# Correlation effects in the non-linear response

In this tutorial, we will show how to include correlation effects in the Second Harmonic Generation. We start from the tutorial on SHG in AlAs, using the same DFT inputs and parameters. Notice that the tutorials on TD-Hartree, TD-DFT, and TD-DPFT require more computational time, if calculations are too slow you can run in parallel or decrease the number of frequencies in your SHG.

## Background

In real-time simulations using TDSE, correlation effects are determined by the choice of the Hamiltonian [math]\displaystyle{ H^{\text{sys}}_{\mathbf k} }[/math] that enters in the equations of motion(EOM).

Yambo can use different models for the system Hamiltonian:

- The independent-particle (IP) model,

[math]\displaystyle{ H^{\text{IP}}_{\mathbf k} \equiv H^{\text{KS}}_{\mathbf k} }[/math]

where [math]\displaystyle{ H^{\text{KS}}_{\mathbf k} }[/math] is the unperturbed KS Hamiltonian and we consider the KS system as a system of independent particles.

- The QP model,

[math]\displaystyle{ H^{\text{QP}}_{\mathbf k} \equiv H^{\text{KS}}_{\mathbf k} + \Delta H_{\mathbf k}. }[/math]

where [math]\displaystyle{ \Delta H_{\mathbf k} }[/math] is the GW correction or scissor operator, estimated from the Many-Body Perturbation Theory (Ref. ^{[1]}), and then applied to the KS eigenvalues.

- The RPA model (or Time-dependent Hartree),

[math]\displaystyle{ H^{\text{RPA}}_{\mathbf k} \equiv H^{\text{KS}}_{\mathbf k} + V_h(\mathbf r)[\Delta\rho], }[/math]

The term [math]\displaystyle{ V_h(\mathbf r)[\Delta\rho] }[/math] is the time-dependent Hartree potential and is responsible for the local-field effects originating from system inhomogeneities, and [math]\displaystyle{ \Delta \rho \equiv \rho(\mathbf r;t)-\rho(\mathbf r;t=0) }[/math] is the variation of the electronic density.

- The Time Dependent-Density Functional Theory (TD-DFT) model, at present only for LDA or GGA functionals:

[math]\displaystyle{ H^{\text{TDDFT}}_{\mathbf k} \equiv H^{\text{KS}}_{\mathbf k} + V_h(\mathbf r)[\Delta\rho] + V_{xc}(\mathbf r)[\Delta\rho] }[/math]

The term [math]\displaystyle{ V_{xc}(\mathbf r)[\Delta\rho] }[/math] is the time-dependent exchange-correlation potential.

- The Time Dependent-Density Polarization Function Theory (TD-DPFT) model, at present different functionals are implemented in Yambo, for more details see Ref.
^{[2]}:

[math]\displaystyle{ H^{\text{TD-DFTP}}_{\mathbf k} \equiv H^{\text{KS}}_{\mathbf k} + V_h(\mathbf r)[\Delta\rho] + V_{xc}(\mathbf r)[\Delta\rho] -\Omega \mathcal{E}^s(t) \cdot \nabla_{\mathbf k} }[/math]

where [math]\displaystyle{ \mathcal{E}^s(t) }[/math] is the Kohn-Sham macroscopic field (see Ref. ^{[2]}, ^{[3]}).

## Prerequisites

You need `yambo_nl`

and `ypp_nl`

compiled in double precision for this tutorial. You must have completed the previous tutorial on the non-linear response to do this one.

## Quasi-particle corrections

Quasi-particle correction can be easily introduced in the real-time dynamics by means of a non-local operator that modifies the Kohn-Sham eigenvalues: [math]\displaystyle{ \Delta \hat H = \sum_{n,\bf k} \Delta_{n,\bf k} |v^0_{n,\bf k}\rangle\langle v^0_{n,\bf k}|. }[/math] The command to turn on quasi-particle corrections is: `yambo_nl -u n -V qp`

:

nloptics # [R NL] Non-linear optics % NLBands 3 | 6 | # [NL] Bands % NLintegrator="INVINT" # [NL] Integrator ("EULEREXP/RK4/RK2EXP/HEUN/INVINT/CRANKNIC") NLCorrelation= "IPA" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/JGM/SEX/HF") NLLrcAlpha= 0.000000 # [NL] Long Range Correction % NLEnRange 1.000000 | 5.000000 | eV # [NL] Energy range % NLEnSteps= 5 # [NL] Energy steps NLDamping= 0.150000 eV # [NL] Damping % Field1_Dir 1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor % Field1_kind= "SOFTSIN" # [RT Field1] Kind(SIN|SOFTSIN|RES|ANTIRES|GAUSS|DELTA|QSSIN) GfnQPdb= "none" # [EXTQP G] Database GfnQP_N= 1 # [EXTQP G] Interpolation neighbours % GfnQP_E 0.900000 | 1.000000 | 1.000000 | # [EXTQP G] E parameters (c/v) eV|adim|adim % GfnQP_Z= ( 1.000000 , 0.000000 ) # [EXTQP G] Z factor (c/v) GfnQP_Wv_E= 0.000000 eV # [EXTQP G] W Energy reference (valence) % GfnQP_Wv 0.00 | 0.00 | 0.00 | # [EXTQP G] W parameters (valence) eV| eV|eV^-1 % GfnQP_Wc_E= 0.000000 eV # [EXTQP G] W Energy reference (conduction) % GfnQP_Wc 0.00 | 0.00 | 0.00 | # [EXTQP G] W parameters (conduction) eV| eV|eV^-1 %

All the parameters of this input are the same of the previous tutorial with the only difference that we introduced a scissor operator of 0.9 eV. Notice that it is possible to import also the quasi-particle corrections calculated within the G_{0}W_{0} approximation by using the flag ` GfnQPdb="E < SAVE/ndb.QP" `

(see Yambo tutorial on GW for more information). Now you can run Yambo and then analyze the results as explained in the previous tutorial. The effect of then scissor-operator is shown in the following figure:

The figure compares results with scissor operator "IPA + scissor" (red dots) with the one obtained in the previous tutorial "IPA". The full converged result is obtained with a 18x18x18 k-point grid and bands between 2 and 10 (violet dashed line). The script and the data to generate this plot can be downloaded here.

## Time-dependent Hartree (RPA with local field effects)

In order to include local field effects in the SHG, we use the Time-dependent Hartree approximation. The input is generated with the usual command where we turned on the Reciprocal Lattice vectors verbosity in order to specify the number of plane waves to use in the calculations. To run a TD-Hartree calculation you just specify `NLCorrelation="HARTREE"`

and then we reduce the number of plane waves to 105, that is enough to have a converged result in this system.

nloptics # [R NL] Non-linear optics % NLBands 3 | 6 | # [NL] Bands % NLintegrator="INVINT" # [NL] Integrator ("EULEREXP/RK4/RK2EXP/HEUN/INVINT/CRANKNIC") NLCorrelation= "HARTREE" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/JGM/SEX/HF") NLLrcAlpha= 0.000000 # [NL] Long Range Correction % NLEnRange 1.000000 | 5.000000 | eV # [NL] Energy range % NLEnSteps= 10 # [NL] Energy steps HARRLvcs= 105 RL # [HA] Hartree RL components EXXRLvcs= 105 RL # [XX] Exchange RL components NLDamping= 0.250000 eV # [NL] Damping % ExtF_Dir

1.000000 | 1.000000 | 0.000000 | # [NL ExtF] Versor

% ExtF_kind= "SOFTSIN" # [NL ExtF] Kind(SIN|SOFTSIN|RES|ANTIRES|GAUSS|DELTA|QSSIN)

In this run we increase the damping to 0.25 eV in such a way to speed up calculations. The result of this calculation is shown in the figure below:

Plot data are available here.

## Time-dependent Density Functional Theory (TD-DFT)

TD-DFT calculations are similar to the TD-Hartree ones. Just change `NLCorrelation="TDDFT"`

and the exchange-correlation functional used in the ground state calculations will be included in the real-time dynamics. Notice that only LDA or GGA functionals are supported by Yambo/Lumen. These functionals usually provide insignificant improvement in respect to the TD-Hartree because they miss the long-range part that will be included in the next tutorial by means of TD-DPFT theory. You can perform TD-DFT calculations as exercise and compare them with the other approximations.
The converged result on a 18x18x18 k-points grid is reported below:

Plot data are available here.

