Angel Rubio

Benasque, September 2004

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

Lectures on Applications of TDDFT

Motivation

I. Illustration of the physics for nano- and bio-structures

II. Extended systems: problems and new developments

Moore's Law

Evolution towards Nanoscience

Nanotechnology    

Paradigm Shift

Current View Think Different

Classical continuum physics Quantum mechanics

Solid state properties Binding properties

Volume dominating Surfaces dominating

Homogeneous Materials Composite-inhomogeneous

Statistical ensemble Individual particles

Miniaturization Self-organization/Assembly

Optical properties of nanostructures

Nanoscience: different scales need to be bridged (multicomponent simulations)

THEORY and EXPERIMENTS on the same object

- spatial scale: 0D - 1D - 2D from molecule-nano-bio-meso-wires-leads

- time scale: femtosecond (electron excitations) picoseconds (phonon) nanosecond (charge-transfer) millisecond >>> bio-structural organization growth; self assembly

FIRST PRINCIPLES TOOLS: SPECTROSCOPIES

MotivationMotivation

Ab-initio approach for non-equilibrium electron dynamics

e-

e-

e-

excited states

Ground state

transport

1. Optical spectroscopy. Photoelectron and time-resolved spectroscopies2. Lifetimes: Electron-electron and electron-phonon coupling (adiabatic/nonadiabatic)4. Photo-induced dynamics (Ultrafast dynamics): e.g. Photoisomerisation in bio-photoreceptors

hv

Time-dependent approach: TDDFT?

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

K. Yamanouchi, Science 295, 1659 (2002)

High-intense short-laser (fs) pulses

Quantum optimally controlled landscapes: time resolved dynamics

Motivation: non-linear phenomena and chemical reactivity

R. J. Levis et al, Science 292, 709 (2001)

Multiphoton <1014 Wcm2 Tunneling<1015 Wcm2

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!!!!!

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

Electronic properties of a many body system: ab-initio approach

Ground state

density n(r)yes, LDA, GGA's

n(r,t) and/or j(r,t)

G1, total energy

G2

yes

Excited states

Corrections toKS eigenvalues?

Yes, non-linear phenomena and time-resolved spectroscopies

Quasiparticle excitationsIP and EAoptical absorption

yes, in some cases!

Scheme

DFT

TDDFT

MBPT

QM

Problems

XC-func.

(r,r') (w)dependence

Comput.

Time Dependent Density Functional Theory (Runge and Gross 1984)

Electron-dynamics first, then e-ion problem

HK-like theorem: v(r,t) <------> � (r,t)The time dependent density determines uniquely the time-dependent external potential and therefore all physical observables

Kohn-Sham formalism:The time dependent density of the interacting system can be calculated as the density of an auxiliary non-interacting system

Same time-dependent

density n(r,t)

i ℏd

dt=H i ℏ

d

dt

i=H

KS [ { j}]i, i=1,⋯N ?

HKS=ℏ2

2 mi∇−

e

c ℏAA

xc

2

Vexternal

Vhartree

Vexchange

Vcorrelation

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 nanostructures and extended systems within TDDFTImplementation:✔ 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 , non-collinear magnetism, periodic systems .........

Linear Response Applications

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

f j− f ii

c r jr ir ' jc r '

i− j−w

[1−v f xc 0]=0

Single-pole approximation

=0

0v f

xc

K ij=∫ dr ir d V xc r d r

jr

– 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)

Linear optical response: Geometry identification

discrimination of C20 isomers through optical spectroscopy

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

• Real-space, real-time TDLDA yields reliable photoabsorption spectra of carbon-clusters.

• Photoabsorption spectra of C20 isomers are significantly different.

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

Optical spectroscopy of gold clusters: relativistic effects.

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)

QM/MM approach

Green Fluorescent Protein (GFP).

(Aequorea victoria: jellyfish)

Towards understanding biomolecular colors: 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

RE=ℑℜE

rotational strength function

ℜE=ℜxℜ

yℜ

z;

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

Excited state electron-ion dynamics

CIS

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

TRANS

Azobenzene: spectroscopy along femtosecond-laser induced photoisomerization

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 the hole and excited electron

2. Automatic monitoring of the nonradiative decay (lifetime, decay path).

|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 i ℏ ∂

∂t

i=H

KS [ { j}]i, i=1,⋯N ?

HKS=ℏ2

2 mi∇−

e

c ℏA

2

Vexternal

Vhartree

Vexchange

Vcorrelation

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

Fat =−⟨t ∣∇ a

H∣t ⟩ t =Det {it }

it t =e

−iHKSt t t /2

e−iH

KSt t /2

it

Femtosecond dynamics: test photodissociation of a dimer (Na2

+).

80fs

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

ω=3.2eV

ω=2.5eV

Time resolved Vibrational Spectroscopy: Raman

Femtosecond dynamics (ongoing work!!!!)

Time resolved Vibrational Spectroscopy: Raman & IR.

Na-dimer

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

=5eVI=1.310

11W cm

2

Femtosecond dynamics: Time-resolved photoelectron spectroscopy

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 spectrum as:

PES p=∑i ∣i

p , t∞∣2

The ultimate biomedical goal of The ultimate biomedical goal of nanotechnology:nanotechnology:

Nanotechnology: a higher form of evolution? (Humility is perhaps appropriate…)

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

Benasque, September 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 effects!!!- Applications to poliacetilene as one-dimensional system

Some Pathologies of the KS-DFT functional

Band-gap problemis discontinuous by a constant when an electron is added to the system(Sham, Schlüter and Perdew, Levy (1983)

Widely separated open-shell atomsAlmbladh and von Barth 1985

Exchange correlation electric field in insulators

Godby and Sham 1984, Gonze, Ghosez and Godby 1995

Some Pathologies of the KS-DFT functionalSome Pathologies of the KS-DFT functionalBan-gap problem(Sham, Schlüter and Perdew, Levy (1983)

Egap=N1KS −N

KSN1KS N1−N1

KS ≡EgapKSxc

Egap=I−A=LUMO−HOM O

xc=[Exc [n ]nr ]N1

−[Exc [n ]nr ]NO1 /N xc-discontinuity

Some Pathologies of the KS-DFT functional

Band-gap problemis discontinuous by a constant when an electron is added to the system(Sham, Schlüter and Perdew, Levy (1983)

Widely separated open-shell atomsAlmbladh and von Barth 1985

Exchange correlation electric field in insulators

Godby and Sham 1984, Gonze, Ghosez and Godby 1995

Some Pathologies of the KS-DFT functionalSome Pathologies of the KS-DFT functionalWidely separated open-shell atoms

Almbladh and von Barth 1985

The KS-effective potential takes a positive value IB-I

A around atom B

Some Pathologies of the KS-DFT functional

Band-gap problemis discontinuous by a constant when an electron is added to the system(Sham, Schlüter and Perdew, Levy (1983)

Widely separated open-shell atomsAlmbladh and von Barth 1985

Exchange correlation electric field in insulators

Godby and Sham 1984, Gonze, Ghosez and Godby 1995

Some Pathologies of the KS-DFT functionalSome Pathologies of the KS-DFT functional

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

EnQ P≃n

KSn∣nKS−vxc∣n

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

E t =−1

c

d Ad t

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

The equation of motion are:

H=∑i

i∣

1

2 mp

e

cA

2

Vion∣

iE

HartreeE

xc

V

8c2 d Adt

2

i ℏ∂∂t

i=[ 1

2 mp

e

cA

2

VionV

HV

xc ]i

d2 A

dt 2=−4e2 n

mA−4c

e

V∑

i

i∣pm∣

i

For the electric field is:d Edt=−4j j=

−e

V∑

i

i∣pm∣

i−

e2

cn Aand

Gonze,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:

fxc=−/q2

=0

0v f

xc

- In the EELS 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 much important

(crucial) in the absorption spectra than in the EELS!!!

EELS−1=1v

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

=0

G W0

G Wv fxc

M

RPA≡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)

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

×W10000×10000

× '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 systems (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 Potential in Molecular Chains

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

Low dimensional systems (1D): polyacetylane

fxc

BSE 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 kernel

Up 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 behavior Small penetration of

the RPA ''forbidden region''

www.tddft.org/programs/octopus

Summary

Ongoing work on applications to:

Time-resolved spectroscopies: pump-probe simulations.

High-harmonic generation from quantum dots.

QM/MM: for excited-state dynamics in biomolecules: photoisomerisation

Control chemical reactivity:Pulse optimization in laser induced reaction

Non-linear effects in One-dimensional systems

Time-dependent Molecular transport !!!!!!!!!

Quantum nuclei, etc......

TDDFT is a powerful 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

Acknowledgments

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)

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