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Time Dependent Density Functional TheoryAn Introduction
Francesco Sottile
Laboratoire des Solides IrradiesEcole Polytechnique, Palaiseau - France
European Theoretical Spectroscopy Facility (ETSF)
Belfast, 29 Jun 2007
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Outline
1 Introduction: why TD-DFT ?
2 (Just) A bit of FormalismTDDFT: the FoundationLinear Response Formalism
3 TDDFT in practice:The ALDA: Achievements and Shortcomings
4 Resources
Time Dependent Density Functional Theory Francesco Sottile
Page 3
Intro Formalism Results Resources
Outline
1 Introduction: why TD-DFT ?
2 (Just) A bit of FormalismTDDFT: the FoundationLinear Response Formalism
3 TDDFT in practice:The ALDA: Achievements and Shortcomings
4 Resources
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Density Functional ... Why ?
Basic ideas of DFT
1 Any observable of a quantumsystem can be obtained fromthe density of the system alone.
2 The density of aninteracting-particles system canbe calculated as the density ofan auxiliary system ofnon-interacting particles.
Importance of the density
Example: atom of Nitrogen (7 electron)
Ψ(r1, .., r7) 21 coordinates
10 entries/coordinate ⇒ 1021 entries8 bytes/entry ⇒ 8 · 1021 bytes
4.7× 109 bytes/DVD ⇒ 2× 1012 DVDs
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Density Functional ... Why ?
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Density Functional ... Why ?
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Density Functional ... Why ?
Basic ideas of DFT
1 Any observable of a quantumsystem can be obtained fromthe density of the system alone.
2 The density of aninteracting-particles system canbe calculated as the density ofan auxiliary system ofnon-interacting particles.
Importance of the density
Example: atom of Carbon (6 electron)
Ψ(r1, .., r6) 18 coordinates
10 entries/coordinate ⇒ 1018 entries8 bytes/entry ⇒ 8 · 1018 bytes
5 · 109 bytes/DVD ⇒ 1013 DVDs
Importance of non-interacting
The Kohn-Sham one-particle equations
Hi (r)ψi (r) = εi (r)ψi (r)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Density Functional ... Why ?
Basic ideas of DFT
1 Any observable of a quantumsystem can be obtained fromthe density of the system alone.
2 The density of aninteracting-particles system canbe calculated as the density ofan auxiliary system ofnon-interacting particles.
Importance of the density
Example: atom of Carbon (6 electron)
Ψ(r1, .., r6) 18 coordinates
10 entries/coordinate ⇒ 1018 entries8 bytes/entry ⇒ 8 · 1018 bytes
5 · 109 bytes/DVD ⇒ 1013 DVDs
Importance of non-interacting
The Kohn-Sham one-particle equations
Hi (r)ψi (r) = εi (r)ψi (r)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Density Functional ... Successfull ?
S. Redner http://arxiv.org/abs/physics/0407137
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Time Dependent DFT ... Why ?
Large field of research concerned withmany-electron systems in time-dependent fields
Different Phenomena
absorption spectra
energy loss spectra
photo-ionization
high-harmonic generation
• photo-emission
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Time Dependent DFT ... Why ?
Large field of research concerned withmany-electron systems in time-dependent fields
Different Phenomena
absorption spectra
energy loss spectra
photo-ionization
high-harmonic generation
• photo-emission
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Time Dependent DFT ... Why ?
Large field of research concerned withmany-electron systems in time-dependent fields
Different Phenomena
absorption spectra
energy loss spectra
photo-ionization
high-harmonic generation
• photo-emission
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Time Dependent DFT ... Why ?
Large field of research concerned withmany-electron systems in time-dependent fields
Different Phenomena
absorption spectra
energy loss spectra
photo-ionization
high-harmonic generation
• photo-emission
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Time Dependent DFT ... Why ?
Large field of research concerned withmany-electron systems in time-dependent fields
Different Phenomena
absorption spectra
energy loss spectra
photo-ionization
high-harmonic generation
• photo-emission
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Time Dependent DFT ... Why ?
Large field of research concerned withmany-electron systems in time-dependent fields
Different Phenomena
absorption spectra
energy loss spectra
photo-ionization
high-harmonic generation
• photo-emission
ω
ω5
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Time Dependent DFT ... Why ?
Large field of research concerned withmany-electron systems in time-dependent fields
Different Phenomena
absorption spectra
energy loss spectra
photo-ionization
high-harmonic generation
• photo-emission
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Time Dependent DFT ... Why ?
