Claudio Attaccalitehttp://abineel.grenoble.cnrs.fr/http://abineel.grenoble.cnrs.fr/

BSE and TDDFT at work

CECAM Yambo School 2013 (Lausanne)CECAM Yambo School 2013 (Lausanne)

Optical Absorption: Microscopic View

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.

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)

Feynman's diagrams andBethe-Salpeter equation

L(1234)=L0(1234)+

L0 1256 [v 57 56 78−W 56 57 68 ]L7834

Intrinsic 4-point equation.It describes the (coupled) propagation 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,

Construction of the BSE 1/2

L0(r1,r2,r '1,r '2∣ω)=2∫ dω '2π

G0(r1,r '2 ;ω+ω ' )G0(r2,r '1 ;ω ' )

G0(r1,r2 ;ω)=∑k , i

ϕk ,i∗ (r1)ϕk , i(r2)

ℏω−ϵk , i+iηsign(ϵk , i−ϵF)

where

We expand L in the independent particle basis

L(r1, r2,r '1,r '2∣ω)= ∑k1, k2, k3,k4

ϕk1(r1)ϕk2

(r2)Lk1, k2, k3,k 4ϕk3(r '1)ϕk4

(r '2)

We start from L0

...integrating in the frequency we get...

Lk1, k2, k '1,k '2

0=δk1, k2

δk '1, k ' 2

2iℏ

f (ϵk2)−f (ϵk ' 2

)

ϵk ' 2−ϵk2

+iδ+ℏω

Construction of the BSE 2/2

Time-dependent Hartree term:

Screened Exchange Coulomb term:

...and now let's solve the equation...

-

H ij , klTD−harteee=2∫ϕi

e(r )ϕ j∗h(r )V (r−r ')ϕk

e (r ')ϕl∗h(r ')

H ij , klTD−SEX=∫ϕi

e (r )ϕ j∗h(r ' )W (r , r ')ϕk

e (r )ϕl∗h(r ')

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 An1n2

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

The frequency term can be separated and an-effective Hamiltonian can be derived without any frequency dependency

L=L0+L0[v−W ]L

[L0−1+[v−W ] ]=L−1 We solve the inverse Bethe-Salpeter

eq., because it is easier

Original BSE

How to transform the BSE in an eigenvalues problem

Using the definition of L0

That is diagonal in the e/h space

We can solve the equation once for all frequency!!!!

...with some linear algebra....

Absoprtion spectra and BSE

BSE calculation in practice

… 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

BSE for charge transfer excitons

donor-acceptor complexes:

benzene, naphthalene, and anthracene

derivatives with the tetracyanoethylene

acceptorX. Blase and C. Attaccalite Apl. Phys. Lett. 99, 171909 (2011)

Exciton analysis

Solving the equations in a smart way ...

R

The BSE can be large...... too large

Tamm-Dancoff approximation

The dielectric constant doesn't require too much information

Let's come back to the original formula

ϵ2(ω ,q )=4 πℑ [⟨P∣ 1ω−H EXC+iη

∣P ⟩]we can write the dielectric constant as

...and ask the help of mathematicians...

∣P ⟩=limq→0 e

iqr

q∣0 ⟩

Lanczos-Haydock method

Lanczos-Haydock algorithm

Lanczos-Haydock performance

What about TDDFT?

TDDFT versus BSE

L(1234)=L0(1234)+

+L0(1256)[v (57)δ(56)δ(78)−W (56)δ(57)δ(68)]L(7834)

χ(12)=χ0(12)+χ0(13)[v (34)+f xc ]χ(42)

BSE

TDDFT

BSE is a 4-points equation => unavoidable

TDDFT is a 2-points equation => that can be rewritten as a 4-point equation

TDDFT in G-space

χG ,G '(q ,ω)=χG ,G '0 (q ,ω)+χG ,G2

0 (q ,ω)(vG2(q)+ f G2, G3

xc (q))χG3 ,G(q ,ω)

ϵG ,G'−1 (q ,ω)=δG ,G '+vG(q)χG ,G '(q ,ω)

ϵM(q ,ω)=

1ϵG=0,G '=0−1 (q ,ω)

Simple static fxc case:

Microscopic dielectric constant:

Macroscopic dielectric constant:

Advantages: 2-points eq. Disadvantages: the eqs. Has to be solved for each frequency

TDDFT in e/h space

Time-dependent exchange correlation function:-

H ij , klTD−hartree=2∫ϕi

e(r )ϕ j∗h(r )V (r−r ' )ϕk

e (r ')ϕl∗h(r ' )

Time-dependent Hartree term:

H ij , klTD−EXC=∫ϕi

e(r )ϕ j∗h(r ) f xc(r , r ')ϕk

e (r ' )ϕl∗h(r ' ) f xc(r , r ')=

∂V xc(r )

∂ρ(r ' )

H ij , klTD−SEX=∫ϕi

e (r )ϕ j∗h(r ' )W (r , r ')ϕk

e (r )ϕl∗h(r ' ) BSE

Beyond the Tamm-Dancoff approx.

Tamm-Dancoff breakdown 1

Tamm-Dancoff breakdown 2

Don't worry! Haydock method still works

SUMMARY

● Optical spectra can be calculated by mean of Green's function theory

●TDDFT and BSE can efficiently be formulated in the e/h space

● By using Lanczos-Haydock approach we do not need to diagonalize the full matrix!

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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: