### Part 3 - University of Missouri

```Tutorial:
Time-dependent density-functional theory
Carsten A. Ullrich
University of Missouri
XXXVI National Meeting on Condensed Matter Physics
Aguas de Lindoia, SP, Brazil
May 13, 2013
Outline
2
PART I:
● The many-body problem
● Review of static DFT
PART II:
● Formal framework of TDDFT
● Time-dependent Kohn-Sham formalism
PART III:
● TDDFT in the linear-response regime
● Calculation of excitation energies
Optical spectroscopy
3
● Uses weak laser as Probe
Photoabsorption cross section
● System Response has peaks at
electronic excitation energies
Na2
Green
fluorescent
protein
Na4
Theory
Energy (eV)
Vasiliev et al., PRB 65, 115416 (2002)
Marques et al., PRL 90, 258101 (2003)
Linear response
4
tickle the system
observe how the
system responds
at a later time
(r, t )
(r, t )
The formal framework to describe the behavior of a system
under weak perturbations is called Linear Response Theory.
5
Linear response theory (I)
Consider a quantum mechanical observable
ground-state expectation value
ˆ
with
0  0 ˆ 0
Time-dependent perturbation:
Hˆ 1 (t )  F (t )ˆ ,
The expectation value of the observable
time-dependent:
ˆ
t  t0
now becomes
 (t )  (t ) ˆ (t ) ,
t  t0
The response of the system can be expanded in powers of
the field F(t):
 (t )  0  1 (t )   2 (t )  3 (t )  ...
= linear + quadratic + third-order + ...
6
Linear response theory (II)
For us, the density-density response will be most important.
The perturbation is a scalar potential V1,
Hˆ 1 (t )   d 3 r V1 (r, t )nˆ (r)
N
where the density operator is
nˆ (r)    (r  rl )
l 1
n(r, t )  (t ) nˆ (r) (t )
The linear density response is

n1 (r, t )   dt  d 3r   nn (r, r, t  t ) V1 (r, t )

 nn (r, r, t  t )  i (t  t ) 0 [nˆ (r, t  t ), nˆ (r)] 0
Linear response theory (III)
7
Fourier transformation with respect to
(t  t )
gives:
n1 (r,  )   d 3 r   nn (r, r,  ) V1 (r,  )
 0 nˆ (r) n n nˆ (r) 0
0 nˆ (r) n n nˆ (r) 0 
 nn (r, r,  )   






i





i

n 1 
n
n


where
n  En  E0
is the nth excitation energy.
►The linear response function has poles at the excitation
energies of the system.
► Whenever there is a perturbation at such a frequency,
the response will diverge (peak in the spectrum)
Spectroscopic observables
8
First-order induced dipole polarization:
p1 ()  α()E()
In dipole approximation, one defines a scalar potential associated with
a monochromatic electric field, linearly polarized along the z direction:
V1 (r, t )  E z sin(t )
This gives
p1, z ( )    d 3 r z n1 (r,  )
And the dynamic
dipole polarization
becomes
2 3
 zz ( )    d r z n1 (r,  )
E
   d 3 r  d 3 r  zz  (r, r,  )
From this we obtain the
photoabsorption cross section:
 zz ( ) 
4
Im  zz ( )
c
9
Spectrum of a cyclometallated complex
F. De Angelis, L. Belpassi, S. Fantacci, J. Mol. Struct. THEOCHEM 914, 74 (2009)
TDDFT for linear response
10
Gross and Kohn, 1985:
n1 (r, t )   d 3 r  dt   (r, t , r, t ) V1 (r, t )
  d 3 r  dt   s (r, t , r, t ) V1, s (r, t )
Exact density response can be calculated as the response of a
noninteracting system to an effective perturbation:
  (t  t )

V1, s (r, t )  V1 (r, t )   dt  d r 
 f xc (r, t , r, t ) n1 (r, t )
 r  r

3
xc kernel:
Vxc [n](r, t )
f xc (r, t , r, t ) 
n(r, t ) n (r )
0
Frequency-dependent linear response
11
n1 (r,  )   d 3 r  r, r,   V1 r,  
  d r  s r, r,   V1, s r,  
3
many-body
response
function:
 r, r,    
m
noninteracting
response
s
function:
0 nˆr  m m nˆr 0
  Em  E0   i
 c.c.( )
exact excitations Ω
 k* (r ) j (r ) *j (r) k (r)
r, r,      f k  f j 
  ( j   k )  i
j ,k
KS excitations
ωKS
 1

V1, s  V1   d r 
 f xc r, r,   n1 r,  
 r  r

3
12
The xc kernel: approximations
1
f xc (r, r,  )   (r, r,  )   (r, r,  ) 
| r  r |
1
s
1
Formally, the xc kernel is frequency-dependent and complex.
Random Phase Approximation (RPA):
frequency-independent
and real
f xcRPA  0
g .s.
2

V
[
n
](
r
)

Exc[n]
xc
f xc (r, r) 

n(r)
n(r)n(r)
2 unif
d
exc (n)
ALDA

(r, r ) 
dn2
nn
0 (r )
 (r  r)
13
Analogy: molecular vibrations
Molecular vibrations are characterized
by the eigenmodes of the system.
Using classical mechanics we can find
the eigenmodes and their frequencies
by solving an equation of the form
 A

2


jk
r m jk  jr  0
j
dynamical coupling
matrix (contains
the spring constants)
frequency of
rth eigenmode
eigenvector
gives the
mode profile
mass tensor
So, to find the molecular eigenmodes and vibrational frequencies
we have to solve an eigenvalue equation whose size depends
on the size of the molecule.
Electronic excitations
14
An electronic excitation can be viewed as an eigenmode of
the electronic many-body system.
This means that the electronic density of the system (atom,
molecule, or solid) can carry out oscillations, at certain
special frequencies, which are self-sustained, and do not
need any external driving force.
n1 (r,  )   d r   (r, r,  ) V1 (r,  )
3
finite density
response: the
eigenmode of
an excitation
diverges when ω
equals one of the
excitation energies
set to zero
(no extenal
perturbation)
Electronic excitations with TDDFT
15
 1

n1 (r,  )   d r   s r, r,    d r 
 f xc (r, r,  ) n1 (r,  )
| r  r |

3
3
Find those frequencies ω where the response equation,
without external perturbation, has a solution with finite n1.
M. Petersilka, U.J. Gossmann, E.K.U. Gross, PRL 76, 1212 (1996)
H. Appel, E.K.U. Gross, K. Burke, PRL 90, 043005 (2003)
We define the following abbreviation:
1
f Hxc (r, r,  ) 
 f xc (r, r,  )
| r  r |
16
Warm-up exercise: 2-level system
Noninteracting response function, where
 jk   j   k :
 k* (r ) j (r ) *j (r) k (r)
 s r, r,      f k  f j 
   jk  i
j ,k
We consider the case of a system with 2 real orbitals, the first
one occupied and the second one empty. Then,
1 (r ) 2 (r ) 2 (r)1 (r) 1 (r ) 2 (r ) 2 (r)1 (r)
 s r, r,   

  21
  21
221
 1 (r ) 2 (r ) 2 (r)1 (r) 2
  212
2-level system
17
n1 (r,  )   d 3r   s r, r,    d 3r  f Hxc (r, r, )n1 (r,  )
221
3
3



 2
d
r

(
r
)

(
r
)

(
r
)

(
r
)
d
r  f Hxc (r, r, )n1 (r,  )
1
2
1
2
2 

  21
Multiply both sides with 1 (r)2 (r) f Hxc (r, r, )
and integrate over r. Then we can cancel terms left and right, and
221
3
3
1 2
d
r
d
r 1 (r)2 (r) f Hxc (r, r, )1 (r)2 (r)
2 

  21
 2  212  221  d 3r  d 3r  1 (r ) 2 (r ) f Hxc (r, r, )1 (r) 2 (r)
    221 12 f Hxc 12
2
2
21
TDDFT correction to
Kohn-Sham excitation
The Casida formalism for excitation energies
18
Excitation energies follow
from eigenvalue problem
(Casida 1995):
A
 *
K
K  X 
  1 0  X 
   
 
* 
A  Y 
 0 1  Y 
Aia ,ia    ii aa    a   i   K ia ,ia 
K ia ,ia 
 1

  d r  d r  r  a r 
 f xc,  r, r,  i  r a  r
 r  r

3
3
*
i
For real orbitals and frequency-independent xc kernel, can rewrite this as
 
ia 
ii

2
2



2


K
Z


Zia 
aa   ai
ai ai  ia ,ia 
ia 
19
The Casida formalism for excitation energies
The Casida formalism gives, in principle, the exact excitation energies
and oscillator strengths. In practice, three approximations are required:
► KS ground state with approximate xc potential
► The inifinite-dimensional matrix needs to be truncated
► Approximate xc kernel (usually adiabatic):
f
xc
V (r )
r, r 
n(r)
stat
xc
advantage: can use any xc functional from static DFT (“plug and play”)
disadvantage: no frequency dependence, no memory
→ missing physics (see later)
20
How it works: atomic excitation energies
LDA + ALDA lowest excitations
Exp.
full matrix
SMA
Vasiliev, Ogut, Chelikowsky, PRL 82, 1919 (1999)
SPA
21
A comparison of xc functionals
Study of various functionals
over a set of ~ 500 organic
compounds, 700 excited
singlet states
D. Jacquemin et al.,
J. Chem. Theor. Comput.
5, 2420 (2009)
Mean Absolute Error (eV)
22
Excited states with TDDFT: general trends
Energies typically accurate within 0.3-0.4 eV
Vibrational frequencies good to 5%
Cost scales as N2-N3, vs N5 for wavefunction methods of
comparable accuracy (eg CCSD, CASSCF)
Available now in many electronic structure codes
challenges/open issues:
● complex excitations (multiple, charge-transfer)
● optical response/excitons in bulk insulators
Single versus double excitations
23
 1

n1 (r,  )   d r   s r, r,    d r 
 f xc (r, r,  ) n1 (r,  )
| r  r |

3
Has poles at KS single
excitations. The exact
response function has
more poles (single, double
and multiple excitations).
3
Gives dynamical corrections to
the KS excitation spectrum.
Shifts the single KS poles to the
correct positions, and creates
new poles at double and
multiple excitations.
► Adiabatic approximation (fxc does not depend on ω): only
single excitations!
► ω-dependence of fxc will generate additional solutions of the
Casida equations, which corresponds to double/multiple excitations.
► Unfortunately, nonadiabatic approximations are not easy to find.
24
Charge-transfer excitations
ZincbacteriochlorinBacteriochlorin complex
(light-harvesting in plants
and purple bacteria)
TDDFT error: 1.4 eV
TDDFT predicts CT states energetically well below local fluorescing
states. Predicts CT quenching of the fluorescence. Not observed!
Charge-transfer excitations: large separation
25

exact
ct
1
 I d  Aa 
R
(ionization potential of donor minus
electron affinity of acceptor plus
Coulomb energy of the charged fragments)
What do we get in TDDFT? Let’s try the single-pole approximation:
ctSPA   La   Hd
 2 d 2 r  d 3r   La (r) Hd (r) f Hxc (r, r,  ) La (r) Hd (r)
The highest occupied orbital of the donor and the lowest unoccupied
orbital of the acceptor have exponentially vanishing overlap!

TDDFT
ct
  
a
L
d
H
For all (semi)local xc approximations,
TDDFT significantly underestimates
charge-transfer energies!
26
Charge-transfer excitations: exchange
a , HF
d , HF
2
3
TDHF





d
r
d
ct
L
H
  r

a , HF
L

d , HF
H
1

R
 La (r ) La (r ) Hd (r) Hd (r)
| r  r |
and use Koopmans theorem!
TDHF reproduces charge-transfer energies correctly. Therefore,
hybrid functionals (such as B3LYP) will give some improvement
over LDA and GGA.
Even better are the so-called range-separated hybrids:
1
f (  | r  r |) 1  f (  | r  r |)


| r  r |
| r  r |
| r  r |
SR GGA
x
Exc  E
LR  HF
x
E
f (x)  e x , f (x)  erfc(x)
E
GGA
c
Excitations in finite and extended systems
27


0 nˆ r   j  j nˆ r 0
 r, r,    lim 
 c.c.   
 0
  E j  E0  i
 j

j
The full many-body response function has poles at the exact excitation
energies:
Im 
Im 
finite
x
xx
x
x
Re 
extended
Re 
► Discrete single-particle excitations merge into a continuum
(branch cut in frequency plane)
► New types of collective excitations appear off the real axis
28
Metals: particle-hole continuum and plasmons
In ideal metals, all single-particle states inside the Fermi sphere
are filled. A particle-hole excitation connects an occupied singleparticle state inside the sphere with an empty state outside.
 pl
From linear response theory, one can show that the plasmon dispersion
goes as
2
(q)   pl  q  ...
4ne 2
 pl 
m
29
Plasmon excitations in bulk metals
Sc
Al
Quong and Eguiluz,
PRL 70, 3955 (1993)
Gurtubay et al., PRB 72, 125114 (2005)
● In general, excitations in (simple) metals very well described by ALDA.
●Time-dependent Hartree (=RPA) already gives the dominant contribution
● fxc typically gives some (minor) corrections (damping!)
●This is also the case for 2DEGs in doped semiconductor heterostructures
30
Plasmon excitations in metal clusters
Yabana and Bertsch (1996)
Calvayrac et al. (2000)
Surface plasmons (“Mie plasmon”) in metal clusters are very well reproduced
within ALDA.
Plasmonics: mainly using classical electrodynamics, not quantum response
Insulators: three different gaps
31
Band gap:
Optical gap:
E g  E g , KS   xc
E
optical
g
 Eg  E
exciton
0
The Kohn-Sham gap
approximates the optical
gap (neutral excitation),
not the band gap!
32
Elementary view of Excitons
Real space:
Mott-Wannier exciton:
weakly bound, delocalized
over many lattice constants
Reciprocal space:
An exciton is a collective
interband excitation:
single-particle excitations are
coupled by Coulomb interaction
33
Excitonic features in the absorption spectrum
● Sharp peaks below the onset of the single-particle optical gap
● Redistribution of oscillator strength: enhanced absorption
close to the onset of the continuum
Optical absorption of insulators
34
Silicon
►absorption edge red shifted
(electron self-interaction)
►first excitonic peak missing
(electron-hole interaction)
Why does ALDA fail?
G. Onida, L. Reining, A. Rubio, RMP 74, 601 (2002)
S. Botti, A. Schindlmayr, R. Del Sole, L. Reining, Rep. Prog. Phys. 70, 357 (2007)
Linear response in periodic systems
35
 GG (k ,  )   sGG (k ,  ) 


G1G 2
sGG 1
(k ,  )

 VG1 (k ) G1G 2  f xcG1G 2 (k ,  )  G 2G (k ,  )
Optical properties are determined by the macroscopic dielectric function:
2
(Complex index of refraction)
mac

~
( )  n
For cubic symmetry,
one can prove that
Therefore, one needs the
inverse dielectric matrix:
 1

 mac ( )  lim  GG (k ,  ) G 0 
k 0 
G0 
1
 G1G (k, )  GG  VG (k)GG (k, )
The xc kernel for periodic systems
36
f Hxc (r, r,  ) 
 e
i (q G )r
qFBZ G,G 
f Hxc ,GG (q,  )e
i (q G)r
TDDFT requires the following matrix elements as input:
K iaG,0iGa0 

qFBZ GG 
ik i ei (q G )r ak a f Hxc ,GG (q,  ) ak a e  i (q G)r ik i
  k a k i q ,G 0  k a k i q ,G0
Most important: long-range
ik i e
but
iqr
f
(q  0) limit of “head” (G  G   0) :
ak a q
 q
0
ALDA
xc , 00
f
exact
xc , 00
(q,  ) q
 const .
0
1
(q,  ) q
 2
0
q
Therefore, no excitons
in ALDA!
37
Long-range xc kernels for solids
● LRC (long-range corrected) kernel
(with fitting parameter α):
f
LRC
xc ,GG 
q   

|qG |
2
 GG 
● “bootstrap” kernel (S. Sharma et al., PRL 107, 186401 (2011)
1

boot
GG  (q,0)vG  (q )
f xc ,GG (q,  ) 
 s 00 (q,0)
● Functionals from many-body theory: (requires matrix inversion)
exact exchange
excitonic xc kernel from
Bethe-Salpeter equation
38
Excitons with TDDFT: “bootstrap” xc kernel
S. Sharma et al., PRL 107, 186401 (2011)
39
Extended systems - summary
► TDDFT works well for metallic and quasi-metallic systems already
at the level of the ALDA. Successful applications for plasmon modes
in bulk metals and low-dimensional semiconductor heterostructures.
► TDDFT for insulators is a much more complicated story:
● ALDA works well for EELS (electron energy loss spectra), but
not for optical absorption spectra
● Excitonic binding due to attractive electron-hole interactions,
which require long-range contribution to fxc
● At present, the full (but expensive) Bethe-Salpeter equation gives
most accurate optical spectra in inorganic and organic materials
(extended or nanoscale), but TDDFT is catching up.
● Several long-range XC kernels have become available
(bootstrap, meta-GGA), with promising results. Stay tuned!
40
The future of TDDFT: biological applications
(TD)DFT can handle big systems (103—106 atoms).
Many applications to large organic systems (DNA, light-harvesting
complexes, organic solar cells) will become possible.
Charge-transfer excitations and van der Waals interactions can
be treated from first principles.
N. Spallanzani, C. A. Rozzi, D. Varsano, T. Baruah, M. R. Pederson, F. Manghi, and
A. Rubio, J. Phys. Chem. (2009)
41
The future of TDDFT: materials science
K. Yabana, S. Sugiyama, Y. Shinohara, T. Otobe, and G.F. Bertsch, PRB 85, 045134 (2012)
Vacuum
Si
Si
● Combined solution of TDKS and Maxwell’s equations
● Strong fields acting on crystalline solids: dielectric breakdown,
coherent phonons, hot carrier generation
● Coupling of electron and nuclear dynamics allows description
of relaxation and dissipation (TDDFT + Molecular Dynamics)
42
The future of TDDFT: open formal problems
► Development of nonadabatic xc functionals
(needed for double excitations, dissipation, etc.)
► TDDFT for open systems: nanoscale transport in
dissipative environments. Some theory exists, but
applications so far restricted to simple model systems
► Strongly correlated systems. Mott-Hubbard insulators,
Kondo effect, Coulomb blockade. Requires subtle xc
effects (discontinuity upon change of particle number)
► Formal extensions: finite temperature, relativistic effects…
TDDFT will remain an exciting field of research
for many years to come!
43
Literature
Time-dependent Density-Functional
Theory: Concepts and Applications
(Oxford University Press 2012)
“A brief compendium of TDDFT”
Carsten A. Ullrich and Zeng-hui Yang
arXiv:1305.1388
(Brazilian Journal of Physics, Vol. 43)
C.A. Ullrich homepage:
http://web.missouri.edu/~ullrichc
[email protected]
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