Time-dependent density functional theory

Size: px

Start display at page:

Download "Time-dependent density functional theory"

Leslie Hubbard
6 years ago
Views:

1 Time-dependent density functional theory E.K.U. Gross Max-Planck Institute of Microstructure Physics Halle (Saale)

2 OUTLINE Phenomena to be described by TDDFT Basic theorems of TDDFT Approximate xc functionals: Exact adiabatic approximation TDDFT in the linear-response regime: -- Optical excitation spectra of molecules -- Excitonic effects in the optical spectra of solids -- Charge-transfer excitations and the discontinuity of the xc kernel

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

3 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

4 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

5 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

6 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

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

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

9 Momentum Distribution of the He 2+ recoil ions

11 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))

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 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

14 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

15 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.

16 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.

17 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

18 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

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

20 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) Investigate cases where no steady state is achieved

21 Ground-State Density Functional Theory 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?

22 Ground-State Density Functional Theory G A 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

23 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 Kohn-Sham Theorem Let ρ o (r) be the ground-state density of interacting electrons

24 HK 1-1 mapping for interacting particles HK 1-1 mapping for non-interacting particles v [ ρ]( r ) ρr v [ ρ]( r) ext 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 ) Note: The KS equations do not follow from the variational principle!!, j s 2

25 Time-dependent density-functional formalism (E. Runge, E.K.U.G., PRL 52, 997 (1984)) Basic 1-1 correspondence: 1-1 ρ ( rt) v rt KS theorem: The time-dependent density determines uniquely the time-dependent external potential and hence all physical observables for fixed initial state. The time-dependent density of the interacting system of interest can be calculated as density of an auxiliary non-interacting (KS) system with the local potential N 2 ρ rt = ϕ j j = 1 3 vs r't' rt v rt d r' rt 2 2 i ϕ j rt = + vs ρ rt ϕj rt t 2m [ ] ( r't) ρ ρ = + + vxc ρ r't' rt r r'

26 Proof of basic 1-1 correspondence between v( rt) and ρ( rt) F: Ψ t ρ rt define maps F: v( rt) Ψ ( t) G potentials v( rt) F wave functions ( t) Ψ ( rt) = Ψ o solve tdse with fixed Ψ t o F G: v rt ρ rt densities ρ( rt) ρ = Ψ( t) ρˆ ( r) Ψ( t) ρ ˆ = ψˆ ψˆ + ( r) ( r) ( r) s s s

27 complete 1-1 correspondence not to be expected! ( ˆ ˆ ˆ ) i Ψ ( t) = T + V( t) + W Ψ t t ( ˆ ˆ ˆ ) i Ψ 't = T + V't + W Ψ 't t ˆ ˆ Ψ ( t o ) =Ψ o Ψ 't ( o ) =Ψ o α i V't Vt Ct 't e t ( t) = + Ψ = Ψ ρ ' rt =ρ ( rt) i.e. no operator with α t = C t { ˆV ( t ) C ( t )} ( rt ) + ρ

28 If G invertible up to within time-dependent function C(t) Ψ= FG 1 ρ fixed up to within time-dependent phase i.e. i e α ( t) [ ] Ψ= Ψ ρ For any observable Ô [ ] ˆ [ ] [ ] ΨOˆ Ψ=ΨρO Ψρ = O ρ is functional of the density

29 THEOREM (time-dependent analogue of Hohenberg-Kohn theorem) The map G: v rt ρ rt defined for all single-particle potentials v( rt) which can be expanded into a Taylor series with respect to the time coordinate around t o is invertible up to within an additive merely time-dependent function in the potential.

30 Proof: to be shown: v( rt) ρ ( rt ) v' rt cannot happen! i.e. vˆ( rt) v' ˆ ( rt) + c( t) ρ( rt ) ρ' ( rt) potential expandable into Taylor series k k 0 v k ( rt ) v' : ( rt ) constant t= t t o j rt ρ j' ( rt) step 1 step 2 ( rt ) ρ' ( rt)

31 Step 1: Current densities j rt t ˆ j r t = Ψ Ψ ˆ 1 2i ( ) s s s s + + with j( r) = ψˆ ( r) ψˆ ( r) ψˆ ( r) ψˆ ( r) s Use equation of motion: ˆ i Ψ t O t Ψ t = Ψ t i Oˆ t t t + Oˆ t,hˆ t Ψ t ˆ i j( rt) ( t) = Ψ j( r ),Hˆ ( t) Ψ( t) t ˆ i j'rt 't = Ψ jr,h't ˆ Ψ't t ˆ note: j( rto) = j' ( rto) = Ψo j( r) Ψo jo( r) ρ =ρ = Ψ ρ Ψ ρ ( rt ) '( rt ) ˆ ( r) ( r) o o o o o

32 ˆ i j( rt) j' ( rt) = Ψ j( r ),Hˆ ( t ) H' ˆ ( t ) Ψ t t= to = Ψ ĵ( r ),V( t ) V' ( t ) Ψ = iρ r v rt v' rt k k t= t t o i j( rt) j' ( rt) 0 t t= t o o o o o o o o o o o o if v( rt) v' ( rt) constant holds for k=0 then j' ( t) j rt t o j' ( rt) j( t) t q.e.d.

33 k t= t t if v( r t k ) v '( r t) constant holds for k>0 use equation of motion k+1 times: o 2 ˆ i j( rt) i ( t) j,h ˆ ( t) = Ψ Ψ( t) t t ˆ t i j,h ˆ t j,h ˆ t,h ˆ t t t = Ψ + Ψ 3 i j rt i t i j,h t j,h t,h t t t t t = = Ψ ˆ + ˆ Ψ k+ 1 k i j ( rt ) j' ( rt ) i o ( r ) i v ( rt ) v' ( rt = ρ ) 0 t t t t = o t o j rt j' ( rt) constant q.e.d.

34 Step 2: densities Use continuity equation: t ρ ( rt ) ρ '( rt) = div j( rt) j' ( rt) k+ 2 k+ 1 ρ( rt ) ρ '( rt) = div j( rt) j' ( rt) t o t k+ 2 t= t k+ 1 t= t = div ρ r v rt v' rt k o k t= t t o o constant remains to be shown: div ρo ( r ) u ( r ) 0 if u ( r ) constant

35 Proof: by reductio ad absurdum div ρo r u r = 0 Assume: ( ) 3 dr ρo r u r 2 with u ( r ) constant = + = 3 dr u ( r ) div ρ o r u r ρo r u r u r ds ρ r u r 0 o 2 contradiction to u r constant

36 The TDKS equations follow (like in the static case) from: i. the basic 1-1 mapping for interacting and noninteracting particles ii. the TD V-representability theorem (R. van Leeuwen, PRL 82, 3863 (1999)). A TDDFT variational principle exists as well, but this is more tricky: R. van Leeuwen S. Mukamel T. Gal G. Vignale..

37 Simplest possible approximation for v [ ρ]( rt) Adiabatic Approximation: v rt : = v [n](r) adiab xc where stat v xc stat xc = xc potential of ground-state DFT xc n =ρ(r't) Example: v rt : = v n ALDA xc hom xc n =ρ(rt) Approximation with correct asymptotic -1/r behavior: time-dependent optimized effective potential (TDOEP) (C. A. Ullrich, U. Gossmann, E.K.U.G., PRL 74, 872 (1995))

38 Assess the quality of the adiabatic approximation by the following steps: Solve 1D model for He atom in strong laser fields (numerically) exactly. This yields exact TD density ρ(r,t). Inversion of one-particle TDSE yields exact TDKS potential. Then, subtracting the laser field and the TD-Hartree term, yields the exact TD xc potential. Inversion of one-particle ground-state SE yields the exact static KS potential, v KS-static [ρ(t)], that gives (for each separate t) ρ(r,t) as groundstate density. Inversion of the many-particle ground-state SE yields the static external potential, v ext-static [ρ(t)], that gives (for each separate t) ρ(r,t) as interacting ground-state density. Compare the exact TD xc potential of step 1 with the exact adiabatic approximation which is obtained by subtraction : v xc-exact-adiab (t) = v KS-static [ρ(t)] v H [ρ(t)] v ext-static [ρ(t)]

39 E(t) ramped over 27 a.u. (0.65 fs) to the value E=0.14 a.u. and then kept constant t = 0 t = 21.5 a.u. t = 43 a.u. Solid line: exact Dashed line: exact adiabatic M. Thiele, E.K.U.G., S. Kuemmel, Phys. Rev. Lett. 100, (2008)

40 4-cycle pulse with λ = 780 nm, I 1 = 4x10 14 W/cm 2, I 2 =7x10 14 W/cm 2 Solid line: exact Dashed line: exact adiabatic M. Thiele, E.K.U.G., S. Kuemmel, Phys. Rev. Lett. 100, (2008)

41 LINEAR RESPONSE THEORY t = t 0 : Interacting system in ground state of potential v 0 (r) with density ρ 0 (r) t > t 0 : Switch on perturbation v 1 (r t) (with v 1 (r t 0 )=0). Density: ρ(r t) = ρ 0 (r) + δρ(r t) Consider functional ρ[v](r t) defined by solution of interacting TDSE Functional Taylor expansion of ρ[v] around v o : [ ] = [ + v 1 ] ρ v rt ρ v rt 0 = ρ[ v 0 ] ( rt) δρ[ v ] ( rt) + v1 ( r't' ) δv ( r't' ) v 0 2 [ ] 1 δ ρ v rt 2 δv r't' δv r"t" 3 d r'dt' v1 r',t' v1 r",t" d r'd r"dt'dt" ρ2 ( rt) v 0 ρo ρ1 ( r) ( rt)

42 ρ 1 (r,t) = linear density response of interacting system [ ] δρ v rt χ ( rt,r't' ) : = = density-density response function of δv r't' interacting system v 0 Analogous function ρ s [v s ](r t) for non-interacting system δρ [ v ] ( rt) [ ] v S,1 vs,1 ( r't' ) δv ( r't' ) = + = + + S S 3 ρs v S rt ρs v S,0 rt ρs v S,0 rt d r'dt' S v S,0 rt,r't' : δρ [ v S] ( rt) ( r't' ) S χ S = = density-density response function of δvs v S,0 non-interacting system

43 GOAL: Find a way to calculate ρ 1 (r t) without explicitly evaluating χ(r t,r't') of the interacting system starting point: Definition of xc potential [ ] = [ ] [ ] [ ] v ρ rt : v ρ rt v ρ rt v ρ rt xc S ext H Notes: v xc is well-defined through non-interacting/ interacting 1-1 mapping. v S [ρ] depends on initial determinant Φ 0. v ext [ρ] depends on initial many-body state Ψ 0. In general, [ ] v = v ρ,φ,ψ xc xc 0 0 only if system is initially in ground-state then, via HK, Φ 0 and Ψ 0 are determined by ρ 0 and v xc depends on ρ alone.

44 [ ] [ ] [ ] ( ) δvxc ρ rt δvs ρ rt δvext ρ rt δ t t' = δρ r't' δρ r't' δρ r't' r r' ρ ρ ρ 0 0 0

45 [ ] [ ] [ ] ( ) δvxc ρ rt δvs ρ rt δvext ρ rt δ t t' = δρ r't' δρ r't' δρ r't' r r' ρ ρ ρ f ( rt,r't' ) 1 χ ( rt, r't' ) χ 1 ( rt,r't' ) W ( rt,r't' ) xc S C

46 [ ] [ ] [ ] ( ) δvxc ρ rt δvs ρ rt δvext ρ rt δ t t' = δρ r't' δρ r't' δρ r't' r r' ρ ρ ρ f ( rt,r't' ) 1 χ ( rt, r't' ) χ 1 ( rt,r't' ) W ( rt,r't' ) xc S C f + W =χ χ 1 1 xc C S

47 [ ] [ ] [ ] ( ) δvxc ρ rt δvs ρ rt δvext ρ rt δ t t' = δρ r't' δρ r't' δρ r't' r r' ρ ρ ρ f ( rt,r't' ) 1 χ ( rt, r't' ) χ 1 ( rt,r't' ) W ( rt,r't' ) xc S C χ S f + W =χ χ χ 1 1 xc C S

48 [ ] [ ] [ ] ( ) δvxc ρ rt δvs ρ rt δvext ρ rt δ t t' = δρ r't' δρ r't' δρ r't' r r' ρ ρ ρ f ( rt,r't' ) 1 χ ( rt, r't' ) χ 1 ( rt,r't' ) W ( rt,r't' ) xc S C χ S f + W =χ χ χ 1 1 xc C S χ f + W χ = χ χ S xc C S

49 χ = χ + χ W + f χ S S C xc Act with this operator equation on arbitrary v 1 (r t) and use χ v 1 = ρ 1 : 3 3 ρ1 ( rt ) = d r'dt'χ S ( rt,r't' ) v1 ( rt ) + d r"dt" WC ( r't',r"t" ) + fxc ( r't', r"t" ) ρ1 r"t" { } Exact integral equation for ρ 1 (r t), to be solved iteratively Need approximation for (either for f xc directly or for v xc ) f xc ( r't', r"t" ) = xc [ ] δv ρ r't' δρ r"t" ρ 0

50 adiab xc Adiabatic approximation [ ] static DFT [ ρ( t) ] v ρ rt := v rt ALDA e.g. adiabatic LDA: LDA v rt : = v ( ρ( rt )) = α 13 xc ALDA ALDA δvxc rt v fxc ( rt,r't' ) = = δ( r r' ) δ( t t' ) δρ r't' ρ r xc In the adiabatic approximation, the xc potential v xc (t) at time t only depends on the density ρ(t) at the very same point in time. xc ρ rt + ALDA xc ρ ρ r 0 0 = δ r r' δ t t' e 2 hom xc 2 n ρ0 ( r)

51 Total photoabsorption cross section of the Xe atom versus photon energy in the vicinity of the 4d threshold. Solid line: self-consistent time-dependent KS calculation [A. Zangwill and P. Soven, PRA 21, 1561 (1980)]; crosses: experimental data [R. Haensel, G. Keitel, P. Schreiber, and C. Kunz, Phys. Rev. 188, 1375 (1969)].

52 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

53 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 ω =χˆ ω v ω 1 1 has poles at the exact excitation energies Ω = E m - E 0

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

55 ( 1ˆ χˆ ) S ω Wˆ ˆ ˆ C + fxc ω ρ1 ω =χs ω v1 ω ρ 1 (ω) for ω Ω (exact excitation energy) but right-hand side remains finite for ω Ω ( 1ˆ χˆ ˆ ˆ ) S ω WC + f xc ω ξω=λωξω hence λ(ω) 0 for ω Ω This condition rigorously determines the exact excitation energies, i.e., ( ˆ ˆ ˆ ˆ ) S C xc 1 χ Ω W + f Ω ξ Ω = 0

56 This leads to the (non-linear) eigenvalue equation (See T. Grabo, M. Petersilka, E. K. U. G., J. Mol. Struc. (Theochem) 501, 353 (2000)) q' ( ) M Ω + ω δ β = Ωβ qq' q qq' q' q Mqq' = α q' d r d r'φ q r + fxc r, r',ω Φq' r' r r' q = j, a double index α = f f q a j Φ r =ϕ r ϕ r * q a j q = ε Equivalent to Casida equations if: ω-dependence of f xc is neglected orbitals are real-valued pure density-response is considered in Casida eqs. ω a ε j

57 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, 1212 (1996) E = KS + K j - k 1 r r' 3 3 K= d r d r' ϕj r ϕj r' ϕk r' ϕ k r + fxc r,r'

58 Excitation energies of CO molecule State Ω expt KS-transition KS KS + K A a I D a' e d 1 Π Σ 2Π Π Σ Π 2Π Σ Σ T. Grabo, M. Petersilka and E.K.U. Gross, J. Mol. Struct. (Theochem) 501, 353 (2000) approximations made: v xc LDA and f xc ALDA

59 Quantum defects in Helium E = 2 n 1 ( µ ) n 2 n [ a.u. ] Bare KS exact x-alda xc-alda (VWN) x-tdoep M. Petersilka, U.J. Gossmann and E.K.U.G., in: Electronic Density Functional Theory: Recent Progress and New Directions, J.F. Dobson, G. Vignale, M.P. Das, ed(s), (Plenum, New York, 1998), p

60 (M. Petersilka, E.K.U.G., K. Burke, Int. J. Quantum Chem. 80, 534 (2000))

61 Failures of ALDA in the linear response regime H 2 dissociation is incorrect: ( 1 + ) ( 1 + ) E Σ E Σ 0(in ALDA) u g R (see: Gritsenko, van Gisbergen, Görling, Baerends, JCP 113, 8478 (2000)) response of long chains strongly overestimated (see: Champagne et al., JCP 109, (1998) and 110, (1999)) in periodic solids, for insulators, f ALDA fxc q, ωρ, = c ρ exact xc q 0 1q 2 whereas, divergent. charge-transfer excitations not properly described (see: Dreuw et al., JCP 119, 2943 (2003))

62 How good is ALDA for solids? optical absorption (q=0) ALDA Solid Argon L. Reining, V. Olevano, A. Rubio, G. Onida, PRL 88, (2002)

63 OBSERVATION: In the long-wavelength-limit (q = 0), relevent for optical absorption, ALDA is not reliable. In particular, excitonic lines are completely missed. Results are very close to RPA. EXPLANATION: In the TDDFT response equation, the bare Coulomb interaction and the xc kernel only appear as sum (W C + f xc ). For q 0, W C diverges like 1/q 2, while f xc in ALDA goes to a constant. Hence results are close to f xc = 0 (RPA) in the q 0 limit. CONCLUSION: Approximations for f xc are needed which, for q 0, correctly diverge like 1/q 2. Such approximations can be derived from many-body perturbation theory (see, e.g., L. Reining, V. Olevano, A. Rubio, G. Onida, PRL 88, (2002)).

64 Excitons in TDDFT

65 TDDFT response equation (a matrix equation) v v f 1 1 ( q, ) 1 0 q, q 1 q xc q, 0 q, ε ω = +χ ω + ω χ ω exact equation f xc ( r, r',t t') δv ( r,t ) δρ( r',t') xc xc kernel χ ( q ω) 0, Kohn-Sham response function RPA is fxc = 0 ( q, ) 1 ( q, ) v( q) 1 v( q) ( q, ) 1 ε0 ω = +χ0 ω χ0 ω 1

66 Bootstrap kernel f bo ot xc ( q, ω) ( q, 0) v( q) ( q, ) ( q, ω= 0) ( q, ω= ) 1 1 ε ω= ε = = ε ω= 0 1 χ 0 ε 1 ( q, ) v v f ( q, ω) ω = 1 +χ0 q, ω q 1 q + xc χ0 q, ω 1 Sharma, Dewhurst, Sanna, EKUG, PRL 107, (2011)

74 Charge transfer excitations and the discontinuity of f xc

75 charge-transfer excitation A R B Ω CT IP CT excitation energy B 3 3 ϕa EA d r d r' 2 A r ϕ r' B r r' 2 (A) In exact DFT IP (A) = ε HOMO Ω CT ε Infinite R (B) (B) EA (B) = ε LUMO xc Finite (but large) R derivative discontinuity 2 B A 3 3 ϕ r ϕ r' A B ε d r d r' MOL MOL r (A) = ε MOL (B) = ε MOL r' 2

76 Ω CT ε 2 ( B) ( A ) ϕ r ϕ r' ε 3 d r 3 A B d r' MOL MOL r r' 2 ~ 1/R

77 Ω CT ε 2 ( B) ( A ) ϕ r ϕ r' ε 3 d r 3 A B d r' MOL MOL r r' 2 ~ 1/R In TDDFT (single-pole approximation) Ω CT ε ε ( B) ( A ) MOL MOL 3 3 * d r d r' ϕ (r') ϕ (r')f (r,r', Ω ) ϕ (r) ϕ A B xc CT A * B (r)

78 Ω CT ε 2 ( B) ( A ) ϕ r ϕ r' ε 3 d r 3 A B d r' MOL MOL r r' 2 ~ 1/R In TDDFT (single-pole approximation) Ω CT ε ε ( B) ( A ) MOL MOL 3 3 * d r d r' ϕ (r') ϕ (r')f (r,r', Ω ) ϕ (r) ϕ A B xc CT A * B (r) Exponentially small Exponentially small

79 Ω CT ε 2 ( B) ( A ) ϕ r ϕ r' ε 3 d r 3 A B d r' MOL MOL r r' 2 ~ 1/R In TDDFT (single-pole approximation) Ω CT ε ε ( B) ( A ) MOL MOL 3 3 * d r d r' ϕ (r') ϕ (r')f (r,r', Ω ) ϕ (r) ϕ A B xc CT A * B (r) Exponentially small Exponentially small CONCLUSIONS: To describe CT excitations correctly v xc must have proper derivative discontinuities f xc (r,r ) must increase exponentially as function of r and of r for ω Ω CT

80 Discontinuity of v xc (r): v () r = v r + + xc xc xc Δ xc is constant throughout space Discontinuity of f xc (r,r ): f ( rr, ) = f ( rr, ) + g ( r) + g ( r ) + xc xc xc xc

81 This exponentially increasing behavour is achieved by the discontinuity: ϕ L r g xc r ~ ~e n r ( s ) 2 2A I r r

82 He-Be neutral diatomic in 1D M. Hellgren, EKUG, PRA 85, (2012)

83 He-Be neutral diatomic in 1D ω =ω + 2 qv+ f ω q ω + 1R CT q x q q x M. Hellgren, EKUG, PRA 85, (2012)

Time-dependent density functional theory

Time-dependent density functional theory E.K.U. Gross Max-Planck Institute of Microstructure Physics Halle (Saale) OUTLINE Phenomena to be described by TDDFT: Matter in weak and strong laser fields Basic