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 II Basic theorems of TDDFT LECTURE III TDDFT in the linear-response regime: Calculation of optical excitation spectra Beyond linear reponse: TD Electron Localization Function

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

Reminder: 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

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

Strong Laser Fields Intensities in the range of 10 13 10 16 W/cm 2 Comparison: Electric field on 1st Bohr-orbit in hydrogen 1 e E = = 4πε a 2 0 0 9 5.1 10 V m 1 I = ε 0cE = 3.51 10 W cm 2 2 16 2 + a 0 e - Three quantities to look at: I. Emitted ions II. Emitted electrons III. Emitted photons

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

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

Momentum Distribution of the He 2+ recoil ions

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))

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 Agostini et al., PRL 42, 1127 (1979) ω

He: Above threshold double ionization M. Lein, E.K.U.G., V. Engel, PRA 64, 23406 (2001) p 2 (a.u.) p 1 (a.u.) p 1 (a.u.)

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 10 15 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 10 15 W/cm 2.

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 + 3.17U p ω ω ω ω ω ω 5 ω

Even harmonic generation due to nuclear motion (a) Harmonic spectrum generated from the model HD molecule driven by a laser with peak intensity 10 14 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, 103901 (2001) H 2

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

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)

Hamiltonian for the complete system of N e electrons with coordinates (r 1 r N e ) r and N n nuclei with coordinates (R 1 R N n ) R, masses M 1 M N n and charges Z 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 First approximation: Clamp the nuclei or treat them classically

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: 10 24 entries 1 byte per entry: 10 24 bytes 10 10 bytes per DVD: 10 14 DVDs 10 g per DVD: 10 15 g DVDs = 10 9 t DVDs

Motivation Functional Theories MBPT RDMFT DFT G(r, r', t t' ) + γ (r,r ') = G(r,r ',0 ) ρ( r) = γ(r, r)

Motivation Functional Theories G(r, MBPT r', t t' Functional: Φ xc [G] or Σ xc [G] ) RDMFT + γ (r,r ') = G(r,r ',0 ) Functional: E xc [γ] DFT ρ( r) = γ(r, Functional: E xc [ρ] or v xc [ρ] r)

Motivation Functional Theories MBPT RDMFT DFT G(r, r', t t' ) Functional: Φ xc [G] or Σ xc [G] easy (e.g. GW) + γ (r,r ') = G(r,r ',0 ) Functional: E xc [γ] difficult ρ( r) = γ(r, r) Functional: E xc [ρ] or v xc [ρ] very difficult

Motivation Functional Theories MBPT RDMFT DFT G(r, r', t t' ) Functional: Φ xc [G] or Σ xc [G] easy (e.g. GW) numerically heavy + γ (r,r ') = G(r,r ',0 ) Functional: E xc [γ] difficult moderate ρ( r) = γ(r, r) Functional: E xc [ρ] or v xc [ρ] very difficult light

Four steps needed Step 1: Basic Theorems, exact features Step 2: Find approximate functionals for ( ) ( ) vxc ρ r't' rt Step 3: Write code that solves the TDKS equations Step 4: Run code for interesting systems/questions

Static Density Functional Theory: Some remarks

compare ground-state densities ρ(r) resulting from different external potentials v(r). ρ(r) v(r) QUESTION: Are the ground-state densities coming from different potentials always different?

A G Ã v(r) Ψ (r 1 r N ) ρ (r) single-particle potentials having nondegenerate ground state ground-state wavefunctions ground-state densities Hohenberg-Kohn-Theorem (1964) G: v(r) ρ (r) is invertible

HOHENBERG-KOHN THEOREM 1 1 1. v(r) ρ(r) one-to-one correspondence between external potentials v(r) and ground-state densities ρ(r) 2. Variational principle Given a particular system characterized by the external potential v 0 (r). Then the solution of the Euler-Lagrange equation δ δρ ( r) E ρ = 0 HK [ ] yields the exact ground-state energy E 0 and ground-state density ρ 0 (r) of this system 3. 3 HK [ ρ= ] [ ρ+ ] ρ( ) vo ( r) E F r dr F[ρ] is UNIVERSAL. In practice, F[ρ] needs to be approximated

Expansion of F[ρ] in powers of e 2 F[ρ] = F (0) [ρ] + e 2 F (1) [ρ] + e 4 F (2) [ρ] + where: F (0) [ρ] = T s [ρ] (kinetic energy of non-interacting particles) ( r) ρ( r' ) 2 2 ( 1 ) e ρ 3 3 e F [ ρ ] = d rd r' Ex 2 + ρ r r' [ ] (Hartree + exchange energies) i= 2 i 2 ( i ( e ) F ) [ ρ ] = Ec [ ρ] (correlation energy) ( r) ρ( r' ) 2 e ρ 3 3 F[ ρ ] = Ts[ ρ+ ] d rd r' Ex Ec 2 + ρ+ ρ r r' [ ] [ ]

By construction, the HK mapping is well-defined for all those functions ρ(r) that are ground-state densities of some potential (so called V-representable functions ρ(r)). QUESTION: Are all reasonable functions ρ(r) V-representable? V-representability theorem (Chayes, Chayes, Ruskai, J Stat. Phys. 38, 497 (1985)) On a lattice (finite or infinite), any normalizable positive function ρ(r), that is compatible with the Pauli principle, is (both interacting and noninteracting) ensemble-v-representable. In other words: For any given ρ(r) (normalizable, positive, compatible with Pauli principle) there exists a potential, v ext [ρ](r), yielding ρ(r) as interacting ground-state density, and there exists another potential, v s [ρ](r), yielding ρ(r) as non-interacting ground-state density. In the worst case, the potential has degenerate ground states such that the given ρ(r) is representable as a linear combination of the degenerate ground-state densities (ensemble-v-representable).

HK 1-1 mapping for interacting particles HK 1-1 mapping for non-interacting particles v [ ρ]( r ) ρr ( ) v [ ρ]( r) ext s Kohn-Sham Theorem Let ρ o (r) be the ground-state density of interacting electrons moving in the external potential v o (r). Then there exists a local potential v s,o (r) such that non-interacting particles exposed to v s,o (r) have the ground-state density ρ o (r), i.e. 2 + vs,o r ϕ j r = ϕ j j r 2 proof: ( ) ( ) ( ) ( ) = [ ]( ) v r v ρ r s,o s o Uniqueness follows from HK 1-1 mapping Existence follows from V-representability theorem, N ( ) ( ) ρo r = ϕj r j (with lowest ) j 2

Define v xc [ρ](r) by the equation ( r' ) ρ 3 v [ ρ]( r) = : v [ ρ]( r) + d r' v ρ r s ext + r r' xc KS equations vh [ ρ]( r) [ ]( ) v s [ρ] and v ext [ρ] are well defined through HK. 2 + vext ρo r + H o + xc ρo r ϕ j r = ϕ j j r 2 v r [ ]( ) v [ ρ ]( r) v [ ]( ) ( ) ( ) o ( ) fixed ρo r = ϕj r to be solved selfconsistently with ( ) ( ) 2 Note: The KS equations do not follow from the variational principle. They follow from the HK 1-1 mapping and the V-representability theorem.

Variational principle gives an additional property of v xc : xc [ ]( ) v ρ r o = δe [ ρ] ( ) xc δρ r ρ o where [ ] [ ] 1 ρ( r ) ρ( r' ) = 2 r - r' [ ] 3 3 Exc ρ : F ρ d rd r' Ts ρ Consequence: Approximations can be constructed either for E xc [ρ] or directly for v xc [ρ](r).