We need a time dependent theory
⇓TDDFT is a promising candidate
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Outline
1 Introduction: why TD-DFT ?
2 (Just) A bit of FormalismTDDFT: the FoundationLinear Response Formalism
3 TDDFT in practice:The ALDA: Achievements and Shortcomings
4 Resources
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Outline
1 Introduction: why TD-DFT ?
2 (Just) A bit of FormalismTDDFT: the FoundationLinear Response Formalism
3 TDDFT in practice:The ALDA: Achievements and Shortcomings
4 Resources
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The name of the game: TDDFT
DFT TDDFTHohenberg-Kohn theorem 1
The ground-state expectationvalue of any physical observableof a many-electrons system is aunique functional of the electron
density n(r)⟨ϕ0
∣∣ O∣∣ϕ0
⟩= O[n]
P. Hohenberg and W. Kohn
Phys.Rev. 136, B864 (1964)
(Fermi, Slater)
Runge-Gross theorem
The expectation value of any physicaltime-dependent observable of a
many-electrons system is a uniquefunctional of the time-dependent electron
density n(r, t) and of the initial stateϕ0 = ϕ(t = 0)⟨
ϕ(t)|O(t)|ϕ(t)⟩
= O[n, ϕ0](t)
E. Runge and E.K.U. Gross
Phys.Rev.Lett. 52, 997 (1984)
(Ando,Zangwill and Soven)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The name of the game: TDDFT
DFT TDDFTHohenberg-Kohn theorem 1
The ground-state expectationvalue of any physical observableof a many-electrons system is aunique functional of the electron
density n(r)⟨ϕ0
∣∣ O∣∣ϕ0
⟩= O[n]
P. Hohenberg and W. Kohn
Phys.Rev. 136, B864 (1964)
(Fermi, Slater)
Runge-Gross theorem
The expectation value of any physicaltime-dependent observable of a
many-electrons system is a uniquefunctional of the time-dependent electron
density n(r, t) and of the initial stateϕ0 = ϕ(t = 0)⟨
ϕ(t)|O(t)|ϕ(t)⟩
= O[n, ϕ0](t)
E. Runge and E.K.U. Gross
Phys.Rev.Lett. 52, 997 (1984)
(Ando,Zangwill and Soven)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The name of the game: TDDFT
DFT TDDFTStatic problem
Second-order differentialequation
Boundary-value problem.
Hϕ(r1, .., rN) = Eϕ(r1, .., rN)
Time-dependent problem
First-order differential equationInitial-value problem
H(t)ϕ(r1, .., rN ; t) = ı~∂
∂tϕ(r1, .., rN ; t)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The name of the game: TDDFT
DFT TDDFTStatic problem
Second-order differentialequation
Boundary-value problem.
Hϕ(r1, .., rN) = Eϕ(r1, .., rN)
Time-dependent problem
First-order differential equationInitial-value problem
H(t)ϕ(r1, .., rN ; t) = ı~∂
∂tϕ(r1, .., rN ; t)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The name of the game: TDDFT
Runge-Gross theorem
The expectation value of any physicaltime-dependent observable of a
many-electrons system is a uniquefunctional of the time-dependent electron
density n(r, t) and of the initial stateϕ0 = ϕ(t = 0)⟨
ϕ(t)|O(t)|ϕ(t)⟩
= O[n, ϕ0](t)
E. Runge and E.K.U. Gross
Phys.Rev.Lett. 52, 997 (1984)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The name of the game: TDDFT
Runge-Gross theorem
Vext(r, t) 6= V ′ext(r, t) ⇐⇒ j(r, t) 6= j′(r, t)
∇ · [n∇Vext ] 6= ∇ · [n∇V ′ext ] ⇐⇒ n(r, t) 6= n′(r, t)
n(r, t) −→ Vext(r, t) + c(t) −→ ϕe ic(t)
⟨ϕ(t)|O(t)|ϕ(t)
⟩= O[n, ϕ0](t)
What about infinite systems?
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The name of the game: TDDFT
Runge-Gross theorem
Vext(r, t) 6= V ′ext(r, t) ⇐⇒ j(r, t) 6= j′(r, t)
∇ · [n∇Vext ] 6= ∇ · [n∇V ′ext ] ⇐⇒ n(r, t) 6= n′(r, t)
n(r, t) −→ Vext(r, t) + c(t) −→ ϕe ic(t)
⟨ϕ(t)|O(t)|ϕ(t)
⟩= O[n, ϕ0](t)
What about infinite systems?
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The name of the game: TDDFT
Runge-Gross theorem
Vext(r, t) 6= V ′ext(r, t) ⇐⇒ j(r, t) 6= j′(r, t)
∇ · [n∇Vext ] 6= ∇ · [n∇V ′ext ] ⇐⇒ n(r, t) 6= n′(r, t)
n(r, t) −→ Vext(r, t) + c(t) −→ ϕe ic(t)
⟨ϕ(t)|O(t)|ϕ(t)
⟩= O[n, ϕ0](t)
What about infinite systems?
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The name of the game: TDDFT
Runge-Gross theorem
Vext(r, t) 6= V ′ext(r, t) ⇐⇒ j(r, t) 6= j′(r, t)
∇ · [n∇Vext ] 6= ∇ · [n∇V ′ext ] ⇐⇒ n(r, t) 6= n′(r, t)
n(r, t) −→ Vext(r, t) + c(t) −→ ϕe ic(t)
⟨ϕ(t)|O(t)|ϕ(t)
⟩= O[n, ϕ0](t)
What about infinite systems?
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The name of the game: TDDFT
Runge-Gross theorem
Vext(r, t) 6= V ′ext(r, t) ⇐⇒ j(r, t) 6= j′(r, t)
∇ · [n∇Vext ] 6= ∇ · [n∇V ′ext ] ⇐⇒ n(r, t) 6= n′(r, t)
n(r, t) −→ Vext(r, t) + c(t) −→ ϕe ic(t)
⟨ϕ(t)|O(t)|ϕ(t)
⟩= O[n, ϕ0](t)
What about infinite systems?
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The name of the game: TDDFT
Runge-Gross theorem
Vext(r, t) 6= V ′ext(r, t) ⇐⇒ j(r, t) 6= j′(r, t)
∇ · [n∇Vext ] 6= ∇ · [n∇V ′ext ] ⇐⇒ n(r, t) 6= n′(r, t)
n(r, t) −→ Vext(r, t) + c(t) −→ ϕe ic(t)
⟨ϕ(t)|O(t)|ϕ(t)
⟩= O[n, ϕ0](t)
What about infinite systems?
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The name of the game: TDDFT
DFT TDDFTHohenberg-Kohn theorem 2
The total energy functional hasa minimum, the ground-state
energy E0, corresponding to theground-state density n0.
n
E[n]
n0
0E
Runge-Gross theorem - No minimum
Time-dependent Schrodinger eq. (initialcondition ϕ(t = 0) = ϕ0), corresponds to astationary point of the Hamiltonian action
A =
∫ t1
t0
dt 〈ϕ(t)| ı ∂
∂t− H(t) |ϕ(t)〉
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The name of the game: TDDFT
DFT TDDFTHohenberg-Kohn theorem 2
The total energy functional hasa minimum, the ground-state
energy E0, corresponding to theground-state density n0.
n
E[n]
n0
0E
Runge-Gross theorem - No minimum
Time-dependent Schrodinger eq. (initialcondition ϕ(t = 0) = ϕ0), corresponds to astationary point of the Hamiltonian action
A =
∫ t1
t0
dt 〈ϕ(t)| ı ∂
∂t− H(t) |ϕ(t)〉
tt 1t
0
H[t]
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The name of the game: TDDFT
DFT TDDFTKohn-Sham equations
»−
1
2· ∇2
i + Vtot (r)
–φi (r) = εi φi (r)
Vtot (r) = Vext (r)+
Zdr′v(r, r′)n(r′)+Vxc ([n], r)
Vxc ([n], r) =δExc [n]
δn(r)
Time-dependent Kohn-Sham equations
»−
1
2∇2 + Vtot (r, t)
–φi (r, t) = i
∂
∂tφi (r, t)
Vtot (r, t) = Vext (r, t) +
Zv(r, r′)n(r′, t)dr′ + Vxc ([n]r, t)
Vxc ([n], r, t) =δAxc [n]
δn(r, t)
Unknown exchange-correlationpotential.
Vxc functional of the density.
Unknown exchange-correlationtime-dependent potential.
Vxc functional of the density atall times and of the initial state.
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The name of the game: TDDFT
DFT TDDFTKohn-Sham equations
»−
1
2· ∇2
i + Vtot (r)
–φi (r) = εi φi (r)
Vtot (r) = Vext (r)+
Zdr′v(r, r′)n(r′)+Vxc ([n], r)
Vxc ([n], r) =δExc [n]
δn(r)
Time-dependent Kohn-Sham equations
»−
1
2∇2 + Vtot (r, t)
–φi (r, t) = i
∂
∂tφi (r, t)
Vtot (r, t) = Vext (r, t) +
Zv(r, r′)n(r′, t)dr′ + Vxc ([n]r, t)
Vxc ([n], r, t) =δAxc [n]
δn(r, t)
Unknown exchange-correlationpotential.
Vxc functional of the density.
Unknown exchange-correlationtime-dependent potential.
Vxc functional of the density atall times and of the initial state.
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The name of the game: TDDFT
Demonstrations, further readings, etc.
R. van LeeuwenInt.J.Mod.Phys. B15, 1969 (2001)
Vxc ([n], r, t) =δAxc [n]
δn(r, t)
δVxc ([n], r, t)
δn(r′, t ′)=
δ2Axc [n]
δn(r, t)δn(r′, t ′)
Causality-Symmetry dilemma
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The name of the game: TDDFT
Demonstrations, further readings, etc.
R. van LeeuwenInt.J.Mod.Phys. B15, 1969 (2001)
Vxc ([n], r, t) =δAxc [n]
δn(r, t)
δVxc ([n], r, t)
δn(r′, t ′)=
δ2Axc [n]
δn(r, t)δn(r′, t ′)
Causality-Symmetry dilemma
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The name of the game: TDDFT
Demonstrations, further readings, etc.
R. van LeeuwenInt.J.Mod.Phys. B15, 1969 (2001)
Vxc ([n], r, t) =δAxc [n]
δn(r, t)
δVxc ([n], r, t)
δn(r′, t ′)=
δ2Axc [n]
δn(r, t)δn(r′, t ′)
Causality-Symmetry dilemma
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The name of the game: TDDFT
Demonstrations, further readings, etc.
