Basics Basics of of time-dependent DFT DFT and and applications

Size: px

Start display at page:

Download "Basics Basics of of time-dependent DFT DFT and and applications"

Rudolph Ramsey
6 years ago
Views:

1 Basics Basics of of time-dependent DFT DFT and and applications E. K. U. GROSS Freie Universität Berlin Arbeitstagung für Theoretische Chemie Mariapfarr, Feb.00

2 PHENOMENA TO BE DESCRIBED WITH TDDFT

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

6 Multiphoton-Ionization (He) λ = 780 nm Walker et al., PRL 73, 17 (1994)

10 1D MODEL Restrict motion of electrons and nuclei to 1D (along polarization axis of laser) Replace in H(r, R) all 3D coulomb interactions by soft 1D interactions (Eberly et al) 1 1 x + y + z α + z α = constant

11 Two goals of 1D calculations 1. Qualitative understanding of physical processes, such as double ionization of He. Exact reference to test approximate xc functionals of time-dependent density functional theory

12 M. Lein, E. K. U. G., V. Engel, J. Phys. B 33, 433 (000)

13 M. Lein, E. K. U. G., V. Engel, J. Phys. B 33, 433 (000)

14 M. Lein, E. K. U. G., V. Engel, PRL 85, 4707 (000)

16 Schnürer et al., PRL 80, 336 (1998)

17 HHG Spectrum S ω ~ e iωt d dt tdψ t dt ( ) () () Ψ HD H (a) Harmonic spectrum generated from the model HD molecule driven by a laser with peak intensity W/cm and wavelength 770 nm. The plotted quantity is proportional to the number of emitted phonons. (b) Same as panel (a) for the model H molecule. T. Kreibich, M. Lein, V. Engel, E. K. U. G., PRL 87, (001)

18 Even-harmonic generation due to beyond-born-oppenheimer dynamics even harmonic generation is parity forbiden + ω not allowed + regard oriented H, D symmetry breaking for HD, not for H

20 H + in 1D v nuc (z) + E 0 z =R crit consequence: ionization-induced Coulomb explosion

21 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 Nn. with Tˆ n Wˆ Uˆ Hˆ = Tˆ ( ) ˆ n R + Wnn( R) + Tˆ () r + Wˆ () r + Uˆ ( R, r) e ee en N n Ne Tˆ i = e = ν M m nn en ν= 1 ν i= 1 = 1 ZZ N n Ne µ ν Wˆ ee = 1, R R µν µ ν jk, µ ν j k Ne N n Z = ν j= 1 ν= 1 r R j ν Time-dependent Schrödinger equation i Ψ ( rrt,, )= HrR (, )+ Vlaser( rrt,, ) ψ rrt,, t r j 1 r Ne Nn Vlaser( r, R, t)= rj ZνRν E f t cosωt () j ( ) ( ) = 1 ν = 1 k

22 Time-dependent Schrödinger equation i t ( )= ( )+ ( ) Ψ rrt,, HrR, V rrt,, ψ rrt,, laser Ne Nn Vlaser( r, R, t)= rj ZνRν E f t cosωt () j ( ) ( ) = 1 ν = 1 E = amplitude of electric field f(t) = temporal envelope of laser pulse Full numerical solution in 3D (barely) feasible for He atom H + H 3 ++ (Dundas,, Taylor, Parker, Smyth, J. Phys. B 3, L31 (1999) ) (Chelkowski, Zuo, Atabek, Bandrauk,, Phys. Rev. A 5, 977 (1995) ) ++ (Zuo, Bandrauk,, Phys. Rev. A 54, 1 (1996) ) For larger systems approximations are inevitable.

23 Time-dependent density-functional formalism (first: electrons only, nuclei fixed) E. Runge,, E.K.U.G., Phys. Rev. Lett. 5,, 997 (1984) HK theorem: 1-1 v ( rt) ρ( rt) The time-dependent density determines uniquely the time-dependent external potential and hence all physical observables KS theorem: The time-dependent density of the interacting system of interest can be calculated as density ρ rt of an auxiliary non-interacting (KS) system with the local potential j N ( ) = ( ) = 1 ϕ j rt ih rt h j vks rt j rt t ϕ ( )= + m [ ρ ]( ) ϕ [ ( )]( )= ( )+ v r'' t rt v rt d r' KS ρ 3 ( ) ρ rt ' r r' + ( ) v xc [ρ(r t )](r t)

24 proof (basic idea): r v ( rt) to be shown that is impossible r v '( rt) r ρ( rt ) i.e. v(r t) v (r t) + c(t) ρ(r t) ρ (r t) r r r r v ( rt) j( rt) ρ( rt) r r r r v ' ( rt) j'( rt) ρ'( rt) use r r i jrt φ t jr ˆ r, Ht ˆ φ t t [ ] () ( ) = () ( ) () equation of motion for j and to show that ρ t r j t r r = div j ( t) r j ' t t= t t= t 0 0 ρ and ρ will become different from each other infinitesimally later that t 0 and ρ(t) continuity equation ρ t t ρ' t 0 0 ρ (t) ρ(t) t t 0 t

25 r Simplest possible approximation for v xc [ρ]( rt ) Adiabatic Local Density Approximation (ALDA) ALDA r hom v t v n xc : xc, stat ( ) = ( ) r n= ρ( t) hom v xc, stat = xc potential of static homogeneous e-gas Approximation with correct asymptotic -1/r behavior: time-dependent optimized effective potential C. A. Ullrich,, U. Gossmann,, E.K.U.G., PRL 74,, 87 (1995)

26 TD optimized effective potential C. A. Ullrich,, U. Gossmann,, E.K.U.G., PRL 74,, 87 (1995) Starting point: A given approximation for in terms of TD orbitals A xc ϕ1... ϕn example: TDHF Ax [ ϕ1... ϕn]= [ ] 1 N jk, = 1 t t dt dr dr' ( ) ( ) ( ) ( ) * * ϕj r', t ϕk r', t ϕj r, t ϕk r, t r r' Determine that very local potential v OPM (r,t) whose orbitals ψ j (r,t) resulting from i ψ t j( r, t)= + νopm( r, t) ψj r, t make the total-action functional stationary, where N t 3 * A ϕ ϕn dt d rϕj r t i * [ 1... ]= (, ) + ϕ j rt, t t 1 j= 1 3 dt drρ( rt, ) ν ( rt, ) 1 t 1 ext ( ) ( ) ( ) ρ rt, ρ r', t dt dr dr' Axc ϕ1... ϕ r r' N ( ) A xc [ ϕ ϕ N ] 1... [ ]

27 Time-dependent DFT in the linear response regime and excited states For times t t 0 : System in ground state of v 0 (r) Density is the ground-state density ρ 0 (r) For times t > t 0 : Total external potential: v (r t) = v 0 (r) + v 1 ( r t) (with v 1 (r t 0 ) = 0) density: ρ (r t) = ρ 0 (r) + δρ(r t) δρ(r t) = ρ 1 (r t) + ρ (r t) + ρ 3 (r t) + linear second order... density response to the perturbation v 1

28 from linear density response ρ 1 calculate: 1. Dynamical polarizability αω e ( )= ρ ( ω ) E z r r, 1 d 3 r. Photo-absorption cross section σω ( )= 4πω c Im α

29 Standard Response Equation ρ 1 3 ( r t) = dt d r χ(r t,r t ) v ( r t ) KS - Alternative full response function of the interacting (inhomogeneous) system very hard to calculate 3 () 1 ρ 1 χ s (r t,r t ) s self-consistent ( r t) = dt d r v ( r t ) response function of the noninteracting (KS) system relatively easy to calculate () 1 () 1 () 1 v ( rt) = v ( rt) + v ( rt) + v ( rt) s 1 H xc 3 3 = v( rt) + drwrr (, ) ρ ( rt ) + dt dr f ρ ( rt xc (r t,r t ) ) 1 f xc (r t,r t ) is the 1 st -order term in a functional Taylor expansion of v xc [ρ](r t) around ρ 0 (r): v xc [ρ](r t) = v xc [ρ 0 ](r t) + dt d 3 r δv xc (r t) δρ(r t ) ρ=ρ0 (ρ(r t ) - ρ 0 (r )) Note: This is an exact representation of the linear density response

30 Total photo-absorption cross section for Xenon above the 4d-threshold (Zangwill & Soven 1980) Approximation used for f xc : Adiabatic LDA ALDA r i.e. v t d hom xc (, )= ( ( )) xc dρ ρε ρ r ρ= ρ( rt, ) ALDA rr r r f r r d hom xc (,'; ω)= δ( ') ( xc dρ ρε ( ρ )) ρ ρ r r hom = δ ' f ρ, q 0, ω 0 xc ( ) ( ) = = no frequency dependence! r = ( ) 0 ρ ρ r = ( ) 0

31 Properties of exact f hom xc q,ω ( ) hom 1. f q d xc ( = 0, ω = 0)= ( xc( )) f 0 (ρ) dρ ρε ρ (compressibility sum rule). f hom xc ( q= 0, ω= 4 )= ρ 5 + 6ρ 13 3 d εxc dρ ρ d εxc dρ ρ ( ρ) 13 ( ρ) 3 f (ρ) (3 rd frequency-moment sum rule) ( )< ( )< 3. f ρ f ρ for all ρ 0 0 (according to best approximations known for ε xc (ρ)) ) 4. hom hom Re fxc ( q, ω)= Re fxc ( q, ω) hom hom Im f ( q, ω)= Im f q, ω xc xc ( )

32 Properties of exact f hom xc q,ω ( ) 5. Kramers - Kronig relations hom hom Re f ( q, ω) f ( q, )= Ρ xc xc hom Im f ( q, ω)= Ρ xc ( ) hom dω' Im fxc q, ω' π ω ω' dω' Re fxc( q, ω' ) fxc q, π ω ω' ( ) 6. hom Im f q, c xc ( = 0 ω), c = 3π ω 3 ω 15 (follows from perturbation expansion of irreducible polarization to nd order in e ) 7. ( ) hom xc = 0 ω f + 3 Re f q, ω c ω (from 5 and 6)

33 Parametrization for long wavelength limit Im f q 0, ω ( ) = xc = αω 1+ βω ( ) 54 with ( ) ( ) ( )= ( ) ( ) 53 αρ Af ρ f 0 ρ ( )= ( ) ( ) 43 βρ Bf ρ f 0 ρ A, B > 0 independent of ρ KK ( ) = + R f q= 0, ω f α π β r e xc E r 1 r r r, , 1 r := 1+ βω with, E, Π = elliptic integrals of nd, 3 rd kind

34 ALDA r = ALDA r =4 E.K.U.G., W. Kohn,, PRL 55,, 850 (1985) N. Iwamoto, E.K.U.G., Phys. Rev. B 35,, 3003 (1987) ALDA

35 Standard linear response formalism H(t 0 ) = full static Hamiltonian at t 0 ( ) = exact many-body Ht m E m 0 m eigenfunctions and energies of system full response function ( ) = lim χ rr,'; ω η + () ( ) 0 ˆ ρ r m m ˆ ρ r' 0 ω ( E E )+ iη 0 m m 0 ( ) ( ) 0 ˆ ρ r' m m ˆ ρ r 0 ω + ( E E )+ iη m 0 The exact linear density response ρ 1 (ω)) = χ ^ (ω)) v 1 (ω) has poles at the exact excitation energies Ω = E m - E 0 goal: Use the TDDFT representation of ρ 1 (ω) to calculate the excitation energies Ω = E m - E 0

36 Excitation energies from TDDFT exact representation of linear density response: ( ) ( )= ( ) ( )+ ( )+ ( ) ( ) ρ ω χˆ ω v ω Wˆ ρ ω fˆ ω ρ ω 1 KS 1 1 xc 1 ^ denotes integral operators i.e. ˆ rr r 3 f ρ f r,' r ρ ' d r' where with χˆ xc KS ( ) ( ) 1 xc 1 rr ( rr,'; ω)= jk, rr Mjk( r,' r ) ω ( εj εk)+ iη r r r * r r * r M (,' )= f f ϕ ϕ ' ϕ ' ϕ ( ) ( ) ( ) ( ) ( ) jk k j j j k k f m = 1 if 0 if ϕ ϕ m m is occupied in KS ground state is unoccupied in KS ground state ε j ε k KS excitation energy

37 ( 1 ˆ χ ˆ ( ω) [ Wˆ + fˆ ( ω) ]) ρ 1( ω )= χ ˆ ( ω ) v1( ω ) KS Clb xc KS ρ 1 (ω) for ω Ω (exact excitation energy) but right-hand side remains finite for ω Ω hence ( 1 ˆ χˆ ( ω ) [ Wˆ + fˆ ( ω ) ]) ξ( ω)= λ( ω) ξ( ω) KS Clb xc λ(ω) 0 for ω Ω This condition rigorously determines the exact excitation energies, i.e., ( ( )[ + ( )]) KS Clb xc ( )= 1ˆ χˆ Ω W ˆ fˆ Ω ξ Ω 0

38 This equation is rigorously equivalent to: (See T. Grabo, M. Petersilka, E. K. U. G., J. Mol. Struc. (Theochem) 501, 353 (000)) q' ( A ( Ω)+ ) = qq ' q qq ' q ' ωδ β Ωβ where 3 3 qq' q' q = () A α d r d r' Φ r q r 1 f r,', r r' r' + ( Ω) Φ xc q' ( ) q = ( jk, ) * Φ q ϕ k ϕ j r r r ()= () () α q = f k f j ωq = εk εj double index

39 Single-pole approximation ( ) j k Expand all quantities about one KS pole ε ε 0 0 Mjk 0 0 ˆχKS ( ω) ω ( ε j ε k )+ i η higher order terms Ω= ( ε ε )+ j k K 0 0 * * K d r d r' ϕ r ϕ r' ϕ r' ϕ r ( ) ( ) ( ) ( ) 3 3 j0 j0 k0 k0 1 fxc r r r r + ( ),' '

40 Excitation energies of CO molecule State Ω expt KS-transition KS KS + K A a I D a' e d 1 Π Σ Π Π Σ Π Π Σ Σ approximations made: vlda xc and f ALDA xc

41 Atom Experimental Excitation Energies 1 S 1 P (in Ry) KS energy differences KS (Ry) KS + K Be Mg Ca Zn Sr Cd from: M. Petersilka, U. J. Gossmann, E.K.U.G., PRL 76, 11 (1996) E = KS + K j - K= d r d r' ϕ ( r) ϕ ( r' ) ϕ ( r' ) ϕ ( r) k f r r r r + ( ),' ' j j k k xc

42 Three approximations necessary a) approximation for v xc to calculate {ϕ j }, {ε j } b) approximation for f xc c) single-pole approximation (truncation of Laurent expansion) Which one is most important? To investigate use (a): exact v xc (from Umrigar) for He and compare ALDA (b): f r r r r d dn n xc (,', ω)= δ ' ε f hom ( ) ( ) xc ρ r ( r,', r ω)= δ Nσ * ϕ r r k σ() ϕ σ( ') r r' ρ r ρ r' TDOEP x only k k xc σσ, ' σσ, ' σ σ ( ) () ( ) (c): Single-pole approximation Mjk 0 0 ˆχ KS ω ( εj εk )+ iη 0 0 Multiple-pole approximation χˆ KS JK, jk, Mjk ω ( εj εk)+ iη

43 Comparison of the excitation energies of neutral helium, calculated from the exact xc potential by using approximate xc kernels. All values are in Hartrees.

46 Failures of ALDA in the linear response regime H dissociation is incorrect: E( 1 1 Σ + u) E( Σ + ) g R 0 (in ALDA) (see: Gritsenko, van Gisbergen, Görling, Baerends, J. Chem. Phys. 113, 8478 (000)) response of long chains strongly overestimated (see: Champagne et al., J. Chem. Phys. 109, (1998) and 110, (1999)) in periodic solids, whereas, for insulators, divergent. ALDA f ( q, ωρ, )= c( ρ) xc f exact xc 0 1 q q

48 Time difference between first and last picture: 1/ period

49 By virtue of time-dependent Hohenberg-Kohn theorem, ALL observables are funtionals of the TD density some observables are easily expressed in terms of the density (no approximations involved) e.g. TD dipole moment 3 d() t ρ r, t zd r = ( ) photon spectrum ~ d( ω) other observables are more difficult to express in terms of the density (involving further approximation) e.g. ionization yields

50 Helium atom, λ = 616 nm, I = 3.5x1014 W/cm

51 physical picture absorbed photons emitted photon

52 Harmonic spectrum of Helium, λ = 616 nm, I = 7.0x1014 W/cm

53 Two-color laser Harmonic spectrum of Helium, λ = 616 nm, I = 7.0x10 14 W/cm Relative phase δ = 0 ω 0 and (ω 0 ) have same intensity total intensity of two-color laser same as intensity of single-color laser

54 Harmonic spectrum of Helium, λ = 616 nm, I = 7.0x1014 W/cm

55 Harmonic spectrum of Helium, λ = 616 nm, I = 3.5x10 14 W/cm ω 0 and ω 0 relative phase = δ min δ = δ max ω 0

56 Two-color laser: ω 0 and (3ω 0 ), I = I 1 + I 3 Harmonic spectrum of Helium, λ = 616 nm, I = 7.0x10 14 W/cm 1 I 1 = I Q+1 I 3 = I Q Q+1

57 Harmonic spectra of Hydrogen, λ = 1064 nm

58 Neon atom λ = 48 nm, I = 3x10 15 W/cm 1 optical cycle = 0.8 fs All valence electrons fully propagated

59 Neon: comparison Adiabatic local density approximation (ALDA) with time-dependent OPM ( TDKLI) Time evolution of the norm of the Ne s orbital (A), the Ne p 0 orbital (B) and the Ne p 1 orbital (C), calculated in the x-only TDKLI and ALDA schemes. Laser parameters: λ = 48 nm,,i = 3x10 15 W/cm, linear ramp over the first 10 cycles. One optical cycle corresponds to 0.8 femtoseconds.

61 Calculation of ionization yields (for He) divide R 3 in: a large analyzing volume A (where ρ(r t) is actually calculated A B and its complement B = R 3 \ A normalization of many-body wave function = dr1 dr ψ ( rr 1 t) + ψ + ψ A A A B B B p (0) (t) p (+1) (t) p (+) (t) pair correlation function grrt ( ): = 1 ψ( rrt) 1 ρ rt ρ rt ( ) ( ) 1 ( 0) p t dr1 drρ rt 1 ρ rtg ρ rrt 1 ( ) p t drρ rt - dr drρ rtρ rtgρ rrt ( ) ()= ( ) ( ) [ ]( ) A A ()= ( ) ( ) ( ) [ ]( ) A A A ()= ( ) + 3 p t 1 d rρ rt A A A 1 1 ( ) ( ) [ ]( ) ρ ρ ρ 3 3 dr1 dr rt 1 rtg rrt 1

62 x-only limit for g[ρ](r 1,r,t); two-electron-system: g [ ρ]( r 1, r, t 1 )= x only resulting ionization probabilities (mean-field expressions: P 0 (t) = N 1s (t) P +1 (t) = N 1s (t) (1- N 1s (t)) P + (t) = (1- N 1s (t)) where: 1 N 1s (t) := d 3 r ρ(r, t) = d 3 r φ 1s (r, t) A A

63 Correlation Contributions [ ]( )= [ ]( ) g ρ r, r, t g ρ r, r, t 1 x only 1 + g c [ρ](r 1,r,t) exactifies the mean-field expressions: P 0 (t) = N 1s (t) + K(t) P +1 (t) = N 1s (t) (1- N 1s (t)) - K(t) P + (t) = (1- N 1s (t)) + K(t) correlation correction: K(t) := 1 d 3 r 1 d 3 r ρ(r 1, t)ρ (r, t) g c [ρ] (r 1, r, t) A A

64 P(He + ) at the end of pulse λ = 780 nm

65 P(He ++ ) at the end of pulse λ = 780 nm

67 The calculation involves two approximate functionals: 1. The xc potential v xc [ρ](r t). The pair correlation function g[ρ](r 1 r t) Which approximation is more critical?

68 1D Helium atom (with soft Coulomb interaction) (Lappas, van Leeuwen, J. Phys. B 31, L49 (1998) P(He + ) exact P(He ++ ) exact P(He + ) with exact density and g=1/ P(He ++ ) with exact density and g=1/

69 Strategy for the construction of improved functionals 1. Derive exact properties of V xc [ρ]. Construct approximate functionals such that they satisfy the rigorous constraints Example Look at system from the point of view of a moving reference frame whose origin is given by x(t). Density as seen from moving frame: ρ (r t) = ρ (r-x(t), t) Galilei invariance implies: V xc [ρ ](r t) = V xc [ρ](r-x(t),t)

70 Exact properties of xc functionals Hˆ = Tˆ + Wˆ + Vˆ ext d r r = Vext() r dt 3 r r 3 r r drρ rtr, drρ rt, V ( rt, ) Ehrenfest theorem for interacting system: i.e. d dt ( ) = ( ) Ehrenfest theorem for H ˆ ˆ ˆ s = T+ V non-interacting (KS) system: d dt ext 3 r 3 r r drρ rtr, drρ rt, V( rt, ) ( ) = ( ) s s s s ➀ ➁ V s KS potential corresponding to V ext ρ = ρ s l.h.s. of ➀ = l.h.s. of ➁ r r 3 3 drρ rt, V (,) rt drρ rt, V(,) rt use: ( ) = ( ) ext r 3 = drρ( rt, ) V + V + V r 3 drρ ( rt, ) VH ( rt, )= 0 r 3 drρ rt, V rt, ( ) ( )= xc s ( ) ext H xc 0

