Many-body Green functions theory for electronic and optical

properties of organic systems(are physicists any good at chemistry?)

Claudio AttaccaliteCNRS/CINAM Campus de Luminy, Case

913 13288 Marseille , France

Excited States Bridging Scales, Marseille, November 7-10 (2016)

Outline

● From the Green’s mill to modern computer codes

● From solids to molecules

● The future…..

Outline

● From the Green’s mill to modern computer codes

● From solids to molecules

● The future…..

What do electrons in solids?

Direct and inverse photo­emission

Electrons moves in bands that correspond to the 

energy to remove or add one electron

How to calculate band-structures?Problems with DFT (and HF)

The Kohn­Sham eigen­systems do not represent the real bands of a solid even if we know the exact Vxc

●  Band width too large for metals●  Band gaps too small●  In materials with d or f orbitals, the Density of States is in disagreement with experiments

●  Metal­insulator transition not described●  The magnetic moments in the transition metal oxides are systematically underestimated

●  Etc...

Green functions Let's "watch" the propagation of an added electron

Green’s function

|ψ0N ⟩ ψ+(r , t)| ψ0

N ⟩ ψ(r ' , t ') ψ+ (r , t)| ψ0N ⟩

● Probability amplitude for propagation of additional electron from (r,t) to (r',t') in a many electron system.

● Generalization of the density matrix (r,r’,t) where fields operators have different times. 

iG (r ' , t ' ; r , t)=⟨ψ0N| ψ(r ' , t ' )ψ+

(r ,t )| ψ0N ⟩θ( t '−t)

Ref:  Quantum Statistics of Nonideal Plasmas, D. Kremp, M. Schlanges and W. D. Kraeft

Which information iscontained in the Green function?

G(r ' ,r ;ω)=∑s

ψN+1(r ´ )[ ψN+1

(r )]∗

ω−ϵsN+1+iη

+∑s

ψN−1(r ´ )[ ψN−1

(r )]∗

ω−ϵsN+1−iη

By Fourier transform and some mathematics….

Poles of Green's function give energies for addition/removing an electron, charged excitations (including ionization energy and electron affinities in molecules)

ρ(r ' , r )=−iG (r ' , r ; t→0 ⁺ )

The t=0 limit is the density matrix!The t=0 limit is the density matrix!

How to calculate Green functions

From the EOM of the annihilation field operator:

Historical note:Green’s functions were originally introduced by the British (miller and) mathematical physicist George Green in the context of the theory of electricity and magnetism. Nowadays, all functions satisfying an inhomogeneous integral­differential equation with a Dirac delta function as source term are called Green functions.

i∂t ψ(r , t)=[ψ(r ,t ), H ]

We obtain a infinite hierarchy of  n­particle Green’s functions similar to the ones of density matrix:

h(1)G(1 ;1' )−i∫ d 2V (1,2)G(1,2 ;1 ' ,2+)=δ(1−1 ' )

G2

G1

G3

Gn

G4

This equations can be closed by introducinga single particle operator  

non­local in space and time, that includecorrelation effects

G=G0+G0 Σ(G)G

Martin-Schwinger Tower

 can be calculated by perturbation theory

Problem with metals and the electron-gas ...

Ref: Many­Body Quantum Theory in Condensed Matter Physics: An Introduction, H. Bruus and K. Flensberg

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Metal Electron gas

Lets use perturbation theory to improve DFT/HF and calculate the Green functions

Problem with metals and the electron-gas ...

Ref: Many­Body Quantum Theory in Condensed Matter Physics: An Introduction, H. Bruus and K. Flensberg

Lets use perturbation theory to improve DFT/HF and calculate the Green functions

It diverges!!

It diverges!!

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Metal Electron gas

How to solve the problem of divergences

We solved this problem many years ago

…. condensed matter physicists discussed with their cousins particle physicists ……...

Sums an infinite series of perturbation terms in such a way to create a screened interaction W, and then start again perturbation theory in W

Self-energy in terms of a screened interaction

Ref: The GW method F. Aryasetiawan, and O. Gunnarsson,  Reports on Progress in Physics, 61(3), 237. (1998)

Where G is the single particle electronic Green's function and W is the screened electron­electron interaction

All correlation effects are included in the self­energy operator  the we approximate as:

Putting together Green’s function and perturbation theory in terms of a screened Coulomb interaction we get the quasi­particle formalism. Namely the mapping of the 

true many­body problem onto a single effective single­particle framework:

Many-body perturbation theory(GW approximation)

Mean-field approaches(DFT, Hartree-Fock, etc.)

Response functionsFrom the derivatives of the Green’s function respect to an external field U 

it is possible to calculate the response functions:

Li , j , k ,l=∂ρi , j

∂U k ,l

=−i∂Gi , j

∂U k , l

L=L0+L0[ v+∂Σ/∂G L]L

The Bethe­Salpeter Equation (BSE)

Similar to TD­DFT Cassida equation

χ=χ0+χ0[ v+f xc ] χ

But does not depend from any functional and naturally include exchange effects 

h ν

+

-

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

A sleeping beautyMany­Body Perturbation Theory within the GW approximation

for the electron gas was presented  by Hedin in 

New method for calculating the one­particle Green's function with application to the electron­gas problem. L. Hedin, Physical Review, 139, A796 (1965)

(from the abstract: …  there is not much new in principle in this paper. ….)

However, we had to wait until 1980 for the first application to semiconductors

by Hanke, Sham, Strinati and Mattausch

Modern implementation of GW by Hybertsen and Louie (1986)

Modern implementation of BSE by Onida et al. (1995)

Some results

Band gaps Optical excitations

G. Onida, L. Reining, and A. Rubio, RMP 74 (2002).

Outline

● From the Green’s mill to modern computer codes

● From solids to molecules

● The future…..

FIESTA and friends

Other implementations using localized orbitals:1) MOLGW (free):  http://www.molgw.org/ 2) TurboMole: http://www.turbomole.com/    3) FHI: https://aimsclub.fhi­berlin.mpg.de/  4) SIESTA:  https://arxiv.org/abs/1105.3360

 French Initiative for Electronic Simulations with Thousands of Atoms 

[http://perso.neel.cnrs.fr/xavier.blase/fiesta/]

Implementation of GW equations and response functions (Bethe­Salpeter equation) using a gaussian basis set

The ab-initio zoo: from mean-field to many-body perturbation theory

Ab-initio formalisms are not all equivalent and there is still “some work” towards reliable and cheap enough approaches able to tackle complex systems.X. Blase, V. Olevano and C. Attaccalite PRB 83, 115105 (2011)

Mean-field approaches(DFT, Hartree-Fock, etc.)

The « exact » many-body problem(QMC, CI, coupled-cluster, etc.)

Many-body perturbation theory(GW, Bethe-Salpeter, etc.)

HOMO-LUMO gap of molecules of interest for organic electronics or photovoltaics

Beyond the scissor operator: level ordering and spacing below the gap

HOMO to (HOMO-3) in cytosine

LDA GW CASPT2 EOM-IP-CCSD

σO + π

ππ

HOMO 6.167 (σO) 8.73 (π) 8.73/8.76 (π) 8.78 (π) 8.8-9.0

HOMO-1 6.172 (π) 9.52 (π’) 9.42/ - (σO) 9.55 (π’) 9.45-9.55

HOMO-2 6.81 (σ) 9.89 (σO) 9.49/ - (π’) 9.65 (σO) 9.89

  LDA            evGW         CASPT2/CCSD(T)   EOM­IP­CCSD      Exp.

π

σO

σO

σO

σ

π’π’

π’-3.72 eV

(Faber, Attaccalite, Olevano, Runge, Blase, PRB 2011)

(DNA nucleobases)

Bethe-Salpeter equations (BSE) for excitonic interactions: comparison to experiment !

BNL = Range-separated hybrid functional

(Stein, Kronik, Baer, JACS 2009)

TDDFT hybrid

Acceptor = TCNE (tetracyanoethylene)Donor = benzene, toluene, o-xylene and naphtalene

(Blase, Attaccalite, APL, 2011)

See also our competitors: Garcia-Lastra and Thygesen, PRL (2011); Baumeier, Andrienko, Rohlfing, JCTC (2012);

F. Bruneval et al. Comp. Phys. Comm. (2016)

A very simple example: dipeptide intramolecular charge-transfer states

(Faber, Boulanger, Duchemin, Attaccalite, Blase, JCP 2013)

TD-DFT versus GW/BSE for charge-transfer excitations

Charge-transfer excitations requiring long-range electron-hole interaction is a well known failure of TD-DFT with local kernels.

Similar situation in solids with extended Wannier excitons.

Excitonic States: (similar to Felix Plasser): from the Electron-Transition Density Matrix electrons are green, holes are gray ϕe (r e)=∫rh

ψ(rh , re)d rh

The future….

Beyond standard approximations

Vibronic coupling(forces?)

Real-time dynamics

Multi-referenceproblems

Multi-referenceproblems

Environment

When a charge arrives onto a molecule, the structural relaxation of the molecule traps the charge and strongly limits the mobility of the carriers (polaronic coupling).  

t1u (3-fold)

The relaxation energy is closely  related to the electron­phonon coupling strength.

LUMO

GW

DFT (LDA)

(courtesy, A. Troisi, Warwick)

Electron-phonon potential

LDA 63 meV GW 109 meVExp. 107 meV

Exps: Wang et al., JCP 2005 ; Hands et al., PRB 2008Theory: Faber et al. J. of Mat. Science 47 (21), 7472 (2012)

Electron-phonon or vibronic coupling in molecular systems

(Implications in superconductivity, inelastic scattering, resonant Raman, etc. ?)

EPC ~ |slope|2

Many-Body Green’s functions and Classical Polarizable Models

I. Duchemin et al., JCP 144, 164106 (2016) and  J. Li et al. JPCL, 7, 2814 (2016)

Hybrid QM/MM scheme merging the many­body Green’s function GW formalism  with classical discrete polarizable models and its 

application to molecular crystals and molecules in solution.

Hybrid QM/MM capture polarization effects not present in standard a delta­SCF DFT scheme.

Beyond standard approximations: self-consistency, higer orders...

C. Faber et al. JCP 139, 194308(2013) and  X. Ren et al. PRB 92, 081102(2015)F. Kaplan et al. JCTC 12, 2528(2016)

Improvements are difficult….

Tetracyanoethylene (TCNE)

Real-time dynamicsMany­body Green’s function theory can be extended to non­equilibrium situations,

and to study real­time dynamics

Non­linear response Pump­probe experiments

Kadanoff, L. P., & Baym, G. A. Quantum statistical mechanics (1962).

Second Harmonic Generation in MoS2 Transient absorption in silicon

M. Grüning, C. Attaccalite PRB 89, 081102 (2014) D. Sangalli & A. Marini  EPL, 110, 47004.  (2015)

Multi-references problems: DMFT + DFT

C. Weber et al. PRL  Phys.  110, 106402 (2013) and PNAS 111, 5790 (2014)

Inclusion of many­body effects results in the correct prediction of similar binding energies 

for oxy­ and carbonmonoxy­myoglobin.

The computed electronic structure of the myoglobin complexes and the nature of the Fe–O2 bonding are validated against experimental spectroscopic observables.

A long-standing problem related to the quantum-mechanical description of the respiration process, namely that DFT calculations predict a strong imbalance between O2

and CO binding, favoring the latter to an unphysically large extent.

Dynimical mean field theory+

DFT

Conclusions

The Green’s mill is still used to do science

Green’s function theory is a powerful tool to study excitations in finite and infinite systems.

The field is rapidly evolving and many application in chemistry are on going, stay  tuned!!

This presentation is available on:  http://attaccalite.com

Ivan Duchemin

CarinaFaber

Acknoledgements

Xavier Blase

Paul Boulanger

Thanks to Martin Spenke for comments on this

presentation

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