applications of tddft for ﬁnite...

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


Introduction - Spectroscopies

Outline





5 Outlook



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.



Linear Response

Optical absorption

Photoemission



Strong fields

K. Yamanouchi, Science 295, 1659 (2002)

J. J. Levis, Science 292, 709 (2001)



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)



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.



The Sternheimer Equation

Hartree-Fock: Coupled Hartree-Fock methodDFT: Density functional perturbation theory



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 )



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


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



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 ...)



Code development - octopus

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

Comput. Phys. Commun. 151, 60–78 (2003)

Phys. Stat. Sol. B 243, 2465–2488 (2006)


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


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


What does not work

Outline





5 Outlook


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...


Some results that work

Outline





5 Outlook


Some results that work Absorption of Nanostructures

Outline





5 Outlook



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



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)


Some results that work Absorption of Biological Systems

Outline





5 Outlook



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


http://faculty.washington.edu/cemills/Aequorea.html

http://faculty.washington.edu/cemills/Aequorea.html


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



Chromophore



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)



GFP: Mutants


Y66W

Y66H

GFP


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



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).


Some results that work Hyperpolarizabilities

Outline





5 Outlook



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)



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)


Some results that work van der Waals coefficients

Outline





5 Outlook



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



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



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



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



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)


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


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


Visualizing Electronic Excitations

Outline





5 Outlook




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



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)



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)



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



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

)



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



LRELF – An example, acrolein

5.64 eV

6.14 eV

5.91 eV



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



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





5 Outlook


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!


Outlook

Collaborators

Xavier AndradeSan Sebastian, Spain

Alberto CastroBerlin, Germany

Angel RubioSan Sebastian, Spain

Hardy GrossBerlin, Germany

Silvana BottiParis, France


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