Angel Rubio -...

Time dependent DFT

Hands-on workshop on EXC!TiNG code, CECAM Lausanne, Switzerland, 15th Noveber 2010

Workshop “Non adiabati

c dynamic

s at surfaces

”, Schloss Reisens

burg October

2007

12th Density Functional Theory Conference, Amsterdam, August 2007

NanoBio Spectroscopy Group and ETSF Scientific Development CentreDpto. Física de Materiales, Universidad del País Vasco,

Centro Física de Materiales CSIC-UPV/EHU San Sebastián, Spainand FHI Max-Planck-Gesellschaft, Berlin, Germany

http://nanobio.ehu.es Email: [email protected]

Angel Rubio

Time dependent DFT

Time dependent DFT



c dynamic

s at surfaces


burg October

2007


OUTLINE

Introduction: Framing the lecture:Excitations; response concepts

Applications:

Optical properties of nanostructures and biomolecules Chlorophylls: photosynthesis (excite state dynamics and energy store/harvesting)

Optimal Control Theory: quantum information High-harmonic generation

Time dependent DFT



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2007


Physical Processes: Spectroscopies

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Time-dependent phenomena!

Photoexcitation and relaxation

Time required for building up of screening?

Introduding a negative bare charge q=-1 in a nanostructure: monitorise the time evolution of the electronic

density (TDDFT calculation)

q=-1

‘Bare particles get dressed’Huber et al.,

Nature 414, 286 (2001)

Time dependent DFT



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Quantum control

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

Time resolved pump-probe techniques

Zewail et al (2005)

Time dependent DFT



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Excitations in bio-systems: encouraging results!!Green fluorescent protein and their mutants (2008, Nobel prize in Chemistry)

(Aequorea victoria: jellyfish)

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

Some biophysical processes

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Theoretical Framework

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Density Functional versus Manybody perturbation theory

groundstate

What about excited-state properties: electron-ion dynamics, spectroscopies

E [R ions]=min ⟨∣He∣ ⟩ Hohenberg-Kohn (1964)

Time dependent DFT



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What is a spectrum?

Time dependent DFT



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Vinduced

= δ n v + fxc

δ n

Hartree

Time dependent DFT



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Time dependent DFT



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Many Body Perturbation Theory (MPBT)

Time dependent DFT



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Time Dependent Density Functional Theory (Runge and Gross 1984):

All observables are functionals of the TD density

?

One-to-one correspondence between the time-dependent density and the external potential, v(r,t) <------> (r,t)

Time-Dependent Density Functional Theory, Lecture Notes in Physics, Vol. 706 (Springer, Berlin, 2006)

Same time-dependent

density n(r,t)

Time dependent DFT



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2007


Time dependent DFT



c dynamic

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burg October

2007


Time dependent DFT



c dynamic

s at surfaces


burg October

2007


Time dependent DFT



c dynamic

s at surfaces


burg October

2007


Time dependent DFT



c dynamic

s at surfaces


burg October

2007


Time dependent DFT



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s at surfaces


burg October

2007


Time dependent DFT



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2007


Time Dependent Density Functional Theory (Runge and Gross 1984):

All observables are functionals of the TD density

i ℏ ddt=H i ℏ d

dti=H KS [ { j} ]i , i=1,⋯N ?

H KS=ℏ

2

2mi∇−

ecℏ

AAxc2

V externalV hartreeV exchangeV correlation

One-to-one correspondence between the time-dependent density and the external potential, v(r,t) <------> (r,t)

Linear and non linear phenomena accessible

Octopus Code http://www.tddft.org

Time-Dependent Density Functional Theory, Lecture Notes in Physics, Vol. 706 (Springer, Berlin, 2006)

Time dependent DFT



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By the Runge-Gross theorem ALL observables are functionals of the TD density

Absorption spectra: from the power spectrum of the TD dipole moment

d(t) = < r n(r,t) > ------ d(w)=FT[d(t)]

........... and current response

Ionisation yields:

Excited state densities and transition probabilities

....................... and many more observables!!!!!!

PES p =∑i ∣i p , t∞∣2=5eV

I=1.3 1011 W cm2

Time dependent DFT



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potential

t (R)| g>

|e>

|ψ(t)>=C1(t)|e>+C2(t)|g> ????

beyond BO Hψ = Eψ

F = mα

Electron dynamics

MD

within BO

i∂

∂ t=H

Ehrenfest's dynamics (non-adiabatic)

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

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TDDFT Lagrangian formulation for BO-MD dynamics: fictitious dynamics as in Car-Parrinello dynamics

L=∑ii∣i

∂∂ t−

12m

pecA

2

−V ion∣i−EHxcV

8 c2

d Adt

2

12∑i

M I R I Ri

The equation of motion without external field for simplicity

i∂

∂ ti=[−ℏ

2

2m∇

2V ionV HV xc]i

J.L. Alonso, X. Andrade, P. Echenique, F. Falceto, D. Prada, AR PRL (2008)

MI R I=−∇ I E [ ,R]

Advantages of this modified Ehrenfest-dynamics:● Time reversibility build in; exact total physical energy conserved● Good scaling: no need to orthogonalise the states during evolution● intrinsicaly parallel, benefit from many developments in CP dynamics ● limits µ=0; exact BO dynamics, µ=1, Ehrenfest, larger µ>>1 desirable for fast simulation ( the ionic ∆t is basicaly µ times the electronic ∆t)

Time dependent DFT



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TDDFT Lagrangian formulation for BO-MD dynamics: fictitious dynamics: performance

Linear response

),( tr

),( t′′r

tickle the systemobserve how thesystem respondsat a later time

( ) ( )tVtttddtn ′′′′′′= ∫∫ ,,,,),( 11 rrrrr χdensity

responseperturbationdensity-density

response function

Time dependent DFT



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TDDFT does the job for you!!!

Vext

however the KS system V

eff

=00v f xc

0 r ,r ' ,w=∑ij f j− f i

icr jr i r ' j

cr '

i− j−wf xcr , r ' ,=

V xcr ,

n r ,

Linear Response

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How to solve the linear response equations

(1) Sum over states: 0 r ,r ' , w=∑ij f j− f i

icr jr i r ' j

cr '

i− j−w

=00v f xc

Cassida's approach

[1−v f xc0]=0

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(2) Time propagation

Propagation Scheme:●Apply a perturbation of the form δvext (r,t) = −κ zδ(t) to the ground state of the system●At t = 0+ the Kohn-Sham orbitals are φ

ι (r,t=0+)= eikz φ

ι (r )

●Propagate these KS wave-functions for a (in)finite time.●The dynamical polarizability can be obtained from α(ω)= − k-1 Σ

ι <φ

ι | z | φ

ι >

Time dependent DFT



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(3) Density Functional Perturbation Theory

Time dependent DFT



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Problems with extended systems in finite fields?

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Macroscopic Polarisation Theory and real-time simulation

E=−1c

d A t d t

i ℏ∂

∂ ti=[ 1

2m p

ecA

2

V ionV HV xc]i

14

d 2 A

dt 2=−e2 n

mA−c

eV∑i

i∣pm∣i =c2 jmac t =−c2 d P t

dt

G.F. Bertsch, J.I. Iwata, AR, K. Yabana, PRB62, 7998 (2000)

A(t) = Aext

(t) + Aind

(t) Emac

= 4 π P(t)

H= H + H

em P=polarisation

What is the link to the “modern theory of polarisation”? (see the recent review by Vanderbilt, Resta)

Time dependent DFT



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Macroscopic Polarisation Theory <-->real-time A(t)

Berry Phase

Time dependent DFT



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H2 molecular chains (infinite long case)

D. Varsano, A. Marini, AR PRL (2008)

Dimerised

equidistant

Metallic

versus

semiconducting

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Illustration: Some applications

Optical properties of nanostructures and biomolecules Chlorophylls: photosynthesis (excite state dynamics and energy store/harvesting)

Optimal control theory: High-harmonic generation

Time dependent DFT



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2007


Towards the color of plants: e.g. Spinach

Fleming et al Nature (2005),.... (Berkeley)

Time dependent DFT



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2007


Journal of Chemical Physics 131, 84101 (2009)

Free standing porphyrins

Time dependent DFT



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Towards the color of plants: The real Chlorophyll molecule

Exp.Gas

phase

LDA

Time dependent DFT



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Charge Transfer excitations

Photovoltaic Hybrids: Grätzell cells

Road to Improving Efficiency: Controlling position of the frontier orbitals

Time dependent DFT



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Photo-excitation of the light-harvesting carotenoid-porphyrin-C60

Time dependent DFT



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Photo-excitation of the light-harvesting carotenoid-porphyrin-C60

Time dependent DFT



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Clamped ions

Carotenoid-porphyrin-C60 photodynamics

Charge on the C60

Dynamical ionsClamped ions

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Optimal control theory

Key question: What is the laser pulse that drives the system intoa predefined goal?

Procedure: Define a target operator Ô and at the end of thelaser interaction (t = T) maximize the functional

W. Zhu, J. Botina, H. Rabitz, JCP 108, 1953 (1998)

Ô = |Φ ><Φ |F F target state

Example: High harmonic generation

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What makes experimental “control” possible

●Existence of laser sources, since the 1960’s.●Femto-second laser sources, which allow for fast processes (avoiding decoherence), and extending the band-width.●High-intensities.●Laser shapers.●Learning-loops algorithms.

Interrogating Molecules?Science, Nov. 2008

Time dependent DFT



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Tailoring HHGHHG consists of the emission of integer multiples of the carrier frequency of a driving laser, due to its highly non-linear interaction with matter. It can be explained with the so-called 3-steps model:

Typically, the HH spectrum (emission intensity vs. photon frequency) consists of a rapid intensity decrease, a plateau, and a cut-off.

Time dependent DFT



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2007


Tailoring HHG

7th 13th

Time dependent DFT



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The combination of MBPT and (TD)DFT is a powerful tool to “predictive” describe the combined

dynamics of electron/ion in response to external electromagnetic fields of large scale nanostructures, biological molecules and extended systems spanning

very different time scales

ETSF

Time dependent DFT



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2007


Acknowledgements

X. Andrade, M. Wanko, A. Akbarim L. Espinosa,N, Helbig, S. Kurth, I. Tokatly, J. Fuks, J. Alberdi, M. A. L. Marques

ETSF , Department of Material Physics, Centro Mixto CSIC-UPV, University of the Basque Country, Donostia, Spain

E. K. U. Gross Halle

A. Castro, J.L. Alonso, P. Echenique Zaragoza

D. Varsano, C. Rozzi, N. Spallazani Modena

K. Thygessen S. Bronsted Nielsen Denmark

Time dependent DFT



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Than

k yo

u!!!!

For more details see:http://nano-bio.ehu.eshttp://etsf.eu

http://etsf.eu/

Time dependent DFT



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2007


Time and size scales

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

−NKSN1

KSN1−N1

KS≡Egap

KSxc

Egap=I−A=LUMO−HOMO

xc=[Exc[n]

n r ]N1

−[Exc [n]

nr ]N O1/N xc-discontinuity






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

atomsAlmbladh and von Barth 1985

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

A

around atom B






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

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