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)


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)

⤷


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′)


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

( )



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

( )

( )



✴ 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′′)

]



[

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



[

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

⤷



[

v

∫

dω

2πL(ω, ω,ω′′) −

∫

dω

2πW (ω′

− ω)L(ω, ω, ω′′)

]



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′′)

]



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′′)

]



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


H2p,exc =

(


coupling(vc)(c′

v′)


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

)

]

⤹


H2p,exc =

(


coupling(vc)(c′

v′)


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)]



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

]


W = v + vχv

⤹


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)]


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

]


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)]


H2p,exc(n

1′n1)(n2n

2′)(ωλ)A(n2n

2′)

λ (ωλ) = ωλA(n

1′n1)

λ (ωλ)


⤹

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)]


+∑

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)⤹



A = ωA

A simple model I.

ωstatic


(vc)(vc), Astatic

1 =1√

2

(

1

1

)

ωstatic


(vc)(vc), Astatic

2 =1√

2

(

1

−1

)

ω1 = ∆εvc + 2v(vc)(vc) − v(cc)(vv)Singlets

Singlet

Triplet

double-excitation


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)]



A = ωA

W(vc)(vc)(ω) = v(cc)(vv) + 2![v(cc)(vc)v(vc)(cc)]

ω − ωstatic1 − ∆εvc

A simple model I.

ωstatic


(vc)(vc), Astatic

1 =1√

2

(

1

1

)

ωstatic


(vc)(vc), Astatic

2 =1√

2

(

1

−1

)

ω2 = ∆εvc − v(cc)(vv)


Triplets

Singlet

Triplet

double-excitation


Wstatic

(vc)(vc)⤹

⤹ω3 = 2∆εvc + 2v(vc)(vc) − W

static

(vc)(vc)


2>> 8"[v(cc)(vc)v(vc)(vv)]



A = ωA



A simple model I.

ωstatic


(vc)(vc), Astatic

1 =1√

2

(

1

1

)

ωstatic


(vc)(vc), Astatic

2 =1√

2

(

1

−1

)



Triplets

Singlet

Triplet

double-excitation

✗


Wstatic

(vc)(vc)⤹

⤹

ω4 = 2∆εvc + 2v(vc)(vc) − Wstatic

(vc)(vc)

ω3 = 2∆εvc + 2v(vc)(vc) − Wstatic

(vc)(vc)


2>> 8"[v(cc)(vc)v(vc)(vv)]



A = ωA



A simple model II.

Eigenvalue eq.

One-electron system[

∆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

One-electron system

Wstatic

(vc)(vc)⤹

[


A = ωA

A simple model II.

Eigenvalue eq.



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)]

[


A = ωA

W(vc)(vc)(ω) = v(cc)(vv) +![v(cc)(vc)v(vc)(cc)]

ω − ωstatic − ∆εvc

A simple model II.

Eigenvalue eq.



ω1 = ∆εvc + v(vc)(vc) − v(cc)(vv)

One-electron system

Wstatic

(vc)(vc)

⤹

⤹

ω2 = 2∆εvc + v(vc)(vc) − Wstatic

(vc)(vc)


2>> 4"[v(cc)(vc)v(vc)(vv)]

[


A = ωA



A simple model II.

Eigenvalue eq.



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)


2>> 4"[v(cc)(vc)v(vc)(vv)]

[


A = ωA



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)


[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− 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)


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

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


+ + +

1

⤹


K = v + i∂Σ

∂G

Σ = + +

1

K = + + +

1


ω1 = ∆εvc + 2v(vc)(vc) − v(cc)(vv)Singlets ✴two-electron system

Tripletω3 = 2∆εvc



+ + +

1

⤹K = v + i∂Σ

∂G

Σ = + +

1

K = + + +

1



⤹

No extra poles WHY?

+ + +

1

⤹K = v + i∂Σ

∂G

Σ = + +

1

K = + + +

1


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




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

⤹


M. Gatti, et al. Phys. Rev. Lett. 99, 057401, (2007)


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




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)




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)



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)


⤹


±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)


fxc = f (1)xc

+ f (2)xcΓ(1, 2) = δ(12) + fxc(1, 4)χ(4, 2)


⤹∫

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


⤹∫

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

A frequency-dependent kernel can create extra poles in the response function

creates double excitations...

...and spurious solutions... related to self-screening

Vertex corrections needed

W (ω)

Conclusions

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