Time-Dependent Electron Localization Functio n

1. Time-Dependent Electron Localization Function Co-workers: Tobias Burnus Miguel Marques Alberto Castro Esa Räsänen Volker Engel (Würzburg)

2. Electron dynamics happens on the femto-second time scale. To probe it we need atto-second pulses. 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? • Those are questions outside the realm of linear-response • theory. To study them we have to propagate in time the • TDSE or -for larger systems- the TDKS equations.

3. 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)

4. = 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 .

5. 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.

6. 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

7. 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

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

9. 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).

10. 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)

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

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

13. INFORMATION ACCESSIBLE THROUGH TDELF How long does it take to break a bond in a laser field? Which bond breaks first, which second, etc, in a collision process? Are there intermediary (short-lived) bonds formed during a collision, which are not present any more in the collision products ?

14. TDELF movies produced from TD Kohn-Sham equations vxc[ (r’t’)](r t) propagated numerically on real-space grid using octopus code 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).

15. R (1) (2) – – + + + 0 Å -5 Å +5 Å x y MODEL S. Shin, H. Metiu, JCP 102, 9285 (1995), JPC 100, 7867 (1996) Nuclei (1) and (2) are heavy: Their positions are fixed

16. Anti-parallel spins Parallel spins M. Erdmann, E.K.U.G., V. Engel, JCP 121, 9666 (2004)

17. Parallel spins M. Erdmann, E.K.U.G., V. Engel, JCP 121, 9666 (2004)

18. Anti-parallel spins TD-ELF is a measure of non-adiabaticity

19. Most commonly used approximation for Adiabatic Approximation e.g. = xc potential of static homogeneous e-gas How restrictive is the adiabatic approximation, i.e. the neglect of memory in the functional vxc[ρ(r’,t’)](r,t) ? Can we assess the quality of the exact adiabatic approximation?

20.  = constant 1D MODEL • Restrict motion of electrons and nuclei to 1D (along polarization axis of laser) • Replace in Hamiltonian all 3D Coulomb interactions by soft 1D interactions (Eberly et al)

21. Two goals of 1D calculations 1. Qualitative understanding of physical processes, such as double ionization of He 2. Exact reference to test approximate xc functionals of time-dependent density functional theory

22. How can we assess the quality of the adiabatic approximation? • 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 • Inversion of one-particle ground-state SE yields exact static KS potential that gives (for each separate t) ρ(r,t) as a ground-state density. This is the exact adiabatic approximation of the TDKS potential.

23. 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, 153004 (2008)

24. 4-cycle pulse with λ = 780 nm, I1= 4x1014W/cm2, I2=7x1014W/cm2 Solid line: exact Dashed line: exact adiabatic M. Thiele, E.K.U.G., S. Kuemmel, Phys. Rev. Lett. 100, 153004 (2008)

25. PRIZE QUESTION No 3 For which kind of processes would you expect that the (exact) adiabatic approximation does not work?

26. By virtue of time-dependent 1-1 correspondence, ALL observables are functionals of the TD density some observables are easily expressed in terms of the density (no approximations involved) e.g. TD dipole moment HHG spectrum obtained from Other observables are more difficult to express in terms of the density (involving further approximation) e.g. ionization yields

27. p(0)(t) p(+1)(t) p(+2)(t) Calculation of ionization yields (for He) M. Petersilka and E.K.U. Gross, Laser Physics 9, 105 (1999). divide |R3 in: a large “analyzing volume” A (where (r t) is actually calculated B A and its complement B = |R3 \ A normalization of many-body wave function pair correlation function

29. 1 2 N1s(t) := d3r (r, t) = d3r|1s (r, t)|2 A A x-only limit for g[](r1,r2,t); two-electron-system: resulting ionization probabilities (mean-field expressions: P0(t) = N1s(t)2 P+1(t) = 2N1s(t) (1- N1s(t)) P+2(t) = (1- N1s(t))2 where:

30. Correlation Contributions + gc[](r1,r2,t) exactifies the mean-field expressions: P0(t) = N1s(t)2 + K(t) P+1(t) = 2N1s(t) (1- N1s(t)) - 2K(t) P+2(t) = (1- N1s(t))2 + K(t) correlation correction: 1 2 K(t) := d3r1 d3r2(r1, t) (r2, t) gc [] (r1, r2, t) A A

31. The calculation involves two approximate functionals: 1. The xc potential vxc[](r t) 2. The pair correlation function g[](r1r2 t) Which approximation is more critical?

32. 1D Helium atom (with soft Coulomb interaction) (Lappas, van Leeuwen, J. Phys. B 31, L249 (1998) P(He+) exact P(He++) exact P(He+) with exact density and g=1/2  P(He++) with exact density and g=1/2

33. Thanks !