An Introduction Laboratoire des Solides Irradiés Ecole Polytechnique, Palaiseau - France European Theoretical Spectroscopy Facility (ETSF) Belfast, 29 Jun 2007
Outline 1 Introduction: why TD-DFT? 2 (Just) A bit of Formalism TDDFT: the Foundation Linear Response Formalism 3 TDDFT in practice: The ALDA: Achievements and Shortcomings 4 Resources
Outline 1 Introduction: why TD-DFT? 2 (Just) A bit of Formalism TDDFT: the Foundation Linear Response Formalism 3 TDDFT in practice: The ALDA: Achievements and Shortcomings 4 Resources
Density Functional... Why? Basic ideas of DFT 1 Any observable of a quantum system can be obtained from the density of the system alone. 2 The density of an interacting-particles system can be calculated as the density of an auxiliary system of non-interacting particles. Importance of the density Example: atom of Nitrogen (7 electron) Ψ(r 1,.., r 7) 21 coordinates 10 entries/coordinate 10 21 entries 8 bytes/entry 8 10 21 bytes 4.7 10 9 bytes/dvd 2 10 12 DVDs
Density Functional... Why?
Density Functional... Why?
Density Functional... Why? Basic ideas of DFT 1 Any observable of a quantum system can be obtained from the density of the system alone. 2 The density of an interacting-particles system can be calculated as the density of an auxiliary system of non-interacting particles. Importance of the density Example: atom of Carbon (6 electron) Ψ(r 1,.., r 6) 18 coordinates 10 entries/coordinate 10 18 entries 8 bytes/entry 8 10 18 bytes 5 10 9 bytes/dvd 10 13 DVDs Importance of non-interacting The Kohn-Sham one-particle equations H i (r)ψ i (r) = ɛ i (r)ψ i (r)
Density Functional... Why? Basic ideas of DFT 1 Any observable of a quantum system can be obtained from the density of the system alone. 2 The density of an interacting-particles system can be calculated as the density of an auxiliary system of non-interacting particles. Importance of the density Example: atom of Carbon (6 electron) Ψ(r 1,.., r 6) 18 coordinates 10 entries/coordinate 10 18 entries 8 bytes/entry 8 10 18 bytes 5 10 9 bytes/dvd 10 13 DVDs Importance of non-interacting The Kohn-Sham one-particle equations H i (r)ψ i (r) = ɛ i (r)ψ i (r)
Density Functional... Successfull? S. Redner http://arxiv.org/abs/physics/0407137
Time Dependent DFT... Why? Large field of research concerned with many-electron systems in time-dependent fields Different Phenomena absorption spectra energy loss spectra photo-ionization high-harmonic generation photo-emission
Time Dependent DFT... Why? Large field of research concerned with many-electron systems in time-dependent fields Different Phenomena absorption spectra energy loss spectra photo-ionization high-harmonic generation photo-emission
Time Dependent DFT... Why? Large field of research concerned with many-electron systems in time-dependent fields Different Phenomena absorption spectra energy loss spectra photo-ionization high-harmonic generation photo-emission
Time Dependent DFT... Why? Large field of research concerned with many-electron systems in time-dependent fields Different Phenomena absorption spectra energy loss spectra photo-ionization high-harmonic generation photo-emission
Time Dependent DFT... Why? Large field of research concerned with many-electron systems in time-dependent fields Different Phenomena absorption spectra energy loss spectra photo-ionization high-harmonic generation photo-emission
Time Dependent DFT... Why? Large field of research concerned with many-electron systems in time-dependent fields Different Phenomena absorption spectra energy loss spectra photo-ionization high-harmonic generation photo-emission ω 5ω
Time Dependent DFT... Why? Large field of research concerned with many-electron systems in time-dependent fields Different Phenomena absorption spectra energy loss spectra photo-ionization high-harmonic generation photo-emission
Time Dependent DFT... Why? We need a time dependent theory TDDFT is a promising candidate
Outline 1 Introduction: why TD-DFT? 2 (Just) A bit of Formalism TDDFT: the Foundation Linear Response Formalism 3 TDDFT in practice: The ALDA: Achievements and Shortcomings 4 Resources
Outline 1 Introduction: why TD-DFT? 2 (Just) A bit of Formalism TDDFT: the Foundation Linear Response Formalism 3 TDDFT in practice: The ALDA: Achievements and Shortcomings 4 Resources
The name of the game: TDDFT DFT Hohenberg-Kohn theorem 1 The ground-state expectation value of any physical observable of a many-electrons system is a unique functional of the electron density n(r) ϕ 0 Ô ϕ 0 = O[n] P. Hohenberg and W. Kohn Phys.Rev. 136, B864 (1964) (Fermi, Slater) Runge-Gross theorem TDDFT The expectation value of any physical time-dependent observable of a many-electrons system is a unique functional of the time-dependent electron density n(r, t) and of the initial state ϕ 0 = ϕ(t = 0) ϕ(t) Ô(t) ϕ(t) = O[n, ϕ 0 ](t) E. Runge and E.K.U. Gross Phys.Rev.Lett. 52, 997 (1984) (Ando,Zangwill and Soven)
The name of the game: TDDFT DFT Hohenberg-Kohn theorem 1 The ground-state expectation value of any physical observable of a many-electrons system is a unique functional of the electron density n(r) ϕ 0 Ô ϕ 0 = O[n] P. Hohenberg and W. Kohn Phys.Rev. 136, B864 (1964) (Fermi, Slater) Runge-Gross theorem TDDFT The expectation value of any physical time-dependent observable of a many-electrons system is a unique functional of the time-dependent electron density n(r, t) and of the initial state ϕ 0 = ϕ(t = 0) ϕ(t) Ô(t) ϕ(t) = O[n, ϕ 0 ](t) E. Runge and E.K.U. Gross Phys.Rev.Lett. 52, 997 (1984) (Ando,Zangwill and Soven)
The name of the game: TDDFT Static problem DFT Second-order differential equation Boundary-value problem. Hϕ(r 1,.., r N ) = Eϕ(r 1,.., r N ) TDDFT Time-dependent problem First-order differential equation Initial-value problem H(t)ϕ(r 1,.., r N ; t) = ı t ϕ(r 1,.., r N ; t)
The name of the game: TDDFT Static problem DFT Second-order differential equation Boundary-value problem. Hϕ(r 1,.., r N ) = Eϕ(r 1,.., r N ) TDDFT Time-dependent problem First-order differential equation Initial-value problem H(t)ϕ(r 1,.., r N ; t) = ı t ϕ(r 1,.., r N ; t)
The name of the game: TDDFT Runge-Gross theorem The expectation value of any physical time-dependent observable of a many-electrons system is a unique functional of the time-dependent electron density n(r, t) and of the initial state ϕ 0 = ϕ(t = 0) ϕ(t) Ô(t) ϕ(t) = O[n, ϕ 0 ](t) E. Runge and E.K.U. Gross Phys.Rev.Lett. 52, 997 (1984)
The name of the game: TDDFT Runge-Gross theorem V ext (r, t) V ext(r, t) j(r, t) j (r, t) [n V ext ] [n V ext] n(r, t) n (r, t) n(r, t) V ext (r, t) + c(t) ϕe ic(t) ϕ(t) Ô(t) ϕ(t) = O[n, ϕ 0 ](t) What about infinite systems?
The name of the game: TDDFT Runge-Gross theorem V ext (r, t) V ext(r, t) j(r, t) j (r, t) [n V ext ] [n V ext] n(r, t) n (r, t) n(r, t) V ext (r, t) + c(t) ϕe ic(t) ϕ(t) Ô(t) ϕ(t) = O[n, ϕ 0 ](t) What about infinite systems?
The name of the game: TDDFT Runge-Gross theorem V ext (r, t) V ext(r, t) j(r, t) j (r, t) [n V ext ] [n V ext] n(r, t) n (r, t) n(r, t) V ext (r, t) + c(t) ϕe ic(t) ϕ(t) Ô(t) ϕ(t) = O[n, ϕ 0 ](t) What about infinite systems?
The name of the game: TDDFT Runge-Gross theorem V ext (r, t) V ext(r, t) j(r, t) j (r, t) [n V ext ] [n V ext] n(r, t) n (r, t) n(r, t) V ext (r, t) + c(t) ϕe ic(t) ϕ(t) Ô(t) ϕ(t) = O[n, ϕ 0 ](t) What about infinite systems?
The name of the game: TDDFT Runge-Gross theorem V ext (r, t) V ext(r, t) j(r, t) j (r, t) [n V ext ] [n V ext] n(r, t) n (r, t) n(r, t) V ext (r, t) + c(t) ϕe ic(t) ϕ(t) Ô(t) ϕ(t) = O[n, ϕ 0 ](t) What about infinite systems?
The name of the game: TDDFT Runge-Gross theorem V ext (r, t) V ext(r, t) j(r, t) j (r, t) [n V ext ] [n V ext] n(r, t) n (r, t) n(r, t) V ext (r, t) + c(t) ϕe ic(t) ϕ(t) Ô(t) ϕ(t) = O[n, ϕ 0 ](t) What about infinite systems?
The name of the game: TDDFT DFT Hohenberg-Kohn theorem 2 The total energy functional has a minimum, the ground-state energy E 0, corresponding to the ground-state density n 0. E[n] TDDFT Runge-Gross theorem - No minimum Time-dependent Schrödinger eq. (initial condition ϕ(t = 0) = ϕ 0 ), corresponds to a stationary point of the Hamiltonian action A = t1 t 0 dt ϕ(t) ı H(t) ϕ(t) t E 0 n 0 n
The name of the game: TDDFT DFT Hohenberg-Kohn theorem 2 The total energy functional has a minimum, the ground-state energy E 0, corresponding to the ground-state density n 0. E[n] TDDFT Runge-Gross theorem - No minimum Time-dependent Schrödinger eq. (initial condition ϕ(t = 0) = ϕ 0 ), corresponds to a stationary point of the Hamiltonian action A = t1 t 0 dt ϕ(t) ı H(t) ϕ(t) t H[t] E 0 n 0 n t 0 t 1 t
The name of the game: TDDFT DFT Kohn-Sham equations TDDFT Time-dependent Kohn-Sham equations» 1 2 2 i + V tot (r) φ i (r) = ɛ i φ i (r) Z V tot (r) = V ext (r)+ dr v(r, r )n(r )+V xc ([n], r) δexc [n] V xc ([n], r) = δn(r)» 1 2 2 + V tot (r, t) φ i (r, t) = i t φ i (r, t) Z V tot (r, t) = V ext (r, t) + v(r, r )n(r, t)dr + V xc ([n]r, t) δaxc [n] V xc ([n], r, t) = δn(r, t) Unknown exchange-correlation potential. V xc functional of the density. Unknown exchange-correlation time-dependent potential. V xc functional of the density at all times and of the initial state.
The name of the game: TDDFT DFT Kohn-Sham equations TDDFT Time-dependent Kohn-Sham equations» 1 2 2 i + V tot (r) φ i (r) = ɛ i φ i (r) Z V tot (r) = V ext (r)+ dr v(r, r )n(r )+V xc ([n], r) δexc [n] V xc ([n], r) = δn(r)» 1 2 2 + V tot (r, t) φ i (r, t) = i t φ i (r, t) Z V tot (r, t) = V ext (r, t) + v(r, r )n(r, t)dr + V xc ([n]r, t) δaxc [n] V xc ([n], r, t) = δn(r, t) Unknown exchange-correlation potential. V xc functional of the density. Unknown exchange-correlation time-dependent potential. V xc functional of the density at all times and of the initial state.
The name of the game: TDDFT Demonstrations, further readings, etc. R. van Leeuwen Int.J.Mod.Phys. B15, 1969 (2001) V xc ([n], r, t) = δa xc[n] δn(r, t) δv xc ([n], r, t) δn(r, t ) = δ 2 A xc [n] δn(r, t)δn(r, t ) Causality-Symmetry dilemma
The name of the game: TDDFT Demonstrations, further readings, etc. R. van Leeuwen Int.J.Mod.Phys. B15, 1969 (2001) V xc ([n], r, t) = δa xc[n] δn(r, t) δv xc ([n], r, t) δn(r, t ) = δ 2 A xc [n] δn(r, t)δn(r, t ) Causality-Symmetry dilemma
The name of the game: TDDFT Demonstrations, further readings, etc. R. van Leeuwen Int.J.Mod.Phys. B15, 1969 (2001) V xc ([n], r, t) = δa xc[n] δn(r, t) δv xc ([n], r, t) δn(r, t ) = δ 2 A xc [n] δn(r, t)δn(r, t ) Causality-Symmetry dilemma
The name of the game: TDDFT Demonstrations, further readings, etc. R. van Leeuwen Int.J.Mod.Phys. B15, 1969 (2001) V xc ([n], r, t) = δa xc[n] δn(r, t) δv xc ([n], r, t) δn(r, t ) = δ 2 A xc [n] δn(r, t)δn(r, t ) Causality-Symmetry dilemma
The name of the game: TDDFT DFT Hohenberg-Kohn theorem The ground-state expectation value of any physical observable of a many-electrons system is a unique functional of the electron density n(r) D ϕ 0 b O ϕ 0 E = O[n] Runge-Gross theorem TDDFT The expectation value of any physical time-dependent observable of a many-electrons system is a unique functional of the time-dependent electron density n(r) and of the initial state ϕ 0 = ϕ(t = 0) D ϕ(t) O(t) ϕ(t) b E = O[n, ϕ 0 ](t) Kohn-Sham equations» 1 2 2 i + V tot(r) φ i (r) = ɛ i φ i (r) Kohn-Sham equations» 1 2 2 + V tot(r, t) φ i (r, t) = i t φ i (r, t)
First Approach: Time Evolution of KS equations [H KS (t)] φ i (r, t) = i t φ i(r, t) occ n(r, t) = φ i (r, t) 2 i φ(t) = Û(t, t 0)φ(t 0 ) U(t, t 0 ) = 1 i t t 0 dτh(τ)û(τ, t 0) A. Castro et al. J.Chem.Phys. 121, 3425 (2004)
First Approach: Time Evolution of KS equations [H KS (t)] φ i (r, t) = i t φ i(r, t) occ n(r, t) = φ i (r, t) 2 i φ(t) = Û(t, t 0)φ(t 0 ) U(t, t 0 ) = 1 i t t 0 dτh(τ)û(τ, t 0) A. Castro et al. J.Chem.Phys. 121, 3425 (2004)
First Approach: Time Evolution of KS equations [H KS (t)] φ i (r, t) = i t φ i(r, t) occ n(r, t) = φ i (r, t) 2 i φ(t) = Û(t, t 0)φ(t 0 ) U(t, t 0 ) = 1 i t t 0 dτh(τ)û(τ, t 0) A. Castro et al. J.Chem.Phys. 121, 3425 (2004)
First Approach: Time Evolution of KS equations Photo-absorption cross section σ σ(ω) = 4πω c Imα(ω) α(t) = drv ext (r, t)n(r, t) in dipole approximation (λ dimension of the system) σ zz (ω) = 4πω c Im α(ω) = 4πω c Im dr z n(r, ω)
First Approach: Time Evolution of KS equations Photo-absorption cross section σ: porphyrin
First Approach: Time Evolution of KS equations Other observables Multipoles M lm (t) = drr l Y lm (r)n(r, t) Angular momentum L z (t) = drφ i (r, t) ı (r ) z φ i (r, t) i
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 V xc [n](r, t) functional of the history 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 in the linear response function!
Outline 1 Introduction: why TD-DFT? 2 (Just) A bit of Formalism TDDFT: the Foundation Linear Response Formalism 3 TDDFT in practice: The ALDA: Achievements and Shortcomings 4 Resources
Linear Response Approach Polarizability interacting system δn = χδv ext non-interacting system δn n i = χ 0 δv tot
Linear Response Approach Polarizability interacting system δn = χδv ext non-interacting system Single-particle polarizability δn n i = χ 0 δv tot χ 0 = ij φ i (r)φ j (r)φ i (r )φ j (r ) ω (ɛ i ɛ j ) hartree, hartree-fock, dft, etc. G.D. Mahan Many Particle Physics (Plenum, New York, 1990)
Linear Response Approach Polarizability interacting system δn = χδv ext non-interacting system δn n i = χ 0 δv tot j unoccupied states χ 0 = ij φ i (r)φ j (r)φ i (r )φ j (r ) ω (ɛ i ɛ j ) i occupied states
Linear Response Approach Polarizability interacting system non-interacting system δn = χδv ext δn n i = χ 0 δv tot Density Functional Formalism δn = δn n i δv tot = δv ext + δv H + δv xc
Linear Response Approach Polarizability χδv ext = χ 0 (δv ext + δv H + δv xc ) ( χ = χ 0 1 + δv H + δv ) xc δv ext δv ext δv H = δv H δn = vχ δv ext δn δv ext δv xc = δv xc δn = f xc χ δv ext δn δv ext with f xc = exchange-correlation kernel
Linear Response Approach Polarizability χδv ext = χ 0 (δv ext + δv H + δv xc ) ( χ = χ 0 1 + δv H + δv ) xc δv ext δv ext δv H = δv H δn = vχ δv ext δn δv ext δv xc = δv xc δn = f xc χ δv ext δn δv ext with f xc = exchange-correlation kernel
Linear Response Approach Polarizability χδv ext = χ 0 (δv ext + δv H + δv xc ) ( χ = χ 0 1 + δv H + δv ) xc δv ext δv ext δv H = δv H δn = vχ δv ext δn δv ext δv xc = δv xc δn = f xc χ δv ext δn δv ext χ = χ 0 + χ 0 (v + f xc ) χ with f xc = exchange-correlation kernel
Linear Response Approach Polarizability χδv ext = χ 0 (δv ext + δv H + δv xc ) ( χ = χ 0 1 + δv H + δv ) xc δv ext δv ext δv H = δv H δn = vχ δv ext δn δv ext δv xc = δv xc δn = f xc χ δv ext δn δv ext χ = [ 1 χ 0 (v + f xc ) ] 1 χ 0 with f xc = exchange-correlation kernel
Linear Response Approach Polarizability χδv ext = χ 0 (δv ext + δv H + δv xc ) ( χ = χ 0 1 + δv H + δv ) xc δv ext δv ext δv H = δv H δn = vχ δv ext δn δv ext δv xc = δv xc δn = f xc χ δv ext δn δv ext χ = [ 1 χ 0 (v + f xc ) ] 1 χ 0 with f xc = exchange-correlation kernel
Linear Response Approach Polarizability χ in TDDFT 1 DFT ground-state calc. φ i, ɛ i [V xc ] 2 φ i, ɛ i χ 0 = ij 3 δv H δn δv xc δn = v = f xc 4 χ = χ 0 + χ 0 (v + f xc ) χ φ i (r)φ j (r)φ i (r )φ j (r ) ω (ɛ i ɛ j ) } variation of the potentials A comment { δvxc f xc = δn any other function
Linear Response Approach Polarizability χ in TDDFT 1 DFT ground-state calc. φ i, ɛ i [V xc ] 2 φ i, ɛ i χ 0 = ij 3 δv H δn δv xc δn = v = f xc 4 χ = χ 0 + χ 0 (v + f xc ) χ φ i (r)φ j (r)φ i (r )φ j (r ) ω (ɛ i ɛ j ) } variation of the potentials A comment { δvxc f xc = δn any other function
Linear Response Approach Polarizability χ in TDDFT 1 DFT ground-state calc. φ i, ɛ i [V xc ] 2 φ i, ɛ i χ 0 = ij 3 δv H δn δv xc δn = v = f xc 4 χ = χ 0 + χ 0 (v + f xc ) χ φ i (r)φ j (r)φ i (r )φ j (r ) ω (ɛ i ɛ j ) } variation of the potentials A comment { δvxc f xc = δn any other function
Linear Response Approach Polarizability χ in TDDFT 1 DFT ground-state calc. φ i, ɛ i [V xc ] 2 φ i, ɛ i χ 0 = ij 3 δv H δn δv xc δn = v = f xc 4 χ = χ 0 + χ 0 (v + f xc ) χ φ i (r)φ j (r)φ i (r )φ j (r ) ω (ɛ i ɛ j ) } variation of the potentials A comment { δvxc f xc = δn any other function
Linear Response Approach Polarizability χ in TDDFT 1 DFT ground-state calc. φ i, ɛ i [V xc ] 2 φ i, ɛ i χ 0 = ij 3 δv H δn δv xc δn = v = f xc 4 χ = χ 0 + χ 0 (v + f xc ) χ φ i (r)φ j (r)φ i (r )φ j (r ) ω (ɛ i ɛ j ) } variation of the potentials A comment { δvxc f xc = δn any other function
Linear Response Approach Polarizability χ in TDDFT 1 DFT ground-state calc. φ i, ɛ i [V xc ] 2 φ i, ɛ i χ 0 = ij 3 δv H δn δv xc δn = v = f xc 4 χ = χ 0 + χ 0 (v + f xc ) χ φ i (r)φ j (r)φ i (r )φ j (r ) ω (ɛ i ɛ j ) } variation of the potentials A comment { δvxc f xc = δn any other function
Finite systems Photo-absorption cross spectrum in Linear Response α(ω) = σ(ω) = 4πω c Imα(ω) drdr V ext (r, ω)δn(r, ω) σ zz (ω) = 4πω c Im drdr zχ(r, r, ω)z
Finite systems Photo-absorption cross spectrum in Linear Response α(ω) = σ(ω) = 4πω c Imα(ω) drdr V ext (r, ω)χ(r, r, ω)v ext (r, ω) σ zz (ω) = 4πω c Im drdr zχ(r, r, ω)z
Finite systems Photo-absorption cross spectrum in Linear Response α(ω) = σ(ω) = 4πω c Imα(ω) drdr V ext (r, ω)χ(r, r, ω)v ext (r, ω) σ zz (ω) = 4πω c Im drdr zχ(r, r, ω)z
Solids Reciprocal space χ 0 (r, r, ω) χ 0 GG (q, ω) G =reciprocal lattice vector q =momentum transfer of the perturbation
Solids Reciprocal space φvk e ı(q+g)r φ ck+q φck+q e ı(q+g )r φvk χ 0 GG (q, ω) = vck ω (ɛ ck+q ɛ vk ) + ıη j unoccupied states i occupied states
Solids Reciprocal space φvk e ı(q+g)r φ ck+q φck+q e ı(q+g )r φvk χ 0 GG (q, ω) = vck ω (ɛ ck+q ɛ vk ) + ıη χ GG (q, ω) = χ 0 + χ 0 (v + f xc ) χ ɛ 1 GG (q, ω) = δ GG + v G (q)χ GG (q, ω)
Solids Reciprocal space φvk e ı(q+g)r φ ck+q φck+q e ı(q+g )r φvk χ 0 GG (q, ω) = vck ω (ɛ ck+q ɛ vk ) + ıη χ GG (q, ω) = χ 0 + χ 0 (v + f xc ) χ ɛ 1 GG (q, ω) = δ GG + v G (q)χ GG (q, ω) ELS(q, ω) = Im { ɛ 1 00 (q, ω) } { } 1 ; Abs(ω) = lim Im q 0 ε 1 00 (q, ω) S.L.Adler, Phys.Rev 126, 413 (1962); N.Wiser Phys.Rev 129, 62 (1963)
Solids Reciprocal space φvk e ı(q+g)r φ ck+q φck+q e ı(q+g )r φvk χ 0 GG (q, ω) = vck ω (ɛ ck+q ɛ vk ) + ıη χ GG (q, ω) = χ 0 + χ 0 (v + f xc ) χ ɛ 1 GG (q, ω) = δ GG + v G (q)χ GG (q, ω) ELS(ω) = limim { ε 1 00 (q, ω) } { } 1 ; Abs(ω) = limim q 0 q 0 ε 1 00 (q, ω) S.L.Adler, Phys.Rev 126, 413 (1962); N.Wiser Phys.Rev 129, 62 (1963)
Solids Absorption and Energy Loss Spectra q 0 ELS(ω) = Im { { } 1 ε 1 00 (ω)} ; Abs(ω) = Im ε 1 00 (ω)
Solids Absorption and Energy Loss Spectra q 0 ELS(ω) = Im { { } 1 ε 1 00 (ω)} ; Abs(ω) = Im ε 1 00 (ω) ε 1 00 (ω) = 1 + v 0χ 00 (ω)
Solids Absorption and Energy Loss Spectra q 0 ELS(ω) = Im { ε 1 00 (ω)} ; Abs(ω) = Im ELS(ω) = v 0 Im { χ 00 (ω) } { ; Abs(ω) = v 0 Im { 1 ε 1 00 (ω) } 1 1+v 0 χ 00 (ω) }
Solids Absorption and Energy Loss Spectra q 0 ELS(ω) = Im { ε 1 00 (ω)} ; Abs(ω) = Im ELS(ω) = v 0 Im { χ 00 (ω) } { ; Abs(ω) = v 0 Im { 1 ε 1 00 (ω) 1 1+v 0 χ 00 (ω) ELS(ω) = v 0 Im { χ 00 (ω) } ; Abs(ω) = v 0 Im { χ 00 (ω) } } }
Solids Absorption and Energy Loss Spectra q 0 ELS(ω) = Im { ε 1 00 (ω)} ; Abs(ω) = Im ELS(ω) = v 0 Im { χ 00 (ω) } { ; Abs(ω) = v 0 Im { 1 ε 1 00 (ω) 1 1+v 0 χ 00 (ω) ELS(ω) = v 0 Im { χ 00 (ω) } ; Abs(ω) = v 0 Im { χ 00 (ω) } } } χ = χ 0 + χ 0 (v + f xc ) χ χ = χ 0 + χ 0 ( v + f xc ) χ { vg G 0 v G = 0 G = 0
Solids Absorption and Energy Loss Spectra q 0 ELS(ω) = Im { ε 1 00 (ω)} ; Abs(ω) = Im ELS(ω) = v 0 Im { χ 00 (ω) } { ; Abs(ω) = v 0 Im { 1 ε 1 00 (ω) 1 1+v 0 χ 00 (ω) ELS(ω) = v 0 Im { χ 00 (ω) } ; Abs(ω) = v 0 Im { χ 00 (ω) } } } Exerciseχ = χ 0 + χ 0 (v + f xc ) χ { } χ = 1 χ 0 + χ 0 ( v + f xc ) Im ε 1 = v { 0 Im { χ } 00 00 vg G 0 v G = 0 G = 0
Solids Abs and ELS (q 0) differs only by v 0 ELS(ω) = Im { { } 1 ε 1 00 (ω)} ; Abs(ω) = Im ε 1 00 (ω) ELS(ω) = v 0 Im { χ 00 (ω) } ; Abs(ω) = v 0 Im { χ 00 (ω) } χ = χ 0 + χ 0 (v + f xc ) χ χ = χ 0 + χ 0 ( v + f xc ) χ { vg G 0 v G = microscopic components 0 G = 0
Solids Microscopic components v v = local field effects χ NLF = χ 0 + χ 0 ( vx + f xc ) χ NLF
Solids Microscopic components v v = local field effects χ NLF = χ 0 + χ 0 ( vx + f xc ) χ NLF Abs NLF = v 0 Im { χ NLF}
Solids Microscopic components v v = local field effects χ NLF = χ 0 + χ 0 ( vx + f xc ) χ NLF Abs NLF = v 0 Im { χ NLF} Abs NLF = Im {ε 00 }
Solids Microscopic components v v = local field effects χ NLF = χ 0 + χ 0 ( vx + f xc ) χ NLF Abs NLF = v 0 Im { χ NLF} Abs NLF = Im {ε 00 } { } 1 Abs = Im ε 1 00
Solids Microscopic components v v = local field effects χ NLF = χ 0 + χ 0 ( vx + f xc ) χ NLF Abs NLF = v 0 Im { χ NLF} Abs NLF = Im {ε 00 } Exercise { } 1 Abs = Im Abs NLF = v ε 1 00 0 Im { χ NLF} = Im {ε 00 }
EELS of Graphite
Absorption of Silicon
Outline 1 Introduction: why TD-DFT? 2 (Just) A bit of Formalism TDDFT: the Foundation Linear Response Formalism 3 TDDFT in practice: The ALDA: Achievements and Shortcomings 4 Resources
TDDFT in practice Practical schema and approximations Ground-state calculation φ i, ɛ i [V xc LDA] χ 0 (q, ω) χ = χ 0 + χ 0 (v + f xc ) χ f xc = 0 RPA f ALDA xc (r, r ) = δvxc(r) δn(r ) δ(r r ) ALDA
Outline 1 Introduction: why TD-DFT? 2 (Just) A bit of Formalism TDDFT: the Foundation Linear Response Formalism 3 TDDFT in practice: The ALDA: Achievements and Shortcomings 4 Resources
ALDA: Achievements and Shortcomings Electron Energy Loss Spectrum of Graphite RPA vs EXP χ NLF = χ 0 + χ 0 v 0 χ NLF χ = χ 0 + χ 0 v χ ELS = v 0 Im { χ 00 } A.Marinopoulos et al. Phys.Rev.Lett 89, 76402 (2002)
ALDA: Achievements and Shortcomings Photo-absorption cross section of Benzene ALDA vs EXP Abs= 4πω c Im drzn(r, ω) K.Yabana and G.F.Bertsch Int.J.Mod.Phys.75, 55 (1999)
ALDA: Achievements and Shortcomings Inelastic X-ray scattering of Silicon ALDA vs RPA vs EXP S(q, ω)= 2 q 2 4π 2 e 2 n Imε 1 00 Weissker et al., PRL 97, 237602 (2006)
ALDA: Achievements and Shortcomings Absorption Spectrum of Silicon ALDA vs RPA vs EXP χ = χ 0 + χ 0 ( v + f ALDA xc ) χ Abs = v 0 Im { χ 00 }
ALDA: Achievements and Shortcomings Absorption Spectrum of Argon ALDA vs EXP χ = χ 0 + χ 0 ( v + f ALDA xc ) χ Abs = v 0 Im { χ 00 }
ALDA: Achievements and Shortcomings Good results Photo-absorption of small molecules ELS of solids Bad results Absorption of solids
ALDA: Achievements and Shortcomings Good results Photo-absorption of small molecules ELS of solids Bad results Absorption of solids Why?
