Time-dependent density functional theory

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1 Time-dependent density functional theory E.K.U. Gross Max-Planck Institute for Microstructure Physics

2 OUTLINE LECTURE I Phenomena to be described by TDDFT LECTURE II Review of ground-state DFT LECTURE III Basic theorems of TDDFT TDDFT in the linear-response regime: Calculation of optical excitation spectra LECTURE IV: TDDFT beyond the regime of linear response TD Electron Localization Function (TDELF) Calculating electronic transport using TDDFT Optimal control theory

3 PHENOMENA TO BE DESCRIBED WITH TDDFT

Time-dependent systems Generic situation: Molecule in laser field Ĥ(t) = + Ŵ Tˆe ee Strong laser (v laser (t) v en ) : Non-perturbative solution of full TDSE required Weak laser (v laser (t) << v en

4 Time-dependent systems Generic situation: Molecule in laser field Ĥ(t) = + Ŵ Tˆe ee Strong laser (v laser (t) v en ) : Non-perturbative solution of full TDSE required Weak laser (v laser (t) << v en ) : Calculate 1. Linear density response ρ 1 (r t) + j, α 2. Dynamical polarizability Z α e 2 r j -R α α + E r j sin ωt e d 3 1 ( ω) = z ρ ( r, ω) E r 3. Photo-absorption cross section σ 4πω c ( ω) = Imα

5 Photo-absorption in weak lasers I 2 I 1 continuum states unoccupied bound states occupied bound states photo-absorption cross section σ(ω) Laser frequency ω No absorption if ω < lowest excitation energy

6 Standard linear response formalism H(t 0 ) = full static Hamiltonian at t 0 ( ) H t m E m 0 m full response function = exact many-body eigenfunctions and energies of system ( ) ( ) ( ) ( ) ( ) ( ) 0 ρˆ r m m ρˆ r 0 0 ρˆ r' m m ρˆ r' 0 χ( r,r'; ω ) = lim η 0 + m ω Em E0 + iη ω+ Em E0 + iη The exact linear density response ρ 1 (ω) = χ (ω) v 1 has poles at the exact excitation energies Ω = E m -E 0

7 Strong Laser Fields Intensities in the range of W/cm 2 Comparison: Electric field on 1st Bohr-orbit in hydrogen 1 e E = = 4πε a V m 1 I= ε 0cE = W cm a 0 e - Three quantities to look at: I. Emitted ions II. Emitted electrons III. Emitted photons

8 I. Emitted Ions Three regimes of ionization, ω depending on Keldysh parameter γ= : (a.u.) E Multiphoton Tunneling Over the barrier γ >> 1 γ 1 γ << 1

9 Multiphoton-Ionization (He) Walker et al., PRL 73, 1227 (1994) λ = 780 nm

10 Momentum Distribution of the He 2+ recoil ions

12 Ψ ( p ) 2 1,p 2,t of the He atom (M. Lein, E.K.U.G., V. Engel, J. Phys. B 33, 433 (2000))

13 Ψ ( p ) 2 1,p 2,t of the He atom (M. Lein, E.K.U.G., V. Engel, J. Phys. B 33, 433 (2000))

14 Wigner distribution W(Z,P,t) of the electronic center of mass for He atom (M. Lein, E.K.U.G., V. Engel, PRL 85, 4707 (2000)) v Laser (z,t) = E z sin ωt I = W/cm 2 λ = 780 nm

15 II. Electrons: Above-Threshold-Ionization (ATI) Ionized electrons absorb more photons than necessary to overcome the ionization potential (IP) Ekin = n+ s ω IP ω Photoelectrons: ( ) Equidistant maxima in intervals of : IP ω Agostini et al., PRL 42, 1127 (1979)

16 He: Above threshold double ionization M. Lein, E.K.U.G., V. Engel, PRA 64, (2001) p 2 (a.u.) p 1 (a.u.) p 1 (a.u.)

of the He model atom by a 250 nm pulse with intensity 10 15 W/cm 2.

17 Role of electron-electron interaction M. Lein, E.K.U.G., and V. Engel, Laser Physics 12, 487 (2002) Two-electron momentum distribution for double ionization of the He model atom by a 250 nm pulse with intensity W/cm 2. Two-electron momentum distribution for double ionization of the He model atom with non-interaction electrons by a 250 nm pulse with intensity W/cm 2.

18 III. Photons: High-Harmonic Generation Emission of photons whose frequencies are integer multiples of the driving field. Over a wide frequency range, the peak intensities are almost constant (plateau). log(intensity) IP U p ω ω ω ω ω ω 5 ω

19 Even harmonic generation due to nuclear motion (a) Harmonic spectrum generated from the model HD molecule driven by a laser with peak intensity W/cm 2 and wavelength 770 nm. The plotted quantity is proportional to the number of emitted phonons. (b) Same as panel (a) for the model H 2 molecule. HD T. Kreibich, M. Lein, V. Engel, E.K.U.G., PRL 87, (2001) H 2

20 Molecular Electronics Dream: Use single molecules as basic units (transistors, diodes, ) of electronic devices left lead L central region C right lead R

21 Molecular Electronics Dream: Use single molecules as basic units (transistors, diodes, ) of electronic devices left lead L central region C right lead R Bias between L and R is turned on: U(t) A steady current, I, may develop as a result. V for large t Calculate current-voltage characteristics I(V)

22 Hamiltonian for the complete system of N e electrons with coordinates (r 1 r ) r N and N e n nuclei with coordinates (R 1 R N ) R, masses n M 1 M and charges Z N n 1 Z. N n with Hˆ = T ˆ (R) + W ˆ (R) N n nn + T ˆ (r) + W ˆ (r) + U ˆ (R,r) e ee en n e n ˆ ν i T ˆ ˆ n = Te = Wnn = Wˆ ee 2 N 2 N 1 ZZ μ ν 2M 2m 2 R R ν= 1 ν i= 1 μ, ν μ ν μ ν 1 1 Z = = 2 r r r R Ne Ne Nn ˆ ν Uen j,k j k j= 1 ν= 1 j ν j k Time-dependent Schrödinger equation ( ) ( ) i Ψ ( r,r,t) = H( r,r) + Vexternal ( r,r,t) ψ r,r,t t

23 Why don t we just solve the many-particle SE? Example: Oxygen atom (8 electrons) Ψ ( r,, ) 1 r 8 depends on 24 coordinates rough table of the wavefunction 10 entries per coordinate: entries 1 byte per entry: bytes bytes per DVD: DVDs 10 g per DVD: g DVDs =10 9 t DVDs

24 ESSENCE OF DENSITY-FUNTIONAL THEORY Every Every observable observable quantity quantity of of a a quantum quantum system system can can be be calculated calculated from from the the density density of of the the system system ALONE ALONE The The density density of of particles particles interacting interacting with with each each other other can can be be calculated calculated as as the the density density of of an an auxiliary auxiliary system system of of non-interacting non-interacting particles particles

Time-dependent density functional theory

Time-dependent density functional theory E.K.U. Gross Max-Planck Institute for Microstructure Physics OUTLINE LECTURE I Phenomena to be described by TDDFT Some generalities on functional theories LECTURE