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.
Electron Localization Function
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)
D  r, r' 
  d r ... d r
3
34 ... N
3
3
N
  r, r', r33 ..., rN  N 
= diagonal of two-body density matrix
= probability of finding an electron with spin  at r
and another electron with the same spin at r'.
P  r, r' :
D  r, r'
  r 
= conditional probability of finding an electron
with spin  at r' if we know with certainty that
there is an electron with the same spin at r .
2
Coordinate transformation
r'
s
r
If we know there is an electron with spin  at r, then
P  r, r  s  is the (conditional) probability of
finding another electron at s , where s is measured
from the reference point r .

Spherical average
2
1
p  r, s  
sin d  dP  r, s , ,  

4 0
0
If we know there is an electron with spin  at r, then p  r, s  is the
conditional probability of finding another electron at the distance s
from r.
Expand in a Taylor series:
p   r, s   p   r, 0  
0
dp   r, s 
The first two terms vanish.
ds
0
1
 s  C  r  s 2
3
s 0
Cσ  r  is a measure of electron localization.
Why? C  r  , being the s2-coefficient, gives the probability of
σ
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.
Cσ  r  small means strong localization at r
C is always ≥ 0 (because p is a probability) and Cσ  r  is not
bounded from above.
Define as a useful visualization of localization
(A.D. Becke, K.E. Edgecombe, JCP 92, 5397 (1990))
ELF 
where
1
1


 C  r  
 uni  
 C  r  
2
23
 3


Cuni r   62  5 3 r   uni r 
5
is the kinetic energy density of the
uniform gas.
Advantage: ELF is dimensionless and
0  ELF  1
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
ELF
Density
E. Räsänen, A. Castro and E.K.U. Gross, Phys. Rev. B 77, 115108 (2008).
For a determinantal wave function one obtains
in the static case (i.e. for real-valued orbitals):
N
C
det

 r    i  r 
i 1
2
1    r  

4   r 
2
(A.D. Becke, K.E. Edgecombe, JCP 92, 5397 (1990))
in the time-dependent case:
N
C
det

 r, t    i  r, t 
i 1
2
1    r, t  
 j  r, t    r, t 

4   r, t 
2
2
T. Burnus, M. Marques, E.K.U.G., PRA (Rapid Comm) 71, 010501 (2005)
Acetylene in a strong laser field
(ħω = 17.15 eV, I = 1.21014 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 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
2
2











i  j rt   
 v KS  rt   j rt 
t
 2m

r ' t 
3
v KS r ' t 'rt   v rt    d r '
 vxc[ (r’t’)](r t)
r  r'
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).
MODEL
S. Shin, H. Metiu, JCP 102, 9285 (1995), JPC 100, 7867 (1996)
R
(1)
(2)
–
+
–
+
+
0Å
-5 Å
+5 Å
x
y
Nuclei (1) and (2) are heavy: Their positions are fixed
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 measure
of non-adiabaticity

Most commonly used approximation for v xc ρrt 
Adiabatic Approximation
v
e.g.
adiab
xc
ALDA
xc
v
vhom
xc,stat
 r t  : v
r t
approx
xc,stat
: v
n 
hom
xc,stat
n ( r t )
  r t 
= 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?
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)
1
1
x y z
2
2
2
 z
2
2
 = 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.
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)
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)
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)
d(t)   ρr, t zd 3 r
e.g. TD dipole moment
HHG spectrum
obtained from
dω
2
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)
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
A
B
and its complement B = |R3 \ A
normalization of many-body wave function
1   d r1  d r2  r1 r2 t   2     
3
A
3
A
2
2
AB
p(0)(t)
pair correlation function
BB
p(+1)(t)
g r1r2 t  :
2
p(+2)(t)
2  r1r2 t 
2
r1t r2 t 
1 3
p t    d r1  d 3r2ρr1t ρr2 t gρr1r2 t 
2A
A
0 
p 1 t    d 3rρr t - d 3r1  d 3 r2ρr1t ρr2 t gρr1r2 t 
A
p
 2 
A
A
1 3
t   1   d rρr t +  d r1  d 3r2ρr1t ρr2 t gρr1r2 t 
2A
A
A
3
x-only limit for g[](r1,r2,t);
1
g x only r1 , r2 , t  
2
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:
N1s(t) :=
1 3
d r (r, t) =
2
A
d3r| 1s (r, t)|2
A
Correlation Contributions
gr1, r2 , t   g x only r1, r2 , t  + 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) :=
1d3r d3r (r , t) (r , t) g [] (r , r , t)
2
1
2
c
1 2
2 1
A
A
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