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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
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 2 / 22
Outline
IntroductionPhenomenaExperimentsCalculationResultsConclusions
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 2 / 22
Outline
IntroductionPhenomenaExperimentsCalculationResultsConclusions
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 2 / 22
Outline
IntroductionPhenomenaExperimentsCalculationResultsConclusions
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 2 / 22
Outline
IntroductionPhenomenaExperimentsCalculationResultsConclusions
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 2 / 22
Outline
IntroductionPhenomenaExperimentsCalculationResultsConclusions
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 2 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 3 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 3 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 3 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 3 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 3 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 3 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 3 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 3 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 3 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 3 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 4 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 4 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 4 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 4 / 22
Second harmonic generation
Frequency doubling: βijk(−2ω;ω, ω)High frequency (green and blue) lasers.
SHG-microscopy.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 5 / 22
Second harmonic generation
Frequency doubling: βijk(−2ω;ω, ω)High frequency (green and blue) lasers.
SHG-microscopy.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 5 / 22
Second harmonic generation
Frequency doubling: βijk(−2ω;ω, ω)High frequency (green and blue) lasers.
SHG-microscopy.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 5 / 22
Optical rectification
Induction of a static (or low frequency) electric field:
βijk(0;ω,−ω)
Generation of THz pulses.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 6 / 22
Optical rectification
Induction of a static (or low frequency) electric field:
βijk(0;ω,−ω)
Generation of THz pulses.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 6 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 7 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 7 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 7 / 22
Experiments
Measured at laser frequencies.Non-resonant.Difficult to measure experimentally.Indirect measurements.Solvent effects.Large error bars.Non-consistent results.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 8 / 22
Experiments
Measured at laser frequencies.Non-resonant.Difficult to measure experimentally.Indirect measurements.Solvent effects.Large error bars.Non-consistent results.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 8 / 22
Experiments
Measured at laser frequencies.Non-resonant.Difficult to measure experimentally.Indirect measurements.Solvent effects.Large error bars.Non-consistent results.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 8 / 22
Experiments
Measured at laser frequencies.Non-resonant.Difficult to measure experimentally.Indirect measurements.Solvent effects.Large error bars.Non-consistent results.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 8 / 22
Experiments
Measured at laser frequencies.Non-resonant.Difficult to measure experimentally.Indirect measurements.Solvent effects.Large error bars.Non-consistent results.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 8 / 22
Experiments
Measured at laser frequencies.Non-resonant.Difficult to measure experimentally.Indirect measurements.Solvent effects.Large error bars.Non-consistent results.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 8 / 22
Experiments
Measured at laser frequencies.Non-resonant.Difficult to measure experimentally.Indirect measurements.Solvent effects.Large error bars.Non-consistent results.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 8 / 22
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)
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 9 / 22
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)
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 9 / 22
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)
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 9 / 22
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)
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 9 / 22
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)
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 9 / 22
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).
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 10 / 22
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).
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 10 / 22
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).
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 10 / 22
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).
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 10 / 22
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).
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 10 / 22
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).
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 10 / 22
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).
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 10 / 22
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).
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 10 / 22
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).
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 10 / 22
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).
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 10 / 22
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).
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 10 / 22
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).
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 10 / 22
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).
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 10 / 22
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).
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 10 / 22
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).
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 10 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 11 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 11 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 11 / 22
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)
. . . . . . . . .
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 12 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 13 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 13 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 13 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 13 / 22
Sternheimer equations
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)
}
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 14 / 22
Sternheimer equations
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)
}
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 14 / 22
Sternheimer equations
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)
}
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 14 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 15 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 15 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 15 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 15 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 15 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 15 / 22
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)〉.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 16 / 22
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)〉.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 16 / 22
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)〉.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 16 / 22
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)
}.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 17 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 18 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 18 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 18 / 22
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.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 18 / 22
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
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
Second Harmonic Generation of Paranitroaniline∗
−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.
∗J. Chem. Phys. 126, 184106 (2007)X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 19 / 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
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
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 22 / 22
Conclusions
TDLDA gives qualitatively correct results for NLOs.Overestimation of values.A tool to predict NLO coefficients.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 23 / 22
Conclusions
TDLDA gives qualitatively correct results for NLOs.Overestimation of values.A tool to predict NLO coefficients.
X. Andrade (EHU/UPV) Non-linear response within TDDFT TDDFT, Benasque 2008 23 / 22