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

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.

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

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

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

is a measure of electron localization. small means strong localization at Why?, being the s 2 -coefficient, gives the probability of finding a second like-spin electron very near the reference electron. If this probability very near the reference electron is low then this reference electron must be very localized.

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)) Advantage: ELF is dimensionless and where is the kinetic energy density of the uniform gas.

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

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

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, (2005)

Acetylene in a strong laser field (ħω = eV, I = 1.2  W/cm 2 ) [Snapshots of TDELF]

Scattering of a high-energy proton from ethylene (E kin (proton) = 2 keV) [Snapshots of TDELF]

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 ?

TDELF movies produced from TD Kohn-Sham equations v xc [  (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).

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

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

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

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

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

1D MODEL 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)  = constant

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

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.

Solid line: exact Dashed line: exact adiabatic 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. M. Thiele, E.K.U.G., S. Kuemmel, Phys. Rev. Lett. 100, (2008)

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

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

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

Calculation of ionization yields (for He) divide |R 3 in: a large “analyzing volume” A (where  (r t) is actually calculated A B and its complement B = |R 3 \ A normalization of many-body wave function p (0) (t) p (+1) (t)p (+2) (t) pair correlation function M. Petersilka and E.K.U. Gross, Laser Physics 9, 105 (1999).

x-only limit for g[  ](r 1,r 2,t); resulting ionization probabilities (mean-field expressions: P 0 (t) = N 1s (t) 2 P +1 (t) = 2N 1s (t) (1- N 1s (t)) P +2 (t) = (1- N 1s (t)) 2 where: N 1s (t) := d 3 r  (r, t) = d 3 r |  1s (r, t) | 2 AA 1212 two-electron-system:

Correlation Contributions + g c [  ](r 1,r 2,t) exactifies the mean-field expressions: P 0 (t) = N 1s (t) 2 + K(t) P +1 (t) = 2N 1s (t) (1- N 1s (t)) - 2K(t) P +2 (t) = (1- N 1s (t)) 2 + K(t) correlation correction: K(t) := d 3 r 1 d 3 r 2  (r 1, t)  (r 2, t) g c [  ] (r 1, r 2, t) AA 1212

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

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