71 An exchange-correlation potential with memory (Dobson, Bünner, E.K.U.G., PRL 79, 1905 (1997) xc force r F xc ( rt) ensured by ansatz: satisfies r 3 ρ( rtf ) xc ( rtd ) r= 1 F xc (r t)= ρ(r t) W xc [ρ](r t) [ ]( ) = ( ) hom W ρ rt dt' Π ρ rt', t t' xc LDA xc ( ) spatially local but temporally non-local violates Galileian invariance Assume that, in the electron liquid, memory resides not with each fixed point r, but rather within each fluid element. Thus the element which arrives at location r at time t remembers what happened to it at earlier times t when it was at the location R(t r,t) given by t' ( )= ( ) ( ) Rt' rt, J R,t ρ R,t W [ n]( rt) xc = dt' Π ( ) t t non local xc ρ R,t', ' hom 0 ( ) satisfies Galileian invariance

73 Molecules in Strong Laser Pulses A time-dependent multi-component DFT

74 What are the right densities to formulate DFT? first attempt: Nn Ne ρ( r) = N d R d rψ( R,) r e 1 NO GOOD: ρ(r)= const for all atoms, molecules and solids second attempt: r' = r R j= 1,..., N where j j CMN e R CMN N n α = = 1 N MR n ν= 1 α M center of mass of the nuclei Nn Ne ρ( r') = N d R d r' Ψ( R,') r e ν α 1 NOT GOOD ENOUGH : ρ(r )= spherical for all atoms, molecules and solids

75 Further transformation to body-fixed coordinate frame required: (α(r), β(r), γ(r)) Euler angles associated with rotation to the principal axes of nuclear inertia tensor r j = D(α, β, γ)(r j -R CMN ) = r j (R, r j ) rotational matrix Nn Ne ρ( r") = N d R d r" Ψ( R,") r e 1 is suitable for DFT formulation (i.e., characterizes the internal structure of the system)

76 Hamiltonian in terms of new coordinates Hˆ = Tˆ ( ) ˆ n R + Wnn( R) + Tˆ (") r + Wˆ (") r ^ e + U en (R, r ) ^ + T MPC (R, r ) ee form-invariant under r r ^ T MPC (R, r ) = mass polarization and Coriolis terms U en ( R, r") N e j= 1 N n = α= 1 Zα 1 D ( R) ( r" j+ RCMN( R) ) R α N n Ne r" Tˆ j MPC( R, r") = 1 R + M α α= 1 α j= 1 Rα r" j Tn ( R) Note: U en is a one-body potential w.r.t. electrons and an N n -body potential w.r.t. nuclei.

77 Densities for the case of diatomic molecule ρ(r, t) electron density (r measured from body-fixed nuclear CM frame) N(R, t) nuclear density (R = internuclear separation) Time-dependent multi-component KS theorem The densities ρ, N of the interactiong system of interest can be calculated as densities of a non-interacting (Kohn-Sham) system: ρ(,) rt = ϕ (,) rt NRt (, ) = χ( Rt, ) i ϕ t j(,) r t = h r + v [ s ρ, N] (,) r t ϕj(,) r t µ e j i χ t ( R, t) = h R + W[ s ρ, N] ( R, t) χ( R, t) µ n j

78 H H v(,) rt = v (,) rt + v (,) rt + v (,) rt s laser ee + v xc [ρ,n](r,t) ZZ H Ws ( R, t) = Wlaser ( R, t) + + Wen ( R, t) R 1 + W xc [ρ,n](r,t) H v (,) r t = d r' ee H en ρ(',) r t r r' v (,) r t = d RN( R,) t u (, r R) H W ( R, t) = d rρ( r, t) u ( r, R) en en en en Note: No Born-Oppenheimer surfaces to be calculated first attempt: Hartree approximation v xc 0 W xc 0

79 Nuclear Dynamics of H + in 1D in a 770 nm Laser pulse with correlation exact with correlation Hartree

81 Stationary two-component DFT some results for the H molecule using the different approximation schemes (electronic interaction is treated within LDA) comparison of the effective nuclear potential T. Kreibich, E. K. U. G., PRL 86, 984 (001)

82 Results for a λ = 8 nm (5.4 ev) 0 fs laser pulse comparison of the nuclear dynamics for different intensities R t 1 χχ drr N R, t ()= ( )

83 Results for a λ = 8 nm (5.4 ev) 0 fs laser pulse comparison of the ionization behavior for different intensities P t 1 drn r, t ion ()= ( ) box

84 comparison of the dissociation behavior for different intensities P t 1 drn R, t diss ()= ( ) box

85 time evolution of the nuclear wavepacket ( )= χ( ) NRt, Rt,

86 Orbital functionals for the static xc energy derived from TDDFT

87 MOTIVATION TDDFT very successfully describes molecular van der Waals coefficients calculated from: C 3h 6 duαa iu αb iu π = ( ) ( ) 0 (see, e.g., van Gisbergen, Snijders, Baerends, J. Chem. Phys. 103, 9347 (1995)) However, what one really wants is a functional E xc [ρ] such that E tot -E fragments ~ R -6 for R (LDA, GGAs, x-only-oep all yield ~ e -α R ) improved description of stretched H description of strongly correlated systems (Mott insulators)

88 ADIABATIC CONNECTION FORMULA ( )= N N + e ( i)+ i= 1 ik, = 1 ri i k ( λ=1 )= N N + e nuc( i) + i= 1 ik, = 1 i k H T v r H T v r = Hamiltonian of fully interacting system Choose v λ (r) such that for each λ the groundstate density satisfies ρ λ (r) = ρ λ=1 (r) Hence v λ=0 (r) = v KS (r) v λ=1 (r) = v nuc (r) 1 r λ λ λ 0 λ 1 Determine the response function χ (λ) (r,r';ω) corresponding to H(λ). Then r i k 1 r k E xc = 1 0 dλ 0 du 3 3 drdr ' π e r r' { ( λ χ ) ( r,'; r iu)+ ρ() r δ( r r' )}

89 Second ingredient : TDDFT ρ = χ v = χ v + W + f ρ 1 s s, 1 s 1 clb xc 1 ρ = χv 1 1 ( [ ] ) ( [ ] ) χv = χ v + W + f χv 1 s 1 clb xc 1 [ ] χ= χ + χ W + f χ s s clb xc and for 0 λ 1 : [ ] λ ( λ ) ( ) ( λ) χ = χ + χ λw + f χ s s clb xc

90 r s -dependent deviation of approximate correlation energies from the exact correlation energy per electron of the uniform electron. M. Lein, E. K. U. G., J. Perdew, Phys. Rev. B 61, (000).

91 truncate after first iteration: χ (λ) χ s + χ s [λ W clb + f xc (λ) ]χ s plug this approoximation into adiabatic connection formula, integrations over λ and ω can be done analytically Orbital functional for E c

92 Resulting v.d.w. coefficients C 6 system Calculated C 6 experiment He-He He-Ne Ne-Ne Li-Li Li-Na Na-Na H-He H-Ne H-Li H-Na Lein, Dobson, EKUG, J. Comp. Chem. ( 99)

93 Resulting Atomic Correlation energies (in a.u.) atom LDA new fctl exact He Be Ne Ar

94 Review articles on time-dependent DFT/excitation energies Density-functional theory of time-dependent phenomena. E. K. U. Gross, J. F. Dobson. M. Petersilka, in: Topics in Current Chemistry, vol. 181, edited by R. Nalewajski (Springer, 1996), p A guided tour of time-dependent DFT. K. Burke, E. K. U. Gross, in:springer Lectures Notes in Physics, vol. 500 (1998), p

95 CO-WORKERS 1D simulations Manfred Lein Volker Engel Time-dependent HK and KS theorems Erich Runge Construction of xc functional with memory Martin Bünner John Dobson Linear-response calculations, excitation energies Martin Petersilka Tobias Grabo Ulrich Gossmann Andrzej Fleszar Rudy Magyar Atoms in strong laser pulses Carsten Ullrich Steffen Erhard Partin Petersilka Heiko Appel Molecules in strong laser pulses Thomas Kreibich Robert van Leeuwen Nikitas Gidopoulos Jan Werschnik Daniele Varsano Miguel Marques Angel Rubio

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