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Time Dependent Density Functional Theory Applications, Limitations and Time Dependent Density Functional Theory Applications, limitations and ... new frontiers Francesco Sottile Laboratoire des Solides Irradi´es Ecole Polytechnique, Palaiseau - France European Theoretical Spectroscopy Facility (ETSF) Vienna, 19 January 2007 1/55 Time Dependent Density Functional Theory Francesco Sottile Time-Dependent Density Functional Theory Applications and results: The ETSF Outline 1 Time-Dependent Density Functional Theory Motivation The theoretical framework Linear response formalism 2 Applications and results: Achievements of RPA and ALDA Problem of solids - new kernels The DP code New Frontiers 3 The ETSF 2/55 Time Dependent Density Functional Theory Francesco Sottile Time-Dependent Density Functional Theory Applications and results: The ETSF Outline 1 Time-Dependent Density Functional Theory Motivation The theoretical framework Linear response formalism 2 Applications and results: Achievements of RPA and ALDA Problem of solids - new kernels The DP code New Frontiers 3 The ETSF 3/55 Time Dependent Density Functional Theory Francesco Sottile Time-Dependent Density Functional Theory Applications and results: The ETSF Outline 1 Time-Dependent Density Functional Theory Motivation The theoretical framework Linear response formalism 2 Applications and results: Achievements of RPA and ALDA Problem of solids - new kernels The DP code New Frontiers 3 The ETSF 4/55 Time Dependent Density Functional Theory Francesco Sottile Time-Dependent Density Functional Theory Applications and results: The ETSF 5/55 Time Dependent Density Functional Theory Francesco Sottile Time-Dependent Density Functional Theory Applications and results: The ETSF Outline 1 Time-Dependent Density Functional Theory Motivation The theoretical framework Linear response formalism 2 Applications and results: Achievements of RPA and ALDA Problem of solids - new kernels The DP code New Frontiers 3 The ETSF 6/55 Time Dependent Density Functional Theory Francesco Sottile Time-Dependent Density Functional Theory Applications and results: The ETSF The name of the game: TDDFT TDDFT: density functional philosophy to the world of the systems driven out of equilibrium, by an external time-dependent perturbation. 7/55 Time Dependent Density Functional Theory Francesco Sottile Time-Dependent Density Functional Theory Applications and results: The ETSF Density Functional Concept ⇒ 2 (important) points 1. 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) 0 0 ϕ Ob ϕ = O[n] P.Hohenberg and W.Kohn Phys.Rev. 136, B864 (1964) 8/55 Time Dependent Density Functional Theory Francesco Sottile Time-Dependent Density Functional Theory Applications and results: The ETSF Density Functional Concept ⇒ 2 (important) points 2. Kohn-Sham equations 1 − ∇2 + V (r) φ (r) = φ (r) 2 i tot i i i Z 0 0 0 Vtot (r) = Vext (r)+ dr v(r, r )n(r )+Vxc ([n], r) Vxc ([n], r) Unknown, stupidity term W.Kohn and L.J.Sham Phys. Rev. 140, A1133 (1965) 8/55 Time Dependent Density Functional Theory Francesco Sottile Time-Dependent Density Functional Theory Applications and results: The ETSF The name of the game: TDDFT DFT TDDFT Hohenberg-Kohn Runge-Gross theorem The ground-state expectation The expectation value of any physical value of any physical observable time-dependent observable of a of a many-electrons system is a many-electrons system is a unique unique functional of the electron functional of the time-dependent density n(r) electron density n(r, t) and of the initial state ϕ0 = ϕ(t = 0) 0 0 ϕ O ϕ = O[n] D E 0 b ϕ(t)|Ob(t)|ϕ(t) = O[n, ϕ ](t) P.Hohenberg and W.Kohn E. Runge and E.K.U. Gross Phys.Rev. 136, B864 (1964) Phys.Rev.Lett. 52, 997 (1984) 9/55 Time Dependent Density Functional Theory Francesco Sottile Time-Dependent Density Functional Theory Applications and results: The ETSF The name of the game: TDDFT DFT TDDFT Kohn-Sham equations Time-dependent KS equations » – » – 1 2 1 2 ∂ − ∇i + Vtot (r) φi (r) = i φi (r) − ∇ + Vtot (r, t) φi (r, t) = i φi (r, t) 2 2 ∂t Z Z 0 0 0 0 0 0 Vtot (r) = Vext (r) + dr v(r, r )n(r ) + Vxc ([n], r) Vtot (r, t) = Vext (r, t) + v(r, r )n(r , t)dr + Vxc ([n]r, t) Vxc ([n], r) = Unknown, stupidity term Vxc ([n], r, t) = Unknown, (even more) stupidity term Unknown exchange-correlation Unknown exchange-correlation time-dependent potential. potential. V functional of the density at V functional of the density. xc xc all times and of the initial state. 10/55 Time Dependent Density Functional Theory Francesco Sottile Time-Dependent Density Functional Theory Applications and results: The ETSF The name of the game: TDDFT DFT TDDFT Kohn-Sham equations Time-dependent KS equations » – » – 1 2 1 2 ∂ − ∇i + Vtot (r) φi (r) = i φi (r) − ∇ + Vtot (r, t) φi (r, t) = i φi (r, t) 2 2 ∂t Z Z 0 0 0 0 0 0 Vtot (r) = Vext (r) + dr v(r, r )n(r ) + Vxc ([n], r) Vtot (r, t) = Vext (r, t) + v(r, r )n(r , t)dr + Vxc ([n]r, t) Vxc ([n], r) = Unknown, stupidity term Vxc ([n], r, t) = Unknown, (even more) stupidity term Unknown exchange-correlation Unknown exchange-correlation time-dependent potential. potential. V functional of the density at V functional of the density. xc xc all times and of the initial state. 10/55 Time Dependent Density Functional Theory Francesco Sottile Time-Dependent Density Functional Theory Applications and results: The ETSF The name of the game: TDDFT Demonstrations, further readings, etc. R. van Leeuwen Int.J.Mod.Phys. B15, 1969 (2001) Linear Response 11/55 Time Dependent Density Functional Theory Francesco Sottile Time-Dependent Density Functional Theory Applications and results: The ETSF Time-Dependent Kohn-Sham equations ∂ H (r, t)φ (r, t) = ı φ (r, t) KS i ∂t i 12/55 Time Dependent Density Functional Theory Francesco Sottile Time-Dependent Density Functional Theory Applications and results: The ETSF Practical computational scheme: Real space - time evolution Evolution of the KS wave functions ∂ H (t)φ(t) = ı φ(t) KS ∂t Z t+∆t −ı H(τ)dτ φ(t + ∆t) = e t φ(t) Approximation for the Vxc ALDA dexc (n) Adiabatic LDA Vxc [n(r, t)] = dn n=n(r,t) 13/55 Time Dependent Density Functional Theory Francesco Sottile Time-Dependent Density Functional Theory Applications and results: The ETSF Practical computational scheme: Real space - time evolution Photo-absorption cross section σ 4πω σ(ω) = Imα(ω) c Z α(t) = − drVext (r, t)n(r, t) in dipole approximation (λ ≫ dimension of the system) Z α(ω) = − drzn(r, ω) 4πω Z σ (ω) = − Im dr z n(r, ω) zz c 13/55 Time Dependent Density Functional Theory Francesco Sottile Time-Dependent Density Functional Theory Applications and results: The ETSF Practical computational scheme: Real space - time evolution Photo-absorption cross section σ: porphyrin I octopus (GPL) http://www.tddft.org/programs/octopus/ 13/55 Time Dependent Density Functional Theory Francesco Sottile Time-Dependent Density Functional Theory Applications and results: The ETSF Practical computational scheme: Real space - time evolution Other observables Multipoles Z l Mlm(t) = drr Ylm(r)n(r, t) Angular momentum X Z Lz (t) = − drφi (r, t) ı (r × ∇)z φi (r, t) i 13/55 Time Dependent Density Functional Theory Francesco Sottile Time-Dependent Density Functional Theory Applications and results: The ETSF Real space - Time evolution approach 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 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! 14/55 Time Dependent Density Functional Theory Francesco Sottile Time-Dependent Density Functional Theory Applications and results: The ETSF Outline 1 Time-Dependent Density Functional Theory Motivation The theoretical framework Linear response formalism 2 Applications and results: Achievements of RPA and ALDA Problem of solids - new kernels The DP code New Frontiers 3 The ETSF 15/55 Time Dependent Density Functional Theory Francesco Sottile −1 Vtot = ε Vext Vtot = Vext + Vind E = ε−1D Dielectric function ε EELS R index ε Abs X-ray Time-Dependent Density Functional Theory Applications and results: The ETSF Linear Response Approach System submitted to an external perturbation 16/55 Time Dependent Density Functional Theory Francesco Sottile E = ε−1D Dielectric function ε EELS R index ε Abs X-ray Time-Dependent Density Functional Theory Applications and results: The ETSF Linear Response Approach System submitted to an external perturbation −1 Vtot = ε Vext Vtot = Vext + Vind 16/55 Time Dependent Density Functional Theory Francesco Sottile Dielectric function ε EELS R index ε Abs X-ray Time-Dependent Density Functional Theory Applications and results: The ETSF Linear Response Approach System submitted to an external perturbation −1 Vtot = ε Vext Vtot = Vext + Vind E = ε−1D 16/55 Time Dependent Density Functional Theory Francesco Sottile EELS R index Abs X-ray Time-Dependent Density Functional Theory Applications and results: The ETSF Linear Response Approach System submitted to an external perturbation −1 Vtot = ε Vext Vtot = Vext + Vind E = ε−1D Dielectric function ε ε 16/55 Time Dependent Density Functional Theory Francesco Sottile EELS R index X-ray Time-Dependent Density Functional Theory Applications and results: The ETSF Linear Response Approach System submitted to an external perturbation
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