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: [email protected]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)
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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: [email protected]
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
✔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
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: [email protected]
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