Angel Rubio

Dpto. de Física de Materiales, Universidad del País Vasco, Donostia International Physics Center (DIPC), and Centro Mixto CSIC-UPV/EHU, Donostia, Spain http://dipc.ehu.es/arubio E-mail: arubio@sc.ehu.es

I. Photoresponse of nanostructures: optical absorption and excite-state dynamics TDDFT Applications II. Optical absorption and electron energy loss spectroscopy of extended systems

Summer School on Time dependent density functional theory and the dynamics of complex systems (Santa Fe, June 2004)

Summer School on Time dependent density functional theory and the dynamics of complex systems (Santa Fe, June 2004)

Time-dependent evolution of a Lecture!!!!!

Fundaments (Gross, Burke, van Leewen, Vignale, Dobson)Time evolution and real-spaceschemes (Berstsch + Reinhard)

Linear response scpectra of molecules, clusters (and bio-molecules) (Furche, Casida, Reinhard, Ullright, Gross)

Non-linear dynamics: ATI spectra (Bandrauk, Gross)

What is left?????

I. Photoresponse of nanostructures: optical absorption and excite-state dynamics

- short review of computer issues: the octopus project - time propagation - electron-ion interaction - Applications to small metal and semiconductor clusters: -optical absortion -photoinduced fragmentations - Applications to biomolecules: - green flourescence protein - azobencene excite state dynamics

Summer School on Time dependent density functional theory and the dynamics of complex systems (Santa Fe, June 2004)

The spatial resolution depends on the wavelength, the spatial localization (STM, SNOM), the inelastic attenuation (PES, PD), the range of the interaction (EELS, MARPE)

1 10 100 10000,1

1

10

100

1000

10000

energía (eV)

LEED

EXAFS, NEXAFSPES, PD, Auger

SNOMphotonic materials

IMFP

electrons

fotones

energy (eV)

wav

elen

gth

(Å)

longitud de onda del electrón (A)

EELS

STEM

STM

NEXAFS

PES, PD,MARPE

o0,1 1 10

10

100

1000

10000

phot

on w

avel

engt

h (Å

)

electron wavelength (Å)

Characterisation tools of the “nanoworld“ Need of the knowledge of response functions!!!!!

Electronic coupling between a solid and an adsorbate governs chemical dynamics at surfaces

The dynamics of electronic excitations play an important role in molecule-surface interactions and reactivity, as in laser-driven surface reactions, and are also critical to technological applications of electronic materials

Photochemistry: Phonon- versus electron- mediated surface reactions:

HOMO

LUMO

Adsorbate

Vacuum level

probe photon

pump photon

EF

M.A.L. Marques, A. Castro, G. Bertsch, AR Comp.Phys.Comm. (2002)C. Rozzi, M.A.L. Marques, A. Castro, E.K.U. Gross A. R. (to be published)

http://www.tddft.org/programs/octopus

The octopus project is aim to the first principle description of the excite state electron-ion dynamics of nanostructres and extended systems within TDDFT

Implementation:✔ Numerical description of functions: real space discretization (1D-3D).✔ Auxiliary use of LCAO/ FFTs. QM/MM for biomolecular structures✔ Electon-ion coupling: pseudopotentials. Spin-Orbit .........

Introduction: standard simulation of excited state dynamics?Introduction: standard simulation of excited state dynamics?

No

Observation of the nonradiative decay!lifetime, decay path

Yes Do MD.

Hellmann-Feynman theorem works

1. No need of level assignment for a hole and an excited electron except at the beginning.

2. Automatic monitoring of the nonradiative decay (lifetime, decay path) without experiences.

|g>

Reaction coordinate

Pot

entia

l

t = 0: Promote the electronic occupations to mimic the excited states. Then perform a SCF calculation

t > 0: Solve � n(t+� t)=exp{- i � tH(t)} � n(t).

|e>

Is the matrix of HKohn-Sham

diagonal?

Our Method: TDDFT✔TDDFT: N-coupled one-particle equations.

i ℏ ∂∂ t

=H iℏ ∂∂ t

i=H KS [ { j }]i , i=1,⋯N ?

H KS=ℏ2

2 mi∇−

ec ℏ

A2

V externalV hartreeV exchange V correlation

Gross et al (1985)-to date

✔Classical description of electromagnetic field. (Absence of radiative-decay channel!)

✔Classical description of nuclei (point particles): Ehrenfest path:

✔Solution of the TD-KS equations by unitary propagation schemes

F a t =−⟨t ∣∇ a H∣t ⟩ t =Det {it }

it t =e−iH KS t t t /2

e−iH KS t t /2it

Propagators for the time-dependent Kohn-Sham equationsIf the Hamiltonian was time-independent:

This is not the case in TDDFT. Another important difficulty is that the Hamiltonian is not known a priori for t>t

0 (it depends on the solution density).

Time-discretization: : short-time propagation:

The largest possible value of the time slice is determined by the maximum frequency that needs to be followed:

Important properties:

Unitarity:

Time reversal symmetry:

Stability:

Some examples...evaluation of

Number of Hamiltonian-wavefunction operations per unit time, as function of the time interval for the Taylor (solid) and Chebyshev (dashed) expansions, and for the Lanczos projection method (dotted).

A. Castro, M.A.L. Marques, A. Rubio, J. Chem. Phys (in press 2004)

Split-operator techniques:Sugino&Miyamoto, Phys. Rev. B 59, 2579 (1999)

Implicit midpoint rule:

Exponential Midpoint Rule:

Other classical techniques: Runge-Kutta, Euler's method, etc.

Enforce time-reversal symmetry: (ETRS)

Magnus expansions:

Error in the dipole moment, for the exponential midpoint (dashed line) and fourth-order Magnus (solid) methods, as a function of the time interval.

Evolution of the dipole moment of Na8

(in the

jellium model), subject to an intense laser field, calculated via the methods indicated in the legend.

G. Onida, L. Reining and AR, Rev. Mod. Phys. 74, 601 (2002)

Nanostructures

How good is the performance of TDDFT and present DFT XC-functionals?

G. Onida, L. Reining and AR, Rev. Mod. Phys. 74, 601 (2002)

Linear optical response: general aspects0r , r ' , w=∑ij

f j− f ii

c r j r ir ' jc r '

i− j−w

[1−v f xc0]=0

Single-pole approximation

=00v f xc

K ij=∫dr ir d V xcr d r

j r

– LDA- thick solid line– EXX- dotted line– Bethe-Salpeter – Experiments

Transport: Current + E-field

Optical response: (Benzene)

7eV

� (HOMO)

M.A.L Marques, A. Castro, G.F. Bertsch and AR, Comp. Phys. Comm. (2002);K. Yabana and G. F. Bertsch, Int. J. Quantum Chem. 75, 55 (1999)

� * (LUMO)

=5eV

I=1.31011W cm2

Femtosecond dynamics: Time-resolved photoelectron spectoscopy

D. Varsano, M.A.L. Marques, H. Appel, E.K.U Gross and AR (to be published)

-The simulation region is divided in two parts: A and B, separated by a smooth mask function- Electrons are not allowed to come back from B to A

We write the photoelectron spectum as:

PES p=∑i∣i p , t∞∣2

Femtosecond dynamics: test photodissociation of a dimer (Na2

+).

80fs

A. Castro, M.L. Marques. J.A. Alonso, G.F. Bertsch and AR , EPJB (2003)

ω=3.2eV

ω=2.5eV

Time resolved Vibrational Spectroscopy: Raman

Photodissociation dynamics: case of He3+

J. Chem. Phys. 103, 3450 (1995).

Photodissociation dynamics: case of He3+

Linear Optical Response. Laser Pulse: (I = 9 1011 W/cm2

A "poor" condensed matter physicist view !!!!

Biological molecules: photoreceptors

QM/MM + TDDFT approach

M.A.L Marques, X. Lopez, D. Varsano, A. Castro, and A. R. Phys. Rev. Lett. 90, 158101 (2003)

F. Gai et al. Science 279, 1886 (1998)

Time Scales:

Multi-scale approaches are needed: time and space

QM/MM Hamiltonian For Local Phenomena

1. Warshel , A.; Levitt, M. J. Mol. Biol., 1976, 103, 227

2. Field, M.J.; Bash, P.A.; Karplus, M. J. Comp. Chem., 1990, 11, 700

Why QM?

