Introduction to linear response theory and TDDFT

1. Claudio Attaccalite
http://abineel.grenoble.cnrs.fhttp://abineel.grenoble.cnrs.f
r/r/
Linear response theory
and TDDFT
CECAM Yambo School 2013 (Lausanne)CECAM Yambo School 2013 (Lausanne)

2. Motivations: Absorption Spectroscopy
-+
-
Many
Body
Effects!!
!
h ν

3. Motivations(II):Absorption
Spectroscopy
Absorption linearly related to the Imaginary part of the
MACROSCOPIC dielectric constant (frequency dependent)

4. Outline
How can we calculate the response of the
system? IP, local field effects and Time
Dependent DFT
Some applications and recent steps forward
Conclusions
Response of the system to a perturbation →
Linear Response Regime

5. Spectroscopy

6. From Maxwell equation to the response
function
D(r ,t)=ϵ0 E(r ,t)+P(r ,t) ∇⋅E(r ,t)=4 πρtot (r ,t)
∇⋅D(r ,t)=4 πρext (r ,t)
From Gauss's law:Materials equations:
Electric
Displacemen
t
Electric Field
Polarization
P(r ,t)=∫χ(t−t ' ,r ,r')E(t ' r')dt ' dr '+∫dt
1
dt
2
χ
2
(...)E(t
1
)E(t
2
)+O(E
3
)
In
general:
For a small perturbation we consider only the first
term, the linear response regimeP(r ,t)=∫χ(t−t ' ,r ,r')E(t ' r')dt ' dr '
In Fourier space:
P(ω)=ϵ0 χ(ω)E(ω)=ϵ0(ϵ(ω)−1)E(ω) D(ω)=ϵ0 ϵ(ω)E(ω)

7. Response Functions
ϵ(ω)=
D(ω)
ϵ0 E(ω)
=
δV ext (ω)
δV tot (ω)
Moving from Maxwell equation to linear
response theory we define
ϵ
−1
(ω)=
δVtot (ω)
δVext (ω)
Vtot (⃗r t)=V ext (⃗r t)+∫dt '∫d ⃗r
'
v(⃗r−⃗r
'
)ρind (⃗r
'
t
'
)
Vtot (⃗r ,t)=V ext (⃗r ,t)+Vind (⃗r ,t ')
where
The induced charge density results in a total
potential via the Poisson equation.
ϵ
−1
(ω)=1−v
δρind
δV ext
ϵ(ω)=1+v
δρind
δVtot
Our goal is to calculate the
derivatives of the induced density
respect to the external potential

8. The Kubo formula 1/2
H=H0+Hext (t)=H0+∫d rρ(r) V ext (r ,t)
We star from the time-dependent
Schroedinger equation:
i
∂ ψ
∂t
=[H0+Hext (t)] ψ(t)
...and search for a solution as product
of the solution for Ho plus an another
function (interaction representation)...
̃ψ(t)=e
i H0 t
ψ(t)
i
∂ ̃ψ(t)
∂t
=e
iH 0 t
Hext(t)e
−iH0 t
̃ψ(t)= ̃Hext (t) ̃ψ(t)
...and we can write a formal solution as: ̃ψ(t)=e
−i∫t 0
t
̃H ext(t)dt
̃ψ(t0)

9. Kubo Formula
(1957)
r t ,r'
t'
=
ind
r ,t
ext r' ,t '
=−i〈[ r ,t r' t ']〉
The Kubo formula 2/2
̃ψ(t)=e
−i∫t0
t
̃H ext(t)dt
̃ψ(t0)=[1+
1
i
∫t0
t
dt ' Hext (t ')+O(Hext
2
)] ̃ψ(t0)
For a weak perturbation we can expand:
And now we can calculate the induced density:
ρ(t)=〈 ̃ψ(t)∣̃ρ(t)∣̃ψ(t)〉≈〈ρ〉0−i∫t0
t
〈[ρ(t), Hext (t ')]〉+O(Hext
2
)
ρind (t)=−i∫t0
t
∫dr〈[ρ(r ,t),ρ(r' t ')]〉ϕext (r' ,t ')
...and finally......and finally...
The linear response to a perturbation is independent on
the perturbation and depends only on the properties of

10. How to calculated the dielectric
constant
i
∂ ̂ρk (t)
∂t
=[Hk +V
eff
, ̂ρk ] ̂ρk (t)=∑i
f (ϵk ,i)∣ψi,k 〉〈 ψi,k∣
The Von Neumann equation
(see Wiki http://en.wikipedia.org/wiki/Density_matrix)
r t ,r
'
t
'
=

ind
r ,t
ext r' ,t '
=−i〈[ r ,t r' t ']〉We want to calculate:
We expand X in an independent particle basis set
χ(⃗r t ,⃗r'
t'
)= ∑
i, j,l,m k
χi, j,l,m, k ϕi, k (r)ϕj ,k
∗
(r)ϕl,k (r')ϕm ,k
∗
(r')
χi, j,l,m, k=
∂ ̂ρi, j, k
∂Vl,m ,k
Quantum Theory of the
Dielectric Constant in Real Solids
Adler Phys. Rev. 126, 413–420 (1962)
What is Veff
?

11. Independent Particle
Independent Particle Veff
= Vext
∂
∂Vl ,m,k
eff
i
∂ρi, j ,k
∂t
= ∂
∂Vl ,m, k
eff
[Hk+V
eff
, ̂ρk ]i, j, k
Using:
{
Hi, j ,k = δi, j ϵi(k)
̂ρi, j, k = δi, j f (ϵi,k)+
∂ ̂ρk
∂V
eff
⋅V
eff
+....
And Fourier transform respect to t-t', we get:
χi, j,l,m, k (ω)=
f (ϵi,k)−f (ϵj ,k)
ℏ ω−ϵj ,k+ϵi ,k+i η
δj ,l δi,m
i
∂ ̂ρk (t)
∂t
=[Hk +V
eff
, ̂ρk ]
χi, j,l,m, k=
∂ ̂ρi, j, k
∂Vl,m ,k

12. Optical Absorption: IP
Non Interacting System
δρNI=χ
0
δVtot χ
0
=∑
ij
ϕi(r)ϕj
*
(r)ϕi
*
(r')ϕj(r ')
ω−(ϵi−ϵj)+ i η
Hartree, Hartree-Fock, dft...
=ℑχ0=∑
ij
∣〈 j∣D∣i〉∣2
δ(ω−(ϵj −ϵi))
ϵ
''
(ω)=
8 π2
ω2 ∑
i, j
∣〈ϕi∣e⋅̂v∣ϕj 〉∣
2
δ(ϵi−ϵj−ℏ ω)
Absorption by independent
Kohn-Sham particles
Particles are interacting!

13. Time-dependent Hartree (local fields)
Time-dependent Hartree
(local fields effects)
Veff
= Vext
+
VH
Vtot r t=V ext r t∫dt '∫d r'
v r−r'
ind r'
t'

The induced charge
density results in a
total potential via
the Poisson equation.
r ,r' ,t−t '=
 r ,t
V ext r ' ,t '
=
 r ,t
 Vtot r ' ' ,t ' '
 Vtot r' ' ,t ' '
 V ext r' ,t '
χ(⃗r t ,⃗r
'
t
'
)=χ0(⃗r t ,⃗r
'
t
'
)+∫dt1 dt2∫d ⃗r1 d ⃗r2 χ0 (⃗r t , ⃗r1 t1)v(⃗r1−⃗r2)χ(⃗r2 t2 ,⃗r
'
t
'
)
ind
Vind
Vtot
0r ,r
'
=
 ind r ,t
V tot r
'
t
'

Screening of the
external perturbation

14. Time-dependent Hartree (local fileds)
PRB 72 153310(2005)

15. Macroscopic Perturbation....
ϵ
−1
(ω)=1+v
δρind
δV ext
ϵ(ω)=1−v
δρind
δV tot
Which is the right equation?

16. ...microscopic observables
Not correct!!

17. Macroscopic averages 1/3
In a periodic medium every function
V(r) can be represented by the Fourier
series:
V (r)=∫dq∑G
V (q+G)ei(q+G)r
or
V (r)=∫dqV (q,r)eiqr
=∫dq∑G
V (q+G)ei(q+G)r
Where: V (q ,r)=∑G
V (q+G)eiGr
The G components describe the oscillation in the cell
while the q components the oscillation larger then L

18. Macroscopic averages 2/4

19. Macroscopic averages 3/4

20. Macroscopic averages 4/4
The external fields is macroscopic,
only components G=0

21. Macroscopic averages and local fields
If you want the macroscopic
response use the first equation
and then invert the dielectric
constant
ϵ
−1
(ω)=1+v
δρind
δV ext
ϵ(ω)=1−v
δρind
δV tot
Local fields are not
enough....

22. What is missing?
Two particle excitations, what is missing?Two particle excitations, what is missing?
electron-hole interaction, exchange, higher order effects......

23. The DFT and TDDFT way

24. DFT versus TDDFT

25. DFT versus TDDFT

26.  V ext=0
 V extV HV xc
q ,=
0
q ,
0
q,vf xc q ,q ,
TDDFT is an exact
theory for neutral
excitations!
Time Dependent DFT
V eff (r ,t)=V H (r ,t)+ V xc (r ,t)+ V ext (r ,t)
Interacting System
Non Interacting System
Petersilka et al. Int. J. Quantum Chem. 80, 584 (1996)
 I= NI=
 I
 Vext
0=
 NI
 V eff
... by
using ...
=
0
1
V H
 V ext

 V xc
V ext

v
f xc 
i
∂ ̂ρk (t)
∂t
=[ HKS , ̂ρk ]=[ Hk
0
+V eff
, ̂ρk ]

27. Time Dependent DFT
Choice of the xc-
functional
...with a good xc-functional
you can get the right spectra!!!

28. Summary
● How to calculate linear response in solids
molecules
● The local fields effects:
time-dependent Hartree
● Correlation problem:
TD-Hartree is not enough!
● Correlation effects can be included by mean
of TDDFT

29. 29

30. References!!!
Electronic excitations: density-functional versus many-body
Green's-function approaches
RMP, vol 74, pg 601, (2002 ) G. Onida, L. Reining, and A. Rubio
On the web:
● http://yambo-code.org/lectures.php
● http://freescience.info/manybody.php
● http://freescience.info/tddft.php
● http://freescience.info/spectroscopy.php
Books: