltonian: W (ω)
H2p,exc(n
1′n1)(n2n
2′)(ωλ) = (εn1
−εn1′
)δn1′
n2δn1n
2′+(fn
1′−fn1
)[
v(n1′
n1)(n2n2′
) − W(n1′
n1)(n2n2′
)(ωλ)]
✴ ω-dependent screening
+∑
λ′
∑
vc,v′c′
v(n2′
n1)(vc)A
(vc),staticλ′ A
∗(v′c′),staticλ′
ωλ − ωstaticλ′ − (εn
2′− εn
1′) + iη
v(v′c′)(n2n1′
)
W(n1′
n1)(n2n2′
)(ωλ) = v(n2′
n1)(n2n1′
)+∑
λ′
∑
vc,v′c′
v(n2′
n1)(vc)A
(vc),staticλ′ A
∗(v′c′),staticλ′
ωλ − ωstaticλ′ − (εn
2′− εn
1′) + iη
v(v′c′)(n2n1′
)
H2p,exc(n
1′n1)(n2n
2′)(ωλ)A(n2n
2′)
λ (ωλ) = ωλA(n
1′n1)
λ (ωλ)
A simple model I.Two-electron system Eigenvalue eq.
(v ↑ c ↑) (v ↓ c ↓)(v ↑ c ↑) ∆εvc + v(vc)(vc) − W(vc)(vc)(ω) v(vc)(vc)
(v ↓ c ↓) v(vc)(vc) ∆εvc + v(vc)(vc) − W(vc)(vc)(ω)
A = ωA
A simple model I.
ωstatic
1 = ∆εvc + 2v(vc)(vc) − W static
(vc)(vc), Astatic
1 =1√
2
(
1
1
)
ωstatic
2 = ∆εvc − W static
(vc)(vc), Astatic
2 =1√
2
(
1
−1
)
Singlet
Triplet
Two-electron system Eigenvalue eq.
Wstatic
(vc)(vc)⤹
(v ↑ c ↑) (v ↓ c ↓)(v ↑ c ↑) ∆εvc + v(vc)(vc) − W(vc)(vc)(ω) v(vc)(vc)
(v ↓ c ↓) v(vc)(vc) ∆εvc + v(vc)(vc) − W(vc)(vc)(ω)
A = ωA
A simple model I.
ωstatic
1 = ∆εvc + 2v(vc)(vc) − W static
(vc)(vc), Astatic
1 =1√
2
(
1
1
)
ωstatic
2 = ∆εvc − W static
(vc)(vc), Astatic
2 =1√
2
(
1
−1
)
ω1 = ∆εvc + 2v(vc)(vc) − v(cc)(vv)Singlets
Singlet
Triplet
double-excitation
Two-electron system Eigenvalue eq.
Wstatic
(vc)(vc)⤹
⤹ω3 = 2∆εvc + 2v(vc)(vc) − W
static
(vc)(vc)
(ωstatic− 2v(vc)(vc) + v(cc)(vv))
2>> 8"[v(cc)(vc)v(vc)(vv)]
(v ↑ c ↑) (v ↓ c ↓)(v ↑ c ↑) ∆εvc + v(vc)(vc) − W(vc)(vc)(ω) v(vc)(vc)
(v ↓ c ↓) v(vc)(vc) ∆εvc + v(vc)(vc) − W(vc)(vc)(ω)
A = ωA
W(vc)(vc)(ω) = v(cc)(vv) + 2![v(cc)(vc)v(vc)(cc)]
ω − ωstatic1 − ∆εvc
A simple model I.
ωstatic
1 = ∆εvc + 2v(vc)(vc) − W static
(vc)(vc), Astatic
1 =1√
2
(
1
1
)
ωstatic
2 = ∆εvc − W static
(vc)(vc), Astatic
2 =1√
2
(
1
−1
)
ω2 = ∆εvc − v(cc)(vv)
ω1 = ∆εvc + 2v(vc)(vc) − v(cc)(vv)Singlets
Triplets
Singlet
Triplet
double-excitation
Two-electron system Eigenvalue eq.
Wstatic
(vc)(vc)⤹
⤹ω3 = 2∆εvc + 2v(vc)(vc) − W
static
(vc)(vc)
(ωstatic− 2v(vc)(vc) + v(cc)(vv))
2>> 8"[v(cc)(vc)v(vc)(vv)]
(v ↑ c ↑) (v ↓ c ↓)(v ↑ c ↑) ∆εvc + v(vc)(vc) − W(vc)(vc)(ω) v(vc)(vc)
(v ↓ c ↓) v(vc)(vc) ∆εvc + v(vc)(vc) − W(vc)(vc)(ω)
A = ωA
W(vc)(vc)(ω) = v(cc)(vv) + 2![v(cc)(vc)v(vc)(cc)]
ω − ωstatic1 − ∆εvc
A simple model I.
ωstatic
1 = ∆εvc + 2v(vc)(vc) − W static
(vc)(vc), Astatic
1 =1√
2
(
1
1
)
ωstatic
2 = ∆εvc − W static
(vc)(vc), Astatic
2 =1√
2
(
1
−1
)
ω2 = ∆εvc − v(cc)(vv)
ω1 = ∆εvc + 2v(vc)(vc) − v(cc)(vv)Singlets
Triplets
Singlet
Triplet
double-excitation
✗
Two-electron system Eigenvalue eq.
Wstatic
(vc)(vc)⤹
⤹
ω4 = 2∆εvc + 2v(vc)(vc) − Wstatic
(vc)(vc)
ω3 = 2∆εvc + 2v(vc)(vc) − Wstatic
(vc)(vc)
(ωstatic− 2v(vc)(vc) + v(cc)(vv))
2>> 8"[v(cc)(vc)v(vc)(vv)]
(v ↑ c ↑) (v ↓ c ↓)(v ↑ c ↑) ∆εvc + v(vc)(vc) − W(vc)(vc)(ω) v(vc)(vc)
(v ↓ c ↓) v(vc)(vc) ∆εvc + v(vc)(vc) − W(vc)(vc)(ω)
A = ωA
W(vc)(vc)(ω) = v(cc)(vv) + 2![v(cc)(vc)v(vc)(cc)]
ω − ωstatic1 − ∆εvc
A simple model II.
Eigenvalue eq.
ωstatic = ∆εvc + v(vc)(vc) − W static
(vc)(vc), Astatic = 1
One-electron system
Wstatic
(vc)(vc)⤹
[
∆εvc + v(vc)(vc) − W(vc)(vc)(ω)]
A = ωA
A simple model II.
Eigenvalue eq.
ωstatic = ∆εvc + v(vc)(vc) − W static
(vc)(vc), Astatic = 1
ω1 = ∆εvc + v(vc)(vc) − v(cc)(vv)
One-electron system
Wstatic
(vc)(vc)
⤹
⤹
(ωstatic− v(vc)(vc) + v(cc)(vv))
2>> 4"[v(cc)(vc)v(vc)(vv)]
[
∆εvc + v(vc)(vc) − W(vc)(vc)(ω)]
A = ωA
W(vc)(vc)(ω) = v(cc)(vv) +![v(cc)(vc)v(vc)(cc)]
ω − ωstatic − ∆εvc
A simple model II.
Eigenvalue eq.
ωstatic = ∆εvc + v(vc)(vc) − W static
(vc)(vc), Astatic = 1
ω1 = ∆εvc + v(vc)(vc) − v(cc)(vv)
One-electron system
Wstatic
(vc)(vc)
⤹
⤹
ω2 = 2∆εvc + v(vc)(vc) − Wstatic
(vc)(vc)
(ωstatic− v(vc)(vc) + v(cc)(vv))
2>> 4"[v(cc)(vc)v(vc)(vv)]
[
∆εvc + v(vc)(vc) − W(vc)(vc)(ω)]
A = ωA
W(vc)(vc)(ω) = v(cc)(vv) +![v(cc)(vc)v(vc)(cc)]
ω − ωstatic − ∆εvc
A simple model II.
Eigenvalue eq.
ωstatic = ∆εvc + v(vc)(vc) − W static
(vc)(vc), Astatic = 1
ω1 = ∆εvc + v(vc)(vc) − v(cc)(vv) ✗
One-electron system
Wstatic
(vc)(vc)
Self-screening (?)
⤹
⤹
ω2 = 2∆εvc + v(vc)(vc) − Wstatic
(vc)(vc)
(ωstatic− v(vc)(vc) + v(cc)(vv))
2>> 4"[v(cc)(vc)v(vc)(vv)]
[
∆εvc + v(vc)(vc) − W(vc)(vc)(ω)]
A = ωA
W(vc)(vc)(ω) = v(cc)(vv) +![v(cc)(vc)v(vc)(cc)]
ω − ωstatic − ∆εvc
Self-screening extra poles???Self-screening problem
[i∂t1− h(1)]G(1, 2) −
∫d3Σ(1, 3)G(3, 2) = δ(1, 2) h(1) = −∇
2/2 + U(1) + vH(1)
Self-screening extra poles???Self-screening problem
[i∂t1− h(1)]G(1, 2) −
∫d3Σ(1, 3)G(3, 2) = δ(1, 2) h(1) = −∇
2/2 + U(1) + vH(1)
Σ(1, 2) = iG(1, 2)v(1+, 2) = −
occ∑
n
φn(x1)φ∗
n(x2)v(x1, x2)HF→
Self-screening extra poles???Self-screening problem
[i∂t1− h(1)]G(1, 2) −
∫d3Σ(1, 3)G(3, 2) = δ(1, 2) h(1) = −∇
2/2 + U(1) + vH(1)
Σ(1, 2) = iG(1, 2)v(1+, 2) = −
occ∑
n
φn(x1)φ∗
n(x2)v(x1, x2)
⤹HF→
[
i∂t1+ ∇
2/2 − U(1)]
G(1, 2)−
∫
dx3v(x1, x3) [φ(x3)φ∗(x3) − φ(x3)φ
∗(x3)]G(1, 2) = δ(1, 2)
Self-screening extra poles???Self-screening problem
Σ = i [Gv + G(W − v)]
[i∂t1− h(1)]G(1, 2) −
∫d3Σ(1, 3)G(3, 2) = δ(1, 2) h(1) = −∇
2/2 + U(1) + vH(1)
GW→
Self-screening extra poles???Self-screening problem
Self-screening
Σ = i [Gv + G(W − v)]
[i∂t1− h(1)]G(1, 2) −
∫d3Σ(1, 3)G(3, 2) = δ(1, 2) h(1) = −∇
2/2 + U(1) + vH(1)
GW→
Self-screening extra poles???Second order self-energy and kernel*
+ + +
1
⤹
* See Davide Sangalli’s poster: Hunting double excitations
K = v + i∂Σ
∂G
Σ = + +
1
K = + + +
1
Self-screening extra poles???Second order self-energy and kernel*
+ + +
1
⤹
* See Davide Sangalli’s poster: Hunting double excitations
K = v + i∂Σ
∂G
Σ = + +
1
K = + + +
1
ω2 = ∆εvc − v(cc)(vv)
ω1 = ∆εvc + 2v(vc)(vc) − v(cc)(vv)Singlets ✴two-electron system
Tripletω3 = 2∆εvc
Self-screening extra poles???Second order self-energy and kernel*
* See Davide Sangalli’s poster: Hunting double excitations
+ + +
1
⤹K = v + i∂Σ
∂G
Σ = + +
1
K = + + +
1
Self-screening extra poles???Second order self-energy and kernel*
* See Davide Sangalli’s poster: Hunting double excitations
⤹
No extra poles WHY?
