Introduction to TDDFT

Transcription

1 Introduction to TDDFT Linear-Response TDDFT in Frequency-Reciprocal space on a Plane-Waves basis: the DP (Dielectric Properties) code Valerio Olevano Institut Néel, CNRS, Grenoble European Theoretical Spectroscopy Facility

2 Résumé Motivation TDDFT Linear-Response TDDFT Frequency-Reciprocal space TDDFT TDDFT on a PlaneWaves basis: the DP code Approximations: RPA, ALDA, new kernels Results Conclusions

3 Condensed Matter Theories Models: Jellium Anderson Hubbard -> DMFT analytic (but C-DMFT medium numerical) Semi-Empirical and Phenomenological Theories: Tight Binding... lightly numerical Ab Initio First-Principles or Microscopical Theories: CI QMC DFT, TDDFT MBPT (GW and Bethe-Salpeter Equation) heavy numerical

4 What DFT can predict (normal error 1~2% in the 99% of cond-mat systems) Atomic Structure, Lattice Parameters (XRD) Total Energy, Phase Stability Electronic Density (STM, STS) Elastic Constants Phonon Frequencies (IR, Neutron scattering, Raman) that is, all Ground State Properties! Vanadium Oxide, VO2 lattice parameters M. Gatti et al., PRL accepted a b c α DFT-LDA nlcc DFT-LDA semic Å Å Å Å Å Å EXP [Longo et al.] ± Å ± Å ± Å ± 0.096

5 What DFT cannot predict Electronic Structure, Bandplot Bandgap, Metal/Insulator/Semiconductor Optical and Dielectric properties that is, all Excited State Properties! You may use DFT to predict all such properties but it cannot be blamed if it does not succeed. F. Bruneval et al., PRL 2006

6 Optical Spectroscopies Lautenschlager et al. PRB 36, 4821 (1987). Optical Absorption Aspnes and Studna PRB 27, 985 (1983). Refraction Index Aspnes and Studna PRB 27, 985 (1983). Reflectance Valerio Olevano, Introduction to TDDFT

7 Energy-Loss Spectroscopies Stiebling Z. Phys. B 31, 355 (1978). Energy Loss (EELS) Sturm et al. PRL 48, 1425 (1982), (ESRF, Grenoble) Schuelke and Kaprolat PRL 67, 879 (1991) (ESRF, Grenoble). Inelastic X-ray Scattering Spectroscopy (IXSS) Valerio Olevano, Introduction to TDDFT Coherent Inelastic Scattering Spectroscopy (CIXS)

8 Why we need ab initio theories to calculate spectra 1)To understand and explain observed phenomena 2)To offer experimentalists reference spectra 3)To predict properties before the synthesis, the experiment

9 Excitations: Charged vs Neutral Charged Excitations N -> N+1 (or N-1) (Photoemission Spectroscopy) e Neutral Excitations N -> N (Optical and Dielectric Spectroscopy) hν c hν c v v Inverse Photoemission Optical Absorption Valerio Olevano, Introduction to TDDFT

10 Excitations: Charged vs Neutral Charged Excitations N -> N+1 (or N-1) Neutral Excitations N -> N c c electron-electron interaction electron-electron interaction + electron-hole interaction v v Inverse Photoemission Optical Absorption Valerio Olevano, Introduction to TDDFT

11 Photoemission Spectroscopy hν hν e- esample tor sample inverse photoemission de tec direct photoemission hν elure Orsay, Sirotti et al. hν c v e- c v Valerio Olevano, Introduction to TDDFT

12 Optical Spectroscopy reflected photon incident photon hν c ωcv v sample transmitted photon Cardona et al. hν dete ctor Valerio Olevano, Introduction to TDDFT

13 Dielectric, Energy-Loss Spectroscopy (EELS) e- c electrons e- ωcv e- v sample Lautenschlager e- c e- e- ect det v ωp plasmon or Valerio Olevano, Introduction to TDDFT

14 Synchrotron Radiation Spectroscopy (IXSS) X c X rays ωcv v sample ESRF, Grenoble c dete ctor v ωp plasmon Valerio Olevano, Introduction to TDDFT

15 Possible ab-initio Theories for the Excited States TDDFT (Time-Dependent Density-Functional Theory) in the Approximations: RPA TDLDA beyond Neutral Excitations MBPT (Many-Body Theory) in the Approximations: Charged Excitations GW BSE (Bethe-Salpeter Equation approach) Neutral Excitations Valerio Olevano, Introduction to TDDFT

16 What is TDDFT? TDDFT is an extension of DFT; it is a DFT with time-dependent external potential: Static Ionic Potential Static Ionic Potential + External Perturbation (E.M. field) v r r Hohenberg-Kohn theorem v r,t r, t Runge-Gross theorem Fundamental degree of freedom: Time-Dependent electronic Density ρ(r,t) (instead of the total many-body wavefunction Ψ(r1,...,rN,t)) Valerio Olevano, Introduction to TDDFT

17 TDDFT milestones Runge and Gross (1984): rigorous basis of TDDFT. Gross and Kohn (1985): TDDFT in Linear Response.

18 The Runge Gross theorem PRL 52, 997 (1984) t i t 1 t = T V 1 t W 1 t i t 2 t = T V 2 t W 2 v 1 t v 2 t c t 1 t 2 t ρ(t) v1 Any observable is functional of the (time-dependent) density: ρ1 o t = t o t =o[ ] t v2 ρ2 t

19 Runge Gross theorem caveats 1) Any observable is functional of the density and of the initial state Ψ0=Ψ(t0) 2) The Runge-Gross theorem has been proven for v(t) Taylor expandable around t0. And if you run into problems already in describing the (nonequilibrium) initial state, you can't go on with TDDFT. Previous demonstrations required periodic v(t) or a small td perturbation (linear response), or later (Laplace transformable switch-on potentials+initial ground-state). But there is no very general prove of the Runge Gross theorem. o t =o [, 0] t v(t) What about this case? t0 v(t) t This other is also not Runge-Gross described t0 t

20 TDDFT vs DFT Zoological Comparative Anatomy

21 DFT vs TDDFT Hohenberg-Kohn: Runge-Gross: v r r The Total Energy: v r,t r, t The Action: t1 =E [ ] H dt t t = A[ ] t i t H t0 are unique functionals of the density. The extrema of the: Total Energy E [ ] =0 r The stationary points of the: Action A[ ] =0 r,t give the exact density of the system: r r,t Valerio Olevano, Introduction to TDDFT

22 DFT vs TDDFT Kohn-Sham: N r = i=1 KS i KS KS N 2 r,t = r v KS r =v r dr ' H Runge-Gross: i =1 r ' E xc [ ] r r ' r KS KS i 2 r,t v KS r, t =v r, t dr ' KS KS i t i r,t =H r i r = i i r KS r ',t A xc [ ] r r ' r, t KS r, t i r, t Kohn-Sham equations ρ(t) ρ(t) W v vks ρ real system Valerio Olevano, Introduction to TDDFT W=0 t Kohn-Sham system ρ t

23 TDDFT further caveats 1) The functional Axc[ρ] is defined only for vrepresentable densities. It is undefined for ρ which do not correspond to some potential v -> problems when variations δa[ρ] with respect to arbitrary densities are required in order to search for stationary points. 2) A functional A[ρ] with ρ(t) on the real time, is hill defined, since we would run in a causality-symmetry paradox -> we can solve the problem by declaring t on the Keldysh contour. Response functions are symmetric on the contour and become causal on physical time. t0 t A[ ] =0 r,t If exists Axc [ ] r,t A s [ ] =v r,t r,t s Then we can define the Lagrange transform: ]= A[ ] dr dt r,t v r, t A[v v] A[ = r, t v r, t A r,t = v r, t v r 't ' v r ' t ' symmetric Van Leeuwen, PRL 80, 1280 (1998). causal (=0 for t'>t)

24 TDDFT in Linear Response Gross and Kohn (1985) If: v r,t =v r v r, t with: v r,t v r Strong (Laser) perturbations excluded! TDDFT = DFT + Linear Response (to the time-dependent perturbation) Valerio Olevano, Introduction to TDDFT

25 Hohenberg-Kohn Theorem for Linear Response TDDFT DFT: TDDFT: v r r v r,t r, t v r v r, t r r, t LR-TDDFT: v r,t r,t Valerio Olevano, Introduction to TDDFT

26 Calculation Scheme DFT calculation (ABINIT code) KS i v r r,, KS i LR-TDDFT calculation (DP code) v r,t r,t,, spectra v Valerio Olevano, Introduction to TDDFT

27 Polarizability χ v ext External Perturbation (variation in the external potential) Induced Density (variation in the density induced by δvext) implicit definition of the polarizability χ: = v ext Polarizability χ Valerio Olevano, Introduction to TDDFT

28 Variation in the Total Potential The variation in the density induces a variation in the Hartree and in the exchange-correlation potentials which screen the external perturbation: vh v H= =v c v xc v xc = = f xc v xc f xc = Exchange-Correlation Kernel definition So that the variation in the total potential (external + screening) is: v tot = v ext v H v xc Total Perturbation Valerio Olevano, Introduction to TDDFT

29 LR-TDDFT Kohn-Sham scheme: the Independent Particle Polarizability χ (0) Let's introduce a ficticious s, Kohn-Sham non-interacting system such that: s= Then, instead of calculating χ, we can more easily calculate the polarizability χ(0) (also χs or χks) of this non-interacting system, called independent-particle polarizability and defined: 0 = v tot Independent Particle Polarizability χ (0) Valerio Olevano, Introduction to TDDFT

30 the Independent Particle Polarizability χ (0) By variation δvtot (= δvs = δvks) of the Kohn-Sham equations, one obtains the Linear-Response variation of the density δρ and then an expression for χ(0 in terms of the Kohn-Sham energies and wavefunctions: 0 r,r ', = ij f i f j i r * j r *i r ' j r ' i j i Independent-Particle Polarizability (Adler-Wiser) In Frequency-Reciprocal space: 0 G G' q, = ij f i f j j e i q G r i i e i q G' r j i j i Valerio Olevano, Introduction to TDDFT

31 χ as a function of χ (0) From: = v ext 0 = v tot The polarizability in terms of the independent-particle polarizability is: 0 0 = v c f xc Coulombian (Local-Fields) Polarizability χ Exchange-Correlation Kernel Also explicitly: 0 0 = 1 v c 1 0 f xc Valerio Olevano, Introduction to TDDFT

32 Dielectric Function 1 v tot = v ext 1 =1 v c 1 te =1 v c f xc 1 G G ' q, definition of the Dielectric Function ε Test-Particle Dielectric Function Test-Electron Dielectric Function In a periodic system Valerio Olevano, Introduction to TDDFT

33 Macroscopic Dielectric Function M r,r ' = r,r ' M q, = q, Macroscopic Dielectric Function εμ LF NLF M 00 = M Macroscopic Dielectric Function εμ Observables A BS= ℑ M 1 EELS= ℑ M

34 TDDFT: fundamental equations Dielectric Function ε 1 =1 v c 0 Observables EELS = -Im ε 1 0 = v c f xc Coulombian (Local-Fields) χ (r, r', ) ( 0) ( fi i j v xc f xc = =? ABS = Im ε Polarizability χ Exchange-Correlation Kernel f j) i (r) *j (r) *i (r') j (r') i εj ω i Independent-Particle Polarizability (Adler-Wiser) Exchange-Correlation Kernel approximations required

35 Local-Fields Effects (LF) tot ext v G = G G ' v G ' 1 G' Effect of the ε non diagonal elements (density inhomogeneities) + v ext NLF M q, = 00 q, 1 ~ v tot Macroscopic Dielectric Function ε without local-fields effects (NLF) Valerio Olevano, Introduction to TDDFT

36 LR-TDDFT Calculation Scheme Résumé DFT DFT DFT KS KS 0 f xc M Valerio Olevano, Introduction to TDDFT

37 dp code (dielectric properties) The thing: Linear-Response TDDFT code in Frequency-Reciprocal Space on PW basis. Purpose: Dielectric and Optical Properties (Absorption, Reflectivity, Refraction, EELS, IXSS, CIXS,.. Systems: bulk, surfaces, clusters, molecules, atoms (through supercells) made of insulator, semiconductor and metal elements. Approximations: RPA, ALDA, GW-RPA, LRC, non-local kernels,, with and without LF (Local Fields). Machines: Linux, Compaq, IBM, SG, Nec (6GFlop), Fujitsu. Libraries: BLAS, Lapack, CXML, ESSL, IMSL, ASL, Goedecker, FFTW. Interfaces: ABINIT, Milan-CP, PWSCF, SPHINGx

38 Frequency-Reciprocal space: why and where could be convenient Reciprocal Space -> Infinite Periodic Systems (Bulk, but also Surfaces, Wires, Tubes with the use of Supercells); Frequency Space -> Spectra. In the case of isolated systems (atoms, molecules) it is more convenient a real space-time approach (e.g. Octopus code) Valerio Olevano, Introduction to TDDFT

39 DP Flow Diagram KS i DFT-code interface KS i G FFT-1 KS i r ij r = *i r j r FFT ij G 0 G G' q, = ij f i f j ij G *ij G' i j INV G G' q, f xc 1 G G ' q, LF M q, NLF M q,

40 DP tricks 1 If we only need 00 =1 v 0 00 that is only 00 then instead of solving (inverting a full matrix): G G ' = v c 1 0 f xc G G ' ' G '' G ' we solve the linear system for only the first column of G' v c O(N2) instead of O(N3) 0 f xc G G ' G' 0= G 0

41 DP Parameters KS i KS i G FFT-1 NG = npwwfn KS i r Nb = nbands ij r = *i r j r FFT ij G 0 G G' q, = ij f i f j ij G *ij G' i j INV G G' q, f xc NG = npwmat 1 G G ' q, LF M q, Nω = nomega NLF M q,

42 DP performances: CPU scaling and Memory usage CPU scaling for CPU scaling for Memory occupation 2 0 : N b N k N N G N r log N r 1 : N NG 2 2 N N G N r N k N b

43 Exchange-Correlation Kernel fxc: the RPA approximation Random Phase Approximation = neglect of the exchange-correlation effects (in the response) RPA f RPA xc =0

44 RPA Approximation RPA (without Local Fields) = sum over independent transitions (application of Fermi s Golden Rule to an independent particle system) confined system solid hν Optical Absorption f c ωcv ωfi hν v i 2 ℑ RPA = vc c D v c v KS wavefunctions Fermi s Golden Rule KS energies Valerio Olevano, Introduction to TDDFT

45 Adiabatic Local Density Approximation (TDLDA) The Adiabatic Local Density Approximation ALDA ALDA f xc f ALDA xc r,r ' = A r r,r ' LDA xc v = =0 local in r-space no memory effect

46 TDDFT: Results Isolated Solids RPA ALDA Energy Loss ok ok Optical Prop ok but.. ok but.. Energy Loss ok ok Optical Prop no no Energy Loss -> Optical Properties -> -Im ε-1 Im ε

47 TDDFT: Results Energy Loss

48 TDDFT and EELS in Solids TDLDA (but also RPA) in good agreement with experiment; Importance of Local-Field (LF) effects.

49 Local Field Effects in EELS A. Marinopoulos et al. PRL 89, (2002) RPA is enough. But when inhomogeneities are present, LocalField effects should be absolutely taken into account. Quantitative Agreement

50 IXSS and CIXS and other synchrotron-radiation spectroscopies Weissker et al. V. Olevano, thesis In Solids all Dielectric Properties related to the Energy-Loss function are well described by TDDFT in RPA with an improvement in ALDA.

51 IXSS synchrotron-radiation spectroscopy Weissker et al. In Solids all Dielectric Properties related to the Energy-Loss function are well described by TDDFT in RPA with an improvement in ALDA.

52 TDDFT: Results Optical Properties

53 TDDFT RPA Optical Properties in Nanotubes RPA is qualitatively able to interpret observed structures in optical spectra

54 LF Effects in in C and BN Nanotubes LF explain depolarization effects both for C and BN Nanotubes in perpendicular polarization optical spectra

55 Optical Absorption in Clusters RPA NLF EXP TDLDA Fair agreement (improvable) of the TDLDA result with the Experiment. RPA LF G. Onida et al, Rev. Mod. Phys. 74, 601 (2002).

56 Optical Properties in Solids: Si The TDLDA cannot reproduce Optical Properties in Solids. We miss both: 1 Self-Energy (electron-electron) effects (red shift of the entire spectrum) 2 Excitonic (electron-hole) effects (underestimation of the low-energy part)

57 GW: optical properties in Solids GW corrects the red-shift but still misses the Excitonic Effects

58 MBPT: GW, BSE and Excitonic Effects c c c hν exciton ωcv hν ωcv v v RPA GW e- P= + hν W ωcv v BSE + + O(2) h+ Polarization Valerio Olevano, CNRS, Intoduction to TDDFT

59 Bethe-Salpeter Equation: optical properties in Solids Almost quantitative agreement of BSE with the experiment

60 Solid Argon: Hydrogen series n=1 n=2 n=3 Exciton ~ Hydrogen atom En ~1/n2 Balmer-like series BSE can reproduce even bound Excitons

61 What can we do to solve the TDDFT kernel problem? If a new Approximation in TDDFT could be established, combining TDDFT s simplicity with MBPT s reliability Hints for this new Approximation: compare critically MBPT with TDDFT fundamental equations.

62 New Approximations: LRC (Long Range Contribution only) Nanoquanta kernel (or mapping BSE on TDDFT)

63 LRC Approximation long-range coulombian 0 0 = v c f xc ALDA: local kernel f LRC xc =limq 0 f 1 =4.6 MT xc = 2 q G Long Range Contribution only Inversely proportional to the screening

64 TDDFT, LRC approximation The LRC approximation makes TDDFT work also on Optical Properties in Solids

65 Nanoquanta kernel G W χ0-1 L. Reining, V. Olevano, A. Rubio and G. Onida, (2001) G. Adragna and R. Del Sole, (2001) F. Sottile, V. Olevano and L. Reining, (2004) A. Marini, R. Del Sole, A. Rubio, (2004) U. Von Barth, N. E. Dahlen, R. Van Leeuwen and G. Stefanucci, (2006) R. Stubner, I. Tokatly and O. Pankratov, (2006)

66 TDDFT vs BSE 0 0 P x 1, x 2, x 3, x 4 = x 1, x 2, x 3, x 4 v c F 0 0 TDDFT P x 1, x 2, x 3, x 4 = P x 1, x 2, x 3, x 4 P v c F F TDDFT F BSE GW P BSE 4-point Coulombian TDDFT-BSE Differences: P = K BSE P 4-point TDDFT P 0 KGW = Self-Energy correction diagonal term x 1, x 2, x 3, x 4 = x 1, x 2 x 3, x 4 f xc x 1, x 3 x 1, x 2, x 3, x 4 = x 1, x 3 x 2, x 4 f xc x 1, x 2

67 Nanoquanta fxc f xc q, G, G ' = n 1 n2 n3 n n1, n2 ; G K n f n fn 1 * 1 1 n 2 n 3 n4 n3, n4 ; G ' 2 static Where: H n2 n1,n2 ;r = n r r 1 Kohn-Sham bilinear GW shift term GW n2 GW n1 K n n n n = n n n n f n f n F 1 F BSE n1 n2 n3 n BSE n1 n2 n3 n4 = dr dr ' n1,n3 ; r W r ',r n2,n4 ;r ' * Excitonic term WARNING: fxc could not exist due to invertibility problems of Φ! PRL 88, (2002).

68 Solid Argon: Bound Excitons n=1 F. Sottile et al. n=2 n=3 The Nanoquanta kernel makes TDDFT reproduce even Bound Excitons

69 TDDFT Excitation Energies and the Casida Equations eigenvalues = poles of χ t ' tt ' at '= 2 at eigenvectors = oscillator strengths 2 Hxc tt' = t tt ' 2 t t ' f tt' t = c v Kohn-Sham excitation energies f tt ' = dr 1 dr 2 *c r 1 v r 1 [ v c r 1,r 2 f xc r 1,r 2, ] *v ' r 2 c ' r 2 Hxc 4-points Hartree+xc kernel

70 DP license: Scientific Software Open Source Academic for Free License Academic, non-commercial, non-military purposes: freedom to: use, copy, modify and redistribute (like GPL) open source (like GPL) cost: for free (in the sense gratis, royalty free) scientific behaviour: request of citation Commercial purposes: excluded by this license, but allowed under separate negotiation and different license

71 Conclusions TDDFT is a valid tool of Condensed Matter Theoretical Physics to calculate from first principles excited-state properties; The agreement with the experiment is good, but the choice of the right xc-approximation with respect to the given excited state property is crucial: RPA with LF is able to reproduce EELS spectra at q=0; TDLDA improves upon RPA on EELS (and also IXSS, CIXS) spectra at high q; TDLDA seems also to improve upon RPA on optical spectra in finite systems; More refined kernels (LRC, MT) are required to reproduce optical spectra in solids, especially in presence of strong excitonic effects and bound excitons. Perspectives: Improve the algorithms, simplify the orbital expressions of the kernel, find a density-functional dependent kernel.