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Exchange and correlation beyond static DFT: Analysis of electronic motion in the time domain

Exchange and correlation beyond static DFT: Analysis of electronic motion in the time domain. E.K.U. Gross. Max-Planck Institute for Microstructure Physics Halle (Saale). Electron dynamics happens on the femto-second time scale. Questions:.

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Exchange and correlation beyond static DFT: Analysis of electronic motion in the time domain

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  1. Exchange and correlation beyond static DFT: Analysis of electronic motion in the time domain E.K.U. Gross Max-Planck Institute for Microstructure Physics Halle (Saale)

  2. Electron dynamics happens on the femto-second time scale Questions: • How much time does it take to break a bond in a laser field? • How long takes an electronic transition from one state to • another? • In a molecular junction, how much time does it take until • a steady-state current is reached (after switching on a bias)? • Is it reached at all?

  3. Perylene at TiO2 surface TiO2 donor bridge acceptor Time-resolved photo-absorption experiment: Frank Willig (HMI Berlin)

  4. OUTLINE THANKS I. Basics of TDDFT Sangeeta Sharma Kay Dewhurst Antonio Sanna Miguel Marques Tobias Burnus Alberto Castro Stefan Kurth Gianluca Stefanucci Angelica Zacarias Elham Khosravi Claudio Verdozzi (Lund) S. Kümmel (Bayreuth) M. Thiele • Some news on excitons II. Analyzing electronic motion in real time • TD Electron Localization Function: • Movie of laser-induced π-π* transition • TD approach to electronic transport • through single molecules III: Adiabatic approximation in TDDFT

  5. Weak laser (vlaser(t) << ven) : Strong laser (vlaser(t) ≥ ven) : Time-dependent systems Generic situation: Molecule in laser field  Zα e2 |rj-Rα|  + E·rj·sin t (t) Non-perturbative solution of full TDSE required Calculate 1. Linear density response 1(r t) 2. Dynamical polarizability 3. Photo-absorption cross section 

  6. continuum states (ω) unoccupied bound states I1 photo-absorption cross section I2 occupied bound states Laser frequency  Photo-absorption in weak lasers No absorption if  < lowest excitation energy

  7. Why don’t we just solve the many-particle SE? Example: Oxygen atom (8 electrons) depends on 24 coordinates rough table of the wavefunction 10 entries per coordinate:  1024 entries 1 byte per entry:  1024 bytes 1010bytes per DVD:  1014 DVDs 10 g per DVD:  1015 g DVDs = 109 t DVDs

  8. 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 ESSENCE OF DENSITY-FUNTIONAL THEORY

  9. Time-dependent density-functional formalism (E. Runge, E.K.U.G., PRL 52, 997 (1984)) Basic 1-1 correspondence: The time-dependent density determines uniquely the time-dependent external potential and hence all physical observables for fixed initial state. KS theorem: 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

  10. LINEAR RESPONSE THEORY t = t0 : Interacting system in ground state of potential v0(r) with density 0(r) t > t0 : Switch on perturbation v1(r t) (with v1(r t0)=0). Density: (r t) = 0(r) + (r t) Consider functional [v](r t) defined by solution of the interacting TDSE Functional Taylor expansion of [v] around vo:

  11. 1(r,t) = linear density response of interacting system = density-density response function of interacting system = density-density response function of non-interacting system Analogous function s[vs](r t) for non-interacting system

  12. 1 () =  () v1 Standard linear response formalism H(t0) = full static Hamiltonian at t0  exact many-body eigenfunctions and energies of system full response function  The exact linear density response has poles at the exact excitation energies  = Em - E0

  13. continuum states (ω) unoccupied bound states I1 photo-absorption cross section I2 occupied bound states Laser frequency  Photo-absorption in weak lasers No absorption if  < lowest excitation energy

  14. Linearization of TDKS equation yields: • Exact integral equation for 1(r t), to be solved iteratively • Need approximation for • (either for fxc directly or for vxc) Equivalent to:

  15. Adiabatic approximation In the adiabatic approximation, the xc potential vxc(t) at time t only depends on the density ρ(t) at the very same point in time. e.g. adiabatic LDA:

  16. 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)]

  17. How good is ALDA for solids? optical absorption (q=0) ALDA Solid Argon L. Reining, V. Olevano, A. Rubio, G. Onida, PRL 88, 066404 (2002)

  18. 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 (WC + fxc). For q 0, WC diverges like 1/q2, while fxc in ALDA goes to a constant. Hence results are close to fxc = 0 (RPA) in the q 0 limit. CONCLUSION: Approximations for fxc are needed which, for q 0, correctly diverge like 1/q2. Such approximations can be derived from many-body perturbation theory (see, e.g., L. Reining, V. Olevano, A. Rubio, G. Onida, PRL 88, 066404 (2002)).

  19. Bootstrap approximation for fxc : S. Sharma, K. Dewhurst, A. Sanna, E.K.U.G., arXiv:1107.0199

  20. WHAT ABOUT FINITE Q?? see: H.C. Weissker, J. Serrano, S. Huotari, F. Bruneval, F. Sottile, G. Monaco, M. Krisch, V. Olevano, L. Reining, Phys. Rev. Lett. 97, 237602 (2006)

  21. Silicon: Loss function Im χ(q,ω)

  22. How can one give a mathematical meaning to intuitive chemical concepts such as • Single, double, triple bonds • Lone pairs Electron Localization Function Note: • Density (r) is not useful! • Orbitals are ambiguous (w.r.t. unitary transformations)

  23. = probability of finding an electron with spin  at and another electron with the same spin at . = diagonal of two-body density matrix = conditional probability of finding an electron with spin  at if we know with certainty that there is an electron with the same spin at .

  24. r' s r Coordinate transformation If we know there is an electron with spin  at , then is the (conditional) probability of finding another electron at , where is measured from the reference point . Spherical average If we know there is an electron with spin  at ,then is the conditional probability of finding another electron at the distances from . Expand in a Taylor series: 0 0 The first two terms vanish.

  25. is a measure of electron localization. Why? , being the s2-coefficient, gives the probability of finding a second like-spin electron very nearthe reference electron. If this probability very nearthe reference electron is low then this reference electron must be very localized. small means strong localization at

  26. C is always ≥ 0 (because p is a probability) and is not bounded from above. Define as a useful visualization of localization (A.D. Becke, K.E. Edgecombe, JCP 92, 5397 (1990)) where is the kinetic energy density of the uniform gas. Advantage: ELF is dimensionless and

  27. ELF A. Savin, R. Nesper, S. Wengert, and T. F. Fässler, Angew. Chem. Int. Ed. 36, 1808 (1997)

  28. ELF Density 12-electron 2D quantum dot with four minima E. Räsänen, A. Castro and E.K.U. Gross, Phys. Rev. B 77, 115108 (2008).

  29. For a determinantal wave function one obtains in the static case (i.e. for real-valued orbitals): (A.D. Becke, K.E. Edgecombe, JCP 92, 5397 (1990)) in the time-dependent case: T. Burnus, M. Marques, E.K.U.G., PRA (Rapid Comm) 71, 010501 (2005)

  30. Acetylene in a strong laser field (ħω = 17.15 eV, I = 1.21014 W/cm2) [Snapshots of TDELF]

  31. Scattering of a high-energy proton from ethylene (Ekin(proton) = 2 keV) [Snapshots of TDELF]

  32. Use TD Kohn-Sham equations (E. Runge, EKUG, PRL 52, 997 (1984)) vxc[ (r’t’)](r t) • propagated numerically on real-space grid using octopus code • www.tddft.org • more TDELF movies • download octopus octopus: a tool for the application of time-dependent density functional theory, A. Castro, M.A.L. Marques, H. Appel, M. Oliveira, C.A. Rozzi, X. Andrade, F. Lorenzen, E.K.U.G., A. Rubio, Physica Status Solidi 243, 2465 (2006).

  33. central region C left lead L right lead R Electronic transport: Generic situation Bias between L and R is turned on: U(t) V for large t A steady current, I, may develop as a result. Goal 1: Calculate current-voltage characteristics I(V) Goal 2: Analyze how steady state is reached, determine if there is steady state at all and if it is unique

  34. Standard approach: Landauer formalism plus static DFT left lead L central region C right lead R Transmission function T(E,V) calculated from static (ground-state) DFT

  35. Chrysazine Relative Total Energies and HOMO-LUMO Gaps Chrysazine (a) Chrysazine (c) 0.0 eV 3.35 eV 1.19 eV 3.77 eV Chrysazine (b) 0.54 eV 3.41 eV

  36. Possible use: Optical switch A.G. Zacarias, E.K.U.G., Theor. Chem. Accounts 125, 535 (2010)

  37. Motivation to develop a time-dependent approach: • Two conceptual issues: •  Assumption that upon switching-on the bias • a steady state is reached •  Steady state is treated with ground-state DFT • One practical issue: • TD external fields, AC bias, laser control, etc, cannot be treated within the static approach

  38. Electronic transport with TDDFT central region C left lead L right lead R vxc[ (r’t’)](r t) TDKS equation (E. Runge, EKUG, PRL 52, 997 (1984))

  39. Electronic Transport with TDDFT central region C left lead L right lead R TDKS equation

  40. is purely kinetic, because KS potential is local are time-independent Propagate TDKS equation on spatial grid with grid points r1, r2, … in region A (A = L, C, R)

  41. L L C R R Next step: Solve inhomogeneous Schrödinger equations , for L, R using Green’s functions of L, R, leads Hence:

  42. r.h.s. of solution of hom. SE r.h.s. of solution of hom. SE L R C insert this in equation Define Green’s Functions of left and right leads: explicity:

  43. Effective TDKS Equation for the central (molecular) region S. Kurth, G. Stefanucci, C.O. Almbladh, A. Rubio, E.K.U.G., Phys. Rev. B 72, 035308 (2005) source term: L → C and R → C charge injection memory term: C → L → C and C → R → C hopping Note: So far, no approximation has been made.

  44. Numerical examples for non-interacting electrons Recovering the Landauer steady state left lead central region right lead U V(x) U Time evolution of current in response to bias switched on at time t = 0, Fermi energy F = 0.3 a.u. Steady state coincides with Landauer formula and is reached after a few femtoseconds

  45. ELECTRON PUMP Device which generates a net current between two electrodes (with no static bias) by applying a time-dependent potential in the device region Recent experimental realization : Pumping through carbon nanotube by surface acoustic waves on piezoelectric surface (Leek et al, PRL 95, 256802 (2005))

  46. Pumping through a square barrier (of height 0.5 a.u.) using a travelling wave in device region U(x,t) = Uosin(kx-ωt) (k = 1.6 a.u., ω = 0.2 a.u. Fermi energy = 0.3 a.u.) Archimedes’ screw: patent 200 b.c.

  47. Experimental result: Current flows in direction opposite to sound wave

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