Download - Linear response theory and TDDFT
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Claudio Attaccalitehttp://abineel.grenoble.cnrs.fhttp://abineel.grenoble.cnrs.fr/r/
Linear response theory and TDDFT
CECAM Yambo School 2013 (Lausanne)CECAM Yambo School 2013 (Lausanne)
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Motivations: Absorption Spectroscopy
-+
-Many Body Effects!!!
h ν
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Motivations(II):Absorption Spectroscopy
Absorption linearly related to the Imaginary part of the MACROSCOPIC dielectric constant (frequency dependent)
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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
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Spectroscopy
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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 )=4πρext (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 '+∫ dt1dt 2χ
2(...)E (t1)E (t 2
)+O (E3)
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)=ϵ0ϵ(ω)E (ω)
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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)=δV tot (ω)
δV ext (ω)
V tot ( r⃗ t)=V ext ( r⃗ t )+∫ dt '∫d r⃗ ' v ( r⃗− r⃗ ')ρind ( r⃗
' t ')
V tot ( r⃗ , t)=V ext ( r⃗ , t)+V ind ( r⃗ , t ')
where
The induced charge density results in a total potential via the Poisson equation.
ϵ−1
(ω)=1−vδρind
δV ext
ϵ(ω)=1+vδρind
δV tot
Our goal is to calculate the derivatives of the induced density respect to the external potential
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The Kubo formula 1/2
H=H 0+H ext (t)=H 0+∫ d rρ(r) V ext (r , t)
We star from the time-dependent Schroedinger equation:
i∂ψ
∂ t=[H 0+H ext (t)] ψ(t)
...and search for a solution as product of the solution for Ho plus an another function (interaction representation)...
ψ̃(t)=ei H 0 t ψ(t )
i∂ ψ̃(t)
∂ t=eiH 0 t H ext(t)e
−iH 0 t ψ̃(t )= H̃ ext (t) ψ̃(t )
...and we can write a formal solution as: ψ̃(t)=e−i∫t 0
t̃H ext(t)dt
ψ̃(t 0)
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Kubo Formula (1957)
r t ,r ' t '=ind r , t ext r ' , t '
=−i ⟨[ r , t r ' t ' ]⟩
The Kubo formula 2/2
ψ̃(t)=e−i∫t 0
t̃H ext(t)dt
ψ̃(t 0)=[1+1i∫t 0
tdt ' H ext (t ')+O (H ext
2 )] ψ̃(t 0)
For a weak perturbation we can expand:
And now we can calculate the induced density:
ρ(t )=⟨ ψ̃(t )∣ρ̃(t)∣ψ̃(t )⟩≈⟨ρ⟩0−i∫t 0
t⟨[ρ(t) , H ext (t ')]⟩+O(H ext
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
the sample
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How to calculated the dielectric constant
i∂ρ̂k (t )
∂ t=[H k+V eff , ρ̂k ] ρ̂k (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 '=
indr , t
ext r ' , t ' =−i ⟨[ r , t r ' t ' ]⟩We want to calculate:
We expand X in an independent particle basis set
χ( r⃗ t , r⃗ ' t ')= ∑i , j , l ,m k
χi , j , l ,m, kϕi , k (r )ϕ j ,k∗ (r )ϕl ,k (r ' )ϕm ,k
∗ (r ')
χi , j , l ,m , k=∂ρ̂i , j , k
∂V l ,m ,k
Quantum Theory of the Dielectric Constant in Real Solids
Adler Phys. Rev. 126, 413–420 (1962)
What is Veff ?
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Independent Particle
Independent Particle Veff = V
ext
∂
∂V l ,m,keff i
∂ρi , j ,k
∂ t= ∂
∂V l ,m, keff [H k+V eff , ρ̂k ]i , j , k
Using: {H i , j ,k = δi , j ϵi(k)
ρ̂i , j , k = δi , j f (ϵi ,k)+∂ ρ̂k
∂V eff⋅Veff+....
And Fourier transform respect to t-t', we get:
χi , j , l ,m, k (ω)=f (ϵi ,k)−f (ϵ j ,k)
ℏω−ϵ j ,k+ϵi ,k+iηδ j ,lδi ,m
i∂ρ̂k (t )
∂ t=[H k+V eff , ρ̂k ]
χi , j , l ,m , k=∂ρ̂i , j , k
∂V l ,m ,k
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Optical Absorption : IP
Non Interacting System
δρNI=χ0δV tot χ
0=∑
ij
ϕi(r)ϕ j*(r)ϕi
*(r ' )ϕj(r ')
ω−(ϵi−ϵ j)+ iη
Hartree, Hartree-Fock, dft...
=ℑχ0=∑ij
∣⟨ j∣D∣i⟩∣2δ(ω−(ϵ j−ϵi))
ϵ''(ω)=
8π2
ω2 ∑i , j
∣⟨ϕi∣e⋅v̂∣ϕ j ⟩∣2δ(ϵi−ϵ j−ℏω)
Absorption by independent Kohn-Sham particles
Particles are interacting!
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Time-dependent Hartree (local fields)
Time-dependent Hartree(local fields effects)
Veff = V
ext + VH
V tot 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 V tot r ' ' , t ' '
V tot r ' ' , t ' '
V ext r ' , t '
χ( r⃗ t , r⃗ ' t ')=χ0( r⃗ t , r⃗ ' t ')+∫ dt1dt2∫ d r⃗1d r⃗2χ0 ( r⃗ t , r⃗1 t1)v (r⃗1−r⃗2)χ(r⃗2 t2 , r⃗' t ')
ind
V indV tot
0r ,r '=
ind r , t
V tot r' t '
Screening of the external perturbation
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Time-dependent Hartree (local fileds)
PRB 72 153310(2005)
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Macroscopic Perturbation....
ϵ−1(ω)=1+v
δρind
δV ext
ϵ(ω)=1−vδρind
δV tot
Which is the right equation?
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...microscopic observables
Not correct!!
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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
orV (r)=∫dqV (q , r)eiqr=∫ dq∑G
V (q+G)ei(q+G )r
Where: V (q ,r )=∑GV (q+G)eiGr
The G components describe the oscillation in the cell while the q components the oscillation larger then L
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Macroscopic averages 2/4
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Macroscopic averages 3/4
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Macroscopic averages 4/4
The external fields is macroscopic, only components G=0
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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+vδρind
δV ext
ϵ(ω)=1−vδρind
δV tot
Local fields are not enough....
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What is missing?
Two particle excitations, what is missing?Two particle excitations, what is missing?electron-hole interaction, exchange, higher order effects......
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The DFT and TDDFT way
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DFT versus TDDFT
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DFT versus TDDFT
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V ext=0 V extV HV xc
q ,=0q ,
0q ,vf xcq , 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
V ext
0=NI
V eff
... by using ...
=01
V H
V ext
V xc
V ext
vf xc
i∂ρ̂k (t )
∂ t=[H KS , ρ̂k ]=[H k
0+V eff , ρ̂k ]
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Time Dependent DFT
Choice of the xc-functional
...with a good xc-functional you can get the right spectra!!!
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Summary
● How to calculate linear response in solidsmolecules
● The local fields effects: time-dependent Hartree
● Correlation problem: TD-Hartree is not enough!
● Correlation effects can be included by mean of TDDFT
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29
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References!!!
Electronic excitations: density-functional versus many-body Green's-function approachesRMP, 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: