Neutral Electronic Excitations: a Many-body approach to the optical absorption spectraAttaccalite

Neutral Electronic Excitations:

a Many-body approach
to the optical absorption spectra
Claudio Attaccalite
http://abineel.grenoble.cnrs.f
r/

Second Les Houches school in computational physics:
ab-initio simulations in condensed matter

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? Time Dependent – DFT and Bethe
Salpeter Equation
Some applications and recent steps forward

Conclusions

Theoretical Spectroscopy
Propagation

Correlation

∂
i
 =H V ext  r ,t 
∂ t1

[

i

]

∂
e iV ext G ij t 1, t 2 = t 1, t 2 ∫  G
∂t1

HARD

i

i

Schrödinger eq.

∂
t =[ HV ext , t ]
∂t

Green's functions

Density Matrix

∂
=T V hV xc V ext 
∂t

TD-DFT

 t 1, t 2 

 2 r , r ,r ,r , 3. ....

∂
2
=V h V xcV ext 1/ 2 [ pA  j  ]  Current-DFT
∂t

i

1
r−r '

V xc , A xc

V xc

Linear Response Regime (I)
The external potential “induces” a
(time-dependent) density
perturbation

Kubo Formula
(1957)

ind

  r ,t 
' '
  t ,  t =
r r
=−i 〈[ r ,t  r ' t ' ]〉
  ext r ' , t ' 

The induced charge
density results in a
total potential via

V tot

(II)
 t =V  t 
r
r
ext

dt ' ∫ d  ' v  − ' ind  ' t ' 
r r r
r
∫

the 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 ' 
Kubo

Formula

  t ,  t = 0  t ,  t ∫∫ dt 1 dt 2∫∫ d r 1 d r 2  0  t , r 1 t 1 v  r 1− r 2   r 2 t 2 , ' t ' 
r r
r r
 
r 
 
 r
' '

' '

'

 0  , =
r r

ind

V ind

V tot

 ind  , t 
r
V tot  ' t ' 
r

Variation of the

charge density w.r.t.

Screening of the

the total potential.

external perturbation

The screening is

described by the
inverse of the

microscopic dielectric
function

V
(III)
 t , t =
r r

−1



' '

 t 
r
 V ext  t 
r
tot

=  − ' ∫ dt ' ' d  ' ' v  − ' '  ' ' , ' 
r r
r
r r
r r

Twofold physical
meaning :

✔

Microscopic level: screening of the interaction
between charge carriers in the system

✔

In the long wave length limit it determines the

macroscopic dielectric function which gives rise to
screening of the external perturbation

The convolution integrals
in real space can be
reduced to products is
Fourier space

−1 ' q ,=1v G q G G ' q ,
GG
G=G '=0

Optical Absorption :
DFT

Time Dependent

1 2
∂
[− ∇ V eff r , t ] i r ,t =i  i r ,t 
2
∂t
N
r , t =∑ ∣ i r ,t ∣2
i=1

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

 I =  NI

using ...

  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 ,

Why does paper turn yellow?
Treasure map
By comparing ultraviolet-visible reflectance spectra of
ancient

and

artificially

aged

modern

papers

with

ab-

initio TD-DFT calculations, it was possible to identify
and

estimate

the

abundance

of

oxidized

functional

groups acting as chromophores and responsible of paper
yellowing.
yellowing

A. Mosca Conte et al.,
Phys. Rev. Lett. 108, 158301
(2012)


(II)

Microscopic View

Elementary process of absorption:
Photon creates a single e-h pair

e
h

2
2

W=
∣〈 i∣e⋅v∣ j 〉∣   i− j −ℏ ~ℑ
∑
ℏ i, j

Non Interacting

Non Interacting

Particles

quasi-particles i , j
GW corrected

i , j

Hartree, HF, DFT

Independent

energies


(III)

Microscopic View

Direct and indirect interactions

between an e-h pair created by a
photon

Summing up all such interaction processes we

get:

L(r 1 t 1 ; r 2 t 2 ; r 3 t 3 ; r 4 t 4 )=L(1,2,3,4)
The equation for L is
the Bethe Salpeter

Equation. The poles are
the neutral excitations.

Derivation of the Bethe-Salpeter
equation (1)
What we want:

 V 1
 1,2=
 U 2
−1

i=r i , t i

... by using ...

V 1=U 1−i ℏ ∫ d3 v 1,3 3

〈 3〉
 1,2= 1,2∫ d3 v 1,3
 U 2
−1

The density is related

to the Green's function by
... by the identity ...

〈1〉=−i ℏ G 1,1 

 G1,2
G2 1,3 ;2, 3 =G1,2G 3,3 −
 U 3




Reducible polarizability

 〈1〉
 1,2=
=i ℏ[G 2 1,2;1 , 2 −G 1,1 G 2,2  ]
 U 2
 1,2=−i ℏ L1,2; 1+ , 2+ 

two-particle correlation function

G. Strinati, Rivista del Nuovo Cimento, 11, 1 (1988)

equation (2)
What we
have:
∂

[i ℏ

∂t

−h 1−U 1]G 1,2−∫ d4  3,4 G  4,2= 1,2 Dyson

equation

 〈 G1,1  〉  〈 1〉
 1,2=−i
=
=〈 1 2〉
 U 2
 U 2
Using
:

 G1,4
 G−1 2,3
= L1,5,4,6=−∫ G 1,2
G 3,4
 U 5,6
 U 5,6
−1

G 1,2=G

0−1

1,2−U 1 1,2− 1,2

Just the Dyson equation for G -1

equation (3)
L=L0+ L0 [ v+ δ Σ ] L
δG

Bethe-Salpeter
Equation!

0

L (1,2,3,4)=G(1,4)G(2,3)
Coulomb term

 1, 2=G1,2v 2,1

=>

Screened Coulomb term

 GW 1,2=−iG 1,2W 2,1

Time-Dependent Hartree-Fock

=>

Standard Bethe-Salpeter equation

(Time-Dependent Screened Hartree-Fock)

 G W 
L= L0 L0 [ v −
]L
G

Feynman's diagrams and
Bethe-Salpeter equation
L= L0 + L0 [ v − W ] L
L(1234)=L0 (1234)+
L0 1256[v 57 56 78− W 56 57 68] L7834
=

Quasihole and
quasielectron

+

Intrinsc 4-point equation.

It describes the (coupled) progation
of

two particles, the electron and the hole
Retardation effects are
W 1,2=W r 1 , r 2  t ! , t 2 
neglected
1

L1,2,3,4=Lr 1, r 2, r 3, r 4 ; t − t 0 =L1,2,3,4, 

Bethe-Salpeter equation (4points - space and time)

L1,2,3,4=Lr 1, r 2, r 3, r 4 ; t − t 0 =L1,2,3,4, 

Should we invert the equation
for L for each frequency???

-

+

-

+

-

+

H

exc
n1 n2 ,n3 n4 

A

n3 n4 


=E  A

n1 n2 


We work in
transition
space...

Effective two particle Hamiltonian
It corresponds to transitions

Pseudo-Hermitian

at positive absorption
frequencies
v.

Tamm Dancoff!!!

It corresponds to transitions
at negative absorption
frequencies
v.

∣〈 v k− q∣e−i q r∣c k〉∣2
∑∑
 M =1−lim v q
q0



vc , k

E  −−i 

Bethe Salpeter Equation
Historical remarks…
1951

1970

1995

First solution of BSE

with dynamical effects:

Shindo approximation
JPSJ 29, 278(1970)

Plane-waves

implementation
G. Onida et al.
PRL 75, 818 (1995)

1974

First applications in solids:
W. Hanke and L.J. Sham PRL 33, 582(1974)
G. Strinati, H.J. Mattausch and W. Hanke
PRL 45, 290 (1980)

… Some results …
Bruneval et al., PRL 97, 267601
(2006)

Strinati et al., Rivista del Nuovo
Cimento 11, 1 (1988)
Albrecht et al., PRL 80, 4510
(1998)

Bruno et al., PRL 98,
036807 (2007)

Tiago et al., PRB 70, 193204

V. Garbuio et al., PRL 97, 137402

Excitons in nanoscale systems
Frenkel excitons

in photosynthesis

Nanotubes/Nanowire
s

Colloidal quantum dots

Excitons in nanoscale systems
Gregory D. Scholes, Garry Rumbles
Nature Materials 5, 683 - 696 (2006)

Beyond Tamm-Dancoff
approximation!
Mixed excitonic-plasmonic excitations
in nanostructures

(Nanoletters, 6, 257(2010))

Excited states of biological chromophores
(J. Chem. Theory Comput., 6, 257–265 (2010))

Ab-initio broadening in BSE
Ab-Initio finite temperature
excitons

A. Marini PRL 101, 106405
(2008).

Ab Initio Calculation of

Optical Spectra of Liquids:
Many-Body Effects in the
Electronic Excitations of
Water

V. Garbuio et al.,

PRL 97, 137402(2006).

Dynamical Excitonic Effects in Metals and
Semiconductors

The inclusion of the full dynamic screening in the BS equation

complicates its numerical solution tremendously, but it is possible to
perform an expansion in the dynamical part of the screened
interaction. First solution of this problem the so-called

Shindo approximation (J. Phys. Soc. Jpn. 29, 278(1970))
Dynamical effects in Sodium
clusters

Dynamical effects
in metals and

semiconductors

A. Marini and R.
Del sole

PRL, 91, 176402
(2003).

G. Pal et al.

EPJ B 79, 327 (2011)

Non-linear response:
frequency and time domain
Second-order response Bethe-Salpeter
equation (PRA, 83, 062122 (2011))

Real-time approach to the optical

properties of solids and nanostructures:

Time-dependent Bethe-Salpeter equation
(PRB, 84, 245110 (2011))

References!!!
Reviews:
●

Application of the Green’s functions method to the study of

the optical properties of semiconductors

Nuovo Cimento, vol 11, pg 1, (1988) G. Strinati
●

Effects of the Electron–Hole Interaction on the Optical Properties

of Materials: the Bethe–Salpeter Equation

Physica Scripta, vol 109, pg 141, (2004) G. Bussi
●

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:

On the web:
●

●

●

●

http://yambo-code.org/lectures.php

http://freescience.info/manybody.php
http://freescience.info/tddft.php

http://freescience.info/spectroscopy.php

Optical Absorption : Microscopic
Limit

δ ρNI =χ 0 δ V tot

0

χ =∑
ij

ϕi (r) ϕ* (r) ϕ* (r ' ) ϕj (r ' )
j
i
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

8π
ϵ (ω)= 2
ω
Particles are interacting!
''

∣〈 ϕi∣e⋅̂ ∣ϕ j 〉∣2 δ (ϵi−ϵ j−ℏ ω)
v
∑
i, j

Neutral Electronic Excitations: a Many-body approach to the optical absorption spectraAttaccalite

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Notes de l'éditeur