Linear response theory
and TDDFT
Claudio Attaccalite
http://abineel.grenoble.cnrs.fr/
CECAM Yambo School 2013 (Lausanne)
Motivations:
+-
hν
Absorption Spectroscopy
Many Body
Effects!!!
Motivations(II):Absorption
Spectroscopy
Absorption linearly related to the Imaginary part of the
MACROSCOPIC dielectric constant (frequency dependent)
Outline
Response of the system to a perturbation →
Linear Response Regime
How can we calculate the response of the system?
IP, local field effects and
Time Dependent DFT
Some applications and recent steps forward
Conclusions
Spectroscopy
From Maxwell equation to the response function
Materials equations:
From Gauss's law:
D(r ,t )=ϵ 0 E (r , t)+ P (r , t)
∇⋅E (r , t)=4 π ρtot (r , t)
∇⋅D(r ,t )=4 π ρext (r , t)
Electric Field
Electric
Displacement
Polarization
In general:
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 )
For a small perturbation we consider only the first term,
the linear response regime
P(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)
Response Functions
Moving from Maxwell equation to linear response
theory we define
D(ω) δ V ext (ω)
ϵ(ω)=
=
ϵ0 E (ω) δ V tot (ω)
δ V tot (ω)
ϵ (ω)=
δ V ext (ω)
−1
where
V tot (⃗r , t )=V ext (⃗r ,t )+V ind (⃗r ,t ' )
The induced charge density results in a total potential
via the Poisson equation.
V tot (⃗r t )=V ext (⃗r t )+∫ dt ' ∫ d ⃗r v (⃗r −⃗r )ρind (⃗r t )
'
δ ρind
ϵ (ω)=1+ v
δV ext
δρind
ϵ(ω)=1−v
δ V tot
−1
'
' '
Our goal is to calculate the derivatives
of the induced density respect to the
external potential
The Kubo formula
1/2
H =H 0 + H ext (t)=H 0 +∫ d r ρ(r) V ext (r , t)
We star from the time-dependent
Schroedinger equation:
∂ψ
i
=[ H 0 + H ext (t)] ψ(t)
∂t
...and search for a solution as product of
the solution for Ho plus an another function
(interaction representation)...
iH t
̃
ψ(t)=e
ψ(t )
0
̃ ) iH t
∂ ψ(t
−iH t
̃ ext (t) ψ(t
̃ )= H
̃ )
i
=e H ext (t)e
ψ(t
∂t
0
0
...and we can write a formal solution as:
t
−i ∫t H̃ext (t)dt
̃
ψ(t)=e
0
̃ 0)
ψ(t
The Kubo formula 2/2
For a weak perturbation we can expand:
t
−i ∫t H̃ext (t)dt
̃
ψ(t)=e
0
1 t
̃ 0 )=[1+ ∫t dt ' H̃ext (t ' )+O ( H̃2ext )] ψ(t
̃ 0)
ψ(t
i
0
And now we can calculate the induced density:
t
̃ )∣ρ(t)
̃ )⟩≈⟨ρ⟩0 −i ∫t ⟨[ρ(t), H ext (t ' )]⟩+O( H 2ext )
ρ(t )=⟨ ψ(t
̃ ∣ψ(t
0
...and finally...
t
ρind (t )=−i ∫t
0
ρind (t )=ρ(t)−ρ0
∫ dr ⟨[ρ(r , t ),ρ(r ' t ')]⟩ V ext (r ' , t ')
Kubo Formula (1957)
ind
δ
ρ
(r , t )
' '
χ (⃗r t ,⃗r t )=
=−i ⟨[δρ(r , t)δρ(r ' t ')]⟩
δ V ext (r ' , t ' )
The linear response to a perturbation is independent on the
perturbation and depends only on the properties of the sample
How to calculated the dielectric constant
We want to calculate:
ind
δ
ρ
(r , t )
' '
χ (⃗r t ,⃗r t )=
=−i ⟨[δ ρ(r ,t )δ ρ(r ' t ' )]⟩
δ V ext (r ' , t ' )
We expand X in an independent particle basis set
χ (⃗r t ,⃗r ' t ' )=
∑
i , j,l ,m k
χ i , j , l , m , k ϕi , k (r )ϕ∗j , k (r ) ϕl , k (r ' )ϕ∗m , k (r ')
∂ ρ̂ i , j , k
χi , j , l , m , k=
∂ V l ,m ,k
Quantum Theory of the
Dielectric Constant in Real Solids
Adler
Phys. Rev. 126, 413–420 (1962)
The Von Neumann equation
(see Wiki http://en.wikipedia.org/wiki/Density_matrix)
∂ ρ̂ k (t )
eff
i
=[ H k +V , ρ̂ k ]
∂t
ρ̂ k (t )=∑i f (ϵk , i )∣ψi , k ⟩ ⟨ ψi , k∣
What is Veff ?
Independent Particle
Independent Particle
∂ ρ̂ i , j , k
χi , j , l , m , k=
∂ V l,m ,k
∂ ρ̂ k (t )
eff
i
=[ H k +V , ρ̂ k ]
∂t
Using:
Veff = Vext
∂ i ∂ρi , j , k = ∂ [ H +V eff , ρ̂ ]
k
k i, j,k
eff
eff
∂t
∂ V l ,m,k
∂ V l ,m, k
{
H i, j ,k =
ρ̂ i , j , k
δi , j ϵi (k)
∂ ρ̂ k
eff
= δi , j f (ϵi , k )+
⋅V
+....
eff
∂V
And Fourier transform respect to t-t', we get:
f (ϵi , k )−f (ϵ j , k )
χ i , j , l , m , k (ω)=
δ j ,l δi , m
ℏ ω−ϵ j , k + ϵi , k +i η
Optical Absorption: IP
δ ρNI =χ 0 δ V tot
0
χ =∑
ij
ϕi (r) ϕ*j (r) ϕ*i (r ' ) ϕj (r ' )
ω−(ϵi −ϵ j )+ i η
Hartree, Hartree-Fock, dft...
Non Interacting System
Absorption by independent
Kohn-Sham particles
=ℑ χ 0 =∑ ∣⟨ j∣D∣i⟩∣2 δ(ω−(ϵ j − ϵi ))
ij
2
8
π
''
2
ϵ (ω)= 2 ∑ ∣⟨ϕi∣e⋅v̂ ∣ϕ j ⟩∣ δ(ϵi −ϵ j −ℏ ω)
ω i, j
Particles are interacting!
Time-dependent Hartree (local fields)
Time-dependent Hartree
(local fields effects)
The induced charge
density results in a
total potential via the
Veff = Vext + VH
V tot r t =V ext r t ∫ dt ' ∫ d r ' v r −r ' ind r ' t ' 
Poisson equation.
 r ,t 
 r ,t   V tot r ' ' ,t ' ' 
 r , r ' , t−t ' =
=
 V ext r ' ,t '   V tot r ' ' ,t ' '  V ext r ' , t ' 
χ (⃗r t ,⃗r ' t ' )=χ 0 (⃗r t ,⃗r ' t ' )+∫ dt 1 dt 2∫ d r⃗1 d r⃗2 χ 0 (⃗r t , r⃗1 t 1 ) v ( r⃗1 −r⃗2) χ ( r⃗2 t 2 ,⃗r ' t ' )
ind
Screening of the
V ind
V tot
external perturbation
'
 0 r ,r =
 ind r , t 
V tot r ' t ' 
Time-dependent Hartree (local fields)
PRB 72 153310(2005)
Macroscopic Perturbation....
δ ρind
ϵ (ω)=1+ v
δV ext
δρind
ϵ(ω)=1−v
δ V tot
−1
Which is the right equation?
...microscopic observables
Not correct!!
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)=∫ dq V (q , r) eiqr =∫ dq ∑G V (q+G) ei(q+G )r
iGr
V
(q
,r
)=
V
(q+G)e
∑G
Where:
The G components describe the oscillation in the cell
while the q components the oscillation larger then L
Macroscopic averages 2/4
Macroscopic averages 3/4
Macroscopic averages 4/4
The external fields is macroscopic,
only components G=0
Macroscopic averages and local fields
If you want the macroscopic
δ ρind
ϵ (ω)=1+ v
δV ext
then invert the dielectric constant
δρind
ϵ(ω)=1−v
δ V tot
response use the first equation and
Local fields are not enough....
−1
What is missing?
Two particle excitations, what is missing?
electron-hole interaction, exchange, higher order effects......
The DFT and TDDFT way
DFT versus TDDFT
DFT versus TDDFT
Time Dependent DFT
[
]
−1 2
∇ +V eff (r , t ) ψi (r , t)=i ∂ ψi (r ,t )
2
∂t
V eff (r ,t )=V H (r , t)+ V xc (r , t)+ V ext (r , t)
Interacting System
Petersilka et al. Int. J. Quantum Chem. 80, 584 (1996)
 I
=
 V ext
 NI
 0=
 V eff
... by using ...
 I =  NI
  V ext = 0  V ext  V H  V xc 
 V H  V xc
= 1


 V ext  V ext
0
v
Non Interacting System
TDDFT is an exact
f xc 
theory for neutral
excitations!
 q ,= 0 q , 0 q , vf xc q ,  q ,
Time Dependent DFT
Choice of the xc-functional
...with a good xc-functional
you
can get the right spectra!!!
Summary
●
How to calculate linear response
in solids and 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
References!!!
On the web:
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http://yambo-code.org/lectures.php
http://freescience.info/manybody.php
http://freescience.info/tddft.php
http://freescience.info/spectroscopy.php
Electronic excitations: density-functional versus many-body
Green's-function approaches
RMP, vol 74, pg 601, (2002 ) G. Onida, L. Reining, and A. Rubio
Books: