Time Dependent Density Functional Theory. Francesco Sottile

Transcription

1 An Introduction Laboratoire des Solides Irradiés Ecole Polytechnique, Palaiseau - France European Theoretical Spectroscopy Facility (ETSF) Belfast, 29 Jun 2007

2 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

3 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

4 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 entries 8 bytes/entry bytes bytes/dvd DVDs

5 Density Functional... Why?

6 Density Functional... Why?

7 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 entries 8 bytes/entry bytes bytes/dvd DVDs Importance of non-interacting The Kohn-Sham one-particle equations H i (r)ψ i (r) = ɛ i (r)ψ i (r)

8 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 entries 8 bytes/entry bytes bytes/dvd DVDs Importance of non-interacting The Kohn-Sham one-particle equations H i (r)ψ i (r) = ɛ i (r)ψ i (r)

9 Density Functional... Successfull? S. Redner

10 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

11 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

12 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

13 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

14 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

15 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

16 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

17 Time Dependent DFT... Why? We need a time dependent theory TDDFT is a promising candidate

18 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

19 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

20 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)

21 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)

22 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)

23 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)

24 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)

25 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?

26 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?

27 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?

28 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?

29 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?

30 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?

31 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

32 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

33 The name of the game: TDDFT DFT Kohn-Sham equations TDDFT Time-dependent Kohn-Sham equations» 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)» 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.

34 The name of the game: TDDFT DFT Kohn-Sham equations TDDFT Time-dependent Kohn-Sham equations» 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)» 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.

35 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

36 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

37 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

38 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

39 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» i + V tot(r) φ i (r) = ɛ i φ i (r) Kohn-Sham equations» V tot(r, t) φ i (r, t) = i t φ i (r, t)

40 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)

41 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)

42 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)

43 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)

44 First Approach: Time Evolution of KS equations Photo-absorption cross section σ: porphyrin

45 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

46 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!

47 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

48 Linear Response Approach Polarizability interacting system δn = χδv ext non-interacting system δn n i = χ 0 δv tot

49 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)

50 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

51 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

52 Linear Response Approach Polarizability χδv ext = χ 0 (δv ext + δv H + δv xc ) ( χ = χ 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

53 Linear Response Approach Polarizability χδv ext = χ 0 (δv ext + δv H + δv xc ) ( χ = χ 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

54 Linear Response Approach Polarizability χδv ext = χ 0 (δv ext + δv H + δv xc ) ( χ = χ 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

55 Linear Response Approach Polarizability χδv ext = χ 0 (δv ext + δv H + δv xc ) ( χ = χ 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

56 Linear Response Approach Polarizability χδv ext = χ 0 (δv ext + δv H + δv xc ) ( χ = χ 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

57 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

58 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

59 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

60 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

61 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

62 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

63 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

64 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

65 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

66 Solids Reciprocal space χ 0 (r, r, ω) χ 0 GG (q, ω) G =reciprocal lattice vector q =momentum transfer of the perturbation

67 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

68 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)

69 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)

70 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)

71 Solids Absorption and Energy Loss Spectra q 0 ELS(ω) = Im { { } 1 ε 1 00 (ω)} ; Abs(ω) = Im ε 1 00 (ω)

72 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)

73 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) }

74 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) } } }

75 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

76 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 } vg G 0 v G = 0 G = 0

77 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

78 Solids Microscopic components v v = local field effects χ NLF = χ 0 + χ 0 ( vx + f xc ) χ NLF

79 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}

80 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 }

81 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

82 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 Im { χ NLF} = Im {ε 00 }

83 EELS of Graphite

84 Absorption of Silicon

85 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

86 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

87 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

88 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, (2002)

89 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)

90 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, (2006)

91 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 }

92 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 }

93 ALDA: Achievements and Shortcomings Good results Photo-absorption of small molecules ELS of solids Bad results Absorption of solids

94 ALDA: Achievements and Shortcomings Good results Photo-absorption of small molecules ELS of solids Bad results Absorption of solids Why?

95 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

96 ALDA: Achievements and Shortcomings Absorption of Silicon f xc = α q 2 L.Reining et al. Phys.Rev.Lett. 88, (2002)

97

98 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

99 Resources Codes (more or less) available for TDDFT Octopus (Marques,Castro,Rubio) -(real space, real time) - mainly finite systems - GPL DP (Olevano,Reining,Sottile) - (reciprocal space, frequency domain) - solides and finite systems - open source for academics Self (Marini) - (reciprocal space, frequency domain) Fleszar code Rehr (core excitations) TDDFT (Bertsch) VASP, SIESTA, ADF, TURBOMOLE TD-DFPT (Baroni)

100 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

101 The DP code The algorithm ε 1 GG (q, ω) = δ GG + v G (q)χ GG (q, ω)

102 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

103 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

104 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)

105 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)

106 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)

107 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)

108 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)

109 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)

110 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)

111 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)

112 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

113 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

114 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

115 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

116 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

117 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

118 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

119 TDDFT vs BSE scaling Scaling of BSE (EXC) H v c k vck Scaling N 2 t N r N 5 at

120 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

121 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)