R. van LeeuwenInt.J.Mod.Phys. B15, 1969 (2001)
Vxc ([n], r, t) =δAxc [n]
δn(r, t)
δVxc ([n], r, t)
δn(r′, t ′)=
δ2Axc [n]
δn(r, t)δn(r′, t ′)
Causality-Symmetry dilemma
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The name of the game: TDDFT
DFT TDDFTHohenberg-Kohn theorem
The ground-state expectation valueof any physical observable of a
many-electrons system is a uniquefunctional of the electron density
n(r)Dϕ0
˛ bO ˛ϕ0
E= O[n]
Kohn-Sham equations
»−
1
2∇2
i + Vtot(r)
–φi (r) = εiφi (r)
Runge-Gross theorem
The expectation value of any physicaltime-dependent observable of a many-electrons
system is a unique functional of thetime-dependent electron density n(r) and of
the initial state ϕ0 = ϕ(t = 0)Dϕ(t)|bO(t)|ϕ(t)
E= O[n, ϕ0](t)
Kohn-Sham equations
»−
1
2∇2 + Vtot(r, t)
–φi (r, t) = i
∂
∂tφi (r, t)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
First Approach: Time Evolution of KS equations
[HKS(t)]φi (r, t) = i∂
∂tφi (r, t)
n(r, t) =occ∑i
|φi (r, t)|2
φ(t) = U(t, t0)φ(t0)
U(t, t0) = 1− i
∫ t
t0
dτH(τ)U(τ, t0)
A. Castro et al. J.Chem.Phys. 121, 3425 (2004)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
First Approach: Time Evolution of KS equations
[HKS(t)]φi (r, t) = i∂
∂tφi (r, t)
n(r, t) =occ∑i
|φi (r, t)|2
φ(t) = U(t, t0)φ(t0)
U(t, t0) = 1− i
∫ t
t0
dτH(τ)U(τ, t0)
A. Castro et al. J.Chem.Phys. 121, 3425 (2004)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
First Approach: Time Evolution of KS equations
[HKS(t)]φi (r, t) = i∂
∂tφi (r, t)
n(r, t) =occ∑i
|φi (r, t)|2
φ(t) = U(t, t0)φ(t0)
U(t, t0) = 1− i
∫ t
t0
dτH(τ)U(τ, t0)
A. Castro et al. J.Chem.Phys. 121, 3425 (2004)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
First Approach: Time Evolution of KS equations
Photo-absorption cross section σ
σ(ω) =4πω
cImα(ω)
α(t) = −∫
drVext(r, t)n(r, t)
in dipole approximation (λ ≫ dimension of the system)
σzz(ω) = −4πω
cIm α(ω) = −4πω
cIm
∫dr z n(r, ω)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
First Approach: Time Evolution of KS equations
Photo-absorption cross section σ: porphyrin
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
First Approach: Time Evolution of KS equations
Other observables
Multipoles
Mlm(t) =
∫drr lYlm(r)n(r, t)
Angular momentum
Lz(t) = −∑
i
∫drφi (r, t) ı (r ×∇)z φi (r, t)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
First Approach: Time Evolution of KS equations
Advantages
Direct application of KS equations
Advantageous scaling
Optimal scheme for finite systems
All orders automatically included
Shortcomings
Difficulties in approximating the Vxc [n](r, t) functional of thehistory of the density
Real space not necessarily suitable for solids
Does not explicitly take into account a “small” perturbation.Interesting quantities (excitation energies) are contained inthe linear response function!
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Outline
1 Introduction: why TD-DFT ?
2 (Just) A bit of FormalismTDDFT: the FoundationLinear Response Formalism
3 TDDFT in practice:The ALDA: Achievements and Shortcomings
4 Resources
Time Dependent Density Functional Theory Francesco Sottile
Page 48
Intro Formalism Results Resources
Linear Response Approach
Polarizability
interacting system δn = χδVext
non-interacting system δnn−i = χ0δVtot
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
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)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
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
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
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
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
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
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
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
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
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
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
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
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
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
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
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
Time Dependent Density Functional Theory Francesco Sottile
Page 58
Intro Formalism Results Resources
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
Time Dependent Density Functional Theory Francesco Sottile
Page 59
Intro Formalism Results Resources
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
Time Dependent Density Functional Theory Francesco Sottile
Page 60
Intro Formalism Results Resources
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
Time Dependent Density Functional Theory Francesco Sottile
Page 61
Intro Formalism Results Resources
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
Time Dependent Density Functional Theory Francesco Sottile
Page 62
Intro Formalism Results Resources
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
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Finite systems
Photo-absorption cross spectrum in Linear Response
σ(ω) =4πω
cImα(ω)
α(ω) = −∫
drdr′Vext(r, ω)δn(r′, ω)
σzz(ω) = −4πω
cIm
∫drdr′zχ(r, r′, ω)z′
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Finite systems
Photo-absorption cross spectrum in Linear Response
σ(ω) =4πω
cImα(ω)
α(ω) = −∫
drdr′Vext(r, ω)χ(r, r′, ω)Vext(r′, ω)
σzz(ω) = −4πω
cIm
∫drdr′zχ(r, r′, ω)z′
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Finite systems
Photo-absorption cross spectrum in Linear Response
σ(ω) =4πω
cImα(ω)
α(ω) = −∫
drdr′Vext(r, ω)χ(r, r′, ω)Vext(r′, ω)
σzz(ω) = −4πω
cIm
∫drdr′zχ(r, r′, ω)z′
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Solids
Reciprocal space
χ0(r, r′, ω) −→ χ0GG′(q, ω)
G =reciprocal lattice vectorq =momentum transfer of the perturbation
S.L.Adler, Phys.Rev 126, 413 (1962); N.Wiser Phys.Rev 129, 62 (1963)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Solids
Reciprocal space
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φ∗ck+q
⟩ ⟨φck+q|e−ı(q+G′)r′|φ∗vk
⟩ω − (εck+q − εvk) + ıη
i
unoccupied states
occupied states
j
S.L.Adler, Phys.Rev 126, 413 (1962); N.Wiser Phys.Rev 129, 62 (1963)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Solids
Reciprocal space
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φ∗ck+q
⟩ ⟨φck+q|e−ı(q+G′)r′|φ∗vk
⟩ω − (εck+q − εvk) + ıη
χGG′(q, ω) = χ0 + χ0 (v + fxc) χ
ε−1GG′(q, ω) = δGG′ + vG(q)χGG′(q, ω)
S.L.Adler, Phys.Rev 126, 413 (1962); N.Wiser Phys.Rev 129, 62 (1963)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Solids
Reciprocal space
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φ∗ck+q
⟩ ⟨φck+q|e−ı(q+G′)r′|φ∗vk
⟩ω − (εck+q − εvk) + ıη
χGG′(q, ω) = χ0 + χ0 (v + fxc) χ
ε−1GG′(q, ω) = δGG′ + vG(q)χGG′(q, ω)
ELS(q, ω) = −Im{ε−100 (q, ω)
}; Abs(ω) = lim
q→0Im
{1
ε−100 (q, ω)
}S.L.Adler, Phys.Rev 126, 413 (1962); N.Wiser Phys.Rev 129, 62 (1963)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Solids
Reciprocal space
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φ∗ck+q
⟩ ⟨φck+q|e−ı(q+G′)r′|φ∗vk
⟩ω − (εck+q − εvk) + ıη
χGG′(q, ω) = χ0 + χ0 (v + fxc) χ
ε−1GG′(q, ω) = δGG′ + vG(q)χGG′(q, ω)
ELS(ω) =− limq→0
Im{ε−100 (q, ω)
}; Abs(ω) = lim
q→0Im
{1
ε−100 (q, ω)
}S.L.Adler, Phys.Rev 126, 413 (1962); N.Wiser Phys.Rev 129, 62 (1963)
Time Dependent Density Functional Theory Francesco Sottile
Page 71
Intro Formalism Results Resources
Solids
Absorption and Energy Loss Spectra q → 0
ELS(ω) = −Im{ε−100 (ω)
}; Abs(ω) = Im
{1
ε−100 (ω)
}
ELS(ω) = −v0 Im{χ00(ω)
}; Abs(ω) = −v0 Im
{χ00(ω)
}
χ = χ0 + χ0 (v + fxc) χ
χ = χ0 + χ0 (v + fxc) χ
vG =
{vG ∀G 6= 00 G = 0
Exercise
Im
{1
ε−100
}= −v0Im
{χ00
}
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Solids
Absorption and Energy Loss Spectra q → 0
ELS(ω) = −Im{ε−100 (ω)
}; Abs(ω) = Im
{1
ε−100 (ω)
}ε−100 (ω) = 1 + v0χ00(ω)
ELS(ω) = −v0 Im{χ00(ω)
}; Abs(ω) = −v0 Im
{χ00(ω)
}
χ = χ0 + χ0 (v + fxc) χ
χ = χ0 + χ0 (v + fxc) χ
vG =
{vG ∀G 6= 00 G = 0
Exercise
Im
{1
ε−100
}= −v0Im
{χ00
}
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Solids
Absorption and Energy Loss Spectra q → 0
ELS(ω) = −Im{ε−100 (ω)
}; Abs(ω) = Im
{1
ε−100 (ω)
}ELS(ω) = −v0Im
{χ00(ω)
}; Abs(ω) = −v0Im
{1
1+v0χ00(ω)
}
ELS(ω) = −v0 Im{χ00(ω)
}; Abs(ω) = −v0 Im
{χ00(ω)
}
χ = χ0 + χ0 (v + fxc) χ
χ = χ0 + χ0 (v + fxc) χ
vG =
{vG ∀G 6= 00 G = 0
Exercise
Im
{1
ε−100
}= −v0Im
{χ00
}
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Solids
Absorption and Energy Loss Spectra q → 0
ELS(ω) = −Im{ε−100 (ω)
}; Abs(ω) = Im
{1
ε−100 (ω)
}ELS(ω) = −v0Im
{χ00(ω)
}; Abs(ω) = −v0Im
{1
1+v0χ00(ω)
}ELS(ω) = −v0 Im
{χ00(ω)
}; Abs(ω) = −v0 Im
{χ00(ω)
}
χ = χ0 + χ0 (v + fxc) χ
χ = χ0 + χ0 (v + fxc) χ
vG =
{vG ∀G 6= 00 G = 0
Exercise
Im
{1
ε−100
}= −v0Im
{χ00
}
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Solids
Absorption and Energy Loss Spectra q → 0
ELS(ω) = −Im{ε−100 (ω)
}; Abs(ω) = Im
{1
ε−100 (ω)
}ELS(ω) = −v0Im
{χ00(ω)
}; Abs(ω) = −v0Im
{1
1+v0χ00(ω)
}ELS(ω) = −v0 Im
{χ00(ω)
}; Abs(ω) = −v0 Im
{χ00(ω)
}
χ = χ0 + χ0 (v + fxc) χ
χ = χ0 + χ0 (v + fxc) χ
vG =
{vG ∀G 6= 00 G = 0
Exercise
Im
{1
ε−100
}= −v0Im
{χ00
}
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Solids
Absorption and Energy Loss Spectra q → 0
ELS(ω) = −Im{ε−100 (ω)
}; Abs(ω) = Im
{1
ε−100 (ω)
}ELS(ω) = −v0Im
{χ00(ω)
}; Abs(ω) = −v0Im
{1
1+v0χ00(ω)
}ELS(ω) = −v0 Im
{χ00(ω)
}; Abs(ω) = −v0 Im
{χ00(ω)
}
χ = χ0 + χ0 (v + fxc) χ
χ = χ0 + χ0 (v + fxc) χ
vG =
{vG ∀G 6= 00 G = 0
Exercise
Im
{1
ε−100
}= −v0Im
{χ00
}Time Dependent Density Functional Theory Francesco Sottile
Page 77
Intro Formalism Results Resources
Solids
Abs and ELS (q → 0) differs only by v0
ELS(ω) = −Im{ε−100 (ω)
}; Abs(ω) = Im
{1
ε−100 (ω)
}ELS(ω) = −v0 Im
{χ00(ω)
}; Abs(ω) = −v0 Im
{χ00(ω)
}χ = χ0 + χ0 (v + fxc) χ
χ = χ0 + χ0 (v + fxc) χ
vG =
{vG ∀G 6= 00 G = 0
microscopic components
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Solids
Microscopic components v
v = local field effects
χNLF = χ0 + χ0 (vX + fxc) χNLF
AbsNLF = −v0 Im{
χNLF}
AbsNLF = Im {ε00}
Abs = Im{
1ε−100
}Exercise
AbsNLF = −v0 Im{χNLF
}= Im {ε00}
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Solids
Microscopic components v
v = local field effects
χNLF = χ0 + χ0 (vX + fxc) χNLF
AbsNLF = −v0 Im{
χNLF}
AbsNLF = Im {ε00}
Abs = Im{
1ε−100
}Exercise
AbsNLF = −v0 Im{χNLF
}= Im {ε00}
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Solids
Microscopic components v
v = local field effects
χNLF = χ0 + χ0 (vX + fxc) χNLF
AbsNLF = −v0 Im{
χNLF}
AbsNLF = Im {ε00}
Abs = Im{
1ε−100
}Exercise
AbsNLF = −v0 Im{χNLF
}= Im {ε00}
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Solids
Microscopic components v
v = local field effects
χNLF = χ0 + χ0 (vX + fxc) χNLF
AbsNLF = −v0 Im{
χNLF}
AbsNLF = Im {ε00}
Abs = Im{
1ε−100
}
Exercise
AbsNLF = −v0 Im{χNLF
}= Im {ε00}
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Solids
Microscopic components v
v = local field effects
χNLF = χ0 + χ0 (vX + fxc) χNLF
AbsNLF = −v0 Im{
χNLF}
AbsNLF = Im {ε00}
Abs = Im{
1ε−100
}Exercise
AbsNLF = −v0 Im{χNLF
}= Im {ε00}
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
EELS of Graphite
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Absorption of Silicon
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Outline
1 Introduction: why TD-DFT ?
2 (Just) A bit of FormalismTDDFT: the FoundationLinear Response Formalism
3 TDDFT in practice:The ALDA: Achievements and Shortcomings
4 Resources
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
TDDFT in practice
Practical schema and approximations
Ground-state calculation → φi , εi [Vxc LDA]
χ0 (q, ω)
χ = χ0 + χ0 (v + fxc) χ
fxc = 0 RPA
f ALDAxc (r, r′) = δVxc (r)
δn(r′)δ(r − r′) ALDA
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Outline
1 Introduction: why TD-DFT ?
2 (Just) A bit of FormalismTDDFT: the FoundationLinear Response Formalism
3 TDDFT in practice:The ALDA: Achievements and Shortcomings
4 Resources
Time Dependent Density Functional Theory Francesco Sottile
Page 88
Intro Formalism Results Resources
ALDA: Achievements and Shortcomings
Electron Energy Loss Spectrum of Graphite
RPA vs EXP
χNLF = χ0 + χ0 v0 χNLF
χ = χ0 + χ0 v χ
ELS = −v0Im{χ00
}
A.Marinopoulos et al. Phys.Rev.Lett 89, 76402 (2002)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
ALDA: Achievements and Shortcomings
Photo-absorption cross section of Benzene
ALDA vs EXP
Abs=−4πωc Im
∫drzn(r, ω)
K.Yabana and G.F.Bertsch Int.J.Mod.Phys.75, 55 (1999)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
ALDA: Achievements and Shortcomings
Inelastic X-ray scattering of Silicon
ALDA vs RPA vs EXP
S(q, ω)=− ~2q2
4π2e2n Imε−100
Weissker et al., PRL 97, 237602 (2006)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
ALDA: Achievements and Shortcomings
Absorption Spectrum of Silicon
ALDA vs RPA vs EXP
χ = χ0 + χ0 (v + f ALDAxc )χ
Abs = −v0Im{χ00
}
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
ALDA: Achievements and Shortcomings
Absorption Spectrum of Argon
ALDA vs EXP
χ = χ0 + χ0 (v + f ALDAxc )χ
Abs = −v0Im{χ00
}
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
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
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
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
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
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
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
ALDA: Achievements and Shortcomings
Absorption of Silicon fxc = αq2
L.Reining et al. Phys.Rev.Lett. 88, 66404 (2002)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Time Dependent Density Functional Theory Francesco Sottile
Page 98
Intro Formalism Results Resources
Outline
1 Introduction: why TD-DFT ?
2 (Just) A bit of FormalismTDDFT: the FoundationLinear Response Formalism
3 TDDFT in practice:The ALDA: Achievements and Shortcomings
4 Resources
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
Resources
Codes (more or less) available for TDDFT
Octopus (Marques,Castro,Rubio) -(real space, real time) -
mainly finite systems - GPL
http://www.tddft.org/programs/octopus/
DP (Olevano,Reining,Sottile) - (reciprocal space, frequency
domain) - solides and finite systems - open source for academics
http://dp-code.org
Self (Marini) - (reciprocal space, frequency domain)
Fleszar code
Rehr (core excitations)
TDDFT (Bertsch)
VASP, SIESTA, ADF, TURBOMOLE
TD-DFPT (Baroni)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The DP code
dp - Dielectric Properties
Reciprocal space
Frequency domain
Planewave basis
Optical absorption
Loss Spectra (EELS,IXS)
Different approximations (RPA, ALDA, NLDA, MT, etc.)