ALDA: Achievements and Shortcomings Good results Photo-absorption of small molecules ELS of solids Bad results Absorption of solids Why? f ALDA xc is short-range f xc (q 0) 1 q 2
ALDA: Achievements and Shortcomings Absorption of Silicon f xc = α q 2 L.Reining et al. Phys.Rev.Lett. 88, 66404 (2002)
Outline 1 Introduction: why TD-DFT? 2 (Just) A bit of Formalism TDDFT: the Foundation Linear Response Formalism 3 TDDFT in practice: The ALDA: Achievements and Shortcomings 4 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)
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, Contributors: Valérie Véniard, Eleonora Luppi non-linear Lucia Caramella Spin Silvana Botti kernel, Wannier functions Margherita Marsili Mapping-Theory kernel Christine Giorgetti metals
The DP code The algorithm ε 1 GG (q, ω) = δ GG + v G (q)χ GG (q, ω)
The DP code The algorithm ε 1 GG (q, ω) = δ GG + v G (q)χ GG (q, ω) χ GG (q, ω) = χ 0 + χ 0 (v + f xc ) χ
The DP code The algorithm ε 1 GG (q, ω) = δ GG + v G (q)χ GG (q, ω) χ GG (q, ω) = χ 0 + χ 0 (v + f xc ) χ φvk e ı(q+g)r φ ck+q φ ck+q e ı(q+g )r φ vk χ 0 GG (q, ω) = vck ω (ɛ ck+q ɛ vk ) + ıη
The DP code The algorithm ε 1 GG (q, ω) = δ GG + v G (q)χ GG (q, ω) χ GG (q, ω) = χ 0 + χ 0 (v + f xc ) χ χ 0 (q, ω) = φvk e ı(q+g)r φ ck+q φ ck+q e ı(q+g )r φ vk GG ω (ɛ ck+q ɛ vk ) + ıη vck φ vk e ı(q+g)r φ ck+q = drφ vk (r)eı(q+g)r φ ck+q (r)
The DP code The algorithm ε 1 GG (q, ω) = δ GG + v G (q)χ GG (q, ω) χ GG (q, ω) = χ 0 + χ 0 (v + f xc ) χ χ 0 (q, ω) = φvk e ı(q+g)r φ ck+q φ ck+q e ı(q+g )r φ vk GG ω (ɛ ck+q ɛ vk ) + ıη vck φ vk e ı(q+g)r φ ck+q = drφ vk (r)eı(q+g)r φ ck+q (r)
The DP code The algorithm ε 1 GG (q, ω) = δ GG + v G (q)χ GG (q, ω) χ GG (q, ω) = χ 0 + χ 0 (v + f xc ) χ χ 0 (q, ω) = φvk e ı(q+g)r φ ck+q φ ck+q e ı(q+g )r φ vk GG ω (ɛ ck+q ɛ vk ) + ıη vck φ vk e ı(q+g)r φ ck+q = drφ vk (r)eı(q+g)r φ ck+q (r)
The DP code The algorithm ε 1 GG (q, ω) = δ GG + v G (q)χ GG (q, ω) χ GG (q, ω) = χ 0 + χ 0 (v + f xc ) χ χ 0 (q, ω) = φvk e ı(q+g)r φ ck+q φ ck+q e ı(q+g )r φ vk GG ω (ɛ ck+q ɛ vk ) + ıη vck φ vk e ı(q+g)r φ ck+q = drφ vk (r)eı(q+g)r φ ck+q (r)
The DP code The algorithm ε 1 GG (q, ω) = δ GG + v G (q)χ GG (q, ω) χ GG (q, ω) = χ 0 + χ 0 (v + f xc ) χ χ 0 (q, ω) = φvk e ı(q+g)r φ ck+q φ ck+q e ı(q+g )r φ vk GG ω (ɛ ck+q ɛ vk ) + ıη vck φ vk e ı(q+g)r φ ck+q = drφ vk (r)eı(q+g)r φ ck+q (r)
The DP code The algorithm ε 1 GG (q, ω) = δ GG + v G (q)χ GG (q, ω) χ GG (q, ω) = χ 0 + χ 0 (v + f xc ) χ χ 0 (q, ω) = φvk e ı(q+g)r φ ck+q φ ck+q e ı(q+g )r φ vk GG ω (ɛ ck+q ɛ vk ) + ıη vck φ vk e ı(q+g)r φ ck+q = drφ vk (r)eı(q+g)r φ ck+q (r)
The DP code The algorithm ε 1 GG (q, ω) = δ GG + v G (q)χ GG (q, ω) χ GG (q, ω) = χ 0 + χ 0 (v + f xc ) χ χ 0 (q, ω) = φvk e ı(q+g)r φ ck+q φ ck+q e ı(q+g )r φ vk GG ω (ɛ ck+q ɛ vk ) + ıη vck φ vk e ı(q+g)r φ ck+q = drφ vk (r)eı(q+g)r φ ck+q (r)
TDDFT vs BSE scaling Scaling of TDDFT (DP) χ GG (q, ω)=χ 0 (q, GG ω)+χ0 (q, ω) [v xc GG G (q)+fg (q, ω)] χ G G G (q, ω)
TDDFT vs BSE scaling Scaling of TDDFT (DP) χ GG (q, ω)=χ 0 (q, GG ω)+χ0 (q, ω) [v xc GG G (q)+fg (q, ω)] χ G G G (q, ω) φ vk e ı(q+g)r φ ck+q φ ck+q e ı(q+g )r φ vk χ 0 GG (q, ω) = vck ω (ɛ ck+q ɛ vk ) + ıη
TDDFT vs BSE scaling Scaling of TDDFT (DP) χ GG (q, ω)=χ 0 (q, GG ω)+χ0 (q, ω) [v xc GG G (q)+fg (q, ω)] χ G G G (q, ω) φ vk e ı(q+g)r φ ck+q φ ck+q e ı(q+g )r φ vk χ 0 GG (q, ω) = vck ω (ɛ ck+q ɛ vk ) + ıη N t
TDDFT vs BSE scaling Scaling of TDDFT (DP) χ GG (q, ω)=χ 0 (q, GG ω)+χ0 (q, ω) [v xc GG G (q)+fg (q, ω)] χ G G G (q, ω) φ vk e ı(q+g)r φ ck+q φ ck+q e ı(q+g )r φ vk χ 0 GG (q, ω) = vck ω (ɛ ck+q ɛ vk ) + ıη N t N G
TDDFT vs BSE scaling Scaling of TDDFT (DP) χ GG (q, ω)=χ 0 (q, GG ω)+χ0 (q, ω) [v xc GG G (q)+fg (q, ω)] χ G G G (q, ω) φ vk e ı(q+g)r φ ck+q φ ck+q e ı(q+g )r φ vk χ 0 GG (q, ω) = vck ω (ɛ ck+q ɛ vk ) + ıη N t N G N G
TDDFT vs BSE scaling Scaling of TDDFT (DP) χ GG (q, ω)=χ 0 (q, GG ω)+χ0 (q, ω) [v xc GG G (q)+fg (q, ω)] χ G G G (q, ω) φ vk e ı(q+g)r φ ck+q φ ck+q e ı(q+g )r φ vk χ 0 GG (q, ω) = vck ω (ɛ ck+q ɛ vk ) + ıη N t N G N G N ω
TDDFT vs BSE scaling Scaling of TDDFT (DP) χ GG (q, ω)=χ 0 (q, GG ω)+χ0 (q, ω) [v xc GG G (q)+fg (q, ω)] χ G G G (q, ω) φ vk e ı(q+g)r φ ck+q φ ck+q e ı(q+g )r φ vk χ 0 GG (q, ω) = vck ω (ɛ ck+q ɛ vk ) + ıη N t N G N G N ω ; N t N r lnn r N ω
TDDFT vs BSE scaling Scaling of TDDFT (DP) χ GG (q, ω)=χ 0 (q, GG ω)+χ0 (q, ω) [v xc GG G (q)+fg (q, ω)] χ G G G (q, ω) φ vk e ı(q+g)r φ ck+q φ ck+q e ı(q+g )r φ vk χ 0 GG (q, ω) = vck ω (ɛ ck+q ɛ vk ) + ıη N t N G N G N ω ; N t N r lnn r N ω Scaling N 4 at
TDDFT vs BSE scaling Scaling of BSE (EXC) H v c k vck Scaling N 2 t N r N 5 at
TDDFT vs BSE scaling Scaling of BSE (EXC) H v c k vck A v c k λ = A λ A vck λ Scaling N 2 t N r N 5 at Scaling N 3 t N 6 at
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 Functional Theory, Springer (2006)