applications of tddft for finite...
TRANSCRIPT
Applications of TDDFT for finite systemsA biased look
Miguel A. L. Marques
1Centre for Computational Physics, University of Coimbra, Portugal2European Theoretical Spectroscopy Facility
July 23, 2007 – San Sebastian
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 1 / 50
Outline
1 Introduction - Spectroscopies
2 What does not work
3 Some results that workAbsorption of NanostructuresAbsorption of Biological SystemsHyperpolarizabilitiesvan der Waals coefficients
4 Visualizing Electronic Excitations
5 Outlook
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 2 / 50
Introduction - Spectroscopies
Outline
1 Introduction - Spectroscopies
2 What does not work
3 Some results that workAbsorption of NanostructuresAbsorption of Biological SystemsHyperpolarizabilitiesvan der Waals coefficients
4 Visualizing Electronic Excitations
5 Outlook
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 3 / 50
Introduction - Spectroscopies
Spectroscopy
Spectroscopy: From the latin spectrum — an appearance, anapparition, from spectare, to behold + the greek skopein — to view.
v
c
unoccupied states
occupied states
Examples: UV/Vis, IR, X-ray, Dichroism, NMR, Raman, etc.
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 4 / 50
Introduction - Spectroscopies
Linear Response
Optical absorption
Photoemission
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 5 / 50
Introduction - Spectroscopies
Strong fields
K. Yamanouchi, Science 295, 1659 (2002)
J. J. Levis, Science 292, 709 (2001)
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 6 / 50
Introduction - Spectroscopies
Time-dependent DFT
One-to-one corespondence between the time-dependent density andthe external potential, n(r , t) ↔ vext(r , t) (Runge-Gross theorem).
The many-body equation is mapped onto the Kohn-Sham equation
−i∂
∂tψi(r , t) =
[−∇
2
2+ vext(r , t) + vHartree[n](r , t) + vxc[n](r , t)
]ψi(r , t)
The current approximations for the xc potential are quite reliable
LDA, GGA, meta-GGA, etc. for the ground-state.
adiabatic approximations to handle (=ignore) time non-locality.
Time-Dependent Density Functional Theory, ed. by M.A.L. Marques et al.,
Lecture Notes in Physics, Vol. 706 (Springer, Berlin, 2006)
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 7 / 50
Introduction - Spectroscopies
Linear response from the KS equations
Apply a perturbation of the form δvext σ(r , t) = −κ0zδ(t) to theground state of the system.At t = 0+ the Kohn-Sham orbitals are
ϕj(r , t = 0+) = eiκ0zϕj(r ) .
Propagate these KS wave-functions for a (in)finite time.The dynamical polarizability can be obtained from
α(ω) = − 1κ0
∫d3r z δn(r , ω) .
This prescription has been used with considerable success tocalculate the photo-absorption spectrum of several finite systems.As it is based on the propagation of the Kohn-Sham equations,this approach can be easily extended to study non-linearresponse.
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 8 / 50
Introduction - Spectroscopies
The Sternheimer Equation
Hartree-Fock: Coupled Hartree-Fock methodDFT: Density functional perturbation theory
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 9 / 50
Introduction - Spectroscopies
The Sternheimer Equation - FrequencyH(0) − εm ± ω + iη
ψ
(1)m (r ,±ω) = −PcH(1)(±ω)ψ
(0)m (r )
with
H(1)(ω) = V (r ) +
∫d3r ′
n(1)(r ′, ω)
|r − r ′|+
∫d3r ′ fxc(r , r ′) n(1)(r ′, ω)
and
n(1)(r , ω) =occ.∑m
[ψ
(0)m (r )
]∗ψ
(1)m (r , ω) +
[ψ
(1)m (r ,−ω)
]∗ψ
(0)m (r )
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 10 / 50
Introduction - Spectroscopies
The Sternheimer Equation - FrequencyH(0) − εm ± ω + iη
ψ
(1)m (r ,±ω) = −PcH(1)(±ω)ψ
(0)m (r )
with
H(1)(ω) = V (r ) +
∫d3r ′
n(1)(r ′, ω)
|r − r ′|+
∫d3r ′ fxc(r , r ′) n(1)(r ′, ω)
and
n(1)(r , ω) =occ.∑m
[ψ
(0)m (r )
]∗ψ
(1)m (r , ω) +
[ψ
(1)m (r ,−ω)
]∗ψ
(0)m (r )
Main advantages:
(Non-)Linear system of equations solvable by standard methods
Only the occupied states enter the equation
Scaling is N2, where N is the number of atoms
Disadvantages:
It is hard to converge close to a resonanceM. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 10 / 50
Introduction - Spectroscopies
Linear Response - Other methods
Dyson equationχ = χ0 + χ0 [v + fxc]χ
Casida’s equationRFq = Ω2
qFq ,
where
Rq,q′ = (εaσ − εiσ)2δqq′ + 2√εaσ − εiσKq,q′(ωn)
√εa′σ′ − εi ′σ′ .
Superoperators and Lanczos methods
〈P1|(ω − L)−1|Q1〉 =1
ω − a1 + b21
ω−a2+···c2
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 11 / 50
Introduction - Spectroscopies
Linear response - Perturbations
Different perturbations are possible:
ElectricV (r ) = r i
(e.g., polarizabilities, absorption,florescence ...)
2 3 4 5Energy (eV)
σ (a
rb. u
nits)
2 3 4 5
xyz
Magnetic
V (r ) = L i
(e.g., susceptibilities, NMR ...)Atomic Displacements
V (r ) =∂v(r )
∂R i
(e.g., phonons ...)
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 12 / 50
Introduction - Spectroscopies
Code development - octopus
http://www.tddft.org/programs/octopus
Comput. Phys. Commun. 151, 60–78 (2003)
Phys. Stat. Sol. B 243, 2465–2488 (2006)
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 13 / 50
Introduction - Spectroscopies
Octopus — Details
Some numbers:
∼10 active developers
∼125 members of octopus-users mailing list
∼70,000 lines of Fortran 90 + ∼20,000 lines of C
What we can calculate:
Ground-state properties
(Hyper)polarizabilities using a variety of numerical methods
Optical spectra / rotatory spectra (dychroism)
Molecules in laser fields
In the works:
Forces in the excited-state / resonant Raman
Electronic transport within TDDFT
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 14 / 50
What does not work
Outline
1 Introduction - Spectroscopies
2 What does not work
3 Some results that workAbsorption of NanostructuresAbsorption of Biological SystemsHyperpolarizabilitiesvan der Waals coefficients
4 Visualizing Electronic Excitations
5 Outlook
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 15 / 50
What does not work
What does not work:
Linear responseTypical absorption energies can be 0.5 eV wrongCharge transfer excitations come out too lowTransition states are often wrong: triplet instabilities...
Nonlinear dynamicsAsymptotics of standard xc functionals are wrong: too muchionization, problems with Rydberg statesNo derivative discontinuityIt is not obvious how to write some observables as functionals ofthe density...
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 16 / 50
What does not work
What does not work:
Linear responseTypical absorption energies can be 0.5 eV wrongCharge transfer excitations come out too lowTransition states are often wrong: triplet instabilities...
Nonlinear dynamicsAsymptotics of standard xc functionals are wrong: too muchionization, problems with Rydberg statesNo derivative discontinuityIt is not obvious how to write some observables as functionals ofthe density...
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 16 / 50
Some results that work
Outline
1 Introduction - Spectroscopies
2 What does not work
3 Some results that workAbsorption of NanostructuresAbsorption of Biological SystemsHyperpolarizabilitiesvan der Waals coefficients
4 Visualizing Electronic Excitations
5 Outlook
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 17 / 50
Some results that work Absorption of Nanostructures
Outline
1 Introduction - Spectroscopies
2 What does not work
3 Some results that workAbsorption of NanostructuresAbsorption of Biological SystemsHyperpolarizabilitiesvan der Waals coefficients
4 Visualizing Electronic Excitations
5 Outlook
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 18 / 50
Some results that work Absorption of Nanostructures
Discriminating the C20 isomers
Real-space, real-time TDLDAyields reliable photo-absorptionspectra of carbon clusters
Spectra of the different C20 aresignificantly different
Optical spectroscopy proposedas an experimental tool to identifythe structure of the cluster
J. Chem Phys 116, 1930 (2002)
5101520
1
2
3
1
2
0.5
1
1.5
0.5
1
1.5
0 2 4 6 8 10 12Energy (eV)
0.5
1
1.5
ring A
BC
bowl
cage
(d)
(e)
(f)
A B C
D
A
B
AB
C DE
AB
C
A
B
C
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 19 / 50
Some results that work Absorption of Nanostructures
CdSe clusters
Cd33Se33 Cd34Se34
cage
0.6
0.4
0.2
2 3 4 5
1/eV
eV
0.6
0.4
0.2
2 3 4 5
1/eV
eV
wur
tzite
0.6
0.4
0.2
2 3 4 5
1/eV
eV
Experiment 80KExperiment 45KTheory
Phys. Rev. B 75, 035311 (2007)
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 20 / 50
Some results that work Absorption of Biological Systems
Outline
1 Introduction - Spectroscopies
2 What does not work
3 Some results that workAbsorption of NanostructuresAbsorption of Biological SystemsHyperpolarizabilitiesvan der Waals coefficients
4 Visualizing Electronic Excitations
5 Outlook
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 21 / 50
Some results that work Absorption of Biological Systems
Aequorea victoria
Aequorea victoria is an abundant jellyfish inPuget Sound, Washington State, from which theluminescent protein aequorin and thefluorescent molecule GFP have been extracted,purified, and eventually cloned. These twoproducts have proved useful and popular invarious kinds of biomedical research in the1990s and 2000s and their value is likely toincrease in coming years.
http://faculty.washington.edu/
cemills/Aequorea.html
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 22 / 50
Some results that work Absorption of Biological Systems
Data Sheet
238 AA protein forming a β-barrel or β-can
Chromophore located inside the β-barrel(shielded)
Info to create the chromophore containedentirely in the gene
High stability: wide pH, T, salt
Long half life: ≈20 years
Resistant to most proteases
Active after peptide fusions: reporterprotein
Availability of chromophores variants
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 23 / 50
Some results that work Absorption of Biological Systems
Chromophore
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 24 / 50
Some results that work Absorption of Biological Systems
Chromophore
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 24 / 50
Some results that work Absorption of Biological Systems
Chromophore
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 24 / 50
Some results that work Absorption of Biological Systems
Optical Absorption
2 3 4 5Energy (eV)
σ (a
rb. u
nits)
2 3 4 5
xyz
[exp1, exp2, neutral (dashes), anionic (dots)]
Excellent agreement withexperimental spectra
Clear assignment of neutral andanionic peaks
We extract an in vivoneutral/anionic ratio of 4 to 1
GFP: Phys. Rev. Lett. 90, 258101 (2003)
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 25 / 50
Some results that work Absorption of Biological Systems
GFP: Mutants
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 26 / 50
Y66W
Y66H
GFP
Some results that work Absorption of Biological Systems
BFP: Spectra
2 3 4 5
arb.
uni
ts
eV
ExperimentGas-phase
In proteinEnv. 1Env. 2
Gas-phase: Chromophore optimized in the gas-phase. Spectrumcalculated using only the chromophore.In protein: Chromophore optimized inside the protein. Spectrumcalculated using only the chromophore.Env. 1: Spectrum calculated using the chromophore, ARG96,HIS148, and GLU222.Env. 1: Spectrum calculated using the chromophore, GLN94,ARG96, HIS148, SER205, GLU222, and a buried watermolecule.
Cationic state (HSP) can be ruled out
Anionic state (HSA) can not be ruledout from our calculations
Most likely candidate is the cis-HSD
Overall agreement with experiment isquite good
Effects due to the protein environmentare fairly small, and tend to canceleach other
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 27 / 50
Some results that work Absorption of Biological Systems
BFP: Temperature Effects
2 3 4 5
arb.
uni
ts
eV
ExperimentHSAHSD At finite temperatures, we observe a large oscillation of the
chromophore
The optical spectra turn out to be quite sensitive to the anglebetween the rings of the chromophore
Taking into account these oscillations leads to a blue shift of≈0.1 eV of the main peak.
Final spectrum is now in excellent agreement with experiments
-40
-20
0
20
40
-40 -20 0 20 40
DH2
DG1
1.3
1.35
1.4
1.45
1.5
1.3 1.35 1.4 1.45 1.5
C β-C
γ
Cα-Cβ
J. Am. Chem. Soc. 127, 12329–12337 (2005).
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 28 / 50
Some results that work Hyperpolarizabilities
Outline
1 Introduction - Spectroscopies
2 What does not work
3 Some results that workAbsorption of NanostructuresAbsorption of Biological SystemsHyperpolarizabilitiesvan der Waals coefficients
4 Visualizing Electronic Excitations
5 Outlook
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 29 / 50
Some results that work Hyperpolarizabilities
First test: SHG
Second harmonic generation ofparanitroaniline: β(−2ω, ω, ω)
−10000
−5000
0
5000
10000
15000
0 1 2 3 4 5
β ||(−
2ω;ω
,ω) [
a.u.
]
2ω [eV]
exp. solv.
This work
6000 5000
4000
3000
2000
1000
0 0.5 1 1.5 2 2.5 3
β ||(−
2ω;ω
,ω) [
a.u.
