Time-dependent density functional theory

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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: Matter in weak and strong laser fields Basic theorems of TDDFT Approximate xc functionals of TDDFT the 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 in molecules and the discontinuity of the xc kernel THANKS Manfred Lein Volker Engel Erich Runge Stephan Kümmel M. Thiele Martin Petersilka Ulrich Gossmann Tobias Grabo Sangeeta Sharma Kay Dewhurst Antonio Sanna Maria Hellgren

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 Weak lasers: Optical spectra of finite systems (atoms, molecules) I 2 I 1 continuum states unoccupied bound states occupied bound states photo-absorption cross section σ(ω) Laser frequency ω

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 Weak lasers: Optical spectra of periodic solids (insulators)

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 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 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

16 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

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

18 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

19 Hamiltonian for the complete system of N e electrons with coordinates (r 1 r ) r and N 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

20 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

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

22 Ground-State Density Functional Theory

23 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?

24 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

25 For any observable Ô [ ] ˆ [ ] [ ] ΨOˆ Ψ=ΨρO Ψρ = O ρ is functional of the density

26 For any observable Ô [ ] ˆ [ ] [ ] ΨOˆ Ψ=ΨρO Ψρ = O ρ is functional of the density For general observables: Need to find approximations for the functionals O[ρ] representing the observables.

27 For any observable Ô [ ] ˆ [ ] [ ] ΨOˆ Ψ=ΨρO Ψρ = O ρ is functional of the density For general observables: Need to find approximations for the functionals O[ρ] representing the observables. Easy observables: E tot [ρ], D[ρ] = ʃ ρ(r) r d 3 r, More difficult: spectroscopic information, gaps,.

28 ESSENCE OF DENSITY-FUNTIONAL THEORY Every observable quantity of a quantum system can be calculated from the density of the system ALONE (Hohenberg, Kohn, 1964) The density of particles interacting with each other can be calculated as the density of an auxiliary system of non-interacting particles (Kohn, Sham, 1965)

29 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

30 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).

31 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'

32 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

33 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.

34 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)

35 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..

36 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)

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) Every approximate ground-state xc potential yields an adiabatic approximation to the time-dependent xc potential Deficiencies of approximate ground-state xc potentials are inherited by the corresponding adiabatic functionals Example : v r t : = v n ALDA xc hom xc n =ρ(rt) has the wrong asymptotic form for large r

38 Two sources of troubles: deficiciences of ground-state xc functional adiabatic approximation

39 Two sources of troubles: deficiciences of ground-state xc functional adiabatic approximation Which problem is more severe? Investigate the best possible adiabatic approximation, i.e. the one that follows from the exact ground-state xc potential

40 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)]

41 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)

42 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)

43 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)

44 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

45 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

46 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.

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 0 0 0

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' ) V ( rt,r't' ) xc S ee

49 [ ] [ ] [ ] ( ) 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' ) V ( rt,r't' ) xc S ee f + V =χ χ 1 1 xc ee S

50 [ ] [ ] [ ] ( ) 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 + V =χ χ χ 1 1 xc ee S

51 [ ] [ ] [ ] ( ) 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 + V =χ χ χ 1 1 xc ee S χ f + V χ = χ χ S xc ee S

52 χ = χ + χ V + f χ S S ee 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" Vee ( 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

53 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)

54 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)].

55 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

56 Discrete excitation energies from TDDFT exact representation of linear density response: ˆ v Vˆ fˆ ρ ω =χ ω ω + ρ ω + ω ρ ω 1 S 1 ee 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

57 ( 1ˆ χˆ ) S ω Vˆ ˆ ˆ ee + 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 Vee + 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 ee xc 1 χ Ω V + f Ω ξ Ω = 0

58 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

59 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'

60 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

61 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

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

63 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)) charge-transfer excitations not properly described (see: Dreuw et al., JCP 119, 2943 (2003)) in the optical spectrum of periodic solids, there are two problems: --absorption edge is wrong (LDA KS gap) --excitons are missing

64 Solid Argon ALDA L. Reining, V. Olevano, A. Rubio, G. Onida, PRL 88, (2002)

65 OBSERVATION: In the long-wavelength-limit (q = 0), relevent for optical absorption, ALDA is not reliable. In particular, excitonic peaks are completely missing. Results are very close to RPA. EXPLANATION: In the TDDFT response equation, the bare Coulomb interaction and the xc kernel only appear as sum (V ee + f xc ). For q 0, V ee 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: exact property xc q 0 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)). f exact 1q 2

66 Excitons in TDDFT

67 TDDFT response equation (a matrix equation) v v f 1 1 ( q, ) 1 S q, q 1 q xc q, S 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) S, Kohn-Sham response function RPA corresponds to: fxc 0 ( q, ) 1 ( q, ) v( q) 1 v( q) ( q, ) 1 εrpa ω = +χs ω χs ω 1

68 Bootstrap kernel (Sharma, Dewhurst, Sanna, EKUG, PRL 107, (2011)) f boot xc ( q, ω) ( q, 0) v( q) ( q, ω 0) 1 ε ω= RPA S ( q, 0) ( q ω ) 1 ε ω= = = ε = 1 χ, = 0 f appr xc ( q, ω) = 1 1 χ ω= χ + ( q, 0) ( q, ω= 0) S 0 f boot xc ( q, ω) appr (, ) v v( q) ( q, ω) 1 1 ε q ω = 1 +χs q, ω q 1 + χ q, ω ( f ) xc S where χ 0 is a single-particle response function that has the right gap, e.g. G 0 W 0, LDA+U, LDA+Scissors, Tran-Blaha functional,

76 Charge transfer excitations and the discontinuity of f xc

77 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 Infinite R Finite (but large) R (A) = ε MOL Ω CT ε (B) (B) EA (B) = ε LUMO xc (B) = ε MOL derivative discontinuity 2 B A 3 3 ϕ r ϕ r' A B ε d r d r' MOL MOL r r' 2

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

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)

80 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

81 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

82 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 M. Hellgren, EKUG, PRA 85, (2012)

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

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

85 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)

86 SFB 450 SFB 658 SPP 1145 SFB 762

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 Basic theorems of TDDFT Approximate xc functionals: