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

Claudio Attaccalitehttp://abineel.grenoble.cnrs.fhttp://abineel.grenoble.cnrs.fr/r/

Neutral Electronic Excitations:

a Many-body approachto the optical absorption spectra

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

Motivations: Absorption Spectroscopy

-+

-Many Body Effects!!!

h ν

Motivations(II):Absorption Spectroscopy

Absorption linearly related to the Imaginary part of the MACROSCOPIC dielectric constant (frequency dependent)

Outline

How can we calculate the response of the system? Time Dependent – DFT and Bethe Salpeter Equation

Some applications and recent steps forward

Conclusions

Response of the system to a perturbation →Linear Response Regime

Spectroscopy

Theoretical Spectroscopy

i∂

∂ t1=HV ext r ,t

Propagation Correlation1

r−r '

[ i ∂

∂ t 1e iV ext ]Gij t 1, t 2=t 1, t 2∫G

i∂∂ t

t =[HV ext ,t ]

t 1,t 2Green's functions

Schrödinger eq.

Density Matrix 2r , r ,r ,r ,3. ....

Current-DFT

TD-DFTi∂∂ t

=TV hV xcV ext V xc

i∂∂ t

=V hV xcV ext 1/2 [ pA j ]2 V xc , Axc

HARD

Linear Response Regime (I)

The external potential “induces” a (time-dependent) density

perturbation

Kubo Formula (1957)

r t ,r ' t '=

indr , t

ext r ' , t ' =−i ⟨[ r , t r ' t ' ]⟩

Linear Response Regime (II)

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 '

Variation of the charge density w.r.t. the total potential.

0r ,r '=

ind r , t

V tot r' t '

r t ,r ' t '= 0r t ,r ' t '∫∫dt1 dt2∫∫ d r1 d r20 r t , r1 t1v r1−r2 r2 t 2 ,r' t '

Screening of the external perturbation

Kubo Formula

ind

V indV tot

Linear Response Regime (III)The screening is

described by the inverse of the

microscopic dielectric function

−1r t ,r ' t ' =V tot r t

V ext r t

=r−r ' ∫ dt ' ' d r ' ' v 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

GG '−1 q ,=1vG qGG ' q ,

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

Fourier spaceG=G '=0

V ext=0 V extV HV xc

q ,=0q ,

0q ,vf xcq , q ,

TDDFT is an exact theory for neutral

excitations!

Optical Absorption : Time Dependent DFT

V eff (r , t )=V H (r , t)+ V xc(r , t)+ V ext (r , t)

Interacting System

Non Interacting System

[−12

∇2V eff r , t ]i r , t =i∂

∂ t ir , t

r , t =∑i=1

N

∣ ir , t ∣2

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

Why does paper turn yellow?Treasure mapTreasure map

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

(2012)

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 oxidized functional groups acting as chromophores and responsible of paper groups acting as chromophores and responsible of paper yellowingyellowing.

Optical Absorption : Microscopic View (II)

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

e

h

W=2ℏ∑i , j

∣⟨i∣e⋅v∣ j ⟩∣2i− j−ℏ~ℑ

Non Interacting Particles

Non Interacting quasi-particles

Independent transitions

i , j

i , j

Hartree, HF, DFTGW corrected

energies

Optical Absorption : Microscopic View (III)

Direct and indirect interactions between an e-h pair created by a

photon

Summing up all such interaction processes we get:

L(r1 t1 ; r2 t 2 ;r3 t 3 ;r4 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)

−11,2=V 1U 2

What we want:

... by the identity ...

V 1=U 1−i ℏ∫ d3 v 1,33

⟨1⟩=−i ℏG 1,1

−11,2=1,2∫d3 v 1,3⟨ 3⟩U 2

... by using ...

G21,3 ;2, 3=G1,2G 3,3−G1,2U 3

1,2= ⟨1⟩U 2

=i ℏ[G21,2;1 ,2−G 1,1G 2,2]

Reducible polarizability Reducible polarizability

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

1,2=−i ℏ L 1,2;1+ ,2+ two-particle correlation functiontwo-particle correlation function

i=r i , t i

The density is related to the Green's function by

Derivation of the Bethe-Salpeter equation (2)

G−11,2=G0−11,2−U 1 1,2−1,2

What we have:

Just the Dyson equation for G-1

What we have:

Using:

1,2=−i ⟨G1,1

⟩

U 2= ⟨1⟩U 2

=⟨1 2⟩

G1,4 U 5,6

=L1,5,4,6=−∫G 1,2G−12,3 U 5,6

G 3,4

[i ℏ∂

∂ t−h 1−U 1]G 1,2−∫d43,4 G 4,2= 1,2 Dyson

equation

Derivation of the Bethe-Salpeter equation (3)

1, 2=G1,2v 2,1 => Time-Dependent Hartree-Fock

Coulomb term

GW 1,2=−iG 1,2W 2,1 => Standard Bethe-Salpeter equation(Time-Dependent Screened Hartree-Fock)

L=L0+ L0[v+δΣδG

]L Bethe-Salpeter Equation!

Screened Coulomb term

L=L0L0[v− GW

G]L

L0(1,2,3,4)=G(1,4)G(2,3)

Feynman's diagrams andBethe-Salpeter equation

L(1234)=L0(1234)+

L0 1256 [v 57 56 78−W 56 57 68 ]L7834

Intrinsc 4-point equation.It describes the (coupled) progation

oftwo particles, the electron and the hole

!

Quasihole and quasielectron

L=L0+ L0[v−W ]L

W 1,2=W r1 , r2 t 1 , t2Retardation effects are

neglected

+ -=

L1,2,3,4 =L r1, r2, r3, r4 ; t −t 0=L1,2,3,4,

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

-

+

-

+

-

+

We work in transition space...

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

H n1n2 ,n3n4 exc A

n3 n4=E A

n1n2

L1,2,3,4 =L r1, r2, r3, r4 ; t −t 0=L1,2,3,4,

Effective two particle Hamiltonian

It corresponds to transitions at positive absorption frequenciesv .

It corresponds to transitions at negative absorption frequenciesv .

Tamm Dancoff!!!

Pseudo

-Herm

itian

M =1−limq0

v q∑

∑vc ,k

∣⟨ v k−q∣e−i qr∣c k⟩∣2

E−−i

1951

Bethe Salpeter Equation Historical remarks…

1970

First solution of BSE with dynamical effects: Shindo approximationShindo approximation

JPSJ 29, 278(1970)

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)

1995

Plane-waves implementationG. Onida et al.

PRL 75, 818 (1995)

… Some results …

V. Garbuio et al., PRL 97, 137402 (2006)

Bruneval et al., PRL 97, 267601 (2006)

Tiago et al., PRB 70, 193204 (2004)

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

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

Albrecht et al., PRL 80, 4510 (1998)

Excitons in nanoscale systems

Excitons in nanoscale systemsGregory D. Scholes, Garry RumblesNature Materials 5, 683 - 696 (2006)

Nanotubes/Nanowires

Colloidal quantum dots

Frenkel excitons in photosynthesis

. . . advances . . .

Mixed excitonic-plasmonic excitations in nanostructures (Nanoletters, 6, 257(2010))

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

Beyond Tamm-Dancoff approximation!

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

WaterV. Garbuio et al.,

PRL 97, 137402(2006).

Dynamical Excitonic Effects in Metals and Semiconductors

Dynamical effects in Sodium clusters

G. Pal et al.EPJ B 79, 327 (2011)

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 approximationShindo approximation (J. Phys. Soc. Jpn. 29, 278(1970))

Dynamical effectsin metals and semiconductorsA. Marini and R.

Del solePRL, 91, 176402

(2003).

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))

Second-order response Bethe-Salpeter equation (PRA, 83, 062122 (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 EquationPhysica Scripta, vol 109, pg 141, (2004) G. Bussi

● 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:

29

DFT meets Many-Body

….. with some algebra......

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

● 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:

37

Optical Absorption : Microscopic Limit

Non Interacting System

δρNI=χ0δV tot χ

0=∑

ij

ϕi(r)ϕ j*(r)ϕi

*(r ' )ϕj(r ')

ω−(ϵi−ϵ j)+ iηHartree, Hartree-Fock, dft...

=ℑχ0=∑ij

∣⟨ j∣D∣i⟩∣2δ(ω−(ϵ j−ϵi))

ϵ''(ω)=8π

2

ω2 ∑

i , j

∣⟨ϕi∣e⋅v̂∣ϕ j⟩∣2δ(ϵi−ϵ j−ℏω)

Absorption by independent Kohn-Sham particles

Particles are interacting!