Angel Rubio. Motivation. I. Illustration of the physics for nano- and bio-structures. II. Extended systems: problems and new developments

Transcription

1 Lectures on Applications of TDDFT Angel Rubio Dpto. de Física de Materiales, Universidad del País Vasco, Donostia International Physics Center (DIPC), and Centro Mixto CSIC-UPV/EHU, Donostia, Spain Motivation I. Illustration of the physics for nano- and bio-structures II. Extended systems: problems and new developments Benasque, September 2004

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3 Evolution towards Nanoscience Moore's Law

4 Nanotechnology Paradigm Shift Current View Think Different Classical continuum physics Quantum mechanics Solid state properties Binding properties Volume dominating Surfaces dominating Homogeneous Materials Composite-inhomogeneous Statistical ensemble Individual particles Miniaturization Self-organization/Assembly

5 Optical properties of nanostructures Nanoscience: different scales need to be bridged (multicomponent simulations) THEORY and EXPERIMENTS on the same object - spatial scale: 0D - 1D - 2D from molecule-nano-bio-meso-wires-leads - time scale: femtosecond (electron excitations) picoseconds (phonon) nanosecond (charge-transfer) millisecond >>> bio-structural organization growth; self assembly FIRST PRINCIPLES TOOLS: SPECTROSCOPIES

6 Motivation Ab-initio approach for non-equilibrium electron dynamics excited states e- transport hv eeground state 1. Optical spectroscopy. Photoelectron and time-resolved spectroscopies 2. Lifetimes: Electron-electron and electron-phonon coupling (adiabatic/nonadiabatic) 4. Photo-induced dynamics (Ultrafast dynamics): e.g. Photoisomerisation in bio-photoreceptors Time-dependent approach: TDDFT?

7 Photochemistry: Phonon- versus electron- mediated surface reactions: probe photon pump photon Vacuum level LUMO EF Electronic coupling between a solid and an adsorbate governs chemical dynamics at surfaces HOMO Adsorbate The dynamics of electronic excitations play an important role in moleculesurface interactions and reactivity, as in laser-driven surface reactions, and are also critical to technological applications of electronic materials

8 Motivation: non-linear phenomena and chemical reactivity Quantum optimally controlled landscapes: time resolved dynamics High-intense short-laser (fs) pulses K. Yamanouchi, Science 295, 1659 (2002) Multiphoton <1014 Wcm2 Tunneling<1015 Wcm2 R. J. Levis et al, Science 292, 709 (2001)

9 Characterisation tools of the nanoworld Need of the knowledge of response functions!!!!! wavelength (Å) 1000 to n SNO ic m M ate rial se fo t X one AF S s 100 elect ro n s 10 IMFP 1 0,1 PES, 10000,N EX AF S PD, A uger LEED energía (ev) energy (ev) 1000 photon wavelength (Å) pho STM EELS STEM PES, PD, MARPE 10 NEXAFS 0, o longitud de onda del electrón (A)(Å) electron wavelength The spatial resolution depends on the wavelength, the spatial localization (STM, SNOM), the inelastic attenuation (PES, PD), the range of the interaction (EELS, MARPE)

10 Electronic properties of a many body system: ab-initio approach Scheme Ground state DFT density n(r) yes, LDA, GGA's Excited states Problems Corrections to KS eigenvalues? XC-func. TDDFT n(r,t) and/or j(r,t) MBPT G1, total energy G2 QM See for a review: yes Yes, non-linear phenomena and time-resolved spectroscopies (r,r') (w) dependence Quasiparticle excitations IP and EA optical absorption Comput. yes, in some cases! G. Onida, L. Reining and AR, Rev. Mod. Phys. 74, 601 (2002)

11 Time Dependent Density Functional Theory (Runge and Gross 1984) Electron-dynamics first, then e-ion problem HK-like theorem: v(r,t) <------> (r,t) The time dependent density determines uniquely the time-dependent external potential and therefore all physical observables Kohn-Sham formalism: The time dependent density of the interacting system can be calculated as the density of an auxiliary non-interacting system d i ℏ =H dt 2 d i ℏ =H, i=1, N KS j i dt i [ { }]? Same timeℏ e dependent V H = i A A V V V KS xc external hartree exchange correlation density n(r,t) 2m cℏ 2

12 The octopus project is aim to the first principle description of the excite state electron-ion dynamics of nanostructures and extended systems within TDDFT Implementation: Numerical description of functions: real space discretization (1D-3D). Auxiliary use of LCAO/ FFTs. QM/MM for biomolecular structures Electon-ion coupling: pseudopotentials. Spin-Orbit, non-collinear magnetism, periodic systems... M.A.L. Marques, A. Castro, G. Bertsch, AR Comp.Phys.Comm. (2002) C. Rozzi, M.A.L. Marques, A. Castro, E.K.U. Gross A. R. (to be published)

13 Linear Response Applications

14 Linear optical response: general aspects = v f 0 0 xc 0 r, r ', w = ij f j f i c c i r j r i r ' j r ' i j w [1 v f xc 0 ] =0 Single-pole approximation LDA- thick solid line EXX- dotted line Bethe-Salpeter d V xc r K ij = dr i r j r d r Experiments Transport: Current + E-field

15 Optical response: (Benzene) * (LUMO) 7eV (HOMO) M.A.L Marques, A. Castro, G.F. Bertsch and AR, Comp. Phys. Comm. (2002); K. Yabana and G. F. Bertsch, Int. J. Quantum Chem. 75, 55 (1999)

16 Linear optical response: Geometry identification discrimination of C20 isomers through optical spectroscopy Real-space, real-time TDLDA yields reliable photoabsorption spectra of carbon-clusters. Photoabsorption spectra of C20 isomers are significantly different. Optical spectroscopy (absorption and emission) is proposed as an experimental tool to identify the cluster structure. A. Castro, M. A. L. Marques, J. A. Alonso, G. F. Bertsch, K. Yabana and AR, J. Chem. Phys. 116, 1930 (2002).

17 Optical spectroscopy of gold clusters: relativistic effects.

18 Biological molecules: photoreceptors A "poor" condensed matter physicist view!!!! QM/MM + TDDFT approach F. Gai et al. Science 279, 1886 (1998) M.A.L Marques, X. Lopez, D. Varsano, A. Castro, and A. R. Phys. Rev. Lett. 90, (2003)

19 Towards understanding biomolecular colors: the GFP case (Aequorea victoria: jellyfish) Green Fluorescent Protein (GFP). T.M.H. Creemers et al, Proc. Natl. Acad. Sci.. USA (1999) QM/MM approach

20 Optical Rotatory Power ℜ j E 0 dt e R E =ℑ i E i t L j t ; ℜ E =ℜ x ℜ y ℜ z ℜ E rotational strength function Guanine Cytosine Absorption Circular dichroism

21 Azobenzene: spectroscopy along femtosecond-laser induced photoisomerization Azobenzene dyes are known to isomerize at the central N-N bond within fractions of picoseconds at high quantum yield [T. Nägele et al, Chem. Phys. Lett. 272, 489 (1997)]. One example is the APB optical trigger: Single-Molecule Optomechanical Cycle HOMO LUMO S. Spörlein et al, Proc. Natl. Acad. Sci., 99, 7998 (2002) T. Hugel et al, Science, 296, 1103 (2002) Y. Yu et al, Nature 425, )

22 Excited state electron-ion dynamics

23 Azobenzene: spectroscopy along femtosecondlaser induced photoisomerization CIS TRANS Next step: QM/MM calculation of the chromophore+peptide system

24 Introduction: standard simulation of excited state dynamics? t = 0: Promote the electronic occupations to mimic the excited states. Then perform a SCF calculation t > 0: Solve n (t+ t)=exp{- i Potential e> g> th(t)} n (t). Hellmann-Feynman theorem works Yes Is the matrix of HKohn-Sham Do MD. diagonal? Reaction coordinate No Observation of the nonradiative decay! lifetime, decay path 1. No need of level assignment for the hole and excited electron 2. Automatic monitoring of the nonradiative decay (lifetime, decay path).

25 Gross et al (1985)-to date Our Method: TDDFT TDDFT: N-coupled one-particle equations. i ℏ =H t? [ { }] i ℏ =H, i=1, N KS j i t i 2 2 ℏ e H = i A V V V KS external hartree exchange 2m cℏ V correlation Classical description of electromagnetic field. (Absence of radiative-decay channel!) Classical description of nuclei (point particles): Ehrenfest path: F t = t H t a a t =Det { t } i Solution of the TD-KS equations by unitary propagation schemes t t =e i ih KS t t t / 2 e ih KS t t / 2 t i

26 Femtosecond dynamics: test photodissociation of a dimer (Na2+). ω=2.5ev 80fs ω=3.2ev Time resolved Vibrational Spectroscopy: Raman A. Castro, M.L. Marques. J.A. Alonso, G.F. Bertsch and AR (2003)

27 Femtosecond dynamics (ongoing work!!!!) Time resolved Vibrational Spectroscopy: Raman & IR. Na-dimer

28 Photodissociation dynamics: case of He3 + J. Chem. Phys. 103, 3450 (1995).

29 Photodissociation dynamics: case of He3 + Linear Optical Response. Laser Pulse: I = W/cm2

30 Femtosecond dynamics: Time-resolved photoelectron spectroscopy -The simulation region is divided in two parts: A and B, separated by a smooth mask function - Electrons are not allowed to come back from B to A We write the photoelectron spectrum as: PES p = p,t i i 2 11 I =1.310 W cm 2 =5 ev D. Varsano, M.A.L. Marques, H. Appel, E.K.U Gross and AR (to be published)

31 The ultimate biomedical goal of nanotechnology:

32 Nanotechnology: a higher form of evolution? (Humility is perhaps appropriate )

33 II. Optical absorption and electron energy loss spectroscopy of extended systems Introduction: how to handle the electron dynamics in extended systems under the influence of an external electromagnetic field? TDDFT: - Problems with standard exchange-correlation functionals - A new fxc derived from Many-body perturbation theory proper description of excitonic effects!!! - Applications to poliacetilene as one-dimensional system G. Onida, L. Reining and AR, Rev. Mod. Phys. 74, 601 (2002) Benasque, September 2004

34 Some Pathologies Pathologies of of the the KS-DFT functional Ban-gap problem Some KS-DFT functional Widely separated open-shell (Sham, Schlüter and Perdew, Levyatoms (1983) Almbladh and von Barth 1985 Egap =I A= LU M O H O M O Band-gap Band-gap problem problem Band-gap problem is discontinuous by by aaa constant constant when when an an electron electron is added to to the the system system isis discontinuous discontinuous by constant when an electron isis added added to the system (Sham, (Sham, Schlüter Schlüterand andperdew, Perdew,Levy Levy(1983) (1983) (Sham, Schlüter and Perdew, Levy (1983) Widely Widely separated separated open-shell open-shell atoms atoms Widely separated open-shell atoms Almbladh Almbladhand andvon von Barth Barth Almbladh and von Barth 1985 Exchange electric field in insulators Exchange correlation correlation Exchange correlationelectric electricfield fieldinininsulators insulators Godby Godbyand andsham Sham 1984, 1984, Gonze, Gonze, Ghosez Ghosezand andgodby Godby Godby and Sham 1984, Gonze, Ghosez and Godby 1995 Egap = [ KS N 1 KS N ] [ KS N 1 KS N 1 N 1 ] KS gap E xc The KS-effective potential takes a positive value IB-IA around atom B Exc [n] Exc [n] xc-discontinuity xc = O 1 /N n r N 1 n r N

35 Hedin Equation's (1965) The GW ''soup''

36 G0 W0 Band Structures of Insulators QP n E n v xc n KS n KS n From "Quasiparticle calculations in solids", W.G. Aulbur, L. Jönsson and J.W. Wilkins, Solid State Physics 54 1 (2000), also available in preprint form at

37 Time-dependent approach for extended systems: a gauge formalism G.F. Bertsch, J.I. Iwata, AR, K. Yabana, PRB62, 7998 (2000) The Hamiltonian of a periodic system in a volume V under a uniform field is H= i i 2 1 e V p A V E E ion i Hartree xc 2 2m c 8 c d A dt 2 The equation of motion are: [ ] 1 e 2 iℏ = p A V V V i ion H xc i t 2m c 2 d A dt 2 = 4 e 2 n e p A 4 c i m i m V i For the electric field is: de = 4 j dt A 1 d t = E c dt 2 j = e p e n A and i i m i V c Gonze,Ghosez,Godby; PRL74, 4035 (95)

38 Lithium Diamond?

39 Non-local fxc for extended systems: Very encouraging results for Si already: TD-CDF: results of P. de Boeij (using Vignale,Kohn functional) TD-EXX: results of A. Goerling What about insulators, bound-excitons, etc...? See for a review: G. Onida, L. Reining and AR, Rev. Mod. Phys. 74, 601 (2002)

40 Why a non-local (static?) fxc for extended systems: f = /q xc - In the EELS spectra fxc is added to the full coulomb that already contains a long range contribution 1 EELS =1 v = v f 0 0 xc - In the absorption spectra fxc is added to the full coulomb that does not contains a long range contribution (q=0) =1 v M = GW 0 GW 0 RPA M 1 /[1 v ] v f 0 G=G '=0 xc The lack of a long range term in fxclda is much important (crucial) in the absorption spectra than in the EELS!!! L. Reining, V. Olevano, AR, G. Onida, PRL88, (2002); S. Botti et al, PRB (2004). 2

41 Density Functional versus Many body perturbation theory Diagrammatic expansion

42 Density Functional Theory and Many-Body Perturbation Theory R. O. Jones and O. Gunnarsson, Rev. Mod. Phys. 61, 689 (1989) Density Functional Theory Exchange-correlation Potential: Real, Local in space, Frequency independent Self-Energy: Complex, Non-local in space, Frequency dependent Many-Body Perturbation Theory G. Onida, L. Reining and AR, Rev. Mod. Phys. 74, 601 (2002) F. Aryasetiawan, Rep. Prog. Phys. 61, (1998)

43 Many-Body approach to the Exchange-Correlation Kernel of TDDFT A diagrammatic approach Hypothesis It exists a ''many-body xc-kernel'' such that the TDDFT and Many-Body polarization functions are identical Consequently TDDFT equation can be used as an equation for the xc-kernel and as a formal solution can be found in terms of an iterative equation for the nth order contribution A. Marini, R. Del Sole and AR, PRL (2003)

44 Many-Body approach to the Exchange-Correlation Kernel of TDDFT TDDFT MBPT Bethe-Salpeter Equation Iterative equation for fxc A. Marini, R. Del Sole and AR, PRL (2003)

45 Bound excitons in TDDFT = TDDFT BSE Experiment TDDFT, scalar fxc A. Marini, R. Del Sole and AR, PRL (2003)

46 How many terms? = = = -2 BSE QP-RPA C C

47 A many-body causal TDDFT kernel Causal/T-ordered Causal/T-ordered fxc BSE TDDFT 1st order TDDFT 2nd order Same agreement between BSE and TDDFT for the finite transferred momentum absorption spectra......and for the off-diagonal elements of the microscopic dielectric function

48 Is TDDFT ''fast'' compared to the BSE? = [# of frequencies] 100 x10000 W ' When the only optical spectra is calculated TDDFT is as time consuming as BSE......but when the full dielectric matrix is needed TDDFT is more favorable than BSE

49 Low dimensional systems (1D): polyacetylane Electric field dependence of the XC Potential in Molecular Chains M. van Faassen et al. PRL (2002) S.J.A. Van Gisbergen PRL (1999) In LDA and GGA xc potential lack of a term counteracting the applied electric field

50 Low dimensional systems (1D): polyacetylane BSE f xc r, r', Isolated infinite Polyacetylene chain

51 What about the description of decaying quasiparticle processes within TDDFT? G. Onida, L. Reining and AR, Rev. Mod. Phys. 74, 601 (2002)

52 Lifetime of quasiparticles h interactions between quasiparticles limit how long the corresponding quantum states retain their identity, i.e., the lifetime of the excitation. In combination with the velocity, this lifetime determines the mean free path, a measure of influence of the excitation Importance of lifetime - screening in an electron gas - surface photochemistry - electron-phonon coupling- electron transfer across interfaces - localization - electron dynamics and energy transfer

53 Heimann et al Experimental lifetimes change quickly with time! Kevan et al Paniago et al Nicolay et al binding energy [mev] from F.Reinert et al., PRB 63 (2001)

54 Excitonic effects (via TDDFT) on the lifetimes of LiF 10'000x10'000 BS kernel Up to 200 G-vectors dielectric function, 256 Q-points in the whole BZ Small broadening stability Linear behavior Small penetration of the RPA ''forbidden region'' The fxc kernel remains stable and robust even with a 10 mev broadening and energies up to 20 ev A BS-based calculation is, in this case,enormously less convenient than using TDDFT

55 Summary TDDFT is a powerful tool to handle the combined dynamics of electron/ion in response to external electromagnetic fields of nanostructures, biological molecules and extended systems. Problem: we need better fxc functionals based on either DFT (or currentdft) or MBPT approaches Ongoing work on applications to: Time-resolved spectroscopies: pump-probe simulations. High-harmonic generation from quantum dots. QM/MM: for excited-state dynamics in biomolecules: photoisomerisation Control chemical reactivity:pulse optimization in laser induced reaction Non-linear effects in One-dimensional systems Time-dependent Molecular transport!!!!!!!!! Quantum nuclei, etc...

56 Acknowledgments A. Castro, A. Marini, X. López, L Wirtz, D. Varsano Department of Material Physics, Centro Mixto CSIC-UPV, University of the Basque Country, and Donostia International Physics Center (DIPC), Donostia, Spain M. A. L. Marques, C.A. Rozzi, and E. K. U. Gross Institut für Theoretische Physik, Freie Universität, Berlin, Germany L. Reining, V. Olevano École Polytechnique, Palaiseau, France S.G. Louie and M.L. Cohen Department of Physics, University of California at Berkeley, USA G.F. Bertsch Physics Department and Institute for Nuclear Theory University of Washinton, Seattle (USA) R. Del Sole and G. Onida INFM e Dipartimento di Fisica dell'università di Roma ``Tor Vergata'', Roma, Italy

57 It is one of the first duties of a professor, in any subject, to exaggerate a little both the importance of his subject and his own importance in it. G.H. Hardy (A Mathematician's Apology) n a h T u o y k arubio@sc.ehu.es

58 The 2-point vertex function Suppose to have a good approximation fot the (TD)DFT potential... No difference with GoWo using ALDA or similar approaches PRL 62, 2718 (1989); PRB 49, 8024 (1994); PRB 56, (1997). BUT IS? PRL 91, (2003);PRL 91, (2003). PRL 88, (2002) etc etc

59 The 3-point vertex function = If Closed expression for the ''on mass-shell'' electronic lifetime