Authors: Valerio Olevano, Lucia Reining, Francesco Sottile
Contributors:
Valerie Veniard, Eleonora Luppi non-linear
Lucia Caramella Spin
Silvana Botti kernel, Wannier functions
Margherita Marsili Mapping-Theory kernel
Christine Giorgetti metals
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The DP code
The algorithm
ε−1GG′(q, ω) = δGG′ + vG(q)χGG′(q, ω)
χGG′(q, ω) = χ0 + χ0 (v + fxc) χ
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φck+q
⟩ ⟨φck+q|e−ı(q+G′)r′ |φvk
⟩ω − (εck+q − εvk) + ıη⟨
φvk|eı(q+G)r|φck+q
⟩=
∫drφ∗vk(r)e
ı(q+G)rφck+q(r)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The DP code
The algorithm
ε−1GG′(q, ω) = δGG′ + vG(q)χGG′(q, ω)
χGG′(q, ω) = χ0 + χ0 (v + fxc) χ
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φck+q
⟩ ⟨φck+q|e−ı(q+G′)r′ |φvk
⟩ω − (εck+q − εvk) + ıη⟨
φvk|eı(q+G)r|φck+q
⟩=
∫drφ∗vk(r)e
ı(q+G)rφck+q(r)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The DP code
The algorithm
ε−1GG′(q, ω) = δGG′ + vG(q)χGG′(q, ω)
χGG′(q, ω) = χ0 + χ0 (v + fxc) χ
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φck+q
⟩ ⟨φck+q|e−ı(q+G′)r′ |φvk
⟩ω − (εck+q − εvk) + ıη
⟨φvk|eı(q+G)r|φck+q
⟩=
∫drφ∗vk(r)e
ı(q+G)rφck+q(r)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The DP code
The algorithm
ε−1GG′(q, ω) = δGG′ + vG(q)χGG′(q, ω)
χGG′(q, ω) = χ0 + χ0 (v + fxc) χ
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φck+q
⟩ ⟨φck+q|e−ı(q+G′)r′ |φvk
⟩ω − (εck+q − εvk) + ıη⟨
φvk|eı(q+G)r|φck+q
⟩=
∫drφ∗vk(r)e
ı(q+G)rφck+q(r)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The DP code
The algorithm
ε−1GG′(q, ω) = δGG′ + vG(q)χGG′(q, ω)
χGG′(q, ω) = χ0 + χ0 (v + fxc) χ
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φck+q
⟩ ⟨φck+q|e−ı(q+G′)r′ |φvk
⟩ω − (εck+q − εvk) + ıη⟨
φvk|eı(q+G)r|φck+q
⟩=
∫drφ∗vk(r)e
ı(q+G)rφck+q(r)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The DP code
The algorithm
ε−1GG′(q, ω) = δGG′ + vG(q)χGG′(q, ω)
χGG′(q, ω) = χ0 + χ0 (v + fxc) χ
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φck+q
⟩ ⟨φck+q|e−ı(q+G′)r′ |φvk
⟩ω − (εck+q − εvk) + ıη⟨
φvk|eı(q+G)r|φck+q
⟩=
∫drφ∗vk(r)e
ı(q+G)rφck+q(r)
Time Dependent Density Functional Theory Francesco Sottile
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Intro Formalism Results Resources
The DP code
The algorithm
ε−1GG′(q, ω) = δGG′ + vG(q)χGG′(q, ω)
χGG′(q, ω) = χ0 + χ0 (v + fxc) χ
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φck+q
⟩ ⟨φck+q|e−ı(q+G′)r′ |φvk
⟩ω − (εck+q − εvk) + ıη⟨
φvk|eı(q+G)r|φck+q
⟩=
∫drφ∗vk(r)e
ı(q+G)rφck+q(r)
Time Dependent Density Functional Theory Francesco Sottile
Page 108
Intro Formalism Results Resources
The DP code
The algorithm
ε−1GG′(q, ω) = δGG′ + vG(q)χGG′(q, ω)
χGG′(q, ω) = χ0 + χ0 (v + fxc) χ
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φck+q
⟩ ⟨φck+q|e−ı(q+G′)r′ |φvk
⟩ω − (εck+q − εvk) + ıη⟨
φvk|eı(q+G)r|φck+q
⟩=
∫drφ∗vk(r)e
ı(q+G)rφck+q(r)
Time Dependent Density Functional Theory Francesco Sottile
Page 109
Intro Formalism Results Resources
The DP code
The algorithm
ε−1GG′(q, ω) = δGG′ + vG(q)χGG′(q, ω)
χGG′(q, ω) = χ0 + χ0 (v + fxc) χ
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φck+q
⟩ ⟨φck+q|e−ı(q+G′)r′ |φvk
⟩ω − (εck+q − εvk) + ıη⟨
φvk|eı(q+G)r|φck+q
⟩=
∫drφ∗vk(r)e
ı(q+G)rφck+q(r)
Time Dependent Density Functional Theory Francesco Sottile
Page 110
Intro Formalism Results Resources
The DP code
The algorithm
ε−1GG′(q, ω) = δGG′ + vG(q)χGG′(q, ω)
χGG′(q, ω) = χ0 + χ0 (v + fxc) χ
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φck+q
⟩ ⟨φck+q|e−ı(q+G′)r′ |φvk
⟩ω − (εck+q − εvk) + ıη⟨
φvk|eı(q+G)r|φck+q
⟩=
∫drφ∗vk(r)e
ı(q+G)rφck+q(r)
Time Dependent Density Functional Theory Francesco Sottile
Page 111
Intro Formalism Results Resources
TDDFT vs BSE scaling
Scaling of TDDFT (DP)
χGG′(q, ω)=χ0GG′(q, ω)+χ0
GG′′(q, ω) [vG′′(q)+f xcG′′G′′′(q, ω)]χG′′′G′(q, ω)
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φ∗ck+q
⟩ ⟨φck+q|e−ı(q+G′)r′ |φ∗vk
⟩ω − (εck+q − εvk) + ıη
NtNGNGNω ; NtNr lnNrNω
Scaling . N4at
Time Dependent Density Functional Theory Francesco Sottile
Page 112
Intro Formalism Results Resources
TDDFT vs BSE scaling
Scaling of TDDFT (DP)
χGG′(q, ω)=χ0GG′(q, ω)+χ0
GG′′(q, ω) [vG′′(q)+f xcG′′G′′′(q, ω)]χG′′′G′(q, ω)
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φ∗ck+q
⟩ ⟨φck+q|e−ı(q+G′)r′ |φ∗vk
⟩ω − (εck+q − εvk) + ıη
NtNGNGNω ; NtNr lnNrNω
Scaling . N4at
Time Dependent Density Functional Theory Francesco Sottile
Page 113
Intro Formalism Results Resources
TDDFT vs BSE scaling
Scaling of TDDFT (DP)
χGG′(q, ω)=χ0GG′(q, ω)+χ0
GG′′(q, ω) [vG′′(q)+f xcG′′G′′′(q, ω)]χG′′′G′(q, ω)
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φ∗ck+q
⟩ ⟨φck+q|e−ı(q+G′)r′ |φ∗vk
⟩ω − (εck+q − εvk) + ıη
Nt
NGNGNω ; NtNr lnNrNω
Scaling . N4at
Time Dependent Density Functional Theory Francesco Sottile
Page 114
Intro Formalism Results Resources
TDDFT vs BSE scaling
Scaling of TDDFT (DP)
χGG′(q, ω)=χ0GG′(q, ω)+χ0
GG′′(q, ω) [vG′′(q)+f xcG′′G′′′(q, ω)]χG′′′G′(q, ω)
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φ∗ck+q
⟩ ⟨φck+q|e−ı(q+G′)r′ |φ∗vk
⟩ω − (εck+q − εvk) + ıη
NtNG
NGNω ; NtNr lnNrNω
Scaling . N4at
Time Dependent Density Functional Theory Francesco Sottile
Page 115
Intro Formalism Results Resources
TDDFT vs BSE scaling
Scaling of TDDFT (DP)
χGG′(q, ω)=χ0GG′(q, ω)+χ0
GG′′(q, ω) [vG′′(q)+f xcG′′G′′′(q, ω)]χG′′′G′(q, ω)
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φ∗ck+q
⟩ ⟨φck+q|e−ı(q+G′)r′ |φ∗vk
⟩ω − (εck+q − εvk) + ıη
NtNGNG
Nω ; NtNr lnNrNω
Scaling . N4at
Time Dependent Density Functional Theory Francesco Sottile
Page 116
Intro Formalism Results Resources
TDDFT vs BSE scaling
Scaling of TDDFT (DP)
χGG′(q, ω)=χ0GG′(q, ω)+χ0
GG′′(q, ω) [vG′′(q)+f xcG′′G′′′(q, ω)]χG′′′G′(q, ω)
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φ∗ck+q
⟩ ⟨φck+q|e−ı(q+G′)r′ |φ∗vk
⟩ω − (εck+q − εvk) + ıη
NtNGNGNω
; NtNr lnNrNω
Scaling . N4at
Time Dependent Density Functional Theory Francesco Sottile
Page 117
Intro Formalism Results Resources
TDDFT vs BSE scaling
Scaling of TDDFT (DP)
χGG′(q, ω)=χ0GG′(q, ω)+χ0
GG′′(q, ω) [vG′′(q)+f xcG′′G′′′(q, ω)]χG′′′G′(q, ω)
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φ∗ck+q
⟩ ⟨φck+q|e−ı(q+G′)r′ |φ∗vk
⟩ω − (εck+q − εvk) + ıη
NtNGNGNω ; NtNr lnNrNω
Scaling . N4at
Time Dependent Density Functional Theory Francesco Sottile
Page 118
Intro Formalism Results Resources
TDDFT vs BSE scaling
Scaling of TDDFT (DP)
χGG′(q, ω)=χ0GG′(q, ω)+χ0
GG′′(q, ω) [vG′′(q)+f xcG′′G′′′(q, ω)]χG′′′G′(q, ω)
χ0GG′(q, ω) =
∑vck
⟨φvk|eı(q+G)r|φ∗ck+q
⟩ ⟨φck+q|e−ı(q+G′)r′ |φ∗vk
⟩ω − (εck+q − εvk) + ıη
NtNGNGNω ; NtNr lnNrNω
Scaling . N4at
Time Dependent Density Functional Theory Francesco Sottile
Page 119
Intro Formalism Results Resources
TDDFT vs BSE scaling
Scaling of BSE (EXC)
Hv ′c ′k′
vck
Av ′c ′k′
λ = AλAvckλ
Scaling N2t Nr . N5
at
Scaling N3t . N6
at
Time Dependent Density Functional Theory Francesco Sottile
Page 120
Intro Formalism Results Resources
TDDFT vs BSE scaling
Scaling of BSE (EXC)
Hv ′c ′k′
vck Av ′c ′k′
λ = AλAvckλ
Scaling N2t Nr . N5
at
Scaling N3t . N6
at
Time Dependent Density Functional Theory Francesco Sottile
Page 121
Intro Formalism Results Resources
TDDFT
Some References
van Leeuwen, Int. J. Mod. Phys. B15, 1969 (2001)
Gross and Kohn, Adv. Quantum Chem. 21, 255 (1990)
Marques and Gross, Annu. Rev. Phys. Chem. 55, 427 (2004)
Botti et al. Rep. Prog. Phys. 70, 357 (2007)
Marques et al. eds. Time Dependent Density FunctionalTheory, Springer (2006)
Time Dependent Density Functional Theory Francesco Sottile