Non-linear response within TDDFT · Non-linear response within TDDFT Xavier Andrade in collaboration with Silvana Botti, Miguel Marques and Angel Rubio European Theoretical Spectroscopy

Non-linear response within TDDFT

Xavier Andradein collaboration with

Silvana Botti, Miguel Marques and Angel Rubio

European Theoretical Spectroscopy Facilityand

Departamento de Fı́sica de MaterialesUniversidad del Paı́s Vasco, Spain

Benasque, September 2008

X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 1 / 22

Outline

IntroductionPhenomenaExperimentsCalculationResultsConclusions


Outline



Outline



Outline



Outline



Outline



Introduction

An external electric field ε on a system: induced electric field.Generated by an induced dipole.

Expansion of the induced dipole

p (ε) = p0 + α ε+ β ε2 + γ ε3 + . . .

Linear response term: αHigher orders:

Intense fields.ε composed of monochromatic fields.Multiphoton processes.Frequency mixing.Biophysical and technological applications.


Introduction



p (ε) = p0 + α ε+ β ε2 + γ ε3 + . . .




Introduction



p (ε) = p0 + α ε+ β ε2 + γ ε3 + . . .




Introduction



p (ε) = p0 + α ε+ β ε2 + γ ε3 + . . .




Introduction



p (ε) = p0 + α ε+ β ε2 + γ ε3 + . . .




Introduction



p (ε) = p0 + α ε+ β ε2 + γ ε3 + . . .




Introduction



p (ε) = p0 + α ε+ β ε2 + γ ε3 + . . .




Introduction



p (ε) = p0 + α ε+ β ε2 + γ ε3 + . . .




Introduction



p (ε) = p0 + α ε+ β ε2 + γ ε3 + . . .




Introduction



p (ε) = p0 + α ε+ β ε2 + γ ε3 + . . .




Second order response

First hyperpolarizability: βijk(ω;ω1, ω2)

Third order of the induced dipole

p(3)i (t) =

∑jk ω

β̄ijk(ω;ω1, ω2) ε1j cos (ω1t) ε2

k cos (ω2t)

=∑jk

[βijk (ω1 + ω2;ω1, ω2) ε1

jε2k cos ((ω1 + ω2) t) +

βijk(ω1 − ω2;ω1, ω2) ε1jε

2k cos ((ω1 − ω2)t)

]

ω = ω1 ± ω2

Zero for systems with inversion symmetry.





p(3)i (t) =

∑jk ω


k cos (ω2t)

=∑jk

[βijk (ω1 + ω2;ω1, ω2) ε1

jε2k cos ((ω1 + ω2) t) +


2k cos ((ω1 − ω2)t)

]

ω = ω1 ± ω2






p(3)i (t) =

∑jk ω


k cos (ω2t)

=∑jk

[βijk (ω1 + ω2;ω1, ω2) ε1

jε2k cos ((ω1 + ω2) t) +


2k cos ((ω1 − ω2)t)

]ω = ω1 ± ω2






p(3)i (t) =

∑jk ω


k cos (ω2t)

=∑jk

[βijk (ω1 + ω2;ω1, ω2) ε1

jε2k cos ((ω1 + ω2) t) +


2k cos ((ω1 − ω2)t)

]ω = ω1 ± ω2



Second harmonic generation

Frequency doubling: βijk(−2ω;ω, ω)High frequency (green and blue) lasers.

SHG-microscopy.




SHG-microscopy.




SHG-microscopy.


Optical rectification

Induction of a static (or low frequency) electric field:

βijk(0;ω,−ω)

Generation of THz pulses.


Optical rectification

Induction of a static (or low frequency) electric field:

βijk(0;ω,−ω)

Generation of THz pulses.


Pockels effect

Variation of the polarizability due to a static electric field:

βijk(0;ω,−ω)

Pockels cell: control polarization of light.Combined with a polarizer: ultrafast shutters.


Pockels effect


βijk(0;ω,−ω)



Pockels effect


βijk(0;ω,−ω)



Experiments

Measured at laser frequencies.Non-resonant.Difficult to measure experimentally.Indirect measurements.Solvent effects.Large error bars.Non-consistent results.


Experiments



Experiments



Experiments



Experiments



Experiments



Experiments



Electric Field Induced Second Harmonic generation

Measure SHG in gas or liquid phase.On average β is zero.A static electric field is applied to (partially) align molecules.

Measured quantity

χ3(0;ω, ω,−2ω) = f0f2ωf2ω

[Eµ

kTβvec(ω, ω,−2ω) + γ(0;ω, ω,−2ω)

]

The quantity to compare

βvecz =

13

3∑i=1

(βzii + βizi + βiiz) (1)




Measured quantity

χ3(0;ω, ω,−2ω) = f0f2ωf2ω

[Eµ


]


βvecz =

13

3∑i=1





Measured quantity

χ3(0;ω, ω,−2ω) = f0f2ωf2ω

[Eµ


]


βvecz =

13

3∑i=1





Measured quantity

χ3(0;ω, ω,−2ω) = f0f2ωf2ω

[Eµ


]


βvecz =

13

3∑i=1





Measured quantity

χ3(0;ω, ω,−2ω) = f0f2ωf2ω

[Eµ


]


βvecz =

13

3∑i=1



Calculation of NLO coefficients

Several methods.Finite differences

β(0; 0, 0) = ∂3E/∂ε3

Simple, only requires total energies.Limited to static response.

Time propagationEfficient method.All response orders are included.Difficult to extract coefficients.One propagation per frequency.

Perturbation theorySum over states

Requires unoccupied states.Bad scaling.

The Sternheimer equation(a.k.a. Density Functional Perturbation Theory).




β(0; 0, 0) = ∂3E/∂ε3









β(0; 0, 0) = ∂3E/∂ε3









β(0; 0, 0) = ∂3E/∂ε3









β(0; 0, 0) = ∂3E/∂ε3









β(0; 0, 0) = ∂3E/∂ε3









β(0; 0, 0) = ∂3E/∂ε3









β(0; 0, 0) = ∂3E/∂ε3









β(0; 0, 0) = ∂3E/∂ε3









β(0; 0, 0) = ∂3E/∂ε3









β(0; 0, 0) = ∂3E/∂ε3









β(0; 0, 0) = ∂3E/∂ε3









β(0; 0, 0) = ∂3E/∂ε3









β(0; 0, 0) = ∂3E/∂ε3









β(0; 0, 0) = ∂3E/∂ε3







Response functions

n(1)(r; t) =∫dt′dr′ χ(r; t, r′; t′)Vext(r′; t′)

n(2)(r; t) =12!

∫dt1dt2dr1dr2 χ

(2)(r; t, r1; t1, r2; t2)Vext(r1; t1)Vext(r2; t2). . . . . . . . .

Response functions for each order.The same for χ0 in terms of Veff .χ0 can be obtained from perturbation theory as a sum over states.


Response functions


n(2)(r; t) =12!

∫dt1dt2dr1dr2 χ




Response functions


n(2)(r; t) =12!

∫dt1dt2dr1dr2 χ




Response functions

n(1)

(r; t) =

Zdt1dr1 χ0(r; t, r1; t1)Vext(r1; t1)

+

Zdt1dt2dr1dr2 χ0(r; t, r1; t1)

δ(t1 − t2)

|r1 − r2|+ fxc(r1; t1, r2; t2)

!n(1)

(r2; t2)

n(2)

(r; t) =1

2

Zdt1dt2dr1dr2 χ

(2)0 (r; t, r1; t1, r2; t2)Vext(r1; t1)Vext(r2; t2)

+1

2

Zdt1dt2dt3dr1dr2dr3 χ0(r; t, r1; t1)Mxc(r1; t1, r2; t2, r3; t3)n

(1)(r2; t2)n

(1)(r3; t3)

+

Zdt1dt2dr1dr2 χ0(r; t, r1; t1)

δ(t1 − t2)

|r1 − r2|+ fxc(r1; t1, r2; t2)

!n(2)

(r2; t2)

. . . . . . . . .


Sternheimer equations

Monochromatic electric field (finite systems)

V = λ re−ηt cos (ωt) =λ

2re−ηt

(eiωt + e−iωt

)(2)

Wave function expansion

|ψk (t)〉 = |ψ(0)k 〉+

λ

2e−ηt

[eiωt|ψ(1)

k (ω)〉+ e−iωt|ψ(1)k (−ω)〉

](3)

The unknowns are the wave function variations.Put 2 and 3 in the TDKS equation.





2re−ηt

(eiωt + e−iωt

)(2)


|ψk (t)〉 = |ψ(0)k 〉+

λ

2e−ηt

[eiωt|ψ(1)

k (ω)〉+ e−iωt|ψ(1)k (−ω)〉

](3)






2re−ηt

(eiωt + e−iωt

)(2)


|ψk (t)〉 = |ψ(0)k 〉+

λ

2e−ηt

[eiωt|ψ(1)

k (ω)〉+ e−iωt|ψ(1)k (−ω)〉

](3)






2re−ηt

(eiωt + e−iωt

)(2)


|ψk (t)〉 = |ψ(0)k 〉+

λ

2e−ηt

[eiωt|ψ(1)

k (ω)〉+ e−iωt|ψ(1)k (−ω)〉

](3)




We get a self-consistent set of equations:

Sternheimer equations{H(0) − εk ± ω + iη

}|ψ(1)k (ω)〉 = −

[r + (vc + fxc)n(1)

]|ψ(0)k 〉

Density variation

n(1)(r, ω) =∑k

fk

{(ψ

(0)k (r)

)∗ψ

(1)k (r, ω) +

(ψ

(1)k (r,−ω)

)∗ψ

(0)k (r)

}





}|ψ(1)k (ω)〉 = −

[r + (vc + fxc)n(1)

]|ψ(0)k 〉

Density variation

n(1)(r, ω) =∑k

fk

{(ψ

(0)k (r)

)∗ψ

(1)k (r, ω) +

(ψ

(1)k (r,−ω)

)∗ψ

(0)k (r)

}





}|ψ(1)k (ω)〉 = −

[r + (vc + fxc)n(1)

]|ψ(0)k 〉

Density variation

n(1)(r, ω) =∑k

fk

{(ψ

(0)k (r)

)∗ψ

(1)k (r, ω) +

(ψ

(1)k (r,−ω)

)∗ψ

(0)k (r)

}


Sternheimer equation: numerical properties

Sternheimer equation{H(0) − εk ± ω + iη

}|ψ(±1)k 〉 = −H(±1)|ψ(0)

k 〉

Mixing scheme for self-consistency.Unoccupied orbitals are not required.Non-Hermitian linear equation.In principle is easier to solve than the ground state.Good scaling with system size.




}|ψ(±1)k 〉 = −H(±1)|ψ(0)

k 〉





}|ψ(±1)k 〉 = −H(±1)|ψ(0)

k 〉





}|ψ(±1)k 〉 = −H(±1)|ψ(0)

k 〉





}|ψ(±1)k 〉 = −H(±1)|ψ(0)

k 〉





}|ψ(±1)k 〉 = −H(±1)|ψ(0)

k 〉



Non-linear response

|ψ(2)k 〉 can be obtained from a Sternheimer equation.

Due to the 2n+1 theorem we don’t need them for βijk(ω1;ω2, ω3).

So we only need |ψ(1)k (±ω1)〉, |ψ(1)

k (±ω2)〉 and |ψ(1)k (±ω3)〉.


Non-linear response




k (±ω2)〉 and |ψ(1)k (±ω3)〉.


Non-linear response




k (±ω2)〉 and |ψ(1)k (±ω3)〉.


Non-linear response

βijk(−1; 2, 3) = −4∑P

∑ζ=±1

{occ.∑m

∫d3r ψ

(1)∗m, i (r,−ζω1)H(1)

j (ζω2)ψ(1)m, k(r, ζω3)

−occ.∑mn

∫d3r ψ(0)∗

m (r)H(1)j (ζω2)ψ(0)

n (r)∫

d3r ψ(1)∗n, i (r,−ζω1)ψ(1)

m, k(r, ζω3)

−23

∫d3r

∫d3r′∫

d3r′′Kxc(r, r′, r′′)n(1)i (r, ω1)n(1)

j (r′, ω2)n(1)k (r′′, ω3)

}.


About the box

For properly converged results, a large region around the systemhas to be properly described.Difficult to treat with localized basis sets.Easier with systematically converged basis sets (plane waves, realspace grids, etc.).Different results in liquid and gas phase.


About the box



About the box



About the box



Second Harmonic Generation of Paranitroaniline∗

C6N2O2H6

Benchmark system for β calculations.

∗J. Chem. Phys. 126, 184106 (2007)X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 19 / 22


C6N2O2H6

Benchmark system for β calculations.



−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

Good agreement with the experiment.


Hyperpolarizability of the H2O

α(0, 0) β‖(0; 0, 0) β‖(−2ω;ω, ω) †

LDA/ALDA 10.51 -25.89 -34.71KLI/ALDA 8.61 -11.75 -14.03LB94 9.64 -17.8 -20.3B3LYP 9.81 -18.54 -24.11HF 8.53 -10.73 -12.52CCSD(T) 9.79 -18.0 -21.1Experimental. 9.81 -22±9

Overestimation of hyperpolarizabilities by LDA/ALDA.Underestimation(?) by KLI/LDA.

†ω = 1.79 [eV]X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 20 / 22

Hyperpolarizability of the H2O

α(0, 0) β‖(0; 0, 0) β‖(−2ω;ω, ω) †

LDA/ALDA 10.51 -25.89 -34.71KLI/ALDA 8.61 -11.75 -14.03LB94 9.64 -17.8 -20.3B3LYP 9.81 -18.54 -24.11HF 8.53 -10.73 -12.52CCSD(T) 9.79 -18.0 -21.1Experimental. 9.81 -22±9

Overestimation of hyperpolarizabilities by LDA/ALDA.Underestimation(?) by KLI/LDA.

†ω = 1.79 [eV]X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 20 / 22

Results for solids‡

EΓ (eV) χ(2)(0; 0, 0) (pm/V)AlSb 1.9 (2.3) 146 (98)GaP 2.0 (2.9) 83 (74)GaAs 1.0 (1.5) 205 (166)GaSb 0.5 (0.8) 617 (838)InP 1.0 (1.4) 145 (287)InAs -0.1 (0.4) — (838)InSb 0.1 (0.2) 957 (1120)ZnS 2.4 (3.8) 33 (61)ZnSe 1.6 (2.8) 65 (156)ZnTe 1.6 (2.4) 122 (184)CdSe 0.8 (1.8) 118 (72)CdTe 1.1 (1.6) 167 (118)

‡Phys. Rev. B 53, 15638 (1996)X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 21 / 22

Second harmonic generation of GaP

χ(2) pm/V ~ω = 0.117 eV ~ω = 0.585 eV ~ω = 0.94 eVTheo. 68 78 103Expt. 74 ± 4 94 ± 20 98 ± 18, 112 ± 12


Conclusions

TDLDA gives qualitatively correct results for NLOs.Overestimation of values.A tool to predict NLO coefficients.


Conclusions



Conclusions



Documents

Non-linear response within TDDFT · Non-linear response within TDDFT Xavier Andrade in collaboration with Silvana Botti, Miguel Marques and Angel Rubio European Theoretical Spectroscopy