+ + +
1
⤹K = v + i∂Σ
∂G
Σ = + +
1
K = + + +
1
From BSE back to TDDFT
I. V. Tokatly, et al. Phys. Rev. B 65, 113107 (2002); F. Bruneval, at al. Phys. Rev. Lett. 94, 186402, (2005)
The frequency-dependent xc kernel
From BSE back to TDDFT
I. V. Tokatly, et al. Phys. Rev. B 65, 113107 (2002); F. Bruneval, at al. Phys. Rev. Lett. 94, 186402, (2005)
The frequency-dependent xc kernel
−iG(1, 1+) = ρ(1)✴ Link GF-DFT
From BSE back to TDDFT
I. V. Tokatly, et al. Phys. Rev. B 65, 113107 (2002); F. Bruneval, at al. Phys. Rev. Lett. 94, 186402, (2005)
The frequency-dependent xc kernel
fxc(1, 2) = f (1)xc
(1, 2) + f (2)xc
(1, 2)
f (1)xc (1, 2) = χ−1
KS(1, 2) − χ−1
0 (1, 2)
f (2)xc
(1, 2) = −i
∫d3d4d5χ−1
0 (1, 5)G(5, 3)G(4, 5)δΣ(3, 4)
δρ(2)
−iG(1, 1+) = ρ(1)✴ Link GF-DFT
⤹
From BSE back to TDDFT
M. Gatti, et al. Phys. Rev. Lett. 99, 057401, (2007)
The frequency-dependent xc kernel
L(1, 2, 1′, 2′) = L0(1, 2, 1′, 2′) +
∫d3d4d5d6L0(1, 4, 1′, 3)Ξ(3, 5, 4, 6)L(6, 2, 5, 2′)
4χ(1, 2, 1′, 2′) = L0(1, 2, 1′, 2′)+
∫d3d4d5d6L0(1, 4, 1′, 3)δ(3, 4)δ(5, 6)fxc(3, 6)4χ(6, 2, 5, 2′)
✴ BSE
✴ TDDFT
From BSE back to TDDFT
M. Gatti, et al. Phys. Rev. Lett. 99, 057401, (2007)
The frequency-dependent xc kernel
L(1, 2, 1′, 2′) = L0(1, 2, 1′, 2′) +
∫d3d4d5d6L0(1, 4, 1′, 3)Ξ(3, 5, 4, 6)L(6, 2, 5, 2′)
4χ(1, 2, 1′, 2′) = L0(1, 2, 1′, 2′)+
∫d3d4d5d6L0(1, 4, 1′, 3)δ(3, 4)δ(5, 6)fxc(3, 6)4χ(6, 2, 5, 2′)
⤹
fxc(1, 2) =
∫d3d4d5d6d7d8χ−1(1, 7)4χ(7, 4, 7, 3)Ξ(3, 5, 4, 6)L(6, 8, 5, 8)χ−1(8, 2)
✴ BSE
✴ TDDFT
χ(1, 2) = 4χ(1, 2, 1, 2) = L(1, 2, 1, 2)
From BSE back to TDDFT
M. Gatti, et al. Phys. Rev. Lett. 99, 057401, (2007)
The frequency-dependent xc kernel
L(1, 2, 1′, 2′) = L0(1, 2, 1′, 2′) +
∫d3d4d5d6L0(1, 4, 1′, 3)Ξ(3, 5, 4, 6)L(6, 2, 5, 2′)
4χ(1, 2, 1′, 2′) = L0(1, 2, 1′, 2′)+
∫d3d4d5d6L0(1, 4, 1′, 3)δ(3, 4)δ(5, 6)fxc(3, 6)4χ(6, 2, 5, 2′)
⤹
χ, L → L0
⤹
fxc(1, 2) =