## Time-dependent Density-Polarization Functional Theory (TD-DPFT)

In this last tutorial on SHG we will include both quasi-particle effects by means of a scissor operator and long-range correction to the response function by means of Time-dependent Density-Polarization Functional Theory. The Lumen input can be generated as usual with the command `yambo_nl -u n -V qp`

:

```
nloptics # [R NL] Non-linear optics
% NLBands
3 | 6 | # [NL] Bands
%
NLintegrator="INVINT" # [NL] Integrator ("EULEREXP/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "LRC" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/JGM")
NLLrcAlpha= 0.350000 # [NL] Long Range Correction
% NLEnRange
0.200000 | 8.000000 | eV # [NL] Energy range
%
NLEnSteps= 10 # [NL] Energy steps
HARRLvcs= 105 RL # [HA] Hartree RL components
EXXRLvcs= 105 RL # [XX] Exchange RL components
NLDamping= 0.150000 eV # [NL] Damping
% ExtF_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%
Field1_kind= "SOFTSIN" # [RT Field1] Kind(SIN|SOFTSIN|RES|ANTIRES|GAUSS|DELTA|QSSIN)
GfnQPdb= "none" # [EXTQP G] Database
GfnQP_N= 1 # [EXTQP G] Interpolation neighbours
% GfnQP_E
0.900000 | 1.000000 | 1.000000 | # [EXTQP G] E parameters (c/v) eV|adim|adim
%
```

The [math]\displaystyle{ \alpha }[/math] parameter for AlAs has been taken from Ref. ^{[4]}. The results is:

Plot data are available here.
Notice that different models for the \( \alpha \) parameter have been proposed in the literature, see Ref. ^{[2]} for a discussion.
Finally, in Yambo we implemented also the Jelium-with gap model NLCorrelation="JGM" that provides an automatic estimation of the \( \alpha \) parameters for each material, see Ref. ^{[2]} for a discussion on this approximation.

## SHG at the TD-BSE level

Second-harmonic generation can be calculated also the TD-BSE level, the procedure is the same of the linear response case: the first one has to calculate the collisions integrals (see appendix of Ref.^{[5]}) and then can run simulation for the non-linear response as it is shown in this tutorial.
Hereafter we will briefly repeat the steps necessary to obtain SHG including excitonic effects.

1) Download DFT wave-functions and input file here: hBN-2D-RT.tar.gz

2) Run the setup

3) Remove symmetries

use the command ypp_nl -y

fixsyms # [R] Remove symmetries not consistent with an external perturbation % Efield1 0.000000 | 1.000000 | 0.000000 | # First external Electric Field % % Efield2 0.000000 | 0.000000 | 0.000000 | # Additional external Electric Field % BField= 0.000000 T # [MAG] Magnetic field modulus Bpsi= 0.000000 deg # [MAG] Magnetic field psi angle [degree] Btheta= 0.000000 deg # [MAG] Magnetic field theta angle [degree] RmTimeRev # Remove Time Reversal

then go in the FixSymm folder and run the setup again

4) Generate collisions integrals generate collision for HSEX using the command: yambo_nl -X s -e -v h+sex

em1s # [R][Xs] Statically Screened Interaction collisions # [R] Collisions dipoles # [R] Oscillator strenghts (or dipoles) DIP_Threads=0 # [OPENMP/X] Number of threads for dipoles X_Threads=0 # [OPENMP/X] Number of threads for response functions RT_Threads=0 # [OPENMP/RT] Number of threads for real-time Chimod= "HARTREE" # [X] IP/Hartree/ALDA/LRC/PF/BSfxc % BndsRnXs 1 | 20 | # [Xs] Polarization function bands % NGsBlkXs= 3000 mHa # [Xs] Response block size % LongDrXs 1.000000 | 0.000000 | 0.000000 | # [Xs] [cc] Electric Field % % COLLBands 4 | 5 | # [COLL] Bands for the collisions % HXC_Potential= "SEX+HARTREE" # [SC] SC HXC Potential HARRLvcs= 1000 mHa # [HA] Hartree RL components EXXRLvcs= 1000 mHa # [XX] Exchange RL components CORRLvcs= 1000 mHa # [GW] Correlation RL components

5) Run the SHG calculation including collisions, with the command: yambo_nl -u n -V qp

nloptics # [R] Non-linear spectroscopy DIP_Threads=0 # [OPENMP/X] Number of threads for dipoles NL_Threads=0 # [OPENMP/NL] Number of threads for nl-optics % NLBands 4 | 5 | # [NL] Bands range % NLverbosity= "high" # [NL] Verbosity level (low | high) NLtime=42.000000 fs # [NL] Simulation Time NLintegrator= "CRANKNIC" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC") NLCorrelation= "SEX" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX") NLLrcAlpha= 0.000000 # [NL] Long Range Correction % NLEnRange 2.000000 | 8.000000 | eV # [NL] Energy range % NLEnSteps= 80 # [NL] Energy steps NLDamping= 0.200000 eV # [NL] Damping (or dephasing) RADLifeTime=-1.000000 fs # [RT] Radiative life-time (if negative Yambo sets it equal to Phase_LifeTime in NL) #EvalCurrent # [NL] Evaluate the current #FrPolPerdic # [DIP] Force periodicity of polarization respect to the external field HARRLvcs= 1000 mHa # [HA] Hartree RL components EXXRLvcs= 1000 mHa # [XX] Exchange RL components GfnQPdb= "none" # [EXTQP G] Database action GfnQP_INTERP_NN= 1 # [EXTQP G] Interpolation neighbours (NN mode) GfnQP_INTERP_shells= 20.00000 # [EXTQP G] Interpolation shells (BOLTZ mode) GfnQP_DbGd_INTERP_mode= "NN" # [EXTQP G] Interpolation DbGd mode % GfnQP_E 3.000000 | 1.000000 | 1.000000 | # [EXTQP G] E parameters (c/v) eV|adim|adim % ............. % Field1_Freq 0.100000 | 0.100000 | eV # [RT Field1] Frequency % Field1_NFreqs= 1 # [RT Field1] Frequency Field1_Int= 1000.00 kWLm2 # [RT Field1] Intensity Field1_Width= 0.000000 fs # [RT Field1] Width Field1_kind= "SOFTSIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN) Field1_pol= "linear" # [RT Field1] Pol(linear|circular) % Field1_Dir 0.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor %

6) Use the command ypp_nl -u to analyze the result

nonlinear # [R] Non-linear response analysis Xorder= 4 # Max order of the response/exc functions % TimeRange 40.000 | -1.00000 | fs # Time-window where processing is done % ETStpsRt= 200 # Total Energy steps % EnRngeRt 0.00000 | 20.00000 | eV # Energy range % DampMode= "NONE" # Damping type ( NONE | LORENTZIAN | GAUSSIAN ) DampFactor= 0.000000 eV # Damping parameter PumpPATH= "none" # Path of the simulation with the Pump only

now you can plot columns 4 and 5 of the o.YPP-X_probe_order_2 file that are the real and imaginary part of Xhi2 along yyy direction. Using gnuplot, just plot

p 'o.YPP-X_probe_order_2' u 1:(sqrt($4**2+$5**2)) t 'SHG in hBN monolayer' w lp

and you will get

If you arrived up here, you have the right to listen to the Yambo song.

## References

- ↑ F. Aryasetiawan, O. Gunnarsson,
*The*, Rep. Prog. Phys.**GW**method**61**, 237 (1998); (arXiv 9712013) - ↑
^{2.0}^{2.1}^{2.2}^{2.3}M. Grüning, D. Sangalli, C. Attaccalite,*Dielectrics in a time-dependent electric field: a real-time approach based on density-polarization functional theory*, Phys. Rev. B**94**, 035149 (2016) - ↑ R. M. Martin and G. Ortiz,
*Functional theory of extended Coulomb systems*, Phys. Rev. B**56**, 1124 (1997) - ↑ S. Botti, F. Sottile, N. Vast, V. Olevano, L. Reining, H. Weissker, A. Rubio, G. Onida, R. Del Sole, and R. W. Godby,
*Long-range contribution to the exchange-correlation kernel of time-dependent density functional theory*, Phys. Rev. B**69**, 155112 (2004) - ↑ C. Attaccalite, M. Grüning, and A. Marini,
*Real-time approach to the optical properties of solids and nanostructures: Time-dependent Bethe-Salpeter equation*, Phys. Rev. B**84**, 245110 (2011)

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