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:
• 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?
Perylene at TiO2 surface
TiO2
donor
bridge
acceptor
Time-resolved photo-absorption experiment: Frank Willig (HMI Berlin)
OUTLINE
I. Basics of TDDFT
• Some news on excitons
II. Analyzing electronic motion in real time
THANKS
Sangeeta Sharma
Kay Dewhurst
Antonio Sanna
• TD Electron Localization Function:
Movie of laser-induced π-π* transition
Miguel Marques
Tobias Burnus
Alberto Castro
• TD approach to electronic transport
through single molecules
Stefan Kurth
Gianluca Stefanucci
Angelica Zacarias
Elham Khosravi
Claudio Verdozzi (Lund)
III: Adiabatic approximation in TDDFT
S. Kümmel (Bayreuth)
M. Thiele
Time-dependent systems
Generic situation:
Molecule in laser field
ˆ 
Hˆ (t)  Tˆ e  W
ee
j ,
e2
 
Zα
+ E ·rj·sin t
|rj-Rα|
Strong laser (vlaser(t) ≥ ven) :
Non-perturbative solution of full TDSE required
Weak laser (vlaser(t) << ven) :
Calculate 1. Linear density response
2. Dynamical polarizability

1(r t)
    
3. Photo-absorption cross section
e
E

3


z

r
,

d
r
 1
    
4 
c
Im 
continuum
states
I1
unoccupied
bound states
I2
occupied
bound states
photo-absorption cross section
Photo-absorption in weak lasers
(ω)
Laser frequency 
No absorption if  < lowest excitation energy
Why don’t we just solve the many-particle SE?
Example: Oxygen atom (8 electrons)


  r1 ,  , r8  depends on 24 coordinates
rough table of the wavefunction
10 entries per coordinate:
1 byte per entry:
1010 bytes per DVD:
10 g per DVD:
 1024 entries
 1024 bytes
 1014 DVDs
 1015 g DVDs
= 109 t DVDs
ESSENCE OF DENSITY-FUNTIONAL THEORY
• 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
Time-dependent density-functional formalism
(E. Runge, E.K.U.G., PRL 52, 997 (1984))
Basic 1-1 correspondence:
v  r t      r t 
1 -1
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
2
N
ρ rt =

j=1
 
 j rt
of an auxiliary non-interacting (KS) system
2
2



i
j rt   
 v S    r t    j  r t 
t
2m



with the local potential
v S    r ' t '    r t   v  r t    d r '
3
 r 't
 v xc   r ' t '    rt 


rr'
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:
ρ  v
 rt   ρ  v0  v1   rt 
 ρ  v0   rt 


1
δρ  v   rt 
δv  r't' 
ρo  r 
v1  r ' t ' d 3 r'dt'
ρ1  rt 
v0
δ ρ  v   rt 
2
  δv  r't' δv  r"t"
2
v1  r ', t ' v1  r ", t " d r'd r"dt'dt"
3

v0
3
ρ2  rt 
1(r,t) = linear density response of interacting system
  rt, r ' t '  :
δρ  v   rt 
δv  r't' 
v0
= density-density response function of
interacting system
Analogous function s[vs](r t) for non-interacting system
ρS  vS   rt   ρS  vS,0  vS,1  rt   ρS  vS,0   rt   
S  rt, r ' t '  :
δρS  vS   rt 
δvS  r't' 
vS,0
δρS  vS   rt 
δvS  r't' 
vS,1  r't' d r'dt' 
3
v S,0
= density-density response function of
non-interacting system
Standard linear response formalism
H(t0) = full static Hamiltonian at t0
H  t0  m  E m m
 exact many-body eigenfunctions
and energies of system
full response function
  r, r ';   
lim
 0

 0 ˆ  r  m m ˆ  r  0
    E  E  i 
 m 0
m 
0 ˆ  r '  m m ˆ  r '  0 


   E m  E 0   i

 The exact linear density response
1 () =  () v1
has poles at the exact excitation energies  = Em - E0
continuum
states
I1
unoccupied
bound states
I2
occupied
bound states
photo-absorption cross section
Photo-absorption in weak lasers
(ω)
Laser frequency 
No absorption if  < lowest excitation energy
Linearization of TDKS equation yields:


3
ρ 1  r t  =  d r'dt'χ S  r t, r't'   v 1  r t  +  d r"dt"  V ee  r't', r"t"  + f xc  r't', r"t"  ρ 1  r"t"  


3
• Exact integral equation for 1(r t), to be solved iteratively
• Need approximation for f xc  r't', r"t" 
(either for fxc directly or for vxc)
δv xc ρ   r't' 
δρ  r"t"
ρ0
Equivalent to:
χ  χ S  χ S  V ee  f xc  χ
Adiabatic approximation
adiab
v xc
 ρ   r t  := v xc
static D FT
 ρ  t   r t 
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:
v
ALD A
xc
 r t  :
 rt 
δρ r't' 
ρ
v
LD A
xc
 ρ  rt  
 α ρ  rt 
ALDA
 f xc
ALDA
 rt, r't' 
δv xc
 δ  r  r'  δ  t  t' 
13
 v xc
ALDA
 ρ r 
0
 δ  r  r' δ  t  t' 

ρ0  r 
 e xc
2
hom
n
2
ρ0  r 
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)]
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)
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)).

1
q,  
 1  v q   q,  
 1
v q  s q,  
1   v  q  f xc  q ,     s  q ,  
Bootstrap approximation for fxc :
f
bootstrap
xc
q,   
q,   0 
s q,   0 

1
S. Sharma, K. Dewhurst, A. Sanna, E.K.U.G., arXiv:1107.0199
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)
Silicon: Loss function Im χ(q,ω)
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
 3  4 ...  N
3
r3 ...  d rN   r  , r' , r3  3 ..., rN  N
3
= 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'
r
s
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 p   r, s

1

2
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
s 
s0
0
1
3
C  r s
2
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))
where
1
ELF 
1

 
 C r  
 u ni  
 C r  
uni
2
C

 r   6  
3
5
2
2 3
5 3 
uni 
  r    r 
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  
2
i 1
1   r 
4
2
  r 
(A.D. Becke, K.E. Edgecombe, JCP 92, 5397 (1990))
in the time-dependent case:
N
C
det

 r, t   
i 1
  i  r, t 
2

1      r, t  
4
   r, t 
2
 j  r, t 
2
  r, t 
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]
Use TD Kohn-Sham equations (E. Runge, EKUG, PRL 52, 997 (1984))

 


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'
2
2
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).
Electronic transport: Generic situation
left lead L
central
region C
Bias between L and R is turned on: U(t)
right lead R
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
Standard approach: Landauer formalism plus static DFT
left lead L
I( V ) 
e
h
central region C
right lead R
 dE T  E , V   f  E    
Transmission function T(E,V) calculated
from static (ground-state) DFT
1

f E   2 
 1,2  E F 
eV
2
Chrysazine
Relative Total Energies and HOMO-LUMO Gaps
OH
O
OH
O
Chrysazine (a)
0.0 eV
3.35 eV
Chrysazine (c)
1.19 eV
3.77 eV
Chrysazine (b)
0.54 eV
3.41 eV
0.1
(a)
(b)
(c)
Current (A)
0.08
0.06
0.04
0.02
0
0
2
4
6
8
Voltage (V)
Possible use: Optical switch
A.G. Zacarias, E.K.U.G., Theor. Chem. Accounts 125, 535 (2010)
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
Electronic transport with TDDFT
central
region C
left lead L
right lead R
TDKS equation (E. Runge, EKUG, PRL 52, 997 (1984))

  

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'
2
2
Electronic Transport with TDDFT
left lead L
central
region C
right lead R
TDKS equation
  L  t    H LL  t 
 
 
i   C  t     H CL  t 
t 
  H t 



t
 R   RL
H LC  t 
H CC  t 
H RC  t 
H LR  t     L  t  


H CR  t     C  t  
H RR  t     R  t  
Propagate TDKS equation on spatial grid

 A  t   vector   r1 , t ,   r2 , t ,   with grid points r1, r2, …
in region A (A = L, C, R)

H AB  t   correspond ing grid - blocks of TDKS Hamiltonia n
H AB  t  for A  B
H CL , H LC , H CR , H RC
H LR  H RL  0
is purely kinetic, because KS
potential is local
are time-independent
  L  t    H LL  t 
 
 
i   C  t     H CL  t 
t 
  H t 



t
 R   RL
H LC  t 
H CC  t 
H RC  t 
H LR  t     L  t  


H CR  t     C  t  
H RR  t     R  t  
Hence:
 



i

H
t

  L  t   H LC  C  t 
LL
 t

i

t
L
 C  t   H CL  L  t   H CC  t  C  t   H CR  R  t 
C
 

 H RR  t    R  t   H RC  C  t 
i
 t

Next step: Solve inhomogeneous Schrödinger equations
L, R using Green’s functions of L, R, leads
R
L , R
for
Define Green’s Functions of left and right leads:
 

 H LL  t   G L  t , t '     t  t ' 
i
 t



ˆ
 L  G L r.h.s. of


ˆ
 R  G R r.h.s. of

explicity:
 L t  
L
R






t
 dt ' G  t , t ' H
L
LC
 

 H RR  t   G R  t , t '     t  t ' 
i
 t


solution of hom. SE


solution of hom. SE

 C  t '   iG
L

 

 H LL  t     0 
i
 t



 

 H RR  t     0 
i
 t


 t , 0  L  0 
0
 R t  
t
 dt ' G  t , t 'H
R
RC
 C  t '   iG
0
insert this in equation C
R
 t , 0  R  0 
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)
i

t
 C  t   H CC  t  C  t 
t

 dt ' H
CL
G L  t , t ' H LC  H CR G R  t , t ' H RC  C  t ' 
0
 iH
CL
G L  t , 0  L 0   iH
CR
G R  t , 0  R 0 
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.
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
ELECTRON PUMP
Device which generates a net current between two
electrodes (with no static bias) by applying a timedependent 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))
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.
Experimental result:
Current flows in direction opposite to sound wave
Simulation: Current goes in direction opposite to the external field !!
G. Stefanucci, S. Kurth, A. Rubio, E.K.U. Gross, Phys. Rev. B 77, 075339 (2008)
Bound state oscillations and memory effects
Analytical: G. Stefanucci, Phys. Rev. B, 195115 (2007))
Numerical: E. Khosravi, S. Kurth, G. Stefanucci, E.K.U.G.,
Appl. Phys. A93, 355 (2008), and Phys. Chem. Chem. Phys. 11, 4535 (2009)
If Hamiltonian of a (non-interacting) biased system in the long-time
limit supports two or more bound states then current has steady, I(S),
and dynamical, I(D), parts:
I( t   )  I
I
(D )
(t) 

bb '
(S)
I
(D)
(t)
sin[(  b   b ' ) t ]
b ,b '
Sum over bound states of biased Hamiltonian
Note: - bb’ depends on history of TD Hamiltonian (memory!)
Questions: -- How large is I(D) vs I(S)?
-- How pronounced is history dependence?
History dependence of undamped oscillations
1-D model:
start with flat potential, switch on constant bias, wait until transients die out, switch
on gate potential with different switching times to create two bound states
note: amplitude of bound-state
oscillations may not be small
compared to steady-state current
Time-dependent picture of Coulomb blockade
Model system
U
Hˆ  t   Hˆ Q D 
Hˆ   Hˆ T  Hˆ bias  t 

  L ,R
ˆ
H
 v ext  nˆ 0   U nˆ 0  nˆ 0 
QD

ˆ t  
H


Hˆ T   



  V cˆ
i 1 , 
cˆ i ,   h.c. 
i 1
 V
†
ˆ
c
cˆ  h.c.
link 1  ,  0 
 ,  i 1
Hˆ bias  t    

 U  t  nˆ
 ,  i 1

i , 

Solve TDKS equations (instead of fully interacting problem):
Hˆ K S  t   Hˆ Q D , K S  t  

Hˆ   Hˆ T  Hˆ bias  t 
  L ,R
ˆ
H
t 
Q D ,K S  
v

KS
 n 0  t   nˆ 0 
n0 t 
 n t
0

v K S  n 0  t    v ext 
1
2
U n 0  t   v xc  n 0  t  
LDA functional for vxc is available from exact Bethe-ansatz solution of
the 1D Hubbard model.
N.A. Lima, M.F. Silva, L.N. Oliveira, K. Capelle, PRL 90, 146402 (2003)
S. Kurth, G. Stefanucci, E. Khosravi, C. Verdozzi, E.K.U.G., Phys. Rev. Lett. 104, 236801 (2010)
Is this Coulomb blockade?
Fingerprint of Coulomb blockade
UL/V
Steady-state density as function of applied bias for KS potential with smoothened discontinuity
S. Kurth, G. Stefanucci, E. Khosravi, C. Verdozzi, E.K.U.G., Phys. Rev. Lett. 104, 236801 (2010)

Most commonly used approximation for v xc ρ  r t 
Adiabatic Approximation
v
adiab
xc
 r t  :
ALD A
e.g. v x c
v
approx
xc ,stat
n 
n  ( r t )
 r t  : v x c ,stat    r t  
hom
hom
v xc , stat = 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?
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)