∫d3d4d5d6d7d8χ−1(1, 7)4χ(7, 4, 7, 3)Ξ(3, 5, 4, 6)L(6, 8, 5, 8)χ−1(8, 2)
✴ BSE
✴ TDDFT
χ(1, 2) = 4χ(1, 2, 1, 2) = L(1, 2, 1, 2)
fxc(1, 2) = −
∫d3d4d5d6χ−1
0(1, 5)L0(5, 4, 5, 3)W (3, 4)L0(3, 6, 4, 6)χ−1
0(6, 2)
From BSE back to TDDFTThe frequency-dependent xc kernel
P. Romaniello et al, to be submitted
fxc(x1, x2,ω) = −
∫
dx3dx4dx5dx6χ−1
0(x1, x5,ω)
[∫
dω′dω′′
(2π)2L0(x5, x4, x5, x3,ω,ω′,ω′′)×
∫
dω′′′dω
(2π)2W (x3, x4,ω
′− ω)L0(x3, x6, x4, x6,ω, ω, ω′′′)
]
χ−1
0(x6, x2, ω)
From BSE back to TDDFTThe frequency-dependent xc kernel
P. Romaniello et al, to be submitted
fxc(x1, x2,ω) = −
∫
dx3dx4dx5dx6χ−1
0(x1, x5,ω)
[∫
dω′dω′′
(2π)2L0(x5, x4, x5, x3,ω,ω′,ω′′)×
∫
dω′′′dω
(2π)2W (x3, x4,ω
′− ω)L0(x3, x6, x4, x6,ω, ω, ω′′′)
]
χ−1
0(x6, x2, ω)
W (x1, x2,ω) = v(x1, x2)+∑
λ
∫
dx1′dx2′v(x1, x1′)
{
∑
vc,v′c′ A(vc),staticλ φ∗
v(x1′)φc(x1′)φv′(x2′)φ∗
c′(x2′)A∗(v′c′),staticλ
ω − ωstaticλ + iη
−
∑
cv,c′v′ A(cv),staticλ φ∗
c(x1′)φv(x1′)φc′(x2′)φ∗
v′(x2′)A∗(c′v′),staticλ
ω + ωstaticλ − iη
}
v(x2′ , x2)
⤹
fxc(x1, x2,ω) = −
∑vc,v′c′
∫dx3dx4dx5dx6χ
−1
0(x1, x5, ω)Lvc
0 (x5, x4, x5, x3,ω)
W vc
v′c′(x3, x4,ω)Lv
′c′
0 (x3, x6, x4, x6,ω)χ−1
0(x6, x2,ω)
Solutions (?)Vertex corrections
F. Bruneval, at al. Phys. Rev. Lett. 94, 186402, (2005)
Γ(1, 2; 3) = δ(13)δ(23) +δΣ(1, 2)
δV (3)
Solutions (?)Vertex corrections
⤹
F. Bruneval, at al. Phys. Rev. Lett. 94, 186402, (2005)
±f (2)xc
χ δΣ
δV=
δΣ
δρχ
Γ(1, 2; 3) = δ(13)δ(23) +δΣ(1, 2)
δV (3)
Γ(1, 2; 3) = δ(13)δ(23) + δ(1, 2)f (2)xc
(1, 4)χ(4, 3) + ∆Γ(1, 2; 3)
∆Γ(1, 2; 3) =
(
δΣ(1, 2)
δρ(4)− δ(1, 2)f (2)
xc(1, 4)
)
χ(4, 3)
Solutions (?)Vertex corrections
⤹∫
d3Σ(1, 3)G(3, 2) = −vH(1)G(1, 2)
fxc = f (1)xc
+ f (2)xcΓ(1, 2) = δ(12) + fxc(1, 4)χ(4, 2)
χ = GGΓ
Σ = GW Γ
Self-screening free
Solutions (?)Vertex corrections
⤹∫
d3Σ(1, 3)G(3, 2) = −vH(1)G(1, 2)
fxc = f (1)xc
+ f (2)xcΓ(1, 2) = δ(12) + fxc(1, 4)χ(4, 2)
χ = GGΓ
Σ = GW Γ
Self-screening free
∆Γ2-point