ReactivityO

OH

CH2

OPOO

CH2

Base

O

OH

Base

O

O

CH2

OP

O

Base

H O

H

O

OH

CH2

OPOO

H

2'

His-H+ 119

O

1'Base

5'

5'1'

His 12

O

3' 2'

4'

O

1'

2'3'

4'

N

HN

5'

HN

NH

2'3'

4'HN

HN

1'

N

NH

5'

His-H+ 119HN

NH

4'

His 12

His 119

N

HN

3'

His-H+ 12

NH3Transphosphorylation

NH3

Hydrolysis

NH3

Photoexcitations Polarization??

Van der Waals Interactions

QM/MM Electrostatics

SCF

QM/MM Boundary

H-Link Atom

Reuter et al, ; J. Phys. Chem. A 2000, 104, 1720.

Green Fluorescent Protein (GFP).

(Aequorea victoria: jellyfish)

QM/MM approach

Green Fluorescent Protein (GFP).

(Aequorea victoria: jellyfish)

HOMO-LUMO

Towards understanding biomolecular colours: the GFP case

T.M.H. Creemers et al, Proc. Natl. Acad. Sci.. USA (1999)

Guanine Cytosine

Absorption

Circulardichroism

ℜ j E ∝∫0

∞dt ei Ei t L j t

R E =ℑℜE

rotational strenght function

ℜE =ℜxℜyℜz;

Optical Rotatory Power

Azobenzene: spectroscopy along femtosecond-laser induced photoisomerization

S. Spörlein et al, Proc. Natl. Acad. Sci., 99, 7998 (2002)

HOMO

LUMO

T. Hugel et al, Science, 296, 1103 (2002) Y. Yu et al, Nature 425, 145 82003)

Azobenzene dyes are known to isomerize at the central N-N bond within fractions of picoseconds at high quantum yield [T. Nägele et al, Chem. Phys. Lett. 272, 489 (1997)].

One example is the APB optical trigger: Single-Molecule Optomechanical Cycle

CIS

Next step: QM/MM calculation of the chromophore+peptide system

TRANS

Azobenzene: spectroscopy along femtosecond-laser induced photoisomerization

Summary

TDDFT is a powerfool tool to handle the combined dynamics of electron/ion in response to external electromagnetic fields of nanostructures, biological

molecules and extended systems.

Problem: we need better fxc functionals based on either DFT (or current-DFT) or MBPT approaches

QUESTIONS?

✔Ground-state DFT?

J(r,t); virtual exc.

✔Temperature

e-phon coupling?

✔Dissipation??

✔Finite bias (resonant)

✔AC transport

✔Role of contacts

✔MOLECULE

(Landauer): Non equilibrium Green's Functions

TDDFT? -----TD-Current-DFT

First principles description of Molecular transport ??????

G=2 e2

hTr [GRwL wG

AwRw]

QUESTION?

Single-pole approximation

K ij=∫dr ir d V xcr d r

j r

Why local approximations for the xc-kernel do not open the gap in solids as it does in clusters?

Angel Rubio

Dpto. de Física de Materiales, Universidad del País Vasco, Donostia International Physics Center (DIPC), and Centro Mixto CSIC-UPV/EHU, Donostia, Spain http://dipc.ehu.es/arubio E-mail: arubio@sc.ehu.es

I. Photoresponse of nanostructures: optical absorption and excite-state dynamics TDDFT Applications II. Optical absorption and electron energy loss spectroscopy of extended systems

Summer School on Time dependent density functional theory and the dynamics of complex systems (Santa Fe, June 2004)

II. Optical absorption and electron energy loss spectroscopy of extended systems

Summer School on Time dependent density functional theory and the dynamics of complex systems (Santa Fe, June 2004)

G. Onida, L. Reining and AR, Rev. Mod. Phys. 74, 601 (2002)

Introduction: how to handle the electron dynamics in extended systems under the influence of an external electromagnetic field?

TDDFT: - Problems with standard exchange-correlation functionals- A new fxc derived from Many-body perturbation theory

proper description of excitonic effecs!!!- Applications to poliacetilene as one-dimensional system

Time-dependent approach for extended systems: a gauge formalismG.F. Bertsch, J.I. Iwata, AR, K. Yabana, PRB62, 7998 (2000)

E t =−1c

d Ad t

The Hamiltonian of a periodic system in a volume V under a uniform field is

The equation of motion are:

H=∑ii∣

12 m

pecA

2

V ion∣iE HartreeE xcV

8c2 d Adt

2

i ℏ∂∂ t

i=[ 12 m

pecA

2

V ionV HV xc]i

d2 Adt2

=−4e2 nmA−4c

eV∑i

i∣pm∣i

For the electric field is:d Edt

=−4j j=−eV∑i

i∣pm∣i−

e2

cn A

andGonze,Ghosez,Godby; PRL74, 4035 (95)

Lithium Diamond

?

See for a review: G. Onida, L. Reining and AR, Rev. Mod. Phys. 74, 601 (2002)

Very encouraging results for Si already:

●TD-CDF: results of P. de Boeij (using Vignale,Kohn functional)

●TD-EXX: results of A. Goerling

●What about insulators, bound-excitons, etc...?

Non-local fxc for extended systems:

Why a non-local (static?) fxc for extended systems:

f xc=−/q2

=00v f xc

- In the EEL spectra fxc

is added to the full coulomb

that already contains a long range contribution

The lack of a long range term in fxc

LDA is relatively weightless in

the EEL but is crucial in the absorption spectra!!!

EEL−1=1v

L. Reining, V. Olevano, AR, G. Onida, PRL88, 0664041 (2002); S. Botti et al, PRB (2004).

=0GW 0

GW v f xc

MRPA≡1/[1−v0]G=G '=0

- In the absorption spectra fxc

is added to the full coulomb

that does not contains a long range contribution (q=0)

M =1−v

Density Functional versus Many­body perturbation theory

Diagrammatic expansion

Density Functional Theory and Many-Body Perturbation Theory

Exchange-correlation Potential: Real, Local in space, Frequency independentDensity Functional Theory

Self-Energy: Complex, Non-local in space, Frequency dependent

Many-Body Perturbation Theory

F. Aryasetiawan, Rep. Prog. Phys. 61, 237-312 (1998)

R. O. Jones and O. Gunnarsson, Rev. Mod. Phys. 61, 689 (1989)

G. Onida, L. Reining and AR, Rev. Mod. Phys. 74, 601 (2002)

=Kohn-Sham states=Plasmons/Electron-hole states

The self-energy physics

The pseudopotential approach

Hedin Equation's (1965)

The GW ''soup''

G0W0 Band Structures of Insulators

From "Quasiparticle calculations in solids", W.G. Aulbur, L. Jönsson and J.W. Wilkins, Solid State Physics 54 1 (2000),

also available in preprint form at http://www.physics.ohio-state.edu/~wilkins/vita/publications.html#reviews

EnQP≃n

KSn∣nKS−vxc∣n

Bethe-Salpeter equation: excitonic effects

=iG0 W0 ;W0=RPA−1 v

G. Onida, L. Reining and AR, Rev. Mod. Phys. 74, 601 (2002)

TDDFT ...

... and Many-Body Perturbation Theory

Many-Body approach to the Exchange-Correlation Kernel of TDDFT

A. Marini, R. Del Sole and AR, PRL (2003)

Hypothesis

It exists a ''many-body xc-kernel'' such that the TDDFT and Many-Body polarization

functions are identical

Consequently TDDFT equation can be used as an equation for the xc-kernel and as a formal solution can be found in terms of an iterative equation for

the nth order contribution

A diagrammatic approach

Many-Body approach to the Exchange-Correlation Kernel of TDDFT

TDDFT

MBPT

Bethe-Salpeter Equation

Iterative equation for fxc

A. Marini, R. Del Sole and AR, PRL (2003)

=

BSETDDFT Experiment

TDDFT, scalar fxc

Bound excitons in TDDFT

A. Marini, R. Del Sole and AR, PRL (2003)

=

-2=

BSE QP-RPA

C

How many terms ?

=

C

A many-body causal TDDFT kernel

Causal/T-ordered

Causal/T-ordered fxc

BSETDDFT 1st orderTDDFT 2nd order

Same agreement between BSE and TDDFT for the finite transferred momentum

absorption spectra...

...and for the off-diagonal elements of the microscopic dielectric function

Is TDDFT ''fast'' compared to the BSE ?

=

[# of frequencies] ×100 x10000×W 10000×10000× '10000×100

When the only optical spectra is calculated TDDFT is as time consuming as BSE...

...but when the full dielectric matrix is needed TDDFT is more favorable than BSE

Low dimensional sytems (1D): polyacetylane

M. van Faassen et al. PRL 88 186401 (2002) S.J.A. Van Gisbergen PRL 83 694 (1999)

Electric field dependence of the XC Potenital in Molecular Chains

In LDA and GGA xc potential lack of a term counteracting the applied electric field

Low dimensional sytems (1D): polyacetylane

f xcBSE r ,r ' ,

Isolated infinite Polyacetylene chain

G. Onida, L. Reining and AR, Rev. Mod. Phys. 74, 601 (2002)

What about the description of decaying quasiparticle processes

within TDDFT?

Lifetime of quasiparticles

interactions between quasiparticles limit how long the corresponding quantum states retain their identity, i.e., the lifetime

of the excitation. In combination with the velocity, this lifetime determines the mean free path, a measure of influence of the excitation

h

Importance of lifetime

- surface photochemistry- electron transfer across interfaces- electron dynamics and energy transfer

- screening in an electron gas- electron-phonon coupling- localization

from F.Reinert et al., PRB 63 (2001) 115415.

Experimental lifetimes change quickly with time!

Heimann et al. 1977

Kevan et al. 1987

Paniago et al. 1995

Nicolay et al. 2000

binding energy [meV]

Excitonic effects (via TDDFT) on the lifetimes of LiF

10'000x10'000 BS kernelUp to 200 G-vectors dielectric

function, 256 Q-points in the whole BZ

Small broadening stability

The fxc kernel remains stable and robust even with a 10 meV broadening and energies up to 20 eV

A BS-based calculation is, in this case,enormously less convenient than using TDDFT

Linear behaviour Small penetration of the

RPA ''forbidden region''

Acknowledgements

A. Castro, A. Marini, X. López, L Wirtz, D. Varsano

Department of Material Physics, Centro Mixto CSIC-UPV, University of the Basque Country, and Donostia International Physics Center (DIPC), Donostia, Spain

L. Reining, V. Olevano

G.F. Bertsch

École Polytechnique, Palaiseau, France

Physics Department and Institute for Nuclear Theory University of Washinton, Seattle (USA)

R. Del Sole and G. OnidaINFM e Dipartimento di Fisica dell'Università di Roma ``Tor Vergata'', Roma, Italy

M. A. L. Marques, C.A. Rozzi, and E. K. U. GrossInstitut für Theoretische Physik, Freie Universität, Berlin, Germany

S.G. Louie and M.L. CohenDepartment of Physics, University of California at Berkeley, USA

Thank you

http://dipc.ehu.es/arubio E-mail: arubio@sc.ehu.es

It is one of the first duties of a professor, in any

subject, to exaggerate a little both the importance of his

subject and his own importance in it.

G.H. Hardy (A Mathematician's Apology)

Optical spectroscopy of gold clusters:assessing relativistic effects.

Free electronsQuinn (1962)

Density of States (DOS) versus Screening

Density of states

More DOS

More phase space available

More probability for the process

Shorter lifetimes

Screening

More DOS

More screening

Weaker interaction

Longer lifetimes

EF

263 rs

-5/2

(E-EF)2

n5/6

(E-EF)2

� � dr dr' φi*(r) φf*(r') ImW(r,r';ω) φi(r') φf(r)

PHYSICS OF LIFETIME

The 2-point vertex functionSuppose to have a good approximation fot the (TD)DFT potential...

PRL 91, 056402 (2003);PRL 91, 256402 (2003). PRL 88, 066404 (2002) etc etc

No difference with GoWo using ALDA or similar approaches PRL 62, 2718 (1989); PRB 49, 8024 (1994); PRB 56, 12832 (1997).

BUT IS ?

=

The 3-point vertex function

If

Closed expression for the ''on mass-shell'' electronic lifetime

A. Castro, M. A. L. Marques, J. A. Alonso, G. F. Bertsch, K. Yabana and AR, J. Chem. Phys. 116, 1930 (2002).

• Photoabsorption spectra of C20 isomers are different.

• Optical spectroscopy (absorption and emission) is proposed as experimental tool to identify cluster structure.