]
2ω [eV]
exp. solv.
exp. gas
This work
LDA/ALDA
LB94/ALDA
B3LYP
CCSD
JCP 126, 184106 (2007)
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 30 / 50
Some results that work Hyperpolarizabilities
Optical Rectification
Optical rectification of H2O: β(0, ω,−ω)
0
5000
10000
15000
20000
25000
0 2 4 6 8 10
−β||(
0;ω
,−ω
) [a.
u.]
ω [eV]
20
30
40
50
60
70
0 1 2 3 4 5
JCP 126, 184106 (2007)
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 31 / 50
Some results that work van der Waals coefficients
Outline
1 Introduction - Spectroscopies
2 What does not work
3 Some results that workAbsorption of NanostructuresAbsorption of Biological SystemsHyperpolarizabilitiesvan der Waals coefficients
4 Visualizing Electronic Excitations
5 Outlook
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 32 / 50
Some results that work van der Waals coefficients
Van der Waals coefficients
Non-retarded regime – Casimir-Polder formula (∆E = −C6/R6):
CAB6 =
3π
∫ ∞
0du α(A)(iu) α(B)(iu) ,
Retarded regime (∆E = −K/R7):
K AB =23c8π2α
(A)(0) α(B)(0)
The polarizability is calculated from
αij(iu) =
∫d3r n(1)
j (r , iu)ri
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 33 / 50
Some results that work van der Waals coefficients
Alternative – Time Propagation
Apply explicitly the perturbation:
δvext(r , t) = −xjκδ(t − t0)
The dynamic polarizability reads, at imaginary frequencies:
αij(iu) = −1κ
∫dt∫
d3r xi δn(r , t)e−ut
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 34 / 50
Some results that work van der Waals coefficients
Alternative – Time Propagation
Apply explicitly the perturbation:
δvext(r , t) = −xjκδ(t − t0)
The dynamic polarizability reads, at imaginary frequencies:
αij(iu) = −1κ
∫dt∫
d3r xi δn(r , t)e−ut
It turns out:
Both Sternheimer and time-propagation have the same scaling
Only a few frequencies are needed in the Sternheimer approach,but ...
2 or 3 fs are sufficient for the time-propagation
In the end, the pre-factor is very similar
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 34 / 50
Some results that work van der Waals coefficients
C6 - Polycyclic Aromatic Hydrocarbons
0 1000 2000 3000 4000 5000N
A x N
B
0
50
100
150
200
250
C6
AB
(a.u./103)
40
45
50
55
60
65
70
C6/N
2
C6/N
2
0
5
10
15
20
25
30
∆α2 /N
2
∆α2/N
2
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 35 / 50
Some results that work van der Waals coefficients
C6 - The characteristic frequency
London approximation
α(iu) =α(0)
1 + (u/ω1)2
which leads to
C6 =3ω1
4α2(0)
0.1 0.2 0.3 0.4ω
1 (Ha)
0.2
0.24
0.28
IP (
Ha)
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 36 / 50
0.34
0.36
0.38
0.4
0.42
0 20 40 60 80 100 120 140
ω1
(a.u
.)
Number of Si atoms
ω1 = 0.34
Some results that work van der Waals coefficients
C3 – Surface-cluster interaction
For a surface-cluster interaction, ∆E = −C3/R3, where
C3 =1
4π
∫ ∞
0du α(iu)
ε(iu)− 1ε(iu) + 1
0.6
0.7
0.8
0.9
0 20 40 60 80 100 120 140
C3/
NS
i (a.
u.)
Number of Si atoms
RPATDLDA
α/q2
α + β ω2/q2
0.62
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 37 / 50
Visualizing Electronic Excitations
Outline
1 Introduction - Spectroscopies
2 What does not work
3 Some results that workAbsorption of NanostructuresAbsorption of Biological SystemsHyperpolarizabilitiesvan der Waals coefficients
4 Visualizing Electronic Excitations
5 Outlook
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 38 / 50
Visualizing Electronic Excitations
Visualizing Electronic Excitations
How to visualize and interpret electron bonds?
We can look at, e.g.,
Electronic density→ Quite featureless
One-particle wave-functions→ Not uniquely defined→ Usually extend over large regions
Electron localization function→ Bonds and lone-pairs are evident
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 39 / 50
Visualizing Electronic Excitations
The time-dependent ELF
The time-dependent electron localization function is defined as
felf =1
1 +[
C(r ,t)Cuni(r ,t)
]2
For a Slater determinant:
Cdet =N∑
i=1
|∇ϕi(r , t)|2 −14
[∇n(r , t)]2
n(r , t)− [j (r , t)]2
n(r , t)
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 40 / 50
Visualizing Electronic Excitations
TDELF – C2H2 in a strong laser field
Laser: ω = 17 eV, T = 8 fs, I = 1.2× 1014 Wcm−2
Phys. Rev. A (Rap. Comm.) 71, 10501 (2005)
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 41 / 50
Visualizing Electronic Excitations
TDELF – H+ + OH− → H2O
0 fs
13.3 fs
23 fs
3.0 fs
15.7 fs
24.8 fs
6.0 fs
18.1 fs
27.2 fs
9.7 fs
20.6 fs
30.4 fs
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 42 / 50
Visualizing Electronic Excitations
TDELF – linear response
The first order variation of the ELF is
C(1)(r) =∑
i
[∇ψ(1)∗ · ∇ψ(0) +∇ψ(0)∗ · ∇ψ(1)
]
− 12∇ρ(0)
ρ(0)· ∇ρ(1) +
14
∣∣∇ρ(0)∣∣2[
ρ(0)]2 ρ(1) + 2
j(0)j(1)
ρ(0)−ρ(1)
[j(0)]2[
ρ(0)]2
And the normalization:
C(1)0 = 6π2
[ρ(0)]2/3
ρ(1)
We obtain
f (1)ELF = −2
[f (0)ELF
]2 C(0)
C(0)0
(C(1)
C(0)− C(0)
C(0)0
C(1)0
C(0)0
)
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 43 / 50
Visualizing Electronic Excitations
LRELF – An example, acrolein
0
0.5
1
1.5
2
2.5
3
3.5
4 4.5 5 5.5 6 6.5 7 7.5 8
σ(ω
) [Å
2 ]
ω [eV]
5.69
5.97
6.18
6.87
7.74
7.46
7.61
x
y
z
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 44 / 50
Visualizing Electronic Excitations
LRELF – An example, acrolein
5.64 eV
6.14 eV
5.91 eV
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 45 / 50
Visualizing Electronic Excitations
LRELF – Another example, pyrrole
0
1
2
3
4
5
6
7
8
9
4.5 5 5.5 6 6.5 7 7.5 8
σ(ω
) [Å
2 ]
ω [eV]
5.48
6.15
7.23x
y
z
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 46 / 50
Visualizing Electronic Excitations
LRELF – Another example, pyrrole
5.43 eV (y) 6.10 eV (z)
7.18 eV (x)M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 47 / 50
Outlook
Outline
1 Introduction - Spectroscopies
2 What does not work
3 Some results that workAbsorption of NanostructuresAbsorption of Biological SystemsHyperpolarizabilitiesvan der Waals coefficients
4 Visualizing Electronic Excitations
5 Outlook
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 48 / 50
Outlook
Outlook
TDDFT has probably no rival currently in what concerns thecalculation of absorption spectra for large systems
Therefore, its use has increased exponentially during the pastyears
We now know fairly well the limitations of existing functionals
Nonlinear dynamics is a very big challenge!
Still some work is required to arrive at theultimate functional!
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 49 / 50
Outlook
Outlook
TDDFT has probably no rival currently in what concerns thecalculation of absorption spectra for large systems
Therefore, its use has increased exponentially during the pastyears
We now know fairly well the limitations of existing functionals
Nonlinear dynamics is a very big challenge!
Still some work is required to arrive at theultimate functional!
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 49 / 50
Outlook
Outlook
TDDFT has probably no rival currently in what concerns thecalculation of absorption spectra for large systems
Therefore, its use has increased exponentially during the pastyears
We now know fairly well the limitations of existing functionals
Nonlinear dynamics is a very big challenge!
Still some work is required to arrive at theultimate functional!
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 49 / 50
Outlook
Outlook
TDDFT has probably no rival currently in what concerns thecalculation of absorption spectra for large systems
Therefore, its use has increased exponentially during the pastyears
We now know fairly well the limitations of existing functionals
Nonlinear dynamics is a very big challenge!
Still some work is required to arrive at theultimate functional!
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 49 / 50
Outlook
Outlook
TDDFT has probably no rival currently in what concerns thecalculation of absorption spectra for large systems
Therefore, its use has increased exponentially during the pastyears
We now know fairly well the limitations of existing functionals
Nonlinear dynamics is a very big challenge!
Still some work is required to arrive at theultimate functional!
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 49 / 50
Outlook
Collaborators
Xavier AndradeSan Sebastian, Spain
Alberto CastroBerlin, Germany
Angel RubioSan Sebastian, Spain
Hardy GrossBerlin, Germany
Silvana BottiParis, France
M. A. L. Marques (Coimbra) TDDFT for finite systems San Sebastian 2007 50 / 50