Time-Dependent Electron Localization FunctionTime-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-responsetheory. 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 FunctionElectron Localization Function

D r, r'P r, r' :

r

= 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

3 4 N

23 33 N 3 3 N N

...

D r, r' d r ... d r r , r' , r ..., r

= probability of finding an electron with spin at and another electron with the same spin at .

r

r'

r'

r

Coordinate transformation

r'

r

sIf 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 .

2

s 0

dp r,s 1p r,s p r,0 s C r s

ds 3

0 0

Expand in a Taylor series:

The first two terms vanish.

Spherical average 2

0 0

1p r, s sin d d P r, s , ,

4

r

P ,r r

s

r

s

s

If we know there is an electron with spin at , then is the conditional probability of finding another electron at the distance s from .

p ,r s

r

r

is a measure of electron localization.

small means strong localization at

σC r

Why? , being the s2-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.

σC r

σC r

r

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))

1 ELF 0 Advantage: ELF is dimensionless and

where

is the kinetic energy density of the uniform gas.

1

1ELF 2

rC

rCuni

rr6

5

3rC uni35322uni

σC r

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, 115108 (2008).

For a determinantal wave function one obtains

2N

2deti

i 1

r1C r r

4 r

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)

2N

2deti

i 1

r, t1C r, t r, t

4 r, t

2

j r, t r, t

Acetylene in a strong laser field (ħω = 17.15 eV, I = 1.21014 W/cm2) [Snapshots of TDELF]

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

INFORMATION ACCESSIBLE THROUGH TDELF

How long does it take to break a bond ina 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

'rr

t'r'rdrtvrt't'rv 3

KSvxc[ (r’t’)](r t)

rt rtvm2

rtt

i jKS

22

j

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).

+ +

-5 Å +5 Å

0 Å

xy

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 spins Parallel 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 measureof non-adiabaticity

Most commonly used approximation for trρvxc

Adiabatic Approximation

adiab approxxc xc,stat

n ( r t )v r t : v n

homstat,xcv = 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?

e.g. ALDA homxc xc,statv r t : v r t

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)

22222

11

zzyx = 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, 153004 (2008)

M. Thiele, E.K.U.G., S. Kuemmel, Phys. Rev. Lett. 100, 153004 (2008)

4-cycle pulse with λ = 780 nm, I1= 4x1014W/cm2, I2=7x1014W/cm2

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

2

ωd

rzdr, tρd(t) 3

Calculation of ionization yields (for He)

divide |R3 in: a large “analyzing volume” A (where (r t) is actually calculated

AB

and its complement B = |R3 \ A

normalization of many-body wave function

B B

2

B A

2

A

2

2123

A

13 2t r r rdrd1

p(0)(t) p(+1)(t) p(+2)(t)

pair correlation function trtr

trr2:trrg

21

2

2121

M. Petersilka and E.K.U. Gross, Laser Physics 9, 105 (1999).

A A

212123

13

A

32

A A

212123

13

A

31

A A

212123

130

trrρgtrρtrρrdrd2

1+trrρd1tp

trrρgtrρtrρrdrd-trrρdtp

trrρgtrρtrρrdrd2

1tp

x-only limit for g[](r1,r2,t);

2

1t,r,rg 21onlyx

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:

N1s(t) := d3r (r, t) = d3r| 1s (r, t)|2

A A

12

two-electron-system:

Correlation Contributions

t,r,rgt,r,rg 21onlyx21 + 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:

K(t) := d3r1 d3r2 (r1, t) (r2, t) gc [] (r1, r2, t)

A A

12

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?

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